# Geometry and Spatial Sense, Grades 4 to 6 2008

```Geometry and
Spatial Sense,
A Guide to Effective Instruction
in Mathematics,
2008
Geometry and
Spatial Sense,
A Guide to Effective Instruction
in Mathematics,
Every effort has been made in this publication to identify mathematics resources and
tools (e.g., manipulatives) in generic terms. In cases where a particular product is used
by teachers in schools across Ontario, that product is identified by its trade name, in the
interests of clarity. Reference to particular products in no way implies an endorsement of
those products by the Ministry of Education.
Contents
Introduction
5
Working Towards Equitable Outcomes for Diverse Students.....................................................5
Accommodations and Modifications.................................................................................................7
The Mathematical Processes............................................................................................................. 10
Addressing the Needs of Junior Learners...................................................................................... 12
The Big Ideas in Geometry and Spatial Sense
14
Overview.................................................................................................................................................. 14
General Principles of Instruction...................................................................................................... 16
Levels of Geometric Thought............................................................................................................ 18
Properties of Two-Dimensional Shapes and Three-Dimensional Figures........................... 19
Geometric Relationships.................................................................................................................... 25
Location and Movement..................................................................................................................... 31
Relating Mathematics Topics to the Big Ideas............................................................................. 37
38
Introduction............................................................................................................................................ 38
Investigating Angle Properties.......................................................................................................... 40
Investigating Congruence................................................................................................................... 44
Investigating Polygon Properties...................................................................................................... 45
53
Introduction............................................................................................................................................ 53
Properties of Prisms and Pyramids................................................................................................. 55
Representing Three-Dimensional Figures in Two Dimensions............................................... 58
61
Introduction............................................................................................................................................ 61
Grid and Coordinate Systems........................................................................................................... 62
Relationships in Transformational Geometry............................................................................... 66
Congruence, Orientation, and Distance......................................................................................... 69
Relationships Between Transformational and Coordinate Geometry.................................. 70
References
73
Learning Activities
75
Introduction to the Learning Activities........................................................................................... 77
79
Two-Dimensional Shapes: Comparing Angles............................................................................. 79
Three-Dimensional Figures: Construction Challenge................................................................ 91
Location: Check Mate........................................................................................................................108
Movement: Hit the Target................................................................................................................. 119
129
Two-Dimensional Shapes: Triangle Sort......................................................................................129
Three-Dimensional Figures: Package Possibilities....................................................................140
Location: City Treasure Hunt...........................................................................................................156
Movement: Drawing Designs..........................................................................................................168
178
Two-Dimensional Shapes: Connect the Dots............................................................................ 178
Three-Dimensional Figures: Sketching Climbing Structures.................................................. 191
Location: Name My Shapes.............................................................................................................213
Movement: Logo Search and Design ..........................................................................................223
Appendix: Guidelines for Assessment 239
Glossary
243
Introduction
Geometry and Spatial Sense, Grades 4 to 6 is a practical guide that teachers will find useful in
helping students to achieve the curriculum expectations outlined for Grades 4 to 6 in the
Geometry and Spatial Sense strand of The Ontario Curriculum, Grades 1–8: Mathematics, 2005.
This guide provides teachers with practical applications of the principles and theories that
are elaborated in A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6, 2006.
This guide provides:
• an overview of each of the three “big ideas”, or major mathematical themes, in the
Geometry and Spatial Sense strand. The overview stresses the importance of focusing on
the big ideas in mathematical instruction to achieve the goal of helping students gain a
deeper understanding of the mathematical concepts;
• three sections that focus on the important curriculum topics of two-dimensional shapes,
three-dimensional figures, and location and movement. Each of these sections provides a
discussion of mathematical models and instructional strategies that have proved effective
in helping students understand the mathematical concepts related to the topics;
• sample learning activities for Grades 4, 5, and 6. These learning activities illustrate how a
learning activity can be designed to:
– focus on an important curriculum topic;
– involve students in applying the seven mathematical processes described in the
mathematics curriculum document and reproduced on pages 10–11 of this document;
– develop understanding of the big ideas in Geometry and Spatial Sense.
This guide also contains a list of the references cited throughout the guide. At the end of the
guide is an appendix that discusses assessment strategies for teachers. There is also a glossary
that includes mathematical and pedagogical terms used throughout the guide.
Working Towards Equitable Outcomes for Diverse Students
All students, whatever their socio-economic, ethnocultural, or linguistic background, must
have opportunities to learn and to grow, both cognitively and socially. When students can
make personal connections to their learning, and when they feel secure in their learning
environment, their true capacity will be realized in their achievement. A commitment to
equity and inclusive instruction in Ontario classrooms is therefore critical to enabling all
students to succeed in school and, consequently, to become productive and contributing
members of society.
To create effective conditions for learning, teachers must take care to avoid all forms of
bias and stereotyping in resources and learning activities, which can quickly alienate
students and limit their learning. Teachers should be aware of the need to provide a variety
of experiences and to encourage multiple perspectives, so that the diversity of the class is
recognized and all students feel respected and valued. Learning activities and resources for
teaching mathematics should be inclusive, providing examples and illustrations and using
approaches that recognize the range of experiences of students with diverse backgrounds,
knowledge, skills, interests, and learning styles.
The following are some strategies for creating a learning environment that acknowledges and
values the diversity of students and enables them to participate fully in the learning experience:
• providing mathematics problems with situations and contexts that are meaningful to all
students (e.g., problems that reflect students’ interests, home-life experiences, and cultural
backgrounds and that arouse their curiosity and spirit of enquiry);
• using mathematics examples drawn from diverse cultures, including those of
Aboriginal peoples;
• using children’s literature that reflects various cultures and customs as a source of mathematical examples and situations;
• understanding and acknowledging customs and adjusting teaching strategies as necessary.
For example, a student may come from a culture in which it is considered inappropriate for
a child to ask for help, express opinions openly, or make direct eye contact with an adult;
• considering the appropriateness of references to holidays, celebrations, and traditions;
• providing clarification if the context of a learning activity is unfamiliar to students (e.g.,
describing or showing a food item that may be new to some students);
• evaluating the content of mathematics textbooks, children’s literature, and supplementary
materials for cultural or gender bias;
• designing learning and assessment activities that allow students with various learning
styles (e.g., auditory, visual, tactile/kinaesthetic) to participate meaningfully;
• providing opportunities for students to work both independently and interdependently
with others;
Geometry and Spatial Sense, Grades 4 to 6
• providing opportunities for students to communicate orally and in writing in their
home language (e.g., pairing English language learners with a first-language peer who
also speaks English);
• using diagrams, pictures, manipulatives, sounds, and gestures to clarify mathematical
vocabulary that may be new to English language learners.
For a full discussion of equity and diversity in the classroom, as well as a detailed checklist
for providing inclusive mathematics instruction, see pages 34–40 in Volume 1 of A Guide to
Effective Instruction in Mathematics, Kindergarten to Grade 6, 2006.
An important aspect of inclusive instruction is accommodating students with special education needs. The following section discusses accommodations and modifications as they
relate to mathematics instruction.
Accommodations and Modifications
The learning activities in this guide have been designed for students with
a range of learning needs. Instructional and assessment tasks are openended, allowing most students to participate fully in learning experiences.
In some cases, individual students may require accommodations and/or
modifications, in accordance with their Individual Education Plan (IEP), to
support their participation in learning activities.
PROVIDING ACCOMMODATIONS
Students may require accommodations, including special strategies,
support, and/or equipment to allow them to participate in learning activities. There are three types of accommodations:
• Instructional accommodations are adjustments in teaching strategies,
including styles of presentation, methods of organization, or the use of
technology or multimedia.
• Environmental accommodations are supports or changes that the student
may require in the physical environment of the classroom and/or the
school, such as preferential seating or special lighting.
• Assessment accommodations are adjustments in assessment activities
and methods that enable the student to demonstrate learning, such as
test questions.
The term accommodations
is used to refer to the
special teaching and
assessment strategies,
human supports, and/or
individualized equipment
required to enable a student
to learn and to demonstrate
learning. Accommodations
do not alter the provincial
curriculum expectations for
Modifications are changes
expectations for a subject
. . . in order to meet a
student’s learning needs.
These changes may involve
developing expectations
that reflect knowledge
and skills required in the
curriculum for a different
increasing or decreasing the
number and/or complexity
curriculum expectations.
(Ontario Ministry of
Education, 2004,
pp. 25–26)
Introduction
Some of the ways in which teachers can provide accommodations with respect to mathematics
learning activities are listed in the following chart.
Instructional Accommodations
• Vary instructional strategies, using different manipulatives, examples, and visuals (e.g., concrete
materials, pictures, diagrams) as necessary to aid understanding.
• Rephrase information and instructions to make them simpler and clearer.
• Use non-verbal signals and gesture cues to convey information.
• Teach mathematical vocabulary explicitly.
• Have students work with a peer.
• Structure activities by breaking them into smaller steps.
• Model concepts using concrete materials, and encourage students to use them when learning
concepts or working on problems.
• Have students use calculators and/or addition and multiplication grids for computations.
• Format worksheets so that they are easy to understand (e.g., use large-size font; an uncluttered
layout; spatial cues, such as arrows; colour cues).
• Encourage students to use graphic organizers and graph paper to organize ideas and written work.
• Provide augmentative and alternative communications systems.
• Provide assistive technology, such as text-to-speech software.
• Provide time-management aids (e.g., checklists).
• Encourage students to verbalize as they work on mathematics problems.
• Reduce the number of tasks to be completed.
• Provide extra time to complete tasks.
Environmental Accommodations
• Provide an alternative workspace.
• Seat students strategically (e.g., near the front of the room; close to the teacher in group settings; with
a classmate who can help them).
• Reduce visual distractions.
• Minimize background noise.
• Provide a quiet setting.
• Provide headphones to reduce audio distractions.
• Provide special lighting.
• Provide assistive devices or adaptive equipment.
Assessment Accommodations
• Have students demonstrate understanding using concrete materials or orally rather than in written
form.
• Have students record oral responses on audiotape.
• Have students’ responses on written tasks recorded by a scribe.
• Provide assistive technology, such as speech-to-text software.
• Provide an alternative setting.
• Provide assistive devices or adaptive equipment.
• Provide augmentative and alternative communications systems.
Geometry and Spatial Sense, Grades 4 to 6
Assessment Accommodations
• Format tests so that they are easy to understand (e.g., use large-size font; an uncluttered layout;
spatial cues, such as arrows; colour cues).
• Provide visual cues (e.g., posters).
• Provide extra time to complete problems or tasks or answer questions.
• Reduce the number of tasks used to assess a concept or skill.
MODIFYING CURRICULUM EXPECTATIONS
Students who have an IEP may require modified expectations, which differ from the regular
grade-level curriculum expectations. When developing modified expectations, teachers
make important decisions regarding the concepts and skills that students need to learn.
Most of the learning activities in this document can be adapted for students who require
modified expectations. The following chart provides examples of how a teacher could
deliver learning activities that incorporate individual students’ modified expectations.
Modified Program
What It Means
Example
Modified learning expectations,
same activity, same materials
The student with modified
expectations works on the
same or a similar activity, using
the same materials.
The learning activity involves
sorting and classifying quadrilaterals (regular and irregular)
by geometric properties related
to symmetry, angles, and
sides using a variety of tools
(e.g., geoboards, protractors).
Students with modified expectations identify and compare
rectangle, rhombus, trapezoid)
and sort and classify them by
geometric properties (e.g.,
sides of equal length, parallel
sides, right angles), using a
variety of tools.
Modified learning expectations,
same activity, different materials
The student with modified
expectations engages in the
same activity, but uses different
materials that enable him/her
to remain an equal participant
in the activity.
The activity involves sketching
different perspectives and
views of three-dimensional
figures. Students with modified expectations may build
three-dimensional figures
from a picture or model, using
interlocking cubes.
(continued)
Introduction
Modified Program
What It Means
Example
Modified learning expectations,
different activity, different
materials
Students with modified expectations participate in different
activities.
Students with modified expectations work on angle activities
that reflect their learning
expectations, using a variety of
concrete materials.
(Adapted from Education for All: The Report of the Expert Panel on Literacy and Numeracy Instruction for
Students With Special Education Needs, Kindergarten to Grade 6, p. 119.)
It is important to note that some students may require both accommodations and
modified expectations.
The Mathematical Processes
The Ontario Curriculum, Grades 1–8: Mathematics, 2005 identifies seven mathematical processes through which students acquire and apply mathematical knowledge and skills. The
mathematical processes that support effective learning in mathematics are as follows:
• problem solving
• connecting
• reasoning and proving
• representing
• reflecting
• communicating
• selecting tools and
computational strategies
The learning activities described in this guide demonstrate how the mathematical processes
help students develop mathematical understanding. Opportunities to solve problems, to
reason mathematically, to reflect on new ideas, and so on, make mathematics meaningful
for students. The learning activities also demonstrate that the mathematical processes are
interconnected – for example, problem-solving tasks encourage students to represent mathematical ideas, to select appropriate tools and strategies, to communicate and reflect on
strategies and solutions, and to make connections between mathematical concepts.
Problem Solving: Each of the learning activities is structured around a problem or an inquiry.
As students solve problems or conduct investigations, they make connections between new
mathematical concepts and ideas that they already understand. The focus on problem solving
and inquiry in the learning activities also provides opportunities for students to:
• find enjoyment in mathematics;
• develop confidence in learning and using mathematics;
• work collaboratively and talk about mathematics;
• communicate ideas and strategies;
• reason and use critical thinking skills;
• develop processes for solving problems;
10
Geometry and Spatial Sense, Grades 4 to 6
• develop a repertoire of problem-solving strategies;
• connect mathematical knowledge and skills with situations outside the classroom.
Reasoning and Proving: The learning activities described in this document provide opportunities for students to reason mathematically as they explore new concepts, develop ideas,
make mathematical conjectures, and justify results. The activities include questions that
teachers can use to encourage students to explain and justify their mathematical thinking,
and to consider and evaluate the ideas proposed by others.
Reflecting: Throughout the learning activities, students are asked to think about, reflect
on, and monitor their own thought processes. For example, questions posed by the teacher
encourage students to think about the strategies they use to solve problems and to examine
mathematical ideas that they are learning. In the Reflecting and Connecting part of each
learning activity, students have an opportunity to discuss, reflect on, and evaluate their
problem-solving strategies, solutions, and mathematical insights.
Selecting Tools and Computational Strategies: Mathematical tools, such as manipulatives, pictorial
models, and computational strategies, allow students to represent and do mathematics. The learning
activities in this document provide opportunities for students to select tools (concrete, pictorial, and
symbolic) that are personally meaningful, thereby allowing individual students to solve problems
and to represent and communicate mathematical ideas at their own level of understanding.
Connecting: The learning activities are designed to allow students of all ability levels to connect
new mathematical ideas to what they already understand. The learning activity descriptions
provide guidance to teachers on ways to help students make connections between concrete,
pictorial, and symbolic mathematical representations. Advice on helping students develop conceptual understanding is also provided. The problem-solving experience in many of the learning
activities allows students to connect mathematics to real-life situations and meaningful contexts.
Representing: The learning activities provide opportunities for students to represent mathematical ideas by using concrete materials, pictures, diagrams, numbers, words, and symbols.
Representing ideas in a variety of ways helps students to model and interpret problem
situations, understand mathematical concepts, clarify and communicate their thinking, and
make connections between related mathematical ideas. Students’ own concrete and pictorial
representations of mathematical ideas provide teachers with valuable assessment information
about student understanding that cannot be assessed effectively using paper-and-pencil tests.
Communicating: Communication of mathematical ideas is an essential process in learning
mathematics. Throughout the learning activities, students have opportunities to express
mathematical ideas and understandings orally, visually, and in writing. Often, students are
asked to work in pairs or in small groups, thereby providing learning situations in which
students talk about the mathematics that they are doing, share mathematical ideas, and ask
clarifying questions of their classmates. These oral experiences help students to organize
their thinking before they are asked to communicate their ideas in written form.
Introduction
11
Addressing the Needs of Junior Learners
Every day, teachers make many decisions about instruction in their classrooms. To make
informed decisions about teaching mathematics, teachers need to have an understanding of
the big ideas in mathematics, the mathematical concepts and skills outlined in the curriculum
document, effective instructional approaches, and the characteristics and needs of learners.
The following chart outlines general characteristics of junior learners, and describes some of the
implications of these characteristics for teaching mathematics to students in Grades 4, 5, and 6.
Characteristics of Junior Learners and Implications for Instruction
Area of
Development
Intellectual
development
Characteristics of Junior Learners
Implications for Teaching Mathematics
Generally, students in the junior grades:
The mathematics program should provide:
• prefer active learning experiences that
allow them to interact with their peers;
• learning experiences that allow students
to actively explore and construct
mathematical ideas;
• are curious about the world around
them;
• are at a concrete, operational stage of
development, and are often not ready
to think abstractly;
• enjoy and understand the subtleties of
humour.
• learning situations that involve the use
of concrete materials;
• opportunities for students to see that
mathematics is practical and important
in their daily lives;
• enjoyable activities that stimulate
curiosity and interest;
• tasks that challenge students to
mathematical ideas.
Physical
development
Generally, students in the junior grades:
The mathematics program should provide:
• experience a growth spurt before
puberty (usually at age 9–10 for girls, at
age 10–11 for boys);
• opportunities for physical movement
and hands-on learning;
• are concerned about body image;
• a classroom that is safe and physically
appealing.
• are active and energetic;
• display wide variations in physical
development and maturity.
12
Geometry and Spatial Sense, Grades 4 to 6
(continued)
Area of
Development
Psychological
development
Characteristics of Junior Learners
Implications for Teaching Mathematics
Generally, students in the junior grades:
The mathematics program should provide:
• are less reliant on praise, but still
respond well to positive feedback;
• ongoing feedback on students’ learning
and progress;
• accept greater responsibility for their
actions and work;
• an environment in which students can
take risks without fear of ridicule;
• are influenced by their peer groups.
• opportunities for students to accept
responsibilities for their work;
• a classroom climate that supports
diversity and encourages all members
to work cooperatively.
Social
development
Generally, students in the junior grades:
The mathematics program should provide:
• are less egocentric, yet require individual attention;
• opportunities to work with others in a
variety of groupings (pairs, small groups,
large group);
• can be volatile and changeable in
regard to friendship, yet want to be part
of a social group;
• can be talkative;
• are more tentative and unsure of
themselves;
Moral and
ethical
development
• opportunities to discuss mathematical
ideas;
• clear expectations of what is acceptable
social behaviour;
• mature socially at different rates.
• learning activities that involve all
students regardless of ability.
Generally, students in the junior grades:
The mathematics program should provide:
• develop a strong sense of justice and
fairness;
• learning experiences that provide
equitable opportunities for participation
by all students;
• experiment with challenging the norm
• begin to consider others’ points of view.
• an environment in which all ideas are
valued;
• opportunities for students to share their
own ideas and evaluate the ideas of
others.
(Adapted, with permission, from Making Math Happen in the Junior Years.
Elementary Teachers’ Federation of Ontario, 2004.)
Introduction
13
The Big Ideas in Geometry and
Spatial Sense
Geometry enables us to describe, analyze, and understand our physical world, so there
is little wonder that it holds a central place in mathematics or that it should be a focus
throughout the school mathematics curriculum.
(Gavin, Belkin, Spinelli, & St. Marie, 2001, p. 1)
Overview
The “big ideas” in the Geometry and Spatial Sense strand in Grades 4 to 6 are as follows:
• properties of two-dimensional shapes and three-dimensional figures
• geometric relationships
• location and movement
The curriculum expectations outlined in the Geometry and Spatial Sense strand for each grade
in The Ontario Curriculum, Grades 1–8: Mathematics, 2005 are organized around these big ideas.
In developing a mathematics program, it is important to concentrate on the big ideas and
on the important knowledge and skills that relate to those big ideas. Programs that are
organized around big ideas and focus on problem solving provide cohesive learning opportunities that allow students to explore mathematical concepts in depth. An emphasis on big
ideas contributes to the main goal of mathematics instruction to help students gain a deeper
understanding of mathematical concepts.
Teaching and Learning Mathematics: The Report of the Expert Panel on Mathematics in Grades 4
to 6 in Ontario, 2004 outlines components of effective mathematics instruction, including a
focus on big ideas in student learning:
When students construct a big idea, it is big because they make connections that allow
them to use mathematics more effectively and powerfully. The big ideas are also critical
leaps for students who are developing mathematical concepts and abilities.
(Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004, p. 19)
14
Students are better able to see the connections in mathematics, and thus to learn mathematics, when it is organized in big, coherent “chunks”. In organizing a mathematics program,
teachers should concentrate on the big ideas in mathematics and view the expectations in
the curriculum policy documents for Grades 4 to 6 as being clustered around those big ideas.
The clustering of expectations around big ideas provides a focus for student learning and for
teacher professional development in mathematics. Teachers will find that investigating and
discussing effective teaching strategies for a big idea is much more valuable than trying to
determine specific strategies and approaches to help students achieve individual expectations.
In fact, using big ideas as a focus helps teachers to see that the concepts presented in the
curriculum expectations should not be taught as isolated bits of information but rather as a
network of interrelated concepts.
In building a program, teachers need a sound understanding of the key mathematical concepts
for their students’ grade level, as well as an understanding of how those concepts connect with
students’ prior and future learning (Ma, 1999). They need to understand the “conceptual structure and basic attitudes of mathematics inherent in the elementary curriculum” (p. xxiv) and to
know how best to teach the concepts to students. Concentrating on developing this knowledge
will enhance effective teaching and provide teachers with the tools to differentiate instruction.
Focusing on the big ideas provides teachers with a global view of the concepts represented
in the strand. The big ideas also act as a “lens” for:
• making instructional decisions (e.g., choosing an emphasis for a lesson or set of lessons);
• identifying prior learning;
• looking at students’ thinking and understanding in relation to the mathematical concepts
addressed in the curriculum (e.g., making note of the ways in which a student solves a
problem in coordinate geometry);
• collecting observations and making anecdotal records;
• providing feedback to students;
• determining next steps;
• communicating concepts and providing feedback on students’ achievement to parents
All learning, especially new learning, should be embedded in well-chosen contexts for
learning – that is, contexts that are broad enough to allow students to investigate initial
understandings, identify and develop relevant supporting skills, and gain experience with
varied and interesting applications of the new knowledge. Such rich contexts for learning
open the door for students to see the “big ideas”, or key principles, of mathematics, such as
pattern or relationship.
(Ontario Ministry of Education, 2005, p. 25)
1.In this document, parent(s) refers to parent(s) and guardian(s).
The Big Ideas in Geometry and Spatial Sense
15
These big ideas are conceptually related and interdependent, and instructional experiences
will often reflect more than one big idea. For example, when students create or analyse
designs made by transforming a shape or shapes (location and movement), they demonstrate an understanding of congruence (geometric relationships), and of how congruence
is connected to the properties of a shape (properties of two-dimensional shapes and threedimensional figures).
Geometry can be thought of as the science of shapes and space, while spatial sense is “an
intuitive feel for one’s surroundings and the objects in them” (National Council of Teachers
of Mathematics, 1989, p. 49). Geometry is an important area of mathematics because it
provides students with a deeper appreciation for the world that surrounds them. Geometric
forms can be found in the natural world as well as in virtually all areas of human creativity and ingenuity.
The skills and concepts developed through the study of Geometry and Spatial Sense
play an important role in other areas of mathematics as well. For example, students use
geometric models to help make sense of number concepts related to multiplication and
fractions (e.g., use an array to show the distributive property of multiplication over addition); students apply spatial relationships to plots and graphs to organize and interpret
data; and students develop an understanding of measurement concepts such as area and
volume by manipulating geometric shapes and figures.
This section describes general principles of effective instruction in Geometry and Spatial
Sense, and describes each of the big ideas in detail. Information about geometry topics, such
as location, transformations, two-dimensional shapes and three-dimensional figures, and
angles, is found in subsequent sections, along with sample grade-level lessons and activities.
General Principles of Instruction
The following principles of instruction are relevant for teaching Geometry and Spatial Sense
in the junior years:
• Varied Lesson Types and Instructional Approaches: Effective mathematics programs
incorporate a variety of lesson types, such as problem-based lessons, games, and
investigations. Within these different kinds of lessons, three approaches to mathematics instruction (guided, shared, and independent) should be used regularly to support
student learning. A detailed discussion of these approaches may be found in Chapter 4:
Instructional Approaches, in Volume 1 of A Guide to Effective Instruction in Mathematics,
• Multiple Representations: Geometric concepts such as symmetry, transformations, and
angles can initially be difficult for junior students to understand. Providing a variety of
representations of geometric concepts, including concrete materials that students can
16
Geometry and Spatial Sense, Grades 4 to 6
manipulate, helps students relate new ideas to prior learnings. As students gain an understanding of geometric concepts through the manipulation of concrete materials, teachers
can introduce other more abstract representations of the ideas (e.g., two-dimensional
representations of three-dimensional figures and dynamic geometry modelling applications).
• Use of Examples and Non-examples: Students benefit from investigating two-dimensional
shapes and three-dimensional figures in a variety of orientations and configurations. In
addition to presenting numerous examples of specific shapes and figures, teachers should
offer non-examples, which have been shown to help students eliminate irrelevant features
and focus their attention on important common properties that define the different classes
of shapes and figures.
Examples
These shapes are all parallelograms.
Non-examples
These shapes are not parallelograms.
• Geometric Terminology: As junior students begin to explore and describe more complex
mathematical ideas and relationships, they become aware of the need for more precise
terms and vocabulary. This “need” for more precise language encourages students to adopt
appropriate geometric vocabulary. By the end of the junior years, students should use appropriate terminology (e.g., parallel, perpendicular, equilateral triangle, face, edge, vertex) to describe
and justify their observations and conjectures about geometric shapes and figures. Students
will also need to describe clearly the motions needed to transform two-dimensional shapes
or three-dimensional figures (e.g., “I rotated the triangle 180° counterclockwise”).
In this guide, the term plane shapes will be used interchangeably with two-dimensional
shapes, and solid figures will be used interchangeably with three-dimensional figures.
The Big Ideas in Geometry and Spatial Sense
17
• Technology: Drawing programs, dynamic geometry computer applications (e.g., The Geometer’s
Sketchpad), and online applets are effective representational tools. These “virtual manipulatives”
can assist students in learning important geometric concepts, allowing students to quickly
and easily manipulate models of geometric shapes and figures in ways that are more difficult
to demonstrate with concrete materials or hand-drawn representations. This flexibility allows
students to focus on reasoning, reflecting, and the problem-solving process. For example, students
can observe how changing one angle of a parallelogram to 90° affects the three other angles when
opposite sides remain parallel. Through this type of dynamic representation, students can develop
a deeper understanding of the relationship between parallelograms and rectangles.
Levels of Geometric Thought
Pierre van Hiele and Dina van Hiele-Geldof explored the development of geometric ideas in
children and adults (Van de Walle & Folk, 2005). In their work, they proposed a five-level
hierarchical model of geometric thinking that describes how individuals think and what types
of geometric ideas they think about at each level of development. Advancement from one
level to the next is more dependent on the amount of a student’s experience with geometric
thought than on the student’s age and level of maturation.
• Level 0: Visualization. Students recognize and identify two-dimensional shapes and threedimensional figures by their appearance as a whole. Students do not describe properties
(defining characteristics) of shapes and figures. Level 0 represents the geometric thinking of
many students in the early primary grades.
• Level 1: Analysis. Students recognize the properties of two-dimensional shapes and threedimensional figures. They understand that all shapes or figures within a class share common
properties (e.g., all rectangles have four sides, with opposite sides parallel and congruent).
Level 1 represents the geometric thinking of many students in the later primary grades and
• Level 2: Informal Deduction. Students use informal, logical reasoning to deduce properties
of two-dimensional shapes and three-dimensional figures (e.g., if one pair of opposite sides
of a quadrilateral is parallel and congruent, then the other sides must be parallel and congruent). Students at this level can use this informal, logical reasoning to describe, explore, and
test arguments about geometric forms and their properties. Level 2 represents the geometric
thinking required in mathematics programs at the intermediate and secondary levels.
• Level 3: Deduction. As students continue to explore the relationships between and among
the properties of geometric forms, they use deductive reasoning to make conclusions
about abstract geometric principles. Level 3 represents the geometric thinking required in
secondary and postsecondary mathematics courses.
• Level 4: Rigour. Students compare different geometric theories and hypotheses. Level 4
represents the geometric thinking required in advanced mathematics courses.
18
Geometry and Spatial Sense, Grades 4 to 6
The levels described by the van Hieles are sequential, and success at one level depends on
the development of geometric thinking at the preceding level. Typically, students at the
primary level demonstrate characteristics of level 0 and are moving toward level 1 of the van
Hieles’ levels of geometric thought.
Students entering the junior grades are most likely functioning in the visualization and
analysis levels (0 and 1) of geometric thought. The goal of the junior program is to provide
instructional activities that will encourage children to develop the thinking and reasoning
skills needed to move toward level 2 of the hierarchy, informal deduction.
Properties of Two-Dimensional Shapes and ThreeDimensional Figures
Both two- and three-dimensional shapes exist in great variety. There are many different
ways to see and describe their similarities and differences. The more ways that one can
classify and discriminate amongst these shapes, the better one understands them.
(Van de Walle & Folk, 2005, p. 324)
OVERVIEW
In the primary grades, mathematics instruction encourages students to focus on geometric
features of two-dimensional shapes and three-dimensional figures. Instructional activities
provide opportunities for students to manipulate, compare, sort, classify, compose, and
decompose these geometric forms. These types of activities help students to identify and to
informally describe some attributes and geometrical properties of two-dimensional shapes
and three-dimensional figures.
In the junior grades, students continue to learn about the properties of two-dimensional
shapes and three-dimensional figures through hands-on explorations and investigations.
The following are key points that can be made about the properties of two-dimensional
shapes and three-dimensional figures in the junior grades:
• Two-dimensional shapes and three-dimensional figures have properties that allow
them to be identified, sorted, and classified.
• Angles are measures of turn, and can be classified by degree of rotation.
• An understanding of polygons and their properties allows students to explore and
investigate concepts in geometry and measurement.
• An understanding of polyhedra and their properties helps develop an understanding
of the solid world we live in, and helps make connections between two- and threedimensional geometry.
The Big Ideas in Geometry and Spatial Sense
19
Identifying, sorting, and classifying shapes and figures according to their properties
In the primary grades, students explore geometric properties by sorting, comparing, identifying, and classifying two-dimensional shapes and three-dimensional figures. In the junior
grades, these kinds of learning experiences help students develop a deeper understanding of
geometric properties.
Identifying Shapes and Figures: Identifying shapes and figures involves more than looking
at their appearance. Students must analyse the properties of the shape or figure (e.g., sides,
angles, parallelism) to identify it accurately. If students have opportunities to view only traditional forms of shapes and figures, they
will experience difficulties in recognizing non-traditional forms.
For example, students who have observed only isosceles trapezoids
may not identify the shape at the right as a trapezoid.
Students may also have difficulty understanding that shapes can be classified in more than
one way (e.g., a square is a rectangle, parallelogram, and quadrilateral) if non-traditional
examples of shapes are not explored.
While it is important that students be able to correctly identify two-dimensional shapes and
three-dimensional figures, it is more important that they be able to discuss the properties
of a shape or figure and justify why it may or may not be identified with a classification or
category. Reasoning and justification should always accompany identification.
Sorting Shapes and Figures: When students sort shapes and figures, they need to think about
the relationships between the geometric forms they are sorting. Shapes may be related to one
another in different ways, sharing a number of properties. Consider these three shapes:
A
B
C
If asked to sort shape C with either shape A or shape B, students may reason differently.
Some might suggest that C and A should be grouped together, since both are hexagons
and have six sides. Another student might group shape C with shape B, since each is a
non-convex or concave polygon (a polygon with at least one non-convex angle). Students’
ability to communicate sorting rules effectively provides teachers with opportunities to
assess understanding of how shapes and figures are related.
In the junior grades, sorting strategies and tools become increasingly complex. Venn diagrams are a useful tool for sorting shapes and figures into more than one category.
20
Geometry and Spatial Sense, Grades 4 to 6
Rectangular Faces
Triangular Faces
Students can create Venn diagrams, using sorting categories of their own choosing, or
teachers can suggest sorting criteria. Sorting shapes and figures encourages students to think
about their properties and about how a shape or figure relates to other shapes and figures.
Classifying Shapes and Figures: Classifying shapes and figures involves an understanding
of the critical properties of a category (i.e., the properties that a shape or figure must possess
in order to belong to a specific category). Teachers can encourage students to determine
the critical properties of a category by asking them to describe the characteristics of a shape
or figure and then to develop a definition. For example, students can describe these two
trapezoids in many ways:
• four sides
• one pair of parallel sides
• one pair of non-parallel sides
• non-parallel sides congruent
• two pairs of congruent angles
• two acute angles, two obtuse angles
• four sides
• one pair of parallel sides
• one pair of non-parallel sides
• two right angles
• one acute angle, one obtuse angle
After comparing the two shapes, students might decide that the critical properties of trapezoids would be those common to both – a four-sided shape with one pair of parallel sides
and one pair of non-parallel sides.
It is interesting to note that in the global mathematics community, there are two different ways to define a trapezoid – as a quadrilateral with only one pair of parallel sides, or a
quadrilateral with at least one pair of parallel sides. Providing students with a third shape,
such as a parallelogram, and having them discuss whether it is also a trapezoid encourages
them to consider which trapezoid definition they find most suitable.
The Big Ideas in Geometry and Spatial Sense
21
The study of geometry presents students with many opportunities for mathematical discussion and debate. Teachers should facilitate rich discussions by encouraging students to focus
on specific properties and attributes of two-dimensional shapes and three-dimensional
figures. The following sections describe the properties relevant to students in the junior years.
INVESTIGATING ANGLE PROPERTIES
Primary students are introduced to the concept of angles as they explore and describe the
attributes and properties of two-dimensional shapes. They learn to identify right angles, and
to describe other angles as smaller than or larger than the right angle. In the junior grades,
students explore the concept of angle and angle properties in greater detail.
It is important for junior students to understand that measuring
angles involves finding the amount of rotation between two
lines, segments, or rays that meet at a common vertex.
Since angles are essentially a measure of rotation, there is
angle
a limit as to how they can be classified – the degree of the
vertex
rotation. In this they are unlike polygons (e.g., triangles and
quadrilaterals), which can be classified by different attributes – length of sides, measure of
angles, and so forth.
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
An angle less than 90°
is an acute angle.
An angle of exactly
90° is a right angle.
An angle more than
90° but less than 180°
is an obtuse angle.
An angle of exactly
180° is a straight
angle.
A fifth classification of angles, known as a reflex angle, is an angle that measures more than
180° and less than 360°.
reflex angle
Although reflex angles are not specifically mentioned in the Ontario curriculum, students
may wonder about the classification of angles larger than 180° in their explorations.
As students develop their understanding of the different classifications of angles, they will
use angles as a property to classify different types of triangles (e.g., acute triangles, obtuse
triangles, and right triangles) and quadrilaterals (e.g., the right angles of rectangles distinguish them as a subclass of parallelogram).
22
Geometry and Spatial Sense, Grades 4 to 6
PROPERTIES OF POLYGONS
In the junior grades, students continue to develop their understanding of the properties of
two-dimensional shapes. Students learn that a two-dimensional shape has length and width,
but not depth. They also learn that shapes can have straight or curved sides.
Junior students focus on specific shapes called polygons. A polygon is a closed shape formed
by three or more straight sides. Included among polygons are triangles, quadrilaterals,
octagons, and so forth.
The properties of polygons examined at the junior level include number of sides, number of
vertices, length of sides, size of angles, parallel lines, diagonals, and lines of symmetry. These
properties are explored in more detail in the chapter “Learning About Two-Dimensional
Exploring the properties of polygons allows students to identify and classify polygons in a
number of ways:
• Regular polygons: all angles are equal, and all sides are the same length.
• Irregular polygons: not all angles are equal, and not all sides are the same length.
Regular pentagon
Irregular pentagons
• Convex polygons: all interior angles are less than 180°.
• Concave or non-convex polygons: at least one interior angle is greater than 180°.
125.5°
104.4°
121.4°
75.9°
68.2°
75.2°
228°
90.4°
105.0°
86.4°
Concave or non-convex hexagon
When students investigate and explore polygon properties rather than focus on recalling
rigid definitions, they are more likely to develop analytical skills and geometric critical
thinking skills.
The Big Ideas in Geometry and Spatial Sense
23
PROPERTIES OF POLYHEDRA
In the primary years, students identify and sort three-dimensional figures according to their
properties. Students’ ability to use appropriate mathematical language in describing these
properties increases as students continue to explore shapes and figures in the junior years.
Students no longer simply say of figures that they “slide” or “roll”, but rather that they have
curved surfaces, faces, edges, and vertices. Students can use more complex mathematical
concepts, such as parallelism, angle measures, and congruence, to sort and classify figures.
In the junior grades, students focus on specific figures called polyhedra. A polyhedron is a threedimensional figure with faces made up of polygons. The polyhedra include prisms and pyramids.
The properties of polyhedra examined at the junior level include number and/or type of
faces, number of edges, number of vertices, parallelism, and perpendicularity. These
properties are explored in more detail in the chapter “Learning About Three-Dimensional
Once students have explored the various properties of polyhedra, they can begin using
appropriate geometric vocabulary to describe the figures.
Figure 1
“Figure 1 has only
rectangular faces. All of
its edges have parallel,
opposite edges, and all of
its faces have congruent
faces that are also parallel.”
Figure 2
“Figure 2 has no faces
parallel to other faces. It
has both rectangular and
triangular faces. Some edges
of the figure run parallel to
other edges; some edges
meet up at one vertex.”
As students explore three-dimensional figures using geometric vocabulary, they begin to
recognize common defining properties among some of the solids.
24
Geometry and Spatial Sense, Grades 4 to 6
Geometric Relationships
With well-designed activities, appropriate tools, and teachers’ support, students can make
and explore conjectures about geometry and can learn to reason carefully about geometric
ideas from the earliest years of schooling. Geometry is more than definitions; it is about
describing relationships and reasoning.
(NCTM, 2000, p. 41)
OVERVIEW
Students experience geometry in the world around them, both at school and at home. Personal
experiences with shapes and solids begin at a very early age, and as students continue to develop
their understanding of geometric concepts and their spatial sense in the junior years, they make
connections between formal school geometry and the environmental geometry that they experience every day. Carefully planned activities will enable students to build on these connections
and identify relationships between and among the various areas of geometry and spatial sense.
The following are key points that can be made about geometric relationships in the
• Plane shapes and solid figures can be composed from or decomposed into other twodimensional shapes and three-dimensional figures.
• Relationships exist between plane and solid geometry (e.g., the faces of a polyhedron are
polygons; views of a solid figure can be represented in a two-dimensional drawing).
• Congruence is a special geometric relationship between two shapes or figures that have
exactly the same size and shape.
COMPOSING AND DECOMPOSING TWO-DIMENSIONAL SHAPES AND THREE-DIMENSIONAL FIGURES
In the primary grades students develop the ability to identify and name common two-dimensional
(plane) shapes and three-dimensional (solid) figures. The ability to identify these shapes and figures,
and later name their properties, is indicative of student thinking in the first two van Hiele levels
(visualization and analysis). The third van Hiele level, informal deduction, involves thinking about
geometric properties without looking at a particular object or shape. Students see relationships
between these properties, and draw conclusions from the relationships. For example, a student
might think:
“If the quadrilateral has two pairs of parallel sides, it must be a parallelogram. If it is a
rhombus, it also has two pairs of parallel sides, but they are all congruent. A rhombus must
be a parallelogram.”
The Big Ideas in Geometry and Spatial Sense
25
In order to prepare students for the informal deduction stage in the later grades, classroom
instruction should focus on the physical relationships between shapes when they are combined
(composed) or taken apart (decomposed). Teachers should use concrete materials and technology to help provide these experiences, in order to develop students’ abilities to visualize.
In the primary grades, composing and decomposing plane shapes provides students with
opportunities to develop spatial awareness and visualization skills. In the junior grades, these
skills continue to develop, but composing and decomposing shapes is applied practically to
the development of measurement concepts and formulas.
convex and concave, can be decomposed into two triangles.
While this discovery alone may not be an important geometric concept, it does illustrate how
students can build upon prior learnings or understandings in order to create new knowledge.
By the end of Grade 6, students will be expected to develop strategies for finding the areas of
rectangles, parallelograms, and triangles. These polygons are closely related, and the relationships between them can help students develop formulas for determining their areas.
This figure represents a continuum for developing area formulas of rectangles, parallelograms, and
triangles. Students can use their knowledge of rectangular area to determine the area of a parallelogram
by decomposing the parallelogram and recomposing it into a rectangle. Since all quadrilaterals can be
decomposed into two triangles, it is also true that any triangle is half of a parallelogram. Understanding
how these polygons are related to each other provides a starting point for the development of area
formulas, and is much more powerful than simply memorizing the formulas.
26
Geometry and Spatial Sense, Grades 4 to 6
Three-dimensional, or solid, figures can also be decomposed into two or more solids.
Exploring this “slicing” of solid figures helps to reinforce the characteristics and properties of
three-dimensional figures. For example, a teacher might present students with a cube made
of modelling clay and ask, “How could you slice this solid to produce a triangular prism?”
Slicing the cube in
this manner will
produce a triangular
prism, though it is not
the only way of slicing
the cube to produce
a triangular prism.
Just as composing and decomposing plane shapes helps students understand and develop
area formulas, so knowing how three-dimensional figures can be decomposed and recomposed can help students better understand and develop volume concepts and formulas.
These types of recomposing and decomposing activities encourage students to begin making
connections between plane shapes and solid figures. In the example above, the student must
think of a “slice” that will produce a rectangular face “on the bottom”, and triangular bases
“on the ends”.
RELATIONSHIPS BETWEEN PLANE SHAPES AND SOLID FIGURES
Children experience many realistic situations that call for the two-dimensional representation
of a three-dimensional object. Understanding how plane shapes relate to solid figures requires
students to use spatial sense – to turn or manipulate objects in one’s mind to obtain different
views of solid objects.
In the primary grades, students experience using different face shapes to sort solids. For
example, a student in Grade 3 might use a Venn diagram to sort solids by rectangular and
triangular faces.
Rectangular Faces
Triangular Faces
The Big Ideas in Geometry and Spatial Sense
27
In the junior grades, students design and construct solids from nets and sketches, and begin
to sketch two-dimensional representations of three-dimensional solids. In order to work
flexibly with solids and their nets, students must not only identify the shapes of the faces
of a solid, but also understand how the faces relate to each other. As an example, consider
these three nets:
A
B
C
Which of the nets will form a cube? Students who simply count the number of faces might
respond that all three will work, since a cube has six square faces. Net B, however, cannot be
formed into a cube, and students might be surprised to find that net C can be formed into a cube.
Students develop spatial sense by visualizing the folding of nets into solids, but they require
meaningful experiences with concrete materials (e.g. paper, Polydron pieces, Frameworks)
to do so. Students can predict whether a net will form a particular solid, then use paper or
Polydron pieces to construct the net and try it out. Teachers should help students make
connections between nets and the properties of three-dimensional solids. For example, these
are nets for pyramids. What common characteristics do they share?
Each of the nets is made of
triangles and one other polygon.
The number of triangles in the
net is the same as the number
of sides the polygon has.
28
Geometry and Spatial Sense, Grades 4 to 6
The relationships between two-dimensional representations of three-dimensional objects
can also be explored using interlocking cubes. Teachers provide students with different
views of a “building”, and students must use the blocks to build the solid and sketch the
three-dimensional view. Consider this example:
front
left
right
rear
The solid that will produce these views would look like the one shown below right when
drawn using isometric grid paper.
These types of activities can be modified in many ways:
• Students can be given the solid and asked to sketch the views.
• Students can be limited in the number of views they are given.
• Students can be given the isometric drawing and asked to
provide sketches of the front, side, top, and rear views.
John Van de Walle (2005) suggests that an answer key can be made
by drawing a top view, and indicating the number of blocks in each “vertical stack”. The
answer key for the example above would look like this:
Top View
Students can make their own challenge cards with answer keys on the back and several
views on the front.
Teachers should encourage students to make conjectures when working with twodimensional representations of three-dimensional figures. Predicting and conjecturing
activities can help students move out of the analysis level of geometric thought and
into the informal deduction phase.
The Big Ideas in Geometry and Spatial Sense
29
Another example of using interlocking cubes would be an activity in which students must
look at an isometric drawing of a figure and determine the number of cubes needed to
build the figure.
Estimate the number of cubes
you would need to build this
figure, and then determine the
exact number. What strategy
did you use to estimate?
What strategy did you use to
determine the exact number?
Although we live in a three-dimensional world, we are often required to represent aspects of
it in two dimensions. It is important for students to see the relationships between plane and
solid geometry in order to develop greater spatial sense and geometric thinking.
CONGRUENCE
Congruence is a special relationship shared by shapes and figures that are the same size and
the same shape. Students develop an understanding of congruence in the primary grades. In
the junior grades, they begin to apply that understanding to classify and categorize shapes
and figures, build three-dimensional figures, and develop measurement formulas.
An important learning for students is that congruence depends on size and shape, but not
orientation. The use of examples and non-examples helps students develop an understanding of congruence.
In the primary grades, students explore congruence by superimposing shapes on one
another. In the junior grades, students can begin to use measurement as a tool for determining congruence. Although formal proofs are not suggested, students can measure the side
lengths and angles of polygons and use these measures to discuss congruence.
Congruence offers opportunities for students to reason geometrically and think critically.
Open-ended questions can allow for engaging student dialogue.
• Can two triangles share identical side lengths and not be congruent? Explain.
• Think of as many quadrilaterals as you can that have only one pair of congruent sides.
• Can a quadrilateral have only three congruent angles? Explain.
• Which solids have only congruent faces?
30
Geometry and Spatial Sense, Grades 4 to 6
Location and Movement
Spatial sense is the intuitive awareness of one’s surroundings and the objects in them.
Geometry helps us represent and describe objects and their interrelationships in space….
Spatial sense is necessary for understanding and appreciating the many geometric aspects
of our world. Insights and intuitions about the characteristics of two-dimensional shapes
and three-dimensional figures, the interrelationships of shapes, and the effects of changes
to shapes are important aspects of spatial sense. Students develop their spatial sense by
visualizing, drawing, and comparing shapes and figures in various positions.
(Ontario Ministry of Education, 2005, p. 9)
OVERVIEW
Spatial reasoning plays an important role in interpreting and understanding the world
around us. As students develop spatial reasoning abilities, they can appreciate the
important role geometry plays in art, science, and their everyday world. Spatial reasoning
includes two important spatial abilities: spatial orientation and spatial visualization. Spatial
orientation is the ability to locate and describe objects in space, and to carry out and
describe transformations of objects. Spatial visualization is the ability to imagine, describe,
and understand movements of two- and three-dimensional objects in space. These skills
play an important role in representing and solving problems in all areas of mathematics
and in real-world situations.
The following are key points that can be made about location and movement in the
• A coordinate grid system can be used to describe the position of a plane shape or solid
object.
• Different transformations can be used to describe the movement of a shape.
USING A COORDINATE GRID SYSTEM TO DESCRIBE POSITION
In the primary grades, students learn to use positional language and simple grid systems to
describe the position of objects and their movement. In the junior grades, students develop
flexibility in spatial reasoning by exploring a variety of coordinate systems.
The Big Ideas in Geometry and Spatial Sense
31
Junior students extend their understanding of the simple grids introduced in the primary
years to include a coordinate system commonly used for road maps and atlases. By convention, in this system a letter is assigned to each column of the grid and a number to each row.
In this system the general location of an object within a specific contained area is described,
not the exact location of the object.
The triangle is
located in E3.
The parallelogram
is in B5 and B6.
The black circle is
located in G4.
What is important for students to understand is that two positional descriptors are provided
to describe the location of an object. One describes the location along a horizontal axis; the
other, the location along a vertical axis. In this case, the horizontal descriptor is a letter, and
the vertical descriptor is a number. These descriptors are sufficient to describe any location
in two dimensions.
As students move into Grade 6, they
y-axis
extend their understanding of map
grid systems to explore the Cartesian
coordinate plane. The Cartesian plane,
named after the French mathematician
René Descartes (1596–1650), is also based
x-axis
on horizontal and vertical positional
descriptors. In the Cartesian plane, both
of these descriptors are numerical. In the
Cartesian system, a pair of numbers is
associated with each point in the plane
rather than an area in the plane. This
allows for a more exact description of the
location of shapes or objects.
The Cartesian coordinate plane uses two axes along which numbers are plotted at regular
intervals. The horizontal axis is known as the x-axis, and the vertical axis as the y-axis. The
location of a point is described by an ordered pair of numbers representing the intersection
of an x and a y “line”. The x- and y-axes divide the plane into four quadrants – the first,
32
Geometry and Spatial Sense, Grades 4 to 6
The First Quadrant of the Cartesian Coordinate Plane
10
9
8
7
B (4, 6)
6
5
C (7, 4)
4
A (2, 3)
3
origin (0, 0)
2
1
x-axis
1
2
3
4
5
6
7
8
9
10
y-axis
In the labelling of points, the x-coordinate is always given first, then the y-coordinate. The
coordinates are written in parenthesis and are separated by a comma (x, y). The origin is
located at the intersection of the x- and y-axes, and is represented as the point (0, 0). The
first quadrant of the coordinate plane is the quadrant that contains all the points with positive x and positive y coordinates.
As students explore locating points and objects on a Cartesian plane, teachers should
provide learning opportunities that help students make connections with prior learnings
about grid systems yet also focus on the differences between the systems. Examples include:
• using grids that have the grid lines drawn in (although in later years students will
explore grids without grid lines, it is recommended that the lines be apparent in initial
investigations);
• plotting points, and having students use ordered pairs to identify their locations;
• using ordered pairs to provide the location of points – the points can be joined to make
a shape;
• using Battleship-like games to reinforce location.
In later grades, students will learn that the coordinate plane can be extended infinitely in four
directions, and that four distinct quadrants are formed. In the junior grades, students should
only be exploring the location of points in the first quadrant, where all integers are positive.
Students in the junior grades will also use the cardinal directions, or north, south, east, and
west, to describe location. Students gain an understanding of these directions by exploring
a variety of maps, and come to understand that on most maps, north is “up” and south is
“down”. East is “right” and west is “left”. Teachers should help students understand why
cardinal directions are more useful than directions based on left, right, forward, and backwards. For example, they might challenge students to describe a situation in which a left
turn could send a person heading east.
The Big Ideas in Geometry and Spatial Sense
33
Using Geometric Transformations to DescribE the Movement of Objects
Many people say they aren’t very good with shape, or they have poor spatial sense. The
typical belief is that you are either born with spatial sense or not. This simply is not true!
We now know that when rich experiences with shape and spatial relationships are provided
consistently over time, children can and do develop spatial sense.
(Van de Walle & Folk, 2005, p. 327)
Young children come to school with an understanding of how objects can be moved, which they
have developed through their play experiences (e.g., assembling puzzles, building with blocks). In
the primary years, students expand this understanding by investigating flips (reflections), slides
(translations), and turns (rotations) through play, exploration, and problem-solving tasks. Primary
learning experiences include kinaesthetic activities (e.g., movement games in the gym), and the
manipulation of concrete materials (e.g., transparent mirrors, or Miras; tangrams; paper folding).
With experience and modelling by the teacher, primary students begin to describe the results of
the transformations in their own everyday language.
In the junior years, students predict the result of a transformation and describe what will
happen to the object as the transformation is performed. As a result of guided investigations, students will eventually be able to look at the original orientation of an object and
the result of a transformation and describe what transformation was performed without
seeing the transformation occur.
Junior students also use transformations (reflections, translations, and rotations) to demonstrate
congruence and symmetry between pairs of shapes and within geometric patterns and designs.
In order to describe clearly the outcomes of their games and explorations, students need to
develop precise mathematical language. Teachers should consistently model transformational vocabulary, including:
• rotate, reflect, translate
• line of reflection
• point of rotation
• directional language (up, down, right, left, clockwise, counterclockwise)
• benchmark turns (1/2, 1/4, 3/4)
34
Geometry and Spatial Sense, Grades 4 to 6
Reflections
A reflection over a line is a transformation in which each point of the original shape has an
image shape that is the same distance from the line of reflection as the original point, but is
on the opposite side of the line.
C'
C
A'
B'
B
In this example, the
points A, B, and C
are exactly the same
distance from the
line of reflection as
points A', B' and C'.
A
In geometry, the prime
( ' ) symbol is used to
label the vertices of an
image of the original
shape after a translation,
rotation, or reflection. If a
second transformation is
described, a double prime
( " ) symbol is used.
Although junior students are not concerned with formal proofs, they may notice that segments drawn to join corresponding points always meet the reflection line at right angles.
The reflection causes a change in the original position and orientation of a shape, but the
reflected image is congruent to the original. In other words, the reflected image is the same
size and shape, but it will be “facing” a different direction, and will be in another position.
It is important to note that the reflection line can be drawn in any direction relative to the
original shape – horizontally, vertically, or diagonally at any angle.
Translations
A translation can be described as a transformation that slides every point of a shape the same
distance in the same direction. During a translation the orientation of the shape does not change
and the image is congruent to the original shape. A translation can occur in any direction.
Translations can be described by the distance and direction of the movement. In the junior
grades, translations can be described using a coordinate grid system. Directional language
(e.g., up, down, right, and left; the cardinal directions) is combined with a number representing the magnitude of the movement.
5 units
4 units
The solid arrow
represents the
translation of 5
units to the right,
and 4 units down.
All of the points
of the pentagon
were translated the
same distance in
the same direction.
The Big Ideas in Geometry and Spatial Sense
35
Although translations represent the movement of an object, Glass (2004) suggests that by
focusing on the path that an object follows rather than on the relationship between the
original shape and its image, students may develop misconceptions about transformations.
In the figure on page 35, the arrow represents the shortest path from the original shape
to its image but does not represent the only path. The image might be a result of multiple
translations. For example, it may first have been translated up 4 units and left 2 units, then
down 8 units and right 7 units. Activities that encourage students to look at the translated
image and describe possible paths it may have “taken” will help develop an understanding
of the relationships between various transformations.
Rotations
A rotation is a transformation that moves every point in a shape or figure around a fixed
point, often called the origin or point of rotation. A rotation creates an image that is congruent
to, and also preserves the orientation of, the original shape.
The point of rotation can be found anywhere on the plane, either outside the shape or
within it.
When the point of rotation is found at a vertex of the shape, the original shape and its
image will share the point.
Rotations are first described by students as fractions of turns. The yellow polygon in the
example immediately above could be described as a rotation of 1/2 turn, either clockwise or
counterclockwise. Students will recognise that every clockwise turn can also be described as
a counterclockwise turn, and with rich explorations will discover that the sum of the two
turns is always 1.
In the junior grades, students are expected to describe rotations in degrees (°). They will
learn that a 1/2 turn can also be described as a 180° rotation, and that 1/4 turn can be
described as a 90° rotation in either a clockwise or counterclockwise direction.
36
Geometry and Spatial Sense, Grades 4 to 6
Relating Mathematics Topics to the Big Ideas
The development of mathematical knowledge is a gradual process. A continuous, cohesive
program throughout the grades is necessary to help students develop an understanding of
the “big ideas” of mathematics – that is, the interrelated concepts that form a framework
for learning mathematics in a coherent way.
(The Ontario Curriculum, Grades 1–8: Mathematics, 2005, p. 4)
In planning mathematics instruction, teachers generally develop learning opportunities
related to curriculum topics, such as two-dimensional shapes and three-dimensional figures.
It is also important that teachers design learning opportunities to help students understand
the big ideas that underlie important mathematical concepts.
When instruction focuses on big ideas, students make connections within and between
topics, and learn that mathematics is an integrated whole, rather than a compilation of
unrelated topics. For example, in a lesson about three-dimensional figures, students can
learn that the faces of three-dimensional figures are two-dimensional shapes, thereby
deepening their understanding of the big idea of geometric relationships.
The learning activities in this guide do not address all topics in the Geometry and Spatial
Sense strand, nor do they deal with all concepts and skills outlined in the curriculum expectations for Grades 4 to 6. They do, however, provide models of learning activities that focus
on important curriculum topics and that foster understanding of the big ideas in Geometry
and Spatial Sense. Teachers can use these models in developing other learning activities.
The Big Ideas in Geometry and Spatial Sense
37
Learning About TwoDimensional Shapes in the
Introduction
Developing an understanding of two-dimensional shapes and their properties is a gradual
process – moving from experiential and visual learning to theoretical and inferential learning. Geometric thinking in the junior years begins to bridge the two kinds of learning.
PRIOR LEARNING
In the primary grades, students learn to recognize and describe geometric properties of
two-dimensional shapes, such as the number of sides, number of vertices, length of sides,
size of angles, and number of parallel lines. Some of these properties are introduced informally (e.g., “Parallel lines are two lines that run side by side in the same direction and stay
the same distance apart”). Learning about geometric properties allows students to develop
the concepts and language they need to analyse and describe two-dimensional shapes.
Experiences in the primary classroom include identifying, comparing, sorting, and classifying shapes. Activities involving both examples and non-examples of shapes help to develop
an understanding of the defining properties of various two-dimensional shapes.
KNOWLEDGE AND SKILLS DEVELOPED IN THE JUNIOR GRADES
In the junior grades, students continue to identify, compare, sort, and classify twodimensional shapes. These investigations include more complex shapes and introduce
additional properties such as symmetry. Junior students become more specific when describing geometric properties in order to expand their geometric vocabulary and allow for more
precision in classifying and identifying two-dimensional shapes. Experiences should include
opportunities to analyse, describe, construct, and classify shapes while considering multiple
properties. For example, students might be required to identify triangles from a group that
includes both scalene and obtuse. Specifically, students in the junior grades explore triangles
and quadrilaterals in depth, investigating side and angle properties as well as symmetry.
38
Instruction that is based on meaningful and relevant contexts helps students to achieve the
curriculum expectations related to two-dimensional geometry, listed in the following table:
Curriculum Expectations Related to Properties of Two-Dimensional Shapes, Grades 4, 5, and 6
By the end of Grade 4,
students will:
By the end of Grade 5,
students will:
By the end of Grade 6,
students will:
Overall Expectation
Overall Expectation
Overall Expectation
three-dimensional figures
and classify them by their
geometric properties, and
compare various angles to
benchmarks.
• identify and classify twodimensional shapes by side
and angle properties, and
compare and sort threedimensional figures.
• classify and construct
polygons and angles.
Specific Expectations
• draw the lines of symmetry
of two-dimensional shapes,
through investigation
using a variety of tools and
strategies;
• identify and compare
laterals (i.e., rectangle,
square, trapezoid, parallelogram, rhombus) and sort
and classify them by their
geometric properties;
• identify benchmark angles
(i.e., straight angle, right
angle, half a right angle),
using a reference tool, and
compare other angles to
these benchmarks;
Specific Expectations
• distinguish among polygons,
regular polygons, and other
two-dimensional shapes;
• identify and classify acute,
right, obtuse, and straight
angles;
Specific Expectations
• sort and classify quadrilaterals by geometric properties
related to symmetry, angles,
and sides, through investigation using a variety of tools
and strategies;
• measure and construct
angles up to 90º, using a
protractor;
• sort polygons according
to the number of lines of
symmetry and the order of
rotational symmetry, through
investigation using a variety
of tools;
• identify triangles (i.e., acute,
right, obtuse, scalene,
isosceles, equilateral), and
classify them according to
angle and side properties;
• measure and construct
angles up to 180° using a
protractor, and classify them
as acute, right, obtuse, or
straight angles;
• construct triangles, using a
variety of tools, given acute
or right angles and side
measurements.
• construct polygons using a
variety of tools, given angle
and side measurements.
• relate the names of the
benchmark angles to their
measures in degrees.
(The Ontario Curriculum, Grades 1–8: Mathematics, 2005)
39
The sections that follow offer teachers strategies and content knowledge to address these
expectations in the junior grades while helping students develop an understanding of twodimensional geometry. Teachers can facilitate this understanding by helping students to:
• investigate angle properties using standard and non-standard tools and strategies;
• investigate the relationship of congruence;
• investigate polygon properties, including line and rotational symmetry, using concrete
materials and technology;
• identify, compare, sort, classify, and construct polygons, including triangles and quadrilaterals, through investigations.
Investigating Angle Properties
It is important for junior students to understand that measuring
angles involves finding the amount of rotation between two
lines, segments, or rays that meet at a common vertex.
angle
By the end of the junior grades, students should be
vertex
comfortable with:
• comparing, classifying, measuring, and constructing angles;
• identifying and analysing angles as properties of geometric shapes.
Investigations should include angles in everyday objects and situations (e.g., angles formed
where walls meet, angles of different golf club faces, angles formed when slicing a pizza).
ANGLE MISCONCEPTIONS
In the study of angles, students may reveal certain misconceptions related to representation
and visualization. Common misconceptions relate to various representations of angles and
their measure or size. Students often associate the size of the angle with the length of the
ray or line segment:
A
B
“ A is smaller
than B. I think
so because the
lines of the angle
are much shorter.”
When students have genuine and meaningful experiences of measuring and comparing
angles that are represented in a variety of ways, they are more likely to understand that an
angle is essentially a measure of rotation.
40
Geometry and Spatial Sense, Grades 4 to 6
The angles below are all equivalent; however, the line segments drawn for each representation
are different. When students are able to compare angle sizes correctly, regardless of the line
segment lengths, they are ready to begin measuring angles accurately.
Probability and spinners offer an excellent opportunity for students to explore angles in
context. Many students might reason that there is a greater probability of landing in the
coloured region of Spinner A, since the area of that coloured sector is larger than the area of
the coloured sector in Spinner B. Probability experiments would yield similar results for both
spinners, however, and students would come to recognize that the measure of the angle for
the coloured sector is the same for each spinner, and the length of the line segment (and
therefore the size of the sector) has no effect on the measure of the angle.
Spinner A
Spinner B
BENCHMARK ANGLES
Just as benchmark numbers like 0, 1/2, and 1 help students develop a
quantitative understanding of fractions, benchmark angles can help
students identify and classify angle measures. Benchmark angles also
help students develop an understanding of angle size. At first, students
describe angles qualitatively (less than a right angle, greater than a
straight angle); eventually they use a numerical value (“The angle is a
little smaller than a right angle, so it’s about 80°”).
Students should be encouraged to use non-standard tools to compare
A
Students can measure
angles using nonstandard units, like the
points of a rhombus.
Angle A is “3 points”.
angles to benchmark angles. A colour tile or square pattern block can
be used to describe angles as less than or greater than a right angle.
41
Important benchmark angles include:
Straight angle
Right angle
Half of a right
angle (45°)
In the later junior grades, students begin using standard units and tools (e.g., protractors)
to measure and construct angles. They also use specific angle measures to construct various
MEASURING ANGLES
When students have had experiences identifying, comparing, and informally measuring
angles using benchmarks, they can begin to measure angles with units and tools such as
protractors. Standard protractors can be very confusing for junior students. There are no
visible angles on the protractor. The unit markings representing angles on the protractor are
very small and appear only on the edge. Most protractors contain two sets of numbers that
run in both directions.
Students need experiences with more “informal” protractors containing larger unit angles
to develop an understanding of how a protractor is used, and they can construct these
informal protractors themselves. For example, they can fold a piece of waxed paper to make
a transparent protractor.
Waxed paper
1st fold
2nd fold
3rd fold
4th fold
Cut and unfold
This transparent protractor can be placed over angles or polygons, and students can see how
to measure the smaller angles in those figures by fitting them within the larger unit angles
of the waxed-paper protractor.
In this case the smaller angles could be referred to as wedges, and the angle above would
measure approximately three and a half wedges.
42
Geometry and Spatial Sense, Grades 4 to 6
Students could cut this full-circle protractor in half to resemble the standard protractor.
Other informal protractors could be constructed from Bristol board, paper, and overhead
transparencies that use outside markings and numbering that is similar to the numbering on
a standard protractor.
1
2
3 4 5
6
7
After these experiences, students can compare their informal protractors with the standard
protractor to understand how this standard tool is used.
CLASSIFYING ANGLES
Since angles are essentially a measure of rotation, or the space between intersecting lines
or segments, there is a limit as to how they can be classified – namely, the degree of the
rotation. In contrast, polygons (e.g., triangles, quadrilaterals) can be classified by different
attributes – length of sides, measure of angles, and so forth.
Acute Angle
Right Angle
An angle less
than 90° is an
acute angle.
An angle of
exactly 90° is a
right angle.
Obtuse Angle
An angle of
more than 90°
but less than
180° is an
obtuse angle.
Straight Angle
An angle of
exactly 180° is a
straight angle.
A fifth classification of angles, known as a reflex angle, is an angle that measures more than
180° and less than 360°.
reflex angle
Although reflex angles are not specifically mentioned in the Ontario curriculum, students
may wonder about the classification of angles larger than 180° in their explorations.
As students develop their understanding of the different classifications of angles, they will use
angles as a property to classify different types of triangles (e.g., acute triangles, obtuse triangles,
and right triangles) and different types of quadrilaterals (e.g., the right angles of rectangles
distinguish them as a subclass of parallelograms).
43
CONSTRUCTING ANGLES
Students gain greater understanding of angle concepts by constructing angles. In their experiences of constructing angles, students should progress from using non-standard materials
to using formal tools like protractors and compasses.
Pattern blocks that identify benchmark angles can be used to construct angles. For example,
a square pattern block can be used to construct an acute or a right angle.
“I draw a line along the bottom of the
square as my first segment. If I want
a right angle, I draw the second
segment down the right-hand side
of the square. If I want an acute
angle, I mark a spot along the top
or left-hand side of the square, and
join it to the bottom line.”
Other non-standard construction techniques include paper folding and using Miras. To
construct angles of a specific measure (e.g., 64°, 125°), students need to be able to use a
protractor or dynamic geometry software.
Investigating Congruence
Congruence is a special relationship between two-dimensional shapes that are the same size
and same shape. In the primary grades students compare shapes and superimpose congruent shapes to show how one fits on top of the other. In the junior grades students continue
these explorations and begin to use measurement as a tool for determining congruence.
An important learning for students is that congruence depends on size and shape, but not orientation. The use of examples and non-examples helps develop an understanding of congruence.
These two line segments are congruent, even
though they are oriented differently, because they
are the same length.
These triangles are not congruent. Although they
are the same shape, they are not the same size.
A parallelogram has two pairs of congruent sides. An
isosceles trapezoid has one pair of congruent sides.
44
Geometry and Spatial Sense, Grades 4 to 6
These two angles are congruent because they are the
same measure. The fact that the rays are different
lengths and the angles “face” opposite directions
65º
65º
does not affect congruence.
Shapes that are transformed by reflection, translation,
or rotation exhibit congruence. The transformed
shape is congruent to the original shape.
An understanding of congruence is beneficial when students encounter other geometric
concepts such as transformations, tiling patterns, and symmetry.
Investigating Polygon Properties
In the junior grades, students continue to develop their understanding of the properties of
two-dimensional shapes. They focus on specific shapes called polygons. A polygon is a closed
shape formed by three or more straight sides. Polygons include triangles, quadrilaterals,
octagons, and so forth.
The properties of polygons are summarized in the following table:
Number of sides
One of the first properties students learn to
consider is the number of sides a shape has.
This information allows students to identify
heptagons, octagons, and so forth.
Number of
vertices
Junior students should recognize that the
point at which two lines meet is called a
vertex. With experience, students will discover
that the number of vertices in a polygon is the
same as the number of sides.
Length of sides
Students learn that the length of sides is an
important property of many two-dimensional
shapes. They recognize that the sides of
squares are of equal length, and that pairs
of opposite sides are equal for all other
parallelograms.
3
2
4
1
5
45
Size of angles
Junior students learn to consider angles as
they identify classes and subclasses of twodimensional shapes. For example, they can
determine that an isosceles triangle has two
identical angles and that an isosceles right
triangle has one 90° angle and, therefore, two
45° angles. Both interior and exterior angles
of polygons can be considered when identifying and classifying polygons.
Parallel lines
Junior students will develop an understanding that parallel lines are always the same
distance apart. They will recognize parallelism
as an essential property in describing the
sides of trapezoids and parallelograms.
Diagonals
A diagonal is a line connecting two nonadjacent vertices. Junior students will
recognize that a polygon can be classified
by the number and nature of its diagonals.
Lines of
symmetry
Students can learn to identify or classify a
polygon by the number of lines of symmetry
it has. For example, squares have four lines of
symmetry; isosceles triangles have one line of
symmetry.
Rotational
symmetry
A shape has rotational symmetry if its position
matches its original position after it has been
rotated less than 360°.
*
*
origninal position
1/2 turn
Exploring the properties of polygons allows students to identify and classify them in a
number of ways:
• Regular polygons: all angles are equal and all sides are the same length.
• Irregular polygons: not all angles are equal, and not all sides are the same length.
Regular pentagon
Irregular pentagons
Junior students continue to become more precise in their description of two-dimensional
properties and to become familiar with the property of symmetry.
46
Geometry and Spatial Sense, Grades 4 to 6
SYMMETRY
In the primary grades students explore lines of symmetry, using paper folding, transparent
tools, and drawings, and in Grade 3 students complete pictures and designs when given
half of the image on one side of a line of symmetry. In the junior grades students expand
their explorations and use lines of symmetry to classify two-dimensional shapes. In Grade 6
students explore the concept of rotational symmetry.
LINES OF SYMMETRY
A line of symmetry divides a shape into two congruent parts that can be matched by
folding the shape in half. Junior students should continue to investigate line symmetry
in two-dimensional shapes, using a variety of tools including paper folding, the Mira, dot
paper, and computer applications. Explorations should include using a variety of methods
to determine the lines of symmetry in a variety of two-dimensional shapes. For example,
students might be asked to find the number of lines of symmetry in the following shapes.
ROTATIONAL SYMMETRY
Rotational symmetry occurs when the position of a shape matches its original position after
the shape has been rotated less than 360°. The order of rotational symmetry refers to the
number of times the position of a shape matches its original position during a complete
For example, a square has rotational symmetry.
Original position
1/4 rotation
1/2 rotation
3/4 rotation
Full rotation
47
When a square is rotated about its centre, its position matches its original position after a 1/4,
1/2, and 3/4 rotation: therefore, a square has rotational symmetry. It has rotational symmetry of
order 4 because its position matches the original position four times during a complete rotation.
An important investigation involves exploring the relationship between the number of sides
of the polygon and its order of rotational symmetry. All regular polygons have an order of
symmetry equal to their number of sides – a square has an order of symmetry of 4, a regular
hexagon has an order of rotational symmetry of 6, and so forth.
Junior students can determine rotational symmetry and the order of rotational symmetry of
two-dimensional shapes by tracing a shape and then rotating that shape within the tracing
to determine if it will fit more than one way.
Original position
1/4 rotation
1/2 rotation
3/4 rotation
Full rotation
Teachers should provide students with opportunities to determine rotational symmetry in
a variety of two-dimensional shapes and have students attempt to create their own twodimensional shapes, given the order of rotational symmetry.
Junior students need to explore and investigate a variety of regular and irregular polygons.
They will focus on two specific classes of polygons: triangles and quadrilaterals.
PROPERTIES OF TRIANGLES
A triangle is a three-sided polygon. Every triangle has three sides and three angles. Through
repeated opportunities to compare, describe, and construct triangles of various orientations
and configurations, junior students will learn to identify important properties of subclasses
of triangles.
Triangles can be classified according to some of their features:
48
Geometry and Spatial Sense, Grades 4 to 6
Angles:
Right Triangles
Acute Triangles
Obtuse Triangles
one right (90°) angle
all angles less than 90°
one angle greater
than 90°
Scalene Triangles
Isosceles Triangles
Equilateral Triangles
all sides different lengths
two congruent sides
all three sides congruent
Sides:
Size:
Similar Triangles
Congruent Triangles
same shape, different size
same size and shape
49
Note that the concept of similarity of polygons is one that is beyond most junior-grade
students and that similarity is not explicitly taught until Grade 7 in the Ontario curriculum.
However, students may informally describe one triangle as having the same “shape” as another
but being smaller or larger than the other. Such informal explorations should be encouraged,
but formal proofs of similarity are developmentally more appropriate for older students.
It is important for junior students to explore triangle attributes (e.g., side length, angle
measure) in combination in order to develop a greater understanding of the properties of
triangles. Probing questions can help guide students in their exploration:
• “Can you construct a triangle that is both a right triangle and an
isosceles triangle?”
• “Can you define/describe an equilateral triangle without using the
word ‘sides’?”
• “Can a triangle have more than one obtuse angle? Why or why not?”
• “Are all scalene triangles also acute triangles? Explain.”
• “Can an equilateral triangle be obtuse?”
A quadrilateral is a four-sided polygon. Every quadrilateral has four sides and four angles.
There are many classifications of quadrilaterals, and students need to learn that quadrilaterals can belong to more than one category. A square, for example, can be classified as a
parallelogram, a rectangle, and a rhombus. Students need opportunities to discuss examples
and non-examples of classes of quadrilaterals in order to help identify defining properties.
The diagram on page 51 represents a classification of quadrilaterals that students in the
junior grades should be familiar with. Each quadrilateral lower in the diagram represents a
special case of a quadrilateral higher in the diagram.
50
Geometry and Spatial Sense, Grades 4 to 6
trapezoid
kite
parallelogram
right trapezoid
rhombus
isosceles trapezoid
rectangle
square
The preceding diagram illustrates that some quadrilaterals can be classified as more than one
shape. For example, a rhombus is a parallelogram, as it has two sets of parallel sides. It is also
a kite, because it has two pairs of congruent adjacent sides.
Some mathematicians view the trapezoid as a special quadrilateral, and define it as a quadrilateral having exactly one pair of parallel sides. This definition is widely accepted in many
countries and mathematics communities. Other mathematicians define a trapezoid as a
quadrilateral having at least one pair of parallel sides. For this reason, the line in the diagram
that joins the trapezoid to the parallelogram and the lines that join the right and isosceles
trapezoids to the rectangle are drawn with dotted lines. Since both definitions are viewed as
mathematically correct, what is important is the students’ reasoning for including or excluding trapezoids in the hierarchy of quadrilaterals.
51
In the early junior grades, students are expected simply to identify trapezoids. In later
grades, they will recognise that trapezoids can be classified like triangles – that is, according
to various properties.
Right trapezoid
Isosceles trapezoid
Like triangles, quadrilaterals can be classified by their size. Quadrilaterals having the same
shape and size are congruent; those whose shape is the same but whose size is different are
similar. Formal proof of congruence or similarity involves careful examination of sides and
angles. While students in the junior grades are not expected to conduct formal proofs, they
can explore the congruence of quadrilaterals concretely:
A
C
B
“I know that A is congruent
to C because when I cut
parallelogram C out, it fits
perfectly on top of A. B is
not congruent. When I cut
it out, it doesn’t fit evenly
on A or C.”
Teachers should use probing questions to encourage students to think about quadrilateral
properties, with the goal of moving them from the analysis phase to the informal deduction
phase. Some examples of questions include the following:
• “A quadrilateral has four congruent sides and at least one right angle. Can it be anything
other than a square?”
• “Is a square a rhombus or a rectangle or both? Why do you think so?”
• “What can you say about the diagonals of a rectangle? How does this compare with what
you can say about the diagonals of a parallelogram, which has no right angles?”
GENERAL INSTRUCTIONAL STRATEGIES
Students in the junior grades benefit from the following instructional strategies:
• having them investigate two-dimensional geometry using concrete materials like pattern
blocks, geoboards, and Power Polygons;
• providing opportunities to analyse properties of shapes using examples and non-examples;
• requiring them to communicate solutions to problems related to two-dimensional
shapes and angles using increasingly accurate terminology;
• providing experiences of determining properties and characteristics of geometric shapes
using dynamic geometry software and other technologies.
52
Geometry and Spatial Sense, Grades 4 to 6
Learning About ThreeDimensional Figures in the
Introduction
The study of three-dimensional figures is closely connected with other mathematical topics
and ideas, including two-dimensional geometry, measurement, and number. As students
investigate the geometric properties and relationships of three-dimensional figures, their
reasoning skills become more complex and their vocabulary more detailed.
PRIOR LEARNING
In the primary grades, students learn to recognize and describe some geometric properties of
three-dimensional figures, such as the number and shape of faces and the number of edges
or vertices. For the most part, they explore these properties concretely, using models of
three-dimensional figures, though they also begin to investigate nets of rectangular prisms.
Learning about geometric properties allows students to develop the concepts and language
they need to analyse and describe three-dimensional figures, and to discover relationships
between two- and three-dimensional geometry. Experiences in the primary classroom include
identifying, comparing, sorting, and classifying figures according to their basic properties,
and making connections between two-dimensional shapes and three-dimensional figures.
KNOWLEDGE AND SKILLS DEVELOPED IN THE JUNIOR GRADES
In the junior grades, students continue to identify, compare, sort, and classify threedimensional figures, with a particular focus on pyramids and prisms. These investigations focus
on making generalizations about a category of figures. As well, students in the junior grades
develop spatial awareness by exploring two-dimensional representations of three-dimensional
figures. These representations include nets; front, side, and top views; and isometric drawings.
Instruction that is based on meaningful and relevant contexts helps students to achieve
the curriculum expectations related to three-dimensional figures, listed in the table on the
following page.
53
Curriculum Expectations Related to Properties of Three-Dimensional Figures, Grades 4, 5, and 6
By the end of Grade 4,
students will:
By the end of Grade 5,
students will:
By the end of Grade 6,
students will:
Overall Expectations
Overall Expectations
Overall Expectation
three-dimensional figures
and classify them by their
geometric properties, and
compare various angles to
benchmarks;
• identify and classify twodimensional shapes by side
and angle properties, and
compare and sort threedimensional figures;
• sketch threedimensional figures, and
construct threedimensional figures from
drawings.
• identify and construct nets of
prisms and pyramids.
Specific Expectations
• construct three-dimensional
figures, using two-dimensional shapes.
Specific Expectations
• identify and describe prisms
and pyramids, and classify
them by their geometric
properties;
• construct a threedimensional figure from a
picture or model of the figure,
using connecting cubes;
Specific Expectations
• distinguish among prisms,
right prisms, pyramids, and
other three-dimensional
figures;
• identify prisms and pyramids
from their nets;
• construct nets of prisms and
pyramids using a variety of
tools.
• build three-dimensional
models using connecting
cubes, given isometric
sketches or different views;
• sketch, using a variety of
tools, isometric perspectives
and different views of threedimensional figures built
with interlocking cubes.
• construct skeletons of threedimensional figures, using a
variety of tools, and sketch
the skeletons;
• draw and describe nets of
rectangular and triangular
prisms;
• construct prisms and
pyramids from given nets;
• construct threedimensional figures, using
only congruent shapes.
(The Ontario Curriculum, Grades 1–8: Mathematics, 2005)
The following sections explain content knowledge related to three-dimensional concepts in
the junior grades, and provide instructional strategies that help students develop an understanding of three-dimensional figures. Teachers can facilitate this understanding by helping
students to:
• investigate the properties of pyramids and prisms in meaningful ways;
• explore relationships between two- and three-dimensional geometry.
54
Geometry and Spatial Sense, Grades 4 to 6
Properties of Prisms and Pyramids
In the primary years, students identify and sort three-dimensional figures according to
their properties. As students continue to explore shapes and figures in the junior years,
their ability to describe these properties increases, as does their ability to use appropriate
mathematical language in so doing.
In the junior grades, students focus on specific figures called polyhedra. A polyhedron is a threedimensional figure with faces made up of polygons. The polyhedra include prisms and pyramids.
The properties of polyhedra are summarized in the following table:
Number or type
of faces
The face of a polyhedron is a flat
surface. Students can describe
polyhedra by the number or shape of
their faces.
5
1
2
3
4
6
A cube has 6 faces.
Number of edges
Junior students should recognize
that the line where two faces meet
is called an edge. Students should
explore the relationships between
the number of edges and faces in
categories of polyhedra.
A triangular prism has 9 edges.
Number of
vertices
Students learn that a vertex is a point
at which two or more edges meet.
Students will explore the relationship
between the number of edges, vertices, and faces of various polyhedra.
A rectangular prism has 8 vertices.
Parallelism
Faces and edges of a figure can be
described as being parallel or nonparallel to other faces and edges.
Only the edges of the base are
parallel in pyramids.
(continued)
55
Perpendicularity
Perpendicularity is a concept that
develops in the later junior grades.
Edges and faces can often be
described as “at right angles” to other
faces and edges.
The base of
this rectangular prism is
perpendicular
to the vertical
faces.
The base of this
prism is not
perpendicular to
its vertical faces..
PRISM PROPERTIES
There are different formal and informal definitions of prisms. Despite the varying definitions,
there is little disagreement as to what a prism is. Students recognise which solids are prisms
and usually use common descriptions – for example, “a solid with two bases that are the
same size and shape”. Essentially, this definition clearly describes prisms; it is the role of the
teacher to guide students to the richer descriptions that include more complex geometric
concepts. An accepted definition of a prism is the following:
A prism is a solid geometric figure whose two ends are parallel and congruent polygons,
called bases. Lines joining corresponding points on the bases are always parallel. The sides of
prisms are always parallelograms.
The ends of this prism are congruent triangles. These
ends (surfaces) are known as the bases of the prism.
The vertices of the bases are connected by parallel lines,
known as edges.
If the vertical edges of the prism are perpendicular to the edges of the base, the prism is
described as a right prism. When the edges do not run perpendicular to each other, the
prism is said to be an oblique prism.
Right prism
56
Geometry and Spatial Sense, Grades 4 to 6
Oblique prism
Prisms are named according to the shape of their bases. Some examples include rectangular
prisms, triangular prisms, hexagonal prisms, and so forth. A special rectangular prism, with
square faces congruent to the base, is known as a cube.
Students almost always wonder how cylinders are related to prisms, since they look and feel
similar to prisms. Although they have been described as “circular prisms”, they are not included
in most accepted definitions of prisms, since their bases are not polygons. On the other hand,
some mathematicians define prisms as special cylinders – cylinders are solids with two parallel
bases connected by parallel elements. When the bases are polygons, the cylinders are prisms.
These differences in classifications underline the fact that definitions in geometry are conventions, and not all conventions are universally accepted. Student discussion and reasoning
are more important in developing geometric thinking than is memorizing definitions.
PYRAMID PROPERTIES
Like prisms, pyramids are a special category of three-dimensional figures with common defining
properties. The Ontario curriculum document formulates the meaning of pyramid as follows:
A pyramid is a polyhedron whose base is a polygon and whose other faces are triangles
that meet at a common vertex.
vertex
The base of this pyramid is a square.
The pyramid has four triangular faces.
triangular faces
square base
Like prisms, pyramids are named by the shape of their base. Examples of pyramids include
square pyramids, hexagonal pyramids, and octagonal pyramids. A tetrahedron is a special type
of triangular pyramid. All of the faces of a tetrahedron are congruent equilateral triangles.
John Van de Walle (2005) considers pyramids to be special cases of cones. He defines a cone
as a solid with a base and a vertex that is not on the base. Edges join the vertex to the vertices of the base, and the base may be any shape at all. When the base is a polygon, the cone
can be classified as a prism. Again, this conventional definition is not universally accepted.
Some consider cones to be pyramids with circular bases and only one face.
Note that pyramids can have a common vertex that is directly over the base, or one that lies
outside the base.
The vertex of each of these
pyramids is found over the base.
The vertex of this pyramid is
found outside the base.
57
Representing Three-Dimensional Figures in Two Dimensions
In the junior grades students are expected to move flexibly between two- and threedimensional representations of a figure. Two-dimensional representations can include nets,
“rectangular” views, and isometric sketches.
A net is a pattern that can be folded into a three-dimensional figure. A net must include
all of the two-dimensional faces of the figure. For example, a net of a cube must have six
squares, a net for a triangular prism must have two triangles and three rectangles, and a net
for a pentagonal pyramid must have one pentagon and five triangles.
In addition to knowing the faces of a figure, students must develop a spatial awareness of how
the faces “fit” together. Consider as an example these two patterns:
Although both of these patterns are composed of the correct number of faces to form a
cube, only the one on the left can actually be folded to form a cube.
Initial experiences with nets should be concrete. Paper folding and Polydron pieces are
examples of manipulatives that serve as an entry point for investigations involving nets
of solids. Polydron shapes can be snapped together and taken apart easily, and folded and
snapped together to form a solid. They are limited, however, in that the number of shapes
is finite. Paper folding takes longer to prepare and is less “user-friendly”, but it has no limits
with respect to the type and size of shapes.
Student experiences with nets should be varied, and should include working from net to
solid and from solid to net. When students are able to “unfold” a solid, they are more likely
to understand the relationships between the faces. These relationships include shared edges
and edge length. Similarly, students should be given examples and non-examples of nets to
fold into solid figures. Consider these two nets for triangular prisms:
Net A
58
Geometry and Spatial Sense, Grades 4 to 6
Net B
Although both nets have the correct number and type of faces found in a triangular prism,
neither can be folded into one. In Net A, the faces are the correct size and shape, but do not
have the correct shared edges. The sides with the red dots are an example of two sides that
will be folded together to form a shared edge. In triangular prisms, the triangle faces are at
opposite ends of the prism and do not share an edge.
Net B also has the correct number and type of faces, but they are not all the correct size. The
sides with the blue dots will be folded together to form an edge, but they are not the same
length, so a shared edge will not be formed.
Explorations that focus on these types of relationships will help students make connections
between two- and three-dimensional geometry and will help to develop their spatial awareness.
Students also use rectangular diagrams to represent solid figures in the junior grades. A
rectangular view is a two-dimensional view of one side of a figure. The view can be from
the front, back, side, top, or bottom, and is a shape or composition of shapes. Various views
of a square pyramid are shown below:
Front view
Side view
Top view
Although it is useful for students to be able to identify solid figures from their various
views, a greater sense of space develops when students explore views of composite figures.
Interlocking cubes are an excellent tool for such explorations.
“Use interlocking cubes
to build a figure that will
produce these views.”
Front view
Side view
Top view
Again, students should work “both ways” when exploring rectangular views of solids – they
should build the solid from different views, and should draw different views of a solid on
grid paper when given a model of the solid.
As students become more confident in their explorations, investigations can focus on
higher-level geometric thinking skills. For example:
• Can a pyramid have a side view that is a rectangle? Explain.
• What does the top view of a prism tell you about the number of rectangular faces it has?
• What is the fewest number of cubes needed to build a solid with these views? (Students
are given three views of a solid made from interlocking cubes.)
59
When planning activities that involve working with rectangular views, it is important to
keep in mind the second van Hiele level – analysis. The focus is on the properties of shapes
and figures, and the ability to generalize from observations. For example, students might
recognize that the number of sides of a prism base is the same as the number of rectangular
faces of the prism. Students are likely to realize this generalization after taking part in a
variety of carefully planned activities.
Isometric diagrams use an isometric grid. An isometric grid shows three axes instead of the
two found in a rectangular grid. One axis runs vertically; the other two axes run “down” at
30° angles to the left and right.
This cube has been drawn using isometric grid paper. Note the
directions of the three axes – one is vertical; two run down
diagonally left and right.
Note that the perimeter of the two-dimensional drawing is a
perfect hexagon, and all the thick lines are of equal length.
Isometric diagrams encourage students to visualize pieces or sections of solids that cannot
be seen. When working from an isometric view and building a solid from interlocking
cubes, students may be required to make assumptions about cubes they may or may not
be able to view directly. For example, in the diagram below, students would have to justify
placing a cube in the space indicated by the arrow.
Is this the top of a cube,
or the side of a cube?
60
Geometry and Spatial Sense, Grades 4 to 6
and Movement in the
Introduction
Spatial sense can be described as the intuitive awareness of one’s surroundings and of the
objects in them. Having spatial sense enables students to accurately describe the location of
objects in their surroundings and of shapes in an abstract plane. Understanding and being
able to visualize the various ways in which shapes and objects can move also contribute to
the development of spatial reasoning skills.
PRIOR LEARNING
In the primary grades, students learn to describe the location of an object using expressions
such as “in front of”, “beside” and “to the left of”. They describe transformations as slides,
flips, and turns, and discuss the results of these transformations by comparing the transformed image with the original shape. In investigating lines of symmetry, primary students
use concrete materials and tools to develop a greater understanding of reflections.
KNOWLEDGE AND SKILLS DEVELOPED IN THE JUNIOR GRADES
Instructional strategies for teaching location and movement in the junior grades should
focus on an understanding of spatial relationships. The location of a point is related to both a
horizontal and a vertical position. Transformations are related to one another – for example,
the position of an image after a rotation can often be replicated by a combination of other
transformations. Although location and movement can be treated as separate components
of a junior curriculum, when they are taught in combination with a focus on relationships,
students are able to make important mathematical connections.
Instruction that is based on meaningful and relevant contexts helps students to achieve the
curriculum expectations related to location and movement, listed in the table on the following page:
61
Curriculum Expectations Related to Location and Movement, Grades 4, 5, and 6
By the end of Grade 4,
students will:
By the end of Grade 5,
students will:
By the end of Grade 6,
students will:
Overall Expectation
Overall Expectation
Overall Expectation
• identify and describe the
location of an object, using
a grid map, and reflect twodimensional shapes.
• identify and describe the
location of an object, using
the cardinal directions, and
translate two-dimensional
shapes.
• describe location in the first
system, and rotate twodimensional shapes.
Specific Expectations
• identify and describe the
general location of an object
using a grid system;
• identify, perform, and
describe reflections using a
variety of tools;
• create and analyse symmetrical designs by reflecting
a shape, or shapes, using a
variety of tools and identify
the congruent shapes in the
designs.
Specific Expectations
• locate an object using the
cardinal directions and a
coordinate system;
• compare grid systems commonly used on maps;
• identify, perform, and
describe translations, using a
variety of tools;
• create and analyse designs
by translating and/or reflecting a shape, or shapes, using
a variety of tools.
Specific Expectations
• explain how a coordinate
system represents location,
and plot points in the first
coordinate plane;
• identify, perform, and
describe, through investigation using a variety of
tools, rotations of 180° and
clockwise and counterclockwise rotations of 90°,
with the centre of rotation
inside or outside the shape;
• create and analyse designs
made by reflecting, translating, and/or rotating a shape,
or shapes, by 90° or 180°.
(The Ontario Curriculum, Grades 1–8: Mathematics, 2005)
The following sections explain content knowledge related to location and movement concepts in the junior grades, and provide instructional strategies that help students develop
an understanding of location and movement. Teachers can facilitate this understanding by
helping students to:
• connect prior learnings about location with formal coordinate geometry;
• explore relationships in the movement of shapes and objects;
• make connections between transformational geometry and coordinate geometry.
Grid and Coordinate Systems
An understanding of the coordinate systems, including the Cartesian plane, is essential to
later learnings, as it is a fundamental aspect of both geometry and algebra. It is important
that junior teachers make connections with prior learnings about location when teaching
62
Geometry and Spatial Sense, Grades 4 to 6
Students’ first experiences with coordinate geometry in the junior grades come in the form
of grids, commonly used in maps. A grid system uses a combination of letters and numbers
to describe a general location of a shape or an object. Because the grid system identifies an
area rather than a point, precise locations cannot be described.
A
B
C
D
E
F
G
1
2
The blue triangle
is located in B2.
3
4
5
6
Students will bring to the junior grades an understanding of grids that comes from both
classroom experiences and non-classroom experiences. The hundreds chart, social studies
maps, data management activities, and games are some examples of students’ exposure to
grids. Teachers should try to make connections with those learnings when introducing grids.
Some examples include:
• exploring location on non-labelled grids. Students may use phrases like “third from the
left and two down” to identify a square. These phrases can later be connected with labels;
• predicting new locations on the basis of a particular movement. For example, if you
started in square B1, and kept moving to the right, how would the label of each new
square change? What if you started in B1 and kept moving down?;
• exploring local and provincial maps and atlases that use similar grid identification systems;
• playing games that require the use of a grid. For example, students might play games
similar to Battleship, in which hidden shapes must be found by guessing grid positions.
Coordinate systems differ from grid systems in that they identify a point rather than an
area. In a coordinate system, the lines are labelled, rather than the area bounded by the lines.
One way to describe or label these lines is to use the cardinal directions with a numbering
system to describe the location of an object in relation to a point.
A
O
B
C
63
In the grid on page 63, point O represents the origin, or starting point. A teacher might ask:
“Describe the position of points A, B, and C in relation to point O. Use cardinal directions
(e.g., north, south) rather than conventional directions (e.g., up, down, right).”
“Point A is 2 units
north and 1 unit
west of the origin.”
“Point B is 3 units
east and 2 units
south of the origin.”
“Point C is 4 units
south and 2 units
west of the origin.”
Many activities can be developed to offer students meaningful first experiences with a
coordinate system. Students will develop fundamental ideas about grids and strategies for
navigating them in these types of activities. These ideas and strategies are an important
component of the mathematics they will learn in later grades (NCTM, 2000).
In the later junior grades, students are formally introduced to the Cartesian plane and learn
to use ordered pairs to identify points.
y-axis
The Cartesian coordinate plane uses
two axes along which numbers are
plotted at regular intervals. The horizontal axis is known as the x-axis, and the
vertical axis as the y-axis. The location
of a point is described by an ordered pair
x-axis
of numbers representing the intersection
of an x and a y “line”. The x- and y-axes
divide the plane into four quadrants – the
first, second, third, and fourth quadrants.
The First Quadrant of the Cartesian Coordinate Plane
10
9
8
7
B (4, 6)
6
5
C (7, 4)
4
A (2, 3)
3
origin (0, 0)
2
1
x-axis
1
2
y-axis
64
Geometry and Spatial Sense, Grades 4 to 6
3
4
5
6
7
8
9
10
The first quadrant of the coordinate plane is the quadrant that contains all the points with
positive x and positive y coordinates. In labelling points, the x-coordinate is always given
first, then the y-coordinate. The coordinates are written in parentheses and are separated by
a comma (x, y). The origin is located at the intersection of the x- and y-axes, and is represented as the point (0, 0).
A common misconception students experience with initial investigations of the Cartesian
plane is confusing the order of the numbers of a point. While some students may rely on
phrases or other “tricks” to help remember which axes the numbers represent (e.g., “You go
in the door first and then up the stairs”), recent developments in dynamic geometry software offer students more meaningful opportunities to develop this understanding.
Consider as an example The Geometer’s Sketchpad. This program allows students to call up
a point on a grid, label it, and display its coordinates. When the point is “dragged” or moved
about the grid, the coordinates change as the point moves.
A sample exploration in which students use The Geometer’s Sketchpad might include
questions like the following:
• Drag point A horizontally along a single line. What do you notice about how the coordinates change each time the point is moved along the line?
• Drag point A vertically along a single line. How do the coordinates change this time?
When students drag a point and see the coordinates change, they may better understand
which ordered pair corresponds with which axis. As a point is moved horizontally but its vertical position remains unchanged, the second number in the ordered pair remains unchanged.
So that number must represent the vertical (or y-axis) position. “Seeing and doing” with
dynamic software is powerful and can make a greater impact than simply plotting points and
listing coordinates. Rich explorations with grid and coordinate systems can help students
develop greater spatial awareness and deepen their understanding of spatial relationships,
while providing foundational knowledge important to mathematics in the later grades.
65
Relationships in Transformational Geometry
Junior students identify, describe, and perform three different types of transformations in
geometry and spatial sense – reflections, translations, and rotations. Although students can
study each transformation independently, it is when they explore the relationships between
transformations that they begin to deepen their understanding and make connections.
A reflection over a line is a transformation in which each point of the original shape,
sometimes called the pre-image, has an image shape that is the same distance from the line of
reflection as the original point, but is on the opposite side of the line.
C'
C
A'
B'
In this example, the points A,
B, and C are exactly the same
distance from the mirror line
as are points A', B' and C.'
B
A
Students’ initial experiences with reflections should be grounded in concrete materials.
Students can physically flip objects across a line of reflection created by folding a piece
of paper, or by using physical shapes like pattern blocks, plastic polygons, or cutouts.
Technology like The Geometer’s Sketchpad allows students to explore reflections dynamically once they are comfortable with the basics of the software.
Teachers should select shapes that are not symmetrical for initial explorations with reflections. Students will more readily see the change of orientation in the reflected image when
shapes are not symmetrical. Block letters work well and will engage students, as they may
choose letters in their name.
Reflection activities should not simply focus on students’ performing the transformation;
they should also require students to determine and describe the transformation that has
taken place. An example of a rich activity is having students determine the location of the
line of reflection, given the original shape and its image.
Draw the mirror line
that will produce
this reflection.
66
Geometry and Spatial Sense, Grades 4 to 6
A translation can be described as a transformation that slides every point of a shape the
same distance in the same direction.
It is useful for students to describe the translation of a shape in multiple ways. In the
diagram above, the pentagon is shown to be directly translated “4 down and 5 to the right”,
but the image may also have resulted from multiple translations. For example, the pentagon
may have moved 5 units right and 3 units up, followed by a second translation of 7 units
down. Activities that allow students to show multiple ways of arriving at a solution help
them develop flexibility in their mathematical thinking and can deepen their understanding
of spatial relationships.
A rotation is a transformation that moves every point in a shape or figure around a fixed
point, often called the origin or point of rotation.
Students should use a number of different tools and resources to explore rotations. Simple
tools, like a shape on a piece of paper and a pencil to hold it down at a rotation point, are
excellent for initial investigations. As students begin to describe rotations by degree of rotation (i.e., fractions or degrees) more sophisticated tools, such as grids or dynamic geometry
software, provide opportunities for deeper learning.
The Geometer’s Sketchpad, for example, allows students to move the point of rotation
and provides immediate feedback about the effect on the rotated image of moving the
point of rotation.
67
Point of rotation
outside the shape
Point of rotation on
a vertex of the shape
Point of rotation
inside the shape
The software also enables students to perform rotations of a specific measure (e.g., 45°). For
example, in the following problem students predict and then determine how many rotations of a certain degree will be needed to transform an image back to the original shape.
68
Geometry and Spatial Sense, Grades 4 to 6
• Construct any shape and place a point of rotation outside the shape. Predict how many
45° rotations it will take before a rotated image will return to its original position.
• Perform repeated 45° rotations until the image returns to the original position. How many
rotations did it take? Why do you think that is? How would the answer change if the
rotations were 60°?
Congruence, Orientation, and Distance
All three transformations (reflection, translation, rotation) have common characteristics. An
original shape or object that is sometimes called the pre-image is transformed by an action or
series of actions applied to the shape, and a resulting image is created. The image is directly
related to the pre-image with respect to congruence, orientation, and distance.
Congruent objects are the same size and shape, and orientation can be thought of as an object’s
position or alignment relative to points of the compass or other specific directions.
Each of the three transformations preserves congruency – the original pre-image is always
congruent to the image. Translations also preserve orientation between the original shape
and its image, but rotations and reflections change the orientation of the original shape.
The yellow hexagon has been
translated up and to the
right. The resulting image is
congruent to the original and
has also maintained orientation
– the “L” faces right, and the
sharp point is on the top.
Here the hexagon has been reflected
over a line. The resulting image is still
congruent to the original shape, but
its orientation has changed. The “L” is
now facing up, and the sharp point of
the hexagon points to the right.
Another important concept that comes from exploring the relationships between trans­
formations is the idea of distance. Consider these probing questions:
• “Measure the distance of points in the original shape and its reflected image to the mirror
line. What do you notice?”
• “Compare the distances between the corresponding points of ∆CDE and its translated
image. What can you tell me about the distances?”
• “Measure the distance from point A and from point A' to the point of rotation. How does
this compare with other original and image points?”
69
Once students fundamentally understand the process of applying a transformation to
a shape, they should begin to explore relationships between the transformations while
considering orientation, congruency, and distance.
One way to explore these relationships is to create, analyse, and describe designs created
by multiple transformations. Look at the design below and the original shape used to
create the design. What transformations occurred in the creation of the design?
Original shape
Design
The generation of multiple responses allows for an examination of relationships between
transformations. Providing students with investigations and explorations that allow for
multiple solutions will also enable students with diverse learning needs and styles to be
successful. An example would be to provide students with an original shape and its image,
and ask them to identify two or more transformations that may have been applied to the
pre-image to produce the resulting image. The number of transformations, as well as the
sophistication of student responses, allows teachers to identify individual learning needs.
By allowing students to share their responses, teachers provide a forum for students to
construct their own learning.
Relationships Between Transformational
and Coordinate Geometry
Although geometric learnings of location and movement are separate and distinct, it is
important to help students discover connections between the two areas of geometry. Welldeveloped spatial sense, which is based on an awareness of the location of objects in one’s
surroundings, also encompasses knowledge of how objects can be moved about the space
to create and organize space.
Using grid systems and coordinate systems as tools for exploring transformations enables
students to make connections between location and movement, and provides a vehicle for
describing resultant images. For example, students can use grid squares to describe a translation
that has occurred, or can use grid squares to perform a translation and describe the position of
the translated image. Using a coordinate plane, students can translate the vertices of a shape to
create an image. Investigations can focus on translating shapes, identifying the translation that
has taken place, and determining whether a translation has been accurately performed.
70
Geometry and Spatial Sense, Grades 4 to 6
Coordinate systems can help students better understand how distance is preserved in transformations. When a point is reflected, the original point and its image are the same distance
from the line of reflection.
10
9
8
J
7
J'
6
Both J and J' are 3
units from the line of
reflection. Both L and
L' are 2 units away.
K
5
K'
4
L'
L
3
2
1
6
5
4
3
2
1
7
9
8
10
Investigating the concept of distance in rotations is difficult to do unless a coordinate
system is used. When a shape is rotated about a point, there is a direct relationship between
the points of the pre-image and the points of the rotated image – corresponding points are
an equal distance from the point of rotation.
10
9
H
8
7
6
P
5
G"
G
4
G'
3
H'
2
H"
1
1
2
3
4
5
6
7
8
9
10
In the figure above, the yellow hexagon was rotated first by a 1/4 turn to the left, resulting in
the green hexagon. It was then rotated 1/2 turn to the right, resulting in the blue hexagon.
In all three shapes, the point G and its rotated image points G' and G" are 2 units away
from the point of rotation P. Points H, H', and H" are 3 diagonal units away from the point
of rotation P.
71
An effective means of investigating this property of distance is to provide students with a
pre-image and its rotated image, and ask them to find the point of rotation.
When students use what they know about location to investigate the movement of
shapes, they are likely to develop a greater understanding of both concepts. Although
each topic can be learned separately, exploring the relationships between the two can
help students develop greater spatial sense and awareness.
72
Geometry and Spatial Sense, Grades 4 to 6
References
Elementary Teachers’ Federation of Ontario. (2004.) Making math happen in the junior years.
Toronto: Author.
Expert Panel on Literacy and Numeracy Instruction for Students With Special Education
Needs. (2005). Education for all: The report of the Expert Panel on Literacy and Numeracy
Instruction for Students With Special Education Needs. Toronto: Ontario Ministry of Education.
Expert Panel on Mathematics in Grades 4 to 6 in Ontario. (2004). Teaching and learning
mathematics: The report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario.
Toronto: Ontario Ministry of Education.
Gavin, M.K., Belkin, L.P., Spinelli, A.M., & St. Marie, J. (2001). Navigating through geometry in
Grades 3–5. Reston, VA: National Council of Teachers of Mathematics.
Glass, B. (2004). Transformations and technology: What path to follow? Mathematics
Teaching in the Middle School, 9(7), 392–397.
Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Erlbaum.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. Reston, VA: Author.
Ontario Ministry of Education. (2004). The individual education plan (IEP): A resource guide.
Toronto: Author.
Ontario Ministry of Education. (2005). The Ontario curriculum, Grades 1–8: Mathematics.
Toronto: Author.
Ontario Ministry of Education. (2006). A guide to effective instruction in mathematics,
Kindergarten to Grade 6. Toronto: Author.
Van de Walle, J., & Folk, S. (2005). Elementary and middle school mathematics: Teaching
Van de Walle, J. (2006). Teaching student-centered mathematics, Grades 3–5. Toronto: Pearson
73
Learning
Activities
Introduction to the Learning
Activities
The following learning activities for Grades 4, 5, and 6 provide teachers with instructional ideas
that help students achieve some of the curriculum expectations related to Geometry and Spatial
Sense. The learning activities also support students in developing their understanding of the big
ideas outlined in the first part of this guide. Learning activities are provided for the following
topics: two-dimensional geometry, three-dimensional geometry, location, and movement.
The learning activities do not address all concepts and skills outlined in the curriculum document, nor do they address all the big ideas – one activity cannot fully address all concepts, skills,
and big ideas. The learning activities demonstrate how teachers can introduce or extend mathematical concepts; however, students need multiple experiences with these concepts to develop
a strong understanding.
Each learning activity is organized as follows:
OVERVIEW: A brief summary of the learning activity is provided.
BIG IDEAS: The big ideas that are addressed in the learning activity are identified. The ways
in which the learning activity addresses these big ideas are explained.
CURRICULUM EXPECTATIONS: The curriculum expectations are indicated for each learning
activity.
ABOUT THE LEARNING ACTIVITY: This section provides guidance to teachers about the
approximate time required for the main part of the learning activity, as well as the materials,
math language, instructional groupings, and instructional sequencing for the learning activity.
and skills addressed in the learning activity.
GETTING STARTED: This section provides the context for the learning activity, activates prior
knowledge, and introduces the problem or task.
WORKING ON IT: In this part, students work on the mathematical task, often in small
groups or with a partner. The teacher interacts with students by providing prompts and
REFLECTING AND CONNECTING: This section usually includes a whole-class debriefing time
that allows students to share strategies and the teacher to emphasize mathematical concepts.
ADAPTATIONS/EXTENSIONS: These are suggestions for ways to meet the needs of all learners
in the classroom.
77
ASSESSMENT: This section provides guidance for teachers on assessing students’ understanding of mathematical concepts.
HOME CONNECTION: This section is addressed to parents or guardians, and includes a task
for students to do at home that is connected to the mathematical focus of the learning activity.
LEARNING CONNECTIONS: These are suggestions for follow-up activities that either extend
the mathematical focus of the learning activity or build on other concepts related to the topic
of instruction.
BLACKLINE MASTERS: These pages are referred to and used throughout the activities and
learning connections.
78
Geometry and Spatial Sense, Grades 4 to 6
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
Two-Dimensional Shapes:
Comparing Angles
This activity is adapted, with permission, from John A. Van de Walle, Teaching StudentCentered Mathematics, Grades 3–5. Toronto: Pearson: 2006, p. 254. The activity is also found
in the Grades 5–8 book in the series.
OVERVIEW
In this learning activity, students use benchmark angles, pattern blocks, and unit angles as a
reference to find angles smaller than, equal to, and larger than 90º. The focus is on gaining a
visual reference for important benchmark angles.
BIG IDEAS
This learning activity focuses on the following big idea:
Properties of two-dimensional shapes and three-dimensional figures: Students develop
a sense of the size of benchmark angles through investigation. Concrete materials and the
classroom environment provide measuring tools for comparing angles to benchmarks.
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectations.
Students will:
• identify benchmark angles (e.g., straight angle, right angle, half a right angle), using a reference tool (e.g., paper and fasteners, pattern blocks, straws), and compare other angles to
these benchmarks (e.g., “The angle the door makes with the wall is smaller than a right angle
but greater than half a right angle.”);
• relate the names of the benchmark angles to their measures in degrees (e.g., a right angle is 90º).
These specific expectations contribute to the development of the following overall
expectation.
Students will:
• identify quadrilaterals and three-dimensional figures and classify them by their geometric
properties, and compare various angles to benchmarks.
79
TIME:
approximately
60 minutes
Materials
• pattern blocks (1 set per each pair of students)
• tagboard or index cards (1 piece per student)
• scissors (1 per small group of students)
• 2D4.BLM1: Pattern Block Angles Chart (1 per student)
• 2D4.BLM2: Angle Search (1 per student)
Math Language
INSTRUCTIONAL
GROUPING:
whole class,
pairs
• ray
• obtuse angle
• angle
• acute angle
• right angle
• unit angle
Instructional Sequencing
This learning activity provides for students an opportunity to reinforce their understanding of angle
concepts and to develop an understanding of benchmark angles. It is intended as an introductory
lesson, and students should have been introduced to the concept of angle prior to the lesson.
Angles can be a difficult geometric concept for students, often because too much emphasis
is placed on measuring angles and not enough on developing an understanding of angles.
Comparing angles to benchmark angles enables students to focus on the angles themselves
and not the measuring of angles.
One common misconception about angles is that degrees are needed to measure angles. A
degree is no more than a small angle. Just as smaller units of length are used to measure a
length or distance, the unit used for measuring an angle must also be an angle.
A physical model of an informal unit (such as a wedge of tagboard or pattern block) can help
students visualize benchmark angles, as well as introduce them to the skills required in measuring angles. Using informal units makes it easier to focus directly on the attribute rather than the
measuring procedure. It is important when measuring angles to begin with informal units and
then move to degrees and formal measuring tools like protractors.
GETTING STARTED
An angle can be thought of as one line segment rotating away from the other. Use two strips
of tagboard or cardboard joined with a paper fastener to form a vertex to demonstrate this
idea. As you rotate one strip of the tagboard, the size of the angle is seen to get larger. Ask
students what the space in between the two line segments is called. Explain to students that
the distance between the two line segments is called an “angle” and that today they will be
looking at different angles found in polygons and in the classroom.
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Geometry and Spatial Sense, Grades 4 to 6
• Show students an orange square pattern block, and then place it on the overhead projector.
Trace the square. Ask a student to identify one angle. Trace the angle, drawing an arc to
kind of angle. Define a square corner as a right angle. Elicit from the class the fact that all
rectangles (including squares) have four right angles.
• Show students a green triangle pattern block, and then place it on the overhead projector.
Trace the triangle and ask a student to identify one angle. Trace the angle, drawing an arc
to indicate the angle between the two line segments. Repeat with the other two angles. Ask
students how these angles compare to those found in the square. Define an angle that is
smaller than a right angle as an acute angle.
• Show students a tan rhombus pattern block, and then place it on the overhead projector. Trace the rhombus. Ask a student to identify an angle larger than a right angle in the
rhombus. Trace the angle, drawing an arc to indicate the angle between the two line
segments. Define an angle that is larger than a right angle as an obtuse angle.
Create an anchor chart on the board or flip chart from the pattern block examples. Draw a
right angle, an acute angle, and an obtuse angle. Invite students to make observations and
name each angle.
WORKING ON IT
Ask the students, working as partners, to identify the number and types of angles found in the
remaining three pattern blocks (trapezoid, blue rhombus, and hexagon). Ask them to use the
angles they have found in the square, triangle, and tan rhombus as tools to help them.
Students are to communicate their findings in 2D4.BLM1: Pattern Block Angles Chart.
The chart includes:
• the pattern block
• the number and type of each angle found in the pattern block
• how they determined each angle
Sample student strategy
I used the square pattern block to help me find the angles in the trapezoid. The two top
angles were outside the square, so they are bigger than 90°, and are obtuse. The two bottom
angles are inside the square, so they are less than 90°, and are acute.
Next, give each student an index card or a small piece of tagboard. Have students use the
acute angle from the tan rhombus pattern block to draw a narrow angle on the tagboard or the
card and then cut it out. The resulting wedge can be used as a unit angle. Students will use this
unit angle to find and identify three other angles found anywhere in the classroom. They will
record their findings in the final three rows of the chart.
Students may use the pattern blocks to help them find relationships between the unit angle
and other angles. For example, students may discover that it takes three unit angles to make
a right angle.
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
81
REFLECTING AND CONNECTING
This activity results in an understanding of benchmark angles – in particular, the benchmark
angle of 90°. Students also begin to develop an understanding of the measurement process
– iterating a unit angle to measure a larger angle.
Ask students to explain their findings. Use probing questions like the following:
• “How did you decide what types of angles there are in each pattern block?”
• “What did you use as a comparison or benchmark angle?”
• “Did you have to do a comparison for every angle in each pattern block? Why or why not?”
• “Could you have used a different benchmark angle? Would that have made things easier or
more difficult?”
Students should realize that using the square, or right angle, as a benchmark angle is the most
efficient way of categorizing angles as acute, right, or obtuse.
Have students share their explorations of environmental angles and their explanations of how they
used the tagboard unit angle to measure or compare angles. Ask questions like the following:
• “How did you use your unit angle?”
• “Did you have any difficulties using it to measure other angles?”
• “How could you describe the size of the angle you measured?”
• “How did you know if the angle you measured was obtuse, right, or acute?”
• “Did anyone use his or her unit angle in a different way?”
• “What were the most common types of angles you could find? Why do you think that is?”
Use the students’ examples to emphasize the focus of the lesson – angles can be classified
using a right angle as a benchmark. Angles larger than 90° are obtuse, and angles smaller than
90° are acute. In later grades, students will learn that a straight angle is 180°, but at Grade 4 it
is enough to generalize about angles larger than 90°.
A unit angle that is iterated a number of times can help students visualize more exact benchmarks. Students may find that it takes about one and one-half of their unit angles to make a
“half of a right angle”, or that one unit angle is a third of a right angle. Even if they do not come
to recognize the fractional representation, or to realize that the actual measure of the unit angle
is 30°, it is important for them to be aware that it takes three unit angles to make a right angle.
Students who struggle making comparisons may need to use a model of a right angle (made
of tagboard) instead of the acute angle from the tan rhombus. Students can immediately see
whether an angle is right, obtuse, or acute if they use a right angle as a benchmark.
As an extension, ask students to compare the kinds of angles they see most frequently. Encourage
students to discuss why there are so many right angles in the classroom and environment.
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Geometry and Spatial Sense, Grades 4 to 6
ASSESSMENT
Observe students as they complete the chart, and assess how well they:
• identify benchmark angles (i.e., acute, right, obtuse) using a reference tool;
• compare angles to benchmark angles;
• classify and sort angles according to benchmarks;
• relate the names of the benchmark angles to their measures in degrees (e.g., a right
angle is 90º).
The charts can be collected and assessed for students’ use of appropriate terminology
in communication.
HOME CONNECTION
Send home 2D4.BLM2: Angle Search. In this task, students are asked to look for angles in
their environment and to record where each angle was found, what kind of angle it is, and why
the angle is important. Make sure students take their unit angles home for this activity, and
LEARNING CONNECTION 1
Materials
• 2D4.BLM3: Quadrilateral Twister Floor Mat
• scissors
Prepare a floor mat, using 2D4.BLM3: Quadrilateral Twister Floor Mat as a template. Prepare
cards from 2D4.BLM4: Quadrilateral Twister Cards, or have students prepare them.
Students work in groups of four. Each student in the group has a role:
• Card Manager
• Player 1 and Player 2
The Card Manager shuffles the cards and holds them. The Card Reader draws a card and reads
the properties to one of the players. Player 1 and Player 2 select the quadrilateral on the mat
to match the card. The Card Reader verifies that each player has chosen the correct shape (the
names of the shapes are on the bottoms of the cards). A player keeps the designated hand or
foot on the selected quadrilateral until a card is drawn with instructions for that hand or foot
to move elsewhere.
The game continues until one of the players can no longer reach a quadrilateral that matches his
or her card. The pairs then change roles and a new game begins. The game may also be played
with the players alternating cards, instead of each player trying to perform the same move.
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
83
LEARNING CONNECTION 2
Materials
• card stock
• paper cut into strips
• 2D4.BLM6: Quadrilaterals Rule Sorting Chart
Students work in pairs for this activity.
This activity provides students with the opportunity to sort quadrilaterals and then discover sorting
rules. The activity will focus the students’ attention on the properties of two-dimensional shapes
and the relationships between the shapes. Copy the shapes from the 2D4.BLM5: Quadrilateral
Shape Cards onto card stock and cut them out. Use strips of paper to make these statements:
• No right angles
• One or more right angles
• One or more acute (smaller than 90˚) angles
• One or more obtuse (larger than 90˚) angles
• All sides the same length
• Two pairs of adjacent sides congruent
• One set of parallel sides
Make blank strips available for students who want to write their own rule. One student sorts
some of the shapes into the two spaces in the first column of 2D4.BLM6: Quadrilaterals Rule
Sorting Chart. The other student uses the sorting rule strips to determine the rule(s). Students
take turns sorting and guessing.
LEARNING CONNECTION 3
Materials
• 10 straws per group
• paper and pencil to record the scores
Students work in groups of two or three for this game. They use straws to create and identify
straight, acute, obtuse, and right angles.
The first student holds ten straws, with the bottom of the straws touching the desk or table
top. When the student opens his or her hand, the straws fall to create angles. Each straight
angle (180°) is worth one point; obtuse angles (more than 90°) and acute angles (less than 90°)
are both worth two points; and right angles (90°) are worth three points. Students take turns
and record their scores for each round. The first student to gain 20 points wins the game.
The straws must be touching to form an angle – close doesn’t count!
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Geometry and Spatial Sense, Grades 4 to 6
Object Measured
Number and Type
of Angles
2D4.BLM1
Pattern Block Angles Chart
How Do You Know?
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
85
2D4.BLM2
Angle Search
Dear Parent/Guardian:
In math this week, we have been learning about angles. Your child has been measuring different
angles in class, using a “unit” angle to determine if the angles are acute, right, or obtuse angles.
Here is an opportunity for your child to demonstrate what he or she has learned.
Ask your child to find different angles in your home. Using the chart below, describe where the
angle was found, what kind of angle it is, and why the angle is important for that object. Try to
find at least one example of each kind of angle (acute, right, obtuse).
Where was the angle
found?
What kind of angle is it?
Thank you for your assistance with this activity.
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Geometry and Spatial Sense, Grades 4 to 6
Why is the angle
important?
2D4.BLM3
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
87
2D4.BLM4
88
Right hand
on a shape
that has:
Right hand
on a shape
that has:
Right hand
on a shape
that has:
Right hand
on a shape
that has:
• two pairs of
congruent
• only one pair
of parallel sides
• two pairs of
parallel sides
• two pairs of
parallel sides
Shape: kite
Shape: trapezoid
Shape: rectangle
Shape: parallelogram
Left hand
on a shape
that has:
Left hand
on a shape
that has:
Left hand
on a shape
that has:
Left hand
on a shape
that has:
• two pairs of
congruent
• only one pair
of parallel sides
• two pairs of
parallel sides
• four sides
Shape: kite
Shape: trapezoid
Shape: rectangle
Right foot on a shape
that has:
Right foot on a shape
that has:
Right foot on a shape
that has:
Right foot on a shape
that has:
• two pairs of
congruent
• only one pair of
parallel sides
• two pairs of
parallel sides
• two pairs of
parallel sides
Shape: kite
Shape: trapezoid
Shape: rectangle
Shape: parallelogram
Left foot on a shape
that has:
Left foot on a shape
that has:
Left foot on a shape
that has:
Left foot on a shape
that has:
• two pairs of
congruent
• only one pair of
parallel sides
• two pairs of
parallel sides
• four sides
Shape: kite
Shape: trapezoid
• four right angles
• no congruent sides
• four right angles
• four right angles
• no congruent sides
• four right angles
Geometry and Spatial Sense, Grades 4 to 6
Shape: rectangle
Grade 4 Learning Activity: Two-Dimensional Shapes – Comparing Angles
2D4.BLM5
89
2D4.BLM6
90
Sorting Space:
I think the rule is/rules are:
Sorting Space:
I think the rule is/rules are:
Geometry and Spatial Sense, Grades 4 to 6
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
Three-Dimensional Figures:
Construction Challenge
OVERVIEW
In this task, students work with Polydron pieces that represent the two-dimensional faces of
three-dimensional figures. Using a given set of these shapes, students try to construct and
name as many three-dimensional figures as they can. This task also provides students with
opportunities to describe and classify prisms and pyramids by their geometric properties as
they sort the three-dimensional figures constructed by each group.
BIG IDEAS
This learning activity focuses on the following big ideas:
Properties of two-dimensional shapes and three-dimensional figures: Students develop an understanding of the properties of prisms and pyramids through the use of two-dimensional shapes.
Geometric relationships: Students investigate the relationships between two-dimensional
shapes and three-dimensional figures. They also develop an understanding of how different
properties of three-dimensional figures (e.g., edges and faces) are related.
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectations.
Students will:
• identify and describe prisms and pyramids, and classify them by their geometric properties
(i.e., shape of faces, number of edges, number of vertices), using concrete materials;
• construct three-dimensional figures (e.g., cube, tetrahedron), using only congruent shapes.
These specific expectations contribute to the development of the following overall expectation.
Students will:
• construct three-dimensional figures, using two-dimensional shapes.
Materials
TIME:
approximately
60 minutes
• triangular prism and square-based pyramid solids (for teacher demonstration)
• 2 demonstration Polydron sets consisting of 4 equilateral triangles and 3 squares
• Polydron sets (for this activity, 1 set consists of: 6 equilateral triangles, 6 squares,
2 hexagons, and 2 pentagons) (1 set per group)
91
• 3D4.BLM1: 3-D Construction Challenge (1 per pair of students)
• chart paper (1 per group of students)
• 3D4.BLM2a–b: What Figures Can You Construct? (1 per student)
Math Language
• three-dimensional figure
• triangular pyramid
• hexagonal prism
• two-dimensional shape
• cube
• face
• triangular prism
• rectangular prism
• edge
• parallel
• pentagonal prism
• vertex
• congruent
• pentagonal pyramid
• vertices
• square-based pyramid
INSTRUCTIONAL
GROUPING:
whole class,
small groups
Instructional Sequencing
This lesson is intended to be delivered in the middle of a geometry unit on three-dimensional
figures. Students should be familiar with the names of three-dimensional figures but do not need
to have had significant experience with their properties. In this activity students investigate
many of the properties of prisms and pyramids.
In Grade 4, students practise constructing three-dimensional figures from two-dimensional
shapes. They build on their experience with prisms and investigate pyramids.
The focus of this lesson is identifying the similarities and differences among and between
prisms and pyramids. Included in these are the following:
• All prisms have congruent parallel bases that are joined by rectangular faces.
• All pyramids have one base with triangular faces that meet at one common vertex
(called an apex).
• The names of both prisms and pyramids are determined by the two-dimensional shapes that
form their bases.
Working with two-dimensional shapes also develops an understanding of how various properties (e.g., edges and faces) are related. For example, two faces can share an edge only if the
sides are adjacent and congruent in length.
These two sides are
congruent in length, and
fold together to form an
edge of this pyramid.
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Geometry and Spatial Sense, Grades 4 to 6
Although these two sides
congruent in length and
will not fold together to
form an edge.
Students have opportunities to represent their ideas physically, share their thinking verbally,
and record their ideas in writing. While they work in their groups, students apply problemsolving strategies to build as many different three-dimensional figures as they can and to test
whether they have found all of the figures that they believe are possible.
Getting Started
Show students a triangular prism. Ask them to name the three-dimensional figure and explain
why it is called a triangular prism. Recall or explain that prisms are named according to their
congruent parallel bases.
triangular base
triangular base
Show students a square pyramid. Again, ask students to name the figure and explain why it is
called a square pyramid.
square base
Allow students a few minutes to discuss with their elbow partner how prisms and pyramids are
named. Elicit responses from the class, reiterating that the shape of the base determines the
name of the pyramid or prism.
Introduce the following scenario:
“A company that makes two-dimensional shapes is trying to sell a small set of the
shapes as a new building toy. The two-dimensional shapes look like these Polydron
pieces. (Show the following seven Polydron pieces: four equilateral triangles and three
squares.) The company wants to be able to tell people about all the different threedimensional figures that can be constructed with their new building set. The company
has asked for your help in constructing all the three-dimensional figures that are possible from the two-dimensional shapes in the set, and asks you to record your findings.”
Ask students to consider the seven Polydron pieces that you just showed them. “What threedimensional figure can be constructed from these two-dimensional shapes?” Invite a student to
come up to build 1 three-dimensional figure from some of the seven Polydron pieces and name it.
Ask: “Is it possible to build another three-dimensional figure from the same seven pieces?”
Invite another student to build and name a second three-dimensional figure using the second
set of the same seven Polydron pieces.
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
93
Ask the students to compare the 2 three-dimensional figures. “How are the two figures alike?
How are they different?” (There are three possible figures that students could construct: a
triangular prism, a triangular pyramid, or a square-based pyramid.) Encourage students to
consider the properties of three-dimensional figures when comparing the two figures (i.e.,
number and shapes of faces, number of edges, number of vertices).
Students may recognize that a third three-dimensional figure can be constructed from the
same seven Polydron pieces. Assure them that they will have an opportunity to construct any
other figures that they believe consist of some of these seven faces when they begin working
in their groups.
Working on It
Arrange students in groups of four. Distribute one copy of 3D4.BLM1: 3-D Construction
Challenge to each pair of students and one set of Polydron pieces (six each of equilateral
triangles and squares, and two each of pentagons and hexagons) for each group.
Instruct students to work with a partner within their groups to build 1 three-dimensional figure
from the two-dimensional shapes. Once both pairs have built one figure, they record the name
of their figure on 3D4.BLM1: 3-D Construction Challenge and share it with the other pair in
their group. The two pairs then discuss and record the similarities of and differences between
the 2 three-dimensional figures that they have just constructed.
Note: It is possible to build seven standard three-dimensional figures from the given Polydron
set (triangular pyramid, triangular prism, square-based pyramid, cube, pentagonal prism,
hexagonal prism, and triangular dipyramid made from six equilateral triangles). A pentagonal
pyramid is possible in theory, but the Polydron edges will not fold far
Assessment Opportunity Circulate to assess students’
problem-solving strategies
as they try to build as many
three-dimensional figures as
possible from the set of twodimensional shapes. If students
believe that they have found
all of the possible figures, ask,
“How do you know?”, to obtain
problem-solving strategies.
enough to construct a pentagonal pyramid without breaking apart. A
hexagonal pyramid is not possible with equilateral triangles, because
the angles of the six triangles that meet total 360º – the triangular faces
would lie flat on the hexagonal base.
Students continue to build and compare different three-dimensional
figures (taking the two figures apart after each comparison and beginning again) until the group members believe that they have constructed
all the three-dimensional figures possible, given the two-dimensional
shapes provided.
Students may also construct non-standard three-dimensional figures by
combining figures. For example, students may combine a triangular prism and a square-based
pyramid by removing the square base of the pyramid and using one of the square faces of
the prism as the base. The resulting three-dimensional figure is an octahedron (named for the
number of faces – eight). Several such combinations can be constructed from the given set
of Polydron shapes. Students should be encouraged to name these figures according to the
number of faces in the figure.
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Geometry and Spatial Sense, Grades 4 to 6
Reflecting and Connecting
Select a few groups to choose 2 three-dimensional figures from those they have constructed
to rebuild and share with the class. Ask group members to be prepared to share how the 2
three-dimensional figures they have chosen are alike and how they are different.
Have each group share its 2 three-dimensional figures and explain the similarities and differences. You may wish to do this orally as a large group, or have groups prepare a poster
outlining the similarities and differences.
Display on the board ledge the three-dimensional figures constructed by
each group. After the members of the second group have shared their
figures, ask the students how they can sort the four figures now on the
ledge. Record the sorting rule on the board, and have the students place
each figure in a group according to the rule. (Students may create more
than two groups, depending on what sorting rule is determined.)
As each new group of students shares, have its members sort their two
Assessment Opportunity
As students present the
similarities and differences
between their two shapes,
assess their ability to express
(represent) their thinking using
the models and appropriate
mathematical language.
figures into the groups on the board ledge and explain why each figure
belongs in the particular group chosen. Allow students to question the sorting if they disagree or
do not understand the group members’ reasoning. Allow groups to suggest a new sorting rule if
it better emphasizes the properties of three-dimensional figures.
After all the groups have shared, review the sorting rule and the figures in each group. Ask:
• “Were any of the three-dimensional figures that you constructed not included in
these groups?”
• “What is the three-dimensional figure and in which group does it belong? Why?”
• “What part of this investigation did you find easy? What part was challenging?”
Connect the activity with the original challenge, asking students if there are enough pieces
in the set to create an appropriate number of three-dimensional figures. Ask students what
pieces they might add to the set, and why.
Some students may have difficulty finding the similarities of and differences between the 2
three-dimensional figures, especially if both figures are prisms or pyramids. Encourage these
students to count the number of faces, edges, and vertices, and/or compare the shapes of the
two-dimensional faces in each figure. You may wish to create/include an anchor chart that can
serve as a starting point for students. List the above strategies – count the number of faces,
and so forth.
Students requiring an extension can be challenged to construct two or more three-dimensional
figures with a specified number of faces, edges, or vertices (e.g., construct two or more threedimensional figures with 8 edges).
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
95
Assessment
Observe students to assess how well they:
• construct three-dimensional figures from two-dimensional shapes;
• identify and describe prisms and pyramids;
• classify prisms and pyramids by geometric properties
As students are constructing and comparing their figures, circulate around the room and ask
students to describe the features of one figure and tell how it is similar to or different from
another figure. Record anecdotal observations about the accuracy of their responses and their
ability to communicate effectively.
Home Connection
Send home 3D4.BLM2a–b: What Figures Can You Construct? In this task, students are given
a set of two-dimensional shapes and asked to name 4 three-dimensional figures that can be
constructed from those shapes. Suggest to students that they may wish to cut out the shapes
and tape them together if it helps them visualize the figures.
Learning Connection 1
Constructing Skeletons
Materials
• 3D4.BLM3: Constructing Skeletons
• round toothpicks (16 per pair of students)
• modelling clay (small block per pair of students)
Have students work in pairs to construct and name as many different three-dimensional skeletons
as possible with the toothpicks and modelling clay provided.
Have students record the names of their constructions as well as the number of toothpicks and
pieces of connective modelling clay required for each figure on 3D4.BLM3: Constructing Skeletons.
Learning Connection 2
Teach an Alien
Materials
• rectangular prism (for teacher demonstration)
• 3D4.BLM4: Teach an Alien (1 per student)
• three-dimensional solid figures (cube, triangular pyramid, triangular prism, square pyramid,
pentagonal prism) for student reference
96
Geometry and Spatial Sense, Grades 4 to 6
Begin by showing students a rectangular prism. Ask them to describe the three-dimensional
figure in such a way that someone who has never seen one (e.g., a reading buddy, younger
sibling) would know what it looks like. Encourage students to use appropriate mathematical
language and to describe the geometric properties of the prism.
Distribute to each student a copy of 3D4.BLM4: Teach an Alien. Have students select one net
from the five shown and identify the three-dimensional figure that can be constructed from that
net. Then, have students write a description of the figure for an alien from another planet who
has never seen that figure before. Student pairs will exchange descriptions and try to build the
figure from each other’s descriptions.
Learning Connection 3
Cube Structures
Materials
• transparency of 3D4.BLM5: 6-Cube Rectangular Prism
• interlocking cubes (8 per student)
• 3D4.BLM6: More Rectangular Prisms (1 per pair of students)
Begin by showing students the first prism, 3D4.BLM5: 6-Cube Rectangular Prism, on the overhead.
Distribute 8 interlocking cubes to each student. Have students try to construct the 6-cube
rectangular prism shown on the overhead.
Distribute 3D4.BLM6: More Rectangular Prisms to each pair of students. Have students each
construct the first prism and compare it with his or her partner’s prism. When both students
agree that their prisms are congruent and match the prism in the picture, they move on to
construct the next rectangular prism.
Once they have accurately constructed all four of the rectangular prisms pictured, students
are asked to work with their partners to construct as many different 12-cube rectangular
prisms as they can.
Learning Connection 4
Identify the Nets
Materials
• 3D4.BLM7a–b: Identify the Nets (1 per student)
• Polydron shapes
Begin by showing students a net for a triangular prism made from Polydron shapes. Ask
students to identify the three-dimensional figure that will be created by folding the net.
Next, show students an arrangement of the same Polydron shapes that will not fold to make
a triangular prism. Ask students whether or not the arrangement is a net, and ask them to
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
97
explain why or why not. When students have recognized (through visualization or folding) that
the arrangement is not a net for a triangular prism, ask them how they could change it to make
a net. Invite a student to come up to do so.
Distribute a copy of 3D4.BLM7: Identify the Nets to each student. Have students work with
a partner to determine which arrangements of two-dimensional shapes are nets (i.e., will fold
to make a three-dimensional figure). Have Polydron shapes available for students for the
activity. Alternatively, enlarge the page on a photocopier for those students who might need
to cut out the arrangements and try folding them to determine whether or not they make
three-dimensional figures.
If students determine that an arrangement is a net, have them identify the three-dimensional
figure that the net will make. Then, have students redraw the arrangements that are not nets
so that they become nets for rectangular or triangular prisms.
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Geometry and Spatial Sense, Grades 4 to 6
3D4.BLM1
3-D Construction Challenge
Names: Name of Your
Figure
Name of Other
Figure
Similarities
Differences
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
99
3D4.BLM2a
What Figures Can You Construct?
Dear Parent/Guardian:
In math this week, we have been learning about constructing three-dimensional figures from
two-dimensional shapes. Your child has been using two-dimensional Polydron shapes in class to
build three-dimensional models of figures.
Here is an opportunity for your child to demonstrate what he or she has learned.
Ask your child to find at least 4 three-dimensional figures that can be constructed from the set
of shapes pictured on the next page. The shapes may be cut out and taped together if it helps
To verify the answers, ask your child to list all of the two-dimensional shapes that form the
faces of each figure that he or she has named.
Name of three-dimensional figure
cube
100
Geometry and Spatial Sense, Grades 4 to 6
Numbers and shapes of all faces
6 squares
3D4.BLM2b
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
101
3D4.BLM3
Constructing Skeletons
Work with a partner.
Using a maximum of 16 toothpicks and 10 pieces of modelling clay for
each skeleton, construct as many skeletons of three-dimensional figures
as you can.
(You will need to take each skeleton apart after you have recorded the
information about it in the table below.)
Name of Figure
102
Number of Edges
(toothpicks)
Geometry and Spatial Sense, Grades 4 to 6
Number of Vertices
(pieces of modelling clay)
3D4.BLM4
Teach an Alien
1. Choose one of these nets of three-dimensional figures.
A.
B.
D.
C.
E.
2. Identify the figure.
Net will fold to make a .
3. Write a description of the figure for an alien from another planet who has
never seen the figure before.
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
103
3D4.BLM5
6-Cube Rectangular Prism
Use 6 connecting cubes to construct a rectangular prism.
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Geometry and Spatial Sense, Grades 4 to 6
3D4.BLM6
More Rectangular Prisms
Construct each of the following rectangular prisms, using the number of connecting cubes indicated. Compare your completed prism with your partner’s
prism. Once you and your partner agree that the two prisms are congruent and
match the prism in the picture, build the next prism.
A.
B.
Use 4 cubes.
Use 5 cubes.
C.
D.
Use 6 cubes.
Use 8 cubes.
Work with your partner to construct as many different 12-cube rectangular
prisms as you can.
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
105
3D4.BLM7a
Identify the Nets
Determine whether each of the arrangements of two-dimensional shapes
below is a net. If it is a net, write the name of the three-dimensional figure that
A.
B.
q Net
q Not a net
q Net
3-D Figure: _________
C.
3-D Figure: _________
D.
q Net
q Not a net
q Net
3-D Figure: _________
E.
q Not a net
3-D Figure: _________
F.
q Net
q Not a net
3-D Figure: _________
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q Not a net
Geometry and Spatial Sense, Grades 4 to 6
q Net
q Not a net
3-D Figure: _________
3D4.BLM7b
For the arrangements that you checked as “Not a net”, redraw the shapes below
so that the arrangements are nets for rectangular or triangular prisms.
Grade 4 Learning Activity: Three-Dimensional Figures – Construction Challenge
107
Grade 4 Learning Activity: Location – Check Mate
Location: Check Mate
Overview
In this learning activity, students play a game in which they determine the location of four
rectangles on a coordinate grid. The game provides an opportunity for students to identify the
location of objects using a grid system.
Big Ideas
This learning activity focuses on the following big idea:
Location and movement: Students describe the position of an object, using a coordinate
grid system.
Curriculum Expectations
This learning activity addresses the following specific expectation.
Students will:
• identify and describe the general location of an object using a grid system (e.g., “The library
is located at A3 on the map.”)
This specific expectation contributes to the development of the following overall expectation.
Students will:
• identify and describe the location of an object, using a grid map, and reflect twodimensional shapes
TIME:
approximately
60 minutes
Materials
• Transparency of the “Your Shapes” grid from Loc4.BLM1a–b: Game Boards
• large copy on chart paper or the board of the “Your Opponents’ Shapes” grid from
Loc4.BLM1a–b: Game Boards,or a learning carpet or 10  10 grid on a carpet and craft sticks
• Loc4.BLM2: Game Instructions (1 per student)
• Loc4.BLM1a–b: Game Boards (1 per student)
• sheets of chart paper or large newsprint (1 per pair of students)
• markers (1 per student)
• Loc4.BLM3: Playing Check Mate (1 per student)
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Math Language
• grid
• rectangle
• cell
• square
• location
• hexagon
• inside
• trapezoid
• outside
• triangle
Instructional Sequencing
This learning activity provides an introduction to the use of a grid system. Students locate
INSTRUCTIONAL
GROUPING:
pairs
objects within cells on a grid. Before starting this activity students should review the different
types of quadrilaterals and their properties.
The ability to use a grid system in describing the location of an object is an important component of spatial reasoning. This activity builds on prior knowledge developed throughout the
location and movement activities established in the Geometry and Spatial Sense document for
the primary grades, and prepares students for the more formal identification of points using a
coordinate grid system.
Getting Started
Prior to the lesson, mark four rectangles on an overhead transparency of the “Your Shapes”
grid from Loc4.BLM1a–b: Game Boards, making at least one of them a square. Students will
attempt to locate the four rectangles while you keep them hidden.
On a large sheet of paper or on the board, make a copy of the “Your Opponents’ Shapes”
grid from Loc4.BLM1a–b: Game Boards as a reference for your students. Review how to read
across the bottom of the grid (naming a letter) before reading up the grid (naming a number)
to identify a cell.
Tell your students that on your overhead grid you have hidden four rectangles for them to find.
Select one student to be the “marker” on the large grid. The other students take turns to call out
locations, at each turn using a letter followed by a number. After a student has identified a cell on
the grid, call out “Check” if any part of a rectangle occupies that cell or “Miss” if the cell lies outside
the rectangles. The marker will then place on the grid a check mark for a hit or an X for a miss. When
the students have located all of the check marks for a particular rectangle, say, “Check Mate”.
Alternatively, the checks and X’s could be recorded on a learning carpet or 10  10 grid on a
carpet, with craft sticks used for making the marks.
Reveal your overhead transparency, indicating how the check marks and X’s match those on
the large grid. One of the students may tell you that you have drawn a square rather than a
rectangle. It is an ideal opportunity for a class review of the square as a special rectangle.
Provide each student with a copy of Loc4.BLM2: Game Instructions, and review the instructions for playing the game.
Grade 4 Learning Activity: Location – Check Mate
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Working on it
Have the class work in groups of four (two versus two) to allow for discussion and collaboration
while playing the game. Provide each student with a copy of Loc4.BLM1a–b: Game Boards.
As students play the game, observe the various strategies they use. Pose questions to help
• “What strategy are you using to hide your rectangles?”
• “What strategy are you using to search for your opponents’ rectangles?”
• “What strategy do you use once you get a ‘check’?”
• “Did you change any of your strategies? Why?”
When students have had a chance to play the game a number of times, provide each pair of
students with markers and a sheet of chart paper or large newsprint. Ask students to record
their game-playing strategies on the paper to show how they hid their rectangles and found
their opponents’ rectangles.
Make a note of pairs who might share their strategies during the Reflecting and Connecting part
of the lesson. Include groups whose methods varied in their degree of efficiency (e.g., hiding
rectangles at random, hiding rectangles in specific locations and orientations, making guesses at
random, using a specific pattern to guess).
Reflecting and Connecting
Once students have had several opportunities to play Check Mate, bring them together to
share their ideas and strategies. Try to order the presentations so that students observe inefficient strategies (picking cells at random) first, followed by more efficient methods.
As students explain their ideas, ask questions to help them to describe their strategies:
• ”How does your strategy work?”
• “Did you change your original strategy?”
• ”Did you know when you would get a second check mark?”
• “Would you change your strategy for hiding rectangles next time?”
Following the presentations, ask students to observe the work that has been posted and to
consider the efficiency of the various strategies. Ask:
• “Which strategy, in your opinion, is an efficient strategy for hiding rectangles?”
• “Which strategy, in your opinion, is an efficient strategy for finding rectangles?”
• “How would you explain the strategies to someone who has never used them?”
Avoid commenting that some strategies are better than others – students need to determine
for themselves which strategies are meaningful and efficient, and which ones they can make
sense of and use.
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Geometry and Spatial Sense, Grades 4 to 6
Refer to students’ work to emphasize geometric ideas:
• A grid system can be helpful in locating specific objects.
• The properties of a two-dimensional shape can help you predict where the shape will be on a
grid (e.g., the number of sides in a rectangle).
Provide students with an opportunity to play additional or related games to try new strategies.
For those students having difficulty with the concept, working on a smaller grid (e.g., 5  5),
with only one rectangle and one square on the grids may be helpful.
Students needing a challenge could use the hexagon, trapezoid, triangle, and square pattern
blocks. They draw around these shapes on their grid paper, making sure that they only draw
around full squares or half squares.
The challenge is greater than in “Check Mate” because when a student “hits” a half square, the
person with the drawing must say, “Top left”, “Top right”, “Bottom left”, or “Bottom right” to
indicate that a half square has a check mark placed in it.
Assessment
Observe students as they play the game and assess how well they:
• locate and mark specific cells on the grid;
• apply appropriate strategies to play the game;
• explain their strategy;
• judge the efficiency of various strategies;
• modify or change strategies to find more efficient ways to solve the problem.
Home Connection
Send home Loc4.BLM3: Playing Check Mate. In the letter, parents are told about the
game played in class and asked to have their child teach them how to play the game.
Photocopy Loc4.BLM2: Game Instructions onto the back of this letter. Attach two copies of
Loc4.BLM1a–b: Game Boards, so that parent and child have their own playing sheets.
LEARNING CONNECTION 1
Hot, Warm, Cold
Materials
• Overhead transparency of the “Your Shapes” grid from Loc4.BLM4: Instructions and Game
Boards for Hot, Warm, Cold
• Large copy of the “Your Opponents’ Shapes” grid from Loc4.BLM4: Instructions and Game
Boards for Hot, Warm, Cold
• Loc4.BLM4: Instructions and Game Boards for Hot, Warm, Cold (1 per student or pair
of students)
Grade 4 Learning Activity: Location – Check Mate
111
This game builds on the concept of locating cells on a grid and reviews the names of the five
quadrilaterals in the Grade 4 geometry expectations. The aim of the game is to be the first to
locate all five quadrilaterals ”hidden” by the other team. Students will benefit from a review of
the quadrilaterals and their properties before beginning this activity.
Prior to the lesson, on an overhead transparency of the “Your Shapes” grid from Loc4.BLM4:
Instructions and Game Boards for Hot, Warm, Cold, follow the directions for marking the five
On a large sheet of paper, or on the board, make a copy of the “Your Opponents’ Shapes” grid
from Loc4.BLM4: Instructions and Game Boards for Hot, Warm, Cold.
Play a practice game. Select one student to be the “marker” on the large grid. The other
students take turns calling out a location on the grid (e.g., D3). When the students miss a
quadrilateral, the marker places an X in the cell named, and you provide clues (as indicated on
Loc4.BLM4: Instructions and Game Boards for Hot, Warm, Cold). When the students have a
hit, say the name of the quadrilateral, and have the marker draw that shape in the cell named.
Mark on the concealed overhead transparency each of the locations that the students name,
saying and drawing the shape of the quadrilateral when that shape is hit, and putting an X in a
cell for a miss. The overhead transparency can then be compared with the large grid at the
end of the game.
Review the instructions on the blackline master for playing the game.
Have the class play the game in two’s or four’s. Pairing the students will allow for discussion
and collaboration. (When students are playing the game for the first time, it might be worthwhile to pair a weaker student with a stronger one.) Each individual or pair receives a copy of
Loc4.BLM4: Instructions and Game Boards for Hot, Warm, Cold (a “Your Shapes” grid for
drawing their quadrilaterals and recording the hits and misses of the other team, and a “Your
Opponents’ Shapes” grid for recording their own hits and misses).
LEARNING CONNECTION 2
Materials
• gridless map of the school neighbourhood or of a town or city (1 per student)
Have students find landmarks on the map and describe the location of each. Lead a discussion
about the locations of the landmarks on the map and the difficulties encountered in describing
those locations to others.
Challenge the students to think of ways in which they could make describing locations on the
map easier. Have them record their ideas on their map, or on separate pieces of paper.
Observe students as they solve the problem, and make note of the different methods that they
use. For example, students might:
112
Geometry and Spatial Sense, Grades 4 to 6
• create a legend to designate landmarks;
• circle landmarks on the map;
• create a grid system to identify cells on the map.
Ask several students to explain their strategies to the class. Include a variety of strategies.
After a variety of strategies have been presented, ask the following questions to help students
evaluate the different methods.
• “Which strategies were easy to use and efficient? Why?”
• “Which strategies are similar? How are they alike?”
• “What strategy would you use if you were to solve this problem again? Why?”
Grade 4 Learning Activity: Location – Check Mate
113
Loc4.BLM1a
Game Boards
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9
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7
6
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3
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A
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B
C
D
E
Geometry and Spatial Sense, Grades 4 to 6
F
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Loc4.BLM1b
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Grade 4 Learning Activity: Location – Check Mate
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Loc4.BLM2
Game Instructions
Instructions for Playing “Check Mate”
1. Making your own game board
Each person draws four rectangles on his or her “Your Shapes” grid, making at
least one of them a square. Your team needs to make sure that your rectangles
are in the same place.
Rectangles must not overlap and need to be traced over the grid lines. Teams
must not see their opponents’ grids.
2. Playing the game
On each team, each person has a copy of the game boards to keep track of the
guesses in the game.
Each team takes turns naming a location where they think their opponents may
have drawn a rectangle. Remember: name the letter before the number. Each
person records hits (√) or misses (X) on the “Your Opponents’ Shapes” grid to
keep track of the team’s guesses.
Each person marks on the “Your Shapes” game board the hits and misses made
by the opponents. If your opponents name a position inside one of your team’s
rectangles, say, “Check”, and make a check mark inside that cell. If your opponents name a position outside your team’s rectangles, say, “Miss”, and make
an X inside that cell. When the other team completely fills one of your team’s
rectangles, say, “Check Mate”. Be sure to mark all hits and misses, so that your
team and your opponents’ team can compare game boards at the end of the
game.
3. Ending the game
When one team has check marks inside all of the other team’s rectangles, the
game is over.
The teams review each other’s sheets to make sure that no mistakes have
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Geometry and Spatial Sense, Grades 4 to 6
Loc4.BLM3
Playing Check Mate
Dear Parent/Guardian:
In class we have been playing a game called Check Mate. A game sheet and
instructions for playing the game are included, but please have your child
explain the game to you, so that you know how well your child understands the
game and the geometry concepts that are necessary for playing the game well.
When you were younger, you might have played games similar to this – for
example, Battleship. If you and your child enjoy Check Mate, similar games are
available in stores and on the Internet.
Game Instructions
1.Each player needs two grids, one titled “Your Shapes” and the other titled
2.Each player needs to draw four rectangles on the “Your Shapes” grid.
Rectangles must not overlap and need to be traced on the grid lines. Do not
show the location of these rectangles to your opponent.
3.The players take turns naming locations where they think their opponent
may have drawn a rectangle. Remember: name the letter before the
number. Each player records a hit or a miss on the “Your Opponents’
4.If your opponent names a position inside one of your rectangles, say “Check”,
a check in that location on his or her “Your Opponents’ Shapes” grid. If your
opponent names a position outside your rectangles, say “Miss”, and mark an
that cell on his or her “Your Opponents’ Shapes” grid. When the other player
completely fills one of your rectangles, say, “Check Mate”.
5.Take turns naming locations on each other’s grids. When one player has
check marks inside all of the other player’s rectangles, the game is over.
Review each other’s sheets to make sure that no mistakes have been made.
Grade 4 Learning Activity: Location – Check Mate
117
Loc4.BLM4
Instructions and Game Boards for Hot, Warm, Cold
This game may be played by individuals or pairs.
Draw five quadrilaterals on your grid: a rectangle, a trapezoid, a parallelogram,
a rhombus, and a square. Make sure that each quadrilateral fits inside one cell.
The pair (or person) playing against you will call out a location, such as D3.
If they hit one of your shapes, say, “Hit”, and tell them the name of the shape.
If they miss one of your shapes, say, “Miss”, but provide a clue for the nearest
shape – for example:
•If they miss by one square, say something like, “Hot, go to the right” or
“Hot, go down”.
•If they miss by two squares, say something like, “Warm, go to the left” or
“Warm, go up ”.
•If they are three or more squares away from any shape, say, “Cold”.
Then it is your turn to call out a location and receive clues.
The first pair (or person) to hit all five of the other pairs’ quadrilaterals
is the winner.
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Geometry and Spatial Sense, Grades 4 to 6
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Grade 4 Learning Activity: Movement – Hit the Target
Movement: Hit the Target
Overview
In this learning activity, students play a game in which they determine the line of reflection
needed to reflect a shape onto a target. The game provides an opportunity for students to
identify and perform reflections and make predictions, and reinforces the concept of using a
grid system to describe the location of an object.
Big IdeaS
This learning activity focuses on the following big idea:
Location and movement: Students develop an understanding of how to use different transformations in describing the movement of an object.
Curriculum Expectations
This learning activity addresses the following specific expectations.
Students will:
• identify and describe the general location of an object using a grid system (e.g., “The library
is located at A3 on the map.”);
• identify, perform, and describe reflections using a variety of tools (e.g., Mira, dot paper,
technology).
These specific expectations contributes to the development of the following overall expectation.
Students will:
• identify and describe the location of an object, using a grid map, and reflect twodimensional shapes.
Materials
TIME:
approximately
60 minutes
• Miras (1 per pair of students)
• rulers (1 per pair of students)
• Mov4.BLM1: Design Reflection (1 per student)
• Mov4.BLM2: Hit the Target (1 per pair of students)
• sheets of chart paper or large newsprint (1 per pair of students)
• markers (1 per pair of students)
• Mov4.BLM3: Scavenger Hunt (1 per student)
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Math Language
• reflection
• rectangle
• congruent
• rhombus
• symmetry
• hexagon
• predictions
• circle
• Mira
• pentagon
• triangle
• parallelogram
• square
INSTRUCTIONAL
GROUPING:
pairs
Instructional Sequencing
This learning activity provides an introduction to the term symmetry and the use of a line of
reflection to create symmetrical designs. Before beginning this activity, students should review
the meaning of congruence.
Rich and appropriate experiences help children connect mathematics with their everyday world
and develop the spatial reasoning skills necessary to be successful in and out of school. This
activity provides opportunities for students to represent and “see” the mathematical ideas
associated with reflections.
Getting Started
Have students draw a design to the left or the right of the line of reflection on Mov4.BLM1:
Design Reflection. Tell students that the design must touch the line in some way. When they
have finished, they should make a reflection drawing on the opposite side of the line. They
should use a Mira or fold the paper to check their work. Lead a brief class discussion to consider the following questions:
• “What strategies did you use to create the reflection?”
• “How can you tell if your second drawing is a reflection?”
• “What is symmetry?”
• “Would you do anything different the next time?”
Introduce Mov4.BLM2: Hit the Target and explain to the students the rules on it.
Working on it
Have students play the game in groups of four (two versus two) to allow for discussion and
collaboration. Provide each student pair with a copy of Mov4.BLM2: Hit the Target, a ruler,
and a Mira. As students play the game, observe the various strategies they use. Pose questions
to help students think about their strategies:
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Geometry and Spatial Sense, Grades 4 to 6
• “What strategy did you use to decide where the line would go?”
• “Did you change your strategy? Why?”
• “Did you use the grid as part of your strategy?”
When students have had a chance to play the game, provide each pair with markers and a sheet of
chart paper or large newsprint. Ask students to record their strategies on the paper to show how
they chose the location for their line of reflection.
Make a note of pairs who might share their strategies during the Reflecting and Connecting part
of the lesson. Include groups whose methods varied in their degree of efficiency (e.g., using
guess and check, finding a cell halfway between the edge of the shape and the target, finding
the cell halfway between the centre of the shape and centre of the target).
Reflecting and Connecting
Once students have had the opportunity to play the game and record their strategies, bring
them together to share their ideas. Try to order the presentations so that students observe
inefficient strategies (e.g., picking at random) first, followed by more efficient methods.
As students explain their ideas, ask questions to help them to describe their strategies:
• ”How does your strategy work?”
• “Did you change your original strategy? Why?”
• ”Did you use the grid in your strategy? How?”
Following the presentations, ask students to observe the work that has been posted and to
consider the efficiency of the various strategies. Ask:
• “Which strategy, in your opinion, is an efficient strategy for finding the line of reflection?”
• “How would you explain the strategy to someone who has never used it?”
Avoid commenting that some strategies are better than others – students need to determine
for themselves which strategies are meaningful and efficient, and which ones they can make
sense of and use.
Refer to students’ work to emphasize geometric ideas:
• A grid system can be helpful in locating specific objects.
• A reflection is the same distance away from the line of reflection as the original object.
Provide for students an opportunity to play additional or related games to try new strategies.
Those students having difficulty with the concept may benefit from working on an enlarged
copy of the game showing one shape and the target.
Students could use plain grids, pattern blocks instead of shapes, and a piece of string as the
target. They could reflect the pattern blocks onto the target from different locations.
Grade 4 Learning Activity: Movement – Hit the Target
121
Assessment
Observe students as they play the game and assess how well they:
• locate and mark lines of reflection on the grid;
• apply appropriate strategies to play the game;
• explain their strategy;
• judge the efficiency of various strategies;
• modify or change strategies to find more efficient ways to solve the problem.
Home Connection
Send home Mov4:BLM3: Scavenger Hunt. This Home Connection task provides an opportunity for parents and students to discuss reflections and symmetry.
LEARNING CONNECTION 1
Symmetry in the Real World
Materials
• Mov4.BLM4: Real-World Symmetry (1 per student)
• overhead transparency of Mov4.BLM4: Real-World Symmetry
• Miras (1 per pair of students)
• scissors (1 per pair of students)
Explain that real-world shapes rarely have perfect symmetry and that in this activity being close
is fine. The discussion about symmetry, rather than the correct answer, is the important component of this learning connection.
Provide each student with a copy of Mov4.BLM4: Real-World Symmetry and have each
student write down the number of lines of symmetry that he or she predicts is in each representation of a real-world shape pictured in the blackline master. The students should make
their predictions by looking at the shapes and visualizing the lines of symmetry. They should
not use Miras, paper folding, or any other means to assist their predictions.
Use an overhead transparency of Mov4.BLM4: Real-World Symmetry, and ask students to share
their predictions for each shape and their reasons for making them. Be sure to elicit whether there
is more than one prediction for any given shape, and ask the reasons for alternative predictions.
Have the students refine their predictions on the basis of feedback from their peers, and take a
“thumbs up” vote to decide on the number of lines of symmetry for each real-world shape. Next to
each of the shapes, record the number of lines of symmetry that the majority of students voted for.
Discuss ways to check for lines of symmetry. Then hand out the Miras. Ask the students to check
for lines of symmetry by using the Miras and also by cutting out the shapes and folding them.
After the students have verified the lines of symmetry by various means, have a discussion
about their answers. Deal with the shapes that created the most controversy regarding the
number of their lines of symmetry.
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Geometry and Spatial Sense, Grades 4 to 6
LEARNING CONNECTION 2
Carmen’s Carpets
Materials
• sample symmetrical tile, carpet, or wallpaper design
• overhead transparency of Mov4.BLM5: Carmen’s Carpets
• Mov4.BLM5: Carmen’s Carpets (1 per student)
Have a picture or an example of a symmetrical tile, carpet, or wallpaper design. Lead a brief
class discussion about the use of symmetry in wallpaper, tile, and carpet designs. Display the
overhead transparency of Mov4.BLM5: Carmen’s Carpets. Discuss the problem presented on
Provide each student with a copy of Mov4.BLM5: Carmen’s Carpets, and have students work
on the solution. Observe them as they solve the problem, and make note of the different
methods they use to determine the number of lines of symmetry in the shapes. For example,
students might use:
• a ruler to measure
• paper folding
• a Mira
• a pattern or rule to determine the number of lines
Some students may jump to the conclusion that the number of sides is equal to the number of
lines of symmetry. Ask other students if they feel this rule is true for all the shapes. Through discussion and further investigation, students may discover that this is true only of regular polygons,
such as the square, the equilateral triangle, the regular pentagon, and the regular hexagon.
Ask several students to explain their strategies to the class. Include a variety of strategies – for
example, using paper folding, using a Mira, and using a pattern if applicable.
After a variety of strategies have been presented, have students evaluate the different
methods by asking the following questions:
• “Which strategies were efficient and easy to use? Why?”
• “Which strategies are similar? How are they alike?”
• “What strategy would you use if you were to solve this problem again? Why?”
Grade 4 Learning Activity: Movement – Hit the Target
123
line of reflection
Mov4.BLM1
Design Reflection
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Geometry and Spatial Sense, Grades 4 to 6
Mov4.BLM2
Hit the Target
Start with the triangle. Each team, at the same time, draws a line on its own grid with its ruler
so that when a Mira is placed on that line, the triangle is reflected onto the target. Each team
is allowed three attempts. Each team checks the other team’s score after each attempt. Your
team puts a check mark in the box below for its highest score of the three attempts.
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First try
Second try
Third try
completely inside the target
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touching the inner circle
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touching the middle circle
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touching the outside circle
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Next, each team draws a line on its own grid with its ruler so that when a Mira is placed on
that line, the trapezoid is reflected onto the target. Scoring is the same as before.
Finally, each team draws a line on its own grid with its ruler so that when a Mira is placed on
that line, the hexagon is reflected onto the target. Scoring is the same as before.
Your team should now have three scores or checks to add up for a total score.
Grade 4 Learning Activity: Movement – Hit the Target
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Mov4.BLM3
Scavenger Hunt
Dear Parent/Guardian:
conduct a scavenger hunt to look for objects that have symmetry. For example:
Picture frames
Water bottle
Before you begin, ask your child to explain the terms line of reflection and
symmetry, so that you know how well he or she understands the geometry
concepts that are necessary for conducting the scavenger hunt.
Objects found in our home that could have a line of reflection are:
1.
2.
3.
4.
5.
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Geometry and Spatial Sense, Grades 4 to 6
Mov4.BLM4
Real-World Symmetry
Decide on the number of lines of symmetry for each of these items. Write your
Grade 4 Learning Activity: Movement – Hit the Target
127
Mov4.BLM5
Carmen’s Carpets
Carmen is a carpet designer. She likes to design carpets that contain the most
symmetrical designs, because these carpets sell the best. She would like to use
the following shapes in the carpets, and she wants to know which of the shapes
have the most lines of symmetry.
128
Geometry and Spatial Sense, Grades 4 to 6
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
Two-Dimensional Shapes:
Triangle Sort
This activity is adapted, with permission, from John A. Van de Walle, Teaching StudentCentered Mathematics: Grades 3–5. Toronto: Pearson, 2006, p. 225. The activity is also found
in the Grades 5–8 book of the series.
OVERVIEW
In this learning activity, students sort and classify triangles into three groups so that no triangle
belongs to more than one group. This problem-solving lesson provides an opportunity for
students to explore the properties of triangles and develop definitions of the different types of
triangles.
BIG IDEAS
This learning activity focuses on the following big ideas:
Properties of two-dimensional shapes and three-dimensional figures: Students use the
properties of triangles to develop definitions for different types of triangles.
Geometric relationships: Students explore the relationships between various triangle properties (e.g., the relationship between side length and angle measure in an isosceles triangle).
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectation.
Students will:
• identify triangles (i.e., acute, right, obtuse, scalene, isosceles, equilateral), and classify them
according to angle and side properties.
This specific expectation contributes to the development of the following overall expectation.
Students will:
• identify and classify two-dimensional shapes by side and angle properties, and compare and
sort three-dimensional figures.
129
TIME:
approximately
60 minutes
Materials
• pictures of triangles
• scissors (a pair per student)
• 2D5.BLM1: Assorted Triangles (1 per pair of students)
• 2D5.BLM2: Triangle Sketching Chart (1 per pair of students)
• 2D5.BLM3: Triangles in the Environment (1 per student)
Math Language
INSTRUCTIONAL
GROUPING:
partners
• right angle (square corner)
• vertex
• acute angle
• isosceles
• obtuse angle
• equilateral
• line segment
• scalene
Instructional Sequencing
This learning activity provides students with an opportunity to explore triangle properties
and relationships in order to develop definitions for the different types of triangles. Prior to
this activity, students should have an understanding of angles in relation to 90º – for example,
larger than 90º is an obtuse angle, smaller than 90º is an acute angle, and 90º is a right angle
(square corner). The activity builds on this knowledge of angles.
Sorting and classifying triangles allows students to develop their own definitions of the different types of triangles on the basis of their observations and understandings. The activity
of sorting and classifying can lead to a discussion of the defining properties of a subclass of
triangles (e.g., an equilateral triangle is literally translated as an “equal-sided” triangle, but
one of its properties is also that it has three equal angles). As students explore triangles and
determine groups and subclasses, they connect the defining properties with their experiences,
rather than receive them handed down as definitions.
Students will recognize that triangles can be classified by two of their properties – their side
lengths (equilateral, isosceles, and scalene) or their angle measures (acute, obtuse, or right).
Investigating which triangles can be classified in more than one way (e.g., a right isosceles
triangle) and which cannot (e.g., a right equilateral triangle) helps deepen students’ understanding of triangle properties.
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Geometry and Spatial Sense, Grades 4 to 6
GETTING STARTED
Place a picture of a triangle on the board or an overhead projector. Invite students to share
what they observe about the triangle with an elbow partner. Open the discussion to the larger
group. Record the students’ observations on the board as they share their discussions. Place
a second (different) triangle beside the first and ask students to share their observations of
this triangle with an elbow partner. Open the discussion to the larger group and record their
observations. Invite students to compare and contrast the two triangles. Explain to students
that in this lesson they will be working with a partner to examine and sort a variety of triangles.
WORKING ON IT
Students will work with a partner for this activity. Provide each pair of students with a copy of
2D5.BLM1: Assorted Triangles and scissors with which to cut out the triangles on the blackline
master. Explain to students that their task is to sort the entire collection of triangles into three
groups so that no triangle belongs to two groups. When they have sorted the triangles, they must
record a name for each group, as well as a description that defines the group of triangles.
When students have created their groups and developed a definition for each, encourage them
to repeat the activity but to use a different sorting criterion. Again, students should name each
group and write a description that defines the group.
As students work, observe how they discuss the properties of the triangles. Do they notice angles,
and if so, how do they describe the angles? Do they notice congruent sides? What kind of language
do they use for recording the descriptions of the groupings? If a student pair is stuck on only
one property (e.g., side length), you may have to hint at the other property (e.g., angle measure).
However, it is best if students struggle somewhat with the activity before you provide them with
hints. Look for student pairs to share their findings during the reflection part of the lesson.
REFLECTING AND CONNECTING
Bring the students back to the larger group. Ask various pairs to share one of the ways in which
they sorted the triangles (either angles measures or side lengths). You may have a student pair
read out the triangle letters in a grouping, and then you may challenge the class to look at the
triangles in that grouping and predict the description before the pair shares it. Ask the class
questions like the following:
• “Which types of triangles did you look for first?”
• “What was the first thing you noticed about some of the triangles?”
• “When you found all the triangles for one group, how did you decide what the next group
would be?”
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
131
Have student pairs read their definitions for the different groups, encouraging them to use
appropriate geometric terminology. Ask other pairs if they sorted their triangles in a similar
manner. Model the appropriate terms (e.g., scalene, isosceles, right), and record terms and
descriptions in an anchor chart for future reference. Alternatively, you might wish to have
students create posters for each type of triangle.
Repeat the process for the sorting of the second property (either angles or side lengths). Ask
questions like the following:
• “Was it difficult to find another way to sort the triangles?”
• “Which property, angles or side lengths, was easier for you to use to find differences in the
triangles? Why?”
• “Can a triangle belong to more than one category?”
• “Did you need to use any tools to help you sort the triangles?”
Note that this last question helps focus in on the properties that are used to categorize triangles – a
ruler measures length, and a protractor measures angles.
Students who are having difficulty with the task should be encouraged to sort according to
only one property. You might also provide them with a ruler as a hint about how to get started.
This activity is also one in which homogeneous grouping could benefit both struggling students
and those who are able to grasp the concepts quickly. There are many triangles to be sorted,
and a dominant student might quickly move them around, preventing a partner from analysing
them and deciding where they should go.
As an extension, provide students with a copy of 2D5.BLM2: Triangle Sketching Chart. Ask
students to sketch a triangle for each of the nine cells. The triangle must match the characteristics of both the column header and the row header. Note that it is impossible to draw two
of the triangles (a right equilateral and an obtuse equilateral). This discovery can lead to an
interesting discussion about the measures of angles in a triangle.
ASSESSMENT
Observe students as they solve the problem, and assess how well they:
• identify triangles (i.e., acute, right, obtuse, scalene, isosceles, equilateral);
• classify triangles according to angle and side properties;
• identify and classify two-dimensional shapes by side and angle properties;
• explain sorting criteria and description criteria.
This lesson may be used appropriately to assess students’ use of geometric terminology,
particularly after the Reflecting and Connecting part of the lesson. It is more important that
students use the terms in context rather than recall a definition by rote.
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Geometry and Spatial Sense, Grades 4 to 6
HOME CONNECTION
In this Home Connection, students look for triangles in their environment. Send home
2D5.BLM3: Triangles in the Environment. Students will use it to record examples of the
types of triangles they see on their way home from school and in their homes.
LEARNING CONNECTION 1
Measuring and Constructing Angles in The
Materials
• The Geometer’s Sketchpad (1 computer per person/per pair)
• 2D5.BLM4: GSP Angles Instructions
• math journals
This activity is designed to provide students with the opportunity to learn some of the features
of The Geometer’s Sketchpad while investigating the properties of angles. It is intended that
students work with partners, or individually if there are enough computers.
Provide each student/pair with a copy of 2D5.BLM4: GSP Angles Instructions. Students follow
the instructions on the page to complete the task.
Have a whole class discussion after the activity. Ask students questions like the following:
• “What aspects of the task did you find challenging?”
• “Up to now, we’ve talked about an angle as being the measure of rotation between two rays
or line segments. Does working with this program change your ideas about that definition?”
• “What is required to measure an angle?” (This will raise an interesting discussion, for in GSP,
only three points are required to form an angle.)
Although this activity is designed to be fairly procedural, it will generate rich discussion about
the properties of angles and the tools required to construct them.
LEARNING CONNECTION 2
Spinning for Triangles
Materials
• 2D5.BLM5: Triangle Construction Spinners (1 per pair of students)
• pencil (1 per pair of students)
• paper clip (1 per pair of students)
• rulers (1 per pair of students)
• protractors (1 per pair of students)
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
133
This activity is a game to be played by two players. Each pair of players makes a spinner with a
copy of 2D5.BLM5: Triangle Construction Spinners, a paper clip, and a pencil.
Players take turns. Player 1 spins on both of the spinners, then constructs a triangle that
satisfies the criteria shown on the spinners (e.g., a triangle with an angle of 120° and no sides
equal). Player 2 checks to see that the construction is accurate. If he or she agrees, Player 1
scores a point. Player 2 takes a turn.
If a player is unable to construct a triangle, the other player may attempt it to steal a point. If
no one is able to construct the triangle, no points are given, and it is the other player’s turn.
The game continues until one player scores 5 points, or the game can be timed.
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Geometry and Spatial Sense, Grades 4 to 6
2D5.BLM1
Assorted Triangles
M
B
D
C
G
O
C
A
E
V
N
F
Q
W
R
A
T
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
135
2D5.BLM2
Triangle Sketching Chart
Equilateral
Right
Acute
Obtuse
136
Geometry and Spatial Sense, Grades 4 to 6
Isosceles
Scalene
2D5.BLM3
Triangles in the Environment
Dear Parent/Guardian:
In math this week, we have been learning about different types of triangles. Your child has
been sorting and classifying different types of triangles in class. Here is an opportunity for your
child to demonstrate what he or she has learned. Ask your child to find different triangles in
the type of triangle, create a sketch, and describe the purpose of the triangle.
Type of Triangle
Sketch
Purpose
Example:
– part of a bridge structure
Equilateral
– must provide strength to the
bridge and support the beams
– it has sides with
the same length
Thank you for your assistance with this activity.
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
137
2D5.BLM4
GSP Angles Instructions
journal.
2. From the Edit menu, choose Preferences.
3. Change the Precision for Angle from hundredths to units.
4. Use the Point tool to place three points on the work area.
5. Click on the Straightedge tool, and join the segments to form two line
segments joined at a vertex.
6. Select the Text tool, and click on each point to label it.
7. Click on the Select tool, and click on the white area to deselect everything.
Then click on the three points in order – A, B, C.
8. From the Measure pull-down menu, choose Angle. What happens?
9. Drag point A around the plane. What happens to the angle measure?
10. Can you move point A so that the angle measure stays the same?
11. Use the Tools and the Measure menu to create a right angle. Describe in
your math journal how you did so.
CHALLENGE: Can you use the compass tool and Animate features to create a
constantly changing angle?
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Geometry and Spatial Sense, Grades 4 to 6
2D5.BLM5
Triangle Construction Spinners
Make two spinners, using this page, a paper clip, and a pencil.
Right
Angle
40˚
Angle
Scalene
Equilateral
120˚
Angle
Isosceles
6 cm
side
No sides
equal
2 sides
7 cm
each
4 cm and
8 cm sides
5 cm
and 12 cm
sides
2 cm
side
Grade 5 Learning Activity: Two-Dimensional Shapes – Triangle Sort
139
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
Three-Dimensional Figures:
Package Possibilities
OVERVIEW
In this investigation, students gain experience identifying pyramids from nets and classifying
these three-dimensional figures by their geometric properties. Students have opportunities to
represent their understanding of three-dimensional figures physically, through drawings, and
verbally, using appropriate mathematical language.
BIG IDEAS
This learning activity focuses on the following big ideas:
Properties of two-dimensional shapes and three-dimensional figures: Students develop an
understanding of the properties of pyramids, specifically looking at faces, edges, and vertices
in two-dimensional representations.
Geometric relationships: Students must be able to use two-dimensional nets to recognize and
represent three-dimensional figures. This activity calls for students to focus on the relationship
between the two-dimensional faces of an object and its three-dimensional construction.
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectations.
Students will:
• identify prisms and pyramids from their nets;
• construct nets of prisms and pyramids, using a variety of tools (e.g., grid paper, isometric dot
paper, Polydrons, computer application).
These specific expectations contribute to the development of the following overall expectation.
Students will:
• identify and construct nets of prisms and pyramids.
TIME:
approximately
60 minutes
Materials
• net of a triangular pyramid made from Polydron pieces
• sample of a box to represent any three-dimensional figure (e.g., tissue box)
140
• transparency of isometric dot paper
• 3D5.BLM1: Pyramid Packages (1 per student)
• Polydron set
• solid square-based pyramid
• sheets of isometric dot paper for recording nets (2 or 3 per pair of students)
• blank overhead transparencies of isometric dot paper (1 per pair of students)
• overhead markers (1 per pair of students)
• large sheet of blank chart paper
• tape
• 3D5.BLM2: Find the Nets (1 per student)
• 3D5.BLM3: Six Nets of a Square Pyramid for teacher reference
Math Language
• net
• tetrahedron
• three-dimensional figure
• isometric dot paper
• two-dimensional shape
• square-based pyramid
• equilateral triangular
• isosceles triangles
faces
• triangular pyramid
• reflection
• rotation
Instructional Sequencing
This lesson can serve as an introduction to a unit on three-dimensional geometry or can be
used after students have had some experiences sorting and classifying three-dimensional
INSTRUCTIONAL
GROUPING:
individual, pairs
figures by their properties. Students should be able to recognize many three-dimensional
figures but do not need to have many prior experiences with nets.
In Grade 5, students are expected to construct nets of prisms and pyramids using a variety of tools.
In this task, students will predict the number of nets possible for a square-based pyramid
and then use two-dimensional Polydron shapes to investigate all of the different nets. After
constructing a square-based pyramid, students will explore different nets by unfolding and
refolding the Polydron faces. If it is assumed that two nets are the same if they can be rotated
or flipped to overlap each other, there are six possible nets for a square-based pyramid. It is
important to give students the time to have a rich discussion about what qualifies as different,
rather than stipulate before the lesson the idea that rotated or flipped nets are not unique.
This task also provides opportunities for students to develop their reasoning, proving, and
reflecting skills as they predict the number of possible nets for the square-based pyramid,
construct the nets, adjust their strategies, and then confirm or revise their predictions.
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
141
Getting Started
Show students the following net for a triangular pyramid constructed from Polydron pieces.
Ask students what an arrangement of two-dimensional faces is called (net). Ask students to
predict the three-dimensional figure that will be created by folding the net. “How do you know?”
Try to elicit properties of the three-dimensional figure that are evident in the net – that is,
the number and shapes of the faces. Ask students to be specific in identifying the twodimensional faces – that is, equilateral triangular faces. Invite a student to test the prediction
by folding the net.
Discuss the two names we use for this particular three-dimensional figure – triangular pyramid
and tetrahedron. Discuss why the figure is named in both of these ways (pyramids are named
for their base; polyhedra in general are named for the number of polygonal faces – tetra
means “four”).
Unfold the net to form the same net you originally showed to the students. On an overhead
transparency, draw the net for the tetrahedron, and display it.
Then ask: “Is it possible to arrange the faces differently and still end up with the same threedimensional figure?” Invite another student to unfold the triangular pyramid to create a net
that is different from the original one.
The student should construct this net:
If the student unfolds the figure to create the same net, only reflected and/or rotated, ask the
class whether the net is, in fact, a different net. If necessary, rotate the net in your hands to
allow the students to see that the net is actually the same one you started the activity with.
Draw the new net (pictured above) on the same overhead transparency.
Next, draw the following arrangement of triangular faces on the transparency:
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Geometry and Spatial Sense, Grades 4 to 6
pyramid? Explain why it is or is not a net for a tetrahedron.” (It is not a net, because two of the
triangles will overlap when the arrangement is folded, and there will be no base.)
Explain to students that there are sometimes many different nets for a given three-dimensional
figure. The triangular pyramid, or tetrahedron, has only the two shown above. Students will
now explore different nets of another pyramid.
Working on It
Introduce the following scenario:
“An electronics company is designing a new video gaming system and wants it to have a
unique shape. The company is considering a square-based pyramid design, but wants to
know what the packaging layout will be in order to determine costs. The company has
asked for your help in determining all of the possible nets for packaging this square-based
pyramid gaming system.”
Show students a package (e.g., cereal box, facial tissue box) and explain that the electronics
company wants the box for their gaming system to be the same shape as the system itself – a
square-based pyramid.
Give each student a copy of 3D5.BLM1: Pyramid Packages. Ask that each student predict the
total number of different nets possible for a square-based pyramid and record both his or her
prediction and reasoning on the handout. (A student may say, for example, that a square-based
pyramid will have three different nets because the triangular pyramid had two nets and the
square pyramid has one more face than the triangular pyramid.) Allow time for students to
think individually, and then have them discuss their predictions with an elbow partner.
Arrange students in pairs. Explain that students will begin by constructing a square-based
pyramid from Polydron pieces.
Have each pair of students determine which Polydron pieces they will need
to construct a square-based pyramid. Display a solid square-based pyramid
for students who may need to refer to it. Students will require a square and
four triangles. To add to the difficulty level, allow students to choose their
own Polydron pieces. Provide choices for students who may require additional
support. Most will choose four equilateral or four isosceles triangles, but two
right-angled triangles, one equilateral triangle, and one isosceles triangle
will also make a square-based pyramid. Note that only the combination of a
square and four equilateral triangles will allow for the creation of all six nets
(see 3D5.BLM3: Six Nets of a Square Pyramid). Invite one of the partners to
collect the necessary pieces from a bin of Polydron pieces.
Assessment Opportunity
Circulate to assess
students’ reasoning and
reflecting by listening to
pairs’ discussions and
For example: “Do you still
the number of possible
nets for the square-based
pyramid is accurate? Why
or why not? How would you
Instruct pairs of students to construct their square-based pyramid. Circulating
to distribute several sheets of isometric dot paper to each pair allows you to
ensure that all pairs have constructed the correct three-dimensional figure.
Allow students to exchange or add Polydron pieces if they have discovered
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
143
that they are missing some. Ask students to then unfold their figure to create a net. Have
students draw the net on the isometric dot paper.
Have students refold their net to form the square-based pyramid again. Ask students to
continue unfolding and refolding to find as many different nets for the square-based pyramid
as they can. Each different net that they can find is to be recorded on their paper.
As students work, circulate with sheets of isometric dot overhead transparencies. Ask each
pair of students to use an overhead marker to record one of their nets – selected by you – on
the transparency. Try to include as many different nets as possible, but also look for nets that
are simply reflections or rotations of each other. These will enhance the discussion during the
Reflecting and Connecting part of the lesson.
Reflecting and Connecting
Bring the class together. Ask pairs of students how many nets they found. (Answers will differ,
depending on whether students found all six unique nets and whether they have reflections
and/or rotations of the same net.)
Ask each pair of students in turn to come up to the overhead projector to share with the class
the net that they recorded on the acetate. Have them explain the geometric properties of the
square-based pyramid and tell how their net can be folded to form that figure (i.e., which sides
of the two-dimensional faces meet to form the edges of the three-dimensional
Assessment Opportunity
figure). Tape each net to a large white piece of chart paper after the net has been
As students present their
net, assess their ability
to communicate how
the net would be folded
to form a solid. Look for
appropriate and accurate
terminology, like edge,
face, and vertex.
shared, so that all of the shared nets can be viewed at once.
As each new pair shares a net, ask students to consider whether the net is truly
unique or whether it can be reflected or rotated to match a net already presented (and displayed on the chart paper).
Ask: “Are any of the nets you found not represented on the chart?” Allow
students to share their drawings of any new nets with the class. If the class
determines that a net is unique, have the student draw the net on a transparency
of isometric dot paper and add it to the chart.
Post the completed chart in the classroom.
Elicit student strategies for finding the unique nets. Ask questions like the following:
• “Did you use any particular strategy for finding all the nets?”
• “How many groups randomly tried different combinations?”
• “Did anyone use a pattern or similar strategy to try to find all of the nets?”
Try to generate a discussion that looks at a systematic method for finding the nets. For
example, some students may have started with the net having one triangle on each of the sides
of the square, and then moved to one triangle on three sides of the square, then two on each
side, and so forth. Ask if this strategy could be applied to predict the number of nets for other
types of polyhedra.
Make a connection with the original scenario; for example, “How might this information help
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Geometry and Spatial Sense, Grades 4 to 6
the toy company?” Ask about the considerations a company might take into account when
designing a package.
Some students may experience difficulty in drawing their nets on the blank paper. Provide
them with graph paper or isometric dot paper to help them with their sketches. They may also
trace the Polydron nets onto a blank sheet of paper if they are unable to create their own.
Challenge students to construct a pentagonal prism and find all of its nets by unfolding and
refolding. Have them record all of the nets they find.
Assessment
Observe students to see how well they:
• identify and construct nets of pyramids using the Polydron pieces.
This lesson focuses on skills and reasoning more than content knowledge. It is appropriate to
assess process skills, like reasoning and communication, during the lesson. Look to see how well
students use accurate and appropriate terms to communicate their findings, and as you circulate
around the room, observe students as they struggle to find additional nets. What is their reasoning? Do they use a system or strategy, or do they randomly piece the polygon faces together?
Home Connection
Send home 3D5.BLM2: Find the Nets. In this homework task, students identify nets of prisms
and pyramids from several arrangements of two-dimensional shapes through visualization or
folding, and then explain why the other arrangements are not nets.
Inform students that they will cut out the nets in class the next day to see if their reasoning
was correct.
Learning Connection 1
Nets in AppleWorks
Materials
• 3D5.BLM4: Drawing Nets in AppleWorks (1 per student)
• computers with AppleWorks (1 per student or pair of students)
Activate students’ prior experience of using the draw tools in AppleWorks by discussing with
the class the various functions of each tool. If students’ experience is limited, they may need
some time to review the different tools in AppleWorks and how they are used.
Distribute a copy of 3D5.BLM4: Drawing Nets in AppleWorks to each student. Following the
instructions provided, students use AppleWorks to create nets for the square-based pyramid
and triangular prism.
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
145
Once students have created their nets, they may save or print them for assessment or use
them to create figures.
Learning Connection 2
Net Characters
Materials
• 3D5.BLM5a–b: Net Characters (1 per student)
• Polydron sets (1 set per group of 4 students)
Arrange students in groups of four. Distribute 3D5.BLM5a–b: Net Characters to each student.
Begin by asking students to predict whether the first character is a net for a prism, a pyramid,
or another three-dimensional figure, and have them name the three-dimensional figure that can
be formed by folding the net.
Next, invite a student to come up to test the prediction with the Polydron shapes. Once the
correct figure has been confirmed, instruct all students to record the name of the figure on
their handouts. (square-based prism).
Students continue to identify the nets of pyramids and prisms from several character nets on
the handout. Allow enough time for each group of students to try to identify each net.
Distribute Polydron sets to each group. Instruct groups to test their predictions for each net by
constructing the net and folding it to create a three-dimensional figure.
Learning Connection 3
Complete the Nets
Materials
• a partially completed net constructed from Polydron shapes, as shown
• 3D5.BLM6a–b: Complete the Nets (1 per student)
• rulers (1 per student)
• Polydron pieces
Begin by showing students the partially completed net constructed from Polydron shapes below.
Tell students that the net is missing one or two faces that you have hidden. Ask them to predict
what the three-dimensional figure is and to tell you what faces are missing. Ask: “How do you
know?” (It is a net for a pentagonal prism with one pentagon and one square face missing.)
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Geometry and Spatial Sense, Grades 4 to 6
Unveil the missing faces and invite a student to come up to add the missing faces to the net.
Ask why the student chose to put the faces where he or she did. Ask the student to then “test”
his or her placement of the faces. If the placement is incorrect, can the students analyse the
Explain to students that they will be given a handout with drawings of six partial nets. They are
to complete the nets (by adding one or two missing faces to each), identify the three-dimensional figure that each net makes when completed, and draw a different net for each figure.
Distribute one copy of 3D5.BLM6a–b: Complete the Nets to each student.
Allow students to work with a partner or in small groups to complete the handout. As a scaffold
for students who struggle with this skill, have Polydron shapes or another appropriate material
available to aid in the “testing” of face placement.
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
147
3D5.BLM1
Pyramid Packages
An electronics company is designing a new video gaming system and wants it
to have a unique shape. The company is considering a square-based pyramid
design, but would like to know all of the packaging layouts for this figure in
order to determine packaging costs. The marketing department manager has
asked for your help in determining all of the possible nets for packaging their
square-based pyramid gaming system.
1. Predict the total number of different nets possible for constructing a squarebased pyramid.
I think there are based pyramid.
different nets possible for constructing a square-
2. Test your prediction by constructing square-based pyramids from Polydron
pieces and unfolding them to create different nets. Draw each net on
isometric dot paper.
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Geometry and Spatial Sense, Grades 4 to 6
3D5.BLM2
Find the Nets
Only some of the following configurations of shapes are nets for three-dimensional figures.
Determine whether each configuration is a net. If it is, name the three-dimensional figure that
the net will form when folded. If a configuration is not a net, explain why it is not a net.
A.
B.
C.
D.
E.
F.
(Hint: This is a net. What threedimensional figure will it form?)
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
149
3D5.BLM3
Six Nets of a Square Pyramid
150
Geometry and Spatial Sense, Grades 4 to 6
3D5.BLM4
Drawing Nets in AppleWorks
1. Follow these steps to create a net for a square-based pyramid:
• Open a new drawing in AppleWorks.
• From the Format menu, choose Rulers – Ruler
Settings. In the new window, click on Graphics
under Ruler Type; click on Centimetres under
Units; and type in 5 for Divisions.
• Click on the rectangle tool,
, from the Tool
Bar at the left of the screen. Draw a square that
is 2 cm by 2 cm near the centre of the page.
This is the base of the pyramid.
• Click on the regular polygon tool,
. Click on Edit – Polygon sides to make sure
the number of sides selected is 3. Click OK.
• Draw a triangle along one side of the square so that one whole side of the
triangle meets one whole side of the square.
• To make a congruent triangle, choose Edit – Duplicate.
• Next, choose Arrange – Rotate 90º to rotate the new triangle so that it can
meet the square along one whole side. Click and drag the new triangle so that it
shares a full side with the square.
• Duplicate the triangle once again. Choose Arrange – Rotate 90º to rotate the
new triangle and drag it so that it also shares a full side with the square.
• Duplicate, rotate, and drag a triangle one more time to complete the net for a
square-based pyramid.
• To move the net, hold down the Shift key and click on each of the five faces.
Then choose Arrange – Group. Now move the net closer to the top of the page.
2. On the bottom half of the page, draw a different net for a square-based pyramid,
using what you have learned about drawing nets in AppleWorks. Save your work
as “Square-based Pyramid Nets”.
3. Open a new AppleWorks drawing (File – New – Drawing). Draw two nets for a
triangular prism. Save your work as “Triangular Prism Nets”.
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
151
3D5.BLM5a
Net Characters
Most of the characters below are nets for prisms and pyramids. Check the
appropriate box below each net character to indicate whether it is a net for a
prism, a pyramid, or another three-dimensional figure.
A.
b.
q Prism
q Prism
q Pyramid
q Pyramid
q Other 3-D figure
q Other 3-D figure
d.
c.
q Prism
q Pyramid
q Other 3-D figure
q Prism
q Pyramid
q Other 3-D figure
e.
152
f.
q Prism
q Prism
q Pyramid
q Pyramid
q Other 3-D figure
q Other 3-D figure
Geometry and Spatial Sense, Grades 4 to 6
3D5.BLM5b
g.
h.
q Prism
q Prism
q Pyramid
q Pyramid
q Other 3-D figure
q Other 3-D figure
I.
J.
q Prism
q Pyramid
q Prism
q Other 3-D figure
q Pyramid
q Other 3-D figure
k.
l.
q Prism
q Pyramid
q Other 3-D figure
q Prism
q Pyramid
q Other 3-D figure
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
153
3D5.BLM6a
Complete the Nets
Complete the partial nets by
adding up to two missing faces.
1.
Name of figure:
2.
Name of figure:
3.
Name of figure:
154
Geometry and Spatial Sense, Grades 4 to 6
Draw a different net for the same figure.
3D5.BLM6b
Complete the partial nets by
adding up to two missing faces.
Draw a different net for the same figure.
4.
Name of figure:
5.
Name of figure:
6.
Name of figure:
Grade 5 Learning Activity: Three-Dimensional Figures – Package Possibilities
155
Grade 5 Learning Activity: Location – City Treasure Hunt
Location: City Treasure Hunt
overview
In this learning activity, students use a coordinate grid system to describe a location on the
intersection of lines. This problem-solving lesson provides students with an opportunity to
explore the concept of describing precise locations.
Big Ideas
This learning activity focuses on the following big idea:
Location and movement: The position of an object can be described using a coordinate grid
system. In Grade 5, students move from describing a location on the basis of spaces (using
numbers and letters to identify an area) to describing a location on the basis of the intersection
of lines (using a coordinate system based on the cardinal directions to describe a specific point).
Curriculum Expectations
This learning activity addresses the following specific expectation.
Students will:
• locate an object using the cardinal directions (i.e., north, south, east, west) and a coordinate
system (e.g., “If I walk 5 steps north and 3 steps east, I will arrive at the apple tree.”).
This specific expectation contributes to the development of the following overall expectation.
Students will:
• identify and describe the location of an object, using the cardinal directions, and translate
two-dimensional shapes.
TIME:
approximately
90 minutes
Materials
• markers (1 per pair of students)
• overhead transparency of Loc5.BLM1: City Treasure Hunt Map
• Loc5.BLM2: City Treasure Hunt Planning Sheet (1 per student)
• Loc5.BLM3: City Treasure Hunt Answer Sheet (1 per student)
• Loc5.BLM4: Planning a Trip (1 per student)
• sheets of chart paper (1 per pair of students)
156
Math Language
• cardinal directions
• north
• grid
• south
• grid lines
• east
• location
• west
• intersection
• axis
• specific
• axes
Instructional Sequencing
This learning activity provides an introduction to the grid system as a means of identifying
specific locations using cardinal directions and intersecting lines. Before beginning this activity
INSTRUCTIONAL
GROUPING:
pairs
students should review cardinal directions and the grid system by exploring the accompanying
learning connections or similar activities.
The key new idea for Grade 5 is location based on grid lines, rather then location based on
spaces. After playing the game Drawing Directions (Learning Connection 2 or a similar activity),
the students should be comfortable with identifying a specific location on a grid. In this activity they will be required to give the east-west directions before the north-south directions to
prepare students for the use of ordered pairs in Grade 6.
Getting Started
Discuss how some cities in North America are designed on a grid system, with avenues often
going east-west, and streets often running north-south.
Use a transparency or poster of Loc5.BLM1: City Treasure Hunt Map to show the streets of
a pretend city. Explain that the students will be required to find five treasures in the city by
Read the instructions one at a time, each time asking a volunteer to come up to label the location and mark the route that must be travelled. Each volunteer should use a different-coloured
marker, so that the routes can be distinguished. Students are to calculate the number of blocks
travelled for each of the instructions, as well as the total distance travelled.
Instructions
Start at the intersection of 10th Street and 2nd Avenue. Label the location S.
1) (first student) Travel 3 blocks west and 4 blocks north. You will find Aunt Agatha’s
artwork. Label it A. Mark your route, the location of the treasure (7th St and 6th Av), and
the number of blocks you have travelled (7).
Grade 5 Learning Activity: Location – City Treasure Hunt
157
2) (next student) Travel 2 blocks west and 4 blocks south. You will find a beautiful bronze
bear. Label it B. Mark your route, the location of the treasure (5th St and 2nd Av), and the
number of blocks you have travelled (6).
3) (next student) Travel 4 blocks east and 2 blocks north. You will find a cherished china
cup. Label it C. Mark your route, the location of the treasure (9th St and 4th Av), and the
number of blocks you have travelled (6).
4) (next student) Travel 6 blocks west and 3 blocks north. You will find a delicate and
delightful diamond. Label it D. Mark your route, the location of the treasure (3rd St and
7th Av), and the number of blocks you have travelled (9).
5) (next student) Travel 3 blocks east and 3 blocks south. You will find an empty but enticing
envelope. Label it E. Mark your route, the location of the treasure (6th St and 4th Av),
and the number of blocks you have travelled (6).
6) (next student) Travel 4 blocks west and 5 blocks north to the finish. Label it F. Mark your
route, the location of the finish (2nd St and 9th Av), and the number of blocks you have
travelled (9).
7) Ask, “What distance has been travelled in total?” (43 blocks).
Working on it
Tell the students that you want them to work in pairs to plan a treasure hunt. Hand out
Loc5.BLM2: City Treasure Hunt Planning Sheet and Loc5.BLM3: City Treasure Hunt
Answer Sheet, one copy of each per student. Review the instructions with the students. As
students plan their treasure hunts, observe the various strategies they use. Pose questions
to help students think about their strategies.
• “What strategy are you using to make your treasure hunt between 40 and 50 blocks long?”
• “What strategy are you using to make sure that the shortest possible route to all the treasures is less than 30 blocks?”
• “Have you modified your strategies? Why?”
If time allows, once students have had a chance to create their treasure hunts, have them copy
their directions without the number of blocks and then trade with another pair and use the
answer sheet to complete the treasure hunt.
After students have had time to complete their treasure hunts, provide each pair with markers
and a sheet of chart paper. Ask students to record the strategies they used to design their
treasure hunt.
Make a note of groups that might share their strategies and solutions during the Reflecting and
Connecting part of the lesson. Include groups whose methods varied in their sophistication.
158
Geometry and Spatial Sense, Grades 4 to 6
Reflecting and Connecting
Reconvene the class. Ask a few groups to share their strategies and post their work. Try to
order the presentations so that students observe inefficient strategies first, followed by more
efficient methods.
As students explain their work, ask questions that probe their thinking:
• “How did you make sure that your treasure hunt would be between 40 and 50 blocks long?”
• “Why did you use that strategy?”
• “How did you make sure that the shortest route to all your treasures would be less than 30
blocks long?”
• “Would you use the same strategies next time? Why or why not?”
• “How would you change your strategy the next time?”
• “Is your strategy similar to another strategy? Why or why not?”
Following the presentations, ask students to observe the work that has been posted and to
consider the efficiency of the various strategies. Ask:
• “Which strategy, in your opinion, is an efficient strategy?”
• “How would you explain this strategy to someone who has never used it?”
Avoid commenting that some strategies are better than others – students need to determine
for themselves which strategies are meaningful and efficient, and which ones they can make
sense of and use.
Refer to students’ work to emphasize important ideas about the location of objects:
• Using intersecting lines on a grid allows us to pinpoint specific locations.
• Cardinal directions (north, south, east, west) can orient people in an intended direction.
For students experiencing difficulty, reduce the grid to 5 × 5 and reduce the number of treasures and blocks.
For more of a challenge, ask students to create a symbol or design with their treasure hunt path.
Assessment
Observe students as they play the game and assess how well they:
• use intersecting lines to locate objects on a coordinate grid;
• apply appropriate strategies to play the game;
• explain their strategy;
• judge the efficiency of various strategies;
• modify or change strategies to find more efficient strategies.
Grade 5 Learning Activity: Location – City Treasure Hunt
159
Home Connection
Send home Loc5.BLM4: Planning a Trip, in which parents are asked to use life experiences to
assist their child’s understanding of location in the real world.
LEARNING CONNECTION 1
What’s on My Mind?
Materials
• Loc5.BLM5: What’s on My Mind? (1 per student)
• Transparency of the Loc5.BLM5: What’s on My Mind? game board
To assess prior knowledge, use the overhead transparency of Loc5.BLM5: What’s on My
Mind? Start at one of the four shapes (hexagon, square, trapezoid, triangle) and provide the
directions – for example, for the first letter of a selected name (e.g., south 1 and east 3). Write
down the directions on the overhead transparency or board, and verbalize the directions.
Encourage students to write down the letters privately to keep track. Continue until someone
guesses the correct name. Review the written directions with the class, writing in the correct
letter next to each of the clues.
Let the student with the correct answer be the next caller. Provide other categories, such as
cities, countries, animals, plants, colours.
Each student receives a copy of Loc5.BLM5: What’s on My Mind? game board and instructions. Students play in pairs.
Before starting, each pair selects its own category. Each player selects a name in that category
and writes down the directions for each letter of the name, as was previously demonstrated on
the overhead projector or board. For example:
M – From the square, west 3, south 3
A – From the triangle, east 1, north 5
R – From the trapezoid, south 3, east 4
Y – From the hexagon, north 4, west 3
LEARNING CONNECTION 2
Drawing Directions
Materials
• Loc5.BLM6: Drawing Directions
• rulers
160
Geometry and Spatial Sense, Grades 4 to 6
This investigation builds on the idea that a coordinate grid system can be used to describe the
position of an object. The focus changes from a location based on cells to identify an area to a
location based on the intersection of lines to identify a specific point.
Organize the students in pairs for this activity. Hand out Loc5.BLM6: Drawing Directions (one
copy per student). Review the instructions for the activity.
Observe the students as they work on this activity, and make note of the different methods
they use (e.g., drawing a rectangle with vertices inside cells, measuring the rectangle with a
ruler, drawing a rectangle with vertices on the intersection of lines).
You may need to use leading questions to encourage some students to label their grids or
use the intersection of lines instead of cells. Choose some groups of students to present their
strategies to the class.
After the students have finished, ask selected students to explain their strategies to the class.
Include a variety of strategies. After a variety of strategies have been presented, ask the following
questions to help students evaluate the different methods:
• “What strategies did you use to create a shape?”
• “Which strategies were easy to use and efficient? Why?”
• “Which strategies are similar? How are they alike? How are they different?”
• “What strategy would you use if you were to solve the problem again? Why?”
Grade 5 Learning Activity: Location – City Treasure Hunt
161
Loc5.BLM1
City Treasure Hunt Map
Avenues
North
10th Av
9th Av
8th Av
7th Av
6th Av
West
East
5th Av
4th Av
3rd Av
2nd Av
1st Av
1st
St
2nd
St
3rd
St
4th
St
5th
St
South
162
Geometry and Spatial Sense, Grades 4 to 6
6th
St
7th
St
8th
St
9th
St
10th
St
Streets
Avenues
Loc5.BLM2
City Treasure Hunt Planning Sheet
North
10th Av
9th Av
8th Av
7th Av
6th Av
West
East
5th Av
4th Av
3rd Av
2nd Av
1st Av
1st
St
2nd
St
3rd
St
4th
St
5th
St
6th
St
7th
St
8th
St
9th
St
10th
St Streets
South
In your City Treasure Hunt you need to provide for:
•
a total travel distance of between 40 and 50 blocks when treasures are collected in order;
•
a total travel distance of less than 30 blocks when treasures are collected in the shortest order;
•
five treasure points;
•
directions that send the treasure hunter east or west by streets, and then north or south by
avenues (e.g., “Travel 3 blocks east and 2 blocks north”).
Start at the intersection of and Distance and direction
and label it S.
Treasure
Label
# of
blocks
1. Travel and . You will find . Label it .
2. Travel and . You will find . Label it .
3. Travel and . You will find . Label it .
4. Travel and . You will find . Label it .
5. Travel and . You will find . Label it .
6. Travel and . You will be at the finish. Label it F.
Grade 5 Learning Activity: Location – City Treasure Hunt
163
Loc5.BLM3
Avenues
North
10th Av
9th Av
8th Av
7th Av
6th Av
West
East
5th Av
4th Av
3rd Av
2nd Av
1st Av
1st
St
2nd
St
3rd
St
4th
St
5th
St
6th
St
7th
St
8th
St
9th
St
10th
St
Streets
South
Label the start S on your grid.
1. Mark a letter ___ on your grid for the location of the first clue.
How many blocks did you travel? ___
2. Mark a letter ___ on your grid for the location of the second clue.
How many blocks did you travel? ___
3. Mark a letter ___ on your grid for the location of the third clue.
How many blocks did you travel? ___
4. Mark a letter ___ on your grid for the location of the fourth clue.
How many blocks did you travel? ___
5. Mark a letter ___ on your grid for the location of the fifth clue.
How many blocks did you travel? ___
6. Label the finish F on your grid. How many blocks did you travel? __
Name the total number of blocks travelled. ___
Collect the treasures in any order. What is the least number of blocks from start to finish? ___
164
Geometry and Spatial Sense, Grades 4 to 6
Loc5.BLM4
Planning a Trip
Dear Parent/Guardian:
In geometry we have been working on how locations can be found in cities
and on maps. The next time you take a trip, perhaps you could ask your child
to plan the route, taking into consideration directions such as north, south,
east, and west. Your child could calculate the number of city blocks for the trip,
or the number of kilometres for a longer trip.
Having your child find specific places on the basis of your directions will also
give him or her a good understanding of distance as well as direction.
Thank you for helping to show that studying location in geometry is important,
and helping your child to make links between school math and real-world math.
Grade 5 Learning Activity: Location – City Treasure Hunt
165
Loc5.BLM5
What’s on My Mind?
North
A
West
V
E
P
X
G W Q
B Y I
T M R
D N S
K
Z
O
L
East
U F C H J
South
Instructions for What’s on My Mind?
Students play in pairs or groups of three.
The students select a category – for example, names of people, countries, or animals.
Each student secretly writes down a word that he or she is thinking about in that category, and
writes the sequence of directions needed to reach the letters of the name. The letters can be
scrambled or out of order to add a challenge.
Getting to each new letter in the secret word should start at a shape, followed by the directions from that shape, for example:
S – From the triangle, east 4 and north 2
A – From the square, west 5 and south 1
M – From the hexagon, west 3 and north 3
The students decide who reads his or her clues first. That student keeps providing directions
for new letters until someone else in the group guesses the name. Each of the other students
in the group is allowed only one guess, so students should not be too quick to say a name. The
person who guesses the name first is the next person to call out the clues.
166
Geometry and Spatial Sense, Grades 4 to 6
Loc5.BLM6
Drawing Directions
North
West
East
South
Instructions for Drawing Directions
Work with a partner.
Draw a shape or simple design on the grid.
Write directions for drawing the shape or design so that someone following
the directions using a blank grid could make the exact same shape or design in
the exact same location without seeing your shape or design.
Grade 5 Learning Activity: Location – City Treasure Hunt
167
Grade 5 Learning Activity: Movement – Drawing Designs
Movement: Drawing Designs
Overview
In this learning activity, students create a picture puzzle with directions containing translations.
The activity provides an opportunity for students to identify and perform translations and to
make predictions; it reinforces the concept of using a grid system to describe the movements
of an object.
Big Ideas
This learning activity focuses on the following big idea:
Location and movement: Students investigate using different transformations to describe the
movement of an object.
Curriculum Expectations
This learning activity addresses the following specific expectation.
Students will:
• identify, perform, and describe translations, using a variety of tools (e.g., geoboard, dot
paper, computer program).
This specific expectation contributes to the development of the following overall expectation:
Students will:
• identify and describe the location of an object, using the cardinal directions, and translate
two-dimensional shapes.
TIME:
approximately
two 60-minute
periods
Materials
• transparency of Mov5.BLM1: Translations
• clear piece of acetate
• Mov5.BLM2: What Am I? (1 per student)
• sheets of tracing paper (1 per student)
• sheets of grid chart paper or large newsprint (2 per student)
• Mov5.BLM4: Translations in the Home (1 per student)
• markers
• ruler or metre stick (1 per pair of students)
168
Math Language
• triangle
• congruent
• trapezoid
• orientation
• pentagon
• position
• hexagon
• size
• circle
• shape
• translation
• prove
Instructional Sequencing
This learning activity provides an introduction to combining translations with a grid system
to locate and move objects to specific locations. Before starting this activity, students should
INSTRUCTIONAL
GROUPING:
pairs
review cardinal directions and the use of a grid – for example, by doing Learning Activity 1 or
a similar activity.
It would be beneficial for students to have experiences with some of the activities in the
Location portion of this guide before attempting this activity. This Drawing Designs learning
activity reinforces the use of a coordinate grid, and provides opportunities for exploring
congruency as it relates to the translation of shapes.
Getting Started
Have a brief discussion of the term slide and about where slides might be seen in the world
(e.g., playground slide, hockey player on ice).
Introduce the term translation (slide), using an overhead transparency of Mov5.BLM1: Translations.
Place a clear piece of acetate on top of the transparency, and use a coloured overhead marker
to trace around the triangle. Mark an arrow in the direction in which you are going to translate
the triangle. Translate (slide) the triangle along the path of the marked arrow.
Ask the students: “What changes?” and “What stays the same?” Discuss why the size and
shape of the triangle stay the same, as does the orientation, but the position of the triangle
changes. Explain that in math we use the term congruent when two shapes have the same size
and shape. Translate the triangle along other paths, asking similar questions.
Ask students if there is a way to describe the direction and length of the translation (number of
squares, cardinal directions).
Distribute copies of Mov5.BLM2: What Am I? and tracing paper. Review the instructions with
the students. Have the students complete the activity in pairs. Observe the various strategies
that students use during the activity (e.g., using an edge of a shape to count the squares,
numbering the grid). After students have completed the activity, lead a class discussion, asking
questions such as the following:
Grade 5 Learning Activity: Movement – Drawing Designs
169
• “What strategies did you use to translate the shapes?”
• “How did you know the shape was in the correct location?”
• “What strategy saved time during the task?”
• “What strategy would you use next time?”
Working on it
Organize students into pairs for this activity.
Hand out a copy of Mov5.BLM3: Design Instructions and one sheet of grid chart paper to each
student. Inform students that they will be using pattern blocks or their own shapes to create
their own design instructions. When they have finished the activity, the shapes should be “mixed
up” on the page and the instructions they write should translate the shapes into their design.
As students work on the activity, observe the various strategies that they use. Pose questions
to help students think about their strategies:
• “What strategy did you use to create your design?”
• “How did you come up with your instructions?”
• “Did you change any of your strategies? Why?”
When students have completed their designs, ask them to record their strategies on another
piece of grid chart paper or large newsprint. Make a note of pairs who might share their strategies during the Reflecting and Connecting part of the lesson. Include groups who used various
methods that range in their degree of efficiency (e.g., using shapes that do not fit easily on
the grid, making the design first and working backwards for the instructions, giving diagonal
instructions, labelling the grid with numbers).
Reflecting and Connecting
Once students have had an opportunity to complete the activity and record their strategies,
bring them together to share their ideas. Try to order the presentations so that students
observe inefficient strategies first, followed by more efficient methods.
As students explain their ideas, ask questions to help them to describe their strategies:
• “How did you choose your shapes?”
• “What strategy did you use to make your design?”
• “How would you explain your strategy to someone who hasn’t done this?”
Avoid commenting that some strategies are better than others – students need to determine
for themselves which strategies are meaningful and efficient, and which ones they can make
sense of and use.
Refer to students’ work to emphasize geometric ideas:
• The movement of an object can be described using translation.
• A translation does not affect the size, shape, or orientation of an object.
• A coordinate grid is helpful for locating objects and describing movement.
170
Geometry and Spatial Sense, Grades 4 to 6
Students experiencing difficulties with this activity may use geoboards (and/or dot paper) to
construct (draw) shapes according to specific oral directions from the teacher or another group
member. They translate these figures according to specific oral directions (e.g., two spaces to
the right and four up). Students can compare their geoboard (dot paper) constructions and
translations with those of their peers to check for accuracy, and can discuss discrepancies.
Assessment
Observe students as they complete the activity and assess how well they:
• apply appropriate strategies for creating a design and the instructions for it;
• translate shapes with accuracy;
• describe the translations (length, direction);
• explain their strategy;
• judge the efficiency of various strategies.
Home Connection
Send home Mov5.BLM4: Translations in the Home. In this home connection students are
asked to look at patterns and designs in their home and/or garden where translations have
been used, and list the two most interesting translations they can find.
LEARNING CONNECTION 1
What Stays the Same?
Materials
• large outside 10 × 10 grid or tiled floor
• string
Place four students on a large 10 ×10 grid or learning carpet to make a shape (e.g., a rectangle).
Each student stands on the grid at a vertex of the rectangle holding the end of a piece of string.
Connect the student “vertices” with one another by string “edges” to make the rectangle.
Give directions. For example: “Everyone move north three units, then everyone move west two
units.” After each move ask the students, “What remains the same?” and “What changes?”
They will find that shape and size remain the same (congruency is preserved), as does orientation. The location changes. If the shape or size does change, talk about strategies to prevent
that from happening (e.g., use a grid, use equal steps).
Use different students, change the shape to a triangle or hexagon, and repeat the procedure.
Working in groups of six or eight, students can take turns giving directions and checking on
what remains the same and what changes.
Grade 5 Learning Activity: Movement – Drawing Designs
171
LEARNING CONNECTION 2
How Many Ways?
Materials
• Mov5.BLM5: How Many Ways? (1 per student)
• overhead transparency of Mov5.BLM5: How Many Ways?
• tracing paper
• Mira
Before beginning the activity, review the following with students:
• For translations, we travel east or west before travelling north or south.
• For reflections, we can talk about a horizontal reflection over an east-west
line (e.g., 6th Avenue), or a vertical reflection over a north-south line
(e.g., 3rd Street).
Have students work in pairs for this activity. Hand out Mov5.BLM5: How Many Ways? and
review the instructions. Have tracing paper and Miras available for student use.
Take up the activity with the whole class, using an overhead transparency of Mov5.BLM5:
How Many Ways? or a large grid of the blackline master drawn on the board or chart paper.
Students volunteer to show different ways of moving the shape from position to position.
Discuss what remains the same and what changes during a reflection and during a translation.
172
Geometry and Spatial Sense, Grades 4 to 6
Mov5.BLM1
Translations
North
West
East
South
Grade 5 Learning Activity: Movement – Drawing Designs
173
Mov5.BLM2
What Am I?
North
West
East
South
Use your tracing paper to trace the shapes. Then translate them, using the
instructions below, and sketch them in the new location.
1. Translate the circle 7 blocks west.
2. Translate the shaded half-moon 9 blocks east and 10 blocks south.
3. Translate the shaded right-angled triangle 4 blocks west and 8 blocks north.
4. Translate the white right-angled triangle 10 blocks east and 1 block south.
5. Translate the white half-moon 5 blocks east and 4 blocks north.
6. Translate the rectangle 7 blocks west and 8 blocks south.
7. Translate the isosceles triangle 3 blocks south.
174
Geometry and Spatial Sense, Grades 4 to 6
Mov5.BLM3
Materials
•pattern blocks or other shapes
•grid chart paper
•markers
•ruler or metre stick
Use a ruler or metre stick and a marker to make a line that divides your grid
chart paper in half.
On the top half of the grid chart paper:
–Create a design, using pattern blocks or your own shapes. Trace your design
lightly in pencil to show the location.
–Move the pattern blocks or shapes to different locations on the grid. Use a
marker to trace each new location.
On the bottom half of the grid chart paper:
–Create a set of instructions that will translate the shapes to make your design.
–Start each instruction with “translate” and use directions and numbers.
For example: Translate the triangle 4 blocks north and 6 blocks east.
Erase the pencil tracing of your design.
Grade 5 Learning Activity: Movement – Drawing Designs
175
Mov5.BLM4
Translations in the Home
Dear Parent/Guardian:
At the moment we are studying geometry in math. Today your child was introduced to a new geometry word, translation.
The word translation in geometry has a very different meaning from the one
used in everyday language, when we talk about a translation from one language into another. In math, translation refers to sliding a shape or an object
along a straight path. For example:
In geometry, people say that the triangle on the left has been translated to
the right.
Ask your child to find some designs in the home or garden in which translations
(slides) have been used to make patterns (e.g., wallpaper, designs on clothes,
rows in the vegetable garden). Your child is also encouraged to look through
magazines or newspapers for translations.
On the back of this letter, or on a separate piece of paper, ask your child to
draw the two most interesting translations found in your home or garden,
describe where they were found, and bring the answers to class tomorrow.
Any time you can give to this activity and to talking with your child about the idea
of translation will benefit your child’s mathematical growth and development.
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Geometry and Spatial Sense, Grades 4 to 6
Mov5.BLM5
How Many Ways?
North
10th Av
A
9th Av
8th Av
7th Av
6th Av
West
East
5th Av
4th Av
C
B
3rd Av
2nd Av
1st Av
1st
St
2nd
St
3rd
St
4th
St
5th
St
6th
St
7th
St
8th
St
9th
St
10th
St
South
Instructions
List all the ways in which you can move the house from:
Position A to Position B
Position B to Position C
Position A to Position C
Grade 5 Learning Activity: Movement – Drawing Designs
177
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
Two-Dimensional Shapes:
Connect the Dots
OVERVIEW
In this learning activity, students use dynamic geometry software to construct and measure
angles. They develop their understanding of benchmark angles, and they construct polygons,
given side length and angle restrictions.
BIG IDEAS
This learning activity focuses on the following big ideas:
Properties of two-dimensional shapes and three-dimensional figures: Students develop
a sense of the size of benchmark angles through investigation. Concrete materials and the
classroom environment provide measuring tools with which to compare angles to benchmarks.
Geometric relationships: Students explore the relationships between various properties of
polygons, and of quadrilaterals in particular. For example, if a polygon has all congruent sides,
it also has congruent angles.
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectations.
Students will:
• measure and construct angles up to 180° using a protractor, and classify them as acute, right,
obtuse, or straight angles;
• construct polygons using a variety of tools, given angle and side measurements.
These specific expectations contribute to the development of the following overall expectation.
Students will:
• classify and construct polygons and angles.
TIME:
approximately
80 minutes
MATERIALS
• computer and data projector
• The Geometer’s Sketchpad dynamic geometry software
www.edu.gov.on.ca/eng/studentsuccess/lms/library.html)
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• number cubes (dice) (1 per pair of students)
• 2D6.BLM1: Angle Cards (1 per pair of students) to be cut out, shuffled, and placed face
down as a deck
• 2D6.BLM2: Shape Cards (1 per pair of students) to be cut out, shuffled, and placed face
down as a deck
• protractors, rulers, and pencils if needed
• 2D6.BLM3: Measuring Angles and Lengths in GSP (as needed)
• 2D6.BLM4: Quadrilateral Angle Properties (1 per student)
MATH LANGUAGE
• point
• protractor
• rhombus
• line segment
• polygon
• trapezoid
• angle
• triangle
• parallelogram
• ray
• rectangle
• vertex
• square
• kite
Instructional Sequencing
INSTRUCTIONAL
GROUPING:
pairs
Students should have had significant experiences constructing and measuring angles up to 90°,
using pencil, paper, ruler, and protractors. They should also have had experience using The
Geometer’s Sketchpad (GSP) software and be familiar with the basic functions of the program.
For GSP-based tutorials to help students learn to use The Geometer’s Sketchpad, refer to
the Curriculum Services Canada website for TIPS resources at http://www.curriculum.org/csc/
In Grade 6 students continue to sort and classify quadrilaterals according to geometric
properties. Students construct polygons, using geoboards, dynamic geometry software, and
protractors. The construction of angles is extended to include angles from 0° to 180°. As
students investigate and construct angles, they begin to apply their knowledge of angles to
the properties of polygons.
In geometry, there is an important distinction between drawing and constructing. The
Geometer’s Sketchpad helps to illustrate this distinction. A constructed shape or geometric
situation will remain unchanged, but a drawn one may not. For example, students may construct an equilateral triangle in GSP and can manipulate it to change its size. However, it will
always remain equilateral, regardless of side length. Students may draw an equilateral triangle
in GSP, using the angle measure or length measure function, but when they move a point on
the triangle, the type of triangle changes. Equal angles and side lengths are not preserved.
In Getting Started, students see some of the features of Sketchpad, but more importantly, they
build on their knowledge of benchmark angles to include 45° and 135° angles – half of a right
angle, and one and a half right angles.
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
179
GETTING STARTED
This part of the lesson can be presented to the whole class as a teacher demonstration, or
students can work with a partner at a computer and work through the activity. If students are
to work on the file, it must be placed on a shared drive to which students have access. This
part of the lesson is presented here as a teacher demonstration.
Using a computer and a data projector, open The Geometer’s Sketchpad. Activate students’
prior use of the program by asking:
• “Who remembers what we’ve used this program for in the past?”
• “Can you tell me what some of the tools on the left are called, and what they do?”
• “What are some of the reasons why we use The Geometer’s Sketchpad in addition to penciland-paper geometry?”
Open the demonstration file called “Gr6AngleDemo.gsp file”. This file is available as a download from www.edu.gov.on.ca/eng/studentsuccess/lms/library.html.
The file shows two segments joined at a vertex to form an angle,
the geometric features they see – points, vertex, segments, and angle. Explain that the action
buttons perform the actions described (e.g., clicking on the “Rotate segment AB in a clockwise
direction” button starts the rotation, and clicking again on the button stops it; clicking on the
“Reset” button returns the sketch to its original appearance). Ask the following questions:
• “What do you think the measure of
ABC is?” (45° – show the angle measure, then hide
it again)
• “What type of angle is it?” (an acute angle)
• “I’m going to rotate segment AB counterclockwise. I want you to tell me to stop when you
think
ABC is a right angle.”
Stop the rotation when the students tell you to, and show the measure of the angle. Hide the
angle again, and rotate segment AB counterclockwise to approximate one and a half right
angles, or 135°.
Continue manipulating
ABC, using the action buttons and asking students to predict the
measures before revealing them to the class. Future useful benchmark angles include 30°,
60°, and 120°.
Consider manipulating
ABC by dragging points instead of using the buttons, to illustrate how
you can:
• create angles of different sizes (by dragging a point);
• make the lengths of the line segments different but have the angle remain the same (e.g., by
dragging point C to make segment BC longer);
• rotate the angle (by dragging point C and point A to create a 45° angle in which segment BC
is not horizontal).
Before pairing students off for Working on It, demonstrate some examples of polygon
constructions by clicking on the “Show a triangle with this interior angle” and “Show a
quadrilateral with this interior angle” buttons. By clicking on the “More” button, students can
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Geometry and Spatial Sense, Grades 4 to 6
attempt to use
ABC to construct their own polygons. When you close the sketch or exit The
Geometer’s Sketchpad, a prompt may appear asking whether you want to save any changes to
the sketch. Select “Don’t Save” in order to keep the sketch in its original form.
WORKING ON IT
Students are to be paired up on a single computer for this activity. Each pair needs a number
cube (die), 2D6.BLM1: Angle Cards, and 2D6.BLM2: Shape Cards.
Students roll the number cube to determine the length of one side of the polygon. Next,
students select an angle card from the shuffled deck to determine the angle, and a shape
card to determine the shape. The challenge is to use the angle and side length at least once in
constructing a polygon with software, or if there are not enough computers, with a protractor,
ruler, and pencil. Students may require 2D6.BLM3: Measuring Angles and Lengths in GSP to
help with some of the features of The Geometer’s Sketchpad.
As students investigate the construction of each polygon, they will begin to discover that
some shapes cannot be created, given the particular side and angle components.
After the students have spent some time on the activity, have them separate the quadrilateral
cards from the other shape cards and use only quadrilaterals as they continue with the activity.
Students can then reflect on the challenges they encounter in creating quadrilaterals.
REFLECTING AND CONNECTING
• “What properties of a polygon/quadrilateral do you need to know in order to construct it?”
• “What polygons/quadrilaterals were easiest to construct?”
• “What did you discover about the polygons/quadrilaterals you tried to construct?”
• “What surprised you about some of the constructions?”
Ask students to identify some of the tasks that were impossible to complete. For example, if
students drew a right-angled triangle card and an angle measure card of 135°, they would find
it impossible to construct the polygon. For each instance, ask students to provide an explanation of why they were unable to do the construction.
Talk with the students about some of the properties of specific quadrilaterals. Refer to side
lengths, interior angles, and the location of lines with respect to one another (e.g., parallel,
• “What can you tell me about all of the quadrilaterals you constructed?”
• “How were they the same and how were they different?”
• “Were there properties that were related? For example, does knowing something about side
length tell you something about the angles?”
If this is the first time the students have used Sketchpad, spend some time discussing the
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
181
During the Working on It some students may benefit from using pencils, rulers, and protractors
for the activities rather than the software. Also, they may need to work with polygons that
have already been partially constructed. For example, they could be given two side lengths
and the included angle and asked to construct a parallelogram.
Challenge students to create polygon features cards (angle measures, side lengths, polygon
names) to create a game that can be played by two or more students.
ASSESSMENT
Observe students to assess how well they:
• measure, construct, and sort acute, right, obtuse, or straight angles;
• describe the construction of polygons with a focus on properties (angle measurement, length
and number of sides).
During the reflection, note how well students make connections to the properties of quadrilaterals after constructing quadrilaterals. Students will be able to perform the construction
task without thinking too deeply about properties and their relationship to each other. Probing
questions can help assess the connections that they make.
• “Why were you unable to construct some of the shapes on the cards?”
• “What could you have changed to be able to make the construction?”
• “Given the length and angle measure, what possible quadrilaterals could you construct?”
HOME CONNECTION
Send home 2D6.BLM4: Quadrilateral Angle Properties. Students are to name each quadrilateral, measure the angles in it, and write a definition of it using angle properties.
Students will need to take protractors home to complete the task. Remind students to be
specific when naming the quadrilaterals (e.g., right trapezoid, isosceles trapezoid).
LEARNING CONNECTION 1
Rotating Polygons
Materials
• 2D6.BLM5: Rotating Polygons (1 per pair of students)
• tracing paper, cut into eighths (12 pieces per pair of students)
• math journals
Have students work with a partner. Provide each pair with a copy of 2D6.BLM5: Rotating
Polygons and 12 pieces of the tracing paper.
Students are to place one of the small pieces of tracing paper over one of the polygons, and
use a ruler to trace it. They then rotate the traced shape over the original polygon, noting how
many times the traced shape matches the orientation of the original polygon.
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Geometry and Spatial Sense, Grades 4 to 6
The partner records in his or her math journal the name of the shape and the number of times
the tracing matched the original. Partners take turns recording and tracing/rotating until they
have completed the process with all of the polygons.
Bring the whole class together for a discussion. Ask which shapes matched the original position
only once during a complete rotation. Explain that those polygons do not have rotational symmetry, because in one rotation, they matched the original position one time.
Identify with the class the polygons with an order of rotational symmetry greater than 1.
Give the student pairs time to discuss those polygons and see if they can find any patterns or
commonalities between them. Students should discover that regular polygons have an order of
rotational symmetry equal to their number of sides.
As an extension, challenge students to create polygons that have rotational symmetry. For
example, ask them to draw a hexagon that is not regular and has rotational symmetry.
LEARNING CONNECTION 2
What’s My Line
Materials
• 2D6.BLM6: Quadrilateral Cards (1 per group of students)
• 2D6.BLM7: What’s My Line Place Mat (1 per group of students)
• Miras (1 per group of students)
• poster paper
Students work in groups of two or three. Provide each group with a copy of 2D6.BLM6:
Quadrilateral Cards and 2D6.BLM7: What’s My Line Place Mat. Students are to cut out the
cards and place each card in the appropriate quadrant of the place mat, depending on the
number and type of lines of symmetry the quadrilateral has. Quadrilaterals with no line of
symmetry can be left off the mat.
Once the quadrilaterals have been sorted, students use Miras to see if they have been placed
correctly in the place mat. Students can also use the Mira to help with sorting.
Have students write a summary on poster paper of their discoveries during the activity. Have
them identify any common characteristics of quadrilaterals with the same number of lines of
symmetry and then write a description for that group (e.g., all rectangles that are not squares
have two lines of symmetry – a horizontal line and a vertical line).
• “Do you think your rule will hold true for all (rectangles, kites, etc.)?”
• “Which quadrilaterals have no line of symmetry? Can you make a generalization about those?”
• “How specific do you need to be when making your rules? For example, can you talk about
all trapezoids, or just specific types?”
Students should recognize that using lines of symmetry is an alternative method of
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
183
2D6.BLM1
Angle Cards
184
45 ˚
90 ˚
25 ˚
30 ˚
120 ˚
60 ˚
135 ˚
15 ˚
55 ˚
115 ˚
180 ˚
25 ˚
35 ˚
100 ˚
40 ˚
Geometry and Spatial Sense, Grades 4 to 6
2D6.BLM2
Shape Cards
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
185
2D6.BLM3
Measuring Angles and Lengths in GSP
To measure an angle: First click
on the select tool and click on
point A then B then C.
Measure
m ABC = 95.10º
A
Select Measure from
and choose Angle.
A
C
C
B
B
Note that you must click on the points in the proper order – the point that is
the vertex must be selected second, or you will not get the Angle option from
B
To measure a length:
First click on the select tool and click
on the line segment (e.g., segment AB).
A
Measure
m AB = 7.20 cm
B
Select Measure from the pulldown menu, and choose Length.
A
Alternatively, you can right-click on the segment, and choose Length from the
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Geometry and Spatial Sense, Grades 4 to 6
2D6.BLM4
Complete the following chart by naming each quadrilateral, measuring and recording the
angles in it, and writing a definition of it using its angle properties. Be specific when you name
each quadrilateral. An example is done for you.
90˚
90˚
Rectangle
90˚
with four 90° angles.
90˚
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
187
2D6.BLM5
Rotating Polygons
188
Geometry and Spatial Sense, Grades 4 to 6
Grade 6 Learning Activity: Two-Dimensional Shapes – Connect the Dots
2D6.BLM6
189
What’s My
Line?
2D6.BLM7
What’s My Line Place Mat
190
Geometry and Spatial Sense, Grades 4 to 6
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
Three-Dimensional Figures:
Sketching Climbing Structures
OVERVIEW
In this task, students build a three-dimensional model, given an isometric sketch. They sketch
the different views of the structure and complete an isometric sketch of the same structure
from a different perspective.
BIG IDEAS
This learning activity focuses on the following big ideas:
Properties of two-dimensional shapes and three-dimensional figures: Students develop their
spatial sense through investigating the properties of three-dimensional figures, specifically
looking at faces, edges, and vertices from different angles.
Geometric relationships: Students must be able to flexibly represent three-dimensional figures
as concrete objects, but also in two-dimensional sketches or drawings. This activity draws upon the
relationship between the two-dimensional faces of an object and its three-dimensional construction.
CURRICULUM EXPECTATIONS
This learning activity addresses the following specific expectations.
Students will:
• build three-dimensional models using connecting cubes, given isometric sketches or different
views (i.e., top, side, front) of the structure;
• sketch, using a variety of tools (e.g., isometric dot paper, dynamic geometry software),
isometric perspectives and different views (i.e., top, side, front) of three-dimensional figures
built with interlocking cubes.
These specific expectations contribute to the development of the following overall expectation.
Students will:
• sketch three-dimensional figures, and construct three-dimensional figures
from drawings.
191
TIME:
approximately
60 minutes
Materials
• interlocking cubes (3 per student)
• 3D6.BLM1: 2 cm Grid Paper (2 sheets per student)
• 3D6.BLM2: Isometric Dot Paper, cut into half sheets (2 half sheets
per student)
• 3D6.BLM3: Sketching Climbing Structures (1 per student)
• 3D6.BLM4: Sketching Isometric Perspectives (1 per student)
• 3D6.BLM5a–b: Teach an Adult to Sketch (1 per student)
Math Language
• congruent
• three-dimensional figure
• orthographic sketches or
• isometric dot paper
views (top, front, side)
• rotate
• vertical line segment
• mirror image
• diagonal line segment
• two-dimensional
• horizontal line segment
representation
INSTRUCTIONAL
GROUPING:
individuals, pairs
• isometric perspective or view
• edge
Instructional Sequencing
This learning activity focuses on the relationships between two-dimensional drawings and
three-dimension figures. Prior to this lesson, students should be familiar with building threedimensional solids from top, front, and side views.
In Grade 6, students learn to draw different views (i.e., top, front, side) and sketch isometric
perspectives of three-dimensional figures. They also require opportunities to build threedimensional models, given isometric sketches or different views of a structure.
These experiences will help students develop an understanding of the relationship between
three-dimensional figures and their two-dimensional representations. Students have opportunities to represent their thinking concretely and through two different types of drawings. They
also have opportunities to make connections between two-dimensional sketches of threedimensional figures in the classroom and the use of these types of two-dimensional sketches in
the real world (e.g., in assembly instructions for toys, furniture, model cars or planes).
Isometric diagrams use an isometric grid. An isometric grid shows three axes instead of the
two found in a rectangular grid. One axis runs vertically; the other two axes run “down” at 30°
angles to the left and right.
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Geometry and Spatial Sense, Grades 4 to 6
GETTING STARTED
Distribute three interlocking cubes and one sheet of 3D6.BLM1: 2 cm Grid Paper to each
student. Show students the following L-shaped structure built from interlocking cubes:
Ask students to build a congruent structure with their three interlocking cubes. Have them
place the structure on the 2 cm grid paper so that the single cube is facing them. Instruct students to label the front, back, left side, and right side of the structure on the paper. (Circulate
as they do so to ensure that all students have the structure oriented in the same way.)
Ask students what they see when they look at the structure from the front view. (Suggest that
students look from the level of their desks to get the complete front view.) Students should see
a vertical rectangle divided in half:
Have students draw the front view next to the “Front” label on their page.
Next, have students rotate their page 90º clockwise so they are looking at the right side view.
Again, ask what they see (mirror image of an “L”). Have students draw this view next to the
“Right Side” label on their page:
Continue having the students rotate their pages 90º clockwise to record the back and left side
views. Finally, have them look down on the structure to see the top view. They should notice
that the top view could be drawn by tracing around the base of the structure.
Explain that the views they have drawn are called orthographic views and that they are
two-dimensional representations of the three-dimensional structure. Ask students to write
“Orthographic Sketches” at the top of the page on which they have just completed the views.
Explain that orthographic sketches are only one type of two-dimensional representation of a
three-dimensional figure. Isometric sketches, or perspective drawings, are also two-dimensional
representations of three-dimensional figures. Make connections to students’ previous experiences
with orthographic and/or isometric sketches/representations. Ask, “When might you see/use this?
How might this kind of sketch be used in the real world?” Provide examples like floor plans and
elevations that architects and designers would use. You may wish to have some examples for the
students to view on the overhead projector.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
193
Distribute one half-sheet of 3D6.BLM2: Isometric Dot Paper to each student. Discuss any
prior experiences students have had of using isometric dot paper (drawing two-dimensional
shapes, drawing nets). Inform students that they are now going to try to sketch an isometric
view, or a three-dimensional view, of the L-shaped structure for which they have already
completed orthographic sketches. Ask students to write “Isometric Sketch” at the top of their
half-page of isometric dot paper.
Have students arrange the structure on their page with the orthographic views so that the
front view is once again facing them. Now have them rotate the page 45º counterclockwise so
that they are seeing the front-left view of the structure.
Ensure that all students have the structure oriented in the same way, and then instruct them to
use the following steps to sketch the isometric perspective. Model the steps on the overhead
projector as you go through them.
Step 1: Connect two dots vertically to represent the closest vertical edge of the structure
(edge of single cube).
Step 2: Draw diagonal line segments to connect three dots up to the left and two dots up to
the right to represent the bottom horizontal edges of the structure. Note that the line segment
that you drew up to the left is twice as long as the one that you drew up to the right because it
represents an edge that is two cubes long.
Step 3: Draw vertical line segments for the remaining four vertical edges.
Step 4: Complete the diagonal line segments to represent the remaining horizontal edges that
are parallel to the bottom edges.
Step 5: Shade the faces of the structure that are visible in the top view only.
The finished drawing should look like this:
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Geometry and Spatial Sense, Grades 4 to 6
WORKING ON IT
Have students work in pairs. Describe the following scenario to the class:
“A company that manufactures climbing equipment for school playgrounds is introducing
new climbing blocks that can be combined to create a variety of different play structures.
The designer of these blocks has sent a sketch of one structure that can be built using six
of the climbing blocks. The company has asked for your help in creating top, front, and
side views of the structure as well as a different isometric perspective to be included in the
assembly instructions for the new climbing blocks.”
Distribute to each student a copy of 3D6.BLM3: Sketching Climbing
Structures, 1 sheet of 3D6.BLM1: 2cm Grid Paper, and a half sheet of
3D6.BLM2: Isometric Dot Paper.
Working with a partner, students are to combine their cubes and use the
Designer’s Sketch shown on 3D6.BLM3: Sketching Climbing Structures to
build a model of the climbing structure. Then have each student sketch top,
front, and side views and draw a different isometric perspective of the same
Assessment Opportunity
Circulate to assess
students’ ability to connect
the two-dimensional
faces of the threedimensional figure with
the orthographic views of
the figure.
arrangement of climbing blocks.
REFLECTING AND CONNECTING
Bring the class together. Refer back to the five views of the L-shaped structure on the
• “What similarities are there between the sketches of the five views and the isometric sketch?
What differences are there?”
• “Which sketches did you find easier to draw? Why?”
• “What did you find easiest about sketching the climbing structure? What did you find most
challenging?”
• “Which type of sketch do you think will be most helpful for assisting people in building the
climbing structure? Why?”
• “Where else are orthographic and isometric drawings used?” (toy/furniture assembly instructions, building plans, model cars/planes, installation instructions)
Ask students to make connections between the two types of sketches – how do the shape and
number of faces of the orthographic sketches relate to the isometric drawing?
Some students may have difficulty drawing the isometric perspective of the climbing structure.
Encourage them to follow the steps used in drawing the L-shaped structure. Try enlarging the
isometric dot paper on a photocopier so that the distance between diagonal dots is equal to
the length of one toothpick and have these students place the toothpicks on the dot paper
until the sketch “looks right” before drawing the lines in pencil. Provide struggling students with
3D6.BLM4: Sketching Isometric Perspectives for step-by-step instructions on how the designer’s
sketch was created. Suggest that they use a similar method for creating a different perspective.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
195
Ask students who finish early or require a challenge to create a different climbing structure using
an additional climbing block (for a total of seven) and to sketch orthographic and isometric views
of the new structure.
ASSESSMENT
Observe students to see how well they:
• build a three-dimensional model, given an isometric sketch of the structure;
• sketch isometric perspectives and different views of three-dimensional figures built with
interlocking cubes.
This lesson is particularly focused on skills, rather than content knowledge. It would be appropriate to focus on the process expectations of representing, communicating, and connecting.
While there are many practical applications of both isometric and orthographic sketching, they
are not always readily evident to the junior learner. Students must stretch their thinking to
make connections with the skills in this lesson.
HOME CONNECTION
Send home 3D6.BLM5a–b: Teach an Adult to Sketch. In this Home Connection activity,
students find a simple polyhedron-shaped object at home and teach a parent or guardian how
to sketch two different isometric perspectives of the object using isometric dot paper.
Learning Connection 1
Drawing Views in The Geometer’s Sketchpad
Materials
• 3D6.BLM6a–b: Drawing Views in The Geometer’s Sketchpad (1 per student)
• computers with The Geometer’s Sketchpad software (1 per student or pair
of students)
• interlocking cubes (6 different-coloured cubes per student or pair of students)
Although it is possible for students with no experience in using The Geometer’s Sketchpad (GSP)
to follow the instructions provided to create top, side, and front views of a three-dimensional
structure, it is recommended that students have a minimum of one period to explore GSP.
For GSP-based tutorials to help students learn to use The Geometer’s Sketchpad, refer to the
Curriculum Services Canada website for TIPS resources (http://www.curriculum.org/csc/library/
Distribute one copy of 3D6.BLM6a–b: Drawing Views in The Geometer’s Sketchpad to each student.
Arrange students in pairs, either to work together on one computer or to work side by side on
individual computers. (Consider pairing students of low reading ability with students who have
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Geometry and Spatial Sense, Grades 4 to 6
Students begin by using six different-coloured interlocking cubes to build a structure from an
isometric sketch provided. Students then follow instructions to draw top, left side, right side,
and front views of the structure using The Geometer’s Sketchpad.
Learning Connection 2
The Great Imitator
Materials
• a three-dimensional structure made from 6 interlocking cubes (for teacher demonstration
– hidden from students’ view)
• 3D6.BLM7: Imitator Demo Structure and Views
• interlocking cubes (10 per student)
Begin by telling students that you have a three-dimensional figure built from six interlocking
cubes hidden behind a book and that you are going to describe the structure to the class.
Students are to try to build a structure congruent to yours on the basis of your description.
Distribute ten interlocking cubes to each student (students will be using up to ten to build their
own structures later on).
Describe the top, side, and front views of your figure in turn. Use precise mathematical language. Ask students to build the figure on the basis of your description. You may wish to use
3D6.BLM7: Imitator Demo Structure and Views or design your own figure.
Have students first share what they built with a partner. Then students share with the whole
class to see whether any of their constructions are congruent to your structure.
Give a round of applause to any students who were able to imitate your structure. Discuss with
students the kind of language and/or description that was useful in helping them create the
Explain that students will now have an opportunity to play a game similar to what the class just
did. The game is called The Great Imitator.
The Great Imitator is played as follows:
Students work with a partner. In each pair, one student (the Creator) builds a structure, using
six to ten interlocking cubes, and hides it from his or her partner (the Imitator). The Creator
then describes the front, top, and side views of the structure, using mathematical language,
while the Imitator tries to recreate the structure. The Imitator may not ask questions. Students
then compare the two structures to see whether they are congruent. Points may be awarded to
students who build a figure that is congruent to the Creator’s figure.
Partners then switch roles. Play the game several times.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
197
Learning Connection 3
Sketches and Views Cards
Materials
• interlocking cubes (5 per student)
• small index cards (2 per student)
• 3D6.BLM1: 2 cm Grid Paper, cut into smaller pieces for gluing onto an index card (1 smaller
piece per student)
• 3D6.BLM2: Isometric Dot Paper, cut into smaller pieces for gluing onto an index card (1
smaller piece per student)
• glue sticks
Explain to students that they are going to create cards for a game that they will then have an
opportunity to play with a partner.
Each student begins by using five interlocking cubes to build a structure. As students build,
distribute two small index cards to each student.
Students then draw three orthographic views (top, side, and front) on grid paper, and the isometric view on a small piece of isometric dot paper. Finally, they cut out the views and glue each to
an index card. The orthographic views go on one card, and the isometric view on another.
Once the students have finished, all of the orthographic views cards and all of the isometric
sketch cards are collected and shuffled separately. (If some students finish well before others,
have them switch cards with a partner and have the partners try to build each other’s structures
from the orthographic views or the isometric sketch.)
To play the game, all students draw a card from the views card pile, build the structure, and
then try to find the matching sketch card for their structure. Alternatively, students draw a card
from the sketch card pile, build the structure, and then try to find the matching views card.
Learning Connection 4
Construction Challenge
Materials
• interlocking cubes (40 per group of students)
• 3D6.BLM8: The 8 Four-Cube Structures (answer page)
• 3D6.BLM9a–c: Construction Challenge (1 per student)
Arrange students into groups of four. Distribute forty interlocking cubes to each group.
Challenge students to build as many different four-cube structures as possible. Remind students
to test for congruent structures by rotating each structure and viewing it from different perspectives to make sure it is unique. (There are 8 unique four-cube structures. It is important to allow
students time to discover this on their own. They will need the experience of visualizing the 8
four-cube structures in different orientations in order to be successful in the activity that follows.)
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Geometry and Spatial Sense, Grades 4 to 6
Once groups of students have discovered what they believe to be all of the possible four-cube
structures, have each group present one structure to the class. Continue having each group
present a structure, making sure to encourage students to test for congruence, until all 8
unique four-cube structures have been shared. Display the 8 structures and make sure each
group has all 8 built. (It may help students to visualize the structures if each of the four-cube
structures is constructed from different-coloured interlocking cubes.)
Distribute to each student one copy of 3D6.BLM9a–c: Construction Challenge.
Have students work in groups to combine two or more of their four-cube structures (as indicated
on the handout) to form other structures, given the isometric sketches of the other structures.
Note: This activity is far more challenging than it first appears. However, the experience is well
worth it for students as they continue to develop their spatial sense and visualization skills.
Encourage groups to share as they successfully construct a figure – have them decompose the
figure into its four-cube parts for their classmates. This will help students to see and copy how
the structure can be formed and provide motivation for students to continue to work on the
other challenges.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
199
3D6.BLM1
2 cm Grid Paper
200
Geometry and Spatial Sense, Grades 4 to 6
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
3D6.BLM2
Isometric Dot Paper
201
3D6.BLM3
Sketching Climbing Structures
A company that manufactures climbing equipment for playgrounds is introducing
new climbing blocks that can be combined to create a variety of different play
structures. The designer of these blocks has forwarded a sketch of one structure
that can be constructed using six of the climbing blocks. The company has asked
for your help in creating top, front, and side views of the structure as well as a
different isometric perspective to be included in the assembly instructions for the
new climbing blocks.
Designer’s Sketch
Climbing Structure A
1. Build a model of Climbing Structure A, using six interlocking cubes.
2. Draw and label the top, front, and side views of the structure on grid paper.
3. Sketch a different isometric perspective of the same structure on isometric dot paper.
202
Geometry and Spatial Sense, Grades 4 to 6
3D6.BLM4
Sketching Isometric Perspectives
1. Draw a line segment to represent the closest
vertical edge of the structure. (Connect two
vertical dots.)
2. Draw diagonal line segments up to the left
and up to the right to represent the bottom
horizontal edges of the structure.
3. Draw vertical line segments for the
remaining vertical edges.
4. Complete the sketch by drawing diagonal
line segments to represent the remaining
horizontal edges that are parallel to the
bottom edges.
5. Shade the faces that are visible in the
top view.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
203
3D6.BLM5a
Dear Parent/Guardian:
We have been learning how to sketch isometric perspectives of three-dimensional
figures in math. Your child has had opportunities to sketch polyhedra (threedimensional figures with polygon faces) built from interlocking cubes using special
paper called isometric dot paper.
In this activity, your child will teach you or another adult at home how to
draw two different isometric perspectives of a polyhedron.
(e.g., a facial tissue box, a cereal box).
Ask your child to show you two different isometric perspectives of this object
and teach you how to draw them using the isometric dot paper provided.
In trying to teach you how to draw the isometric perspectives, your child will
be reinforcing the learning that occurred in the classroom and gaining confidence in his or her ability to sketch three-dimensional figures.
204
Geometry and Spatial Sense, Grades 4 to 6
3D6.BLM5b
Name of Object:
Name of Object:
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
205
3D6.BLM6a
Drawing Views in The Geometer’s Sketchpad
Build this three-dimensional structure, using six different-coloured interlocking cubes.
Follow the instructions below to draw and label top, left side, right side, and
2. Prepare the screen:
•To create a grid, click on Graph – Define Coordinate System.
•Next, click on the two blue lines (axes) and two red dots. Choose
Display – Hide Objects to hide these objects.
•To create dotted grid lines, click on one point of intersection of any two
grid lines. The lines are now highlighted pink. Choose Display – Line
Width – Dotted. Click anywhere on the screen other than a dot to
deselect the dots.
3. Create a label:
•To create a label for your drawing, choose the Text Tool from the tool
bar at the left of the screen. Click on a dot near the top left corner of
the screen and drag to create a text box. Type: “Top View” (or “Left Side
View”, “Right Side View”, “Front View” for subsequent drawings).
4. Draw the view:
•From the Graph menu, choose Snap Points.
•Next, choose the Straightedge Tool from the tool bar. Click on a dot
on the screen and drag to draw one line segment in the view. Continue
drawing line segments by clicking one dot and dragging to another dot
until you have completed the squares in the view.
Top View
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Geometry and Spatial Sense, Grades 4 to 6
3D6.BLM6b
•Select the Selection Arrow Tool from the tool bar. Click the four
corners of one square in your view in clockwise order. Select Construct
– Quadrilateral Interior to fill the square. Choose Display – Color to
select a colour to match the cube in the view.
•Add colour to the remaining squares in the view according to the coloured
6. Follow steps 2–4 to complete the three other views (left side, right side,
7. Save your work as “Orthographic Views of a 6-Cube Structure”.
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
207
3D6.BLM7
Imitator Demo Structure and Views
top view
front view
left side view
208
Geometry and Spatial Sense, Grades 4 to 6
rear view
right side view
3D6.BLM8
The 8 Four-Cube Structures
Here are the eight possible structures that can be constructed from four
interlocking cubes:
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
209
3D6.BLM9a
Construction Challenge
Use the four-cube structures indicated to construct the new structures shown in
each question.
1. Use
and
to construct these:
210
Geometry and Spatial Sense, Grades 4 to 6
3D6.BLM9b
2. Use
and
to construct these:
Grade 6 Learning Activity: Three-Dimensional Figures – Sketching Climbing Structures
211
3D6.BLM9c
3. Use
and
to construct these:
212
Geometry and Spatial Sense, Grades 4 to 6
and
Grade 6 Learning Activity: Location – Name My Shapes
Location: Name My Shapes
overview
In this learning activity, students create a shape puzzle, using a coordinate grid, and describe
precise locations, using ordered pairs.
Big IdeaS
This learning activity focuses on the following big idea:
Location and movement: Students investigate using a coordinate grid system to describe the
position of an object.
Curriculum Expectations
This learning activity addresses the following specific expectation.
Students will:
• explain how a coordinate system represents location, and plot points in the first quadrant of
a Cartesian coordinate plane.
This specific expectation contributes to the development of the following overall expectation.
Students will:
• describe location in the first quadrant of a coordinate system, and rotate twodimensional shapes.
TIME:
approximately
60 minutes
Materials
• overhead transparency of Loc6.BLM1: Blank 10 × 10 Grid
• Loc6.BLM1: Blank 10 × 10 Grid (1 per student)
• Loc6.BLM2: Name My Shapes Instructions (1 per student)
• Loc6.BLM3: Name My Shapes (2 per student)
• sheets of chart paper or large newsprint (1 per pair of students)
• markers (1 per pair of students)
• Loc6.BLM4: Introduction to Name My Shapes (1 per student)
213
Math Language
• rectangle
• coordinate grid
• isosceles triangle
• location
• equilateral triangle
• vertical axis
• horizontal axis
• pentagon
• axes
• hexagon
• ordered pair
• vertex
• origin
• vertices
• triangle
INSTRUCTIONAL
GROUPING:
pairs
Instructional Sequencing
This learning activity provides an introduction to the use of ordered pairs and a grid system in
locating precise points in the first quadrant of a coordinate plane. Students may benefit from a
review of two-dimensional shapes and their properties before the activity.
Using a coordinate system to represent the location of two-dimensional shapes assists students
in recognizing important properties of these shapes while they learn how to plot points in
the first quadrant of a Cartesian coordinate plane. Recognizing and analysing patterns in the
coordinate plane will be particularly important in later grades when students examine graphical
representations of various algebraic functions.
Getting Started
Have an enlarged version of Loc6.BLM1: Blank 10 × 10 Grid on the board, or make a transparency for use on the overhead projector. Ask students how a specific location is found on a
coordinate grid (i.e., counting along the horizontal axis, then up the vertical axis – this concept
should be a review of Grade 5 location skills). Introduce the idea of using an ordered pair to
designate a specific location. Demonstrate how the notation (6, 2) means 6 along the horizontal
axis, followed by 2 up the vertical axis. Ask, “What does (2, 6) mean?” Discuss the similarities
of and differences between ordered pairs that have their numbers in reverse order. Also,
discuss the point (0, 0) – the origin.
Once the students are comfortable locating points on the coordinate grid, provide each
student with a copy of Loc6.BLM1: Blank 10 × 10 Grid, and write the following ordered pairs
on the board or the overhead projector:
(5, 7) (7, 5) (5, 3) (3, 5)
Students plot these points on their grids, and join the four points in order with a ruler. They
record the name of the shape drawn on their grid.
Ask volunteers to plot the four points on the board or the overhead projector.
214
Geometry and Spatial Sense, Grades 4 to 6
Ask other members of the class to justify why the locations are correct or incorrect.
Once the four points have been plotted and the points connected in order with a ruler, have
students name the shape.
Working on it
Provide each student with one copy of Loc6.BLM2: Name My Shapes Instructions and two
copies of Loc6.BLM3: Name My Shapes. Explain the rules of the game, and have the students
play the game. As students play the game, observe the various strategies they use. Pose questions to help students think about their strategies:
• “Did you find a strategy to locate the vertices of your shapes quickly?”
• “What strategy did you use to name the shapes?”
• “Were any of you able to predict what shape it might be just from the ordered pairs?”
• “Did you change any of your strategies? Why?”
When students have had a chance to play the game, provide each pair with markers and a
sheet of chart paper or large newsprint. Ask students to record their strategies or any problems they encountered while playing the game.
Make a note of pairs who might share their strategies during the Reflecting and Connecting part
of the lesson. Include pairs who used different strategies or who disagreed about a shape name.
Reflecting and Connecting
Once students have had the opportunity to play the game and record their strategies, bring
them together to share their ideas. Try to order the presentations so that students observe a
variety of strategies and any unsolved solutions regarding the names of shapes that the whole
class could attempt to resolve.
Following the presentations, ask students to observe the work that has been posted and to
consider the efficiency of the different strategies. Ask:
• ”Which strategy, in your opinion, is an efficient strategy for locating the vertices of shapes?”
• “How would you explain the strategy to someone who has never used it?”
Avoid commenting that some strategies are better than others – students need to determine
for themselves which strategies are meaningful and efficient, and which ones they can make
sense of and use.
Refer to students’ work to emphasize geometric ideas:
• A coordinate grid system is helpful for locating and plotting specific points.
• Using ordered pairs is an efficient way to identify points on a coordinate grid.
Grade 6 Learning Activity: Location – Name My Shapes
215
Groups of four can select and refine a drawing for presentation to the class.
Students could create a computer version of this activity, using The Geometer’s Sketchpad.
Assessment
Observe students as they play the game and assess how well they:
• locate and mark the vertices of shapes on the coordinate grid;
• create ordered pairs to represent the location of the vertices;
• plot points on a grid from ordered pairs;
• apply appropriate strategies to play the game;
• judge the efficiency of various strategies.
Home ConnectioN
Send home one copy of Loc6.BLM4: Introduction to Name My Shapes, two copies of
Loc6.BLM2: Name My Shapes Instructions, and a suitable number of copies of Loc6.BLM3:
Name My Shapes. Parents have their child explain the game Name My Shapes and play the
game once or twice with their child. Parents are asked to complete a short questionnaire at the
bottom of the letter.
LEARNING CONNECTION 1
Find the Missing Point
Materials
• Loc6.BLM1: Blank 10 × 10 Grid
• Overhead transparency of Loc6.BLM1: Blank 10 × 10 Grid
Provide two points of a triangle and tell the students that the shape you have started to plot is
an isosceles triangle. Have the students write down the coordinates of the third point. (There
Have a volunteer place his or her answer on an overhead transparency of Loc6.BLM1:
Blank 10 × 10 Grid or a large grid drawn on the board, and explain why it is correct.
Ask if anyone has a different answer that he or she thinks is correct, and repeat the previous
directions. Continue until no more solutions are provided.
Offer different challenges – for example: two points of a square, three points of a parallelogram, four points of a regular hexagon.
Students can develop their own challenges, to be solved on grid paper or on The
216
Geometry and Spatial Sense, Grades 4 to 6
LEARNING CONNECTION 2
Coordinate Challenge
Materials
• Loc6.BLM5: Coordinate Challenge (1 per student)
• Loc6.BLM1: Blank 10 × 10 Grid (1 per student)
• overhead transparency or chart-sized copy of Loc6.BLM1: Blank 10 × 10 Grid
Hand out Loc6BLM1: Blank 10 × 10 Grid (one per student). Ask students to plot the following
points: (1, 9) (5, 2) (5, 9) (1, 2). Ask the students to join the points and name the shape.
Have a volunteer place his or her answer on the overhead transparency of Loc6.BLM1: Blank
10 × 10 Grid, or on a large grid drawn on the board, and explain why it is correct.
Ask the students to find and record the perimeter and area of the shape on the grid. Ask for
a volunteer to record the perimeter of the shape on the overhead grid and explain why it is
correct. Ask for another volunteer to record the area of the shape on the overhead grid and
explain why it is correct. During the explanations allow time for discussion and clarification.
Distribute Loc6.BLM5: Coordinate Challenge (one per student). Review the instructions, and
have the students complete the challenge.
Have volunteers record their answers on an overhead transparency, and allow for discussions
Grade 6 Learning Activity: Location – Name My Shapes
217
Loc6.BLM1
Blank 10  10 Grid
10
9
8
7
6
5
4
3
2
1
218
1
2
3
4
Geometry and Spatial Sense, Grades 4 to 6
5
6
7
8
9
10
Loc6.BLM2
Name My Shapes Instructions
1. Work with a partner as a team of two.
2. Draw a design together, using a total of four shapes. The four shapes can
be regular or irregular but need to be chosen from the following:
Triangle
Pentagon
Hexagon
You may use the same shape more than once, but your design may have a
total of only four shapes. Your shapes can overlap if you want them to.
3. Under your design, use ordered pairs to write the locations of the vertices
for each of the shapes. Then write the name of the shape, being as accurate
as you can (e.g., irregular pentagon, isosceles triangle).
4. When you have finished drawing your design, find another team to challenge.
Do not show the other team your design.
5. Choose one team to start reading the points for their design. That team
read the ordered pairs for the first shape while the other team plot and
connect the points, and name the shape on their blank game sheets. Then
switch and have the other team read the ordered pairs for their first shape
while the first team draw and name the shape on their blank game sheets.
Keep going until both teams have finished all four shapes.
6. After you have finished, compare your drawings and shape names with
those of the other team. Make note of any shape names you could not
agree on.
Grade 6 Learning Activity: Location – Name My Shapes
219
Loc6.BLM3
Name My Shapes
10
9
8
7
6
5
4
3
2
1
220
1
2
3
4
5
6
7
8
Shape one
Shape two
Coordinates:
Coordinates:
Name: Name: Shape three
Shape four
Coordinates:
Coordinates:
Name: Name: Geometry and Spatial Sense, Grades 4 to 6
9
10
Loc6.BLM4
Introduction to Name My Shapes
Dear Parent/Guardian:
In class we have been playing a game called Name My Shapes. (Two copies of
the game board and instructions are attached to this letter.)
The game is designed to help your child plot ordered pairs and review twodimensional shapes, such as hexagons, parallelograms, and trapezoids.
Over the next few days, please find a time to let your child explain how to play
the game and then play a game or two with you. Please fill out the questionnaire below, to let me know how well your child can explain the mathematical
ideas in this game.
Thank you.
QUESTIONNAIRE
Name of person completing the questionnaire Please circle one answer below each question:
1. very well
was able to explain the game.
well
somewhat well
not very well
2. understood how to plot ordered pairs on the
grid, and could tell me how to plot them.
very well
well
3. very well
somewhat well
not very well
knew the names of most of the shapes we used.
well
somewhat well
4. Some directions and /or shapes sure about are:
not very well
was not too
Grade 6 Learning Activity: Location – Name My Shapes
221
Loc6.BLM5
Coordinate Challenge
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
Draw your solutions to the following questions on the grid above, and label
each shape as instructed. List the coordinates for the vertices of each shape,
using ordered pairs.
1.Draw a rectangle with an area of 9 cm2 . Label it ABCD. The vertices are
located at: A (
)
B(
)
C(
)
D(
)
2.Draw a right-angled isosceles triangle. Label it EFG. The vertices are located at:
E(
)
F(
)
G(
)
3.Draw a square with a perimeter of 12 cm. Label it HIJK. The vertices are
located at: H (
)
I(
)
J(
)
K(
)
4.Draw a hexagon with one vertex at (1, 1). Label it, using letters of your
choice, and list where all the vertices are located:
5.Draw an obtuse-angled triangle with a perimeter of less than 15 cm. Label it
XYZ. The vertices are located at: X (
)
Y(
)
Z(
)
222
Geometry and Spatial Sense, Grades 4 to 6
Grade 6 Learning Activity: Movement – Logo Search and Design
Movement: Logo Search and Design
Overview
In this learning activity, students analyse and design logos, using different transformations and
kinds of symmetry.
Big Ideas
This learning activity focuses on the following big idea:
Location and movement: Students can use different transformations to describe the movement of an object.
Curriculum ExpectationS
This learning activity addresses the following specific expectations.
Students will:
• identify, perform, and describe, through investigation using a variety of tools (e.g., grid
paper, tissue paper, protractor, computer technology), rotations of 180° and clockwise and
counterclockwise rotations of 90°, with the centre of rotation inside or outside the shape.
• create and analyse designs made by reflecting, translating, and/or rotating a shape, or
shapes, by 90° or 180°;
These specific expectations contribute to the development of the following overall expectation.
Students will:
• describe location in the first quadrant of a coordinate system, and rotate twodimensional shapes.
Materials
TIME:
approximately
two 60-minute
periods
• logo samples
• sheets of chart paper or large newsprint (1 per small group of students; 1 per pair of students)
• Mov6.BLM1a–b: Logos (1 per student)
• Mov6.BLM2: Logo Design (1 per pair of students)
• geoboards, Miras, pattern blocks
• markers or crayons (1 per pair of students)
223
Math Language
INSTRUCTIONAL
GROUPING:
individual and
pairs or groups
of three
• symmetry
• rotational symmetry
• reflection
• shapes
• rotation
• properties
Instructional Sequencing
This learning activity provides an opportunity to explore the use of rotational and line symmetry in logo design. It is important for students to have completed the Learning Connection
“Alphabet Rotations” or some other rotation activities, before completing this main activity, so
that students have some prior knowledge of rotational symmetry.
It is important to offer students real-world experiences that provide opportunities for them to
visualize rotations and rotational symmetry. Students experience more difficulty in visualizing
the rotation of objects than they do in reflecting similar objects about a line of symmetry.
Getting Started
Before the lesson, collect some examples of logos containing rotational and/or line symmetry.
Using several of the examples, have a brief whole-class discussion with your students about
logos. Have other examples ready for students to examine in small groups. Distribute the
logos and ask students to jot down on chart paper or large newsprint ideas of what makes an
effective logo.
As students are working, you may want to ask:
• “What do you like about the logo your group is examining?”
• “Why do you think the company chose this logo?”
• “What are some of the characteristics or properties of the logo?”
Select some students to present their observations about the logos they were examining. Try
to choose students who will provide a variety of observations, especially those who notice
symmetry in the logo.
Discuss how many logos use some form of symmetry through the use of rotations and reflections, and show examples of symmetry.
Provide each student with a copy of Mov6.BLM1a–b: Logos. Tell the students that they will
have homework tonight but they should ask parents/guardians to help if they can (the parent
letter explains the assignment). Explain that tonight’s homework is to search through magazines, newspapers, Internet sites, and so forth, to find a few logos that are visually appealing
and contain examples of symmetry, one of which must be of rotational symmetry. Ask students
to cut out logos (with their parents’ permission) or copy them as accurately as possible. Then
students should fill out the questions on the back of the parent letter and bring the answers
and logos back to school.
224
Geometry and Spatial Sense, Grades 4 to 6
Working on it
On another day ask a few volunteers to show their most interesting logo to the class, and have
them explain the mathematical symmetry and the shapes found in the logo. You may want to
create a Logo bulletin board with the logos students have brought in.
Organize the students into pairs for the next activity. Hand out copies of Mov6.BLM2: Logo
Design, and go over the instructions with the students.
As the students are working, observe the various strategies that they use. Pose questions to
help students think about their strategies:
• “Have you thought of a strategy to find out if an object or shape has symmetry?”
• “Did you change your strategy? Why?”
• “Did you use any tools in the classroom as part of your strategy?”
As students are completing their logos, provide each pair with chart paper or large newsprint
and markers to record the strategies that they used to create and check their logos.
Make note of pairs who might share their strategies during the Reflecting and Connecting
portion of the lesson. Include groups who used different strategies for creating designs with
line and rotational symmetry (e.g., guess and check, rotating their design, using a Mira, using
dot paper and a line of reflection).
Reflecting and Connecting
After students have completed their designs and recorded their strategies, bring them
together to share their ideas. Try to order the presentations so that the students observe
strategies varying in efficiency.
As students explain their ideas, ask questions to help them to describe their strategies:
• “How did you select the shapes in your logo?”
• “Where is the line and rotational symmetry in this logo?”
• “What strategies did you use to include symmetry in your logo?”
• “Why do you think many logos have symmetry?”
Avoid commenting that some logos and/or strategies are better than others – students need
to determine for themselves which strategies are meaningful and efficient, and which ones they
can make sense of and use.
Refer to students’ work to emphasize geometric ideas:
• Designs can be made by reflecting or rotating shapes.
• Real-world objects contain line and rotational symmetry.
• Reflections and rotations can create symmetrical designs.
A “gallery walk” at the conclusion of this activity provides all students with an opportunity to
display their work, see the logos produced by their peers, and explain their designs to other
class members.
Grade 6 Learning Activity: Movement – Logo Search and Design
225
Groups can present their design to the school administration and/or team/club members,
explaining why they believe their logo should be adopted.
Assessment
Observe students as they work on and explain their logo designs, and assess how well they:
• explain why they selected the mathematical shapes and tools to design their logos;
• show their understanding of the symmetries associated with their logos;
• explain the strategies they used to create line and rotational symmetry in their logos;
• judge the efficiency of various strategies.
Student reflections on the reverse of Mov6.BLM6 Logos can be used for diagnostic purposes,
to provide information on students’ strengths and difficulties.
Home Connection
See Mov6.BLM1a–b: Logos for a letter to be sent home to parents as part of the Getting
Started portion of the lesson.
LEARNING CONNECTION 1
Regular Shapes and Rotational Symmetry
Materials
• Mov6.BLM3a–d: Rotating Regular Shapes (1 per student)
• 2 large demonstration copies of each shape on Mov6.BLM3a–d: Rotating Regular Shapes
• pieces of tracing paper (4 per student)
• paper fasteners (1 per student)
• rulers (1 per pair of students)
• protractors (1 per pair of students)
Distribute copies of Mov6.BLM3a–d: Rotating Regular Shapes (one copy per student). Ask,
“Why are these two-dimensional shapes called regular?” Allow students to examine the shapes
in pairs to determine possible patterns. Facilitate a discussion through which students discover
that all the sides and all the angles must be congruent for a shape to be called regular.
Have each student trace the shapes from Mov6.BLM3a–d: Rotating Regular Shapes. Students
cut out the shapes from the four pages of the blackline master and the shapes from their
tracing paper. Discuss how a centre of rotation allows you to turn a regular polygon onto
itself exactly. There can be no overlapping parts when this occurs. Demonstrate with the large
equilateral triangles.
226
Geometry and Spatial Sense, Grades 4 to 6
Ask students to work with a partner to see if they can find a way to mark the centre of rotation
for the equilateral triangle. Allow time for experimentation, and ask for volunteers to share
their findings. (The centre of rotation can be found by bisecting one angle of the triangle
through folding, repeating this process with a second angle, and finding where the two fold
lines meet. Bisecting a third angle will verify that the centre of rotation is correct. Demonstrate
with a large equilateral triangle if none of the students discover the strategy.)
Label the centre of rotation for each of the large equilateral triangles and place one triangle
exactly on top of the other, using a paper fastener to hold the triangles in place. Label the angles
A, B, and C. Slowly rotate the top copy until it lies exactly over the bottom triangle. Have the students tell you when each overlying occurs. Keep rotating the triangle until it returns to its original
position. Ask, “How many times does one equilateral triangle lie exactly over the other?” (3)
Instruct the students to work in pairs or groups of three, repeating the process with the
square, the pentagon, and the regular hexagon. Students should record their findings and look
for patterns.
Have a discussion to see which patterns the students have found. You want them to recognize
that the number of rotations corresponds to the number of sides. However, the students might
discover other interesting ideas or patterns.
Ask if their patterns or ideas would be true for other regular polygons. What about irregular
polygons? Have the students cut out other shapes to test their conjectures.
conjectures in that environment, exploring more complex regular polygons.
LEARNING CONNECTION 2
Alphabet Rotations
Materials
• Mov6.BLM4: Rotating the Alphabet (1 per student)
• tracing paper (1 piece per student)
Students use Mov6.BLM4: Rotating the Alphabet to discover which of the capital letters of
the alphabet have rotational symmetry. Students work individually to decide on the letters that
they believe have rotational symmetry. At this time they are not permitted to use any aids and
must base their conjectures on perception.
Students then work in pairs, checking their conjectures with tracing paper to prove which of
the letters have rotational symmetry.
Students could also discover, and then check, which of the numbers and the lower case letters
of the alphabet have rotational symmetry.
Grade 6 Learning Activity: Movement – Logo Search and Design
227
LEARNING CONNECTION 3
Materials
• Mov6.BLM5: Quadrilaterals, Questions, and Symmetry (1 per student)
• Mira (1 for every two students)
• sheets of tracing paper (1 per student)
• Miras (1 per pair of students)
Select a shape that is not on the sheet, such as a hexagon, to explain how the worksheet is to
be completed. The emphasis is on reasoning, followed by proving the various symmetries of
Students work individually to complete the chart on Mov6.BLM5: Quadrilaterals, Questions,
and Symmetry. At this point, they are NOT to use any tools, such as Miras or tracing paper.
Students then work in pairs, using tracing paper and Miras to verify their conjectures.
During a whole-class discussion, students reflect on what they discovered. Those who are
interested might explore the symmetries of other two-dimensional shapes.
228
Geometry and Spatial Sense, Grades 4 to 6
Mov6.BLM1a
Logos
Dear Parent/Guardian:
In geometry we are exploring the different forms of symmetry that large
companies use when designing their logos.
Students are being asked to search through magazines, newspapers, the
Internet, and so forth, to find a few logos that are visually appealing, are not
offensive, and contain examples of symmetry (especially rotational symmetry).
symmetry before he or she begins to look for logos.
If your child finds logos in magazines or newspapers, he or she can either cut
them out or copy them. Logos that your child finds through an Internet search
can be printed.
To search the Internet, at home or at the public library, your child can go into a
search engine like Google and type “most famous company logos”.
After finding the logos, your child should complete the questions on the back
of this letter.
Thank you for helping your child to make connections between geometry in
school and geometry in the real world.
Grade 6 Learning Activity: Movement – Logo Search and Design
229
Mov6.BLM1b
Logos (Reverse side)
Student Reflections
Please complete the following questions about the two or three most interesting
logos that you found.
1. What I found most interesting about these logos is:
2. The line symmetry used in one of the logos is (horizontal, vertical, diagonal).
3. Something I found interesting/unusual about the line symmetry is:
4. The rotational symmetry used in one of the logos is (90°, 180°, other).
5. Something I found interesting/unusual about the rotational symmetry is:
6. The transformations used in logo #1 are:
7. The transformations used in logo #2 are:
8. Some mathematical shapes I found in my logos are:
9. Something I still don’t understand about symmetry or transformations is:
230
Geometry and Spatial Sense, Grades 4 to 6
Mov6.BLM2
Logo Design
1. Work with a partner.
2. Design a logo for our classroom or school.
3. Your logo must contain an example of line symmetry and an example of
rotational symmetry.
4. You may use any of the manipulatives and/or paper provided.
5. You need to work as a team and be able to explain the strategies you used
to design and create your logo, and you need to be able to show that your
logo contains an example of line symmetry and an example of rotational
symmetry.
Grade 6 Learning Activity: Movement – Logo Search and Design
231
Mov6.BLM3a
Rotating Regular Shapes
232
Geometry and Spatial Sense, Grades 4 to 6
Mov6.BLM3b
Grade 6 Learning Activity: Movement – Logo Search and Design
233
Mov6.BLM3c
234
Geometry and Spatial Sense, Grades 4 to 6
Mov6.BLM3d
Grade 6 Learning Activity: Movement – Logo Search and Design
235
Mov6.BLM4
Rotating the Alphabet
Which capital letters have rotational symmetry?
A B C D
E F G H I
J K L M N
O P Q R
S T U V
W X Y Z
236
Geometry and Spatial Sense, Grades 4 to 6
Mov6.BLM5
Some quadrilaterals have lines of symmetry; others do not. Some quadrilaterals have rotational
symmetry; others do not.
precise name you know, and write the name in the first column. Then use your imagination to
speculate whether the shape has lines of symmetry or rotational symmetry.
In the last row, draw a different quadrilateral, name it, and complete the last two columns.
Name
Shape
Lines of
symmetry
Rotational
symmetry
Yes __
Yes ___
How many? ____
No ___
No __
Yes __
Yes ___
How many? ____
No ___
No __
Yes __
Yes ___
How many? ____
No ___
No __
Yes __
Yes ___
How many? ____
No ___
No __
Yes __
Yes ___
How many? ____
No ___
No __
Yes __
Yes ___
How many? ____
No ___
No __
Once you have completed the chart, ask for some tracing paper and a Mira.
Work with a partner to verify your predictions. If you disagree with each other, take turns
explaining why and try to come to an agreement.
Grade 6 Learning Activity: Movement – Logo Search and Design
237
Appendix: Guidelines for
Assessment
There are three types of assessment: assessment for learning, assessment as learning, and
assessment of learning.
Assessment for learning involves teachers observing the knowledge, skills, experience, and
interests their students demonstrate, and using those observations to tailor instruction to
meet identified student needs and to provide detailed feedback to students to help them
improve their learning.
Assessment as learning is a process of developing and supporting students’ metacognitive
skills. Students develop these skills as they monitor their own learning, adapt their thinking,
and let the ideas of others (peers and teachers) influence their learning. Assessment as learning helps students achieve deeper understanding.
Assessment of learning is summative. It includes cumulative observations of learning and
involves the use of the achievement chart to make judgements about how the student has
done with respect to the standards. Assessment of learning confirms what students know
and are able to do, and involves reporting on whether and how well they have achieved the
curriculum expectations.
Teachers use assessment data, gathered throughout the instruction–assessment–instruction
cycle, to monitor students’ progress, inform teaching, and provide feedback to improve student
learning. Effective teachers view instruction and assessment as integrated and simultaneous
processes. Successful assessment strategies – those that help to improve student learning – are
thought out and defined ahead of time in an assessment plan.
An assessment profile, developed by the teacher for each student, can be an effective way
of organizing assessment data to track student progress. Students can also maintain their
own portfolios, in which they collect samples of their work that show growth over time.
Creating an Assessment Plan
To ensure fair and consistent assessment throughout the learning process, teachers should
work collaboratively with colleagues to create assessment and instructional plans. Ideally,
such planning should start with learning goals and work backwards to identify the assessment and instructional strategies that will help students achieve those goals.
239
Guiding Questions
Planning Activities
“What do I want students
to learn?”
Teachers begin planning by identifying the overall and specific expectations from the Ontario curriculum that will be the focus of learning in a
given period. The expectations may need to be broken down into specific,
incremental learning goals. These goals need to be shared with students,
before and during instruction, in clear, age-appropriate language.
“How will I know they
have learned it?”
Teachers determine how students’ learning will be assessed and evaluated.
Both the methods of assessment and the criteria for judging the level of
performance need to be shared with students.
“How will I structure the
learning?”
Teachers identify scaffolded instructional strategies that will help students
achieve the learning goals and that integrate instruction with ongoing
assessment and feedback.
An assessment plan should include:
• clear learning goals and criteria for success;
• ideas for incorporating both assessment for learning and assessment as learning into each
series of lessons, before, during, and after teaching and learning;
• a variety of assessment strategies and tools linked carefully to each instructional activity;
• information about how the assessment profiles will be organized;
• information about how the students’ assessment portfolios will be maintained.
Feedback
When conducting assessment for learning, teachers continuously provide timely, descriptive,
and specific feedback to students to help them improve their learning. At the outset of instruction,
the teacher shares and clarifies the learning goals and assessment criteria with the students.
Effective feedback focuses the student on his or her progress towards the learning goals. When
providing effective feedback, teachers indicate:
• what good work looks like and what the student is doing well;
• what the student needs to do to improve the work;
• what specific strategies the student can use to make those improvements.
Feedback is provided during the learning process in a variety of ways – for example, through
written comments, oral feedback, and modelling. A record of such feedback can be maintained in an assessment profile.
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Geometry and Spatial Sense, Grades 4 to 6
Assessment Profile
An assessment profile is a collection of key assessment evidence, gathered by the teacher
over time, about a student’s progress and levels of achievement. The information contained
in the profile helps the teacher plan instruction to meet the student’s specific needs. An
extensive collection of student work and assessment information helps the teacher document the student’s progress and evaluate and report on his or her achievement at a specific
point in time.
The assessment profile also informs the teacher’s conversations with students and parents
about the students’ progress. Maintaining an assessment profile facilitates a planned, systematic
approach to the management of assessment information.
Assessment profiles may include:
• assessments conducted after teaching, and significant assessments made during teaching;
• samples of student work done in the classroom;
• samples of student work that demonstrates the achievement of expectations;
• teacher observation and assessment notes, conference notes;
• EQAO results;
• results from board-level assessments;
• interest inventories;
• notes on instructional strategies that worked well for the student.
Student Assessment Portfolio
A portfolio is a collection of work selected by the student that represents his or her improvement
in learning. It is maintained by the student, with the teacher’s support. Assembling the portfolio enables students to engage actively in assessment as learning, as they reflect on their
progress. At times, the teacher may guide students in the selection of samples that show how
well they have accomplished a task, that illustrate their improvement over a period of time,
or that provide a rationale for the teacher’s assessment decisions. Selections are made on the
basis of previously agreed assessment criteria.
Student assessment portfolios can also be useful during student/teacher and parent/teacher
conferences. A portfolio may contain:
• work samples that the student feels reflect growth;
• the student’s personal reflections;
• self-assessment checklists;
• information from peer assessments;
• tracking sheets of completed tasks.
Appendix: Guidelines for Assessment
241
Students should not be required to assign marks, either to their own work or to the work of
their peers. Marking is part of the evaluation of student work (i.e., judging the quality of the
work and assigning a mark) and is the responsibility of the teacher.
Assessment Before, During, and After Learning
Teachers assess students’ achievement at all stages of the instructional and assessment cycle.
Assessment before new instruction identifies students’ prior knowledge, skills, strengths, and
needs and helps teachers plan instruction.
Effective assessment during new instruction determines how well students are progressing and
helps teachers plan required additional instruction. The teacher uses a variety of assessment
strategies, such as focused observations, student performance tasks, and student self- and
peer assessment, all based on shared learning goals and assessment criteria. As noted earlier,
the teacher provides students with feedback on an ongoing basis during learning to help
them improve.
During an instructional period, the teacher often spends part of the time working with small
groups to provide additional support, as needed. The rest of the time can be used to monitor
and assess students’ work as they practise the strategies being learned. The teacher’s notes
from his or her observations of students as they practise the new learning can be used to
provide timely feedback, to develop students’ assessment profiles, and to plan future lessons.
Students can also take the opportunity during this time to get feedback from other students.
Assessment after new learning has a summative purpose. As assessment of learning, it involves
collecting evidence on which to base the evaluation of student achievement, develop teaching practice, and report progress to parents and students. After new learning, teachers assess
students’ understanding, observing whether and how the students incorporate feedback into
their performance of an existing task or how they complete a new task related to the same
learning goals. The assessment information gathered at this point, based on the identified
curriculum expectations and the criteria and descriptors in the achievement chart, contributes to the evaluation that will be shared with students and their parents during conferences
and by means of the grade assigned and the comments provided on the report card.
242
Geometry and Spatial Sense, Grades 4 to 6
Glossary
acute angle. An angle whose measure is
Ontario curriculum are properties of two-
between 0° and 90°.
dimensional shapes and three-dimensional
acute triangle. A triangle whose angles all
measure less than 90°.
adjacent angle. One of two angles that have
a common vertex and common side.
angle. A shape formed by two rays or two
line segments with a common endpoint.
Measuring angles involves finding the amount
figures, geometric relationships, and location
and movement.
cardinal directions. The four main points
of the compass: north, east, south, and west.
circle. A two-dimensional shape with a
curved side. All points on the side of a circle
are equidistant from its centre.
of rotation between two rays or line segments
Cartesian coordinate plane. A plane that
with a common endpoint or vertex.
contains an x-axis (horizontal) and a y-axis
(vertical), which are used to describe the location of a point. Also called coordinate plane.
angle
vertex
compose. Order or arrange parts to form
a whole. In geometry, two-dimensional
attribute. A quantitative or qualitative
shapes and three-dimensional figures can
characteristic of a shape, an object, or an
occurrence; for example, colour, size, thick-
decompose.
ness, or number of sides. An attribute may
(geometric).
axis. A reference line used in a graph or
coordinate system.
base. In three-dimensional figures, the face
that is usually seen as the bottom (e.g., the
square face of a square-based pyramid). In
prisms, the two congruent and parallel faces
are called bases (e.g., the triangular faces of a
triangular prism).
big ideas. In mathematics, the important
concave polygon. A polygon containing at
least one interior angle greater than 180°.
cone. A three-dimensional figure with a
circular base and a curved surface that tapers
to a common point.
congruent. Having the same size and shape.
conjecture. An unproven mathematical
theorem; a conclusion based on incomplete
information.
convex polygon. A polygon whose interior
angles are all less than 180°.
concepts or major underlying principles. For
coordinate grid. A plane that contains
example, in this document, the big ideas
horizontal and vertical lines that intersect to
that have been identified for Grades 4 to 6 in
form squares or rectangles. In a coordinate
the Geometry and Spatial Sense strand of the
grid system (for example, a road map),
243
the representation of an object within the
first quadrant. In the coordinate plane, the
squares or rectangles describes the location
quadrant that contains all the points with
of the object.
positive x and positive y coordinates.
coordinate plane. See Cartesian coordinate
y-axis
plane.
coordinates. See ordered pair.
coordinate system. A system that identifies
x-axis
a point rather than an area. In a coordinate
system, the lines are labelled, rather than the
area bounded by the lines.
cube. A right rectangular prism with six
congruent square faces. A cube is one of the
Platonic solids. Also called hexahedron.
flip. See reflection.
cylinder. A three-dimensional figure with
Frameworks. Commercially produced
two parallel and congruent circular faces and
learning tools that help students learn about
a curved surface.
geometric figures. Frameworks is a system of
decompose. Separate a whole into parts.
In geometry, two-dimensional shapes and
three-dimensional figures can be decomposed into smaller shapes and figures. See
interlocking frames for geometric constructions. The frames in this system match the
dimensions, constructions, and features of
geometry. The study of mathematics
also compose.
diagonal. A line segment joining two vertices of a polygon that are not next to each
other (i.e., that are not joined by one side).
that deals with the spatial relationships,
properties, movement, and location of twodimensional shapes and three-dimensional
figures. The name comes from two Greek
words meaning earth and measure.
diagonal
grid. A network of regularly spaced lines
that cross one another at right angles to
form squares or rectangles.
edge. The intersection of a pair of faces in a
grid system. A system that uses a combina-
three-dimensional figure.
tion of a grid and letters and numbers to
equilateral triangle. A triangle with three
equal sides.
face. One of the polygons that make up a
polyhedron.
figure. See three-dimensional figure.
244
Geometry and Spatial Sense, Grades 4 to 6
describe a general location of a shape or an
object. Because the grid system identifies an
area rather than a point, it cannot be used to
hexagon. A polygon with six sides.
integer. Any one of the numbers …, −4, −3,
line segment. The part of a line between
−2, −1, 0, 1, 2, 3, 4,….
two points on a line.
irregular polygon. A polygon that does not
A
regular polygon.
magnitude. An attribute relating to size or
isometric drawing. A perspective drawing
of a three-dimensional figure.
isometric dot paper. Dot paper used for
creating isometric drawings. The dots
are formed by the vertices of equilateral
triangles. Also called triangular dot paper or
triangle dot paper.
isosceles trapezoid. A trapezoid that has
two sides of equal length.
isosceles triangle. A triangle that has two
sides of equal length.
kite. A quadrilateral that is not a parallelogram, but has two pairs of equal sides that
quantity.
manipulative. An object that students
handle and use in constructing their own
understanding of mathematical concepts
and skills and in illustrating their understanding. Some examples are geoboards,
geometric solids, and pattern blocks.
Mira. A transparent mirror used to locate
reflection lines, reflection images, and lines
of symmetry, and to determine congruency
and line symmetry.
net. A pattern that can be folded to make a
three-dimensional figure.
A net of a cube
line. A geometric figure that has no thickness but its length goes on infinitely in two
directions.
mirror in the form of a perpendicular bisector so that corresponding points are the
same distance from the line.
C'
C
B
obtuse angle. An angle that measures more
than 90° and less than 180°.
obtuse triangle. A triangle with one angle
line of reflection. A line that acts as a
A
B
B'
A'
In this example,
the points A, B,
and C are exactly
the same distance
from the mirror
line as are points
A', B' and C'.
that measures more than 90° and less than
180°.
octagon. A polygon with eight sides.
order of rotational symmetry. The number
of times the position of a shape coincides
with its original position during one complete rotation about its centre. For example,
a square has rotational symmetry of order 4.
line of symmetry. A line that divides a
ordered pair. Two numbers, in order, that
shape into two congruent parts that can be
are used to describe the location of a point
matched by folding the shape in half. See
on a plane, relative to a point of origin
also symmetry.
(0, 0); for example, (2, 6). On a coordinate
plane, the first number is the horizontal
Glossary
245
coordinate of a point, and the second is
Polydron. Commercially produced learn-
ing tools that help students learn about
coordinates.
geometric figures. Polydron is a system of
orientation. The relative physical position
or direction of something. The orientation
of a shape may change following a rotation
polygon. A closed shape formed by three or
or reflection. The orientation of a shape does
more line segments; for example, triangle,
not change following a translation.
origin. The point of intersection of the verti-
polyhedron. A three-dimensional figure that
cal and horizontal axes of a Cartesian plane.
has polygons as faces.
The coordinates of the origin are (0, 0).
positional descriptor. A symbol that repre-
parallel. Extending in the same direction,
sents the position of an object on a grid or a
remaining the same distance apart. Parallel
coordinate system.
lines or parallel shapes never meet, because
they are always the same distance apart.
parallelism. The state of being parallel.
parallel lines. Lines in the same plane that
prism. A three-dimensional figure with two
bases that are parallel and congruent. A prism
is named by the shape of its bases; for example,
rectangular prism, triangular prism.
property. An attribute that remains the same
do not intersect.
opposite sides are parallel.
parallel sides. On a shape, sides that would
never intersect.
pentagon. A polygon with five sides.
for a class of objects or shapes. A property
of any parallelogram, for example, is that its
pyramid. A three-dimensional figure whose base
is a polygon and whose other faces are triangles
that meet at a common vertex. A pyramid is
perpendicularity. The state of being perpen-
named by the shape of its base; for example,
dicular or having right angles.
square-based pyramid, triangle-based pyramid.
The base
of this
rectangular
prism is
perpendicular
to its vertical
faces.
The base
of this
prism is not
perpendicular
to its vertical
faces.
plane shape. See two-dimensional shape.
quadrant. One of the four regions formed
by the intersection of the x-axis and the
y-axis in a coordinate plane.
y-axis
point of intersection. The point at which
x-axis
two or more lines intersect.
point of rotation. The point about which a
shape is rotated.
246
Geometry and Spatial Sense, Grades 4 to 6
quadrilateral. A polygon with four sides.
ray. A line that has a starting point but no
endpoint.
rectangle. A quadrilateral in which opposite
sides are equal, and all interior angles are
right angles.
rectangular prism. A three-dimensional
figure with two parallel and congruent
rectangular faces. The four other faces are
also rectangular.
reflection. A transformation that flips a
shape over an axis to form a congruent
rotational symmetry. A geometric property
of a shape whose position coincides with
its original position after a rotation of less
than 360° about its centre. For example,
the position of a square coincides with its
original position after a 1/4 turn, a 1/2 turn,
and a 3/4 turn, so a square has rotational
rotational symmetry.
slide. See translation.
solid figure. See three-dimensional figure.
spatial sense. An intuitive awareness of
one’s surroundings and the objects in them.
shape. A reflection image is the mirror image
sphere. A three-dimensional figure with a
that results from a reflection. Also called flip.
curved surface. All points on the surface of
regular polygon. A closed figure in which
all sides are equal and all angles are equal.
reflex angle. An angle that measures more
than 180° and less than 360°.
rhombus. A parallelogram with equal sides.
Sometimes called diamond.
right angle. An angle that measures 90°.
right prism. A prism whose rectangular faces
are perpendicular to its congruent bases.
right trapezoid. A trapezoid with at least
one right angle.
a sphere are equidistant from its centre. A
sphere looks like a ball.
square. A rectangle with four equal sides
and four right angles.
square-based pyramid. A three-dimensional
figure with a base that is square and four
triangular faces.
straight angle. An angle that measures 180°.
symmetry. In a two-dimensional shape, the
property of having two parts that match
exactly, either when one half is a mirror image
of the other half (line symmetry) or when one
part can take the place of another if the shape
right triangle. A triangle with exactly one
right angle.
symmetry and rotational symmetry.
rotation. A transformation that turns a
three-dimensional figure. A figure that has
shape about a fixed point to form a congru-
length, width, and depth; a figure that is three-
ent shape. A rotation image is the result of a
dimensional. Also called figure or solid figure.
rotation. Also called turn.
Glossary
247
transformation. A change in a figure that
results in a different position, orientation,
or size. The transformations include
the translation (slide), reflection (flip),
reflection, rotation.
translation. A transformation that moves
every point on a shape the same distance,
in the same direction, to form a congruent
shape. A translation image is the result of a
translation. Also called slide.
trapezoid. A quadrilateral with one pair of
parallel sides, or a quadrilateral with at least
one pair of parallel sides.
triangle. A polygon with three sides.
triangle-based pyramid. A threedimensional figure with a triangular
base and three triangular faces.
turn. See rotation.
two-dimensional shape. A shape that has
length and width but no depth.
Venn diagram. A diagram consisting of
overlapping and/or nested shapes used to
show what two or more sets have and do not
have in common.
vertex. The common endpoint of the two line
x-axis. The horizontal number line on the
Cartesian coordinate plane.
y-axis. The vertical number line on the
Cartesian coordinate plane.
248
Geometry and Spatial Sense, Grades 4 to 6