Module 1 :Number Systems

Module 1 :Number Systems
Module 1
Junior Secondary
Mathematics
Number Systems
Science, Technology and Mathematics Modules
for Upper Primary and Junior Secondary School Teachers
of Science, Technology and Mathematics by Distance
in the Southern African Development Community (SADC)
Developed by
The Southern African Development Community (SADC)
Ministries of Education in:
• Botswana
• Malawi
• Mozambique
• Namibia
• South Africa
• Tanzania
• Zambia
• Zimbabwe
In partnership with The Commonwealth of Learning
COPYRIGHT STATEMENT
The Commonwealth of Learning, October 2001
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted in any form, or by any means, electronic or mechanical, including
photocopying, recording, or otherwise, without the permission in writing of the publishers.
The views expressed in this document do not necessarily reflect the opinions or policies of
The Commonwealth of Learning or SADC Ministries of Education.
The module authors have attempted to ensure that all copyright clearances have been
obtained. Copyright clearances have been the responsibility of each country using the
modules. Any omissions should be brought to their attention.
Published jointly by The Commonwealth of Learning and the SADC Ministries of
Education.
Residents of the eight countries listed above may obtain modules from their respective
Ministries of Education. The Commonwealth of Learning will consider requests for
modules from residents of other countries.
ISBN 1-895369-65-7
SCIENCE, TECHNOLOGY AND MATHEMATICS MODULES
This module is one of a series prepared under the auspices of the participating Southern
African Development Community (SADC) and The Commonwealth of Learning as part of
the Training of Upper Primary and Junior Secondary Science, Technology and
Mathematics Teachers in Africa by Distance. These modules enable teachers to enhance
their professional skills through distance and open learning. Many individuals and groups
have been involved in writing and producing these modules. We trust that they will benefit
not only the teachers who use them, but also, ultimately, their students and the communities
and nations in which they live.
The twenty-eight Science, Technology and Mathematics modules are as follows:
Upper Primary Science
Module 1: My Built Environment
Module 2: Materials in my
Environment
Module 3: My Health
Module 4: My Natural Environment
Junior Secondary Science
Module 1: Energy and Energy
Transfer
Module 2: Energy Use in Electronic
Communication
Module 3: Living Organisms’
Environment and
Resources
Module 4: Scientific Processes
Upper Primary Technology
Module 1: Teaching Technology in
the Primary School
Module 2: Making Things Move
Module 3: Structures
Module 4: Materials
Module 5: Processing
Junior Secondary Technology
Module 1: Introduction to Teaching
Technology
Module 2: Systems and Controls
Module 3: Tools and Materials
Module 4: Structures
Upper Primary Mathematics
Module 1: Number and Numeration
Module 2: Fractions
Module 3: Measures
Module 4: Social Arithmetic
Module 5: Geometry
Junior Secondary Mathematics
Module 1: Number Systems
Module 2: Number Operations
Module 3: Shapes and Sizes
Module 4: Algebraic Processes
Module 5: Solving Equations
Module 6: Data Handling
ii
A MESSAGE FROM THE COMMONWEALTH OF LEARNING
The Commonwealth of Learning is grateful for the generous contribution of the
participating Ministries of Education. The Permanent Secretaries for Education
played an important role in facilitating the implementation of the 1998-2000
project work plan by releasing officers to take part in workshops and meetings and
by funding some aspects of in-country and regional workshops. The Commonwealth of
Learning is also grateful for the support that it received from the British Council (Botswana
and Zambia offices), the Open University (UK), Northern College (Scotland), CfBT
Education Services (UK), the Commonwealth Secretariat (London), the South Africa
College for Teacher Education (South Africa), the Netherlands Government (Zimbabwe
office), the British Department for International Development (DFID) (Zimbabwe office)
and Grant MacEwan College (Canada).
The Commonwealth of Learning would like to acknowledge the excellent technical advice
and management of the project provided by the strategic contact persons, the broad
curriculum team leaders, the writing team leaders, the workshop development team leaders
and the regional monitoring team members. The materials development would not have
been possible without the commitment and dedication of all the course writers, the incountry reviewers and the secretaries who provided the support services for the in-country
and regional workshops.
Finally, The Commonwealth of Learning is grateful for the instructional design and review
carried out by teams and individual consultants as follows:
•
Grant MacEwan College (Alberta, Canada):
General Education Courses
•
Open Learning Agency (British Columbia, Canada):
Science, Technology and Mathematics
•
Technology for Allcc. (Durban, South Africa):
Upper Primary Technology
•
Hands-on Management Services (British Columbia, Canada):
Junior Secondary Technology
Dato’ Professor Gajaraj Dhanarajan
President and Chief Executive Officer
ACKNOWLEDGEMENTS
The Mathematics Modules for Upper Primary and Junior Secondary Teachers in the
Southern Africa Development Community (SADC) were written and reviewed by teams
from the participating SADC Ministries of Education with the assistance of The
Commonwealth of Learning.
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CONTACTS FOR THE PROGRAMME
The Commonwealth of Learning
1285 West Broadway, Suite 600
Vancouver, BC V6H 3X8
Canada
National Ministry of Education
Private Bag X603
Pretoria 0001
South Africa
Ministry of Education
Private Bag 005
Gaborone
Botswana
Ministry of Education and Culture
P.O. Box 9121
Dar es Salaam
Tanzania
Ministry of Education
Private Bag 328
Capital City
Lilongwe 3
Malawi
Ministry of Education
P.O. Box 50093
Lusaka
Zambia
Ministério da Eduação
Avenida 24 de Julho No 167, 8
Caixa Postal 34
Maputo
Mozambique
Ministry of Education, Sport and
Culture
P.O. Box CY 121
Causeway
Harare
Zimbabwe
Ministry of Basic Education,
Sports and Culture
Private Bag 13186
Windhoek
Namibia
iv
COURSE WRITERS FOR JUNIOR SECONDARY MATHEMATICS
Ms. Sesutho Koketso Kesianye:
Writing Team Leader
Head of Mathematics Department
Tonota College of Education
Botswana
Mr. Jan Durwaarder:
Lecturer (Mathematics)
Tonota College of Education
Botswana
Mr. Kutengwa Thomas Sichinga:
Teacher (Mathematics)
Moshupa Secondary School
Botswana
FACILITATORS/RESOURCE PERSONS
Mr. Bosele Radipotsane:
Principal Education Officer (Mathematics)
Ministry of Education
Botswana
Ms. Felicity M Leburu-Sianga:
Chief Education Officer
Ministry of Education
Botswana
PROJECT MANAGEMENT & DESIGN
Ms. Kgomotso Motlotle:
Education Specialist, Teacher Training
The Commonwealth of Learning (COL)
Vancouver, BC, Canada
Mr. David Rogers:
Post-production Editor
Open Learning Agency
Victoria, BC, Canada
Ms. Sandy Reber:
Graphics & desktop publishing
Reber Creative
Victoria, BC, Canada
v
TEACHING JUNIOR SECONDARY MATHEMATICS
Introduction
Welcome to the programme in Teaching Junior Secondary Mathematics! This series of six
modules is designed to help you to strengthen your knowledge of mathematics topics and
to acquire more instructional strategies for teaching mathematics in the classroom.
Each of the six modules in the Junior Secondary Mathematics series provides an
opportunity to apply theory to practice. Learning about mathematics entails the
development of practical skills as well as theoretical knowledge. Each mathematics topic
includes an explanation of the theory behind the mathematics, examples of how the
mathematics is used in practice, and suggestions for classroom activities that allow students
to explore the mathematics for themselves.
Each module also explores several instructional strategies that can be used in the
mathematics classroom and provides you with an opportunity to apply these strategies in
practical classroom activities. Each module examines the reasons for using a particular
strategy in the classroom and provides a guide for the best use of each strategy, given the
topic, context and goals.
The guiding principles of these modules are to help make the connection between
theoretical maths and the use of the maths; to apply instructional theory to practice in the
classroom situation; and to support you, as you in turn help your students to apply
mathematics theory to practical classroom work.
Programme Goals
This programme is designed to help you to:
•
strengthen your understanding of mathematics topics
•
expand the range of instructional strategies that you can use in the mathematics
classroom
Programme Objectives
By the time you have completed this programme, you should be able to:
•
develop and present lessons on the nature of the mathematics process, with an
emphasis on where each type of mathematics is used outside of the classroom
•
guide students as they work in teams on practical projects in mathematics, and help
them to work effectively as a member of a group
•
use questioning and explanation strategies to help students learn new concepts and
to support students in their problem solving activities
•
guide students in the use of investigative strategies on particular projects, and thus to
show them how mathematical tools are used
•
guide students as they prepare their portfolios about their project activities
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The relationship between this programme and your local mathematics curriculum
The mathematics content presented in these modules includes some of the topics most
commonly covered in the mathematics curricula in southern African countries. However, it
is not intended to cover all topics in any one country’s mathematics curriculum
comprehensively. For this, you will need to consult your national or regional curriculum
guide. The curriculum content that is presented in these modules is intended to:
•
provide an overview of the content in order to support the development of
appropriate teaching strategies
•
use selected parts of the curriculum as examples for application of specific teaching
strategies
•
explain those elements of the curriculum that provide essential background
knowledge, or that address particularly complex or specialised concepts
•
provide directions to additional resources on the curriculum content
How to Work on this Programme
As is indicated in the programme goals and objectives, the programme provides for you to
participate actively in each module by applying instructional strategies when exploring
mathematics with your students and by reflecting on that experience. In other words, you
“put on your student uniform” for the time you work on this course. There are several
different ways of doing this.
Working on your own
You may be the only teacher of mathematics in your school, or you may choose to work on
your own so you can accommodate this programme within your schedule. If this is the case,
these are the recommended strategies for using this module:
1. Establish a schedule for working on the module: choose a date by which you plan to
complete the first module, taking into account that each unit will require between six to
eight hours of study time and about two hours of classroom time for implementing your
lesson plan. (Also note that each module contains two to four units.) For example, if you
have two hours a week available for study, then each unit will take between three and
four weeks to complete. If you have four hours a week for study, then each unit will
take about two weeks to complete.
2. Choose a study space where you can work quietly without interruption, for example, a
space in your school where you can work after hours.
3. If possible, identify someone who is interested in mathematics or whose interests are
relevant to mathematics (for example, a science teacher in your school) with whom you
can discuss the module and some of your ideas about teaching mathematics. Even the
most independent learner benefits from good dialogue with others: it helps us to
formulate our ideas—or as one learner commented, “How do I know what I’m thinking
until I hear what I have to say?”
vii
Working with colleagues
If you are in a situation where there are other teachers of mathematics in your school or in
your immediate area, then it is possible for you to work together on this module. You may
choose to do this informally, perhaps having a discussion group once a week or once every
two weeks about a particular topic in one of the units. Or, you may choose to organise more
formally, establishing a schedule so that everyone is working on the same units at the same
time, and you can work in small groups or pairs on particular projects. If you and several
colleagues plan to work together on these modules, these are the recommended steps:
1. Establish and agree on a schedule that allows sufficient time to work on each unit, but
also maintains the momentum so that people don’t lose interest. If all of you work
together in the same location, meeting once a week and allocating two weeks for each
unit, this plan should accommodate individual and group study time. If you work in
different locations, and have to travel some distance to meet, then you may decide to
meet once every two weeks, and agree to complete a unit every two weeks.
2. Develop and agree on group goals, so that everyone is clear about the intended
achievements for each unit and for each group session.
3. Develop a plan for each session, outlining what topics will be covered and what
activities will be undertaken by the group as a whole, in pairs or in small groups. It may
be helpful for each member of the group to take a turn in planning a session.
Your group may also choose to call on the expertise of others, perhaps inviting someone
with particular knowledge about teaching or about a specific mathematics topic, to speak
with the group.
Your group may also have the opportunity to consult with a mentor, or with other groups.
Colleagues as feedback/resource persons
Even if your colleagues are not participating directly in this programme, they may be
interested in hearing about it and about some of your ideas as a result of taking part. Your
fellow teachers of mathematics may also be willing to take part in discussions with you
about the programme.
Working with a mentor
As mentioned above, you may have the opportunity to work with a mentor, someone with
expertise in mathematics education who can provide you with feedback about your work. If
you are working on your own, your communication with your mentor may be by letter mail,
telephone or e-mail. If you are working as a group, you may have occasional group
meetings or teleconferences with your mentor.
viii
Using a learning journal
Whether you are working on your own or with a group, use a learning journal. The learning
journal serves a number of different purposes, and you can divide your journal into
compartments to accommodate these purposes. You can think of your journal as a “place”
with several “rooms” where you can think out loud by writing down your ideas and
thoughts.
In one part of your journal, keep your assignments and lesson plans. In another, you can:
•
keep notes and a running commentary about what you are reading in each unit
•
write down ideas that occur to you about something in the unit, and note questions
about the content or anything with which you disagree
•
record ideas about how to use some of the content and strategies in the classroom
•
record your answers to the “Reflection” questions that occur in most units
If you consistently keep these notes as you work through each unit, they will serve as a
resource when you work in the classroom, since you will have already put together some
ideas about applying the material there. This is also the section of the journal for your notes
from other resources, such books or articles you read or conversations with colleagues.
Resources available to you
Although these modules can be completed without referring to additional resource
materials, your experience and that of your students can be enriched if you use other
resources as well. There is a list of resource materials for each module provided at the end
of that module.
Preliminary knowledge required for this Mathematics Programme
If it has been some time since you reviewed these principles, you may want to check your
background knowledge and study these concepts by reviewing the appropriate modules in
Science and Mathematics.
Before you begin, you should ensure that you are sufficiently familiar with the basic
concepts of mathematics:
•
fractions, especially the manipulation of proper fractions
•
basic algebraic equations
•
general skill with problem solving, graphing, and relating mathematics to everyday
situations—a “maths sense”
ix
ICONS
Throughout each module, you will find the following icons or symbols that alert you to a
change in activity within the module.
Read the following explanations to discover what each icon prompts you to do.
Introduction
Rationale or overview for this part of the course.
Learning Objectives
What you should be able to do after completing
this module or unit.
Text or Reading Material
Course content for you to study.
Important—Take Note!
Something to study carefully.
Self-Marking Exercise
An exercise to demonstrate your own grasp of
the content.
Individual Activity
An exercise or project for you to try by yourself
and demonstrate your own grasp of the content.
Classroom Activity
An exercise or project for you to do with or
assign to your students.
Reflection
A question or project for yourself—for deeper
understanding of this concept, or of your use of it
when teaching.
Summary
Unit or Module
Assignment
Exercise to assess your understanding of all the
unit or module topics.
Suggested Answers to
Activities
Time
Suggested hours to allow for completing a unit or
any learning task.
Glossary
Definitions of terms used in this module.
x
CONTENTS
Module 1: Number systems
Module 1 – Overview ..................................................................................................... 2
Unit 1: Number sense, numerals and powers ............................................................... 4
Section A A reflection on teaching mathematics ....................................................... 5
Section B The general aim of learning about numbers............................................... 6
Section C Activities to try in the classroom to enhance number sense in pupils ......... 8
Section D Numbers, numerals, powers ..................................................................... 11
Answers to self mark exercises Unit 1....................................................... 24
Unit 2: Sequences ........................................................................................................... 31
Section A Introduction .............................................................................................. 32
Section B Finding the nth term formula .................................................................... 33
Section C Investigating the nth term of a sequence.................................................... 34
Section D Using the method of differences to find the nth term ................................. 36
Section E Using differences to find the next term in the sequence............................. 39
Section F Using the method of differences to find the nth term if
the second differences are constant ........................................................... 42
Section G Games in the learning of mathematics....................................................... 46
Section H Problem solving method in sequences ...................................................... 51
Answers to self mark exercises Unit 2....................................................... 66
Unit 3: Polygonal numbers ............................................................................................ 69
Section A Triangular, square, pentagonal, hexagonal numbers .................................. 70
Section B Relationships among polygonal numbers .................................................. 76
Section C Activities to try in the classroom to enhance understanding
of figurative numbers and their relationships............................................. 81
Answers to self mark exercises Unit 3....................................................... 84
Unit 4: Rational and irrational numbers ...................................................................... 86
Section A The growing number system ..................................................................... 87
Section B Natural numbers, counting numbers and whole numbers........................... 88
Section C The number system: notation convention used .......................................... 88
Section D Denseness of the numbers......................................................................... 90
Section E Proving that 2 is irrational ...................................................................... 91
Section F Rational and irrational numbers in the classroom ...................................... 94
Answers to self mark exercises Unit 4....................................................... 97
References ................................................................................................................... 100
Glossary ....................................................................................................................... 102
Appendices .................................................................................................................. 104
Module 1
1
Number systems
Module 1
Number systems
Introduction to the module
“God made the whole numbers, all else is the work of man” (Kronecker).
What Kronecker is trying to express in this statement is that properties of
numbers can be derived from the basic properties of the whole numbers 0, 1,
2, 3, ......
This module looks at the structure of the number system and different classes
of numbers that can be identified within the system. The emphasis is on
clearly understanding the concepts as a teacher, and on pupil-centred
teaching methods to create a class environment in which pupils can learn the
concepts at their level.
Aim of the module
The module aims at:
(a) having you reflect on your present methods in the teaching of numbers in
the number system
(b) enhancing your content knowledge so that you may set activities with
more confidence on the topic to your pupils
(c) making your teaching of number more effective by using a pupil centred
approach and methods such as group discussion, games and
investigations
Structure of the module
In Unit 1 you will learn about number sense and how to enhance number
sense in pupils. You will distinguish between numbers and numerals and look
at powers. In Unit 2 you will learn about the technique of using differences
for predicting more terms in a sequence and for finding an expression for the
general term in the sequence. Unit 3 looks at polygonal numbers and uses an
investigative method to discover multiple relationships among these
numbers. In Unit 4 you will learn more about the irrational numbers. In all
units you will be required to try out activities with your pupils.
Module 1
2
Number systems
Objectives of the module
When you have completed this module you should be able to create a
learning environment for your pupils to enhance their:
(a) number sense
(b) understanding of classes of numbers (squares, cubes, even, odd, primes,
factors, multiples, polygonal numbers)
(c) understanding and use of the technique of using difference to analyse
sequences
(d) understanding of rational and irrational numbers
by making use of a pupil-centred approach in general and using games,
group discussion and investigations to enhance pupils’ learning.
Module 1
3
Number systems
Unit 1: Number sense, numerals and powers
Introduction to Unit 1
Pupils frequently can manipulate numbers using algorithms without a real
awareness of what these numbers represent, their relative size and multiple
relationships. In this unit you first look at what number sense is and how to
enhance number sense in your pupils. You will study some classes of whole
numbers such as powers, factors, prime numbers and multiples.
Purpose of Unit 1
The aim of this unit is to widen your knowledge and to reflect on and try out
activities in the classroom that allow your pupils to enhance their feel for
numbers and number operations and their understanding of powers, factors,
prime numbers and multiples.
Objectives
When you have completed this unit you should be able to:
•
state and explain the general objective for pupils who are learning about
numbers and their relationships
•
develop and use activities to enhance number sense in pupils
•
justify using activity-based learning methods for learning of mathematics
•
distinguish between a number and a numeral
•
develop and use investigative activities with numbers in the classroom
•
represent even numbers, odd numbers, consecutive numbers, consecutive
odd and even numbers, square numbers, cubes and multiples of a whole
number in algebraic form
•
illustrate, give examples of and explain power, base, index, factors,
multiples
•
give examples of and explain prime numbers
•
illustrate, give examples of and explain rectangular numbers and oblong
numbers
•
use algebra to prove relationships among whole numbers
Time
To study this unit will take you about four hours. Trying out and evaluating
the activities with your pupils in the class will be spread over the weeks you
have planned to cover the topics.
Module 1: Unit 1
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Number sense, numerals and powers
Unit 1: Number sense, numerals and powers
Section A: A reflection on teaching mathematics
All teachers have their ideas (more or less outspoken) about what the nature
of mathematics is (Is it a true, unchangeable body of knowledge? Is it a
product of mankind and hence ever changing and expanding?), what the
most effective way is to teach mathematics (Telling and demonstrating to
pupils? Setting tasks for pupils to work on cooperatively?) and how pupils
best learn mathematics (Listening to the teacher and following the examples?
Working in groups on a task and discussing?). It is important for you to
reflect on your present classroom practices and ideas as some of the module
ideas might vary from what you are doing presently. You will be asked to try
out activities in your classroom, evaluate the effectiveness of the activities,
and to report on the feelings of the pupils. Cockcroft, in the report
Mathematics Count stated that in the teaching of mathematics the following
activities should take place:
1. Exposition
2. Consolidation
3. Discussion
4. Practical activities
5. Problem solving
6. Investigations
My teaching approach and methods
Write down how you teach the concepts listed in your syllabus under
“Number”. Make an outline of the lessons and the main activities in each
lesson when covering the topic.
Describe the main features of your lessons and justify their inclusion.
What do you consider the most effective way of teaching the concepts under
“Number”. Justify your choice.
Make a list of the activities that take place in your lessons on “Number”.
Are the Cockcroft six well covered? Justify.
Keep your reflection as you will need it for your final assignment on this
module. Now continue to work through this module.
Module 1: Unit 1
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Number sense, numerals and powers
Section B: The general aim of learning about numbers
The aim of number activities is to develop a number sense in the pupils.
When asked to find mentally the cost of 64 items costing P0.50 each, one
1
student recalled that 0.50 = , and took half of 64 and concluded that the
2
cost is P32.- .
Another student mentally multiplied 5 × 64, taking 5 times 4 remembering
the 0 and ‘carrying’ the 2. Then took 5 × 6, getting 30 and added the 2 he
carried, giving him 32. Taking the place of the point into account he gave as
his correct response P32.- .
A third child gave an answer P32 and explained that 10 items cost P5 and
60 items therefore cost 6 × P5 = P30. The remaining 4 items cost P2. The
total for 64 items is then P30 + P2 = P32.
The first and third pupil used their knowledge about number relationships
efficiently, the second applied mentally the standard algorithm for
multiplication which was not very convenient to do mentally.
In calculating the area of a rectangular room in a house measuring 3 m by
4.5 m a student gave the answer 144 m2.
Looking at the answer and being aware of possible areas of rooms the
student should have realized that the answer is unreasonably big. Using
approximations to estimate the answer the student should have realized that
the answer should be in the order of 3 m × 5 m = 15 m2.
7.5% interest was paid on an investment of P250.- Using a calculator a
student gave P 187.5 as the answer for the interest received.
This student failed to enter the figures correctly (ignoring the point and using
75% instead of 7.5%) and never questioned the display on the calculator.
Similar situations as described above are probably familiar to you as a
teacher. Apart from the first and third pupil in the first example, the others
did not show ‘number sense’. As teachers we are to aim at developing
number sense in our pupils with the emphasis on the processes and strategies
used in solving problems rather than on mechanical applications of
algorithms and rules to get “the right” answer.
Number sense is characterized by
(1) understanding, representing and using numbers in a variety of equivalent
forms in real-life and mathematical problem situations.
(2) having an awareness of multiple relationships among numbers.
(3) recognizing the relative magnitude of numbers.
(4) knowing the relative effect of operating on numbers.
(5) possessing referents for numbers—especially small and large
numbers—in the environment. This means having in mind situations one
can relate numbers of various magnitude to; for example, knowing that at
Module 1: Unit 1
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Number sense, numerals and powers
assembly in the morning the number of pupils is about 650, that the
height of a door is about 2 m, etc.
Self mark exercise 1
1. One-quarter can be represented in many different ways. Write down
some ways to represent one-quarter.
2. Write down at least ten questions involving whole numbers that all have
an answer of 36.
2
1
3. Is greater or smaller than ? How did you find out quickly?
5
2
4. Dividing a number N by another number will always result in a quotient
Q smaller than the number N. True or false? Justify your answer.
5. To count from 1 to 1 000 000 (a million) reciting one number per second
will take you not more than an hour. True or false? Justify your answer.
6. Explain how each of the above questions relate to one or more of the
characteristics of number sense.
Check your answers at the end of this unit.
Module 1: Unit 1
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Number sense, numerals and powers
Section C: Activities to try in the classroom to enhance
number sense in pupils
There is overwhelming research evidence that pupils come to a better and
lasting understanding of mathematics when they are actively involved in
activities that allow group discussion. Discussion in small groups enhances
concept building and understanding. Knowledge is, first and foremost, the
subjective knowledge of the learner. The learner initially tries to make sense
of the new information in the context of the knowledge, experience and
beliefs he or she already has. This personal constructed knowledge is
expressed and compared with the knowledge of others to come to shared and
agreed knowledge.
The activities for class use suggested below are directed to you as a teacher.
They are not a complete lesson plan, but are some ideas for developing
number sense in your pupils. You can use the ideas and extend them for use
in the classroom. The main characteristic of the suggested activities is that
they promote understanding and generate rich discussion among the pupils in
small groups and between the pupils and the teacher. Before taking an
activity to the classroom, work through the activity yourself.
Pupils activity 1: Establishing referent objects for whole numbers
To the teacher:
Pupils are fascinated by large and small numbers. For large numbers pupils
need real life situations as referents, i.e., something they can refer to, or
relate the number to. Knowing the number of pupils in their school might
help them to estimate the number of people attending a football match. The
object of this activity is to see how many pupils are aware of the size of
some commonly used referents of real life situations. It will also lead to
discussion on strategies that can be used to find approximate answers to the
questions.
Present the following questions to pupils working together in groups of four.
Allow discussion to come to an agreed estimate from the group. Write all the
estimates on the board and allow research to find the value that answers each
question. To extend the activity, encourage the pupils to come up with
additional questions.
To develop a feeling for “How big is a million?” a school project to collect
one million beer/soft drink tins could be started.
Questions set to pupils:
1. What is the population of the world?
2. What is the population of your country?
3. What is the population of your town/village?
4. What is the number of pupils in the your school?
5. What is the Government’s budget this year?
6. How many children in the age 12 - 14 are there in the country?
Module 1: Unit 1
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Number sense, numerals and powers
7. How many pupils completed Junior Secondary School last year?
8. How many kilograms of rice are to be cooked in the school to feed all
pupils?
9. How far is the Earth from the Sun?
10. What is the diameter of a hair on your head?
11. Can one million Pula in P 10 notes fit in a normal suitcase?
12. How many grains of rice are there in a 1 kg packet?
Pupils activity 2: Using context to determine reasonable values
To the teacher:
Pupils working in groups of four, should discuss and insert the missing
values in the activity below. As an extension ask pupils in one group to write
a story with numerical data structured as the example below. Another group
can then place the appropriate values in the place in the story. A newspaper
article or text in a book with lots of numerical data, blanking out the values
and writing them above the article for pupils to insert at the appropriate place
could be used as another activity.
Ask pupils to complete this newspaper story by inserting at the appropriate
place 180, hundreds, fourth, 24th, 400, six, 526, 616 and seventeen into the
following story:
PTA Meeting Attracts ________ Participants
An audience of about ________ attended the end of year Chorus organised
by the PTA on Monday, November ______. The parents, teachers, and
friends heard _____ songs and saw _____ musical skits. After the
entertainment, last year’s school-attendance award was presented to Opela.
She attended all _______ days of the last school year. It was her ______ year
of perfect attendance. Then the parents and pupils had snacks. ______ fat
cakes were eaten and _____ glasses of soft drinks consumed.
Pupils activity 3: Verifying mathematical statements
To the teacher:
Pupils are to learn to consider whether or not published data are reasonable.
Obtain some facts from a published source. Pupils in their groups are to
verify the accuracy of the statement based on the given fact. Give different
questions to each group. The groups are to present their findings in a report
to the class.
FACT: Driving on a dust road stirs up 81 000 tonnes of dust.
If you cover a football field with this dust the layer will be about
16 m deep.
FACT: The world population is about 6 billion (6 000 000 000).
If all the people line up shoulder to shoulder along the equator, the
line will go 75 000 times around the world.
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Number sense, numerals and powers
FACT: Every day about 2 million tennis balls are rejected after use.
You will need 100 classrooms to pack in all these tennis balls.
FACT: A spaceship is launched to Mars and travels at 12 800 km/h.
It will cover the 80 million km journey in about 6 months.
FACT: The speed of light is 3.0 × 108 m/s.
It will take light from the Sun about 5 minutes to reach the Earth
(average distance Sun - Earth is 150 million km).
Unit 1, Assignment 1
1. Choose one of the above three pupils activities and work it out in more
detail for use in your class. Take into account the level of understanding
of your pupils (differentiate the activity if necessary for the different
levels of pupil achievement) and use the local environment as context.
2. Try your activity out in the classroom.
3. Write an evaluation of the lesson in which you presented the activity.
Some questions you might want to answer could be: What were the
strengths and weaknesses? What aspects of the activity needs
improvement? How was the reaction of the pupils? What did you learn
as a teacher from the lesson? Could all pupils participate? Were your
objective(s) attained? Was the timing correct? What did you find out
about pupils’ number sense? What further activities are you planning to
strengthen pupils’ number sense? Were you satisfied with the outcome
of the activity? Was the activity different from what you usually do in
the class with your pupils? How did it differ?
Present your assignment to your supervisor or study group for discussion.
Module 1: Unit 1
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Number sense, numerals and powers
Section D: Numbers, numerals and powers
The following exercise will help you recall some facts related to numbers.
Self mark exercise 2
1. What is a prime number? List the first five.
2. List the first five even numbers.
3. List the first five odd numbers.
4. What are the first five multiples of 6?
5. List the factors of 24.
6. On the dotted line place ALL the possible correct statements choosing
from
is a factor of
divides
is not a factor of
does not divide
is a multiple of
is divisible by
is not a multiple of
is not divisible by
a. 3 ........................................................ 12
b. 24 ...................................................... 4
c. 2 ........................................................ 36
d. 72 ...................................................... 18
e. 5 ........................................................ 7
f. 6 ......................................................... 42
g. 121 .................................................... 11
h. 16 ...................................................... 4
7. List the first 6 square numbers.
8. How many factors have a square number? Try 16, 25, and 36.
9. List the first 5 cube numbers.
10. What is meant by HCF and LCM? Illustrate these concepts using the
numbers 24 and 56.
Check your answers at the end of this unit.
Numbers and Numerals
As a teacher you should be aware of the difference between a number and
the representation of that number called the numeral. A number is an
abstract concept. You can have four cows, four children playing together,
four bags of cement. All these examples have in common the idea of
‘fourness’: there are four elements in each set. A numeral is any symbol used
to represent that concept. The concept “four” can be represented by numerals
Module 1: Unit 1
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Number sense, numerals and powers
4, IV, 16 , 3.99999.... , ////, 400% and many others. You may notice that
one concept in mathematics may have several representations. It is important
to present pupils with multiple representations of a concept so that they may
get the concept clear.
Algebraic format of numbers
In proofs and justification it is often useful to represent numbers in algebraic
format.
The even whole numbers are 2, 4, 6, 8, .... or in general 2n for any natural
value of n.
If 2n is even then the next number 2n + 1 will be odd.
Consecutive numbers are numbers following each other in a sequence. For
example 2n and 2n + 2 are consecutive even numbers, and so are 2n – 2 and
2n; consecutive odd numbers are 2n – 1 and 2n + 1, or 2n + 1 and 2n + 3.
Three consecutive numbers can be represented by n – 1, n, n + 1 or by n,
n + 1, n + 2 or by n – 2, n – 1, n, or by 3n, 3n + 1, 3n + 2 etc. The
representation used generally does not matter although taking n – 1, n, n + 1
often leads to simpler algebra. The context of the problem might suggest a
particular representation.
The multiples of three 3, 6, 9, 12, 15, ... can in general be represented by 3n
for natural n.
3n – 1, 3n, 3n + 1 are three consecutive numbers. So are 3n, 3n + 1 and
3n + 2. This last format shows also that every natural number is a threefold,
a threefold plus 1, leaving remainder 1 when divided by three or a threefold
plus 2, leaving a remainder 2 when divided by three.
Using algebra to prove statements about whole numbers
(i) Suppose we want to verify the statement:
The sum of the squares of five consecutive numbers is divisible by 5.
We could proceed as follows:
The numbers n – 2, n – 1, n, n + 1 and n + 2 are five consecutive
numbers for any natural value of n greater than or equal to three.
The sum of their squares is
(n – 2)2 + (n – 1)2 + n2 + (n + 1)2 + (n + 2)2
= n2 – 4 n + 4 + n2 – 2 n + 1 + n2 + n2 + 2 n + 1 + n2 + 4 n + 4
= 5n2 + 10 = 5(n2 + 2).
The expression, sum of the squares of five consecutive numbers which is
equal to 5(n2 + 2), has a factor 5 and is hence divisible by 5.
If you would have taken n, n + 1, n + 2, n + 3, n + 4 to represent five
consecutive numbers, the same result would have emerged:
Module 1: Unit 1
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Number sense, numerals and powers
The sum of their squares is
n2 + (n + 1)2 + (n + 2)2 + (n + 3)2 + (n + 4)2
= n2 + n2 + 2n + 1 + n2 + 4n + 4 + n2 + 6n + 9 + n2 + 8n + 16
= 5n2 + 20n + 30 = 5(n2 + 4n + 6).
The expression has again a factor 5 and hence the sum of the squares of
five consecutive numbers is divisible by 5.
(ii) To prove that the sum of three consecutive even numbers is a multiple of
6 you could work as follows:
Let the three even consecutive be 2n – 2, 2n and 2n + 2 (n > 1).
The sum is 2n – 2 + 2n + 2n + 2 = 6n, which is clearly a multiple of six.
Powers and indices
24, 42, 57, 153 are powers. Powers are a short way of writing down a
repeated multiplication of the same factor.
24 = 2 × 2 × 2 × 2, the product of 4 factors of 2.
42 = 4 × 4, the product of 2 factors of 4.
57 = 5 × 5 × 5 × 5 × 5 × 5 × 5, the product of 7 factors of 5.
np is the product of p values of n.
In 24, 4 is called the index of the power (plural, ‘indices’) and 2 is called the
base.
2 is the same as 21, although the index 1 is not written in final expressions,
but sometimes used in computations to help as a reminder.
The index shows in the above cases how many times the base number
appears as a factor in the product.
We restrict ourselves here to powers with both n and p being natural
numbers, although n could be any real number and the same statement would
still apply provided p is a natural number. For example:
(1.2)3 = 1.2 × 1.2 × 1.2, the product of 3 factors of 1.2
1
1
1
1
1
1
(-5 )4 = (-5 ) × (-5 ) × (-5 ) × (-5 ), the product of 4 factors of (-5 )
3
3
3
3
3
3
Module 1: Unit 1
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Number sense, numerals and powers
Factors of natural numbers
Problem 1
A group of 18 people went to a dinner party. The people in the party were to
be seated at tables each containing the same number of people. How many
arrangements are possible?
How many factors has 18?
Did you find six possible arrangements and six factors?
The six possible arrangements are
1 table with 18 persons
2 tables with 9 persons
3 tables with 6 persons
6 tables with 3 persons
9 tables with 2 persons
18 tables with 1 person
The six factors of 18 are: 1, 2, 3, 6, 9 and 18. The factors occur in pairs:
1 and 18, 2 and 9, 3 and 6 as
18 = 1 × 18 = 2 × 9 = 3 × 6 = 6 × 3 = 9 × 2 = 18 × 1.
Note that 1 × 18 and 18 × 1 are not the same in meaning (1 table with 18
persons is not the same as 18 tables with 1 person each) although they have
the same value.
Similarly 2 × 9 and 9 × 2 have different meanings, but the same value.
2 × 9 = 9 + 9 is the sum of two terms of nines and
9 × 2 = 2 + 2 + 2 + 2+ 2 + 2 + 2 + 2 + 2 is the sum of nine terms of twos.
Module 1: Unit 1
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Number sense, numerals and powers
Problem 2
a. Copy and complete a table for the numbers 1 to 100
Number
Factors
Number of factors
1
1
1
2
1, 2
2
3
1, 3
2
4
1, 2, 4
3
5
6
....
b. List the numbers with exactly 2 factors. What name is given to these
numbers?
c. List all the numbers with exactly 3, 5, 7, ... factors. What is the name
of these numbers?
d. Represent the numbers in c algebraically.
Check your answers at the end of this unit.
Primes, squares and rectangular numbers
The numbers with exactly 2 factors (1 and the number itself) are called
prime numbers.
2, 3, 5, 7, 11, 13, 17, 19, ... are prime numbers.
There is no pattern in this sequence, i.e., there is no expression that can
generate all the prime numbers.
The numbers 1, 4, 9, 16, 25, ..... with an odd number of factors (1, 3, 5, 7, ...)
are called square numbers. If represented using a dot pattern, the dots can
be placed to represent a square. Each square number is the result of
multiplying a natural number by itself—1 × 1, 2 × 2, 3 × 3, 4 × 4, ...—which
can also be written as 12, 22 , 32, 42, ...
Module 1: Unit 1
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Number sense, numerals and powers
Numbers with three or more factors are called rectangular numbers as,
when using dots, they can form a rectangular pattern. The diagram below
illustrates the rectangular numbers 6 and 8. The pattern is always to have at
least two rows (or columns).
Self mark exercise 3
1. Illustrate, using a dot pattern, that 12 is a rectangular number. How many
different patterns can you make?
2. Is a square number a rectangular number? Justify your answer.
3. Explain why the following dot pattern is not a correct way to represent
the rectangular number 6.
4. a. Explain that of any three consecutive numbers, one of the three must
be a threefold, i.e., a multiple of three.
b. Explain that of any four consecutive numbers, one must be a
fourfold, i.e., a multiple of four.
5. Justify the following statements using examples and give an algebraic
proof.
a. The product of three consecutive odd numbers is divisible by 3.
b. The sum of three consecutive odd numbers is divisible by 3.
c. The product of two consecutive even numbers is always the
difference between a square number and 1.
d. The square of any whole number is either a multiple of three or has
a remainder 1 when divided by three.
e. The square of any odd number is one more than a multiple of 4.
f. The difference between two consecutive square numbers is an odd
number.
g. If the square of a whole number is even then the whole number must
be even.
Check your answers at the end of this unit.
Module 1: Unit 1
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Number sense, numerals and powers
Rectangles that are not squares are called oblongs. The rectangular numbers
could be placed into two groups: square numbers and oblong numbers.
Cube numbers
The numbers 13 = 1, 23= 8, 33 = 27, 43 = 64, 53 = 125, ... are called cube
numbers because they can be represented using unit cubes (cubes with an
edge of length 1 unit) to build cubes as illustrated.
Module 1: Unit 1
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Number sense, numerals and powers
Self mark exercise 4
1. Illustrate using (i) square grid paper (ii) isometric paper that 27 is a cube
number.
2. Is a cuboid a cube or a cube a cuboid? Justify your answer.
3. How would you define cuboidal numbers? Illustrate some cuboidal
numbers using square grid paper and isometric paper.
4. The number 60 is a cuboidal number. In how many different ways could
you illustrate this?
5. a. A wooden 3 cm × 3 cm × 3 cm cube is painted blue on the outside
and cut into 27 unit cubes (1 cm × 1 cm × 1 cm). How many of the
unit cubes will have (i) no face painted (ii) 1 face painted (iii) two
faces painted (iv) three faces painted?
b. What if the original cube measured 4 cm × 4 cm × 4 cm or
5 cm × 5 cm × 5 cm or ... was painted at the outside and cut into unit
cubes?
Tabulating your results might be helpful.
Dimensions
2×2×2
3×3×3
Total number
of unit cubes
8
27
Number of unit cubes with... faces painted blue
0
1
2
3
0
0
0
8
..
..
..
..
c. Can you generalize for an n cm × n cm × n cm cube?
d. What are the dimensions of a cube with 3750 unit cubes painted at
one face?
e. What are the dimensions of a cube with 1440 unit cubes painted on
two faces?
f. What are the dimensions of a cube with 64 000 unpainted unit cubes?
Check your answers at the end of this unit.
Module 1: Unit 1
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Number sense, numerals and powers
Investigating patterns in powers
Problem 3
The last digit (unit digit) of the powers of the natural numbers follow
patterns. In this investigation you are to find and describe those patterns.
1. Copy and complete the ‘patterns in the powers chart’ table partly
illustrated below.
Patterns in the powers chart
N1
1
2
3
4
5
6
7
8
9
10
N2
1
4
N3
1
8
N4
1
16
N5
1
32
N6
1
64
..
..
Units digits
1
2, 4, 8, 6
2. a. Look at the rows: what are the last digits of powers of 2, 3, ... etc.?
b. In what cycle are the last digits of the powers of each number
repeating?
c. How are the indices of the powers of the numbers related to the last
digit?
3. Use the pattern to find the last digit of 21999, 31999, ......101999 and
19991999.
4. Look at the columns: what are the possible last digits of square numbers?
cube numbers? n4?
5. Use your pattern to decide whether the number 345 678 965 232 can be a
square number, a cube, a fourth power, etc.
6. Justify the following statements (as a challenge: try to prove them
deductively).
a. All cubes are multiples of four, or of four added or subtracted 1
(i.e., one more or less than a multiple of four).
b. All cubes are multiples of seven, or of seven added or subtracted 1.
7. Set some questions of your own on powers and investigate them.
Check your answers at the end of this unit.
Module 1: Unit 1
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Number sense, numerals and powers
Teacher’s notes on Pattern in the powers chart
You just investigated patterns in the chart of powers of the natural numbers.
The investigation can be presented to pupils, for example starting as a
challenge.
Pupils activity 4
What is the last digit of 210?
What of 2100?
Be sure pupils understand the question. Expand 210. What is the digit in the
unit place? (Use calculator to obtain the answer.)
[210 = 1024 so the digit in the unit place is 4].
Allow pupils time to explore the problem. Trying the calculator for 2100 will
not work as the calculator will give an approximate result in scientific
notation (1.2676506 × 1030). Suggest looking at smaller exponents to find a
pattern.
The unit digits repeat in a cycle of four: 2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6, 2, ......
Pupils are to observe where each power of 2 occurs in the cycle: 2
corresponds to the exponents 1, 5, 9, ... or for the exponents that have
remainder 1 when divided by 4 (or are 1 more than a multiple of 4).
Similarly 4 occurs for the exponents 2, 6, 10, ... or for the exponents that
have a remainder 2 on division by 4; 8 will occur for the exponents that have
a remainder 3 when divided by 4 and 6 will occur for exponents that have a
remainder 0 (or are multiples of four) when divided by 4.
So 2100 will end in 6, 2103 ends in 8, etc.
Extension
The original question, restricted to the powers of 2, can now be extended for
powers of all whole numbers 1 through 10. Let pupils investigate these
powers. An amazing pattern emerges. The longest cycle for the units digit is
four.
Tabulating the results lead to other patterns to be considered: endings of
square numbers [1, 4, 9, 6, 5, 6, 9, 4, 1, 0 i.e., a square number can never end
in 2, 3, 7 or 8], cube numbers [all digits 0 to 9 can occur as units].
Which powers are the same? (For example 24 = 42 = 16; 26 = 43 = 64 Why?
This can lead to the generalization [2n]m = 2nm).
The other exponential rules can be investigated from the table. Also
questions such as:
Does 23 × 32 = 66? {No} Why ?
Does 23 × 22 = 26? {No} Why ?
Does 23 × 33 = 63 ? {Yes} Why ?
can be explored.
Module 1: Unit 1
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Number sense, numerals and powers
Patterns in the powers chart
N1
N2
N3
N4
N5
N6
1
1
1
1
1
1
1
2
4
8
16
32
64
2, 4,
8, 6
3
9
27
81
243
729
3, 9,
7, 1
4
16
64
256
1024
4096
4, 6
5
25
125
625
3125
15625
5
6
36
216
1296
7776
46656
6
7
49
343
2401
16807
117649
7, 9,
3, 1
8
64
512
4096
32768
262144
8, 4,
2, 6
9
81
729
6561
59049
531442
9, 1
10
100
1000
10 000
100 000
1 000 000
0
..
..
Units
digits
Some further questions for pupil investigation:
What is the digit in the units place for 17100, 2550, 311000?
Investigate the last two digits of powers of numbers.
Module 1: Unit 1
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Number sense, numerals and powers
You have completed this section and should be able to answer the following
questions.
Self mark exercise 5
1. Give two different meanings of the word (i) ‘square’ and (ii) ‘cube’.
2. Explain why square numbers have always an odd number of factors
while all other numbers have an even number of factors.
3. Explain the different meaning of 3 × 6 and 6 × 3.
4. Give five different representations (numerals) of ‘tenness’.
5. Is n2 + n + 41 a prime number for all natural values of n? Investigate.
6. A pupil wrote 23 = 6 and 33 = 9. What error did the pupil make? Which
remedial steps would you take to help the pupil to overcome the
problem?
7. Explain why 1 is NOT a prime number, while 2 is.
8. Give a dot illustration of a oblong number and a square number.
9. How many factors has the rectangular number 120? In how many
different ways could you make a dot pattern to illustrate that 120 is a
rectangular number?
10. How many factors has the rectangular number 121? In how many
different ways could you make a dot pattern to illustrate that 121 is a
rectangular number?
11. Comparing the rectangular numbers 120 and 121, describe their
difference.
12. Explain, using examples, the meaning of power, base and index.
13. Is 225 736 192 672 a square number? How can you quickly tell?
Check your answers at the end of this unit.
Module 1: Unit 1
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Number sense, numerals and powers
Unit 1: Practice activity
1. Choose one of the previous three problem solving activities and work it
out in more detail for use in your class as a group task. Take into
account the level of understanding of your pupils (differentiate the
activity if necessary for the different levels of pupils’ achievement).
2. Try your activity out in the classroom with a form 2 or 3. Pupils should
work together in groups of 4 and come up with ONE agreed response.
3. Write an evaluation of the lesson in which you presented the activity.
Some questions you might want to answer could be: What was / were
the objectives of your lesson? How did you structure the activity set to
the groups and why? What was the criteria you used to form the
groups? What were the strengths and weaknesses of the activity? How
was the reaction of the pupils? Did they like the activity? Did they like
to work as a group? Were there any problems you have to take into
account when using a similar method? What did you learn as a teacher
from the lesson? Could all pupils participate? Were your objective(s)
attained? Was the timing correct? What did you find out about pupils’
number sense? What further activities are you planning to strengthen
pupils’ number sense? Were you satisfied with the outcome of the
activity? Were the objectives achieved? How do you know? Was the
activity different from what you usually do in the class with your
pupils? How did it differ?
Present your assignment to your supervisor or study group for discussion.
Summary
Unit 1 discussed the challenges that one faces when trying to teach
mathematics in a way that causes students to acquire mathematics skills
(rather than the simpler skills of moving symbols correctly on the page). It
presented examples of teaching through pictures, diagrams and everyday
objects, as well as through math symbols. The overall aim—which should
also be your overall aim in your classroom—was to show students that the
manipulations they learn to do with numbers correspond to the way objects
in the world also behave.
Module 1: Unit 1
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Number sense, numerals and powers
Unit 1: Answers to self mark exercises
Self mark exercise 1
1.
,
1 2
, , and many others.
4 8
2. The emphasis is on examples in context. Number, calculations are to
relate to something to make sense to pupils. E.g.
a) What is the cost of 9 bars of chocolate at P4 each?
b) In a youth group there are an equal number of boys and girls. If there
are 18 girls in the group, how many are there in the group altogether?
c) A rectangular card measures 12 cm by 3 cm. What is its area?
d) Tiles measure 9 cm by 4 cm. What is the length of the side of the
smallest square that can be covered by these tiles without having to
cut any?
3.
1
2
< . Convert to decimals or percent, or fractions with common
2
5
4
5
denominator (
,
).
10 10
4. Divide -4 by 2, the quotient -2 is greater than -4.
divide 4 by 0.5, the quotient 8 is greater than 4, Hence dividing n by a
number might give as a quotient a result greater than n.
5. It takes 1 000 000 seconds which is about 277.8 hours (1 dp) or 11.6
days (1 dp).
6. Question 1 relates to (1) & (2); Question 2 relates to (1); Question 3 to
(3); Question 4 to (4) and question 5 to (3) & (5).
Self mark exercise 2
1. Prime numbers are numbers with exactly two factors (1 and the number
itself)
2, 3, 5, 7, 11, ..
2. 2, 4, 6, 8, 10, ..
3. 1, 3, 5, 7, 9, ..
4. 6, 12, 18, 24, 30, ..
5. 1, 2, 3, 4, 6, 8, 12, 24
Module 1: Unit 1
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Number sense, numerals and powers
6. (a)(c)(f) is a factor of, divides, is not divisible by, is not a multiple of
(b)(d)(g)(h) is a multiple of, is divisible by, is not a factor of, does not
divide
(e) is not a factor of, does not divide, is not divisible by, is not a multiple
of
7. 1, 4, 9, 16, 25, 36, ..
8. an odd number (3, 5, 7, ..) but never even number of factors
9. 1, 8, 27, 64, 125, ..
10. HCF highest common factor of 24 and 56 is 8
LCM lowest common multiple of 24 and 56 is 168, i.e., first number
being multiple of both
Problem 2
1b. prime number have 2 factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,
43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97
c. 1 factor has only 1
Numbers with odd number of factors are square numbers.
3 factors have 4, 9, 25, 49
5 factors have 16, 81
7 factors has 64
9 factors have 36, 100
Numbers with 4, 6, 8, ..or more even number of factors are oblong
numbers.
4 factors have 6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51,
55, 57, 58, 62, 65, 69, 74,, 77, 82, 85, 86, 91, 93, 94, 95
6 factors have 12, 18, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 98, 99
8 factors have 24, 30, 40, 42, 54, 56, 66, 70, 78, 88
10 factors have 48, 80, 84
12 factors have 72, 90, 96
d. Square numbers can be represented algebraically as n2, where n is a
whole number.
Self mark exercise 3
1. 12 = 2 × 6 = 3 × 4 = 4 × 3 = 6 × 2. Four different patterns if you consider
3 × 4 and 4 × 3 as different patterns, 2 × 6 and 6 × 2.
2. Except for 1, all square numbers are rectangular as they have at least 3
factors.
3. By convention a rectangular number dot pattern needs at least 2 rows
(columns) and NOT one row (or column).
Module 1: Unit 1
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Number sense, numerals and powers
4a. If you divide a number by three the remainder is either 0, 1 or 2.
If n, n + 1, n + 2 are three consecutive numbers and you divide n by three
then
(i) if remainder 0 then n is multiple of 3 (ii) if remainder 1 then n + 2 will
be multiple of 3, if remainder 2 then n + 1 must be multiple of 3. Hence
one of the three numbers is multiple of 3.
4b. Same reasoning as in 4a. If n, n + 1, n + 2, n + 3 are four consecutive
numbers then when dividing n by 4 remainder is 0, 1, 2 or 3. If remainder
0, n is fourfold, if remainder 1 then n + 3 is fourfold. Complete the
argument.
5a. Taking the consecutive numbers 2n, 2n + 1, 2n + 2, 2n + 3, 2n + 4,
2n + 5, the underlined ones represent three consecutive odd numbers.
Use similar reasoning as in question 4. If 2n + 1 is divided by 3
remainder is 0, 1 or 2. If 0 then 2n + 1 is a threefold, if remainder 1 then
2n + 3 will be threefold and if remainder 2 then 2n + 5 is a threefold.
Hence one of the three consecutive odd numbers is a threefold and hence
their product.
(2n + 1)(2n + 3)(2n + 5) must have a factor 3 i.e., is a threefold.
5b. Three consecutive odd numbers can be represented by 2n – 1, 2n + 1,
2n + 3; their sum is 6n + 3 = 3(2n + 1), i.e., is a threefold.
5c. 2n(2n + 2) = 4n2 + 4n = (2n + 2)2 – 4 = (2n + 2)2 – 22 = 4[(n + 1)2 – 1]
This is the difference of the square of a number and 1.
E.g., 4 × 6 = 24 = 52 – 12, 6 × 8 = 48 = 72 – 1
5d. n2 = n × n. Remember from question 4 that n is either a threefold, a
threefold + 1 or a threefold + 2.
If n = 3m (a threefold) n2 = 3m × 3m which is clearly a threefold.
If n = 3m + 1 (a threefold + 1), n2 = (3m + 1)(3m + 1) = 9m2 + 6m + 1 =
3[3m2 + 2m] + 1, which when divided by 3 gives remainder 1.
If n = 3m = 2 (a threefold + 2),
n2 = (3m + 2)(3m + 2) = 9 m 2 + 12 m + 4 = 3(3m 2 + 4m + 1) + 1, which
gives again remainder 1 when divided by 3.
5e. (2n + 1)2 = 4n2 + 4n + 1 = 4[n2 + n] + 1, which is a fourfold + 1.
5f. (n + 1)2 – n2 = 2n + 1, which is an odd number.
5g. If n2 is even it must contain a factor 2 and so n2 = 2M, n × n = 2M. But
if n × n has a factor 2 it means n must have a factor 2, i.e., is even.
Module 1: Unit 1
26
Number sense, numerals and powers
Self mark exercise 4
1.
2. A cube is a special cuboid, i.e., a cuboid with all edges equal in length.
3. Cuboid numbers are numbers that can be express as product of three
factors (> 1).
For example 12 = 2 × 2 × 4 which can make a cuboid with length of
sides 2 cm by 2 cm by 4 cm.
4. 60 = 3 × 4 × 5 = 2 × 5 × 6 = 2 × 3 × 10 = 2 × 2 × 15.
This gives 4 different ways taking the combinations 3 × 4 × 5, 3 × 5 × 4,
4 × 3 × 5, 4 × 5 × 3, 5 × 3 × 4, 5 × 4 × 3 as ‘the same’.
5a. (i) 1 (ii) 6 (iii) 12 (iv) 8
5b. 4 × 4 × 4: (i) 8 (ii) 24 (iii) 24 (iv) 8
(iv) 8
5 × 5 × 5: (i) 27 (ii) 54 (iii) 36
5c. n × n × n:
(i) (n – 2)3
(ii) 6(n – 2)2
(iii) 12(n – 2)
(iv) 8
5d. 27
5e. 122
5f. 42
Module 1: Unit 1
27
Number sense, numerals and powers
Problem 3
1. 2. Patterns in the powers chart
N1
N2
N3
N4
N5
N6
N7
N8
N9
Units
digits
1
1
1
1
1
1
1
1
1
1
2
4
8
16
32
64
128
256
512
2, 4, 8,
6
3
9
27
81
243
729
2187
6561
19683 3, 9, 7,
1
4
16
64
256
1064
4256
17024
68096
272384
5
25
125
625
3125
15625
78125 390625
6
36
216
1296
7776
46656
6
7
49
343
2401
18807
131649
7, 9, 3,
1
8
64
512
4096
32768
262122
8, 4, 2,
6
9
81
729
6461
58149
523341
9, 1
10
100
1000 10 000
100 000
1 000 000
0
all
1, 4, 6, 5, 9,
0
Last 1, 4, 6,
digits 5, 9, 0
all
6, 1, 5, 0
all
0, 1, 5, 6
4, 6
5
all
3. Last digit: 8, 7, 4, 5, 6, 3, 2, 9, 0
19991999 last digit 9
4. See table
5. Number could be power of number to ODD index, e.g., n3, n5, n7, etc.
6a. Any number is fourfold, fourfold + 1, fourfold + 2 or fourfold + 3, i.e., if
you divide a natural number by 4 the remainder will be 0, 1, 2 or 3.
Look at the cubes of 4n, 4n + 1, 4n + 2, 4n + 3.
(4n)3 = 64n3 which is multiple of 4
(4n + 1)3 = 64n3 + 48n2 + 12n + 1 = 4[16n3 + 12n2 + 3n] + 1= 4M + 1,
fourfold + 1
(4n + 2)3 = 64n3 + 96n2 + 48n + 8 = 4[ ... ] , a fourfold
(4n + 3)3 = 64n3 + 144n2 + 108n + 27 = 4[16n3 + 36n2 + 27n + 7] – 1,
fourfold – 1
Module 1: Unit 1
28
Number sense, numerals and powers
6b. The first cubes are 1, 8 = 7 + 1, 27 = 4 × 7 – 1, 64 = 9 × 7 + 1,
125 = 18 × 7 – 1, 216 = 31 × 7 – 1, 343 = 49 × 7, 512 = 73 × 7 + 1,
729 = 104 × 7 + 1, ..
The pattern suggests that each cube is a multiple of seven or of seven
added or subtracted 1.
For a formal prove, work along the lines as in 6a. Each natural number
when divided by 7 leaves remainder 0, 1, 2, 3, 4, 5 or 6, hence can be
written as 7m, 7m + 1, 7m + 2, 7m + 3, 7m + 4, 7m + 5 or 7m + 6.
Work now the cube of each of these numbers. For example:
(7m + 2)3 = 73m3 + 3.72m2 + 32.
7m + 8 = 7[49m3 + 21m2 + 9 m + 1] + 1 = 7M + 1, a sevenfold added 1.
Work the remaining six possibilities similarly.
7. For example: Investigate the last 2 digits of the power of N.
Investigate the sum of the digits of the powers of N.
Self mark exercise 5
1. Square - (1) rectangle with equal sides (2) number multiplied by itself
n × n = n2
Cube (1) cuboid with all edges equal in length (2) product n × n × n = n3
2. Factors occur in pairs 6 = 2 × 3 = 1 × 6, 8 = 1 × 8 = 2 × 4, etc. So in
general a number will have an even number of factors. However a square
number has one set of equal factors, e.g.:
9 = 1 × 9 = 3 × 3, 16 = 1 × 16 = 2 × 8 = 4 × 4, hence the total number of
factor will end up being an odd number.
3. 3 × 6 = 6 + 6 + 6 6 × 3 = 3 + 3 + 3 + 3 + 3 + 3
4. 10, X, \\\\\ \\\\\, 100 , ***** *****,
20
,2×5
2
5. Did you try 41? 82? or other multiples of 41?
6. Pupil took 23 = 2 × 3 and 33 = 3 × 3, confusing raising to a power with
multiplication of base and index.
Go with the pupil through the four steps of remediation:
(i) ask the pupil to explain how he/she worked the question
(ii) create mental conflict by asking the pupil to use the calculator to
find 23, 33
(iii) develop the correct concept by going to the meaning (convention) of
23 = 2 × 2 × 2, product of three factors two, 33 as product of three
factors three, encouraging the pupil to write out the ‘in-between-step’
and to verbalize, e.g., (2.4)4 = 2.4 × 2.4 × 2.4 × 2.4, product of 4
factors 2.4
Module 1: Unit 1
29
Number sense, numerals and powers
(iv) set some drill and practice questions to consolidate the correct
concept (include fractions / decimals)
7. One (1) has only 1 factor while by definition prime numbers have exactly
2 factors. Two (2) meets this criteria for prime numbers as it has as
factors 1 and 2.
8. Below square number 9 and oblong number 10
9. 120 has 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Dot patterns 7 (or 14 if you take 2 × 60 and 60 × 2 as different).
N.B. 1 x120 = 120 × 1 are by convention NOT taken as representations
of rectangular numbers.
10. 121 has as factors 1, 11, 121. Only one dot pattern is possible 11 × 11.
11. Differences you could mention are
120 is even, 121 is odd
120 is a rectangular number (and NOT square), 121 is a square number
120 has 16 factors (even number), 121 only 3 (odd number)
120 uses three different digits, 121 only 2
12. In an, a is called the base, n the index and an is called a power.
13. See table in P3; square numbers end in 1, 4, 6, 5, 9 or 0 NEVER in 2.
Module 1: Unit 1
30
Number sense, numerals and powers
Unit 2: Sequences
Introduction to Unit 2
In Unit 1, you did some investigations. In most of them a pattern of numbers
emerged. In problem solving and investigations, looking for a pattern is
one of the strategies to consider. Patterns are very common around us. Think
of tile patterns, patterns on cloth, patterns used in decoration, patterns in
knitting, etc. These patterns can frequently be described using numbers.
Number patterns are therefore studied in mathematics and various techniques
have been developed to describe in general terms the pattern in a sequence of
numbers.
Purpose of Unit 2
The aim of this unit is to familiarize you with:
•
the method of differences to analyse sequences, and to find an expression
for the nth term of a sequence
•
the use of games to consolidate pupils’ concepts
•
the use of problem solving activities as a learning method
Objectives
When you have completed this unit you should be able to:
•
express the pattern in a sequence of numbers in words
•
use the method of differences to analyse sequences of numbers
•
use the method of differences to find a linear or quadratic expression for
the general term in a sequence
•
give simple examples of sequences with a clear pattern where the
expression for the nth term is not linear or quadratic
•
use problem solving and investigative activities in the classroom, related
to number patterns and sequences
•
justify the use of games in consolidation of concepts
•
evaluate the effectiveness of using games in pupils’ learning
Time
To study this unit will take you about 6 hours. Trying out and evaluating the
activities with your pupils in the class will be spread over the weeks you
have planned to cover the topic.
Module 1: Unit 2
31
Sequences
Unit 2: Sequences
Section A: Introduction
Patterns occur all around us. The calendar of May 1999 is shown.
S
M
T
W
T
F
S
1
8
15
22
29
2
3
4
5
6
7
9
10
11
12
13
14
16
17
18
19
20
21
23
24
25
26
27
28
30
31.
Take any square of four numbers. For example the ‘eleven’ square:
11
12
18
19
11 + 19 = 12 + 18 = 30 = 2 × 11 + 8
Repeat for other squares. Is the diagonal sum always equal to twice the
‘starting’ number of the square plus 8?
Here is another number pattern:
3 × 11 = 33
33 × 11 = 363
333 × 11 = 3663
3333 × 11 = 36663
How does the pattern continue? Can you explain?
Here you have a ‘growing’ dot-line pattern:
What would be the next dot-line pattern? Is there a relationship between the
number of dot and the number of line segments?
Module 1: Unit 2
32
Sequences
Reflection
Before you study this unit think about the following questions and write
down your responses.
•
Was a lot of attention paid to patterns and sequences while you were
studying?
•
Why yes or no? What was covered?
•
Do you consider the topic to be important or not? Justify your
answer.
•
Are you setting activities (and which) to your pupils related to
patterns and sequences? Why? or Why not?
•
Can you list situations in real life in which patterns / sequences are
involved?
While working through this unit refer back to your ‘reflection notes’.
Section B: Finding an nth term formula
You may have found it hard in the painted cube problem to come up with the
general expression for the n cm × n cm × n cm cube. In this section we look
at a technique that is frequently very useful when trying to find a general
algebraic expression for the nth term in a sequence. It is called the method
of differences. The method will help you in the next unit when we look at
figurative numbers or polygonal numbers. This are numbers that can be
illustrated with dot patterns of regular polygons.
The numbers in the sequence are called terms. 4 is the value of the first term
t1. We write t1 = 4. 8 is the value of the second term represented by t2, i.e.,
t2 = 8. The next term in the sequence is found by adding four to the previous
term. The ‘rule’ can be expressed as: start with 4 and continue to add four.
The general term in the sequence, the nth term is tn = 4n. Check that for
n = 1, 2, 3, 4, 5, ... you obtain the terms of the sequence.
Module 1: Unit 2
33
Sequences
Section C: Investigating the nth term of a sequence
In this section you are going to study how the pattern in a sequence relates to
the expression for the nth term.
Problem 1
Study the following sequence
3, 7, 11, 15, 19, ............., 4n – 1, ...
tn = 4n – 1
8, 15, 22, 29, 36, ............., 7n + 1 , ...
30, 25, 20, 15, 10, ........... , -5n + 35, ..
tn = 7n + 1
tn = -5n + 35
1. a) What are the (first) differences between consecutive terms in the first
sequence?
Answer _____
b) Express the ‘rule’ for the sequence in words:
________________________________________________
c) Where do you find the constant first difference in the rule for the
nth term of the sequence?
________________________________________________
2. a) What are the differences between consecutive terms in the second
sequence? Answer _____
b) Express the ‘rule’ for the sequence in words:
________________________________________________
c) Where do you find the constant first difference in the rule for the
nth term of the sequence?
________________________________________________
3. a) What are the differences between consecutive terms in the third
sequence?
Answer _____
b) Express the ‘rule’ for the sequence in words:
________________________________________________
c) Where do you find the constant first difference in the rule for the
nth term of the sequence?
________________________________________________
Check your answers at the end of this unit.
Module 1: Unit 2
34
Sequences
The constant first differences in each sequence we find in our rule as the
coefficient of n.
In the first sequence, the constant first differences were 4 and the rule for the
nth term started with 4n + ...
Similarly the constant first differences for the second sequence were 7, for
the third -5, the nth term started with respectively 7n ... and -5n....
How to find the rule?
Consider the following sequence:
Sequence
75
66
(First) differences
-9
57
-9
48
...
-9
We expect now the rule to be of the form:
-9n + (number)
As the first number is 75, taking n = 1
-9 × 1 + (number) = 75
Solving for (number): (number) = 75 + 9 = 84
Our rule for the nth term in the sequence is: tn = -9n + 84
Check that this gives the correct terms: substitute for n the values 1, 2, 3, ...
As our rule is of the form
y = px + q
(when y is plotted against x it will give us pair of corresponding points that,
if joined, would be on a straight line) we say that it is a linear rule.
The graph below illustrates the sequence given by tn = 2n – 3
n
1
2
3
4
5
...
-1
tn
1
3
5
7
...
Note that the graph consists of isolated points. Connecting the points would
be meaningless as there is no 1.5th or 1.6th term.
If the terms would be connected the points are on the line with equation
y = 2x – 3, which is a linear equation and hence the sequence is also said to
be a sequence with a linear rule.
Module 1: Unit 2
35
Sequences
Section D:
Using the method of differences to find the nth term
Looking at differences between consecutive terms is a method that can lead
to a formula for the nth term of a sequence. You are first going to study
linear rules for the nth term. Rules that are of the form tn = an + b (a ≠ 0)
Self mark exercise 1
Using the method of differences to find the nth term.
1. Find rules that generate the terms of the following sequences.
Write the rules as tn = ..........
Test the rule by finding the next few terms.
a) 6, 10, 14, 18, .......
tn = ___________
b) 2, 7, 12, 17, .........
tn = ___________
c) 52, 49, 46, 43, .........
tn = ___________
d) -5, -1, 3, 7, ...........
tn = ___________
e) -2, -4, -6, -8, .........
tn = ___________
f) 2.5, 3, 3.5, 4, 4.5, ....
tn = ___________
1 1 3
g) 6, 5 , 4 , 2 , ..........
4 2 4
tn = ___________
2. a. A sequence of equilateral triangles is made by placing them side by
side as illustrated.
Find the perimeter if the ‘train’ is one, two, three, four, ....... equilateral
triangles long. Complete the following table
Number of triangles
1
2
3
4
5
6
Perimeter
3
n
Self mark exercise 1 continued on next page
Module 1: Unit 2
36
Sequences
Self mark exercise 1 continued
2
b. Now consider a “train” using squares.
Find the perimeter if the ‘train’ is one, two, three, four, ....... squares
long. Complete the following table.
Number of squares
Perimeter
1
2
3
4
5
6
4
n
2
c. Now draw “trains” with regular pentagons, regular hexagons, regular
heptagons. Find the perimeter if the ‘train’ is one, two, three, four,
....... regular shapes long. Complete the following tables.
Record your results in a table.
Number of pentagons
Perimeter
1
2
3
4
5
6
n
Record your results in a table.
Number of hexagons
Perimeter
1
2
3
4
5
6
n
6
Self mark exercise 1 continued on next page
Module 1: Unit 2
37
Sequences
Self mark exercise 1 continued
Record your results in a table.
Number of heptagons
Perimeter
1
2
3
4
5
6
n
2
d. Try to generalize and find the perimeter of a train of n regular
polygons with p - sides. Summarizing your results in a table as
outlined below might be of help.
Regular shape
Perimeter of ‘train’ of
n shapes
equilateral triangle
square
pentagon
hexagon
heptagon
octagon
nonagon
decagon
p-gon
n+2
2n + 2
..
..
..
..
..
..
..
Check your answers at the end of this unit.
Module 1: Unit 2
38
Sequences
Section E: Using differences to find the next term in the
sequence
The method of differences can be extended and used to predict more
numbers in the sequence assuming the differences stay constant.
Working from the last row ( only 2’s appear there) upwards, we deduce that
after the 10 in the second row we get 12, and adding 33 + 12 in the first row,
gives the next term in the sequence 45.
Which number do you expect to follow after 45?
Did you find 14 in the second row and hence 45 + 14 = 59 as the next term
in the sequence.
Check that the next term in the sequence is 75.
The method of difference can help you to find more terms in a sequence
even if you do not know the expression for the nth term. In more difficult
situations it might only be the third, fourth or fifth differences that stay
constant.
In the next activity you will find how the constant difference relates to the
type of expression for the nth term and how you can use difference tables to
predict more terms of the sequence.
Module 1: Unit 2
39
Sequences
Problem 2: Investigating constant differences
1. The terms of a sequence are given by tn = 2n – 3
Write down the first 6 terms, and the differences. Which differences are
constant?
2. The terms of a sequence are given by tn = 3n2 – 2.
Write down the first 6 terms, and the differences. Which differences are
constant?
3. The terms of a sequence are given by tn = 2n3 – 2.
Write down the first 6 terms, and the differences. Which differences are
constant?
4. Looking at your results to question 1, 2 and 3 can you make a conjecture
as to how the difference method can be used to predict whether the
highest power of n in a rule is n, n2 or n3?
Check your conjecture by considering the sequences with nth term.
a) tn = 2n2 + 1
b) tn = n3 – n2
c) tn = n2 – 2n
d) tn = n3 – 5n
Write down the first 6 terms and the differences, until the differences
stay constant.
5. Write down some more expression for the nth term of a sequence.
Write down the first 6 terms and the difference, until the differences stay
constant. Check the conjecture you made.
6. Now complete the following statement relating the constant second or
third difference to the expression for the nth term tn
If the first differences in a sequence are constant then _______________
If the 2nd differences in a sequence are constant then _______________
If the 3rd differences in a sequence are constant then _______________
If the p-th differences in a sequence are constant then _______________
Check your answers at the end of this unit.
Module 1: Unit 2
40
Sequences
Self mark exercise 2
Using differences to find the next term in the sequence.
1. Use the difference method to find the next three terms of these
sequences
a) 2, 5, 10, 17, 26, ........
The next 3 terms are ____, ____, ____
b) 1, 7, 17, 31, 49, .......
The next 3 terms are ____, ____, ____
c) 0, 9, 24, 45, 72, .......
The next 3 terms are ____, ____, ____
d) 9, 15, 25, 39, 57, .......
The next 3 terms are ____, ____, ____
e) -4, 3, 22, 59, 120, ......
The next 3 terms are ____, ____, ____
f) 0, 7, 26, 63, 124, .......
The next 3 terms are ____, ____, ____
2. To make a 1 × 1 square using toothpicks you will need 4 toothpicks.
For a 2 by 2 square, made of 4 smaller squares, you will need 12
toothpicks.
a) How many toothpicks do you need to make a 3 by 3 square?
A 4 by 4 square?
b) Tabulate your results:
Side of the square
1
2
3
4
Number of toothpicks
4
12
Use the method of differences to find the number of toothpicks needed
for a 8 by 8 square.
Answer: ______________ toothpicks
3. Lines are drawn to intersect all other lines in each diagram and the
number of points of intersection are counted to form a sequence. Find
the next four terms in the sequence using the method of differences.
Check your answers at the end of this unit.
Module 1: Unit 2
41
Sequences
Unit 2: Practice activity
1. Choose one or more of the above four activities P1, P2 or Self mark
exercise 1 or 2 and work it out in more detail for use in your class as a
group task. Take into account the level of understanding of your pupils
(differentiate the activity if necessary for the different levels of pupils’
achievement).
2. Try your activity out in the classroom with a form in which the topic is to
be covered. Pupils should work together in groups of 4 to allow
discussion and actively constructing and reconstructing their knowledge.
As a group they should come up with ONE agreed response.
3. Write an evaluation of the lesson in which you presented the activity.
Some questions you might want to answer could be: What was / were the
objectives of your lesson? How did you structure the activity set to the
groups and why? Was is an open or closed task? What was the criteria
you used to form the groups? Did the activity work well? What was the
main strength or weakness of the activity ? How was the reaction of the
pupils? Did they like the activity? Did they like to work as a group?
Were there any problems you have to take into account when using a
similar method? What did you learn as a teacher from the lesson? Could
all pupils participate? Were your objective(s) attained? Was the timing
correct? What did you find out about pupils’ problem solving
techniques? What further activities are you planning to strengthen pupils’
problem solving techniques? Were you satisfied with the outcome of the
activity? Were the objectives achieved? How do you know? Was the
activity different from what you usually do in the class with your pupils?
How did it differ?
Present your assignment to your supervisor or study group for discussion.
Section F: Using the method of difference to find the
nth term if the second differences are constant
Not all relationships are linear. For most of our pupils in the age range
12 - 14 linear relationships can be mastered with confidence. However a few
challenging questions or problems should make them aware that the method
has its limitation as several relationships are not linear. High achievers can
be challenged with quadratic and even other type of relationships. In this
section you will learn that the method of differences cannot only help to
predict the next terms in a sequence, but also help you to find the rule for the
nth term. We will be looking at sequences with second differences constant.
You found in the previous activity that the expression for the nth term, tn, is
quadratic if second differences are constant.
Module 1: Unit 2
42
Sequences
Problem 3
Investigating quadratic rules for patterns.
1. A quadratic rule for a sequence is of the format tn = an2 + bn + c.
There is a relationship between the second row of differences and a.
Investigate to find this relationship.
Start your investigation by writing down the first six terms of the
following sequences and the two rows of differences.
tn = 2n2 + 3
tn = 3n2 + 5n
tn = n2 – 2n + 4 tn = 4n2 + n + 1
a) Can you make a conjecture? Can you find a relationship between the
differences and the coefficient a of n2?
2. Thato investigated further to find b and c in tn = an2 + bn + c. She wrote
as part of her investigation:
t1
= a × 12 + b × 1 + c =
t2
= a × 22 + b × 2 + c = 4a + 2b + c
a+ b+c
t 3 = a × 32 + b × 3 + c = 9a + 3b + c
t4
= a × 42 + b × 4 + c = 16a + 4b + c
Thato used the difference pattern to find quadratic rules for sequences.
She found a first from the 2nd differences line, then she found b from
the first difference line and finally she found c using the first term t1.
Consider the following sequences write down the first 6 terms, first
differences and second differences. Try to follow Thato’s method to
find the rule for the nth term in each sequence.
(i) 3, 4, 7, 12, ....
(ii) 0, 8, 22, 42, ...
(iii) 5, 11, 21, 35, ...
(iv) 8, 22, 42, 68, ...
3. Can you describe how quadratic rules for patterns can be found?
________________________________________________________
________________________________________________________
Check your answers at the end of this unit.
Module 1: Unit 2
43
Sequences
Finding the quadratic rule for the sequence
Worked example
Find the quadratic rule for the sequence:
-4, 3, 16, 35, .......
Terms
4
1st differences
3
7
2nd differences
16
13
6
35
19
6
The rule will be of the format tn = an2 + bn + c.
Then using the differences table in the previous investigation you find that:
2a = 6
a=3
3a + b = 7
9+b=7
a + b + c = -4
3 + (-2) + c = -4
b = -2
c = -5
The rule is tn = 3n2 – 2n – 5.
Sometimes the quadratic rule can be written down easily. Look at these
examples:
1, 4, 9, 16, 25 ...
tn = n2
2, 5, 10, 17, 26, ...
tn = n2 + 1
2, 8, 18, 32, 50, ...
tn = 2n2
If the rule cannot be written down easily then the above method can be used
to find the quadratic rule.
Module 1: Unit 2
44
Sequences
Self mark exercise 3
Finding the quadratic rule for the sequence.
1. Write down the rule for these sequences.
a) 10, 40, 90, 160, ...... tn = ____________
b) -9, -6, -1, 6, .....
tn = ____________
c) -2, 1, 6, 13, ......
tn = ____________
d) 21, 24, 29, 36, ....
tn = ____________
e) 3, 6, 11, 18, .....
tn = ____________
f) 5, 20, 45, 80, .....
tn = ____________
2. Find the quadratic rule for these sequences.
a) 3, 9, 19, 33, 51, .....
tn = ____________
b) 4, 11, 22, 37, 56, ..... tn = ____________
c) 5, 14, 27, 44, 65, ..... tn = ____________
d) 2, 13, 32, 59, 94, ..... tn = ____________
e) -2, 7, 22, 43, 70, ..... tn = ____________
3. Diagonals are drawn from each vertex of a convex polygon to every
other vertex, forming a sequence according to the number of sides of the
polygon. Find the next four terms in the sequence and find an expression
for the number of diagonals for an n-sided polygon.
Number of sides (n)
3
4
5
6
number of diagonals (d)
0
2
5
....
...
...
4. To make a 1 × 1 square using toothpicks you will need 4 toothpicks.
For a 2 by 2 square, made of 4 smaller squares you will need 12
toothpicks.
How many do you need to make a n by n square? Answer: __________
Check your answers at the end of this unit
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Section G: Games in the learning of mathematics
In this module, and others, games are used for the consolidation of concepts.
In mathematics pupils are to be able to recall instantly certain facts such as
the multiplication tables of 1 to 12, the first 10 powers of 2. Pupils are to
apply the four basic rules on whole numbers in the range 1 - 100 , ...... and
also the squares of the numbers 1 - 15. In the past books used to give pages
with drill and practice exercises to consolidate these, and other, concepts.
Nowadays emphasis is more on processes, on relational understanding, but
still consolidation is needed. For consolidation games, short challenges and
puzzles form an excellent medium.
Reflection
Stop reading for a moment to think about the use of games and puzzles
in the learning of mathematics.
Write down some of your thoughts.
Here are some questions that might guide your thinking.
When you were a pupil yourself at secondary school were games used in
the mathematics lesson?
Can you think of some games that would be useful in the learning of
mathematics?
What advantages can you see in using games as a mean to consolidate
concepts?
Do you see major disadvantages? Could these be overcome? If yes, how?
If no, why not?
Would you like to use (more) games in the mathematics lesson?
Now continue to read and compare with what you wrote down.
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Advantages of using games
Games can:
1. Develop a positive attitude towards mathematics.
Pupils need to experience: success, excitement, satisfaction, enthusiasm,
self-confidence, interest, enjoyment, active involvement. Few media are
more successful than games in providing all of these experiences.
2. Consolidate mathematical concepts, facts, vocabulary, notation.
Concepts, facts, vocabulary, mathematical notation need consolidation. The
traditional drill and practice episodes in a lesson are not, in general, very
motivating, while a game-like environment might consolidate the concepts in
a enjoyable and motivating way. In particular games can be used to
consolidate mathematical facts, vocabulary, notation.
3. Develop mental arithmetic skills.
Despite the fact that calculators are a tool in the learning of mathematics, this
does not dismiss the need for pupils to know basic number facts and
approximate sums, differences, products and quotients. Games can address
specifically these important aspects for natural numbers, integers, (decimal)
fractions and percent. For example a set of dominos can be designed to
consolidate the equivalence of fractions, conversion of fractions to percent or
the four basic operations with integers.
4. Develop strategic thinking.
Games can encourage pupils to devise winning strategies. Can the person
(i) playing first (ii) playing second always win? What is the ‘best’ move in a
given situation? For example the game of nought and crosses. Is there a
winning strategy?
5. Promote discussion between pupils and between teacher and pupil(s).
When certain games are used in the mathematics class as a learning activity,
there is a need to discussion: What mathematics did you learn? Is it a good
game? Can it be improved? are some of the questions to look at.
6. Encourage co-operation among pupils.
Some games requires a group playing against another group or a pair of
pupils against another pair. Such games can enhance co-operation among the
group or the pupils paired. They have to co-operate in order to ‘win’ the
game.
7. Contribute to the development of communication skills.
In a game the rules need to be explained. Pupils can explain the rules to
others orally, formulate rules in writing, describing strategies used to each
other—activities enhancing communication.
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8. Stimulate creativity and imagination.
If pupils have been playing a game for some time they can be encouraged to
make a similar new game for themselves or for younger brothers and sisters.
Pupils frequently devise new rules to add to or to replace the basic rules to
make the game more challenging to them once the basic rules have been
mastered. Pupils can also be challenged to devise variations and extensions
to the game. These activities call on pupils’ creativity and imagination.
9. Serve as a source for investigational work.
Games can form a source for investigational work by analysing the game and
answering questions such as: Is there a best move? Can the first player
always win? How many possible moves are possible? Is it a fair game? What
is the maximum score I can make? What would happen if ... ? This can lead
to looking at simpler cases first, tabulating results, making conjectures,
testing hypotheses i.e., leading to investigational work.
Below find examples of a game, and some challenges and puzzles that can
be used with pupils.
Pupils activity 1: Square routes
Game for 2 players.
Required: game board and 2 different coloured counters.
Rules: Pupils take turns in moving their counter from start to end, stepping
on tiles with square numbers only. They write down the route (sequence of
square numbers followed). Counters cannot be placed on the same field. The
pupil reaching END first wins the game. In the next game a pupil is not
allowed to take the same route as in the previous game!
After having played the game a number of times the two players pool the
routes followed and investigate how many different routes there are all
together.
The pair of players finding most routes are the ‘class pair winners’.
Objectives:
(i) consolidation of the squares of the whole numbers 1 – 15
(ii) develop the problem solving technique of systematic counting
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How many routes can you make from start to end stepping on tiles with
square numbers only?
NB: the game can be adapted for other type of numbers: prime numbers,
cubes, etc.
Pupils activity 2: Challenges
To the teacher:
Small problem solving and investigative type of questions allow teacher and
pupils to assess the level the concept has been understood, as the challenges
do not call on standard procedures or recall of facts only. Challenges can be
used as introduction to lessons to motivate pupils or as ‘gap fillers’ - to fill
some ‘left over’ minutes at the end of a lesson or topic. Challenges (pasted
on a notice board) can also be set at the beginning of the week as a contest
within a class or between classes. The underlying idea in all cases is to make
mathematics enjoyable and interesting. This requires that challenges should
be set for various levels of pupil achievement. Lower achievers need
challenges just as higher achievers do.
1. Square pairs
Objective: consolidation of the squares of the one digit numbers 1 to 9.
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Sequences
The diagram is made up of shapes like these
. The number
in the square, when squared, gives a two digit number. These are written
in the triangles. For example: with 9 in the square the shape will look
like
as the square of 9 is 81.
The column of the triangles add up to the totals shown: 13, 37, 36 and
18. The columns of the squares all add up to the same total. What is this
total?
2
a. Find a triangle ABC with each angle being a square number.
b. Find a quadrilateral with all angles being a square number of degrees.
3. In 112 = 121, both 11 and 121 are palindromic numbers, as they read the
same from left to right as from right to left. Is the square of a palindromic
number always palindromic? What about cubes, fourth powers. ... ?
Pupils activity 3: Puzzles
The distinction between a challenge and a puzzle is not very clear, but in
general one could say that in puzzles a ‘trial and error’ is suggested while in
challenges other strategies suggest themselves:
1. The square of 12 is 144. If you reverse the digits in both numbers
(reading from right to left) you get another true statement: the square of
21 is 441.
Find other two digit numbers with this property.
Is there a three digit number with this property?
2. Find digits A, B and C such that (BC)2 = ABC.
3. Find digits A, B and C such that AB × CA = ABCA.
Unit 2: Practice activity
1. Find or design (i) a game (ii) 5 challenging questions and (iii) 5 puzzles
related to the topic you are presently covering in the classroom.
2. Try out your game, challenges and puzzles in the classroom.
3. Write an evaluation of the lessons in which you presented the activities.
Some questions to consider in the evaluation: What where the
objectives? Did the activities succeed? If not what were the problems
encountered and how do you envisage to avoid these next time? What
was the reaction of the pupils? What did you learn from the activities
as a teacher? Do you consider the use of games, challenges and puzzles
an effective way to assist pupils in the learning of mathematics?
Justify your answer.
Present your assignment to your supervisor or study group for discussion.
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Sequences
Section H: Problem solving method in sequences
Some mathematicians say that mathematics is about describing the patterns
among numbers and shapes. It is true that in many investigations and
problem solving situations data are generated, starting from simple cases,
and placed in a sequence. Finding the pattern in the sequence is the next step
pupils have to master.
In this unit you have learned how the method of difference may be used for
finding more terms in a sequence and for finding an expression for the nth
term in some sequences (linear and quadratic). How can some of the things
you learned be presented to pupils in the classroom? Below you find three
suggestions for activities you might use in the classroom. All use a problem
solving, investigative approach. The cooperative model: think - pair - share
could be used. In this model the pupil first thinks and goes through (part) of
the question individually. Next the pupil pairs with a neighbour to compare
their findings and thinking. In the next step the agreed ideas of the pair are
shared with other pairs (or the whole class). Work through the following
activities yourself.
Pupils activity 1: Pascal’s triangle
Objective: using problem solving techniques. Start with a simpler case,
tabulate results, look for a pattern and extend the pattern.
a. Find the sum of the numbers in the 50th row
Number of row
1st
2nd
3rd
4th
5th
6th
Sum of all the numbers
1
2
4
.
.
.
50th
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b. Find the sum of all the numbers in a Pascal triangle with 50 rows.
Start with a Pascal triangle with 1 row. Next a triangle with 2 rows, etc.
Tabulate results and look for a pattern.
Number of rows
1
2
3
4
5
6
Sum of all the numbers
1
3=1+2=4–1
7 = 1+ 2 + 4 = 8 – 1
.
.
.
Pupils activity 2: Paths, patterns
Objective:
(i) Develop techniques for finding patterns in sequences and use the
pattern to predict more terms in the sequence.
(ii) Develop the technique of differences for linear relationships.
Pupils work in groups in order to compare and discuss individually
completed worksheets.
Three worksheets for pupils follow.
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Sequences
Worksheet 1
Paths
1. The diagram illustrates a path surrounding a flower bed paved with
square slabs.
a. If the flower bed is 1 m by 1 m the number of square slabs needed is ___.
b. If the flower bed is 2 m by 2 m the number of square slabs needed is ___.
c. On squared paper make a scale drawing of a path around a 3 m by 3 m
flower bed.
d. How many square slabs are needed for a 3 m by 3 m flower bed?
Needed are _______ square slabs.
e. What happens if the flower bed increases in size to 4 m by 4 m,
5 m by 5 m, etc. How many square slabs will be needed in each case?
Record your results in a table.
Size flower bed
Number of square slabs
1×1
8
2×2
3×3
4×4
5×5
6×6
f. What pattern do you notice in the last column?
g. Look at the differences between consecutive numbers of square slabs
and see if you will be able to extend your table.
How many slabs are needed for a 7 m by 7 m flower bed? ______
A 10 m by 10 m flower bed? _______
A 50 m by 50 m flower bed? ________
h. How many slabs are needed for an n by n flower bed? _________
Compare your answers with the students in your group. Discuss
differences and agree on a rule to find the number of square slabs
needed for any square flower bed.
Your rule:
The number of slabs needed for an n by n flower bed is _______________.
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Sequences
2a. What happens if the path is 2 slabs wide?
Draw, on square cm paper, the first three paths. The first one is
illustrated in the diagram.
Record your results in a table.
Size flower bed
Number of square slabs
1×1
24
2×2
3×3
4×4
5×5
6×6
b. Look at the differences between consecutive numbers of square slabs
and see if you will be able to extend your table.
How many slabs are needed for a 7 m by 7 m flower bed? ______
c. A 10 m by 10 m flower bed? _______
d. A 50 m by 50 m flower bed? ________
e. An n by n flower bed?
Compare your answers with the students in your group. Discuss
differences and agree on a rule to find the number of square slabs
needed for any square flower bed.
Your rule:
The number of slabs needed for an n by n flower bed is _______________ .
3a. What number of slabs will be needed if the path is 3 slabs wide?
Draw some of the paths surrounding the flower bed and tabulate your
results.
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Sequences
Record your results in a table.
Size flower bed
Number of square slabs
1×1
48
2×2
3×3
4×4
5×5
6×6
b. Look at the differences between consecutive numbers of square slabs
and see if you will be able to extend your table.
How many slabs are needed for a 7m by 7m flower bed? ______
c. A 10 m by 10 m flower bed? _______
d. A 50 m by 50 m flower bed? ________
e. An n by n meter flower bed?
Compare your answers with the students in your group. Discuss
differences and agree on a rule to find the number of square slabs
needed for any square flower bed.
Your rule:
The number of slabs needed for an n by n flower bed is _______________ .
4. Did you find the following results in your tables?
8, 12, 16, 20, 24, 28, 32, 36, ......................... 4n + 4 = 4(n + 1)
24, 32, 40, 48, 56, 64, 72, ............................ 8n + 16 = 8(n + 2)
48, 60, 72, 84, 96, 108, ............................... 12n + 36 = 12(n + 3)
Look at the number of square slabs needed for an n by n flower bed with
width 1, 2 and 3 slabs.
a. What do you think will be the number of slabs needed surrounding an n
by n flower bed if the path is 4 slabs wide ?
The number of square slabs needed is ____________.
b. What if the path is 5 slabs wide?
The number of square slabs needed is ____________.
c. What if the path is 6 slabs wide?
The number of square slabs needed is ____________.
d. What if the path is s slabs wide?
The number of square slabs needed is ____________.
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Sequences
Worksheet 2
Using differences to find linear rules of sequences
Introduction.
Perhaps you had some difficulties in finding the expressions for the number
of slabs surrounding a n metre by n metre flower bed.
You should have found: 4n + 4 = 4(n + 1), for a path of width 1 slab
8n + 16 = 8(n + 2), for a path of width 2 slabs
12n + 36 = 12(n + 3), for a path of width 3 slabs
16n + 64 = 16(n + 4), for a path of width 4 slabs
20n + 100 = 20(n + 5), for a path of width 5 slabs
...
...
4sn + 4s2 = 4s(n + s), for a path of width s slabs.
In a table.
Size of paths
Number of square slabs needed to
surround an n by n flower bed
1 slab wide
4(n + 1)
2 slabs wide
8(n + 2)
3 slabs wide
12(n + 3)
4 slabs wide
16(n + 4)
5 slabs wide
20(n + 5)
...
...
s slabs wide
4s(n + s)
Let’s look how these rules can be found.
You found the following sequences:
8, 12, 16, 20, 24, 28, 32, 36, ......................... 4n + 4 = 4(n + 1)
24, 32, 40, 48, 56, 64, 72, ............................. 8n + 16 = 8(n + 2)
48, 60, 72, 84, 96, 108, ................................ 12n + 36 = 12(n + 3)
1a. What is the difference between terms in the first sequence?
Answer _____
b. Where do you find this constant difference back in the rule for the
number of slabs needed to surround an n metre by n metre flower bed?
_________________________________________________________
c. What is the difference between terms in the second sequence?
Answer ______
d. Where do you find this constant difference back in the rule for the
number of slabs needed to surround an n metre by n metre flower bed?
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Sequences
e. What is the difference between terms in the third sequence?
Answer _____
f. Where do you find this constant difference back in the rule for the
number of slabs needed to surround an n meter by n meter flower bed?
_________________________________________________________
The constant differences seem to be important as we find them back in
our rule as the coefficient of n.
How to find the rule.
Consider the following sequence:
7
(First) differences
10
3
13
3
16
3
....
3
We expect now the rule to be of the form:
3n + (number)
As the first number is 7, taking for n = 1
3 × 1 + (number) = 7
Solving for (number): (number) = 7 – 3 = 4
Our rule for the nth term in the sequence is
tn = 3n + 4
Check that this gives the correct terms.
As our rule is of the form y = ax + b (when y is plotted against x it gives a
LINE graph) we say that tn = 3n + 4 is a linear rule.
Module 1: Unit 2
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Sequences
Worksheet 3
Using differences to find linear rules of sequences
1. Find rules that generate the terms of the following sequences. Write the
rules as tn = ________
Test the rule by finding the next few terms.
a. 3, 6, 9 ,12, ...
tn = ___________
b. 4, 9, 14, 19, ...
tn = ___________
c. 8, 14, 20, 26, ...
tn = ___________
d. -6, -2, 2, 6, ...
tn = ___________
e. -4, -6, -8, -10 ...
tn = ___________
2. The following illustrates a growing pattern of unit squares. In the first
pattern there are 4 unit squares, in the second 6, etc.
a. Complete this table:
pattern number
1
2
3
4
5
6
number of squares
4
6
.
.
.
.
b. How many unit squares are there in the 10th diagram if the pattern
continues?
c. How many unit squares are there in the 100th diagram if the pattern
continues?
d. How many unit squares are there in the 500th diagram if the pattern
continues?
e. How many unit squares s are there in the nth diagram if the pattern
continues?
s = ___________
f. Investigate the perimeter of each pattern and find an expression for
the perimeter (P) of the nth diagram.
3. A path is made using white and black tiles. The following diagram
illustrates paths of length 2, 3, 4 and 5 metres long.
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a. Draw the next two patterns on square cm paper.
b. Complete this table:
Path length
No. of black squares
No. of white squares
2
1
5
3
2
4
5
6
7
8
9
p
c. Find a formula relating the number of white tiles (W) and the number of
black tiles (B).
W = ______________
d. If there are 72 black tiles how many white tiles are needed?
e. If there are 209 white tiles in a path how many black tiles are there?
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Sequences
4. A ‘growing’ shape is made using tiles. The following diagram illustrates
the first four shapes.
a. Draw the next two patterns on square cm paper.
b. Complete this table:
Pattern
number
No. of square tiles
used
1
2
2
5
3
4
5
6
7
8
n
c. Find a formula relating the number of tiles (tn) needed to make the nth
pattern.
tn = ______________
d.
Module 1: Unit 2
If there are 78 tiles what pattern number can be made?
60
Sequences
5. Repeat question 4 a, b, and c for the next ‘growing’ tiles patterns.
6. Match sticks are used to make patterns. In the following patterns find:
a. Number of match sticks in the perimeter of the nth pattern.
b. How many match sticks are needed to make the nth pattern?
Record your results in a table.
Pattern number
1
No. of match sticks in the perimeter
4
No. of matches to make the pattern
Module 1: Unit 2
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2
3
n
7
Sequences
Pupils activity 3: Power sequences - The Tower of Hanoi & spread of
rumours
To the teacher:
On the pupils’ worksheet which follows, the problem is stated without
suggesting the steps. For weaker pupils you might have to adapt the
worksheet.
Objective: To develop and use the 4 Polya steps (1. understand the question
2. make a plan 3. carry out the plan 4. evaluate the solution) in problem
solving (Polya, 1957).
Notes to the teacher on Tower of Hanoi
Pose the problem first for a 2 disc situation. Be sure pupils understand how
the discs are to be moved. A model is needed. Use 2 paper circles of
different sizes; instead of the three needles use three pieces of paper to mark
the position of the first, second and third needle. Coins or jar lids could also
be used for the discs. Let pupils work on the 2 disc model and extend to a 3,
4 and 5 disc model. Some might find the 31 moves needed for the 5 discs
case.
Now tell the story of the tower. Ask pupils to guess and give an explanation
of their guess. Write down the guesses on the board. Suggest that they record
the results obtained so far and start looking for a pattern.
Tower of Hanoi
No. of discs
No. of moves
1
1
2
3
3
7
4
15
5
31
.
.
.
.
n
2n – 1
Pupils should recognize the same pattern as in the Pascal triangle and rumour
problem. It is the same pattern as the sum of the powers of 2. This needs
further exploring: how many times is each disc moved? Number the discs
and let pupils record the number of moves of each disc. Pupils are to be
encouraged to keep record of the moves of each disc. (The bottom disc will
make only one move, the disc above the bottom one will make 2 moves etc.
leading to the summation of the powers of two.)
The number of moves for the 64 golden discs in the temple complex is
264 – 1 = 18 446 744 073 709 551 615 moves and takes the same amount of
minutes; converted to years this gives about 3.5 × 1015 years, a long time
before the end of the world!
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Notes to the teacher about the spread of rumours
Pose the problem. Be sure pupils understand how the rumour is spread. You
may want to act it out in the classroom.
Ask pupils to guess and for an explanation of their guess. Write down the
guesses on the board. Having pupils guess before formally pursuing the
problem gets them interested in the problem: they have interest in the
outcome! (Did I guess right?)
Allow pupils to explore the problem in groups and ensure that the groups
understand the problem. Suggest that they keep record of their findings in an
organized way. If a group finishes early set some extension questions. (On
what day will more than 10 000 be told the rumour? On what day will more
than 100 000 have heard the rumour?)
After the groups have finished, call on different groups to discuss their
solution. A group will surely have used a pattern approach. Set up a table on
the chalk board. As you generate the table, ask pupils to predict the next
entry. Stop the pattern on day 12. Encourage pupils to look for a pattern by
examining the second column as the powers of 2. Pupils should also express
the generalization in words.
Day
No. of new people hearing
the rumour
Total number of people who
heard the rumour
1
2 = 21
3 = 4 – 1 = 22 – 1
2
4 = 22
7= 8 – 1= 23 – 1
3
4
8 = 23
16 = 24
15 = 16 – 1= 24 – 1
31= 32 – 1= 25 – 1
5
32 = ....
63 = ....
6
64 = ....
127 = ....
.
.
.
10
1024 = ....
2047 = ....
11
2048 =....
4095 = ....
12
4096 =....
8191= ....
.
.
.
.
.
n
Module 1: Unit 2
.
2n
63
.
2n + 1 – 1
Sequences
Worksheet
4
The Tower of Hanoi & Rumours
The Tower of Hanoi
The Tower of Hanoi is in the great temple of Benares in India. The story
goes that there are three diamond needles and sixty-four golden discs,
graduated in size. The largest disc is at the bottom, the smallest at the top on
one of the needles. (See the illustration for a tower with only five discs.)
The monks from the temple are to move the discs one at the time to another
needle. A larger disc can never be placed on a smaller disc. What is the
fewest number of moves necessary to move the entire stack of discs to
another needle, so that they are again arranged from largest at the bottom to
smallest at the top?
When all the discs have been transferred the world will come to an end. If
the monks started at the beginning of mankind, how close are we to having
the world end? (Assume, as the discs are very large and heavy, it takes one
minute per move).
Rumours
On the 5th of October a rumour is started that pupils in JSSs will have to pay
school fees as from the next school year. On the first day—October 5th—
one pupil tells the rumour to two other pupils with the instruction that each is
to spread the rumour to two more pupils the next day and that each of these
pupils is to repeat the rumour to two more pupils on the third day, and so on.
So on the first day three pupils heard the rumour, on the second day four
more will have heard, on the third day eight more will have heard and so on.
How many new pupils will be hear the rumour on the tenth day? The 50th
day? The nth day?
How long will it take before all pupils in your school have heard the
rumour ?
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Self mark exercise 4
1. A square paper is folded repeatedly. Investigate (i) the number of regions
formed and (ii) the area of the smallest region.
Check your answers at the end of this unit.
Unit 2, Assignment 3
1. Choose one of the pupils’ three activities suggested in section G. Work
the activity out in more detail so you can use it in your class with your
pupils. Take into account the level of understanding of your pupils
(differentiate the activity if necessary for the different levels of pupils’
achievement).
2. Try out your activity in the classroom.
3. Write an evaluation of the lesson in which you presented the activity.
Some questions you might want to answer could be: What were the
strengths and weaknesses? What needs improvement? How was the
reaction of the pupils? What did you learn as a teacher from the lesson?
Could all pupils participate? Were your objective(s) attained? Was the
timing correct? What did you find out about pupils’ investigative
abilities? What further activities are you planning to strengthen pupils’
problem solving and investigative work? Were you satisfied with the
outcome of the activity?
Present your assignment to your supervisor or study group for discussion.
Summary
This unit has covered the traditional subject of sequences. It has also
introduced the much more recent teaching concepts of:
•
teaching maths through games
•
having pupils work on problems jointly, then present how they did it to
the class (“Think Pair Share”)
When guided well by the teacher, approaches like those can have remarkable
benefits to the quality of learning in a mathematics classroom.
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Unit 2: Answers to self mark exercises
Problem 1
1a. 4
b. Rule: multiply the number of the term by 4 and subtract 1
c. coefficient of n
2a. 7
b. Rule: multiply the number of the term by seven and add 1
c. coefficient of n
3a. -5
b. Rule: multiply the number of the term by -5 and add 35
c. coefficient of n.
Self mark exercise 1
1a. tn = 4n + 2
e. tn = -2n
b. tn = 5n – 3
f. tn = 0.5n + 2
2a. tn = n + 2 b. tn = 2n + 2
d. tn = (p – 2)n + 2
c. tn = -3n + 55
3
3
g. tn = - n + 6
4
4
d. tn = 4n – 9
c. tn = 3n + 2, tn = 4n + 2, tn = 5n + 2
Problem 2
1. -1, 1, 3, 5, 7, 9
Constant 1st differences (of 2)
2.
2nd differences are constant (6)
3.
3rd differences are constant (12).
4/5/6. If the first difference is constant the rule starts with an.
tn = an + b (a ≠ 0).
If the second difference is constant the rule starts with an2.
tn = an2 + bn + c (a ≠ 0)
If the third difference is constant the rule starts with an3.
tn = an3 + bn2 + cn + d (a ≠ 0)
If the pth difference is constant the rule will start with anp (a ≠ 0).
Module 1: Unit 2
66
Sequences
Self mark exercise 2
1a. 37, 50, 65 b. 71, 97, 127 c. 105, 144, 189
e. 211, 338, 507
2a/b. 24, 40
d. 79, 105, 135
f. 215, 342, 511
For 8 × 8 needed 144
3. 0, 1, 3, 6, 10, 15, 21, 28, 36
Problem 3
1a. Conjecture (i) the second difference is constant (ii) the coefficient of n2
is half the second difference.
2. (i) n2 – 2n + 4 (ii) 3n2 – n – 2
(iii) 2n2 + 3
(iv) 3n2 + 5n
3. The general term is of the form tn = an2 + bn + c.
(i) Make a difference table
(ii) a is half the second difference
(iii) Use the first difference row to find b
(iv) Use the first term to find c
Self mark exercise 3
1a. tn = 10n2 b. tn = n2 – 10
e. tn = n2 + 2 f. tn = 5n2
c. tn = n2 – 3
2a. tn = 2n2 + 1 b. tn = 2n2 + n + 1
e. tn = 3n2 – 5
3. tn =
d. tn = n2 + 20
c. tn = 2n2 + 3n
d. tn = 4n2 – n – 1
1
n(n – 3)
2
4. tn = 2n(n + 1)
Module 1: Unit 2
67
Sequences
Self mark exercise 4
Number of folds
Module 1: Unit 2
Number of regions
Area of smallest region
0
1
1
2
2
4
..
..
..
n
2n
1
2n
68
1
1
2
1
4
Sequences
Unit 3: Polygonal numbers
Introduction to Unit 3
The number of dots required to make ‘growing’ dot patterns of (regular)
polygons forms a sequence. The numbers in the sequence and the numbers in
different sequences have multiple relationships which can be illustrated
geometrically and proved algebraically. Figurative numbers therefore form a
rich topic to explore both algebra and geometry.
Purpose of Unit 3
The aim of this unit is to:
•
apply the technique to analyse sequences developed in Unit 2 to
figurative numbers
•
link geometric and algebraic representation of numbers
•
offer a motivating context for working with algebra
Objectives
When you have completed this unit you should be able to:
•
list and illustrate the first five figurative numbers (triangular numbers,
square numbers, pentagonal numbers, ..., p-gonal numbers)
•
express in algebraic form the nth polygonal number using the method of
differences
•
illustate relationships among polygonal using geometrical dot patterns
•
prove algebraically relationships among polygonal numbers
•
set activities for pupils to learn about figurative numbers and their
relationships
Time
To study this unit will take you about four hours. Trying out and evaluating
the activities with your pupils in the class will be spread over the weeks you
have planned to cover the topic.
Module 1: Unit 3
69
Polygonal numbers
Unit 3: Polygonal numbers
Section A: Triangular, square, pentagonal,
hexagonal numbers
If we make dot patterns of ‘growing’ patterns of regular polygons, the
number of dots needed form sequences. These sequence of numbers are
called triangular numbers, square numbers, pentagonal numbers, hexagonal
numbers, etc. In general we could speak about p-gonal numbers, where p
represent the number of sides of the regular polygon (the polygons don’t
really have to be regular, but the patterns look more attractive when placed
in regular polygonal arrays).
Triangular numbers can be illustrated with a dot pattern giving a triangle.
Below are two formats. The first places the dots to form right-angled
isosceles triangles, the second pattern places the dots to form equiangular
triangles.
The triangular numbers are 1, 3, 6, 10, 15, 21, ...
Circles or squares can be used as well in illustrating the triangular numbers.
See the following diagram. Joining centers of ‘corner’ circles or squares
suggest the ‘triangular’ form.
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70
Polygonal numbers
Self mark exercise 1
1. The triangular numbers are 1, 3, 6, 10, 15, 21, ... List the next 3
triangular numbers.
2. Continue the dot patterns of the previous page by drawing the dot
patterns to represent the next 3 triangular numbers in each of the
sequences.
3. Express in words how the terms in the sequence are obtained.
4. t1 = 1, t2 = 3, t3 = 6, .... Can you find an expression for tn, the nth
triangular number in the sequence?
Check your answers at the end of this unit.
Did you use the method of differences or did you use another method?
Did you see that
2×3
3× 4
4×5
1× 2
t1 = 1 =
, t2 = 3 =
, t3 = 6 =
, t4 = 10 =
, ....
2
2
2
2
n(n + 1) 1
= n(n+1).
continuing the pattern gives as for tn =
2
2
Here is a sequence of figurative number patterns.
Looking at the pentagons you can see that the second pentagonal numbers
(5) are represented by the vertices of a pentagon with side one unit. The next
pentagonal number (12) is the previous pentagon and an enlargement of it by
factor 2 (the bigger pentagon with side of two units). In the next step the
original one unit sided pentagon is enlarged to a pentagon with side 3 units.
Hexagonal number dot patterns are made in a similar way.
Module 1: Unit 3
71
Polygonal numbers
Self mark exercise 2
1. Find expressions for the nth triangular, square, pentagonal number and
hexagonal number, drawing more dot patterns if needed to find more
terms of the sequence.
2. Express in words the rule to find the next triangular, square, pentagonal
and hexagonal number in a sequence once you found the first 5 or 6
terms.
3. Make dot patterns in the form of ‘growing’ heptagon and octagons to
represent heptagonal and octagonal numbers.
4. Using the method of differences find at least the first 6 terms in each
sequence of heptagonal and octagonal numbers.
5. Find an expression for the nth term in each of these sequences.
6. Place all your results in a table. Look for patterns in the rows and in the
columns.
Check your answers at the end of this unit.
Module 1: Unit 3
72
Polygonal numbers
The table you were to complete on the previous page should look as below.
Type of
number
Number
of sides
First terms in the
sequence
nth term: tn
triangular
p=3
1, 3, 6, 10, 15, 21,
28, ...
square
p=4
1, 4, 9, 16, 25, 36,
49, ...
pentagonal
p=5
1, 5, 12, 22, 35, 51,
70,...
hexagonal
p=6
1, 6, 15, 28, 45, 66,
91, ...
heptagonal
p=7
1, 7, 18, 34, 55, 81,
112, ...
octagonal
p=8
1, 8, 21, 40, 65, 96,
133, ...
1
n(n + 1)
2
1
n2 = n(2n + 0)
2
1
n(3n – 1)
2
1
n(4n – 2) = n (2n – 1)
2
1
n(5n – 3)
2
1
n(6n – 4) = n(3n – 2)
2
nonagonal
p=9
decagonal
p = 10
p-gonal
p
You are now to study the table in more detail to find the many patterns and
relationships in order to complete the next and last row: the data for the
nonagonal, decagonal and p-gonal numbers, the number of dots that can be
placed such that they form polygons with p sides. The following questions
should guide you.
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73
Polygonal numbers
Self mark exercise 3
1
a. What is the first term in each sequence?
b. What will be the first nonagonal, decagonal, p-gonal number? Write
them in the table.
2. Now look at the second term of each sequence: 3, 4, 5, 6, ..... and
compare also with the number of sides of the polygon in each case.
Assuming the pattern continues you can write down the second
nonagonal, decagonal number and p-gonal number.
3. Now move to the third term in each sequence. The third terms form the
sequence 6, 9, 12, 15, 18, 21 ... This allows you to find the third
nonagonal and decagonal number.
4. Lets look for the third p-gonal number. Here you must be careful because
in the sequence 6, 9, 12, 15, ... the general term would be 3n + 3
(Check!). But the first term 6 goes together with p = 3, the second term 9
goes with p = 4. Tabulated we have:
Complete the table.
5. Following the method as used in 3 and 4 find the first seven nonagonal,
decagonal and p-gonal numbers.
Self mark exercise 3 continued on next page
Module 1: Unit 3
74
Polygonal numbers
Self mark exercise 3 continued from previous page
6. Finally look at the last column, the nth term in each sequence. Tabulate
the values of p and the nth terms
All nth terms start with ......
Now look at the expression in the brackets. The coefficient of n is ... less
than the value of p.
Finally the last number in the brackets 1, 0, -1, -2, ... forms a sequence
with a constant difference of -1 so the general term starts with -p. The
format is -p + (...). To find what is on the dots in the brackets remember
that for p = 3 you must get 1.
Now you can find what is to be put on the place of the dots in the
brackets: ....
Using all this information you can now complete the table for the nth
term of the nonagonal numbers and decagonal numbers and also for the
nth term in the sequence of p-gonal numbers.
Check your answers at the end of this unit.
Module 1: Unit 3
75
Polygonal numbers
Section B: Relationships among polygonal numbers
The relationships among the polygonal numbers are numerous. Apart from
deriving relationships using algebra (deductive knowledge, illustrated below
under III) relationships can also be illustrated (inductive knowledge) using
dot, circular or square patterns. This is done below in part I and II.
Let’s look at a few relationships first before you try some on your own.
Statement:
The sum of two consecutive triangular numbers is a square number.
I. Inductive method using patterns
The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Tabulate sum of 2 consecutive triangular numbers:
1 + 3 = 4 = 22
3 + 6 = 9 = 32
6 + 10 = 16 = 42
10 + 15 = 25 = 52
...
...
...
Write down some more lines in this pattern.
The conjecture is: the sum of two consecutive triangular numbers is a
square number.
II. Inductive method using representations of the triangular numbers by
patterns of unit squares.
The diagrams above illustrate that two consecutive patterns representing
two consecutive triangular numbers can be arranged to form a square
pattern. The conjecture is therefore: the sum of two consecutive
triangular numbers is a square number.
Module 1: Unit 3
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Polygonal numbers
III. Algebraic proof
The nth triangular number is tn =
1
n(n + 1)
2
The next triangular number tn + 1 is obtained by replacing our n in the
previous line by (n + 1).
1
1
Hence tn + 1 = (n + 1) [(n + 1) + 1] = (n + 1)(n + 2)
2
2
The sum of the two consecutive triangular numbers is
1
1
common factor (n + 1) outside
tn + tn+1 = n(n + 1) + (n + 1)(n + 2)
2
2
the brackets
1
(n + 1) [n + n + 2]
2
1
= (n + 1)(2 n + 2)
2
1
= (n + 1) × 2 × (n + 1)
2
= (n + 1)2
=
simplifying
common factor 2 outside brackets
simplifying
a square number!
Statement: Every hexagonal number is a triangular number.
I. Inductive verification.
The hexagonal numbers are 1, 6, 15, 28, 45, ... ... hn =
1
n(4n – 2)
2
Comparing with the triangular numbers
1
n(n + 1)
2
you can note that the first hexagonal number = first triangular number
h1 = t1
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ....
tn =
the second hexagonal number = third triangular number
h2 = t3
the third hexagonal number = fifth triangular number
h3 = t5
h4 = t7
h5 = t9
Or generalizing hn = t2n–1. The nth hexagonal number is equal to the
(2n – 1)th triangular number. So each hexagonal number is a triangular
number.
Module 1: Unit 3
77
Polygonal numbers
II. Using patterns to illustrate that each hexagonal number equals to a
triangular number. (“Pushing in the circles from the bottom of the
hexagon”)
III. Algebraic proof
1
n(4n – 2)
(factorise)
2
1
= n × 2 × (2n – 1) (rearrange)
2
The nth hexagonal number is hn =
=
1
(2 n − 1)(2 n)
2
(writing as
triangular number)
=
1
(2 n − 1) [(2n – 1) + 1]
2
This is the expression for the (2n – 1)th triangular number.
Statement:
Every pentagonal number is one-third of a triangular number.
I. The statement is equivalent to saying that 3 times a pentagonal number is
a triangular number.
The pentagonal numbers are 1, 5, 12, 22, 35, ...
Three times these numbers give the sequence 3, 15, 36, 66, 105, ... these
are respectively the triangular numbers t2, t5, t8, t11, ...
The pattern is therefore
3 × p1 = t2
3 × p2 = t 5
3 × p3 = t 8
3 × p4 = t11
Or generalizing 3 × pn = t3n–1
Module 1: Unit 3
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Polygonal numbers
II. Geometrically flatten the ‘roof’ of each pentagonal number to make rows
of dots / circles to make an equilateral trapezoid (bucket shape). Three of
these fit together to make a triangle.
This is illustrated for the pentagonal number 12.
III. Algebraic proof
3×
1
1
1
n(3n – 1) = (3n − 1)(3n) = (3n − 1) [(3n – 1) + 1]
2
2
2
3 times the nth pentagonal number gives the (3n – 1)th triangular number.
Module 1: Unit 3
79
Polygonal numbers
Self mark exercise 4
1. Illustrate (inductively) and prove (deductively) that doubling any
triangular number (except the first) gives a rectangular number.
2. Find the first four dodecagonal (12 sided polygonal) numbers.
3. A p-gonal number is 40. Find p.
4. Illustrate and prove that pentagonal numbers (except the first) are the
sum of a triangular number and a square number.
5. The diagram illustrates the first three Hex numbers—centred hexagonal
numbers.
a. Find and illustrate the next two hex numbers.
b. Use the method of differences to find an expression for the nth hex
number.
6. The diagram illustrates the first three central square numbers.
a. Find and illustrate the next two central square numbers.
b. Use the method of differences to find an expression for the nth
central square number.
7. Prove and illustrate that each central square number is the sum of two
consecutive square numbers.
Check your answers at the end of this unit.
Module 1: Unit 3
80
Polygonal numbers
Section C: Activities to try in the classroom to enhance
understanding of figurative numbers and their
relationships
You have been studying some figurative numbers. You might have
discovered that the topic is inexhaustible and that many more questions
could be looked at. Figurative numbers, and number patterns in general, are
a rich ground for investigations. Here follows your assignment. You are
asked to try out some of the activities in the classroom. The activities
suggested are described below your assignment.
Unit 3: Practice activity
1. Try out the ‘handshake’ activity in the classroom (Worksheet 1).
2. Write an evaluation of the lesson in which you presented the activity.
Some questions you might want to answer could be: What were the
strengths and weaknesses? What needs improvement? How was the
reaction of the pupils? What did you learn as a teacher from the lesson?
Could all pupils participate? Were your objective(s) attained? Was the
timing correct? What did you find out about pupils’ investigative
abilities? What further activities are you planning to strengthen pupils’
problem solving and investigative work? Were you satisfied with the
outcome of the activity?
3. Use the information on figurative numbers presented in the previous
section B to develop investigative activities for your class. Evaluate the
activity.
Present your assignment to your supervisor or study group for discussion.
Pupils activity: Handshakes
Objectives of the activity:
•
to enhance problem solving strategies
•
to use acting out as a strategy to collect data
•
to relate a ‘real life’ situation to triangular numbers
To the teacher:
The triangular numbers appear in the ‘handshake’ activity. As we do not
expect pupils to be able to generate rules for the nth term of a sequence that
is not linear, assist pupils to find the general rule. For some groups of pupils,
continuing the pattern might be enough.
The pupils predict, collect data and compare their predictions with the
outcomes. The introduction to the class can be short. The predicted outcomes
are placed on the board.
Module 1: Unit 3
81
Polygonal numbers
Pupils work in groups of 4 – 6 counting the number of handshakes by
forming sub-groups of 2, 3, 4, .. pupils. Groups report back on different
strategies used in solving the problem: starting with simpler case, tabulating
data, looking for patterns.
The completed table will look as follows
Number of people
2
3
4
5
6
7
n
Number of handshakes 1
3
6
10
15
21
1
n(n – 1)
2
Explanations considered might be:
i) If there are 3 people the first person will shake hands with 2 people (and
sit down), the next person will shake hands with 1 person.
1
Total: 1 + 2 = 3 = (2 × 3)
2
1
For four persons: 1 + 2 + 3 = 6 = (3 × 4)
2
1
For five persons 1 + 2 + 3 + 4 = 10 = ( 4 × 5)
2
The pattern generalizes to:
For n persons 1+ 2 + 3 + ... + (n – 1) =
1
[(n – 1) × n]
2
(ii) One handshake is exchanged between 2 people. If there are 10 people
each will shake hands with nine others: 10 × 9, but now the handshakes
have been counted double—A with B and B with A have been counted
1
separately—so we must take half of the number: × 10 × 9.
2
1
This generalizes to × n × (n – 1).
2
Summary
Polygonal numbers provide a highly visual aid to learning many concepts of
sequences and other algebra. They also provide easily managed settings for
problem solving skills to be learned and used, both singly and in groups.
You should find that in every class where you teach using polygonal patterns
and numbers, more students grasp algebraic concepts more rapidly than you
otherwise would expect.
Module 1: Unit 3
82
Polygonal numbers
Worksheet
1
Handshakes
Problem.
Suppose each of you in class shakes the hand of each other, once. How
many handshakes are there in all?
Guess: ______________________________________________________
Should we do it and keep count of all handshakes? Could you remember
whose hand you had already shaken?
Answer: _____________________________________________________
Should we write out each possibility, then count them?
Answer: _____________________________________________________
Form groups of 2 pupils, 3 pupils, 4 pupils, etc. and count the number of
handshakes if all pupils in a group shakes the hand of each other, once.
Record your results in a table.
Number of people
2
3
4
5
6
7
n
Number of handshakes
Let’s go back to the first question of the worksheet:
Suppose each of you in class shakes the hand of each other, once. How
many handshakes are there in all?
Answer: _____________________________________________________
Describe how you found your answer.
_____________________________________________________________
_____________________________________________________________
How was your guess? Overestimated? Underestimated? Can you explain the
difference?
_____________________________________________________________
Challenge: Can you find the number of handshakes for n people?
Module 1: Unit 3
83
Polygonal numbers
Unit 3: Answers to self mark exercises
Self mark exercise 1
1. 28, 36, 45
3. add 2, add 3, add 4, .. to the previous term
4. tn =
1
n(n + 1)
2
Self mark exercise 2 & 3
Answer to question can be obtained from the table
Module 1: Unit 3
Type of
number
triangular
Number of First terms in
sides
the sequence
p=3
1, 3, 6, 10, 15, 21,
28, ...
square
p=4
1, 4, 9, 16, 25, 36,
49, ...
pentagonal
p=5
1, 5, 12, 22, 35,
51, 70,...
hexagonal
p=6
1, 6, 15, 28, 45,
66, 91, ...
heptagonal
p=7
1, 7, 18, 34, 55,
81, 112, ...
octagonal
p=8
1, 8, 21, 40, 65,
96, 133, ...
nonagonal
p=9
1, 9, 24, 46, 75,
111, 154, ...
decagonal
p = 10
1, 10, 27, 52, 85,
126, 175, ...
1
n (n + 1)
2
1
n2= n (2n + 0)
2
1
n (3n – 1)
2
1
n (4n – 2) = n (2n – 1)
2
1
n (5n – 3)
2
1
n (6n – 4) = n(3n – 2)
2
1
2 n(7n – 5)
1
2 n(8n – 6) = n(4n – 3)
p-gonal
p
1, p, 3p – 3,
6p – 8, 10p – 15
1
2 n[(p – 2)n – p + 4]
84
nth term: tn
Polygonal numbers
Self mark exercise 4
1. The nth triangular number is
1
n (n + 1).
2
Algebraic proof
2 × triangular number = 2 ×
1
n (n + 1) = n(n + 1). n(n + 1) is a
2
rectangular number.
1
2. tn = n(10n – 8) is the nth dodecagonal number.
2
3. p = 8 or p = 40
4. 1, 5, 12, 22, 35, 51, 70, ... are the pentagonal numbers
1, 1 + 4, 3 + 9, 6 + 16, 10 + 25, 15 + 36, 21 + 49, .. the pentagonal
numbers written as sum of a triangular number (1, 3, 6, 10, 15, 21, ...)
and a square number (4, 9, 16, 25, 36, 49, ...)
Drawing of patterns is left to you.
Algebraic proof:
1
1
1
1
n(3n – 1) = n[n – 1 + 2n] = n(n – 1) + n(2n) = tn-1 + sn (the
2
2
2
2
(n – 1)st triangular number plus the nth square number).
pn =
5b. nth hex number 3n2 – 3n + 1
6b. nth central number 2n2 – 2n + 1
7. 1, 5, 13, 25, 41, ... are the central numbers.
You can write them as 1, 1 + 4, 4 + 9, 9 + 16, 16 + 25, .. i.e., sum of two
square numbers.
Algebraic proof:
2n2 – 2n + 1 = n2 + n2 – 2n + 1 = n2 + (n – 1)2 being the sum of the nth
and (n – 1)st square number.
Module 1: Unit 3
85
Polygonal numbers
Unit 4: Rational and irrational numbers
Introduction to Unit 4
Counting and using numerals to represent numbers most likely belong to
some of the earliest activities of mankind. When we speak of our numeration
system we may be referring to either of two distinct ideas: the written
symbols or the number words (at times accompanied by gestures)
(Zaslavvsky, 1973). The structure of the number system we use now
developed through the ages and each new discovery to extend the system
initially met with great resistance; e.g., irrational numbers, negative numbers
and complex numbers all were initially unacceptable to ‘mainstream’
mathematics. This unit looks at part of the structure of the number system.
Purpose of Unit 4
The aim of this unit is to:
•
review and extend your knowledge on the structure of the number system
in order give you more confidence in teaching of numbers.
Objectives
When you have completed this unit you should be able to:
•
explain the structure of the number system (whole numbers, integers,
rational numbers, irrational numbers, real numbers, complex numbers)
•
explain the difference between rational and irrational numbers,
illustrating with examples
•
explain and illustrate the denseness of rational and irrational numbers on
the number line
•
explain why the integers are not dense on the number line
•
change recurring decimal fractions to rational numbers
•
set and justify activities for classroom use to enhance understanding of
the number system
Time
To study this unit will take you about 4 hours. Trying out and evaluating the
activities with your pupils in the class will be spread over the weeks you
have planned to cover the topic.
Module 1: Unit 4
86
Rational and irrational numbers
Unit 4: Rational and irrational numbers
Section A: The growing number system
The classification of numbers and the structure of the number system are far
from straightforward for pupils. The natural or counting numbers will be
accepted by pupils as a natural abstraction of the child’s experience with
counting of objects and ordering them. The extensions of the system are far
less straightforward and form a conceptual leap. Negative integers, rational
numbers, irrational numbers and complex numbers were created by people to
describe new situations.
At primary school pupils are introduced to the counting numbers 1, 2, 3, 4,
.... Next the 0 (no members in the set) is added. Historically the introduction
of a numeral to represent the absence of certain values in a place value
system was a great step forward. It dates back to the second century BC
(India) and was introduced to the West through the Arabic mathematician
Al-Khwarizmi (AD 680). Babylonian, Egyptian and Roman numerals have
no symbol for zero and are much harder to calculate with.
Later, rational numbers (referred to as fractions) were introduced as part of a
whole: a whole is divided into equal parts, each part is a fraction of the
whole. Decimals (finite such as 4.56, 0.08934 or recurring such as
0.33333..., 1.27272727... ) and percents are not extending the number system
of the rational numbers; they are different ways of representing rational
numbers.
For example:
2
1
1
= 0.4 = 40% or
= 0.3333... = 33 %
5
3
3
Showing the need to extend the number line with the negative integers (and
rational numbers) is readily accepted (to make it possible to solve equations
such as 2x + 5 = 1 and 2x + 3 = 2). Historically, negative numbers were
greatly resisted as ‘not real’. What does -3 apples mean? Nowadays pupils
are used to temperature scales including negative numbers (temperature
below zero on a Celsius scale).
To introduce irrational numbers, explaining that these cannot be expressed as
rational numbers, is too difficult for most children in the age range 12 - 16.
The discovery of irrational numbers is credited to Pythagoras who found that
the diagonal of a square is not a rational multiple of its side. For example
p
2 cannot be expressed as a rational number (with p and q integers and
q
q ≠ 0), its decimal expansion does not terminate or become periodic
(recurring).
At a higher level, algebra led to the introduction of complex numbers, in
order to find a solution to equations such as x2 = -1. The number
i = − 1 was invented to deal with this situation. The square root of negative
numbers are called ‘ imaginary’, however they are just as real as any other
number invented by the human mind. In the physical sense, 2 or π are not
real either, as we cannot measure with an infinite precision. Complex
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87
Rational and irrational numbers
numbers are generally introduced after the age of 16 as it requires a fair level
of abstract thinking.
Section B: Natural numbers, counting numbers and
whole numbers
Conventions differ as to what are considered to be natural numbers. Some
authors take the natural numbers to be the numbers represented by the
numerals 1, 2, 3, 4, , ... and the whole numbers as 0, 1, 2, ... . Other authors
do not distinguish between the whole numbers with 0 and without 0. In this
unit we will use the convention to denote the natural numbers (or counting
numbers) by N and take them to be 1, 2, 3, 4,... . The whole numbers will be
indicated with W and be the natural numbers N and 0;
i.e., W = {0, 1, 2, 3, 4, ...}
Section C: The number system: notation convention
used
The following notation will be used throughout,
N, the natural numbers, counting numbers or positive integers: 1, 2, 3, 4, ...
W, the whole numbers or non negative integers: 0, 1, 2, 3, 4, ....
Z the integers ..... -4, -3, -2, -1, 0, 1, 2, 3, ........ The Z is from the German
word for numbers: Zahlen.
ℜ the real numbers, rational numbers together with the irrational numbers.
Q, the rational numbers, all the numbers that can be written in the form
p
, with p and q integers and q ≠ 0. The Q is from quotient.
q
Irrational numbers are numbers that cannot be written as rational numbers;
their decimal format leads to an infinite non recurring decimal fraction.
Examples of irrational numbers are
Module 1: Unit 4
•
roots that cannot be expressed as a finite or recurring decimal:
5
, 5 + 3 , 6 11 , 3 2
5,
7
•
numerals involving π:
π, 3π + 2, π
•
many trigonometric values:
sin 20°, cos 45°
•
non-terminating and non-recurring decimals:
0.12112111211112111112....
0.123456789101112131415161718192021....
88
Rational and irrational numbers
Some numerals look like irrationals but are NOT!
For example the following are NOT irrational numbers:
121 which is a representation of the rational number 11.
12 ×
3 which is representing the rational number 6.
sin π a representation of the rational number 0.
2 24 ÷
6 representing the rational number 4.
Recurring decimal fractions are rational numbers. Here you can see how to
change a recurring fraction to its rational format.
Let us look at the recurring decimal 0.1212121212 .....
Let us call it rational format x, them x = 0.1212121212 .....
Because the recurring part consists of two digits we multiply by 100 and we
obtain
100x = 12.1212121212....
We had
x=
Subtracting,
0.1212121212 .....
99x = 12. Hence x =
12
4
=
99 33
Here is another example
2.213434343434...
Use the same procedure as above. Note that the recurring digits 34 are
preceded by a non-recurring part 2.21; we therefore multiply by 10 000.
if
then
Subtracting,
x=
2.213434343434 ..
10 000 x = 22134.34343434 ..
10 000 x – x = 22132.13
9999x =
2213213
100
x=
2213213
999900
C, the complex numbers, are any numbers that can be expressed in the form
a + ib, where a and b are real numbers and i = −1 . a is called the real part
of the complex number and ib the imaginary part. For example 3 – 2i, 4i,
3 (= 3 + 0i), 0.5 (= 0.5 + 0i) are examples of complex numbers.
Module 1: Unit 4
89
Rational and irrational numbers
The structure of the number system can be illustrated with the following
labelled Venn diagram. A typical example of each type of number is given in
the diagram.
Note the inclusive nature of the structure: all integers are rational numbers as
+4 and -3 can be expressed as a rational number. For example
−
−
+4 = 4 = 12 = 16 = ... and -3 = 3 = 27 = ...
−
−
1
3
1
4
9
At a higher level: whole numbers, rational numbers, real numbers are also
complex numbers with the imaginary part equal to 0.
Section D: Denseness of the numbers
Between two consecutive whole numbers, for example 11 and 12, NO other
whole number can be found. This is expressed by saying: the whole numbers
are NOT dense on the number line. If you represent the whole numbers by
points on the number line there are ‘gaps’, sections not filled with numbers.
The rational numbers on the other hand are dense on the number line:
between any two rational numbers you can always find infinitely many other
rational numbers (for example the average of the two rational numbers will
be between them).
11
12
23 34 59
and
you can find the rational numbers
,
,
, .... and
13
13
26 39 65
infinitely more.
Between
By now we could think that the number line is ‘full’, no gaps left to place
any other number. However there are still the irrationals! They are also
dense on the number line: between any two rational numbers infinitely many
other irrationals can be found and also between any two irrational numbers
infinitely many other irrational numbers can be found.
Module 1: Unit 4
90
Rational and irrational numbers
For example between
11
12
and
you can find the irrational numbers
13
13
1
1
1
73 ,
3,
19 , ...
10
2
10
Check this using your calculator.
Between 17 and 18 = 3 2 you can find the rational numbers 4.13, 4.15,
1
1
4.152, ... and the irrational numbers
70 ,
71 , ... Check this using your
2
2
calculator.
Cantor, a German mathematician living at the beginning of the 20th century
(he died in 1918), was able to prove that the vast majority of numbers are
irrational. The rational numbers are only a small collection of numbers
compared to the irrational numbers. He expressed this by introducing
different forms of ‘infinity’. The rational numbers are ‘countably infinite’
while the irrational numbers are ‘uncountably infinite’ as their order of
infinity is higher.
Section E: Proving that
2 is irrational
Remember that in self mark exercise 3 you showed that if n2 is an even
number, then also n is even. This we will use in the following proof by
contradiction.
A proof by contradiction starts by assuming that what is to be proved is not
true. In this case you assume that 2 is a rational number. From that
assumption you proceed to come to a statement that contradicts the
assumption. The conclusion is then that the assumption ( 2 being rational)
cannot be true, so that 2 must be irrational.
Carefully go over the following proof ensuring every step is clear and
understood. It is rather abstract algebraic reasoning (not appropriate for
pupils in the age range 12 - 14, with exception of the very high achievers).
You might have to read this section more than once.
m
where m and n are
n
m
whole numbers with no common factor other than 1. That is to say that
is
n
in its simplest form and cannot be simplified.
Assume
2 is rational so that it can be written as
If
2 =
m
n
then
2=
m2
n2
2n2 = m2
(squaring both sides)
(multiplying both sides by n2)
This tells you that 2n2 is divisible by 2, as it starts by a factor 2. So also m2,
which is equal to 2n2 must be divisible by 2 and be an even number.
Module 1: Unit 4
91
Rational and irrational numbers
Since m2 is even so also is m. (This you have proved in Unit 1, Section D.)
So it must be possible to write m = 2p for a certain whole number p.
Substituting m = 2p gives
2n2 = (2p)2
2n2 = 4p2
(expanding)
n2 = 2p2
(dividing both sides by 2)
Now the same argument applies again: Since n2 is even (as 2p2 is even) so
must be n.
You have now deduced that both m and n are even numbers. Both m and n,
being even, have a factor 2. This contradicts the assumption that m and n had
only 1 as common factor. Our assumption ( 2 being rational) must have
been incorrect. Hence 2 is NOT rational, that is 2 is irrational.
Self mark exercise 1
1. Are ‘positive integers’ and ‘non-negative integers’ referring to the same
set of numbers?
2. Why is it stated in the definition of the rational numbers
p
, that q ≠ 0?
q
3. What is the difference between fractions and rational numbers? Illustrate
with examples.
4. Do you know numbers that are NOT rational? What are they called?
Illustrate with examples.
5. How will you classify the percents?
6. To what number system(s) do the decimals belong?
7. Are the integers dense on the number line? Explain.
8. Classify the following numerals as integers, rational numbers, irrational
6 π +1
numbers and/or real numbers: -2.1, 25 , ,
, π 2 , sin 90°,
2
π
cos 45° ÷ 2 , 2.25 , 0, 1.3 (recurring fraction), 0.123123123.....,
1
(
6 )2
4
1
9. Find three numbers between 3 and 3 which are (i) rational
2
(ii) irrational.
10. Find three numbers between 10 and 11 which are (i) rational
(ii) irrational.
Self mark exercise 1 continued on next page
Module 1: Unit 4
92
Rational and irrational numbers
Self mark exercise 1 continued
11. p is a positive integer such that p = 23.4 (1 decimal point). Find the
value(s) of p and explain why p is irrational.
12. Investigate whether the following statements are true or false, justifying
your answers, illustrating with examples and non examples.
a. The sum of any two irrational numbers is an irrational number.
b. The sum of any rational and any irrational number is an irrational
number.
c. The product of any two irrational numbers is an irrational number.
d. The product of any rational and any irrational numbers is an
irrational number.
13. Write the following recurring decimals as fractions: 0.7, 0.34, 0.25,
0.341, 0.9
14. Prove that
3 is an irrational number.
15. Is 4 irrational? What happens if you attempt to prove by contradiction
that 4 is irrational?
16. Give two examples of irrational numbers p and q (different from each
p
other) such that is a rational number.
q
17. State for the following types of numbers whether they are always
rational, possibly rational or never rational. Justify your answer.
a. finite decimals
b. infinite decimals
c. square root of whole numbers
18. A pupil in an attempt to define an ‘irrational number’ said “An irrational
number is a number which, in its decimal form, goes on and on.”
a. What example would you present to the pupil to show that the
definition is not correct?
b. How is the definition to be ‘refined’ to make it correct?
19. π is an irrational number. Various rational approximations to π are in
use. Using your calculator approximation of π find the percent error
when using the following historical rational approximations.
22
1
(i)
(ii) 3
(iii) 10
7
8
Check your answers at the end of this unit.
Module 1: Unit 4
93
Rational and irrational numbers
Section F: Rational and irrational numbers in the
classroom
What do you want pupils in the age range 12 - 14 to learn about the number
system? In the sections above: what was new to you? What in the above
section could be used in the classroom for all / some pupils? What do you
consider to be definitely inappropriate to present to pupils and why? (“It is
difficult” is generally NOT a good argument as many pupils can handle
‘difficult’ concepts provided they are presented at their level with sufficient
guidance from the teacher as facilitator.)
Reflection
1. Write down on a piece of paper what you consider can be learned by
pupils in the age range 12 - 14 about the number system.
2. For each of the concepts listed in the above question 1, suggest a
teaching method that can be used to assist pupils’ learning.
Did you include the following?
1. Given any number, pupils should be able to identify it as belonging to the
natural numbers, integers, rational numbers, irrational numbers.
2. Pupils should be able to give examples and non-examples of numbers
belonging to the natural numbers, integers, rational numbers, irrational
numbers.
3. Pupils should be able to relate the rational numbers to either terminating
decimal numbers or non-terminating recurring decimal numbers.
4. Pupils should be able to find rational / irrational numbers between two
given rational / irrational numbers using a calculator.
5. Pupils should be able to explain that the rational numbers are “dense on
the number line”: between any two rational numbers there exist infinitely
many other rational numbers.
6. Pupils should be able to identify non-terminating, non-repeating
decimals with irrational numbers and be able to give examples.
7. Pupils should be aware of the ‘inclusive’ nature of the number system
e.g., all integers are rational numbers, all natural numbers are integers,
etc.
8. Pupils should be able to state the rational equivalent of some simple
1
2
1
recurring decimal fractions, e.g., 0.3 = , 0.6 = , 0.1 = .
3
3
9
9. Pupils should be able to give π as an example of an irrational number and
22
show awareness of 3.14,
being approximations to π by appropriate
7
use of ≈ symbol.
Module 1: Unit 4
94
Rational and irrational numbers
Some suggestions for pupils activities
1. Investigate rational numbers and terminating decimals. Which rational
numbers have a terminating decimal format? Expected outcome: all
fractions with denominators, when factorised, of the form 2p × 5q
p and q being whole numbers (0, 1, 2, ...)
2. Investigate patterns in recurring decimals e.g., the recurring decimal
1 2 3 4 5
6
format for , , , , and . Expected outcome: pupils discover that
7 7 7 7 7
7
in all decimal formats the very same digits (142857) appear but start at
different digits.
3. Investigate non-terminating, non-recurring decimals. For example:
0.797797779777797777797777779...
How many digits precede the one hundredth 9?
0.12112211122211112222...
Among the first 1000 digits how many are 1, how many 2?
Expected outcome: enhancing understanding of non-terminating, nonrecurring decimals and enhancing of problem solving strategies.
Unit 4, Assignment 1
1. Using the objectives for pupils, the ideas you learned yourself in this unit
and the suggestions above develop an activity based lesson on (i) number
systems: classification of numbers and (ii) irrational numbers. Pupils
should be able, in their groups, to learn about number systems and
irrational numbers.
2. Try out your lessons with your pupils.
3. Write an evaluation of the lessons. Some questions you might want to
answer could be: What were the strengths and weaknesses? What needs
improvement? How was the reaction of the pupils? What did your learn
as a teacher from the lesson? Could all pupils participate? Were your
objective(s) attained? Was the timing correct? What did you find out
about pupils’ investigative abilities? What further activities are you
planning to strengthen pupils’ understanding of the number system and
more especially about the nature of irrational numbers. Were you
satisfied with the outcome of the activity?
Present your assignment to your supervisor or study group for discussion.
Module 1: Unit 4
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Rational and irrational numbers
Summary
You have come to the end of the first module. It is expected that you have
reviewed and increased your knowledge of number systems, classes of
numbers, number sequences and the multiple relationships among numbers.
Apart from having increased your own knowledge, you should have worked
with teaching methods that might not have been part of your practice before
you started with this module. The experience in the classroom with a pupil
centred approach using, among others, games, challenging questions and
problem solving / investigation activities should have widened your
classroom practice and methods. The task of a teacher is in the first place to
create an environment for the pupils in which they can learn by doing
mathematics. Your final module assignment is to assess the progress you
have made.
Module 1, Assignment
1. Read again what you wrote at the start of this module “My teaching
approach and methods”. If you were to write now again on this would
there be any changes? Justify the changes in your teaching approach and
practice or justify why there is no need for you to make any changes.
2. Pupils learn by doing mathematics (practical activities, problem solving,
investigations) and discussion with each other and the teacher on how
they work the tasks set. Design a sequence of lessons on “Numbers” to
enhance problem solving skills in your pupils. Write out the lesson plans
and worksheets, taking into account the wide range of achievement
levels of your pupils in your class. Each and every pupil should be able
to develop problem solving skills at their own level.
3. Write an evaluation of the lessons. Some questions you might want to
answer could be: What were the strengths and weaknesses? What needs
improvement? How was the reaction of the pupils? What did you learn
as a teacher from the lessons? Could all pupils participate? Were your
objective(s) attained? Was the timing correct? What did you find out
about pupils’ investigative abilities? What further activities are you
planning to strengthen pupils’ problem solving skills?
4. Investigate: How many factors do the different classes of numbers you
encountered in this module have (square numbers, cubes, figurative
numbers, ...)? Which numbers have a square number of factors? Which
numbers have a prime number of factors?
Present your assignment to your supervisor or study group for discussion.
Module 1: Unit 4
96
Rational and irrational numbers
Unit 4: Answers to self mark exercises
Self mark exercise 1
1. Positive integers are 1, 2, 3, 4, 5, ...
Non-negative integers are 0, 1, 2, 3, 4, 5, ...
2. Dividing by 0 is undefined.
3. Rational numbers are numbers that can be expressed in the form
where a and b are integers and b ≠ 0.
Fractions are numbers that can be expressed in the form
are real numbers and b ≠ 0.
a
b
a
where a and b
b
All rational numbers are fractions (as the real numbers include the
rational numbers) but not all fractions are rational numbers. For example,
π
2
,
are fractions but NOT rational numbers.
2
2
4. (i) irrational numbers such as
such as -2
3 , e, π, sin 60° (ii) complex numbers
5. Percents are rational numbers. % represents the denominator 100.
6. Terminating and recurring decimals are rational numbers, nonterminating and non-recurring decimals are irrational. See for example
the text.
7. No, as between two consecutive integers you cannot find another integer.
8. Rational are:
-2.1 = - 21 ,
10
2.25 = 1.5, cos 45° ÷
0.123123123.....=
Integers are:
2 =
1
1
, 1.3 = 1 ,
2
3
3
41 1
,(
6 )2 =
8
333 4
25 = 5,
6
= 3, sin 90° = 1, 0 integer
2
Real / irrational are
π+1
π2
π
N.B. integers are also rational numbers and real numbers; rational
numbers are also real numbers.
9. For example: (i) Rational 3
1
1 1 1
,3 ,3 ,3
(ii) Irrational 10 ,
10 5 3 4
11 , 3π
Module 1: Unit 4
97
Rational and irrational numbers
10. (i) Rational 3.2, 3.24, 3.3 (ii) Irrational
3
1
1
42 ,
95 , 1020
2
3
11. 546, 547, 548 or 549, none of these being a perfect square.
12 a. not true, e.g., 2 + (- 2 ) = 0, if restricted to positive irrationals it is
a true statement.
b. true, e.g.,
1
+
2
c. not true, e.g.,
2 is irrational
2 ×
2 = 2,
3×
12 = 6
0
× 2 = 0, if 0 is excluded from the rational numbers
2
the statement is true.
d. not true, e.g.,
34
7
23
169
, 0.34 =
, 0.25 =
, 0,341 =
, 0.9 = 1
99
9
90
495
m
14. Assume 3 is rational so that it can be written as
where m and n are
n
whole numbers with no common factor other than 1. That is to say that
m
is in its simplest form and cannot be simplified.
n
m
3=
If
n
13. 0.7 =
then
3=
m2
n2
(squaring both sides)
3n2 = m2 (multiplying both sides by n2)
This tells you that 3n2 is divisible by 3, as it starts by a factor 3. So also
m2, which is equal to 3n2 must be divisible by 3.
Since m2 is divisible by 3 so also is m.
So it must be possible to write m = 3p for a certain whole number p.
Substituting m = 3p gives
3n2 = (3p)2
3n2 = 9p2 (expanding)
n2 = 3p2 (dividing both sides by 3)
Now the same argument applies again: Since n2 is divisible by 3 (as 3p2
is divisible by 3) so must be n.
You have now deduced that both m and n are divisible by 3. Both m and
n, being divisible by 3, have a factor 3 in common. This contradicts the
assumption that m and n had only as common factor 1. Our assumption
( 3 being rational) must have been incorrect. Hence 3 is NOT
rational, that is 3 is irrational.
Module 1: Unit 4
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Rational and irrational numbers
m
where m and n are
n
whole numbers with no common factor other than 1. That is to say that
m
is in its simplest form and cannot be simplified.
n
m
4 =
If
n
m2
Then 4 = 2
(squaring both sides)
n
4n2 = m2
(multiplying both sides by n2)
15. Assume
4 is rational so that it can be written as
This tells you that 4n2 is divisible by 4, as it starts with a factor 4. So
also m2, which is equal to 4n2 must be divisible by 4.
At this point the proof breaks down as we cannot conclude:
Since m2 is divisible by 4 so also is m, because m could be even and then
m2 has a factor 4. We can only conclude that m is even—NOT
necessarily a multiple of 4.
If you would continue now:
So it must be possible to write m = 2p for a certain whole number p.
Substituting m = 2p gives
4n2 = (2p)2
4n2 = 4p2 (expanding)
n2 = p2 (dividing both sides by 4)
Which does not lead to a contradiction.
16.
2
4 2π
2 12
= , 3π =
3
3
3 3
17 a. always rational
b. possibly rational (if recurring)
c. possibly rational, e.g., 4
18 a. 0.333... =
1
3
b. to be added: ‘and is not recurring’
19. (i) 0.04% (2 dp) (below)
(ii) 0.53% (2 dp) (above)
(iii) 0.66% (2 dp) (below)
Module 1: Unit 4
99
Rational and irrational numbers
References
Cockcroft, W.H. (1982) Mathematics Counts. London HMSO.
Polya , G. (1957) How To Solve It. A New Aspect of Mathematical Aspect.
2nd Edition. Princeton University Press. ISBN 0691 0235 65
Zaslavsky, C. (1979) Africa Counts. Lawrence Hill Books.
ISBN 1556 5207 51
Additional References
The following books have been used in developing this module and contain
more ideas for the classroom.
UB-Inset, University of Botswana, Patterns and Sequences, 1997
NCTM, Developing Number Sense, 1991, ISBN 087 353 3224
NCTM, Patterns and functions, 1991, ISBN 087 353 3240
ATM, Numbers Everywhere, 1972, ISBN 090 009 5172
Kirby D, Games in the teaching and learning of mathematics, 1992,
ISBN 052 142 3201
NCTM, Understanding rational numbers and properties, 1994,
ISBN 087 353 3259
NCC, Mathematics programmes of study, 1992, ISBN 187 267 6898
NCTM, Teaching and learning of algorithms in school mathematics, 1998,
ISBN 087 353 4409
London, R., Nonroutine Problems: Doing Mathematics, 1989,
ISBN 093 976 5306
Mathematical Challenges, Jason Publications, 1989.
Spotline on Understanding: Strategies for solving problems, Jason
Publications, 1996, ISBN 093 976 65691
Further Reading
The following series of books is highly recommended to use with the text in
this module. The Maths in Action Books (book 3, book 4 and book 5 to be
published in 2000) with the accompanying Teacher’s Files are based on a
constructive approach to teaching and learning. The student books allow
differentiation within the classroom. The books are activity based: learning
by doing and discovery. The Teacher’s File contains materials which may be
photocopied: worksheets, games and additional exercises for students.
OUP/Educational Book Service, Maths in Action Book 1
ISBN 019 571776 7, P92
Module 1
100
References
OUP/EducationalBook Service, Maths in Action Book 2
ISBN 019 ... (published 1999)
OUP/Educational Book Service, Maths in Action Teacher’s File Book 1
ISBN 019 ...(published 1999)
OUP/Educational Book Service, Maths in Action Teacher’s File Book 2
ISBN 019 … (published 1999)
Module 1
101
References
Glossary
Base
in n p , n is called the base
Complex numbers
any number that can be expressed as a + bi where
a and b are real and i = -1
Consecutive numbers
two whole numbers following each other, e.g.,
n and n + 1 are two consecutive whole numbers,
2n and 2n + 2 are two consecutive even numbers
Counting numbers
see whole numbers
Cube numbers
natural numbers that can be represented by a cube
pattern of unit cubes; numbers of the form n3
Directed numbers
see integers
Factors
factors of a natural number N are natural numbers
that divide into N
Figurative numbers
natural numbers that can be represented in a
geometric dot pattern e.g., a triangle, square,
rectangle, pentagon, etc.
Fraction
any number that can be expressed as
p
, where p
q
and q are real and q ≠ 0
Module 1
Index
in the expression n p , p is the index of the
power n p
Integers
the positive, negative and 0 whole numbers
Irrational numbers
numbers that are not rational
Linear expression
expressions of the form ax + b
Multiples
multiples of a natural number N are the numbers
pN where p is a natural number
Natural numbers
1, 2, 3, 4, ... Also called the Counting numbers,
they do not include zero or the negative numbers.
Number sense
a feel for size of numbers and having referent
objects related to numbers, being aware of
multiple relationships among numbers and the
effect of operations on numbers
Numeral
representation of a number
Oblong numbers
rectangular numbers that are not square numbers
Polynomial numbers
natural numbers that can be represented in a
polygonal dot pattern
102
Glossary
Power
expression of the form n p . If p is natural number
it is the product p factors n.
Quadratic expression
expression of the form ax2 + bx + c
Rational numbers
any number that can be expressed as
p
(with p
q
and q integers and q ≠ 0)
Module 1
Real numbers
set of rational and irrational numbers
Recurring decimal
decimal in which one or a string of digits keeps
on repeating e.g., 0.3333..., 0.123 434 343 4 ...
Rectangular numbers
natural numbers with at least three factors and
hence can be represented in a rectangular dot
pattern
Square numbers
natural numbers that can be represented by a
square dot pattern 1, 4, 9, 16, ...
Triangular numbers
natural numbers that can be represented by
triangular dot patterns 1, 3, 6, 10, ...
Whole numbers
the numbers 0, 1, 2, 3, 4, ...
103
Glossary
Appendix 1
Module 1
104
Appendices
Appendix 2
Module 1
105
Appendices
Appendix 3
Module 1
106
Appendices
Appendix 4
Module 1
107
Appendices
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