Book 3: Getting Started
Getting Started
Getting Started
Introduction
The purpose of this booklet is to provide guidelines for organising your
assessment information and developing your classroom programme.
NumPA provides a wealth of diagnostic assessment information about
students. There needs to be an appropriate link between this data and the
learning experiences you provide for your students through the classroom
programme. Your planning should also meet the achievement aims and
objectives of Mathematics in the New Zealand Curriculum (MiNZC).
This book consists of three main parts:
• Organising your NumPA information
• Developing the number component of your classroom programme
• Choosing from useful formats to simplify the task of planning and
ongoing assessment.
Successful numeracy teachers emphasise connections. These connections
can be between the ideas in mathematics, between mathematics and other
learning areas, and between mathematics and the real-life experiences of
students. Book 1: The Number Framework and the NumPA data give a direct
window into students’ number knowledge and strategy application. This
provides clear direction for teaching and allows you to choose appropriate
learning experiences for your students.
Organising Test Information
There are three clear steps to organising your NumPA information:
1. Analysis of knowledge hot spots
2. Assigning strategy stages to students
3. Grouping for instruction.
Analysis of Hot Spots
Examining the Knowledge section of the NumPA will give you a collective
picture of your students’ number knowledge as well as highlighting areas
of strength and weakness for individual students. From this data, you will
be able to identify hot spots, which are problematic areas of knowledge
that are common to many students in your class. These hot spots should
form the core of your whole class knowledge teaching at the warm-up
phase of each lesson.
You should choose suitable Knowledge Learning Activities (from
Book 4: Teaching Number Knowledge) to target these hot spots during your
warm-up or group teaching time (refer to the models for daily numeracy
lessons on pages 4–5 of this book). Pages 18 and 19 provide possible
formats for recording knowledge hot spots in your class grouping sheet
for NumPA.
It is quite common to see a mismatch between a student’s strategy stage
and their knowledge. This suggests that either the student has more
knowledge than they are able to use or the student has powerful strategies
but lacks the knowledge to apply them to different numbers. This is why
the identification of knowledge hot spots is crucial for quality teaching.
1
Getting Started
Assigning a Strategy Stage
Your initial grouping of students should be by their dominant strategy
stage. Stages 0–6 describe the transition from counting to part-whole
addition and subtraction strategies. Responses to the Operational Strategy
Windows of NumPA will dictate which strategy stage to assign to your
students. Addition and subtraction form the basis for identifying strategy
stages 0–6 because students need to be able to add and subtract fluently
before they can derive multiplication and division facts independently.
For example, a student might be stage 5 for addition and subtraction and
stage 6 for multiplication and division. You should assign the student as
stage 5 and focus initially on addition and subtraction. This student
understands how to derive multiplication facts but lacks the addition and
subtraction strategies to do so efficiently.
Strategy stage 7 is assigned to students who have reached stage 6 for
addition and subtraction and have demonstrated high-level strategies for
multiplication or division (Advanced Multiplicative stage).
Strategy stage 8 is assigned to students who have reached stage 6 for
addition and subtraction, stage 7 for multiplication and division, and have
demonstrated high-level strategies for solving problems with fractions,
ratios, and proportions (Advanced Proportional stage).
Grouping for Instruction
Group your students for instruction by their assigned strategy stages. You
can record this information onto a class grouping sheet for NumPA. Pages
18 and 19 provide a format for this purpose and a hypothetical example is
provided on page 20. Choose the page that has the relevant strategy stages
for your class and write each student’s name in the relevant column. The
progress of a student can be shown by highlighting their name through to
their new stage. This constitutes a useful record of achievement.
In situations where there is a wide range of strategy stages within your
class, you need to consider either cross-grouping between classes or
compromising by putting together students from close strategy stages to
reduce the number of teaching groups. For example, you may have groups
at these strategy stages:
Stage 2
Counting from One on Materials
Stage 3
Counting from One by Imaging
Stage 4
Advanced Counting
Stage 5
Early Additive Part-Whole
2
Getting Started
Students at stages 3 and 4 could be combined, provided that the advanced
counting students are only counting on for simple addition problems.
Such a combined group could be termed “early advanced counters”.
Students from stages 4 and 5 could be combined provided that the
advanced counting students are imaging counting on to solve addition
and subtraction problems. The main obstacle to these students doing
partitioning is likely to be a lack of knowledge, for example, of the “teen”
code.
Another grouping scenario may be that most of the students are at one
strategy stage. In this case, it may be necessary to form two or more
groups, according to both the knowledge and the strategies they
demonstrate.
For example, you have 20 students grouped at stage 4, advanced counting.
Eight students in the group solve problems by using counting on with
their fingers, and 12 students solve problems of counting on and back by
imaging in their heads. There is a clear division between these students,
and this enables you to confidently divide them into two groups. The
students using counting on by imaging are more ready for the transition
to part-whole thinking than their counting-on-with-fingers counterparts.
Finger counters are likely to need work on imaging before the part-whole
transition is successful.
Developing Your Number Programme
Number lessons can have many structures. These structures include:
whole-class knowledge lessons, a combination of both knowledge and
strategy teaching, and pure strategy teaching. Over time, a balanced
programme should contain both knowledge and strategy teaching.
Whole-class instruction has the benefit of involving all students. It is very
effective in simplifying management and preparation. It is therefore most
suited to situations where there is a common goal for instruction. This
approach is typically used to develop students’ thinking in the knowledge
section of the Number Framework. Students can learn to count together,
recognise numerals, and practise relevant basic facts effectively in wholeclass situations.
Your identification of knowledge hot spots provides good mathematical
content for whole-class instruction. For example, you may have identified
and recorded on the class grouping sheet for NumPA (on page 18 or 19)
that most of your students are lacking in a similar concept of number
knowledge, like backwards counting.
Using the knowledge-only lesson model enables in-depth teaching of the
knowledge needed by most students in the class. This will enhance further
strategy development for all students.
3
Getting Started
Knowledge-only Lesson
Warm-up
10 minutes
Practice
30 minutes
Warm-down
5 minutes
•
Whole-class knowledge activities
•
Hot spot focus is shared with students as
learning outcomes.
•
One or two short activities related to
numeral recognition, counting (including
both sequencing and ordering), grouping, or
basic facts are shared with the whole class.
•
Activity-based practice in pairs or small
groups or as a whole class
•
Activities may be organised as stations for
students to visit.
•
Students share their thinking with the class.
•
The teacher summarises learning outcomes
from today’s lesson, making connections to
previous lessons and existing knowledge.
•
The students reflect on their learning
through teacher questioning.
Strategy-only Lesson
Whole-class instruction has both strengths and weaknesses as an approach
to developing students’ number strategies. When problem-solving
strategies are shared, the students provide models for others to adopt and
adapt. They can encounter strategies they were previously unaware of.
A disadvantage of whole-class teaching is that the students at different
strategy stages solve operational problems at greatly differing speeds and
levels of sophistication. This can make management very difficult unless
open-ended problems are used. Optimum strategy development is more
likely to occur when the students are working in groups with others who
are functioning at strategy stages close to their own.
Below, in table form, is a model for managing strategy teaching group
rotation. In this model, each group gets two lessons on consecutive days,
and three lessons during the week.
Mon
Group 1
Group 2
Group 3
Teacher
Teacher
Tues
Teacher
Teacher
Wed
Thurs
Class Together
Teacher
Teacher
Teacher
Teacher
Class Together
Fri
A combination of:
- Assessment
- Individual help
- Activity introduction
Teacher
A model for a lesson based on this mixed knowledge and strategy
teaching is shown on the opposite page.
4
Getting Started
Knowledge and Strategy Mixed Lesson
Warm-up
10 minutes
Knowledge hot
spot focus
identified through
analysis
•
•
Direction of
Groups
•
Group
Teaching 1
•
2 minutes
15 minutes
Based on strategy
groups identified
on class grouping
sheet for NumPA
(see pages 18 and
19)
•
•
•
•
•
Redirection
of Groups
•
Group
Teaching 2
•
3 minutes
15 minutes
Warm-down
5 minutes
•
•
Whole-class knowledge activities (hot spot
focus), for example, counting, place value
modeling, fraction recognition
Short activities related to numeral
recognition, counting – sequencing and
ordering, grouping, or basic facts
The teacher ensures that students are ready
to move to their first activity.
A task board (see page 11) may be used to
guide students to the activities.
The teacher provides in-depth teaching for
one group of students.
Group teaching is aimed at key knowledge
or transition from one strategy stage to
another.
Learning outcomes are shared with the
students.
The teaching group moves to a suitable
reinforcement activity at the completion of
this session.
The other students are engaged in
independent activities.
The teacher ensures that the students are
ready to move onto their next activity.
A task board (see page 11) may be used to
guide students to the activities.
As for Group Teaching 1, but with a new
group of students
The teaching group will be set up to begin
tomorrow’s lesson with a suitable
reinforcement activity.
Possible activities include:
• Sharing examples of students’ work;
• Discussing a key idea from the unit;
• Demonstrating independent activities;
• Checking work;
• Students reflecting on their learning
through teacher questioning;
• Students write up their work for that
session.
5
Getting Started
A Model for Strategy Teaching
Book 4 of the Numeracy series, Teaching Number Knowledge, contains
knowledge activities. Knowledge is defined as those things that students
need to recall automatically. Conventional teaching methods like playing
games, individual practice, and rehearsing number sequences can help
students to increase their set of automatically recalled facts.
In books 5, 6, and 7, Teaching Addition, Subtraction, and Place Value; Teaching
Multiplication and Division; and Teaching Fractions, Decimals, and Percentages
respectively, the activities are designed to develop mental strategies.
As an example, to distinguish between knowledge and strategy, consider
how students may choose to work out 8 + 9. Knowing that 8 + 8 is 16,
students use a strategy to work out that the answer to 8 + 9 is one more
than 16.
Compared with learning knowledge like counting and basic facts, learning
to strategise is more complex. To help
Existing
you to plan for student learning, the
Knowledge &
Strategies
strategy booklets incorporate a teaching
model. The model used for strategy
teaching activities is shown in this
diagram.
Using Materials
Existing knowledge and strategies are
prerequisites for developing more
advanced strategies.
In the Using Materials phase, students
are presented with problems to be
solved with the support of materials.
Using Imaging
Using Number Properties
New
Knowledge &
Strategies
In the Using Imaging phase, the materials are shielded from the students,
and they are encouraged to image what actions they would take when
using these materials. For students who are struggling with this imaging,
the backwards direction of the arrow indicates that you should “fold
back” to allow access to materials. This working between Using Materials
and Using Imaging may occur for some time until students have developed
the ability to image the problems.
Using Number Properties focuses the students’ attention on solving
problems by discarding imaging in favour of working with the numbers
and operations as ideas. This is generally achieved by using numbers that
are too large or complex to image. Once again, for students who are not
making the abstractions, the backwards direction of the arrow indicates
that you should “fold back” to posing problems that can be imaged.
In the process of developing more advanced strategies, new knowledge
and strategies have been created.
6
Getting Started
This model applies to all strategy teaching. It should always be your aim
to introduce important mathematical ideas by using materials and to
progress to Using Imaging then Using Number Properties as soon as your
students are able. When applying the model, you should listen and
observe to see whether individual students are making the connections at
each phase. If these connections are not being made, it is suggested you
should not attempt to push on to the next phase. Either drop back to the
previous phase or end the lesson. Similar activities can be revisited at a
future time.
An example of a strategy lesson being applied is provided on pages 13–17.
Planning Numeracy Lessons
When filling in your mathematics long-term plan, remember that in the
first four years of schooling, the emphasis should be on the number
strand. In the middle and upper primary years, the emphasis may change
depending on the needs of individual students identified by NumPA.
Number will also be integrated into the teaching of other strands.
Theme units are an excellent way to work across strands and to connect
mathematics to other learning areas and to daily life experiences.
Your specific coverage of number operations should be clearly
documented on a long-term plan. This plan may cover one term or a full
year. It should be seen as a statement of intent that may legitimately vary
in response to student needs and progress.
A variety of unit structures can be used. Number units may be generic and
deal with the achievement objectives from Mathematics in the New Zealand
Curriculum in an integrated way, or units may target a specific operational
domain, like addition and subtraction or fractions. Theme units might be
used to develop a collection of key number ideas.
Unit Planning
You need to identify (highlight) the following aspects in planning number
units:
• Key knowledge outcomes
• Strategy outcomes
• Activities for learning these outcomes.
It is important that the knowledge and strategy outcomes are aligned.
Choosing a small number of outcomes is preferable to inadequate
coverage of many outcomes.
Pages 21 to 37 provide possible unit planning formats for number. These
formats are organised by strategy stages and operation, consistent with
the layout of Book 1: The Number Framework. They provide links to the
relevant achievement objectives from the curriculum statement. Each
planning form has a reference bar that spans two stages. This suggests that
the outcomes on the form are suitable for students who need to make the
transition between these stages. The outcomes are also suitable for
broadening students who are already working at the higher stage.
7
Getting Started
For example, if the bar highlights AC and EA, the
outcomes are suitable for transition of Advanced
Counting students to Early Additive Part-Whole,
or for broadening Early Additive Part-Whole students.
AC
EA
You can use a single planning format for more than one unit of teaching.
An existing plan can be highlighted using a different colour to indicate a
new teaching period. Once the unit plans are complete, you can use a
weekly plan format or a planning diary to record the day-to-day details.
There is a risk of over-planning. Some teachers wisely fill in weekly plans
to Wednesday or Day 3. This allows for flexibility for the rest of the week,
particularly when the students have not completed activities or new
learning needs are identified. Pages 38 and 39 contain possible weekly
planning formats.
You might find the Numeracy Planning Assistant on the NZmaths website
useful. You can access it at www.nzmaths.co.nz/numeracy
8
Getting Started
Finding Your Way around Unit Plan
Templates
Relevant strategy stages and
operational domain
Links to MiNZC
Mathematical
Processes and
Achievement
Objectives
Key
Knowledge
Required
Knowledge
needed by the
students to
meet the
strategy
outcomes
Knowledge
Being
Developed
Knowledge
taught
alongside the
strategy
outcomes
Strategy Learning Outcomes to explain
what the students are learning within the
operational domain
Knowledge
Activities
Referenced
activities from Book
4: Teaching Number
Knowledge and
Enriching the
Number Framework
with Beginning
School Mathematics
Strategy Activities
Referenced activities from
Book 5: Teaching Addition,
Subtraction, and Place Value;
Book 6: Teaching Multiplication
and Division; Book 7: Teaching
Fractions, Decimals, and
Percentages; Enriching the
Number Framework with
Beginning School Mathematics
and the Figure It Out series
9
Getting Started
Ongoing Assessment
Give priority to assessing students’ progress through strategy stages and
in the key knowledge required. You may wish to use checklists of the key
knowledge required to help you to identify potential blocks to students
making strategy transitions. Progress across strategy stages can be
documented easily on the class grouping sheet for NumPA (see pages
18–19) by highlighting a student’s name through to the next stage.
Where possible, you should use non-obtrusive methods of assessment.
Group teaching provides many opportunities for discussions with
students that can form the basis of judgments about achievement.
Snapshots can be used to assess students’ progress. Such snapshots consist
of mini-interviews, where you ask a problem, such as those suggested in
NumPA, that might produce a range of responses from the students.
Possible assessment portfolios can be developed. See pages 40–48 for a
sample of these portfolios. They list learning outcomes for both strategy
and knowledge and are organised by stage. They may be used in different
ways:
• As individual portfolio assessments
• As group assessment sheets
• As student self or peer assessment sheets
• As goal-setting sheets.
You may also use these assessment sheets as a teaching coverage
guideline.
Managing Independent Activities
The success of small-group strategy teaching depends on the ability of the
rest of the class to engage productively in independent activities.
Routines need to be established for:
• introducing “new” activities;
• resolving student uncertainty without involving the teacher;
• indicating to the students which activities to attempt and monitoring
their completion (for example, name tags, check sheets);
• caring for and maintaining materials (for example, class monitors);
• checking answers and written recording.
You need to have some system for informing your students about their
independent work for the lesson. This can take the form of a task board
(see page 11) or a group box, which is a box that is assigned to a particular
group that contains instructions and activities.
Working in pairs during independent activity time encourages students to
engage intensively with others. Students tend to sustain their effort longer
when they are working in pairs than they do individually. There is
tremendous potential for students to learn by interacting with each other,
provided that the abilities of the students in each pair are similar.
10
Getting Started
Individual work is useful in establishing whether the student can solve
problems independently.
The types of activities that are appropriate for independent work are:
• practice that is related to the most recent strategy lesson;
• games and puzzles that are suitable for the student’s strategy stage;
• knowledge-practice activities;
• problem-solving or knowledge-based worksheets;
• students creating their own problems for classmates;
• exploring activities or equipment related to a new unit of work.
A Model for Informing Students
about Independent Group Work
Maths Task Board
Group names and or
shapes may be
shifted along to
show each daily
programme.
The icons for
activities may be
on labels that can
be moved around
the task board.
Teachers may use
pictorial
representations or
writing to further
explain the
activities.
T
Pr
KA
T
Pr
G
T = Teaching
Pr = Practice
G = Games
KA = Knowledge Activity
KA before T
allows students
to revise key
knowledge or
knowledge
being developed
before a strategy
teaching session.
The key to
recommended
activities during a
number lesson
The task board may be drawn up on a whiteboard or made out of
cardboard and laminated. You may choose to use other existing task
boards (for example, a reading task board) and change the symbols
or pictures when it is mathematics time.
11
Getting Started
Classroom Resources
Materials Masters
Material Masters are photocopiable resources from which you can make key pieces of
cardware that are referred to in the teaching books of this folder. They can be
downloaded as PDF files from the NZ Maths website (nzmaths.co.nz). Several
commercial firms also produce this cardware. Purchasing it is often a more cost
effective option than making it yourself.
If a teaching activity requires a Material Master, it will be referenced using a code
number. For example, Material Master 5-2 is first introduced in Book 5: Teaching
Addition, Subtraction, and Place Value. The suffix -2 indicates that it is the second material
master referred to in that book. You will require these Material Masters in order to
teach many of the strategy development and knowledge lessons effectively.
Ministry Publications Available in Schools
• Figure It Out series. Currently available: two number books at level 2, nine books at
levels 2–3, nine books at level 3, eight books at levels 3–4, six number books at years
7–8. Under development: two number books at level 3, two number books at levels
3–4, one theme book at levels 3–4, 14 books at years 7–8 (including two on number
sense and four on algebra). The available books are referenced throughout the
planning forms on pages 21–37.
• www.nzmaths.co.nz This website is funded by the Ministry of Education. It
contains units of work on all strands of MiNZC, links to useful websites throughout
the world, has a “Bright Sparks” section for extending able students, and includes a
Numeracy Planning Assistant for planning units of work for students working at
stages one to six of the Number Framework.
• Connected series
• Problem Solving CD
• Beginning School Mathematics (BSM)
• Development Band Mathematics
Useful Hardware
The list below provides a basic set of equipment for numeracy. This hardware should
be available in every numeracy project class.
• Dice (dotted and numbered 1–6) and/or blank cubes (for making your own dice)
• Slavonic abacus
• Tens frames
• Number lines and number strips, used with counters and pegs
• Strings of 100 beads in groups of five beads of each colour and supermarket tags
• Transparent counters and other types of novelty counters (for example, teddies,
fruit)
• Iceblock sticks and pipe cleaners (or rubber bands)
• Beans, film canisters, and plastic containers (to package hundreds)
• Play money
• Interlocking cubes, preferably Unifix or similar
• Numeral cards
• Ice-cream containers
• Hundreds board and thousands book
• Calculators, overhead projector, and student models
• Counting flip boards
• Dominoes
• Fraction kits, circular and strip models.
12
Getting Started
Applying the Strategy Teaching Model
The lesson illustrated below is “Subtraction in Parts” from Book 5: Teaching Addition and
Subtraction and Place Value, pages 26–27. This lesson is not a script to be imitated
rigorously, but simply one possible lesson using the strategy teaching model detailed in
this book, on page 6. Using the same activity with different groups of students will
require you to take different instructional paths. The lesson agenda should be
determined by the responses of students. An important point is not to focus on lesson
differences but rather the central place the teaching model occupies in strategy lessons.
Student’s Objective for Lesson
I am learning to subtract by splitting numbers into parts instead of counting down.
Prior Knowledge and Strategies
Before attempting to develop students’ advanced part-whole ideas, the teacher knows
or establishes that the students can:
•
identify any number from 1 to 10 on tens frames instantly;
•
instantly recall addition of single-digit numbers up to a total of 10 and know the
related subtraction facts;
•
identify tens and ones in any two-digit number.
Materials Required
•
A metal plate with two tens frames drawn on it and magnetized counters (or use
two blank tens frames [Material Master 4-6] and counters), bundled sticks (ones
and tens) or beans in film canisters (ones and tens).
Lesson Transcript
Actions/Words
Commentary
Knowledge Check Phase
The teacher asks sufficient questions
Teacher (T): “I am going to ask you some
like these to assure herself that the
questions. Cross your arms when you know the
students know the basic facts adding
answer.
to 10 and their related subtraction
Six plus what makes 10? Seven plus what makes
facts. John’s knowledge/instant recall
10? 10 minus seven gives what?”
is not good enough. The teacher notes
The teacher continues with similar problems.
this for future teaching. All the
The teacher opens one film canister of beans
students can produce 36 in tens and
and shows the students that there are 10 beans in
ones. The teacher decides to proceed
it.
with the lesson.
T: “I am going to write a number on the board.
I want each of you to make this number of beans using the canisters.”
The teacher writes 36 on the whiteboard.
The students attempt to show three canisters and six loose beans.
Using Materials Phase
The teacher organises the students into groups of
four.
T: “Our problem to solve is: Brian has 14 oranges,
and he eats six of them. How many are left?
Firstly how could we write this on the board as a
take-away? Andrew?”
A: “14 minus six”.
The teacher begins the teaching with
the Using Materials part of the
teaching model. She is conscious that
she will not supply methods of
solution but rather lets the students
discuss how they might solve the
problem. When getting the students
to explain their solutions, she asks
questions and links the answers to the
materials.
13
Getting Started
The teacher writes 14 – 6 on the board.
T: “On this tens board, I am putting 10 green magnetic counters. To make 14 altogether,
how many yellow counters do I need? Sarah?”
S: “Four.”
T: “Good. So these 14 counters stand for oranges. Look carefully at the counters and
work out how many are left when you take away six counters.
Discuss this in your groups.
No, Sarah, I don’t want anyone to touch the counters. Just look at them and imagine
taking away six. There could be more than one way of doing this.”
Group discussions follow.
T: “Who thinks they have got the answer? Moana?”
M: “You need to take off two.”
T: “Why do I need to take off two more?”
M: “Because four and two makes six.”
T: “I see. Well, Moana, when you take off two more, how many green counters will be
left?”
M: “Eight, I think.”
T: “How do you know there will be eight?”
M: “Because if you take two away that leaves eight.”
T: “Can you come out the front and show everyone how you did it?”
Moana comes to the front.
T: “What colour did you take away first?”
M: “Yellow.”
T: “Why did you choose the yellow counters?”
M: “Because there would be 10 left.”
T: “Okay, take the yellow ones off the board.”
Moana removes the four yellow counters.
T: (asking the other students) “How many more counters does Moana need to take?”
Z: “Two more … that would leave eight.”
T: “Well done, Zoe. Does everyone else understand that? Moana, take the two counters
off to show us.”
Moana removes two of the green counters.
T: “Did anyone do it a different way?
Charles, okay. Can you come out to the front?
Can you show us which six oranges you ate?”
Charles takes six green counters off the left-hand
tens frame.
T: “So how do you know the answer is eight?”
C: “Because four and four is eight.”
T: “Where are your four and four, Charles?”
Charles points at the four green counters and the
four yellow counters.”
A set of similar problems follows, with
the students having access to materials.
14
The teacher encourages multiple
solutions because she is trying to help
the students construct the fact that
part/whole thinking is superior to
counting on and is not concerned
about any particular part/whole
reasoning at this stage.
The teacher observes which students
are making the connections. She
notices that Helen insists on using
counting by ones methods and is
unlikely to benefit from the next Using
Materials problems. She decides to
leave Helen in the group for today.
Getting Started
T: “Let’s try another one.
In your groups get out 34 beans.”
The students get three canisters of 10 and four loose beans.
The teacher writes 34 – 5 on the whiteboard.
T: “Tara has 34 beans, and she eats five. How many beans does she have left?”
All groups solve the problem by opening one canister and discuss the answer.
T: “Do you all agree on the answer?
Could any of your group explain how you worked out the answer? Explain your
method to each other.”
More discussion occurs within the groups.
T: “Who can tell me the answer? Natalie?”
N: “It is 29.”
T: “Yes. Can you show everyone using the
canisters?”
Natalie removes the four loose beans, opens a
canister, and removes one more.
The teacher repeats the previous
teaching using the beans with bigger
numbers. She notices in the group
discussion that Helen continues to use
her fingers to count back. She notes
that after three examples everyone
else appears to be making sense of the
part/whole reasoning so she then
moves on to the Using Imaging phase
of the teaching model.
T: “How many are left in the canister?”
N: “Nine.”
T: “How do you know that, Natalie? Did you
count them?”
N: “No, I know 10 take away one is nine.”
T: “Does everyone understand that? Has anyone got another way?”
There is no reply. The students do more examples using materials.
Using Imaging Phase
T: “Let’s do another one. This time I am going to hide the beans, and you are going to
imagine how to solve the problem.”
The teacher writes 43 – 5 on the board and shields
43 beans from the students’ view using an icecream container.
T: “There are 43 beans under here. What do they
look like?”
A: “Four canisters and three more beans.”
T: “Yes (lifting the ice-cream container and
replacing it), you’re right. I’m going to take away
five. Imagine what I will do. What is 43 – 5?
Discuss this in your groups.”
Discussion occurs within the groups.
T: “Okay, what is the answer?”
S: “38.”
In the Using Imaging phase, the
teacher deliberately shields the
materials from the students to
encourage them to think about the
groups of beans (and hence numbers)
that can be rearranged mentally to
solve subtraction problems. Once she
has asked for the answer, the teacher
folds back to materials to allow any
students who could not do the
imaging to connect with the materials.
After three examples, the teacher
notices that seven of the eight
students seem to be able to reason
correctly with imaging. She infers
they have a chance of making
connections at the Using Number
Properties phase. Helen is not coping
even at the Using Materials level.
15
Getting Started
T: “Yes, that’s right. Show everyone how you worked it out. Use the beans.”
Sarah removes three beans, opens a canister, and removes two more.
T: “So why is the answer 38?”
S: “I took two out of the canister. That leaves eight.”
T: “Where is the 30 part of 38?”
S: “There are three canisters and eight.”
T: “I see, so three tens is 30, and eight is 38. Good.”
The students attempt several more problems, using imaging of the materials.
Using Number Properties
T: “Now this time we are not going to use any
beans.”
The teacher writes 75 – 8 on the whiteboard.
T: “I have 75 lollies, and I eat eight of them. How
many are left? Discuss this in your groups.
Remember if your classmates can’t work out the
answer, your job is to teach them.”
Discussion follows.
T: “Okay, who thinks they have the answer?”
N: “67.”
J: “66.”
T: “Hmmm … you can’t both be right. John, how
did your group work it out?”
J: “Um … First we took off five.”
T: “Why did you take off five?”
J: “Because that made 70.”
T: “But the problem is to take off eight. How many
more did you have to take off?”
J: “Three … Oh, the answer is 67.”
T: “How do you know that?”
J: “Well, 70 take off three is 67.”
T: “Great, so you all agree now?”
M: “Yes, we got 67.”
S: “But I got 77?”
The teacher notes the number size has
been increased to the point where
imaging is difficult, and she
encourages the students to reason
directly on the numbers. So 75 is seen
as seven tens and five ones, and the
method used in Using Imaging is
abstracted to act on numbers as
abstract ideas.
The teacher collects various answers.
She concludes from John’s answer of
66 that either John has made an
arithmetic slip or he is having
difficulty applying his recall of basic
facts to larger numbers. She suspects
that Sarah’s answer of 77 is basically
from correct reasoning but she has
lost track of the fact that the step
10 – 3 decomposes a 10 so there are
only 6 tens left.
The teacher encourages the students
to fold back to imaging to understand
why the answer is 67.
The teacher writes all the answers 66, 67, and 77 on the board.
T: “Well, let’s have a look. Imagine how many tens and ones we would need to make
75.”
The teacher points at 75 – 8 on the whiteboard.
M: “Seven tens and five ones.”
T: “So we have seven tens and five ones, right. How can we remove eight?”
S: “Take away five ones first.”
T: “Stop for a moment. How many are left?”
S: “70.”
T: “How many more to go?
16
Getting Started
S: “Three.”
The teacher writes 70 – 3 on the whiteboard.
T: “What is 70 take away three?”
M: “67.”
T: “Because?”
S: “You take three out of one of the tens.”
T: “Leaving how many tens?”
S: “Six ... I know what I did now.”
T: “Well done Sarah, good thinking. Now, let’s
try this one.”
The teacher writes 85 – 9 on the whiteboard, and
the lesson continues.
The teacher observes that Andrew,
Sarah, Moana, and Charles have
understood Using Number Properties,
at least for today. John would be able
to cope but doesn’t know or apply his
basic facts of subtraction well enough.
Zoe and Natalie can cope with
imaging but have not made the
transition to number properties.
Helen cannot understand part/whole
at any level.
The teacher decides to put the Helen
with the Advanced Counting group.
With the rest she plans to use other
activities on part/whole thinking and
repeat the use of the teaching model.
17
Getting Started
18
Class Grouping Sheet for NumPA
Teacher
School
Operation Domain
Knowledge
Hot Spots
Emergent
One-to-One Counting
Counting from One
Materials
Counting from One
Imaging
Advanced Counting
Early Additive
Class Grouping Sheet for NumPA
Teacher
School
Operation Domain
Knowledge
Hot Spots
Counting from One
Advanced Counting
Early Additive
Advanced Additive
Advanced
Multiplicative
Advanced Proportional
Getting Started
19
Teacher
Knowledge
Hot Spots
Ms Print
School
Tapworth Primary
Operation Domain
Addition/Subtraction
One more/one less than the number, up to 1 000
Recognition of fractions symbols and diagrams
Groupings with fives and tens, tens/hundreds in whole numbers to 1 000
Counting from One
Advanced Counting
Early Additive
Advanced Additive
Skye
Tania
Leon
Allan
Zoe
Jane
Bruce
Vey-un
Tui
Carla
Leyton
Rawiri
Damion
Toru
Kahu
Maria
Advanced
Multiplicative
Advanced Proportional
Getting Started
20
Class Grouping Sheet for NumPA
Getting Started
Transition Emergent to One to One Counting
Addition and Subtraction
Mathematical Processes (Level 1)
•
Problem Solving: Use equipment
appropriately when exploring
mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical Ideas:
Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 1)
•
Make up, tell, and record number
stories, up to nine, about given
objects and sequence pictures.
•
Form a set of up to 20 objects.
•
Read and write any two-digit whole
number.
•
Rote count to at least 50.
Knowledge Being Developed
The students are learning to:
•
Identify all of the numbers in the range 0–10.
•
Say the forwards and backwards number word sequences
in the range 0–10.
•
Say the number before and after a given number in the
range 0–10.
•
Order the numbers in the range 0–10.
•
Instantly recognise patterns to five, including finger
patterns.
Exploring Computation and
Estimation (Level 1)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Model and explain addition
calculations with a sum of up to
20.
•
Using up to 20 objects, model
and explain subtraction
calculations.
E
CA
AC
EA
AA
AM
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Tens Frames: page 2
Pipe Cleaner Numbers: page 4
Number Fans: page 4
Counting: page 12
Number Line Flips: page 16
Grouping Based
Fabulous Fives: page 23
AP
BSM (for Emergent to One-to-one Counting):
2-1-1, 2-1-2, 2-1-4, 2-1-21, 2-1-84,
2-3-7, 2-3-8, 2-3-24, 2-3-55, 3-1-2, 3-1-3, 3-1-4, 3-1-7, 3-1-21, 3-1-22,
3-1-23, 3-1-46, 3-1-47, 3-3-9, 3-3-22, 3-3-48.
BSM (for One-to-one Counting):
2-1-5, 3-1-43, 4-1-5, 4-1-22, 4-1-23, 4-1-44, 4-1-45, 4-1-48, 4-1-83,
4-3-5, 4-3-6, 4-3-21, 4-3-44, 4-3-84, 4-3-46, 4-3-24, 4-3-25, 4-3-51,
5-1-3, 5-1-6, 5-1-7, 5-1-8, 5-1-21, 5-1-22, 5-1-46, 5-1-83, 5-3-5, 5-3-22,
5-3-45, 5-3-46, 5-3-82, 5-3-6, 5-3-48, 5-3-7, 5-3-23, 5-3-49, 5-3-83,
6-1-3, 6-1-21, 6-1-4, 6-1-43, 6-1-5, 6-1-6, 6-1-44, 6-1-7, 6-1-45,
6-1-46, 6-1-81, 6-1-82, 6-1-9, 6-2-21, 6-2-22, 6-2-9, 6-2-48, 6-3-3,
7-1-1, 7-1-41, 7-1-42, 7-1-81, 7-1-2, 7-1-43, 7-1-3, 7-1-44, 7-1-46,
7-2-7, 7-2-48, 7-2-49, 7-2-82, 7-3-4, 8-2-8, 8-2-47, 8-2-48, 8-2-83,
9-2-18, 9-2-61, 9-2-85.
Strategy Learning Outcomes
The students are learning to:
•
Count a set of objects in the range 1–10.
•
Form a set of objects in the range 1–10.
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Lucky Dip: page 1
Match It Up: page 1
Counting as We Go: page 2
How Many Now?: page 2
How Many …?: page 2
Loud and Soft: page 2
Tick Tock: page 3
Before and After: page 3
Clapping: page 3
Walk the Bridge: page 3
Up or Down: page 4
Ordering Numerals: page 4
Dice Groups: page 4
How Many Cubes?: page 4
Caterpillar Legs: page 5
Petals and Flower Centres: page 5
Feed the Elephant: page 5
Birthday Cake: page 5
Turtles 5 and …: page 5
Facts to 10: page 6
BSM (for One-to-one Counting)
2-1-5, 4-1-23, 4-3-21, 4-3-44, 4-3-46, 5-1-7.
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
21
Getting Started
Transition One-to-One Counting to Counting from One on Materials
Addition and Subtraction
Mathematical Processes (Level 1)
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 1)
•
Make up, tell, and record number
stories, up to nine, about given
objects and sequence pictures.
•
Form a set of up to 20 objects.
•
Read and write any two-digit whole
number.
•
Rote count to at least 50.
Exploring Computation and
Estimation (Level 1)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Model and explain addition
calculations with a sum of up to
20.
•
Using up to 20 objects, model
and explain subtraction
calculations.
E
CA
AC
EA
AA
AM
Key Knowledge Required for Addition and Subtraction
Knowledge Activities
The students can:
•
Rote count to 10, at least.
•
Order all of the numbers in the range 0–10.
•
Say the forwards and backwards number word sequences
in the range 0–10.
•
Say the number before and after a given number in the
range 0–10.
•
Instantly recognise patterns to five, including finger
patterns.
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Tens Frames: page 2
Pipe Cleaner Numbers: page 4
Number Fans: page 4
Counting: page 12
Number Line Flips: page 16
Grouping Based
Fabulous Fives: page 23
BSM (for One-to-one Counting):
2-1-5, 3-1-43, 4-1-5, 4-1-22, 4-1-23, 4-1-44, 4-1-45, 4-1-48, 4-1-83,
4-3-5, 4-3-6, 4-3-21, 4-3-44, 4-3-84, 4-3-46, 4-3-24, 4-3-25, 4-3-51,
5-1-3, 5-1-6, 5-1-7, 5-1-8, 5-1-21, 5-1-22, 5-1-46, 5-1-83, 5-3-5, 5-3-22,
5-3-45, 5-3-46, 5-3-82, 5-3-6, 5-3-48, 5-3-7, 5-3-23, 5-3-49, 5-3-83,
6-1-3, 6-1-21, 6-1-4, 6-1-43, 6-1-5, 6-1-6, 6-1-44, 6-1-7, 6-1-45,
6-1-46, 6-1-81, 6-1-82, 6-1-9, 6-2-21, 6-2-22, 6-2-9, 6-2-48, 6-3-3,
7-1-1, 7-1-41, 7-1-42, 7-1-81, 7-1-2, 7-1-43, 7-1-3, 7-1-44, 7-1-46,
7-2-7, 7-2-48, 7-2-49, 7-2-82, 7-3-4, 8-2-8, 8-2-47, 8-2-48, 8-2-83,
9-2-18, 9-2-61, 9-2-85.
Knowledge Being Developed for Addition and Subtraction
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Pipe Cleaner Numbers: page 4
Counting: page 12
Card Ordering: page 13
Number Line Flips: page 16
The students are learning to:
Counting Based
•
Identify numbers in the range 0–20.
•
Say the forwards and backwards number word sequences
in the range 0–20.
•
Say the number before and after a given number in the
range 0–20.
•
Order numbers in the range 0–20.
•
Record the results of counting and operations using
symbols, pictures, and diagrams.
BSM (for Counting from One on Materials):
2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4,
6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9,
8-1-4, 8-1-53.
Strategy Learning Outcomes
The students are learning to:
•
Solve simple addition problems to 20 by counting all the
objects.
•
Solve simple subtraction problems from 20 by counting all
the objects.
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Adding and Subtracting with Counters: page 7
Adding and Subtracting with One Hand: page 7
Fly Flip: page 8
Using Fives: page 8
Challenging Hand Problems: page 9
Teens and Fingers: page 9
Ones and Tens: page 10
More Ones and Tens: page 11
BSM (for Counting from One on Materials):
6-1-3, 6-1-9, 6-1-46, 6-1-47, 6-1-81, 6-1-82, 7-1-6, 7-1-52, 7-1-53,
7-1-84, 7-1-85, 7-3-4, 7-3-7, 7-3-50, 7-3-51, 7-3-52, 7-3-82, 8-1-5,
8-1-8, 8-1-46, 8-1-51, 8-1-52, 8-3-7, 8-3-8, 8-3-9, 8-3-50, 8-3-51.
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
22
AP
Getting Started
Transition Counting from One on Materials to Counting from One by Imaging
Addition and Subtraction
Mathematical Processes (Level 1)
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 1)
•
Make up, tell, and record number
stories, up to nine, about given
objects and sequence pictures.
•
Form a set of up to 20 objects.
•
Read and write any two-digit whole
number.
•
Rote count to at least 50.
Exploring Computation and
Estimation (Level 1)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Model and explain addition
calculations with a sum of up to
20.
•
Using up to 20 objects, model
and explain subtraction
calculations.
AC
EA
AA
AP
Knowledge Activities
The students can:
•
Identify numbers in the range 0–20.
•
Say the forwards and backwards number word sequences
in the range 0–20.
•
Say the number before and after a given number in the
range 0–20.
•
Order numbers in the range 0–20.
Teaching Number Knowledge
Counting Based
Pipe Cleaner Numbers: page 4
Knowledge Being Developed for Addition and Subtraction
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Number Fans: page 4
Counting: page 12
Card Ordering: page 13
Lucky Dip: page 15
Number Line Flips: page 16
Grouping Based
Fabulous Fives: page 23
Using the Slavonic Abacus to Reinforce Five Grouping : page 35
Patterns to 10: page 36
BSM (for Counting from One by Imaging):
2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4,
6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9,
8-1-4, 8-1-53.
Strategy Learning Outcomes
The students are learning to:
•
Solve simple addition problems by counting all the objects
in their head (by imaging).
•
Solve simple subtraction problems by counting all the
objects in their head (by imaging).
CA
AM
Key Knowledge Required for Addition and Subtraction
The students are learning to:
Counting Based
•
Identify numbers in the range 0–20, at least.
•
Say the forwards and backwards number word sequences
in the range 0–20, at least.
•
Say the number before and after a given number in the
range 0–20, at least.
•
Order numbers in the range 0–20, at least.
Grouping Based
•
Recall groupings with five.
•
Recall groupings within five and 10.
•
Instantly recognise patterns to 10 (doubles and five based),
including finger patterns.
•
Recall doubles to 10.
•
Record the results of counting and operations using
symbols, pictures, and diagrams.
E
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Using One Hand: page 11
Using Tens Frames: page 12
Both Hands: page 12
Imaging with Tens Frames: pages 13
What’s Hidden?: page 14
Imaging Many Hands: page 14
Making Tens: page 14
Crossing the Five Barrier: page 15
Fingers Again: page 16
Ten Sweets per Packet: page 16
BSM (for Counting from One by Imaging):
6-1-3, 6-1-9, 6-1-46, 6-1-47, 6-1-81, 6-1-82, 7-1-6, 7-1-52, 7-1-53,
7-1-84, 7-1-85, 7-3-4, 7-3-7, 7-3-50, 7-3-51, 7-3-52, 7-3-82, 8-1-5,
8-1-8, 8-1-46, 8-1-51, 8-1-52, 8-3-7, 8-3-8, 8-3-9, 8-3-50, 8-3-51.
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
23
Getting Started
Transition Counting from One on Materials to Counting from One by Imaging
Multiplication and Division/Fractions
Mathematical Processes (Level 1)
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 1)
•
Make up, tell, and record number
stories, up to nine, about given
objects and sequence pictures.
•
Form a set of up to 20 objects.
•
Read and write any two-digit whole
number.
•
Rote count to at least 50.
Exploring Computation and Estimation
(Level 1)
•
Make sensible estimates and check
the reasonableness of answers.
•
Find by practical means, one half
and one quarter of a shape, and a
half of a set of objects.
Knowledge Being Developed for Mult/Div, Fractions
Knowledge Activities
Teaching Number Knowledge
Counting Based
Counting: page 12
Skip-counting on the Number Line: page 12
Beep: page 13
Grouping Based
Fabulous Fives: page 23
Using the Slavonic Abacus to Reinforce Five Grouping : page 35
Patterns to 10: page 36
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Number Fans: page 4
Card Ordering: page 13
Lucky Dip: page 15
Number Line Flips: page 16
Teaching Fractions, Decimals and Percentages
Refer to Tasks for Key Knowledge on page 1.
Strategy Learning Outcomes
Strategy Activities
Teaching Fractions, Decimals, and Percentages
Fair Shares: page 2
BSM (for Counting from One by Imaging):
8-3-6, 8-3-47, 8-3-48, 8-3-49, 8-3-82.
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
24
AC
EA
AM
Knowledge Activities
The students are learning to:
•
Solve simple multiplication and division problems by
counting all the objects.
•
Find halves and quarters of shapes or sets of objects to 20
by equal sharing of the objects.
•
Find halves, and quarters of shapes and objects, e.g., half a
glass of water.
CA
AA
Key Knowledge Required for Multiplication/Division and
Fractions
The students can:
Counting Based
•
Identify numbers in the range 0–20, at least.
•
Say the forwards and backwards number word sequences
in the range 0–20, at least.
•
Say the number before and after a given number in the
range 0–20, at least.
•
Order numbers in the range 0–20, at least.
Grouping Based
•
Recall doubles to 10.
The students are learning to:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–20 for twos and fives.
Grouping Based
•
Instantly recognise patterns to 10 (doubles and five-based),
including finger patterns.
•
Recall doubles to 10.
•
Record the results operations using symbols, pictures, and
diagrams.
E
AP
Getting Started
Transition Counting from One by Imaging to Advanced Counting
Addition and Subtraction
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 2)
•
Read any three-digit whole number.
•
Explain the meaning of the digits in
two- or three-digit whole numbers.
•
Order any set of three or more whole
numbers (up to 99).
•
Write and solve comparison problems.
Exploring Computation and
Estimation (Level 2)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Recall the basic addition and
subtraction facts.
•
Mentally perform calculations
involving addition and
subtraction.
•
Write and solve story problems
that involve whole numbers
using addition and subtraction.
Knowledge Activities
The students can:
Counting Based
•
Identify numbers in the range 0–20, at least.
•
Say the forwards and backwards number word sequences
in the range 0–20, at least.
•
Say the number before and after a given number in the
range 0–20, at least.
•
Order numbers in the range 0–20, at least.
Grouping Based
•
Instantly recognise patterns to 10 (doubles and five based),
including finger patterns.
Teaching Number Knowledge
Counting Based
Number Mat and Lily Pads: page 2
Counting: page 12
Grouping Based
Fabulous Fives: page 23
Using the Slavonic Abacus to Reinforce Five Grouping : page 35
Patterns to 10: page 36
BSM (for Counting from One by Imaging):
2-1-5, 6-1-3, 6-1-4, 6-1-7, 6-1-9, 6-1-21, 6-1-46, 6-3-22, 6-3-3, 6-3-4,
6-3-5, 6-3-7, 6-3-49, 7-1-1, 7-1-2, 7-1-41, 7-1-42, 7-1-43, 7-3-9,
8-1-4, 8-1-53.
Knowledge Being Developed for Addition and Subtraction
The students are learning to:
Counting Based
•
Say the forwards and backwards number word sequences
in the range 0–100, at least.
•
Say the number before and after a given number in the
range 0–100, at least.
•
Identify all of the numbers in the range 1–100, at least.
Grouping Based
•
Recall groupings with 10 and the patterns of teens.
•
Recall groupings within 20.
•
Recall the number of tens within decades.
•
Recall the names for 10, e.g., 4 + 6 and 5 + 5, and the teen
facts, e.g. 16 = 10 + 6.
•
Recall the doubles to 20, and the corresponding halves.
•
Record the results of mental addition and subtraction
using equations.
Knowledge Activities
Teaching Number Knowledge
Counting Based
“Teen” and “Ty” Numbers: page 3
Number Fans: page 4
Counting: page 12
Card Ordering: page 13
Lucky Dip: page 15
Number Line Flips: page 16
Squeeze - Guess My Number: page 17
Hundreds Boards and Thousands Book: page 17
Bead Strings: page 18
Grouping Based
Slavonic Abacus: page 24
Tens and Ones: page 25
Up to Ten: page 33
Double Trouble: page 34
Number Boggle: page 35
Tens Frames Again: page 36
BSM (for Advanced Counting):
9-1-4, 9-1-6, 9-1-12, 9-1-42, 9-1-83, 9-3-9, 9-3-48, 9-3-84, 10-1-3,
10-1-4, 10-1-42, 10-1-47, 10-3-6, 10-3-9, 10-3-11, 10-3-56, 11-1-6,
11-1-47, 11-1-48, 11-1-84, 11-3-4, 11-3-8, 11-3-9, 11-3-13, 12-1-1,
12-1-5, 12-1-44, 12-1-45, 12-1-84.
Figure It Out
Number, Level 2, Book 1, pages 4–5
The students are learning to:
•
Solve addition problems by counting on in their head from
the largest number, using supporting materials, if
necessary, but progressing to imaging the count on
number.
•
Solve subtraction problems by counting back from the
largest number in their head, using supporting materials, if
necessary, but progressing to imaging the count back
number.
•
Solve addition and subtraction problems by counting on in
ones and tens, e.g., 43 + 32 as 43, 53, 63, 73, 74, 75.
CA
AC
EA
AA
AM
Key Knowledge Required for Addition and Subtraction
Strategy Learning Outcomes
E
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Number Tiles: page 17
The Number Strip: page 18
The Bears’ Picnic: page 19
Frog Jumps: page 19
The Bigger Number First: page 20
Change Unknown: page 20
Counting Back: page 21
Adding Tens: page 21
Subtracting Tens: page 22
Adding Ones and Tens: page 23
Subtracting Ones and Tens: page 23
The Missing Ones and Tens: page 24
The Thousands Book: page 24
BSM (for Advanced Counting):
9-1-8, 9-1-12, 9-1-13, 9-1-51, 9-1-52, 9-1-53, 9-3-11, 9-3-12, 9-3-13,
9-3-48, 10-1-6, 10-1-41, 10-1-42, 10-1-47, 10-1-49, 10-3-8, 10-3-11,
10-3-53, 10-3-56, 10-3-86, 11-1-3, 11-1-7, 11-1-47, 11-1-48, 11-1-83,
11-3-4, 11-3-10, 12-1-1, 12-1-5, 12-1-44.
Figure It Out
Basic Facts Levels 2–3: pages 1, 3, 5, 21
Number Levels 2–3: page 1
Basic Facts Level 3: page 1
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
25
AP
Getting Started
Transition Counting from One by Imaging to Advanced Counting
Multiplication and Division/Fractions
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 2)
•
Write and solve story problems that
involve halves, quarters, thirds, and
fifths.
Exploring Computation and Estimation
(Level 2)
•
Make sensible estimates and check
the reasonableness of answers.
•
Write and solve story problems that
involve whole numbers, using
addition, subtraction, multiplication,
or division.
CA
AC
EA
AA
AM
Key Knowledge Required for Multiplication/Division
and Fractions
Knowledge Activities
The students can:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–20 for twos and fives.
Grouping Based
•
Instantly recognise patterns to 10 (doubles and five-based),
including finger patterns.
•
Recall doubles to 10.
Teaching Number Knowledge
Grouping Based
Fabulous Fives: page 23
Using the Slavonic Abacus to Reinforce Five Grouping: page 35
Patterns to 10: page 36
Teaching Multiplication and Division
Refer to Tasks for Key Knowledge on page 1
Teaching Fractions, Decimals, and Percentages
Refer to Tasks for Key Knowledge on page 1.
Knowledge Being Developed for Multiplication/Division and
Fractions
The students are learning to:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–100 for twos, fives, and tens.
•
Identify the symbols for halves, quarters, thirds, and fifths.
Grouping Based
•
Recall the number of tens in decades, e.g., four tens in 40.
Knowledge Activities
Teaching Number Knowledge
Fraction Pieces: page 6
Counting: page 12
Skip-counting on the Number Line: page 12
Beep: page 13
Using Calculators: page 15
Strategy Learning Outcomes
The students are learning to:
•
Solve multiplication problems by skip-counting in twos,
fives, and tens, for example, 4 x 5 as 5, 10, 15, 20, by
tracking the count with their fingers, if necessary, but
progressing to tracking the count by imaging.
•
Find simple fractions of shapes and lengths, starting with
halves and quarters.
•
Find a fraction of a number by sharing out the
objects equally, moving towards anticipating the sharing
by imaging or skip-counting, for example, 1 of 12 by
4
sharing 12 objects into four sets or by trial skip-counting 3,
6, 9, 12.
Strategy Activities
Teaching Multiplication and Division
Introduction: page 1
Number Strips: page 2
Teaching Fractions, Decimals, and Percentages
Fair Shares: page 2
BSM (for Advanced Counting Multiplication):
9-1-45, 9-1-46, 9-1-47, 9-1-83, 9-1-84, 9-3-10, 9-3-53, 10-1-4, 11-1-6,
11-3-12, 11-3-13, 11-3-54.
BSM (for Advanced Counting Fractions):
9-2-18, 9-2-61, 9-2-85, 9-3-10, 9-3-53, 11-2-18, 11-2-59.
Figure It Out
Multiplication:
Basic Facts, Levels 2–3: pages 7, 9
Fractions:
Number, Levels 2–3: pages 17, 18
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
26
E
AP
Getting Started
Transition Counting from Advanced Counting to Early Additive
Addition and Subtraction
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 2)
•
Read any three-digit whole number.
•
Explain the meaning of the digits in
two- or three-digit whole numbers.
•
Order any set of three or more
whole numbers (up to 99).
Exploring Computation and Estimation
(Level 2)
•
Make sensible estimates and check
the reasonableness of answers.
•
Recall the basic addition and
subtraction facts.
•
Mentally perform calculations
involving addition and subtraction.
•
Write and solve story problems
that involve whole numbers, using
addition, subtraction,
multiplication, or division.
Knowledge Activities
The students can:
Grouping Based
•
Recall groupings with 10 and the patterns of teens.
•
Recall groupings within 20.
•
Recall the number of tens within decades.
•
Recall the names for 10, e.g., 4 + 6 and 5 + 5, and the teen
facts, e.g., 16 = 10 + 6.
•
Recall the doubles to 20, and the corresponding halves.
Teaching Number Knowledge
Counting Based
“Teen” and “Ty” Numbers: page 3
Grouping Based
Up to Ten: page 33
Double Trouble: page 34
Number Boggle: page 35
Tens Frames Again: page 36
BSM (for Advanced Counting):
9-1-4, 9-1-6, 9-1-12, 9-1-42, 9-1-83, 9-3-9, 9-3-48, 9-3-84, 10-1-3,
10-1-4, 10-1-42, 10-1-47, 10-3-6, 10-3-9, 10-3-11, 10-3-56, 11-1-6,
11-1-47, 11-1-48, 11-1-84, 11-3-4, 11-3-8, 11-3-9, 11-3-13, 12-1-1,
12-1-5, 12-1-44, 12-1-45, 12-1-84.
Figure It Out
Number, Level 2, Book 1, pages 4–5
Knowledge Being Developed for Addition and Subtraction
The students are learning to:
Counting Based
•
Identify all of the numbers in the range 0–1 000.
•
Say the forwards and backwards number word sequences
by ones, tens, and hundreds in the range 0–1 000.
•
Say the number 1, 10, or 100 more or less than a given
number
•
Order numbers in the range 0–1 000.
Grouping Based
•
Recall the number of groupings of tens that can be made
from a three-digit number.
•
Recall the number of tens and hundreds in centuries and
thousands.
•
Recall addition and subtraction facts to 20.
•
Round three-digit whole numbers to the nearest 10, or
hundred.
•
Identify the multiples of 10 that add to 100.
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Fans: page 4
Place Value Houses: page 5
Number Hangman: page 5
Counting: page 12
Card Ordering: page 13
Arrow Cards: page 14
Lucky Dip: page 15
Number Line Flips: page 16
Squeeze – Guess My Number: page 17
Hundreds Boards and Thousands Book: page 17
Bead Strings: page 18
Who is the Richest?: page 19
Grouping Based
Slavonic Abacus: page 24
Tens and Ones: page 25
Close to 100: page 25
Nudge: page 26
Estimating: page 27
Traffic Lights: page 27
Zap: page 28
Number Mats and Number Fans: page 36
Loopy: page 39
Addition Flash Cards: page 39
BSM (for Early Additive Part-Whole):
11-3-13, 11-3-54, 11-3-55, 12-1-3, 12-1-4, 12-1-7, 12-1-9, 12-1-46,
12-1-43, 12-1-48, 12-1-50, 12-1-86, 12-1-87, 12-3-6, 12-3-7, 12-3-45,
12-3-46, 12-3-49, 12-3-65, 12-3-81, 12-3-82, 12-3-83.
Figure It Out
Number, Level 2, Book 1, page 1
Book 2, pages 2, 4–6
The students are learning to:
•
Solve adding and subtracting problems in their head by
working out the answer from basic facts they know,
e.g., 8 + 7 as 8 + 8 – 1 (using doubles),
or (5 + 3) + (5 + 2) = 5 + 5 + 5 (using fives),
or 10 + 5 (making tens).
•
Solve adding and subtracting problems with two- and
three-digit numbers using groupings of 10 and 100,
for example, 43 + 25 as (40 +20) + (3 +5),
or (43 + 20) + 5 (standard place value partitioning),
or 39 + 26 as 40 +25 ( tidy numbers with compensation)
CA
AC
EA
AA
AM
Key Knowledge Required for Addition and Subtraction
Strategy Learning Outcomes
E
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Make Ten: page 25
Grouping then Subtracting: page 26
Subtraction in Parts: page 26
Up over the Tens: page 27
Adding in Parts: page 28
Comparisons: page 29
More Comparisons: page 29
How Many Ten-dollar Notes?: page 30
BSM (for Early Additive Part-Whole)
11-3-12, 11-3-54, 11-3-55, 12-1-6, 12-1-9, 12-1-46, 12-1-47, 12-1-52,
12-1-55, 12-1-56, 12-1-85, 12-1-89, 12-3-8, 12-3-13, 12-3-47,
12-3-52, 12-3-53, 12-3-81.
Figure It Out
Number, Level 2, Book 1, pages 2–3, 6–11, 16, 22–24
Number, Level 2, Book 2, pages 1, 3, 7–15
Basic Facts, Levels 2–3: pages 12, 15, 19
Number, Levels 2–3: page 2 Basic Facts, Level 3: pages 4, 5, 6, 7, 8
Number, Level 3: page 19
Number, Years 7–8, Link, Book One, pages 1–2, 6–8, 15–17
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
27
AP
Getting Started
Transition Advanced Counting to Early Additive
Multiplication and Division/Fractions
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 2)
•
Write and solve story problems that
involve halves, quarters, thirds, and
fifths.
Knowledge Activities
Knowledge Being Developed for Multiplication/Division and
Fractions
The students are learning to:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–100 for twos, threes, fives, and tens at least.
•
Identify the symbols for halves, quarters, thirds, fifths, and
tenths including fractions greater than 1.
Knowledge Activities
Teaching Multiplication and Division
Refer to Tasks for Key Knowledge on page 4.
Teaching Fractions, Decimals, and Percentages
Refer to Tasks for Key Knowledge on page 4.
Teaching Number Knowledge
Creating Fractions: page 6
Counting: page 12
Fraction Pieces: page 6
Skip-counting on the Number Line: page 12
Beep: page 13
Using Calculators: page 15
Order fractions with like denominators, e.g., 1 and 2 .
4
4
Grouping Based
•
Recall groupings of two in numbers to 20, groupings of five
in numbers to 50, and groupings of 10 in numbers to 1 000.
•
Automatically recall the multiplication and division facts
for the multiples of 2, 5, and 10.
•
Record the results of mental multiplication calculations
using equations and diagrams.
Strategy Learning Outcomes
The students are learning to:
•
Use repeated addition and adding and subtracting from
known facts to solve multiplication problems with twos,
threes, fours, fives, and tens at least.
•
Use repeated addition and adding and subtracting from
known addition or subtraction facts to solve simple
division problems of these types:
sharing , e.g., 16 lollies among four friends
sets of, e.g., 16 lollies in packets of four
•
Find fractions of shapes and lengths, including fractions
greater than one, e.g., 5 of a circle.
•
•
2
Put fractions in order from smallest to largest.
Find a fraction of a set by skip-counting, by using repeated
addition, or by adding and subtracting from known
addition or subtraction facts, for example, 4 + 4 + 4 = 12,
so 1 of 12 is 4.
3
Strategy Activities
Teaching Multiplication and Division
Introduction: pages 4–5
Animal Arrays: page 5
Pirate Crews: page 7
Biscuit Boxes: page 8
Twos, Fives, and Tens: page 10
Teaching Fractions, Decimals, and Percentages
Wafers: page 5
Animals: page 7
Fraction Circles: page 8
Hungry Birds: page 11
BSM (for Early Additive Part-Whole Multiplication):
11-3-12, 11-3-55, 12-3-13.
BSM (for Early Additive Part-Whole Fractions):
12-3-7, 12-3-49, 12-3-50, 12-3-83, 12-3-84.
Figure It Out
Multiplication:
Number, Level 2, Book 1, pages 12–15, 19
Number, Level 2, Book 2, pages 16–19
Basic Facts, Levels 2–3 pages: 10, 14, 20, 22
Number, Levels 2–3 page 13
Fractions:
Number, Level 2, Book 1, pages 17–18, 21
Number, Level 2, Book 2, pages 20–21, 24
Number, Levels 2–3, page 19
Number, Level 3, page 9
Number, Years 7–8, Link, Book One, pages 20–24
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
28
E
CA
AC
EA
AA
Key Knowledge Required for Multiplication/Division and
Fractions
The students can:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–100 for twos, fives, and tens.
•
Identify the symbols for halves, quarters, thirds, and fifths.
Grouping Based
•
Recall the number of tens in decades, e.g., four tens in 40.
•
Exploring Computation and Estimation
(Level 2)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Write and solve story problems
that involve whole numbers
using addition, subtraction,
multiplication, or division.
AM
AP
Getting Started
Transition Early Additive to Advanced Additive
Addition and Subtraction
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify objects.
•
Communicating Mathematical Ideas:
Devise and follow a set of instructions
to carry out a mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explain the meaning of the
digits in any whole number.
Exploring Computation and Estimation
(Levels 2 and 3)
•
Make sensible estimates and check
the reasonableness of answers.
•
Mentally perform calculations that
involve addition and subtraction.
•
Write and solve story problems that
involve whole numbers using
addition, subtraction, multiplication,
or division.
E
CA
AC
EA
AA
Key Knowledge Required for Addition and Subtraction
The students can:
Counting Based
•
Identify all of the numbers in the range 0–1 000.
•
Say the forwards and backwards number word sequences by
ones, tens, and hundreds in the range 0–1 000.
•
Say the number 1, 10, or 100 more or less than a given number
•
Order numbers in the range 0–1 000.
Grouping Based
•
Recall the number of groupings of tens that can be made from a
three-digit number.
•
Recall the number of tens and hundreds in centuries and
thousands.
•
Recall addition and subtraction facts to 20.
•
Round three-digit whole numbers to the nearest ten or hundred.
Knowledge Activities
Teaching Number Knowledge
AM
Counting Based
Counting: page 12
Grouping Based
AP
Slavonic Abacus: page 24
Tens and Ones: page 25
Number Mats and Number Fans: page 36
Loopy: page 39
Addition Flash Cards: page 39
BSM (for Early Additive Part-Whole):
11-3-13, 11-3-54, 11-3-55, 12-1-3, 12-1-4, 12-1-7, 12-1-9,
12-1-46, 12-1-43, 12-1-48, 12-1-50, 12-1-86, 12-1-87, 12-3-6,
12-3-7, 12-3-45, 12-3-46, 12-3-49, 12-3-65, 12-3-81, 12-3-82,
12-3-83.
Figure It Out
Number, Level 2, Book 1, page 1
Book 2, page 2, 4-6
Knowledge Being Developed for Addition and Subtraction
The students are learning to:
Counting Based
•
Identify all of the numbers in the range 0–1 000 000.
•
Say the forwards and backwards whole-number word sequences
by ones, tens, hundreds, and thousands in the range
0–1 000 000, including finding numbers that are 10, 100, and 1 000
more or less than a given number.
•
Order whole numbers in the range 0–1 000 000.
Grouping Based
•
Recall addition and subtraction facts to 20.
•
Recall groupings within 1000, e.g., 240 + 760.
•
Record the results of mental calculation using addition and
subtraction equations and diagrams.
•
Recall how many tens and hundreds there are in four-digit
numbers.
•
Round whole numbers to the nearest ten, hundred, or thousand.
•
Carry out column addition and subtraction with whole numbers
of up to four digits.
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Fans: page 4
Place Value Houses: page 5
Number Hangman: page 5
Card Ordering: page 13
Arrow Cards: page 14
Lucky Dip: page 15
Rocket-Where Will It Fit?: page 16
Number Line Flips: page 16
Squeeze - Guess My Number: page 17
Hundreds Boards and Thousands Book: page 17
Bead Strings: page 18
Who is the Richest?: page 19
Grouping Based
Close to 100: page 25
Nudge: page 26
Estimating: page 27
Traffic Lights: page 27
Zap: page 28
Swedish Rounding: page 30
Bridges: page 37
Bowl a Fact: page 37
Figure It Out
Number, Years 7–8, Link, Book Two, pages 4, 16
The students are learning to:
•
Choose appropriately from a full range of strategies to solve
addition and subtraction problems mentally, including:
Compensation, e.g., 632 – 179 as 632 – 180 + 1
Place Value, e.g., 273 – 106 as (273 – 100) - 6
Compatible Numbers, e.g., 47 + 86 + 53 as (47 + 53) + 86
Reversibility, e.g., 903 – 798 as 798 +
= 903.
Equal Additions, e.g., 754 – 529 as 755 – 530.
Decomposition, e.g., 82 – 48 as 712 - 48.
Strategy Activities
Teaching Addition, Subtraction, and Place Value
Saving Hundreds: page 31
Jumping the Number Line: page 32
Don’t Subtract – Add!: page 33
Problems like … page 33
How Many Tens and Hundreds?: page 34
Problems like … page 35
When One Number Is Near a Hundred: page 36
Problems like … page 36
Equal Additions: page 37
People’s Ages: page 38
A Balancing Act: page 38
Near Doubles: page 39
Three or More at a Time: page 40 Problems like … page 40
A Standard Written Form for Addition: page 41
Large Numbers Roll Over: page 41
Decomposition: page 42
Mixing the Methods: page 43
Mental or Written?: page 44 Estimation as a Check: page 44
Figure It Out
Basic Facts, Levels 2–3: page 4
Number, Levels 2–3: pages 6, 10, 11, 12
Basic Facts, Level 3: page 9
Number, Level 3: pages 6, 18, 20
Number, Levels 3–4: pages 13, 14
Number, Years 7–8, Link, Book One, pages 9–10, 14, 18–19
Number, Years 7–8, Link, Book Two, pages 8, 10–12, 15
Number Sense, Years 7–8, Link, Book One, page 15
Strategy Learning Outcomes
•
Use pencil and paper or calculators to work out addition and
subtraction answers where the numbers are large or untidy.
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
29
Getting Started
Transition Early Additive to Advanced Additive
Multiplication and Division
Mathematical Processes
•
Problem Solving: Devise and use
problem solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explore number patterns showing
multiples.
Exploring Computation and
Estimation (Levels 2 and 3)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Recall the basic multiplication
facts.
•
Write and solve problems that
involve whole numbers, using
addition, subtraction,
multiplication, or division.
•
Write and solve problems that
involve whole numbers and
decimals and that require a
choice of one or more of the four
arithmetic operations.
Key Knowledge Required for Multiplication and Division
The students can:
Counting Based
•
Say the forwards and backwards skip-counting sequences
in the range 0–100 for twos, threes, fives, and tens at least.
Grouping Based
•
Recall groupings of two in numbers to 20, groupings of five
in numbers to 50, and groupings of 10 in numbers to 1 000.
•
Automatically recall the multiplication and division facts
for the multiples of two, five, and 10.
•
Recall the number of groupings of tens that can be made
from a three-digit number.
•
Recall the number of tens and hundreds in centuries and
thousands.
Knowledge Activities
Knowledge Being Developed for Multiplication and Division
The students are learning to:
Grouping Based
•
Recall groupings of twos, threes, fives, and tens that are in
numbers to 100 and the resulting remainders.
•
Recall groupings of 10 and 100 that can be made from a
four- digit number.
•
Recall all the basic multiplication and division facts.
•
Recall multiplication facts for squares to 100.
•
Record the results of mental calculation using
multiplication and division equations and diagrams.
Knowledge Activities
Teaching Number Knowledge
Skip-counting on the Number Line: page 12
Beep: page 13
Using Calculators: page 15
Zap: page 28
Tens in Hundreds and More: page 29
Number Mats and Number Fans: page 36
Bowl a Fact: page 37
In and Out: page 38
Multiplication Madness: page 39
Loopy: page 39
Strategy Learning Outcomes
The students are learning to:
•
Solve multiplication and division problems from other
facts I know, using a variety of strategies, including:
Doubling, e.g., 4 x 6 as 2 x 6 x 2
Adding and subtracting, e.g., 9 x 7 as 10 x 7 – 7
Reversing, e.g., 56 ÷ 8 as 8 x = 56
Rounding, e.g., 19 x 4 as 20 x 4 – 4
•
Multiply by tens, hundreds, thousands, and other
multiples of ten.
•
Change the order of the factors to make a multiplication
problem easier, e.g., 26 x 3 as 3 x 26.
•
Solve division problems of the types:
equal sharing
finding the number of equivalent sets
Teaching Number Knowledge
Counting: page 12
Teaching Multiplication and Division
Refer to Tasks for Key Knowledge on page 12.
Strategy Activities
Teaching Multiplication and Division
Introduction: page 12
Fun with Fives: page 13
Multiplying Tens: page 14
A Little Bit More/A Little Bit Less: page 16
Turn Abouts: page 18
Long Jumps: page 19
Goesintas: pages 21
Figure It Out
Basic Facts, Levels 2–3: pages 11, 13, 15, 17, 18, 23, 24
Number, Levels 2–3: pages 14, 15, 16
Basic Facts, Level 3: pages 10, 11, 12, 15, 16, 17, 18, 19, 21, 22, 23,
24
Number, Level 3: page 7
Basic Facts, Levels 3–4: page 20
Number, Years 7–8, Link, Book One, pages 3–5, 11–13
Number, Years 7–8, Link, Book Two, pages 2–3
Number Sense, Years 7–8, Link, Book One, pages 7–9, 13
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
30
E
CA
AC
EA
AA
AM
AP
Getting Started
Transition Early Additive to Advanced Additive
Fractions and Decimals
Mathematical Processes
•
Problem Solving: Devise and use
problem solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explain the meaning of the digits in
decimal numbers with up to three
decimal places.
•
Order decimals with up to three
decimal places.
Exploring Computation and
Estimation (Level 3)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Write and solve problems that
involve whole numbers and
decimals that require a choice
of one or more of the four
arithmetic operations.
•
Solve practical problems that
require finding fractions of
whole number and decimal
amounts.
Key Knowledge Required for Fractions and Decimals
Knowledge Activities
The students can:
Counting Based
•
Identify the symbols for halves, quarters, thirds, fifths, and
tenths including fractions greater than 1.
Teaching Number Knowledge
Creating Fractions: page 6
Counting: page 12
Fraction Pieces: page 6
Teaching Fractions, Decimals, and Percentages
Refer to Tasks for Key Knowledge on pages 13–14.
•
Order fractions with like denominators, e.g., 1 and 2 .
4
4
Knowledge Activities
Teaching Number Knowledge
Number Fans: page 4
Place Value Houses: page 5
Fraction Pieces: page 6
Creating Fractions: page 6
More Geoboard Fractions: page 7
Non-Unit Fractions: page 7
Packets of Lollies: page 8
Reading Decimal Fractions: page 9
Skip-counting on the Number Line: page 12
Beep: page 13
Card Ordering: page 13
Arrow Cards: page 14
Lucky Dip: page 15
Using Calculators: page 15
Rocket – Where Will It Fit?: page 16
Number Line Flips: page 16
Squeeze – Guess My Number: page 17
Bead Strings: page 18
Super Liquorice: page 21
Zap: page 28
Strategy Learning Outcomes
The students are learning to:
•
Use a variety of multiplication and division strategies to
solve problems that involve finding a fraction of a whole
number amount, for example, 1 of 27 from 27 ÷ 3 = 9.
•
3
Use multiplication and division to compare the size of
fractions with whole numbers, especially fractions greater
than one,
e.g., Position of 17 on a numbers line showing 0–10?
•
•
3
Use multiplication and division to create equivalent ratios,
e.g., 2:3 as 10:15.
Use symmetry, area, and volume to find simple fractions of
shapes, lengths, and objects.
CA
AC
EA
AA
AM
AP
Grouping Based
•
Recall groupings of two in numbers to 20, groupings of five
in numbers to 50, and groupings of 10 in numbers to 1 000.
•
Automatically recall the multiplication and division facts
for the multiples of two, five, and 10.
Knowledge being Developed for Fractions and Decimals
The students are learning to:
Counting Based
•
Identify decimals to three places.
•
Identify symbols for any fraction, including tenths,
hundredths, thousandths, and those greater than 1.
•
Say the forwards and backwards word sequences for
halves, quarters, thirds, fifths, and tenths.
•
Order unit fractions for halves, quarters, thirds, fifths, and
tenths.
Grouping Based
•
Recall the number of tenths and hundredths in decimals to
two places.
•
Round decimals with up to two places to the nearest whole
number.
E
Strategy Activities
Teaching Fractions, Decimals, and Percentages
Birthday Cakes: page 14
Fractional Blocks: page 16
Seed Packets: page 18
Trains: page 20
Figure It Out
Number, Level 2, Book 2, pages 22–23
Number, Levels 2–3: pages 20, 21, 22, 23, 24
Number, Level 3: pages 10, 11
Number, Levels 3–4: page 3
Number, Years 7–8, Link, Book Two, pages 1, 22–23
The Numeracy Planning Assistant at www.nzmaths.co.nz/numeracy may be useful.
31
Getting Started
Transition Advanced Additive to Advanced Multiplicative
Addition and Subtraction
Mathematical Processes
•
Problem Solving: Devise and use
problem solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explain the meaning of the digits in
any whole number.
•
Explain the meaning of the digits in
decimal numbers with up to three
decimal places.
•
Order decimals with up to three
decimal places.
Exploring Computation and
Estimation (Levels 2 and 3)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Mentally perform calculations
that involve addition and
subtraction.
•
Write and solve problems that
involve whole numbers and
decimals that require a choice of
one or more of the four
arithmetic operations.
Key Knowledge Required for Addition and Subtraction
Knowledge Activities
The students can:
Counting Based
•
Identify decimals to three places.
•
Identify symbols for any fraction, including tenths,
hundredths, thousandths, and those greater than 1.
Grouping Based
•
Recall the number of tenths and hundredths in decimals to
two places.
•
Round decimals with up to two places to the nearest whole
number.
Teaching Number Knowledge
Counting Based
Lucky Dip: page 15
Number Line Flips: page 16
Hundreds Boards and Thousands Book: page 17
Bead Strings: page 18
Who is the Richest?: page 19
Grouping Based
Estimating: page 27
Traffic Lights: page 27
Bridges: page 37
Bowl a Fact: page 37
Figure It Out
Number, Years 7–8, Link, Book Two, pages 4, 16
Knowledge Being Developed for Addition and Subtraction
The students are learning to:
Counting Based
•
Say the forwards and backwards decimal word sequences
by thousandths, hundredths, tenths, ones, tens, etc.,
starting at any whole number in common usage.
•
Say the number one-thousandth, one-hundredth, onetenth, one, and ten, etc., before and after any given whole
number in common usage.
•
Order decimals to three places.
Grouping Based
•
Recall the number of groupings of tens, hundreds, and
thousands that can be made from a number of up to seven
digits.
•
Round whole numbers and decimals, with up to two
places, to the nearest whole number, or tenth.
•
Record the results of mental calculations using equations,
and diagrams, for example, empty number line.
•
Carry out column addition and subtraction for whole
numbers.
Knowledge Activities
Teaching Number Knowledge
Counting Based
Number Fans: page 4
Place Value Houses: page 5
Number Hangman: page 5
Card Ordering: page 13
Arrow Cards: page 14
Rocket-Where Will It Fit?: page 16
Squeeze - Guess My Number: page 17
Grouping Based
Close to 100: page 25
Nudge: page 26
Zap: page 28
Swedish Rounding: page 30
Figure It Out
Number, Years 7–8, Link, Book Two, pages 17–19
Number, Years 7–8, Level 4, Book Three, pages 13, 15, 22, 24
Number, Years 7–8, Level 4, Book Four, page 3
Number, Years 7–8, Level 4, Book Four, pages 12–13, 24
Strategy Learning Outcomes
The students are learning to:
•
Use a broad range of mental strategies to solve addition
and subtraction problems with whole numbers and
decimals, including:
Tidy numbers, e.g., 3.1 – 2.79 as (3.1 – 2.8) + 0.01
Place value, e.g., 3.06 + 2.7 as (3.06 + 2) + 0.7
Reversibility and commutativity,
e.g., 6.7 – 4.9 as 4.9 +
= 6.7, and 2.3 + 4.8 as 4.8 + 2.3.
Exploring equal addition, e.g., 4.2 – 2.8 as 4.4 – 3.0
Exploring decomposition, e.g., 7.1 – 2.8 as 6.11 – 2.8.
Using negative numbers, e.g., 64 – 38 as 4 – 8 = - 4,
60 – 30 = 30, 30 – 4 = 26.
•
Use a pencil and paper or a calculator to add and subtract
decimals and whole numbers where the numbers are
difficult or untidy.
32
Strategy Activities
Teaching Fractions, Decimals, and Percentages
Pipe Music with Decimals: page 24
Candy Bars: page 30
Figure It Out
Number, Level 3: page 5
Number, Levels 3–4: pages 6, 7
Basic Facts, Levels 3–4: page 19
Number, Years 7–8, Level 4, Book Three, page 12
Number, Years 7–8, Level 4+, Book Six, page 18
Number Sense, Years 7–8, Link, Book One, pages 15, 21
Number Sense, Years 7–8, Book Two, pages 1, 7–8, 11
E
CA
AC
EA
AA
AM
AP
Getting Started
Transition Advanced Additive to Advanced Multiplicative
Multiplication and Division
Mathematical Processes
•
Problem Solving: Devise and use
problem solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explain the meaning of the digits in
any whole number.
Exploring Computation and
Estimation (Level 3)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Recall the basic multiplication
facts.
•
Write and solve problems that
involve whole numbers and
decimals and require a choice
of one or more of the four
arithmetic operations.
Knowledge Activities
The students can:
Grouping Based
•
Recall groupings of twos, threes, fives, and tens that are in
numbers to 100 and the resulting remainders.
•
Recall groupings of 10 and 100 that can be made from a
four- digit number.
•
Recall all the basic multiplication and division facts.
•
Recall multiplication facts for squares to 100.
•
Record the results of mental calculation using
multiplication and division equations and diagrams.
Teaching Number Knowledge
Number Mats and Number Fans: page 36
Bowl a Fact: page 37
In and Out: page 38
Multiplication Madness: page 39
Loopy: page 39
Teaching Multiplication and Division
Refer to Tasks for Key Knowledge on page 23.
Knowledge Being Developed for Multiplication and Division
The students are learning to:
Grouping Based
•
Recall the number of groupings of tens, hundreds, and
thousands that can be made from a number of up to seven
digits.
•
Recall multiplication and division facts to 10 x 10, and the
corresponding division facts.
•
Record the results of mental calculations using equations
and diagrams, for example, empty number line.
•
Carry out a short written algorithm for multiplication and
division of a three-digit whole number by a single-digit
number.
Knowledge Activities
Teaching Number Knowledge
Skip-counting on the Number Line: page 12
Beep: page 13
Using Calculators: page 15
Zap: page 28
Tens in Hundreds and More: page 29
Dividing? Think About Multiplying First: page 39
Multiplication Flash Cards: page 40
Figure It Out
Number, Years 7–8, Level 4, Book Three, pages 8–9
Number, Years 7–8, Level 4, Book Five, pages 6–7
The students are learning to:
•
Use a diverse range of strategies to solve problems
involving multiplication and division with whole numbers
including:
Compensating from tidy numbers
Place value
Reversibility and commutativity,
e.g., 84 ÷ 7 as 7 x
= 84 , or 2.37 x 6 as 6 x 2.37.
Proportional adjustment, for example, doubling and
halving.
Changing the starting number, the dividing number, or
both numbers, e.g. 201 ÷ 3 as (99 ÷ 3) + (99 ÷ 3) + (3 ÷ 3).
•
Solve division problems that have remainders and express
the answer in fraction, decimal, or whole number form,
e.g., 76
•
÷ 5 as 15 15 , or 15.2,
or 15 r 1.
Use written working forms or calculators where the
numbers are difficult and/or untidy.
CA
AC
EA
AA
AM
Key Knowledge Required for Multiplication and Division
Strategy Learning Outcomes
E
AP
Strategy Activities
Teaching Multiplication and Division
Introduction: pages 23–25
Cut and Paste: page 25
Multiplication Smorgasboard: page 27
Proportional Packets: page 29
Royal Cooking Lessons: page 31
Remainders: page 33
Paper Power: page 35
Cross Products: page 38
Figure It Out
Basic Facts, Level 3: pages 13, 14, 20
Number, Level 3: pages 4, 21, 22, 23, 24
Number, Levels 3–4: pages 15, 16, 17, 18, 20
Basic Facts, Levels 3–4: pages 3, 12, 13, 15, 16, 17
Number, Years 7–8, Link, Book Two, pages 6–7, 13–14
Number, Years 7–8, Level 4, Book Three, pages 7–10, 16, 23
Number, Years 7–8, Level 4, Book Five, pages 1, 2 , 5, 17
Number Sense, Years 7–8, Link, Book One, pages 1–3, 10–12, 14,
17, 20, 23–24
Number Sense, Years 7–8, Book Two, pages 2, 16, 24
33
Getting Started
Transition Advanced Additive to Advanced Multiplicative
Fractions, Decimals, Proportions, and Ratios
Mathematical Processes
•
Problem Solving: Devise and use
problem solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 3)
•
Explain the meaning of the digits in
decimal numbers with up to three
decimal places.
•
Order decimals with up to three
decimal places.
Key Knowledge Required for Fractions, Decimals,
Proportions, and Ratios
The students can:
Counting Based
•
Identify decimals to three places.
•
Identify symbols for any fraction, including tenths,
hundredths, thousandths, and those greater than 1.
•
Say the forwards and backwards word sequences for
halves, quarters, thirds, fifths, and tenths.
•
Order unit fractions for halves, quarters, thirds, fifths, and
tenths.
Grouping Based
•
Recall the number of tenths and hundredths in decimals to
two places.
•
Round decimals with up to two places to the nearest whole
number.
Knowledge Activities
Knowledge Being Developed for Fractions, Decimals,
Proportions, and Ratios
The students are learning to:
Grouping Based
•
Order decimals to three places, for example, 6.25 and 6.3.
•
Order fractions, including halves, quarters, thirds, fifths,
and tenths.
•
Recall equivalent fractions for halves, thirds, quarters,
fifths, and tenths with numbers to 100 and with 1 000.
•
Recall fraction ↔ decimal ↔ percentage conversions for
halves, thirds, quarters, fifths, and tenths.
•
Record the results of mental calculations using equations
and diagrams, for example, empty number line.
•
Round whole numbers and decimals with up to two places
to the nearest whole number or tenth.
Knowledge Activities
Strategy Learning Outcomes
The students are learning to:
•
Use mental strategies based on multiplying and dividing to
solve problems with fractions, decimals, proportions, and
ratios, including:
Finding equivalent fractions, decimals, and percentages,
e.g., 3 = 0.75 = 75%
-
4
Using unit fractions,
e.g., 5 of 72 by 1 of 72 = 9, 5 x 9 = 45.
-
34
8
8
Place value,
e.g., 6 x 3.4 = (6 x 3) + (6 x 0.4).
Compensating from tidy numbers or fractions,
e.g., 2.9 x 6.3 as 3 x 6.3 = 18.9, 0.1 x 6.3 = 0.63,
18.9 – 0.63 = 18.27.
Exploring Computation and
Estimation (Level 3)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Recall the basic multiplication
facts.
•
Write and solve problems that
involve whole numbers and
decimals that require a choice
of one or more of the four
operations.
•
Solve practical problems that
require finding fractions of
whole number and decimal
amounts.
E
CA
AC
EA
AA
Teaching Number Knowledge
Fraction Pieces: page 6
Rocket – Where Will It Fit?: page 16
Number Line Flips: page 16
Squeeze – Guess My Number: page 17
Bead Strings: page 18
Zap: page 28
Figure It Out
Number, Years 7–8, Link, Book Two, pages 9, 20
Number, Years 7–8, Level 4, Book Three, pages 2-3
Number, Years 7–8, Level 4, Book Five, pages 16
Teaching Fractions, Decimals, and Percentages
Refer to Tasks for Key Knowledge on pages 22–23.
Teaching Number Knowledge
Number Fans: page 4
Place Value Houses: page 5
Creating Fractions: page 6
More Geoboard Fractions: page 7
Non-Unit Fractions: page 7
Packets of Lollies: page 8
Reading Decimal Fractions: page 9
More Reading of Decimal Fractions: page 10
Linking Money and Decimal Fractions: page 10
Card Ordering: page 13
Arrow Cards: page 14
Using Calculators: page 15
Who Has More Cake?: page 20
Super Liquorice: page 21
Who Wins?: page 21
Nudge: page 26
Sensible Rounding: page 30
Locating Decimal Fractions: page 31
Digits on the Move: page 32 The Same But Different: page 33
Figure It Out
Number, Years 7–8, Link, Book Two, pages 21, 24
Number, Years 7–8, Level 4, Book Three, pages 1, 4, 17–19
Number, Years 7–8, Level 4, Book Five, pages 11, 19, 23
Strategy Activities
Teaching Fractions, Decimals, and Percentages
Introduction: pages 22–24
Deci-mats: page 27
Candy Bars: page 30
Figure It Out
Number, Level 3: pages 12, 13, 14
Number, Levels 3–4: pages 4,–7, 9–12, 22–24
Basic Facts, Levels 3–4: pages 22, 23
Number Sense, Years 7–8, Link, Book One, pages 6, 16, 19, 22
Number Sense, Years 7–8, Book Two, pages 4, 6, 10, 18-19, 22-23
AM
AP
Getting Started
Transition Advanced Additive to Advanced Multiplicative
Number Patterns and Relationships
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 4)
•
Explain the meaning of negative
numbers.
•
Explain the meaning and evaluate
powers of whole numbers.
Exploring Computation and
Estimation (Level 4)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Explain satisfactory algorithms
for addition, subtraction, and
multiplication.
•
Demonstrate knowledge of the
conventions for order of
operations.
CA
AC
EA
AA
AM
AP
Key Knowledge Required for Number Patterns
Knowledge Activities
The students can:
•
Recall all the basic multiplication and division facts.
•
Recall multiplication facts for squares to 100.
•
Recall groupings of 10 and 100 that can be made from a
four-digit number.
Teaching Number Knowledge
Skip-counting on the Number Line: page 12
Beep: page 13
Using Calculators: page 15
Zap: page 28
Tens in Hundreds and More: page 29
Dividing? Think About Multiplying First: page 39
Multiplication Flash Cards: page 40
Knowledge Being Developed for Number Patterns
The students are learning to:
•
Recall the number of groupings of tens, hundreds, and
thousands that can be made from a number of up to seven
digits.
•
Recall multiplication and division facts to 10 x 10.
•
Record the results of mental calculations using equations
and diagrams, for example, empty number line.
•
Carry out short multiplication and division of a three-digit
whole number by a single-digit number.
Knowledge Activities
Teaching Number Knowledge
Arrow Cards: page 14
Strategy Learning Outcomes
The students are learning to:
•
Find out whether a whole number is prime or non-prime.
•
Solve problems that involve the ordering, and addition and
subtraction of integers.
•
Find powers and square roots of whole numbers,
particularly squares and cubes.
•
Use factorials to solve problems.
•
Find general rules for finding unknown members of a
repeating sequence of numbers or objects, for example,
- recursion rules,
e.g., 6, 10, 14, 18, … (+ 4 to the previous number)
- function rules,
e.g., 1
2
3
4
…
4
7
10
13
…
Bottom number equals (top number x 3) + 1
•
Show and interpret relationships using equations, tables,
and graphs.
•
Solve number problems that have one or more
“unknowns”.
E
Strategy Activities
Figure It Out
Number, Years 7–8, Level 4, Book Four, pages 1–7 (Prime Nos)
Number, Years 7–8, Level 4, Book Four, pages 8–13, 15, 21
(Integers)
Number, Years 7–8, Level 4, Book Four, pages 14, 16-20, 22–24
Number, Years 7–8, Level 4, Book Five, page 10 (Powers and
square roots)
Number, Years 7–8, Level 4, Book Five, pages 4, 21 (Factorials)
Number Sense, Years 7–8, Book Two, pages 5, 9, 12
Algebra, Levels 3–4, pages 1–24
Algebra, Years 7–8, Link, Book One, pages 1–24
Algebra, Years 7–8, Book Two, pages 1–9, 11–13, 15, 20–21
Algebra, Years 7–8, Book Three, pages 1–7, 10–17, 20–21
Algebra, Years 7–8, Level 4+, Book Four, pages 6–7
35
Getting Started
Transition Advanced Multiplicative to Advanced Proportional
Fractions, Decimals, Proportions, and Ratios
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 4)
•
Find fractions equivalent to the one
given.
•
Express a fraction as a decimal and
vice versa.
•
Express a decimal as a percentage
and vice versa.
•
Express quantities as fractions or
percentages of a whole.
Exploring Computation and
Estimation (Level 4)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Find a given fraction or
percentage of a quantity.
•
Write and solve problems
involving decimal multiplication
and division.
•
Explain satisfactory algorithms
for addition, subtraction, and
multiplication.
Key Knowledge Required for Fractions and Decimals
Knowledge Activities
The students can:
Grouping Based
•
Order decimals to three places, for example, 6.25 and 6.3.
•
Order fractions, including halves, quarters, thirds, fifths,
and tenths.
•
Recall equivalent fractions for halves, thirds, quarters,
fifths, and tenths with numbers to 100 and with 1 000.
•
Recall fraction ↔ decimal ↔ percentage conversions for
halves, thirds, quarters, fifths, and tenths.
•
Record the results of mental calculations using equations
and diagrams, for example, empty number line.
•
Round whole numbers and decimals with up to two places
to the nearest whole number or tenth.
Teaching Number Knowledge
Place Value Houses: page 5
More Geoboard Fractions: page 7
Non-Unit Fractions: page 7
Packets of Lollies: page 8
Reading Decimal Fractions: page 9
Card Ordering: page 13
Arrow Cards: page 14
Using Calculators: page 15
Super Liquorice: page 21
Rocket – Where Will It Fit?: page 16
Squeeze – Guess My Number: page 17
Figure It Out
Number, Years 7–8, Link, Book Two, pages 21, 24
Number, Years 7–8, Level 4, Book Three, pages 1, 4, 17–19
Number, Years 7–8, Level 4, Book Five, pages 11, 19, 23
Teaching Fractions, Decimals and Percentages
Refer to Tasks for Key Knowledge on pages 32-33.
Knowledge Being Developed for Fractions and Decimals
The students are learning to:
Counting Based
•
Say the forwards and backwards decimal word sequences
by thousandths, hundredths, tenths, ones, and tens,
starting at any decimal number.
•
Order fractions, decimals, and percentages.
Grouping Based
•
Recall the number of tenths, hundredths, and onethousandths in numbers of up to three decimal places.
•
Recall what happens when a whole number or decimal is
multiplied or divided by the power of 10.
•
Recall fraction ↔ decimal ↔ percentage conversions for
fractions in common use, e.g., eighths, tenths, twentieths.
•
Record the results of mental calculations using equations.
•
Carry out column addition and subtraction for whole
numbers and decimals to three places.
•
Carry out short multiplication and division of whole
numbers and decimals by a single-digit number.
•
Carry out multiplication of a three- or four-digit whole
number by a two-digit whole number.
Knowledge Activities
Teaching Number Knowledge
More Geoboard Fractions: page 7
Non-Unit Fractions: page 7
Packets of Lollies: page 8
Reading Decimal Fractions: page 9
More Reading of Decimal Fractions: page 10
Linking Money and Decimal Fractions: page 10
Measurement and Zeros: page 11
Who Has More Cake?: page 20
Little Halves and Big Quarters: page 20
Who Wins?: page 21
Who Gets More?: page 22
Equivalent Fractions, Decimals, and Percentages: page 22
Difficult Fractions to Percentages: page 23
The students are learning to:
Choose appropriately from a broad range of mental strategies to
solve problems with fractions and decimals, ratios, and
proportions, including:
Applying reversibility to find fractions, decimals and
percentages, e.g., 13.6 ÷ 0.4 = , as 0.4 x
= 13.6
Applying equivalent ratios and proportions by finding
common factors,
Strategy Activities
Teaching Fractions, Decimals, and Percentages
Introduction: pages 32–34
Hot Shots: page 34
Mixing Colours: page 37
Folding Fractions and Decimals: page 39
Figure It Out
Number, Levels 3–4: pages 19, 21
Basic Facts, Levels 3–4: pages 8, 9
Number, Years 7–8, Level 4, Book Three, pages 5–6, 11, 14, 20–21
Number, Years 7–8, Level 4, Book Five, pages 8–9, 14–15, 20–22
Number, Years 7–8, Level 4, Book Six, pages 10–11, 17, 21
Number Sense, Years 7–8, Book Two, pages 13–15, 20–21
Strategy Learning Outcomes
e.g., 28 ÷ 42 =
-
42
-
6
3
Finding proportional relationships between measures,
e.g., 12 → 15 as
-
% as 28 = 4 = 2 = 66.6%
→ 25 by 12 is 4 of 15, 20 is 4 of 25
5
5
Using proportional adjustment,
e.g., 7.2 ÷ 0.3 as 8 x 0.9 = 7.2 so 24 x 0.3 = 7.2
Converting from ratios to proportions and vice versa,
e.g., 3 : 5 out of a total of 96 as 3 : 5 is 3 , 3 of 96 is 36
so 36 : 60
36
8
8
E
CA
AC
EA
AA
AM
AP
Getting Started
Transition Advanced Multiplicative to Advanced Proportional
Number Patterns and Relationships
Mathematical Processes
•
Problem Solving: Devise and use
problem-solving strategies to explore
situations mathematically.
•
Use equipment appropriately when
exploring mathematical ideas.
•
Logic and Reasoning: Classify
objects.
•
Communicating Mathematical
Ideas: Devise and follow a set of
instructions to carry out a
mathematical activity.
•
Record and talk about the results of
mathematical exploration.
Exploring Numbers (Level 4)
•
Explain the meaning of negative
numbers.
•
Explain the meaning and evaluate
powers of whole numbers.
Exploring Computation and
Estimation (Level 4)
•
Make sensible estimates and
check the reasonableness of
answers.
•
Explain satisfactory algorithms
for addition, subtraction, and
multiplication.
•
Demonstrate knowledge of the
conventions for order of
operations.
E
CA
AC
EA
AA
AM
AP
Key Knowledge Required for Number Patterns
Knowledge Activities
The students can:
•
Order integers.
•
Recall square numbers to 100.
•
Recall multiplication facts to 10 x 10, and the
corresponding division facts.
•
Record the results of calculations using equations.
Teaching Number Knowledge
Arrow Cards: page 14
Knowledge Being Developed for Number Patterns
The students are learning to:
•
Order integers and positive and negative decimals
•
Recall square numbers to 100, and the corresponding
square roots.
•
Recall the prime numbers to 20, at least.
•
Express whole numbers in standard form.
Knowledge Activities
The students are learning to:
•
Express whole numbers as the product of prime numbers
(prime factorisation).
•
Solve problems involving the addition, subtraction, and
multiplication of integers.
•
Solve problems with powers, including exponential
growth, multiplication of powers with the same base (e.g.,
16 x 8), estimating and finding square roots of whole
numbers.
•
Work in number bases other than 10.
•
Find general rules for finding unknown members of a
repeating sequence of numbers or objects, including nonlinear repeating patterns, for example,
- recursion rules,
e.g., 1
4
9
16
25…
+3
+5
+7
+9
- function rules,
e.g., 1
2
3
4
…
2
4
8
16
…
Bottom number equals (2 to the power of top number, 2n)
•
Show and interpret relationships using algebraic
expressions and equations, tables, and graphs.
•
Solve problems that involve one or more unknowns,
including equations with letters,
e.g., r = (4 x t) + 3, if r = 51 what is t?
Strategy Activities
Figure It Out
Number, Years 7–8, Level 4+, Book Six, pages 2–3, 7 (Prime Nos)
Number, Years 7–8, Level 4+, Book Six, pages 14–16, 22–24
(Integers)
Number, Years 7–8, Level 4+, Book Six, pages 4–5, 8, 18–20
(Powers and square roots)
Number, Years 7–8, Level 4+, Book Six, pages 12–13 (Bases)
Strategy Learning Outcomes
Number, Years 7–8, Level 4+, Book Six, pages 6, 9 (General rules)
Algebra, Years 7–8, Level 4, Book Two, pages 10, 14, 16–19, 22–24
Algebra, Years 7–8, Level 4, Book Three, pages 8–9, 18–19, 22–24
Algebra, Years 7–8, Level 4+, Book Four, pages 1–5, 8–24
37
Week:
Room:
Hot Spot Focus:
SLO:
Monday
G
Tuesday
Pr
Wednesday
T
Thursday
G
Friday
Pr
T
KA
Pr
T
KA
T
G
Pr
T
G
Pr
T
KA
Pr
T
Pr
T
G
Pr
T
KA
Pr
T
KA
Pr
Warm-down:
SLO = Strategy Learning Outcome, T = Teaching, Pr = Practice, G = Games, KA = Knowledge Activity
Getting Started
38
Weekly Number Plan
Getting Started
Weekly Number Plan
Week:
Room:
Hot Spot
Focus
SLO:
SLO:
SLO:
Day 1
G
T
Pr
T
Pr
KA
Warm-down:
Day 2
Pr
G
T
KA
T
Pr
Warm-down:
Day 3
T
Pr
G
Pr
KA
T
Warm-down:
Day 4
G
T
Pr
T
Pr
KA
Warm-down:
Day 5
Pr
G
T
KA
T
Pr
Warm-down:
Name:
SLO = Strategy Learning Outcome, T = Teaching, Pr = Practice, G = Games, KA = Knowledge Activity
39
Getting Started
Name:
E
Date
achieved
I can …
Emergent
I am learning to …
Knowledge
• Read
0 1 2 3 4 5, first, and then …
0 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5, first, and then …
1 2 3 4 5 6 7 8 9 10
5 4 3 2 1 0, first, and then …
10 9 8 7 6 5 4 3 2 1 0
The number after a number
• Count
• Count
• Say
3, 4, 5, ___
• Say
The number before a number
___2, 3, 4
• Order
5 3 1 2 4 0, first, and then …
5 3 8 1 2 9 7 10 4 6 0
Strategy
• Count
a set of objects
1
• Get
3
4
5
a set of objects, like seven teddies
1
40
2
2
3
4
5
6
7
Getting Started
E
Name:
CA
Date
achieved
I can …
One-to-one Counting
I am learning to …
Knowledge
• Read
• Count
0 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
• Count
10
9 8 7 6 5 4 3 2 1 0
• Say
The number after a number
• Say
4, 5, 6 ___
The number before a number
• Order
___3, 4, 5
5 3 8 1 2 9 7 10 4 6 0
• Know
Patterns to five
Strategy
• Join
Groups of objects together
and
• Split
A number of objects into groups
41
Getting Started
Name:
CA
Date
achieved
I can …
Counting from One on Materials
I am learning to …
Knowledge
• Read
• Count
• Count
• Say
• Order
• Read
0 1 2
11 12 13
3 4 5 6 7
14 15 16 17
1 2 3 4 5 6 7 8 9
… 20
8 9 10
18 19 20
10
11
20 19 18 17 16 15 14 13 12 11 10
…0
The number after a number between
0 and 20
4, 5, 6 _____ … 11, 12, 13 ____
The number before a number between
0 and 20
____ 4, 5, 6 … ____ 11, 12, 13
Numbers to 20
8 17 3 14 11 as 3 8 11 14 17
1
1
2
4
• Know
Groups within five
+
+
Groups with five
Strategy
• Solve +
and –
problems
up to 10
by
42
12
Counting all the objects
and
3
+
2
Getting Started
CA
Name:
Counting from One by Imaging
I am learning to …
Knowledge
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
• Read
•
Count
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
•
Count
20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0
•
Say
The number after a number in the range 1-20
Say
14, 15, 16, ______
The number before a number in the range 1-20
•
•
Order
•
Know
•
Know
•
Know
•
Skipcount
forwards
Strategy
• Solve +
and problems
to 10 by
•
Share
objects
equally to
make
1
’s and 41 ’s
2
Date
achieved
I can …
_____14, 15, 16
7, 15, 12, 6, 18, 13, 5, 19, 10, 1, 8, 16, 9, 3, 11, 2, 14, 17, 20, 0, 4
1
1
2
4
Groupings within 10
5+5
6+4
3+7
Doubles to 10
2+2
3+3
4+4
5+5
In twos
2 4 6 8 10 12 14 16 18 20
In fives
5 10 15 20 25 30 35 40 45 50
In tens
10 20 30 40 50 60 70 80 90 100
Counting all the objects in my head
4+3=
Share 12 muffins amongst four
1, 2, 3, 4,
5, 6, 7
1
4
of 12
43
Getting Started
CA
Name:
AC
Advanced Counting
I am learning to …
Knowledge
• Read
Any number up to 100, for example, 17, 26, 38, 47, 53, 74,
86, 99
• Count
Forwards from any number up to 100, for example, 34, 35,
36 …
• Count
Backwards from any number from 100, for example, 47,
46, 45 …
• Say
The number after a number in the range 1-100
54, 55, 56, ______
The number before a number in the range 1-100
• Say
• Order
• Count
• Read
_____54, 55, 56
Numbers to 100, for example, 37, 26, 55, 73, 22, 34, 45
Forwards and backwards in twos, fives, and tens
Unit fractions
1
2
• Know
10 + 5 = 15
• Know
• Know
Strategy
• Solve +
and problems
by
• Solve x
problems
by
• Find
1
2
and 41 of
sets
44
1
4
1
3
1
5
1
10
Teen numbers
10 + 8 = 18 10 +
= 16
Tens in decades, for example, “How many tens in 60?”
Tens that add to 100, for example, 30 + 70
Doubles to 20 and halves
1
1
3 + 3 6 - 3 2 of 6 7 + 7 14 - 7
of 14
2
Counting on or back from the largest number, in my head.
16 + 3
17, 18, 19
Skip-counting in twos, fives, or tens
4x5
Share 12 muffins among four
5, 10, 15, 20
1
4
of 12
Date
achieved
I can …
Getting Started
AC
Name:
Date
achieved
I can …
Early Additive Part-Whole
I am learning to …
Knowledge
• Read
• Count
Numbers to 1 000
333
479
983
Forwards by ones, tens, and hundreds up to 1 000
Backwards by ones, tens, and hundreds from 1 000
• Say
The number one more, 10 more, 100 more than numbers to 1 000
The number one less, 10 less, 100 less than numbers to 1 000
• Order
• Know
• Order
Numbers from 0 – 1 000
58
376
1
1
2
1
4
1
3
1
5
837
1
10
Fractions with the same denominators
1
4
>
• Skipcount
Forwards and backwards in threes
3 6 9 12 15
30
• Round
Three-digit numbers to the nearest 10 or 100
246
250 (nearest 10)
Addition and subtraction facts to 20
12 + 8 = 20
11 + 9 = 20
20 – 12 = 8
20 – 9 = 11
20 – 8 = 12
20 – 11 = 9
• Know
Strategy
• Solve +
and –
problems
in my head
by
• Use
repeated
addition to
solve x
problems
by
• Find a
fraction of
a number
by
3
4
Using doubles, for example, 8 + 7 as 8 + 8 – 1
Using fives, for example, 8 + 7 as 5 + 3 + 5 + 2
Using making tens, for example, 8 + 7 as 10 + 5
Using making tens, for example, 19 + 6 as 20 + 5,
29 + 8 as 30 + 7
Using place value, for example,
33 + 16 as 30 + 10 + 3 + 6
Twos 2 + 2 + 2 + 2 = 4 x 2
Threes 3 + 3 + 3 + 3 + 3 = 5 x 3
Fours 4 + 4 + 4 = 3 x 4
Fives 5 + 5 + 5 + 5 + 5 = 5 x 5
Tens 10 + 10 = 2 x 10
Using repeated addition or subtraction,
for example,
1
3
of 12 as 4 + 4 + 4
for example, 12 – 2 – 2 – 2 = 6,
6 – 2 – 2 – 2 = 0,
1
3
of 12 is 2 + 2
45
EA
Getting Started
EA
Name:
Advanced Additive Part-Whole
I am learning to …
Knowledge
Numbers to 1 000 000, for example, 1 374; 98 765; 763 104
• Read
and
order
Decimals up to three places, for example, 0.764; 0.14; 0.8
• Read
Symbols for any fraction, for example, 1 1
• Say
• Know
• Order
• Know
• Know
• Round
• Know
• Record
Strategy
• Solve +
and –
problems
by
• Solve x
and ÷
problems
by
• Solve
problems
with
fractions
46
3
4
5
13
3
The number 1 000 more/1 000 less
How many tens and hundreds are in four-digit numbers
Fractions with the different denominators, for example,
1
4
2
3
4
5
Groups within 1 000, for example, 240 + 760
How many twos, threes, fives, and tens in numbers to 100 and any
remainders, for example, threes in 17
Whole numbers to the nearest 10, 100, and 1 000,
for example, 5 508
6 000
Decimals to the nearest whole number,
for example, 3.49
3
Multiplication facts for squares to 100,
for example, 5 x 5, 9 x 9
Results of calculations using empty number lines and written algorithms
Using compensation from tidy numbers
725 - 389 as 725 - 400 + 11 = 336
Using place value
376 + 431 as 300 + 400 + 70 + 30 + 6 + 1 = 807
Using compatible numbers
35 + 37 + 65 as (35 + 65) + 37 = 100 + 37 = 137
Using reversibility
814 - 789 =
as 789 +
= 814
Using equal additions
72 - 37 as 75 - 40 (add three to both numbers)
Using decomposition
83 - 28 as renaming 83 so 87 13 - 28
Using doubling, for example, 2 × 6 = 12 so 4 × 6 = 24
Deriving facts, for example, 2 × 6 = 12 so 3 × 6 = 12 + 6 = 18
Using reversibility, for example, 7 × 4 = 28 so 28 ÷ 4 = 7
Using proportional adjustment,
for example, 3 × 12 is the same as, 6 × 6 = 36 (doubling and halving),
or 24 ÷ 4 = 6 so 24 ÷ 8 = 3
Mentally, using known multiplication and division facts,
for example,
1
3
of 36 as, 3 × 12 = 36 so,
1
3
of 36 = 12
AA
Date
achieved
I can…
Getting Started
AA
Name:
Advanced Multiplicative Part-Whole
Date
achieved
I can …
I am learning to …
Knowledge
•
•
•
•
•
Count
Say
Order
Order
Know
• Know
Forwards and backwards in 0.001s, 0.01s, 0.1s, ones, tens
Number 0.001, 0.01, 0.1, 1, 10 before/after any whole number
Decimals up to three places, for example, 6.25 and 6.3
Fractions including halves, thirds, quarters, fifths, tenths
Groupings of 10, 100, 1 000, made from a number of up to
seven digits, for example, tens in 47 562
Equivalent fractions and proportions for 1 s, 1 s, 1 s, 1 s, 1 s
2
3
4
with denominators of 10, 100, 1 000, for example,
• Round
• Recall
• Recall
Whole numbers and decimals to the nearest 1 or
x and ÷ facts to 10 x 10, 100 ÷ 10
Fraction decimal percentage conversions for
and
Strategy
• Solve +
and problems by
using
• Solve x
and ÷
problems
using
• Solve
problems with
fractions,
decimals,
proportions,
and ratios,
using
1
4
1
10
s, for example,
3
4
1
2
s,
= 0.75 = 75%
5
=
1
10
1
s, 1
3
4
10
25
100
s,
1
5
s,
Compensation from tidy numbers,
e.g., 3.2 + 1.95 as 3.2 + 2 – 0.05
Place value, e.g., 8.65 – 4.2 = (8 – 4) + (0.6 – 0.2) + 0.05
or 8.65 – 4 = 4.65 then 4.65 – 0.2 = 4.45
Reversibility and commutativity,
e.g., 6.03 – 5.8 =
as 5.8 + = 6.03 (reversibility)
or
+ 3.98 = 7.04 as 3.98 + = 7.04 (commutativity)
Equal additions, e.g., 7.24 – 3.8 as 7.44 – 4.0 = 3.44
Using negatives, e.g., 6.4 – 2.5 as 0.4 – 0.5 is -0.1;
6.0 – 2.0 = 4.0; 4.0 – 0.1 = 3.9
Decomposition, e.g., 9.25 – 6.83 as 8. 125 – 6.83
Compensation from tidy numbers, e.g., 6 x 998 as,
(6 x 1 000) – (6 x 2) or 56 ÷ 4 using (60 ÷ 4) – 1
Place value, e.g., 28 x 7 as (20 x 7) + (8 x 7)
or 72 ÷ 4 as (40 ÷ 4) + (32 ÷ 4)
Reversibility, e.g., 96 ÷ 6 as 6 x
= 96
and commutativity, e.g., 17 x 6 as 6 x 17
Proportional adjustment, e.g., 4 x 18 as 8 x 9
or 81 ÷ 3 as (81 ÷ 9) x 3
Written working forms or calculators where the numbers are
difficult and/or untidy
Unit fractions, e.g., 4 x 18 as ( 1 x 18) x 4
9
9
Place value, e.g., 3.4 x 8 as (3 x 8) + (0.4 x 8)
= 24 + 3.2 = 27.2
Compensating from tidy numbers or fractions, e.g., 83 x 28 as
1
2
of
3
4
x 28
or 1.9 x 3.4 as (2 x 3.4) – (0.1 x 3.4)
Using equivalent fractions and ratios,
e.g., 40% of 35 as 2 of 35 = 14
5
47
AM
Getting Started
AM
Name:
Advanced Proportional Part-Whole
I am learning to …
Knowledge
• Count
Forwards and backwards in 0.001s, 0.01s, 0.1s, ones, tens,
etc.
• Say
The number 0.001, 0.01, 0.1, 1, 10 before/after decimal
numbers
• Order
Fractions, decimals, and percentages, e.g., 40%, 3 , 0.5
• Know
• Know
• Round
• Recall
Strategy
• Solve x and
÷ problems
with fractions
and decimals
by
• Find
• Solve
problems with
ratios and
proportions
by
48
5
Number of tenths, hundredths, and thousandths that are
in numbers up to three decimal places, e.g., tenths in 45.6.
What happens when a whole number or decimal is x or ÷
by a power of 10, e.g., 4.5 x 100; 67.3 ÷ 10
Decimals to the nearest 100, 10, 1,
1
10
,
1
100
Fractional, decimal, and percentage conversions for
commonly used fractions, e.g., 81 = 0.125 = 12.5%
Conversion between fractions and decimals,
e.g., 0.75 x 2.4 as 43 x 2.4
Place value, e.g., 0.15 x 3.6, as (0.1 x 3.6) + (0.05 x 3.6)
Doubling and halving, etc., e.g., 7.2 ÷ 0.4 as (7.2 ÷ 0.8) x 2
Commutativity, e.g., 48 x 0.125 as 0.125 x 8 = 81 of 8 = 1
Multiplying numerators and denominators,
e.g., 43 x 25 as 43 xx 25
Fractions, decimals, and percentages of given amounts,
e.g., 65% of 24 as 50% of 24 is 12, 10% of 24 is 2.4,
and 5% is 1.2 so 12 + 2.4 + 1.2 = 15.6
Finding equivalent ratios with a common factor,
e.g., 21:28 as
:8 as 21:28 is 3:4 so 6:8
18
e.g., 27 = 23 so 23 = 10
15
Finding a multiplier between the units,
e.g., 18 out of 27 as 10 out of 15 by multiplying 15 by
2
3
Date
achieved
I can …
AP
Book 3
Numeracy Professional Development Projects 2003 (Draft)
NUMERACY AND THE MATHEMATICS CURRICULUM
Numeracy arises out of effective mathematics teaching.
All the strands within Mathematics in the New Zealand
Curriculum are important in the pathway to numeracy.
Number is central to this pathway, although the relative
emphasis on this strand changes with the stages of
schooling:
The general aims of mathematics education in New
Zealand define the features of school programmes that
contribute to the development of numerate people.
Such programmes:
•
help students to develop a belief in the value of
mathematics and its usefulness to them, to nurture
confidence in their own mathematical ability, to
foster a sense of personal achievement, and to
encourage a continuing and creative interest in
mathematics;
•
in the first four years of schooling, the main
emphasis should be on the number strand;
•
in the middle and upper primary years of schooling,
the emphasis is spread across the strands of the
curriculum;
•
towards the end of compulsory schooling, number
sense becomes a tool for use across the other
strands.
develop in students the skills, concepts,
understandings, and attitudes which will enable
them to cope confidently with everyday life;
•
help students to develop a variety of approaches
to solving problems involving mathematics, and to
develop the ability to think and reason logically;
•
At all stages, students should:
•
develop an understanding of numbers, the ways
they are represented, and the quantities for which
they stand;
•
help students to achieve the mathematical and
statistical literacy needed in a society which is
technologically oriented and information rich;
•
develop accuracy, efficiency, and confidence in
calculating – mentally, on paper, and with a
calculator;
•
provide students with the mathematical tools, skills,
understandings, and attitudes they will require in
the world of work;
•
develop the ability to estimate and to make
approximations, and to be alert to the
reasonableness of results and measurements.
•
provide a foundation for those students who may
continue studies in mathematics or other learning
areas where mathematical concepts are central;
•
help to foster and develop mathematical talent.
(Mathematics in the New Zealand Curriculum, page 31)
(Mathematics in the New Zealand Curriculum, page 8)
These achievement aims enable students to develop
the ability and inclination to use mathematics to solve
problems in a range of contexts.
Numeracy Professional Development Projects 2003 (Draft)
Published by the Ministry of Education.
PO Box 1666, Wellington, New Zealand.
Copyright © Crown 2003. All rights reserved.
Enquiries should be made to the publisher.
ISBN 0 478 27243 X
Dewey number 372.7
Topic Dewey number 510
Item number 27243
Although the groundwork is laid in mathematics, other
curriculum areas also provide opportunities for numeracy
learning. In addition, the home, early childhood settings,
and the community assist in the development of
numeracy.
ACKNOWLEDGMENTS
The Ministry of Education wishes to acknowledge the
following people and organisations for their contribution
towards the development of this draft handbook.
Holmes (Dunedin College of Education), Errolyn Taane
(Dunedin College of Education), Malcolm Hyland
(Ministry of Education), Ro Parsons (Ministry of
Education).
THE PARTICIPANTS:
The New Zealand numeracy project personnel –
facilitators and principals, teachers and children
from more than eight hundred New Zealand schools
who contributed to this handbook through their
participation in the numeracy development projects
in 2000, 2001, and 2002.
THE NUMERACY REFERENCE
GROUP:
Professor Derek Holton, convenor (The University
of Otago), Professor Megan Clark (Victoria University),
Dr Joanna Higgins (Wellington College of Education),
Dr Kay Irwin (Auckland University), Dr Gill Thomas
(Dunedin College of Education), Dr Jenny Young
Loveridge (The University of Waikato), Dr Glenda
Anthony (Massey University), Tony Trinick (Auckland
College of Education), Garry Nathan (Auckland College
of Education), Graham Cochrane (Education Review
Office), Eleanor Burt (Christchurch College of Education),
Dr Joanna Wood (New Zealand Association of
Mathematics Teachers), Peter Hughes (Auckland
College of Education), Vince Wright (The University
of Waikato School Support Services), Geoff Woolford
(Parallel Services), Kevin Hannah (Christchurch College
of Education), Chris France (School Trustees'
Association), Julie Hepburn (NZPF), Jo Jenks (Early
Childhood Division, Wellington College of Education),
Gary Sweeney (New Zealand Association of
Intermediate and Middle Schools), Diane Leggatt (NZEI
Te Riu Roa), Sului Mamea (Pacific Island Advisory
Group, Palmerston North), Sally Peters (The University
of Waikato School of Education), Andrew Kear (PPTA),
Ro Parsons (Ministry of Education), Malcolm Hyland
(Ministry of Education).
PUBLISHING:
Kathy Campbell, Jocelyn Cranefield (Learning Media
Limited), Kirsty Farquharson (Learning Media Limited),
Jan Kokason (Learning Media Limited), Dr Gill Thomas
(Dunedin College of Education), Joe Morrison (Maths
Technology Limited), Bronwen Wall (Learning Media
Limited).
In addition, the Ministry of Education wishes to
acknowledge Professor Bob Wright (Southern Cross
University, Lismore, NSW), Dr Noel Thomas (Charles
Sturt University, Bathurst, NSW), Dr Keono Gravemeier
(Freudenthaal Institute, Utrecht, Netherlands),
Jim Martland (The University of Manchester, UK).
The Ministry of Education also wishes to acknowledge
The New South Wales Department of Education and
Training for permission to trial Count Me In Too in 2000
through a one-year arrangement. The findings from
the use of this pilot project informed the development
of the numeracy policy in New Zealand.
Count Me In Too is the registered Trade Mark of the
Crown in the Right of the State of New South Wales
(Department of Education and Training). Copyright of
the Count Me In Too Professional Development Package
materials (1997–2002), including the Learning
Framework in Number and the Schedule for Early
Number Assessment, is also held by the Crown in the
Right of the State of New South Wales (Department of
Education and Training) 2002.
The cover design is by Dave Maunder (Learning Media
Limited) and Base2 Communication Design Ltd. All
other illustrations are by Noel Eley and James Rae.
THE WRITERS AND REVIEWERS:
Vince Wright (The University of Waikato), Peter Hughes
(Auckland College of Education), Lynn Tozer (Dunedin
College of Education), Sarah Vokes (Bayview School),
Gaynor Terrill (The University of Waikato School of
Education), Carla McNeill (The University of Waikato
School of Education), Professor Derek Holton (The
University of Otago), Dr Gill Thomas (Dunedin College
of Education), Bruce Moody (mathematics consultant),
Lynne Petersen (Dominion Road School), Marilyn
All illustrations copyright © Crown 2003.
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