the numeracy development projects - nzmaths · the number framework helps teachers, parents, and...

THE NUMERACY DEVELOPMENT PROJECTS

Teachers are key figures in changing the way in whichmathematics and statistics is taught and learned inschools. Their subject matter and pedagogicalknowledge are critical factors in the teaching ofmathematics and statistics understanding. The effectiveteacher of mathematics and statistics has a thoroughand deep understanding of the subject matter to betaught, how students are likely to learn it, and thedifficulties and misunderstandings they are likely toencounter.

The focus of the Numeracy Development Projects is toimprove student performance in mathematics throughimproving the professional capability of teachers. Todate, almost every teacher of year 1 to 6 children andthe majority of teachers of year 7 and 8 children havehad the opportunity to participate.

A key feature of the projects is their dynamic andevolutionary approach to implementation. This ensuresthat the projects can be informed by developingunderstanding about mathematics learning and effectiveprofessional development and that flexibility in approachand sector involvement is maximised.

The projects continue to build on the findings andexperience associated with the numeracy professionaldevelopment projects that operated in 2002–2007.These projects made an important contribution to whatwe know about:

• children’s learning and thinking strategies inearly mathematics;

• effective identification of, and response to,children’s learning needs;

• the characteristics of professional developmentprogrammes that change teaching practice;and

• effective facilitation.

Such findings continue to inform the modification andfurther development of the projects. Nationalco-ordinators and facilitators from each region provideongoing feedback about aspects of the projects.

Note: Teachers may copy these notes for educational

purposes.

This book is also available on the New Zealand Maths website,

at www.nzmaths.co.nz/Numeracy/2008numPDFs/pdfs.htm

Published by the Ministry of Education.PO Box 1666, Wellington, New Zealand.

Copyright © Crown 2008. All rights reserved.Enquiries should be made to the publisher.

ISBN 978 0 7903 1668 0Dewey number 372.7Topic Dewey number 510

Item number 31668

Numeracy Professional Development Projects 2008

Note: Book 1: The Number Framework was modified in2007 in response to research findings and facilitatorfeedback. A paragraph on calculation methods (seepage 14) was added in 2008.

www.nzmaths.co.nz/Numeracy/2008numPDFs/pdfs.htm

The Number Framework

1

The Number FrameworkIntroductionThe Number Framework helps teachers, parents, and students to understand the requirements of the Number and Algebra strand of the mathematics and statistics curriculum learning area. The Framework embodies most of the achievement aims and objectives in levels 1 to 4. In the two main sections of the Framework, the distinction is made between strategy and knowledge. The strategy section describes the mental processes students use to estimate answers and solve operational problems with numbers. The knowledge section describes the key items of knowledge that students need to learn.

It’s important that students make progress in both sections of the Framework. Strong knowledge is essential for students to broaden their strategies across a full range of numbers, and knowledge is often an essential prerequisite for the development of more advanced strategies. For example, a student is unlikely to solve 9 + 6 as 10 + 5 if he or she does not know the “ten and” structure of teen numbers. Similarly, using more advanced strategies helps students to develop knowledge. For example, a student who uses doubling of the multiplication by two facts to work out the multiplication by four facts is likely to learn these facts through appropriate repetition.

Strategy

creates new knowledge through use

Knowledge provides the foundation for strategies

The strategy section of the Framework consists of a sequence of global stages. The diagram below presents the strategy stages as an inverted triangle. Progress through the stages indicates an expansion in knowledge and in the range of strategies that students have available. The triangle also suggests that students build new strategies on their existing strategies and that these existing strategies are not subsumed. Students frequently revert to previous strategies when presented with unfamiliar problems, or when the mental load gets high.

Advanced ProportionalAdvanced Multiplicative–Early Proportional

Advanced Additive–Early MultiplicativeEarly Additive

............................................................................................................................................................................................................................................

Advanced CountingCounting from One

by ImagingCounting from One

on MaterialsOne-to-one Counting

Emergent

Coun

ting

Part

-Who

le


2

The global stages make it easier to identify and describe the types of mental strategies used by students. An important aim of teaching is helping students to develop more sophisticated strategies. Students appear to be very consistent in their approach to operating with numbers. Their consistency will help you to anticipate what strategies students are likely to use across a broad range of problems and to plan appropriate learning activities and questions.

An Overview of the Strategy Section of the FrameworkEach strategy stage is described below. Links are made to the views that students hold about numbers and how numbers can be manipulated to solve operational problems. Experience with using the Numeracy Project Assessment (NumPA) will help you to understand these stages.

The table below gives the structure of the strategy section of the Framework. Each stage contains the operational domains of addition and subtraction, multiplication and division, and proportions and ratios.

Operational Domains Stages Addition Multiplication Proportions Subtraction Division Ratios

Zero Emergent

One One-to-one Counting

Two Counting from One on Materials

Three Counting from One by Imaging

Four Advanced Counting

Five Early Additive Part-Whole

Six Advanced Additive– Early Multiplicative Part-Whole

Seven Advanced Multiplicative– Early Proportional Part-Whole

Eight Advanced Proportional Part-Whole

Students are sometimes between stages. That is, they display characteristics of one stage given a certain problem but use a more-or-less advanced strategy given a different problem. Gaps in key knowledge are usually the reason. For example, a student may use doubles to solve 8 + 7 as 7 + 7 + 1 but count on to work out 9 + 5 as the teen number code is not known.

A close stage match in progress across the three domains is most common and should be seen as a desirable growth path for students. However, for some students, their progress across the operational domains is not in phase. For example, on NumPA, some students understand how to derive multiplication answers from known facts but do so by counting on or back. They are unlikely to derive multiplication answers independently in this way. For addition and subtraction, they are rated at the Advanced Counting stage, but they are rated at the Advanced Additive–Early Multiplicative stage for multiplication and division.

Part

-Who

le

Co

unti

ng


3

Strategy StagesThis section contains descriptions of the strategies that students use at each stage. Some specifi c examples are given to illustrate student thinking. There are many other examples of strategies that students may use at each stage.

Stage Zero: EmergentStudents at this stage are unable to consistently count a given number of objects because they lack knowledge of counting sequences and/or the ability to match things in one-to-one correspondence.

Stage One: One-to-one CountingThis stage is characterised by students who can count and form a set of objects up to ten but cannot solve simple problems that involve joining and separating sets, like 4 + 3.

Stage Two: Counting from One on MaterialsGiven a joining or separating of sets problem, students at this stage rely on counting physical materials, like their fi ngers. They count all the objects in both sets to fi nd an answer, as in “Five lollies and three more lollies. How many lollies is that altogether?”

Stage Three: Counting from One by ImagingThis stage is also characterised by students counting all of the objects in simple joining and separating problems. Students at this stage are able to image visual patterns of the objects in their mind and count them.

Stage Four: Advanced Counting (Counting On)

1, 2, 3, 5, 9,

Students at this stage understand that the end number in a counting sequence measures the whole set and can relate the addition or subtraction of objects to the for-ward and backward number sequences by ones, tens, etc. For example, instead of counting all objects to solve 6 + 5, the student recognises that “6” represents all six objects and counts on from there: “7 , 8 , 9, 10, 11.”


4

Students at this stage also have the ability to co-ordinate equivalent counts, such as “10, 20, 30, 40, 50,” to get $50 in $10 notes. This is the beginning of grouping to solve multiplication and division problems.

Stage Five: Early Additive Part-WholeAt this stage, students have begun to recognise that numbers are abstract units that can be treated simultaneously as wholes or can be partitioned and recombined. This is called part-whole thinking.

A characteristic of this stage is the derivation of results from related known facts, such as fi nding addition answers by using doubles or teen numbers.

The strategies that these students commonly use can be represented in various ways, such as empty number lines, number strips, arrays, or ratio tables.

For example:

(i) Compensation from known factsExample: 7 + 8: 7 + 7 is 14, so 7 + 8 is 15.

Number line:

+7 +7 +1

0 7 14 15

Number strip:

(ii) Standard place value partitioningExample: 43 + 35 is (40 + 30) + (3 + 5) = 70 + 8.

Number line:

+40 +30 +3 +5

0 40 70 73 78

Number strip:

7

7

8

7 1

114

40 30 5

530 3

3

40

43 35


5

Stage Six: Advanced Additive–Early Multiplicative Part-Whole Students at this stage are learning to choose appropriately from a repertoire of part-whole strategies to solve and estimate the answers to addition and subtraction problems. They see numbers as whole units in themselves but also understand that “nested” within these units is a range of possibilities for subdivision and recombining.

Simultaneously, the effi ciency of these students in addition and subtraction is refl ected in their ability to derive multiplication answers from known facts. These students can also solve fraction problems using a combination of multiplication and addition-based reasoning.

For example, 6 � 6 as (5 � 6) + 6;

or 43 of 24 as 4

1 of 20 is 5 because 4 � 5 = 20, so 43 of 20 is 15, and 4

1 of 4 is 1 because 4 � 1 = 4, so 4

3 of 4 is 3. Therefore 43 of 24 is 15 + 3, namely 18.

Here are some examples of the addition and subtraction strategies used by Advanced Additive Part-Whole students:

(i) Standard place value with tidy numbers and compensationExample: 63 – 29 = □ as 63 – 30 + 1 = □.

Number line: –30

33 34 63

Number strip:

+1

? 29

30

?63

29

1

2933

34

1


6

(ii) Reversibility Example: 53 – 26 = □ as 26 + □ = 53. 26 + (4 + 20 + 3) = 53, so 53 – 26 = 27.

Number line:

+4 +3

26 30 50 53

Number Strip:

Advanced Additive students also use addition strategies to derive multiplication facts. Their strategies usually involve partitioning factors additively. Here are two examples of such strategies:

(i) Doubling Example: 3 � 8 = 24, so 6 � 8 = 24 + 24 = 48.

Number line:

3 x 8 3 x 8

0 24 48

Array:

+20

26 20

?53

26

4

2653

?

533

2653

27

8

3

8

3

3

8

24 24

24

48

8

6so so


7

(ii) Compensation Example: 5 � 3 = 15, so 6 � 3 = 18 (three more; compensation using addition)

Number line:

5 x 3 +3

0 15 18

Array:

Example: 10 � 3 = 30, so 9 � 3 = 27 (three less; compensation using subtraction)

Number line:

Array:

5

3

15 5

3

15

31

6

3

18

3 3 3

10 9 27 9 27

31

3 27 30

-3

10 x 3

30


8

Stage Seven: Advanced Multiplicative–Early Proportional Part-Whole Students at this stage are learning to choose appropriately from a range of part-whole strategies to solve and estimate the answers to problems involving multiplication and division. These strategies require one or more of the numbers involved in a multiplication or division to be partitioned, manipulated, then recombined.

For example, to solve 27 � 6, 27 might be split into 20 + 7 and these parts multiplied then recombined, as in 20 � 6 = 120, 7 � 6 = 42, 120 + 42 = 162, or 2 � 27 = 54, 3 � 54 = 162. The fi rst strategy partitions 27 additively, the second strategy partitions 6 multiplicatively.

A critical development at this stage is the use of reversibility, in particular, solving division problems using multiplication. Advanced Multiplicative Part-Whole students are also able to solve and estimate the answers to problems with fractions using multiplication and division. For example, to solve 3

2 of □ = 18, 21 of 18 = 9, □ = 3 � 9 = 27 (using unit fractions). Students,

at this stage, can also understand the multiplicative relationship between the numerators and denominators of equivalent fractions, e.g. 4

3 = 10075 .

Here are some strategies used by students at the Advanced Multiplicative stage to solve multiplication and division problems:(i) Doubling and halving or trebling and dividing by three (thirding)

Example: 4 � 16 as 8 � 8 = 64 (doubling and halving)

Double number line:

0 16 32 48 64

0 8 16 24 32 40 48 56 64

Array:

(ii) Halving and halving Example: 72 ÷ 4 as 72 ÷ 2 = 36, 36 ÷ 2 = 18 (dividing by four is the same as dividing by two

twice).

Number line:

÷2 ÷2

0 18 36 72

Array:

16

4

16

4

4

8

8

8

÷2?

4 72

?

2 36 118÷2

(Not drawn to scale)


9

(iii) Reversibility and place value partitioningExample: 72 ÷ 4 as 10 � 4 = 40, 72 – 40 = 32, 8 � 4 = 32, 10 + 8 = 18, so 18 � 4 = 72

Number line:

+32

10 x 4 8 x 4

0 40 72

Array (reversibility)


Advanced Multiplicative students also apply their strengths in multiplication and division to problems involving fractions, proportions and ratios. Generally, their strategies involve using equivalent fractions. Here are some examples of the strategies they use:

(i) Multiplying withinExample: Every packet in the jar has 8 nuts in it. Three of the 8 nuts are peanuts. The jar contains 40 nuts altogether, all in packets. How many of the nuts are peanuts? (3:8 as □:40, 5 � 8 = 40 so 5 � 3 = □)

Double number line:

0 8 40 nuts in the jar

0 3 15 peanuts

?

4

10

4

? 10

4

8

40 32 40 32

18

4 7272

5 x 8

5 x 3


10

(ii) Using unit fractions and conversion from percentagesExample: The tracksuit is usually $56, but the shop has a 25% off sale. How much does Mere pay?25% is 4

1 , so Mere pays 43 of $56. 4

1 of 56 is 56 ÷ 4 = 14. 43 = 3 � 4

1 , 3 � 14 = 42

Double number line:

0 14 28 42 56 cost of tracksuit ($)

0 25 50 75 100 percentage

Array:


(ii) Using the distributive property Example: Albert has 32 matchbox toys. Five-eighths of them are sports cars. How many of the matchbox toys are sports cars? Method 1: 8

5 of 32 as 21 of 32 is 16 and 8

1 of 32 is 4, so 8

5 of 32 is 16 + 4 = 20

Double number line:

0 4 16 20 32

0 1 whole

Array:

4 3 1 3 1

14 14 1425%3 x 14=75%

4275%

1 3

56

18

12

58

1 x 4

4 x 4 5 x 4

8

4 32and

4 4 4

4 1 5

20

41

43

21

16

1 x 14


11

Method 2: 81 of 32 is 4, so 8

5 of 32 is 5 � 4 = 20

Double number line:

0 4 20 32

0 1 whole

Array:

Stage Eight: Advanced Proportional Part-WholeStudents at this stage are learning to select from a repertoire of part-whole strategies to solve and estimate the answers to problems involving fractions, proportions, and ratios. These strategies are based on fi nding common factors and include strategies for the multiplication of decimals and the calculation of percentages.

These students are able to fi nd the multiplicative relationship between quantities of two different measures. This can be thought of as a mapping. For example, consider this problem: “You can make 21 glasses of lemonade from 28 lemons. How many glasses can you make using 8 lemons?”

To solve the problem, students may need to fi nd a relationship between the number of lemons and the number of glasses. This involves the creation of a new measure, glasses per lemon. The relationship is that the number of glasses is three-quarters the number of lemons. This could be recorded as: 21:28 is equivalent to □ :8? 21 is 4

3 of 28 ⇒ 6 is 43 of 8.

Double number line:

0 8 28 lemons 0 6 21 glasses

Some problems can be solved by fi nding the relationship within units.Example: Of every 10 children in the class, 6 are boys. There are 25 children in the class. How many of them are boys?

Double number line:

0 6 15 boys 0 10 25 total class

18

58

1 x 4

5 x 4

8

4 4

1 5

32 4

5

4 20

43 x

43 x

x 2 21

x 2 21


12

Advanced Proportional students may use either within or between measures as strategies to solve a fractional multiplication problem.

Example: It takes 10 balls of wool to make 15 beanies. How many balls of wool make 6 beanies?

Within measures: 2 21 � 6 = 15, so 2 2

1 � □ = 10.

Double number line:

0 6 12 15 beanies

0 4 8 10 balls of wool

Between measures: 32

� 15 = 10, so 32 � 6 = 4.

Double number line:

0 6 15 beanies

0 4 10 balls of wool

wool beanies

10 15

□ 6� 2 2

1� 2 2

1

2 21 x 6

2 21 x 4

wool beanies

10 15 □ 6

32

�

32

�

32 x 3

2 x


13

Example: Cayla’s Clothing Shop is giving a discount. For a $75 pair of jeans, you only pay $50. What percentage discount is that?

The discount is $75 – $50 = $25. As a fraction of the original price of $75, this discount is 3

1. So the percentage discount is 33.3%

Ratio tables:

Within strategy Between strategy

Full price Discount price FP DP FP DP

75 50 75 50 75 50

25 16.6

100 ? 100 66.6 100 66.6%

Number lines:

0 50 75 price

0 ? 100 percentage

0 25 50 75 price

0 33.3 (31) 66.6 (3

2) 100 percentage

÷ 3 ÷ 3

x 131 or

x 4 x 4

x 131

6632 %

...

32 x

..

.


14

An Overview of the Knowledge Section of the FrameworkThe knowledge section of the Framework outlines the important items of knowledge that students should learn as they progress through the strategy stages. This knowledge plays a critical role in students applying their available strategies with profi ciency and fl uency across all the numbers and problem types that they may encounter.

It is important to avoid a “culture of limits” when exposing students to the teaching of number knowledge. The same items of knowledge can be taught to students across a range of strategy stages. For example, students can come to read and write decimals long before their strategies are suffi ciently developed for them to use decimals in operational problems.

In the Framework, knowledge is categorised under four content domains: Number Identifi cation, Number Sequence and Order, Grouping/Place Value, and Basic Facts. Progressions for written recording are indicated on pages 18–22 in a fi fth column.

Written recording can be seen as a thinking tool, a communication tool, and a refl ective tool. It is critical that, at every stage, students engage in building meaning through recording their mathematical ideas in the form of pictures, diagrams, words, and symbols. Making explicit the links between oral and written forms is fundamental to the sound development of the meaning and structure of the language of mathematics. This oral and written language underpins all understanding. The informal jottings of students are to be encouraged as a way to capture their mental processes so that their ideas can be shared with others. Students should not be exposed to standard written algorithms until they use part-whole mental strategies. Premature exposure to working forms restricts students’ ability and desire to use mental strategies. This inhibits their development of number sense. However, in time, written methods must become part of a student’s calculation repertoire.

Calculators and computers can also allow students to fi nd answers to calculations that may be beyond their mental and written capacity. Access to this technology allows students to model real world problems in which the numbers may be otherwise prohibitive, to explore patterns in number relationships, and to concentrate on conceptual understanding without the cognitive load of calculation. In using such tools, other mathematical thinking is required, including evaluating the validity of the mathematical model chosen for solving the problem and checking the “reasonableness” of the answer produced.

Counting needs to be developed with all students. This counting should extend past counting whole numbers by ones, twos, and other simple multiples into recognising patterns in powers of ten, in decimals, and in fractions. The Framework also acknowledges the importance of backwards counting sequences, particularly in the early years.

The Framework encourages greater use of students’ natural inclination to use groupings of fi ves. There is signifi cant emphasis on using fi nger patterns in the early years and in using multiples of fi ve in later work, for example, 35 + 35 = 70, 500 + 500 = 1000.

Basic fact knowledge is critical. The Number Framework emphasises that the process of deriving number facts using mental strategies is important in coming to know and apply these facts. It also demands that students come to know a broader range of facts than previously, including groupings of “benchmark” numbers, and that they have knowledge of factors of numbers and decimal and fraction conversions at the higher stages.

You may teach me the standard written forms after I’ve learned to partition the numbers.


15

The

Num

ber Fr

amew

ork

– St

rate

gies

Ope

ration

al D

omains

Glob

al S

tage

Add

ition

and

Subt

ract

ion

Multiplicat

ion

and

Division

Prop

ortion

s an

d Ra

tios

Counting

Zero

: Em

erge

ntEm

erge

ntTh

e st

uden

t is

unab

le to

cou

nt a

giv

en s

et o

r for

m a

set

of u

p to

ten

obje

cts.

One

: One

-to-

one

Coun

ting

One

-to-

one

Coun

ting

The

stud

ent i

s ab

le to

cou

nt a

set

of o

bjec

ts b

utis

una

ble

to fo

rm s

ets

of o

bjec

ts to

sol

ve s

impl

ead

ditio

n an

d su

btra

ctio

n pr

oble

ms.

One

-to-

one

Coun

ting

The

stud

ent i

s ab

le to

cou

nt a

set

of o

bjec

ts b

utis

una

ble

to fo

rm s

ets

of o

bjec

ts to

sol

ve s

impl

em

ultip

licat

ion

and

divi

sion

pro

blem

s.

Une

qual S

haring

The

stud

ent i

s un

able

to d

ivid

e a

regi

on o

r set

into

two

or fo

ur e

qual

par

ts.

Two:

Cou

nting

from

One

on

Mat

erials

Coun

ting

– f

rom O

neTh

e st

uden

t sol

ves

sim

ple

addi

tion

and

subt

ract

ion

prob

lem

s by

cou

ntin

g al

l the

obje

cts,

e.g

., 5

+ 4

as 1

, 2, 3

, 4, 5

, 6, 7

, 8, 9

. T

hest

uden

t nee

ds s

uppo

rtin

g m

ater

ials

, lik

efin

gers

.

Coun

ting

– f

rom O

neTh

e st

uden

t sol

ves

mul

tiplic

atio

n an

d di

visi

onpr

oble

ms

by c

ount

ing

one

to o

ne w

ith th

e ai

dof

mat

eria

ls.

Equa

l Sha

ring

The

stud

ent i

s ab

le to

div

ide

a re

gion

or s

et in

togi

ven

equa

l par

ts u

sing

mat

eria

ls. W

ith s

ets

this

is d

one

by e

qual

sha

ring

of m

ater

ials

. With

shap

es s

ymm

etry

(hal

ving

) is

used

.

Thre

e: C

ount

ing

from

One

by

Imag

ing

Coun

ting

– f

rom O

neTh

e st

uden

t im

ages

all

of th

e ob

ject

s an

dco

unts

them

.Th

e st

uden

t doe

s no

t see

ten

as a

uni

t of a

nyki

nd a

nd s

olve

s m

ulti-

digi

t add

ition

and

subt

ract

ion

prob

lem

s by

cou

ntin

g al

l the

obje

cts.

Coun

ting

– f

rom O

neTh

e st

uden

t im

ages

the

obje

cts

to s

olve

sim

ple

mul

tiplic

atio

n an

d di

visi

on p

robl

ems,

by

coun

ting

all t

he o

bjec

ts, e

.g.,

4 �

2 a

s 1,

2, 3

, 4,

5, 6

, 7, 8

. For

pro

blem

s in

volv

ing

larg

ernu

mbe

rs th

e st

uden

t will

stil

l rel

y on

mat

eria

ls.

Equa

l Sha

ring

The

stud

ent i

s ab

le to

sha

re a

regi

on o

r set

into

give

n eq

ual p

arts

by

usin

g m

ater

ials

or b

yim

agin

g th

e m

ater

ials

for s

impl

e pr

oble

ms,

e.g.

,of

8.

With

set

s th

is is

don

e by

equ

al s

hari

ng o

fm

ater

ials

or b

y im

agin

g. W

ith s

hape

ssy

mm

etry

is u

sed

to c

reat

e ha

lves

, qua

rter

s,ei

ghth

s, e

tc.

1 2

Four

: Adv

ance

dCo

unting

Coun

ting

On

The

stud

ent u

ses

coun

ting

on o

r cou

ntin

g ba

ckto

sol

ve s

impl

e ad

ditio

n or

sub

trac

tion

task

s,e.

g., 8

+ 5

by

8, 9

, 10,

11,

12,

13

or 5

2 –

4 as

52,

51, 5

0, 4

9, 4

8. I

nitia

lly, t

he s

tude

nt n

eeds

supp

ortin

g m

ater

ials

but

late

r im

ages

the

obje

cts

and

coun

ts th

em.

The

stud

ent s

ees

10as

a c

ompl

eted

cou

nt c

ompo

sed

of 1

0 on

es.

The

stud

ent s

olve

s ad

ditio

n an

d su

btra

ctio

nta

sks

by in

crem

entin

g in

one

s (3

8, 3

9, 4

0, ..

.),te

ns c

ount

s (1

3, 2

3, 3

3, ..

.), a

nd/o

r aco

mbi

natio

n of

tens

and

one

s co

unts

(27,

37,

47, 4

8, 4

9, 5

0, 5

1).

Skip-c

ount

ing

On

mul

tiplic

atio

n ta

sks,

the

stud

ent u

ses

skip

-cou

ntin

g (o

ften

in c

onju

nctio

n w

ith o

ne-

to-o

ne c

ount

ing)

, e.g

., 4

� 5

as

5, 1

0, 1

5, 2

0.Th

e st

uden

t may

trac

k th

e co

unts

usi

ngm

ater

ials

(eg.

fing

ers)

or

by im

agin

g.


16

1 2

Five

: Ea

rly

Add

itive

Part

-Who

le

Ope

ration

al D

omains

Part-Whole

Early

Add

ition

and

Subt

ract

ion

The

stud

ent u

ses

a lim

ited

rang

e of

men

tal

stra

tegi

es to

est

imat

e an

swer

s an

d so

lve

addi

tion

or s

ubtr

actio

n pr

oble

ms.

The

sest

rate

gies

invo

lve

deri

ving

the

answ

er fr

omkn

own

basi

c fa

cts,

e.g

., 8

+ 7

is 8

+ 8

– 1

(dou

bles

) or 5

+ 3

+ 5

+ 2

(fiv

es) o

r 10

+ 5

(mak

ing

tens

). T

heir

str

ateg

ies

with

mul

ti-di

git

num

bers

invo

lve

usin

g te

ns a

nd h

undr

eds

asab

stra

ct u

nits

that

can

be

part

ition

ed,

e.g.

, 43

+ 25

= (4

0 +

20) +

(3 +

5) =

60

+ 8

= 68

(sta

ndar

d pa

rtiti

onin

g) o

r 39

+ 26

= 4

0 +

25 =

65

(rou

ndin

g an

d co

mpe

nsat

ion)

or

84 –

8 a

s 84

– 4

– 4

= 7

6 (b

ack

thro

ugh

ten)

.

Multiplicat

ion

by R

epea

ted

Add

ition

On

mul

tiplic

atio

n ta

sks,

the

stud

ent u

ses

aco

mbi

natio

n of

kno

wn

mul

tiplic

atio

n fa

cts

and

repe

ated

add

ition

,e.

g., 4

� 6

as

(6 +

6) +

(6 +

6) =

12

+ 12

= 2

4.Th

e st

uden

t use

s kn

own

mul

tiplic

atio

n an

dre

peat

ed a

dditi

on fa

cts

to a

ntic

ipat

e th

e re

sult

of d

ivis

ion,

e.g

., 20

÷ 4

= 5

bec

ause

5 +

5 =

10

and

10 +

10

= 20

.

1 3

Frac

tion

of

a Num

ber

by A

ddition

The

stud

ent f

inds

a fr

actio

n of

a n

umbe

r and

solv

es d

ivis

ion

prob

lem

s w

ith re

mai

nder

sm

enta

lly u

sing

hal

ving

, or d

eriv

ing

from

know

n ad

ditio

n fa

cts,

e.g.

,of

12

is 4

bec

ause

3 +

3 +

3 =

9, s

o 4

+ 4

+ 4

= 12

;e.

g., 7

pie

s sh

ared

am

ong

4 pe

ople

(7 ÷

4) b

ygi

ving

eac

h pe

rson

1 p

ie, a

ndpi

e, th

enpi

e.1 4

Six:

Adv

ance

dAdd

itive

(Ear

lyMultiplicat

ive)

Part

-Who

le

Adv

ance

d Add

ition

and

Subt

ract

ion

ofW

hole N

umbe

rsTh

e st

uden

t can

est

imat

e an

swer

s an

d so

lve

addi

tion

and

subt

ract

ion

task

s in

volv

ing

who

lenu

mbe

rs m

enta

lly b

y ch

oosi

ng a

ppro

pria

tely

from

a b

road

rang

e of

adv

ance

d m

enta

lst

rate

gies

,e.

g., 6

3 –

39 =

63

– 40

+ 1

= 2

4 (r

ound

ing

and

com

pens

atin

g) o

r39

+ 2

0 +

4 =

63, s

o 63

– 3

9 =

24 (

reve

rsib

ility

)or

64

– 40

= 2

4 (e

qual

add

ition

s)e.

g., 3

24 –

86

= 30

0 –

62 =

238

(sta

ndar

d pl

ace

valu

e pa

rtiti

onin

g) o

r32

4 –

100

+ 14

= 2

38 (r

ound

ing

and

com

pens

atin

g).

Der

ived

Multiplicat

ion

The

stud

ent u

ses

a co

mbi

natio

n of

kno

wn

fact

san

d m

enta

l str

ateg

ies

to d

eriv

e an

swer

s to

mul

tiplic

atio

n an

d di

visi

on p

robl

ems,

e.g.

, 4 �

8 =

2 �

16

= 32

(dou

blin

g an

d ha

lvin

g),

e.g.

, 9 �

6 is

(10

� 6

) – 6

= 5

4 (r

ound

ing

and

com

pens

atin

g),

e.g.

, 63

÷ 7

= 9

beca

use

9 �

7 =

63

(rev

ersi

bilit

y).

Frac

tion

of

a Num

ber

by A

ddition

and

Multiplicat

ion

The

stud

ent u

ses

repe

ated

hal

ving

or k

now

nm

ultip

licat

ion

and

divi

sion

fact

s to

sol

vepr

oble

ms

that

invo

lve

findi

ng fr

actio

ns o

f a s

etor

regi

on, r

enam

ing

impr

oper

frac

tions

, and

divi

sion

with

rem

aind

ers,

e.g.

,of

36,

3 �

10

= 30

,36

– 3

0 =

6, 6

÷ 3

= 2

, 10

+ 2

= 12

e.g.

,=

5(u

sing

5 �

3 =

15)

e.g.

, 8 p

ies

shar

ed a

mon

g 3

peop

le (8

÷ 3

) by

givi

ng e

ach

pers

on 2

pie

s an

d di

vidi

ng th

ere

mai

ning

2 p

ies

into

thir

ds(a

nsw

er: 2

++

= 2

).Th

e st

uden

t use

s re

peat

ed re

plic

atio

n to

sol

vesi

mpl

e pr

oble

ms

invo

lvin

g ra

tios

and

rate

s,e.

g. 2

:3 ➝

4:6

➝ 8

:12

etc.

1 3

1 3 16 3

1 31 3

2 3

Glob

al S

tage

Add

ition

and

Subt

ract

ion

Multiplicat

ion

and

Division

Prop

ortion

s an

d Ra

tios


17

Seve

n: A

dvan

ced

Multiplicat

ive

(Ear

lyPr

opor

tion

al)

Part

-Who

le

Ope

ration

al D

omains

Glob

al S

tage

Add

ition

and

Subt

ract

ion

Multiplicat

ion

and

Division

Prop

ortion

s an

d Ra

tios

Part-Whole

Add

ition

and

Subt

ract

ion

of D

ecim

als

and

Inte

gers

The

stud

ent c

an c

hoos

e ap

prop

riat

ely

from

abr

oad

rang

e of

men

tal s

trat

egie

s to

est

imat

ean

swer

s an

d so

lve

addi

tion

and

subt

ract

ion

prob

lem

s in

volv

ing

deci

mal

s, in

tege

rs, a

ndre

late

d fr

actio

ns. T

he s

tude

nt c

an a

lso

use

mul

tiplic

atio

n an

d di

visi

on to

sol

ve a

dditi

onan

d su

btra

ctio

n pr

oble

ms

with

who

lenu

mbe

rs.

e.g.

, 3.2

+ 1

.95

= 3.

2 +

2 –

0.05

= 5.

2 –

0.05

= 5.

15 (c

ompe

nsat

ion)

;e.

g., 6

.03

– 5.

8 =

□ a

s 6.

03 –

5 –

0.8

= 1

.03

– 0.

8=

0.23

(sta

ndar

d pl

ace

valu

e pa

rtiti

onin

g) o

r as

5.8

+ □

= 6

.03

(rev

ersi

bilit

y)e.

g., □

+ 3

.98

= 7.

04 a

s 3.

98 +

□ =

7.0

4,□

= 0

.02

+ 3.

04 =

3.0

6 (c

omm

utat

ivity

)e.

g.,

+=

(+

) +=

1(p

artit

ioni

ngfr

actio

ns)

e.g.

, 81

– 36

= (9

� 9

) – (4

� 9

) = 5

� 9

(usi

ngfa

ctor

s)e.

g., 2

8 +

33 +

27

+ 30

+ 3

2 =

5 �

30

(ave

ragi

ng)

e.g.

, + 7 –

- 3 =

+ 7 +

+ 3 =

+ 10

(equ

ival

ent

oper

atio

ns o

n in

tege

rs)

3 45 8

3 83 4

2 83 8

Adv

ance

d M

ultiplicat

ion

and

Division

The

stud

ent c

hoos

es a

ppro

pria

tely

from

abr

oad

rang

e of

men

tal s

trat

egie

s to

est

imat

ean

swer

s an

d so

lve

mul

tiplic

atio

n an

d di

visi

onpr

oble

ms.

The

se s

trat

egie

s in

volv

e pa

rtiti

onin

gon

e or

mor

e of

the

fact

ors

in m

ultip

licat

ion,

and

appl

ying

reve

rsib

ility

to s

olve

div

isio

npr

oble

ms,

par

ticul

arly

thos

e in

volv

ing

mis

sing

fact

ors

and

rem

aind

ers.

The

par

titio

ning

may

be a

dditi

ve o

r mul

tiplic

ativ

e.e.

g., 2

4 �

6 =

(20

� 6

) + (4

� 6

) (pl

ace

valu

epa

rtiti

onin

g) o

r 25

� 6

– 6

(rou

ndin

g an

dco

mpe

nsat

ing)

e.g.

, 81

÷ 9

= 9,

so

81 ÷

3 =

3 �

9 (p

ropo

rtio

nal

adju

stm

ent)

e.g.

, 4 �

25

= 10

0, s

o 92

÷ 4

= 2

5 –

2 =

23(r

ever

sibi

lity

and

roun

ding

with

com

pens

atio

n)e.

g., 9

0 ÷

5 =

18 s

o 87

÷ 5

= 1

7 r 2

(rou

ndin

g an

ddi

visi

bilit

y)e.

g., 2

01 ÷

3 a

s 10

0 ÷

3 =

33 r

1, 2

00 ÷

3 =

66

r 2,

201

÷ 3

= 67

(div

isib

ility

rul

es)

Eigh

t: A

dvan

ced

Prop

ortion

alPa

rt-W

hole

Add

ition

and

Subt

ract

ion

of F

ract

ions

The

stud

ent u

ses

a ra

nge

of m

enta

l par

titio

ning

stra

tegi

es to

est

imat

e an

swer

s an

d so

lve

prob

lem

s th

at in

volv

e ad

ding

and

sub

trac

ting

frac

tions

, inc

ludi

ng d

ecim

als.

The

stu

dent

isab

le to

com

bine

ratio

s an

d pr

opor

tions

with

diffe

rent

am

ount

s. T

he s

trat

egie

s in

clud

e us

ing

part

ition

s of

frac

tions

and

“on

es”,

and

find

ing

equi

vale

nt fr

actio

ns.

e.g.

, 2–

1 =

1 +

(–

) = 1

+ (

–) =

1(e

quiv

alen

t fra

ctio

ns)

e.g.

, 20

coun

ters

in ra

tio o

f 2:3

com

bine

d w

ith60

cou

nter

s in

ratio

8:7

giv

es a

com

bine

d ra

tioof

1:1

.

8 122 3

3 42 3

3 41 12

9 12

3 4

Multiplicat

ion

and

Division

of D

ecim

als/

Multiplicat

ion

of F

ract

ions

The

stud

ent c

hoos

es a

ppro

pria

tely

from

ara

nge

of m

enta

l str

ateg

ies

to e

stim

ate

answ

ers

and

solv

e pr

oble

ms

that

invo

lve

the

mul

tiplic

atio

n of

frac

tions

and

dec

imal

s. T

hest

uden

t can

als

o us

e m

enta

l str

ateg

ies

to s

olve

sim

ple

divi

sion

pro

blem

s w

ith d

ecim

als.

The

sest

rate

gies

invo

lve

the

part

ition

ing

of fr

actio

nsan

d re

latin

g th

e pa

rts

to o

ne, c

onve

rtin

gde

cim

als

to fr

actio

ns a

nd v

ice

vers

a, a

ndre

cogn

isin

g th

e ef

fect

of n

umbe

r siz

e on

the

answ

er, e

.g.,

3.6

� 0

.75

=�

3.6

= 2

.7(c

onve

rsio

n an

d co

mm

utat

ivity

);e.

g.,

�=

□ a

s�

=so

�=

so�

==

e.g.

, 7.2

÷ 0

.4 a

s 7.

2 ÷

0.8

= 9

so 7

.2 ÷

0.4

= 1

8(d

oubl

ing

and

halv

ing

with

pla

ce v

alue

).

3 4

3 42 3

2 32 3

1 41 3

1 41 2

2 126 12

1 12

Frac

tion

s, R

atios,

and

Pro

port

ions

by

Multiplicat

ion

The

stud

ent u

ses

a ra

nge

of m

ultip

licat

ion

and

divi

sion

str

ateg

ies

to e

stim

ate

answ

ers

and

solv

e pr

oble

ms

with

frac

tions

, pro

port

ions

, and

ratio

s. T

hese

str

ateg

ies

invo

lve

linki

ng d

ivis

ion

to fr

actio

nal a

nsw

ers,

e.g

., 11

÷ 3

=

= 3

e.g.

, 13

÷ 5

= (1

0 ÷

5) +

(3 ÷

5) =

2Th

e st

uden

t can

als

o fin

d si

mpl

e eq

uiva

lent

frac

tions

and

rena

me

com

mon

frac

tions

as

deci

mal

s an

d pe

rcen

tage

s.e.

g.,

of 2

4 as

of 2

4 =

4, 5

� 4

= 2

0or

24

– 4

= 20

e.g.

, 3:5

as

□ :

40, 8

� 5

= 4

0, 8

� 3

= 2

4so

□ =

24.

e.g.

,=

= 75

% =

0.7

5

11 32 3

5 6

75 100

3 4

1 21 2

1 6

Frac

tion

s, R

atios,

and

Pro

port

ions

by

Re-u

nitising

The

stud

ent c

hoos

es a

ppro

pria

tely

from

abr

oad

rang

e of

men

tal s

trat

egie

s to

est

imat

ean

swer

s an

d so

lve

prob

lem

s in

volv

ing

frac

tions

, pro

port

ions

, and

ratio

s. T

hese

stra

tegi

es in

volv

e us

ing

com

mon

fact

ors,

re-u

nitis

ing

of fr

actio

ns, d

ecim

als

and

perc

enta

ges,

and

find

ing

rela

tions

hips

bet

wee

nan

d w

ithin

ratio

s an

d si

mpl

e ra

tes.

e.g.

, 6:9

as

□ :2

4, 6

� 1

= 9,

□ �

1=

24,

□ =

16

(bet

wee

n un

it m

ultip

lyin

g);

or 9

� 2

= 24

, 6 �

2=

16 (w

ithin

uni

tm

ultip

lyin

g)e.

g., 6

5% o

f 24:

50%

of 2

4 is

12,

10%

of 2

4 is

2.4

so 5

% is

1.2

, 12

+ 2.

4 +

1.2

= 15

.6 (p

artit

ioni

ngpe

rcen

tage

s).

2 32 3

3 5


18

The

stud

ent s

ays:

•th

e nu

mbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, i

n th

e ra

nge

0–20

;•

the

num

ber b

efor

e an

daf

ter a

giv

en n

umbe

r in

the

rang

e 0–

20;

•th

e sk

ip-c

ount

ing

sequ

ence

s, fo

rwar

ds a

ndba

ckw

ards

, in

the

rang

e0–

20 fo

r tw

os a

nd fi

ves.

The

stud

ent o

rder

s:•

num

bers

in th

e ra

nge

0–20

.

The

stud

ent k

now

s:•

grou

ping

s w

ithin

5,

e.g.

, 2 a

nd 3

, 4 a

nd 1

;•

grou

ping

s w

ith 5

,e.

g., 5

and

1, 5

and

2, .

..;•

grou

ping

s w

ithin

10,

e.g.

, 5 a

nd 5

, 4 a

nd 6

, ...

etc;

The

stud

ent i

nsta

ntly

reco

gnis

es:

•pa

ttern

s to

10

(dou

bles

and

5-ba

sed)

, inc

ludi

ng fi

nger

patte

rns.

The

stud

ent i

dent

ifies

:•

all o

f the

num

bers

in th

era

nge

0–10

.

Stag

eNum

ber

Iden

tifica

tion

Num

ber

Sequ

ence

Grou

ping

/Place

Value

Basic

Fact

sW

ritt

en R

ecor

ding

and

Ord

er

Stage Zero: Emergent

Stages One, Two, Three: Counting from One

The

Num

ber Fr

amew

ork

– Kn

owledg

e

The

stud

ent s

ays:

•th

e nu

mbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, i

n th

e ra

nge

0–10

at l

east

;•

the

num

ber b

efor

e an

daf

ter a

giv

en n

umbe

r in

the

rang

e 0–

10.

The

stud

ent o

rder

s:•

num

bers

in th

e ra

nge

0–10

.

The

stud

ent i

nsta

ntly

reco

gnis

es:

•pa

ttern

s to

5, i

nclu

ding

finge

r pat

tern

s.

The

stud

ent i

dent

ifies

:•

all o

f the

num

bers

in th

era

nge

0–20

.

The

stud

ent r

ecal

ls:

•ad

ditio

n an

d su

btra

ctio

nfa

cts

to fi

ve, e

.g.,

2 +

1,3

+ 2,

4 –

2, .

.. et

c;•

doub

les

to 1

0,e.

g., 3

+ 3

, 4 +

4, .

.. et

c.

The

stud

ent r

ecor

ds:

•th

e re

sults

of c

ount

ing

and

oper

atio

ns u

sing

pic

ture

san

d di

agra

ms.

The

teac

her a

nd th

e st

uden

tre

cord

:•

the

resu

lts o

f ope

ratio

nsus

ing

sym

bols

, e.g

., th

ere

adin

g of

five

and

two

is s

even

is re

cord

ed a

s5

and

2 is

7, 5

plu

s 2

equa

ls 7

, 5 +

2 =

7.

.........

......... .........

.........

The

stud

ent r

ecor

ds:

•nu

mer

als

to m

atch

the

sets

they

form

.


19

The

stud

ent i

dent

ifies

:•

all o

f the

num

bers

in th

era

nge

0–10

0;•

sym

bols

for h

alve

s,qu

arte

rs, t

hird

s, a

nd fi

fths.

Stag

eNum

ber

Iden

tifica

tion

Num

ber

Sequ

ence

Grou

ping

/Place

Value

Basic

Fact

sW

ritt

en R

ecor

ding

and

Ord

erStage Four: Advanced Counting

Stage Five: Early Additive

The

stud

ent s

ays:

•th

e nu

mbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, i

n th

e ra

nge

0–10

0;•

the

num

ber b

efor

e an

daf

ter a

giv

en n

umbe

r in

the

rang

e 0–

100;

•th

e sk

ip-c

ount

ing

sequ

ence

s, fo

rwar

ds a

ndba

ckw

ards

, in

the

rang

e0–

100

for t

wos

, fiv

es, a

ndte

ns.

The

stud

ent o

rder

s:•

num

bers

in th

e ra

nge

0–10

0.

The

stud

ent k

now

s:•

grou

ping

s w

ith 1

0, e

.g.,

10an

d 2,

10

and

3, ..

. and

the

patte

rn o

f “-te

ens”

;•

grou

ping

s w

ithin

20,

e.g.

, 12

and

8, 6

and

14;

•th

e nu

mbe

r of t

ens

inde

cade

s, e

.g.,

tens

in 4

0,in

60.

The

stud

ent r

ecal

ls:

•ad

ditio

n an

d su

btra

ctio

nfa

cts

to 1

0,e.

g., 4

+ 3

, 6 +

2, 7

– 3

, ...;

•do

uble

s to

20

and

corr

espo

ndin

g ha

lves

,e.

g., 6

+ 6

, 7 +

7,

of 1

4;•

“ten

and

” fa

cts,

e.g

., 10

+ 4

,7

+ 10

•m

ultip

les

of 1

0 th

at a

dd to

100,

e.g

., 30

+ 7

0, 4

0 +

60.

The

stud

ent i

dent

ifies

:•

all o

f the

num

bers

in th

era

nge

0–10

00;

•sy

mbo

ls fo

r the

mos

tco

mm

on fr

actio

ns,

incl

udin

g at

leas

t hal

ves,

quar

ters

, thi

rds,

fifth

s, a

ndte

nths

;•

sym

bols

for i

mpr

oper

frac

tions

, e.g

.,.

1 42 4

3 4

5 4

The

stud

ent s

ays:

•th

e nu

mbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, b

y on

es, t

ens,

and

hund

reds

in th

e ra

nge

0–10

00;

•th

e nu

mbe

r 1, 1

0, 1

00be

fore

and

afte

r a g

iven

num

ber i

n th

e ra

nge

0–10

00;

•th

e sk

ip-c

ount

ing

sequ

ence

s, fo

rwar

ds a

ndba

ckw

ards

, in

the

rang

e0–

100

for t

wos

, thr

ees,

fives

, and

tens

.

The

stud

ent o

rder

s:•

num

bers

in th

e ra

nge

0–10

00;

•fr

actio

ns w

ith li

kede

nom

inat

ors,

e.g

.,,

,,

... e

tc.

The

stud

ent k

now

s:•

grou

ping

s w

ithin

100

,e.

g., 4

9 an

d 51

(par

ticul

arly

mul

tiple

s of

5,

e.g.

, 25

and

75);

•gr

oupi

ngs

of tw

o th

at a

rein

num

bers

to 2

0,e.

g., 8

gro

ups

of 2

in 1

7;•

grou

ping

s of

five

innu

mbe

rs to

50,

e.g.

, 9 g

roup

s of

5 in

47;

•gr

oupi

ngs

of te

n th

at c

anbe

mad

e fr

om a

thre

e-di

git

num

ber,

e.g.

, ten

s in

763

is 7

6;•

the

num

ber o

f hun

dred

s in

cent

urie

s an

d th

ousa

nds,

e.g.

, hun

dred

s in

800

is 8

and

in 4

000

is 4

0.

The

stud

ent r

ound

s:•

thre

e-di

git w

hole

num

bers

to th

e ne

ares

t 10

or 1

00e.

g., 5

61 ro

unde

d to

the

near

est 1

0 is

560

and

to th

ene

ares

t 100

is 6

00.

The

stud

ent r

ecal

ls:

•ad

ditio

n fa

cts

to 2

0 an

dsu

btra

ctio

n fa

cts

to 1

0,e.

g., 7

+ 5

, 8 +

7, 9

– 6

, ...;

•m

ultip

licat

ion

fact

s fo

r the

2, 5

, and

10

times

tabl

esan

d th

e co

rres

pond

ing

divi

sion

fact

s;•

mul

tiple

s of

100

that

add

to10

00, e

.g.,

400

and

600,

300

and

700.

............................................................................................................... .....................................

.............................................................................

1 2

The

stud

ent r

ecor

ds:

•th

e re

sults

of m

enta

lad

ditio

n an

d su

btra

ctio

n,us

ing

equa

tions

,e.

g., 4

+ 5

= 9

, 8 –

3 =

5.

The

stud

ent r

ecor

ds:

•th

e re

sults

of a

dditi

on,

subt

ract

ion,

and

mul

tiplic

atio

n ca

lcul

atio

nsus

ing

equa

tions

,e.

g., 3

5 +

24 =

59,

4 �

5 =

20,

and

diag

ram

s, e

.g.,

anem

pty

num

ber l

ine.


20

Stag

eNum

ber

Iden

tifica

tion

Num

ber

Sequ

ence

Grou

ping

/Place

Value

Basic

Fact

sW

ritt

en R

ecor

ding

and

Ord

erTh

e st

uden

t say

s:•

the

who

le n

umbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, b

y on

es, t

ens,

hund

reds

, and

thou

sand

sin

the

rang

e 0–

1 00

0 00

0;•

the

num

ber 1

, 10,

100

, 100

0be

fore

and

afte

r a g

iven

who

le n

umbe

r in

the

rang

e0–

1 00

0 00

0;•

forw

ards

and

bac

kwar

dsw

ord

sequ

ence

s fo

r hal

ves,

quar

ters

, thi

rds,

fifth

s, a

ndte

nths

, e.g

.,,

, 1,

,, e

tc.

•th

e de

cim

al n

umbe

r wor

dse

quen

ces,

forw

ards

and

back

war

ds, i

n te

nths

and

hund

redt

hs.

The

stud

ent o

rder

s:•

who

le n

umbe

rs in

the

rang

e 0–

1 00

0 00

0;•

unit

frac

tions

for h

alve

s,th

irds

, qua

rter

s, fi

fths,

and

tent

hs.

2 34 3

5 31 3

The

stud

ent k

now

s:•

grou

ping

s w

ithin

100

0,e.

g., 2

40 a

nd 7

60,

498

and

502,

...;

•gr

oupi

ngs

of tw

o, th

ree,

five,

and

ten

that

are

innu

mbe

rs to

100

and

find

sth

e re

sulti

ng re

mai

nder

s,e.

g., t

hree

s in

17

is 5

with

2re

mai

nder

, fiv

es in

48

is 9

with

3 re

mai

nder

.•

grou

ping

s of

10

and

100

that

can

be

mad

e fr

om a

four

-dig

it nu

mbe

r,e.

g., t

ens

in 4

562

is 4

56 w

ith2

rem

aind

er, h

undr

eds

in78

94 is

78

with

94

rem

aind

er.

•te

nths

and

hun

dred

ths

inde

cim

als

to tw

o pl

aces

,e.

g. te

nths

in 7

.2 is

72,

hund

redt

hs in

2.8

4 is

284

.

The

stud

ent r

ound

s:•

who

le n

umbe

rs to

the

near

est 1

0, 1

00, o

r 100

0.•

deci

mal

s w

ith u

p to

two

deci

mal

pla

ces

to th

ene

ares

t who

le n

umbe

r,e.

g., r

ound

s 6.

49 to

6,

roun

ds 1

9.91

to 2

0.

The

stud

ent r

ecal

ls:

•ad

ditio

n an

d su

btra

ctio

nfa

cts

up to

20,

e.g.

, 9 +

5, 1

3 –

7;•

mul

tiplic

atio

n ba

sic

fact

sup

to th

e 10

tim

es ta

bles

(10

� 1

0) a

nd s

ome

corr

espo

ndin

g di

visi

onfa

cts;

•m

ultip

licat

ion

basi

c fa

cts

with

tens

, hun

dred

s an

dth

ousa

nds,

e.g.

, 10

� 1

00 =

100

0,10

0 �

100

= 1

0 00

0

The

stud

ent:

•re

cord

s th

e re

sults

of

calc

ulat

ions

usi

ng a

dditi

on,

subt

ract

ion,

mul

tiplic

atio

n,an

d di

visi

on e

quat

ions

,e.

g., 3

49 +

452

= 35

0 +

451

= 80

1,e.

g., 4

5 ÷

9 =

5,•

dem

onst

rate

s th

eca

lcul

atio

n on

a n

umbe

rlin

e or

with

a d

iagr

am.

The

stud

ent p

erfo

rms:

•co

lum

n ad

ditio

n an

dsu

btra

ctio

n w

ith w

hole

num

bers

of u

p to

four

digi

ts.

Stage Six: Advanced Additive–Early Multiplicative

The

stud

ent i

dent

ifies

:•

all o

f the

num

bers

in th

era

nge

0–1

000

000;

•de

cim

als

to th

ree

plac

es;

•sy

mbo

ls fo

r any

frac

tion

incl

udin

g te

nths

,hu

ndre

dths

, tho

usan

dths

,an

d im

prop

er fr

actio

ns.

........... .........

e.g.

,47

6+

285

761

1 1

7000

– 58

664

14

69

91


21

The

stud

ent s

ays:

•th

e de

cim

al w

ord

sequ

ence

s, fo

rwar

ds a

ndba

ckw

ards

, by

thou

sand

ths,

hun

dred

ths,

tent

hs, o

nes,

tens

, etc

.;•

the

num

ber

one-

thou

sand

th,

one-

hund

redt

h, o

ne-te

nth,

one,

ten,

etc

. bef

ore

and

afte

r any

giv

en w

hole

num

ber.

The

stud

ent o

rder

s:•

deci

mal

s to

thre

e pl

aces

,e.

g., 6

.25

and

6.3;

•fr

actio

ns, i

nclu

ding

hal

ves,

thir

ds, q

uart

ers,

fifth

s,te

nths

.

Stag

eNum

ber

Iden

tifica

tion

Num

ber

Sequ

ence

Grou

ping

/Place

Value

Basic

Fact

sW

ritt

en R

ecor

ding

and

Ord

erTh

e st

uden

t kno

ws:

•th

e gr

oupi

ngs

of n

umbe

rsto

10

that

are

in n

umbe

rsto

100

and

find

s th

ere

sulti

ng re

mai

nder

s, e

.g.,

sixe

s in

38,

nin

es in

68;

•th

e gr

oupi

ngs

of 1

0, 1

00,

and

1000

that

can

be

mad

efr

om a

num

ber o

f up

tose

ven

digi

ts, e

.g.,

tens

in47

562

, hun

dred

s in

782

894,

thou

sand

s in

2 78

5 67

1;•

equi

vale

nt fr

actio

ns fo

rha

lves

, thi

rds,

qua

rter

s,fif

ths,

and

tent

hs w

ithde

nom

inat

ors

up to

100

and

up to

100

0, e

.g.,

1 in

4is

equ

ival

ent t

o 25

in 1

00 o

r25

0 in

100

0.

The

stud

ent r

ound

s:•

who

le n

umbe

rs a

ndde

cim

als

with

up

to tw

opl

aces

to th

e ne

ares

t who

lenu

mbe

r or

, e.g

., ro

unds

6.49

to 6

.5 (n

eare

st te

nth)

.

1 10

The

stud

ent r

ecal

ls:

•di

visi

on b

asic

fact

s up

to th

e10

tim

es ta

bles

, e.g

., 72

÷ 8

.•

frac

tion

<—>

deci

mal

<—

>pe

rcen

tage

con

vers

ions

for

halv

es, t

hird

s, q

uart

ers,

fifth

s, a

nd te

nths

,e.

g.,

= 0

.75

= 75

%.

The

stud

ent k

now

s:•

divi

sibi

lity

rule

s fo

r 2, 3

, 5,

9, a

nd 1

0,e.

g., 4

71 is

div

isib

le b

y 3

sinc

e 4

+ 7

+ 1

= 1

2;•

squa

re n

umbe

rs to

100

and

the

corr

espo

ndin

g ro

ots,

e.g.

, 72 =

49,

√49

= 7

.

The

stud

ent i

dent

ifies

:•

fact

ors

of n

umbe

rs to

100

,in

clud

ing

prim

e nu

mbe

rs,

e.g.

, fac

tors

of 3

6 =

{1, 2

, 3,

4, 6

, 9, 1

2, 1

8, 3

6};

•co

mm

on m

ultip

les

ofnu

mbe

rs to

10,

e.g

., 35

, 70,

105,

... a

re c

omm

onm

ultip

les

of 5

and

7.

3 4

The

stud

ent r

ecor

ds:

•th

e re

sults

of c

alcu

latio

nsus

ing

equa

tions

,e.

g., 6

� 2

8 =

168,

and

diag

ram

s, e

.g.,

empt

ynu

mbe

r lin

e.

The

stud

ent p

erfo

rms:

•co

lum

n ad

ditio

n an

dsu

btra

ctio

n fo

r who

lenu

mbe

rs;

•sh

ort m

ultip

licat

ion

and

divi

sion

of a

thre

e-di

git

who

le n

umbe

r by

a si

ngle

-di

git n

umbe

r.

e.g.

,47

3�

837

84

5 2

784

754

885 2

)

Stage Seven: Advanced Multiplicative–Early Proportional

........... .........


22

Stag

eNum

ber

Iden

tifica

tion

Num

ber

Sequ

ence

Grou

ping

/Place

Value

Basic

Fact

sW

ritt

en R

ecor

ding

and

Ord

er

1 10

The

stud

ent s

ays:

•th

e de

cim

al w

ord

sequ

ence

s, fo

rwar

ds a

ndba

ckw

ards

, by

thou

sand

ths,

hun

dred

ths,

tent

hs, o

nes,

tens

, etc

.,st

artin

g at

any

dec

imal

num

ber;

•th

e nu

mbe

ron

e-th

ousa

ndth

,on

e-hu

ndre

dth,

one

-tent

h,on

e, te

n, e

tc. b

efor

e an

daf

ter a

ny g

iven

dec

imal

num

ber.

The

stud

ent o

rder

s:•

frac

tions

, dec

imal

s, a

ndpe

rcen

tage

s.

The

stud

ent k

now

s:•

the

num

ber o

f ten

ths,

hund

redt

hs, a

ndon

e-th

ousa

ndth

s th

at a

re in

num

bers

of u

p to

thre

ede

cim

al p

lace

s,e.

g., t

enth

s in

45.

6 is

456

,hu

ndre

dths

in 9

.03

is 9

03,

thou

sand

ths

in 8

.502

is85

02;

•w

hat h

appe

ns w

hen

aw

hole

num

ber o

r dec

imal

is m

ultip

lied

or d

ivid

ed b

ya

pow

er o

f 10,

e.g.

, 4.5

� 1

00, 6

7.3

÷ 10

.

The

stud

ent r

ound

s:•

deci

mal

s to

the

near

est 1

00,

10, 1

,, o

r ,

e.g.

, rou

ndin

g 52

34 to

the

near

est 1

00 g

ives

520

0.

1 100

The

stud

ent r

ecal

ls:

•fr

actio

n <—

> de

cim

al <

—>

perc

enta

ge c

onve

rsio

ns fo

rgi

ven

frac

tions

and

deci

mal

s,e.

g.,

= 1.

125

= 11

2.5%

.

The

stud

ent k

now

s:•

divi

sibi

lity

rule

s fo

r 2, 3

, 4,

5, 6

, 8, a

nd 1

0, e

.g.,

5 63

2 is

divi

sibl

e by

8 s

ince

632

isdi

visi

ble

by 8

,e.

g., 7

56 is

div

isib

le b

y 3

and

9 as

its

digi

tal r

oot i

s 9;

•si

mpl

e po

wer

s of

num

bers

to 1

0, e

.g.,

24 = 1

6, 5

3 = 1

25

The

stud

ent i

dent

ifies

:•

com

mon

fact

ors

ofnu

mbe

rs to

100

, inc

ludi

ngth

e hi

ghes

t com

mon

fact

or,

e.g.

, com

mon

fact

ors

of 4

8an

d 64

= {1

, 2, 4

, 8, 1

6};

•le

ast c

omm

on m

ultip

les

ofnu

mbe

rs to

10,

e.g

., 24

isth

e le

ast c

omm

on m

ultip

leof

6 a

nd 8

.

9 8

The

stud

ent r

ecor

ds:

•th

e re

sults

of c

alcu

latio

nsus

ing

equa

tions

,e.

g.,

� 2

8 =

21, a

nddi

agra

ms,

e.g

., do

uble

num

ber l

ine.

The

stud

ent p

erfo

rms:

•co

lum

n ad

ditio

n an

dsu

btra

ctio

n fo

r who

lenu

mbe

rs, a

nd d

ecim

als

toth

ree

plac

es;

•sh

ort m

ultip

licat

ion

and

divi

sion

of w

hole

num

bers

and

deci

mal

s by

sin

gle-

digi

t num

bers

, e.g

.,

•m

ultip

licat

ion

of th

ree-

or

four

-dig

it w

hole

num

bers

by tw

o-di

git w

hole

num

bers

, e.g

.,

Stage Eight: Advanced Proportional......... .........

3 4

763

� 4

968

6730

520

3738

7

2 1

5 214

.65

458

.60

)1 2

2

ACKNOWLEDGMENTSThe Ministry of Education wishes to acknowledge thefollowing people and organisations for their contributiontowards the development of this handbook.

THE PARTICIPANTS:The New Zealand numeracy project personnel –facilitators and principals, teachers, and children frommore than two thousand New Zealand schools whocontributed to this handbook through their participationin the numeracy development projects from 2000–2006.

Professor Derek Holton, convenor (The University ofOtago), Professor Megan Clark (Victoria University ofWellington), Dr Joanna Higgins (Victoria University ofWellington College of Education), Dr Gill Thomas (MathsTechnology Limited), Associate Professor Jenny Young-Loveridge (The University of Waikato), AssociateProfessor Glenda Anthony (Massey University), TonyTrinick (The University of Auckland Faculty ofEducation), Garry Nathan (The University of Auckland),Paul Vincent (Education Review Office), Dr JoannaWood (New Zealand Association of MathematicsTeachers), Peter Hughes (The University of AucklandFaculty of Education), Vince Wright (The University ofWaikato School Support Services), Geoff Woolford(Parallel Services), Kevin Hannah (Christchurch Collegeof Education), Chris Haines (School Trustees’Association), Linda Woon (NZPF), Jo Jenks (VictoriaUniversity of Wellington College of Education, EarlyChildhood Division), Bill Noble (New ZealandAssociation of Intermediate and Middle Schools), DianeLeggatt of Karori Normal School (NZEI Te Riu Roa),Sului Mamea (Pacific Island Advisory Group, PalmerstonNorth), Dr Sally Peters (The University of WaikatoSchool of Education), Pauline McNeill of ColumbaCollege (PPTA), Dr Ian Christensen (He Kupenga Hao ite Reo), Liz Ely (Education Review Office), Ro Parsons(Ministry of Education), Malcolm Hyland (Ministry ofEducation).

THE ORIGINAL WRITERS,REVIEWERS, AND PUBLISHERS:Vince Wright (The University of Waikato School SupportServices), Peter Hughes (The University of AucklandFaculty of Education), Sarah Martin (Bayview School),Gaynor Terrill (The University of Waikato School ofEducation), Carla McNeill (The University of WaikatoSchool of Education), Professor Derek Holton (TheUniversity of Otago), Dr Gill Thomas (Maths TechnologyLimited), Bruce Moody (mathematics consultant), LynnePetersen (Dominion Road School), Marilyn Holmes(Dunedin College of Education), Errolyn Taane (DunedinCollege of Education), Lynn Tozer (Dunedin College of

Education), Malcolm Hyland (Ministry of Education), RoParsons (Ministry of Education), Kathy Campbell(mathematics consultant), Jocelyn Cranefield, KirstyFarquharson, Jan Kokason, Bronwen Wall (LearningMedia Limited), Joe Morrison, Andrew Tagg (MathsTechnology Limited).

REVISED EDITION 2007:People who contributed to the 2007 revision of pages 4–13 include Associate Professor Jenny Young-Loveridge(The University of Waikato), Malcolm Hyland (Ministry ofEducation), Gaynor Terrill and Vince Wright (TheUniversity of Waikato School Support Services), othernumeracy facilitators throughout New Zealand, andSusan Roche (Learning Media Ltd).

In addition, the Ministry of Education wishes toacknowledge Professor Bob Wright (Southem CrossUniversity, Lismore, NSW), Dr Noel Thomas (CharlesSturt University, Bathhurst, NSW), Dr KoenoGravemeijer (Freudenthal Institute, Utrecht,Netherlands), Jim Martland (The University ofManchester, UK), and Susan Lamon (MarquetteUniversity, USA).

The Ministry of Education also wishes to acknowledgeThe New South Wales Department of Education andTraining for permission to trial Count Me In Too in 2000through a one-year arrangement. The findings from theuse of this pilot project informed the development of thenumeracy policy in New Zealand.

Count Me In Too is the registered Trade Mark of theCrown in the Right of the State of New South Wales(Department of Education and Training). Copyright ofthe Count Me In Too Professional DevelopmentPackage materials (1997-2002), including the LearningFramework in Number and the Schedule for EarlyNumber Assessment, is also held by the Crown in theRight of the State of New South Wales (Department ofEducation and Training) 2002.

The cover design is by Dave Maunder (Learning MediaLimited) and Base Two Design Ltd. All other illustrationsare by Noel Eley and James Rae.

All illustrations copyright © Crown 2008 except: Digitalimagery of the conch copyright © 2000 PhotoDisc, Inc.

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