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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 Professional Development Projects 2008Published by the Ministry of Education.PO Box 1666, Wellington 6140, 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
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.
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
The Number Framework
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
The Number Framework
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.”
The Number Framework
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
The Number Framework
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
The Number Framework
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
The Number Framework
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
The Number Framework
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)
The Number Framework
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)
(Not drawn to scale)
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
The Number Framework
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:
(Not drawn to scale)
(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
The Number Framework
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
The Number Framework
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
The Number Framework
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
..
.
The Number Framework
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.
The Number Framework
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.
The Number Framework
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
The Number Framework
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
The Number Framework
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
.
The Number Framework
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.
The Number Framework
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 MultiplicativeTh
e st
uden
t ide
ntifi
es:
•al
l of t
he n
umbe
rs in
the
rang
e 0–
1 00
0 00
0;•
deci
mal
s to
thre
e pl
aces
;•
sym
bols
for a
ny fr
actio
nin
clud
ing
tent
hs,
hund
redt
hs, t
hous
andt
hs,
and
impr
oper
frac
tions
.
........... .........
e.g.
,47
6+
285
761
1 1
7000
– 58
664
14
69
91
The Number Framework
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
........... .........
The Number Framework
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.
THE NUMERACY REFERENCE GROUP: