The%20key%20ideas%20and%20strategies%20that%20underpin%20Multiplicative%20Thinking

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The key ideas and strategies that underpin
Multiplicative Thinking
Presented by Dianne Siemon
Support for this project has been provided by the
Australian Research Council, RMIT University, the
Victorian Department of Education and Training,
and the Tasmanian Department of Education.
TASMANIAN Department of Education
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KEY IDEAS AND STRATEGIES

• Early Number (counting, subitising,
  part-part-whole, trusting the count, composite
  units, place-value)
• Mental strategies for addition subtraction
  (count on from larger, doubles/near doubles,
  make-to-ten)
• Concepts for multiplication and division (groups
  of, arrays/regions, area, Cartesian Product,
  rate, factor-factor-product)
• Mental strategies for multiplication and division
  (eg, doubles and 1 more group for 3 of anything,
  relate to 10 for 5s and 9s facts)
• Fractions and Decimals (make, name, record,
  rename, compare, order via partitioning)

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COUNTING
Jenni can count to 100 ...
To count effectively, children not only need to
know the number naming sequence, they need to
recognise that

• counting objects and words need to be in
  one-to-one correspondence
• three means a collection of three no matter
  what it looks like
• the last number counted tells how many.

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SUBITISING PART-PART-WHOLE
But can Jenni read numbers without counting?
To develop a strong sense of number, children
also need to be able to

• recognise collections up to five without counting
  subitising) and
• name numbers in terms of their parts
  (part-part-whole knowledge).

Eg, for this collection see 3 instantly but
also see it as a 2 and a 1 more
5
Eg, How many?
Close your eyes. What did you see?
6
Try this
7
and this
What difference does this make?
8
Try this
9
and this
What did you notice?
10
What about this?
Would colour help? How? Why?
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But what about?
How do you feel?
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The numbers 0 to 9 are the only numbers most of
us ever need to learn ... it is important to know
everything there is to know about each number.
For this collection, we need to know

• it can be counted by matching number names to
  objects one, two, three, four, five, six,
  seven, eight and that the last one says, how
  many
• it can be written as eight or 8 and
• it is 1 more than 7 and 1 less than 9.

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But we also need to know 8 in terms of its parts,
that is,





8 is
2 less than 10 6 and 2 more 4 and 4 double 4 3
and 3 and 2 5 and 3, 3 and 5
Differently configured ten-frames are ideal for
this
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TRUSTING THE COUNT
This recently recognised capacity builds on a
number of important early number ideas.
Trusting the count has a range of meanings

• initially, children may not believe that if they
  counted the same collection again, they would get
  the same result, or that counting is a strategy
  to determine how many.

• Ultimately, it is about having access to a range
  of mental objects for each of the numerals, 0 to
  9, which can be used flexibly without having to
  make, count or see these collections physically.

See WA Department of Education, First Steps in
Mathematics
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Trusting the count is evident when children

• know that counting is an appropriate response to
  How many ? questions

• believe that counting the same collection again
  will always produce the same result irrespective
  of how the objects in the collection are arranged

• are able to subitise (ie, identify the number of
  objects without counting) and invoke a range of
  mental objects for each of the numbers 0 to ten
  (including part-part-whole knowledge)

• work flexibly with numbers 0 to ten using
  part-part-whole knowledge and/or visual imagery
  without having to make or count the numbers and
• are able to use small collections as composite
  units when counting larger collections (eg, count
  by 2s, or 5s)

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MENTAL STRATEGIES FOR ADDITION
Pre-requisites

• Children know their part-part-whole number
  relations (eg, 7 is 3 and 4, 5 and 2, 6 and 1
  more, 3 less than 10 etc)
• Children trust the count and can count on from
  hidden or given
• Children have a sense of numbers to 20 and beyond
  (eg, 10 and 6 more, 16)

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1. Count on from larger for combinations
involving 1, 2 or 3 (using commutativity)
For example, for 6 and 2, THINK 6 7, 8 for 3
and 8, THINK 8 9, 10, 11 for 1 and 6, THINK
6 7 for 4 and 2, THINK 4 5, 6
This strategy can be supported by ten-frames,
dice and oral counting
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For example





Cover 5, count on
Cover 4, count on
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2. Doubles and near doubles
For example, for 4 and 4, THINK double 4,
8 for 6 and 7, THINK 6 and 6 is 12, and 1 more,
13 for 9 and 8, THINK double 9 is 18, 1 less,
17 for 7 and 8, THINK double 7 is 14, 1 more, 15
This strategy can be supported by ten-frames and
bead frames (to 20) can be used to build doubles
facts
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For example










Ten-frames
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For example
Count 6 and 6 is 12, and 1 more, 13
Bead Frame (to 20) Double-decker bus scenario
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3. Make to ten and count on
For example, for 8 and 3, THINK 8 10, 11 for
6 and 8, THINK 8 10, 14 for 9 and 6, THINK
9 10, 15 for 7 and 8, THINK double 7 is 14, 1
more, 15
Ten-frames and bead frames (to 20) can be used to
bridge to ten, build place-value facts (eg 10 and
6 more , sixteen)
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For example










For 8 and 6
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For example










Think 10 and 4 more ... 14
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MENTAL STRATEGIES FOR SUBTRACTION
For example, for 9 take 2, THINK 9 8, 7
(count back) for 6 take 3, THINK 3 and 3 is 6
(think of addition) for 15 take 8, THINK 15,
10, 7 (make back to 10)
Or for 16 take 9, THINK 16 take 8 is 8, take 1
more, 7 (halving) 16, 10, 7 (make back to
10) 9, 10, 16 7 needed (think of
addition) 16, 6, add 1 more, 7 (place-value)
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CONCEPTS FOR MULTIPLICATION
Establish the value of equal groups by

• exploring more efficient strategies for counting
  large collections using composite units and
• sharing collections equally.

Explore concepts through action stories that
involve naturally occurring equal groups, eg,
the number of wheels on 4 toy cars, the number of
fingers in the room, the number of cakes on a
bakers tray ...., and stories from Childrens
Literature, eg, Counting on Frank or the Doorbell
Rang
See Booker et al, pp.182-201 pp.221-233
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1. Groups of
3 fours ... 4, 8, 12
4 threes ... 3, 6, 9, 12
Focus is on the group. Really only suitable for
small whole numbers, eg, some sense in asking
How many threes in 12? But very little sense in
asking How many groups of 4.8 in 34.5?
Strategies make-all/count-all groups, repeated
addition (or skip counting).
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2. Arrays
Rotate and rename
4 threes ... THINK 6 and 6
3 fours ... THINK 8, 12
Focus on product (see the whole, equal groups
reinforced by visual image), does not rely on
repeated addition, supports commutativity (eg, 3
fours SAME AS 4 threes) and leads to more
efficient mental strategies
Strategies mental strategies that build on from
known, eg, doubling and addition strategies
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3. Regions
Rotate and rename
4 threes ... THINK 6 and 6
3 fours ... THINK 8, 12
Continuous model. Same advantages as array idea
(discrete model) establishes basis for
subsequent area idea.
Note For whole number multiplication continuous
models are introduced after discrete this is
different for fraction models!
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4. Area idea
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3
3 by 1 ten and 4 ones
3 by 1 ten ... 3 tens
3 by 4 ones ... 12 ones
Think 30 ... 42
Supports multiplication by place-value parts and
the use of extended number fact knowledge, eg, 4
tens by 2 ones is 8 tens ... Ultimately, 2-digit
by 2-digit numbers and beyond
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The Area idea (extended)
33
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Supports multiplication by place-value parts, eg,
2 tens by 3 tens is 6 hundreds... Ultimately,
that tenths by tenths are hundredths and
(2x4)(3x3) is 6x218x12
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5. Cartesian Product
Eg, lunch options
4 different types of filling
2 different types of fruit
3 different types of bread
3 x 4 x 2 24 different options
Supports for each idea and multiplication by 1
or more factors
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6. Rate
Eg, 5 sweets per bag. 13 bags of sweets. How many
sweets altogether? Eg, Jason bought 3.5 kg of
potatoes at 2.95 per kg. How much did he spend
on potatoes?
These problems require thinking about the unit.
In this case, 1 bag and 1 kg respectively
Eg, Samanthas snail travels 15 cm in 3 minutes.
Annas snail travels 37 cm in 8 minutes. Which is
the speedier snail?
This problem involves rate but actually asks for
a comparison of ratios which requires
proportional reasoning.
Rate builds on the for each idea and underpins
proportional reasoning
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MENTAL STRATEGIES FOR MULTIPLICATION
The traditional multiplication tables are not
really tables at all but lists of equations which
count groups, for example
This is grossly inefficient
1 x 3 3 2 x 3 6 3 x 3 9 4 x 3 12 5 x 3
15 6 x 3 18 7 x 3 21 8 x 3 24 9 x 3 27 10
x 3 30 11 x 3 33 12 x 3 36
1 x 4 4 2 x 4 8 3 x 4 12 4 x 4 16 5 x 4
20 6 x 4 24 7 x 4 28 8 x 4 32 9 x 4 36 10
x 4 40 11 x 4 44 12 x 4 48
3 fours not seen to be the same as 4 threes
... 10s and beyond not necessary
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More efficient mental strategies build on
experiences with arrays and regions
Eg, 3 sixes? ... THINK double 6 ... 12, and 1
more 6 ... 18
And the commutative principle
3
6
Eg, 6 threes? ... THINK 3 sixes ... double 6,
12, and 1 more 6 ... 18
3
6
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This involves a shift in focus
From a focus on the number IN the group
A critical step in the development of
multiplicative thinking appears to be the shift
from counting groups, for example, 1 three, 2
threes, 3 threes, 4 threes, ... to seeing the
number of groups as a factor, For example, 3
ones, 3 twos, 3 threes, 3 fours, ... and
generalising, for example, 3 of anything is
double the group and 1 more group.
To a focus on the number OF groups
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Mental strategies for the multiplication facts
from 0x0 to 9x9

• Doubles and doubles reversed (twos facts)
• Doubles and 1 more group ... (threes facts)
• Double, doubles ... (fours facts)
• Same as (ones and zero facts)
• Relate to ten (fives and nines facts)
• Rename number of groups (remaining facts)

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An alternative multiplication table

This actually represents the region idea and
supports efficient, mental strategies (read
across the row), eg,
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
6 ones, 6 twos, 6 threes, 6 fours, 6 fives, 6
sixes, 6 sevens, 6 eights, 6 nines
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The region model implicit in the alternative
table also supports the commutative idea

X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
Eg, 6 threes?
THINK .
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The region model implicit in the alternative
table also supports the commutative idea

X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
Eg, 6 threes?
THINK 3 sixes This halves the amount of
learning
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Doubles Strategy (twos)
2 ones, 2 twos, 2 threes, 2 fours, 2 fives ...
2 fours ... THINK double 4 ... 8
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
2 sevens ... THINK double 7 ... 14
7 twos ... THINK double 7 ... 14
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Doubles and 1 more group strategy (threes)
3 ones, 3 twos, 3 threes, 3 fours, 3 fives ...
3 eights THINK double 8 and 1 more 8 16 , 20, 24
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
9 threes ... THINK?
3 twenty-threes THINK?
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Doubles doubles strategy (fours)
4 ones, 4 twos, 4 threes, 4 fours, 4 fives ...
4 sixes THINK double 4 ... 8 double again, 16
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
8 fours ... THINK?
4 forty-sevens THINK?
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Same as strategy (ones and zeros)
1 one, 1 two, 1 three, 1 four, 1 five, ...
1 of anything is itself ... 8 ones, same as 1
eight
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
Cannot show zero facts on table ... 0 of
anything is 0 ... 7 zeros, same as 0 sevens
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Relate to tens strategy (fives and nines)
5 ones, 5 twos, 5 threes, 5 fours, 5 fives ... 9
ones, 9 twos, 9 threes, 9 fours, 9 fives ...
5 sevens THINK half of 10 sevens, 35
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
8 fives ... THINK?
9 eights THINK less than 10 eights, 1 eight
less, 72
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Rename number of groups (remaining facts)
6 sixes, 6 sevens, 6 eights ... 7 sixes, 7
sevens, 7 eights ... 8 sixes, 8 sevens, 8 eights
...
6 sevens THINK 3 sevens and 3 sevens, 42 ... OR
5 sevens and 1 more 7
X 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81
8 sevens THINK 7 sevens is 49, and 1 more 7, 56
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CONCEPTS FOR DIVISION
1. How many groups in (quotition)
How many fours in 12?
1 four, 2 fours, 3 fours
12 counters
Really only suitable for small collections of
small whole numbers, eg, some sense in asking
How many fours in 12? But very little sense in
asking How many groups of 4.8 in 34.5?
Strategies make-all/count-all groups, repeated
addition
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Quotition (guzinta) Action Stories
24 tennis balls need to be packed into cans that
hold 3 tennis balls each. How many cans will be
needed? Sam has 48 marbles. He wants to give his
friends 6 marbles each. How many friends will
play marbles?
How many threes?
How many sixes?
Total and number in each group known Question
relates to how many groups.
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2. Sharing (partition)
18 sweets shared among 6. How many each?
3 in each group
18 counters
More powerful notion of division which relates to
array and regions models for multiplication and
extends to fractions and algebra
Strategy Think of Multiplication eg, 6 whats
are 18? ... 6 threes
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Partition Action Stories
42 tennis balls are shared equally among 7
friends. How many tennis balls each? Sam has 36
marbles. He packs them equally into 9 bags. How
many marbles in each bag?
THINK 7 whats are 42?
THINK 9 whats are 36?
Total and number of groups known Question
relates to number in each group.
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28 7 4
groups of 7
7 groups or parts
Q 7 shares, how many in each share? PARTITION
Q How many 7s in 28? QUOTITION
This supports arrays, regions and division more
generally, in particular, fractions and ratios
This suggests a count of 7s, only practical for
small whole numbers
7 whats are 28?
28 7
Meaning 28 divided by 7 What does 28 sevenths
imply?
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MENTAL STRATEGY FOR DIVISION
Does 7 represent the number in each group or the
number of groups?

• Think of multiplication

Eg, 56 divided by 7? THINK 7 whats are 56?
7 sevens are 49, 7 eights are 56 So, 56 divided
by 7 is 8
4 sixes are 24, 24 divided by 4 is 6, 24 divided
by 6 is 4, 1 quarter of 24 is 6, 1 sixth of 24
is 4
Work with fact families What do you know if you
know that 6 fours are 24?
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FRACTIONS AND DECIMALS
2 5
Traditional practices (eg, shade to show only
require students to count to 2 and colour!

Students do not necessarily attend to the number
of parts, or the equality of parts and the unit
is assumed.
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Introducing Fractions
Young children come to school with an intuitive
sense of proportion based on fair shares and a
working knowledge of what is meant by, half and
quarter.

• Youve got more than me, thats not fair!
• half of the apple, the glass is half full
• a quarter of the orange,
• 3 quarters of the pizza

This is a useful starting point, but much more is
needed before children can be expected to work
with fractions formally
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Initial ideas
In Prep to Year 3, children need to be exposed to
the language and concepts of fractions through
real-world examples. These occur in two forms
CONTINUOUS 3 quarters of the pie 2 thirds of the netball court 5 eighths of the chocolate bar left Continuous models are infinitely divisible DISCRETE Half a dozen eggs 2 thirds of the marbles Discrete models are collections of whole
Note language only, no symbols
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Use real-world examples AND non-examples to
ensure students understand that EQUAL parts are
required.
Cut plasticene rolls and pies into equal and
unequal parts discuss fair shares
Share jelly-beans or smarties equally and
unequally discuss fair shares
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The consequences of not appreciating the need for
equal parts.
They know how to play the game but what do they
really know?
Work Sample from SNMY Project 2003-2006 Male,
Year 5
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Explore paper folding, what do you notice as the
number of parts increases?
Fold a sheet of newspaper in half. Repeat until
it cant be folded in half again discuss what
happens to the number of parts and the size of
the parts
Halve paper strips of different lengths, compare
halves how are they the same? How are they
different?
The size of the part depends upon the whole and
the number of parts
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Formalising Fraction Knowledge

1. Prior knowledge and experience - informal
   experiences, fraction language, key ideas
2. Partitioning the missing link in building
   fraction knowledge and confidence, strategies for
   making, naming and representing fractions
3. Recording common fractions and decimal fractions
   problems with recording, the fraction symbol,
   decimal numeration (to tenths)
4. Consolidating fraction knowledge comparing,
   ordering/sequencing, counting, and renaming.

Equal parts
As the number of parts increases, the size of the
part decreases
The number of parts names the part
The numerator tells how many, the denominator
tells how much
Links to multiplication and division
60
Partitioning

Counting and colouring parts of someone elses
model is next to useless - students need to be
actively involved in making and naming their own
fraction models. Partitioning (making equal
parts) is the key to this

• develop strategies for halving, thirding and
  fifthing
• generalise to create diagrams and number lines
• use to make, name, compare, order, and rename
  mixed and proper fractions including decimals.

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Explore partitioning informally through paper
folding, cutting and sharing activities based on
halving using a range of materials, eg,
plasticene rolls and icy-pole sticks
paper streamers
rope and pegs
Kindergarten Squares
Smarties
62
The halving strategy
For example,
Explore paper folding with coloured paper
squares, paper streamers and newspaper.
63
Both shapes are 1 half
How are they different? How are they the same?
64
Explore make and name as many fractions in the
halving family as you can
8 equal parts, eighths
How many different designs can you make which are
3 quarters red and 1 quarter yellow?
65
For example, make a poster
2 and 3 quarters
Write down as many things as you can about your
fraction. How many different ways can you find to
name your fraction?
Its bigger than 2 and a half ... Smaller than 3
.... Its 11 quarters ... Its 5 halves and 1
quarter ... It could be 2 and 3 quarter slices of
bread ...
66
Extend partitioning to diagrams
Ask What did the first fold do? It cut the top
and bottom edges in half
Estimate 1 half
67
Ask What did the second fold do? It cut the top
and bottom edges in half again
68
Ask What did the third fold do? It cut the side
edges in half.
How would you describe this strategy using paper
streamers?
69
The thirding strategy
Think 3 equal parts ... 2 equal parts 1 third
is less than 1 half ... estimate
Halve the remaining part
Fold kindergarten squares or paper streamers into
3 equal parts
70
Use to draw diagrams, for example,
Apply thirding strategy to top and bottom edge,
halving strategy to side edges to get sixths
71
The fifthing strategy
Think 5 equal parts ... 4 equal parts 1 fifth
is less than 1 quarter ... estimate
Then halve and halve again
Fold kindergarten squares or paper streamers into
5 equal parts
72
Use to draw diagrams, for example,
Apply fifthing strategy to top and bottom edge,
halving strategy to side edges to get tenths
Apply to number line
4 5
73
No. of parts Name
1 whole
2 halves
3 thirds
4 quarters (fourths)
5 fifths
6 sixths
8 eighths
9 ninths
10 tenths
12 twelfths
15 fifteenths
Notice
Halving family
Halving and Thirding
Thirding family
Halving and Fifthing
Fifthing family
Thirding and Fifthing
As the number of parts increases, the size of the
parts gets smaller the number of parts, names
the part
74
Explore strategy combinations to recognise that
Thirds by quarters give twelfths
quarters
thirds
What other fractions can be generated by halving
and thirding or by fifthing and thirding?
Thirds by fifths give fifteenths
thirds
fifths
tenths










What other fractions can be generated by fifthing
and halving?
Tenths by tenths give hundredths
tenths
75
Use real-world examples to explore the difference
between how many and how much
Young children expect numbers to be used to say
how many
This tells how many tens
This tells how many ones
34
Informally describe and compare
Is it a big share or a little share? Would you
rather have 2 thirds of the pizza or 3 quarters
of the pizza? Why? How could you convince me?
Construct fraction diagrams to compare more
formally
76
Recording common fractions
Introduce recording once key ideas have been
established through practical activities and
partitioning

• equal shares - equal parts
• fraction names are related to the total number of
  parts (denominator idea the more parts there
  are, the smaller they are)
• the number of parts required tells how many
  (numerator idea the only counting number)

Explore non-examples
This tells how much
This tells how many
77
Introduce the fraction symbol

2 out of 5
2 5
2 fifths
2 5
This number tells how many
This number names the parts and tells how much
Make and name mixed common fractions
78

• Recognise
• different meanings for ordinal number names, eg,
  third can mean third in line, the 3rd of April
  or 1 out of 3 equal parts
• that the out of idea only works for proper
  fractions and recognised wholes, eg,

third 3rd
3 out of 4
Note this idea does not work for improper
fractions, eg, 10 out of 3 is meaningless! But
10 thirds does make sense, as does 10 divided
into 3 equal parts
79
Introducing Decimals
Recognise decimals as fractions use halving and
fifthing partitioning strategies to make and
represent tenths
Halves by fifths are tenths 7 out of ten parts, 7
tenths
fifths
halves
Fifth then halve each part or halve then fifth
each part, 2 and 4 tenths
2
2.4
3
Name decimals in terms of their place-value
parts, eg, two and four tenths NOT two point
four
Why is this important?
80
Recognise tenths as a new place-value part

1. Introduce the new unit 1 one is 10 tenths
2. Make, name and record ones and tenths

ones tenths
one and 3 tenths
1 ? 3

The decimal point shows where ones begin
3. Consolidate compare, order, count forwards
and backwards in ones and tenths, and rename
Note Money and MAB do not work Why?
81
Extend decimal place-value
Recognise hundredths as a new place-value part

1. Introduce the new unit 1 tenth is 10 hundredths
2. Show, name and record ones, tenths hundredths

via partitioning
hundredths
tenths
ones
5.0 5.3 5.4
6.0
5 ? 3 7
5.30
5.37 5.40
3. Consolidate compare, order, count forwards
and backwards, and rename
82
Establish links between tenths and hundredths,
and hundredths and per cent











7 10
0.7 is 7 tenths or
0.75 is 7 tenths, 5 hundredths 75 hundredths 75
per cent, 75, or
Recognise per cent benchmarks 50 is a half,
25 is a quarter, 10 is a tenth, 33 is 1
third
75 100
83
Consolidating decimal place-value

1. Compare decimals which is larger, which is
   smaller, why?
2. Order decimal fractions on a number line, eg,
3. Count forwards and backwards in place-value
   parts, eg,
4. Rename in as many different ways as possible, eg,

Which is longer, 4.5 metres or 4.34 metres? Which
is heavier, 0.75 kg or 0.8 kg?
Order from smallest to largest and place on a 0
to 2 number line (rope) 3.27, 2.09, 4.9,
0.45, 2.8
2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 5.23, 5.43,
5.63, 5.83,
4.23 is 4 ones, 2 tenths, 3 hundredths 4 ones,
23 hundredths 42 tenths, 3 hundredths 423
hundredths
84
Extending Fraction Decimal Ideas
By the end of primary school, students are
expected to be able to
Requires partitioning strategies, fraction as
division idea and region idea for multiplication

• rename, compare and order fractions with unlike
  denominators
• recognise decimal fractions to thousandths

Requires partitioning strategies, place-value
idea that 1 tenth of these is 1 of those, and the
for each idea for multiplication
85
Renaming Common Fractions
Use paper folding student generated diagrams
to arrive at the generalisation
1 3
2 6
3 4
9 12
3 parts
9 parts
4 parts
12 parts
If the total number of parts increase by a
certain factor, the number of parts required
increase by the same factor
86
Comparing common fractions
Which is larger 3 fifths or 2 thirds?
But how do you know? ... Partition
fifths
thirds
THINK thirds by fifths ... fifteenths
87
Comparing common fractions
Which is larger 3 fifths or 2 thirds?
3 5
9 15

2 3
10 15

THINK thirds by fifths ... fifteenths
88
Extend decimal place-value
Recognise hundredths as a new place-value part

1. Introduce the new unit 1 hundredth is 10
   thousandths
2. Show, name and record ones, tenths, hundredths
   and thousandths

via partitioning
5.0 5.3 5.4
6.0
hundredths
thousandths
tenths
ones
5.30
5.37 5.38 5.40
5 ? 3 7 6
5.370 5.376
5.380
3. Consolidate compare, order, count forwards
and backwards, and rename
89
Compare, order and rename decimal fractions
Some common misconceptions

• The more digits the larger the number (eg, 5.346
  said to be larger than 5.6)
• The less digits the larger the number (eg, 0.4
  considered to be larger than 0.52)
• If ones, tens hundreds etc live to the right of
  0, then tenths, hundredths etc live to the left
  of 0 (eg, 0.612 considered smaller than 0.216)
• Zero does not count (eg, 3.01 seen to be the same
  as 3.1)
• A percentage is a whole number (eg, do not see
  that 67 is 67 hundredths or 0.67)

90
Compare, order and rename decimal fractions

1. Is 4.57 km longer/shorter than 4.075 km?
2. Order the the long-jump distances 2.45m, 1.78m,
   2.08m, 1.75m, 3.02m, 1.96m and 2.8m
3. 3780 grams, how many kilograms?
4. Express 7¾ as a decimal

ones tenths
hundredths thousandths
2 9 0 7
1
Use Number Expanders to rename decimals
91
Consolidating fraction knowledge

1. Compare mixed common fractions and decimals
   which is bigger, which is smaller, why?
2. Order common fractions and decimal fractions on a
   number line
3. Count forwards and backwards in recognised parts
4. Rename in as many different ways as possible.

Which is bigger? Why? 2/3 or 6 tenths ... 11/2
or 18/16

92
For example,

(Gillian Large, Year 5/6, 2002)
93

(Gillian Large, Year 5/6, 2002)
94
Games

• For example,
• Make a Whole
• Target Practice
• Fraction Concentration

(Make a Whole Game Board, Vicki Nally, 2002)
95
Make a Whole
(Vicki Nally, 2002)
96
(Vicki Nally, 2002)
97
Make a Model, eg, a Think Board
(Gillian Large, 2002)