Transcript The key ideas and strategies that underpin Multiplicative

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
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)
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’.
SUBITISING & PART“But can Jenni read
PART-WHOLE:
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 (partpart-whole knowledge).
Eg, for this collection see “3” instantly
but also see it as a “2 and a 1 more”
Eg, How many?
Close your eyes. What did you see?
Try this:
…and this:
What difference does this make?
Try this:
… and this:
What did you notice?
What about this?
Would colour help? How? Why?
But what about?
How do you feel?
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.
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
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
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-partwhole 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)
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)
1. Count on from larger for combinations
involving 1, 2 or 3 (using commutativity)
For example,
for 6 and 2, THINK:
for 3 and 8, THINK:
for 1 and 6, THINK:
for 4 and 2, THINK:
6 … 7, 8
8 … 9, 10, 11
6…7
4 … 5, 6
This strategy can be supported by tenframes, dice and oral counting
For example:
Cover 5, count on
Cover 4, count on
2. Doubles and near doubles
For example,
for 4 and 4, THINK:
for 6 and 7, THINK:
for 9 and 8, THINK:
for 7 and 8, THINK:
double 4, 8
6 and 6 is 12, and 1 more, 13
double 9 is 18, 1 less, 17
double 7 is 14, 1 more, 15
This strategy can be supported by tenframes and bead frames (to 20) can be
used to build doubles facts
For example:
Ten-frames
For example:
Count: 6 and 6 is 12, and 1 more, 13
Bead Frame (to 20)
Double-decker bus scenario
3. Make to ten and count on
For example,
for 8 and 3, THINK:
for 6 and 8, THINK:
for 9 and 6, THINK:
for 7 and 8, THINK:
8 … 10, 11
8 … 10, 14
9 … 10, 15
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)
For example:
For 8 and 6 …
For example:
Think: 10 … and 4 more ... 14
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)
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
baker’s tray ...., and stories from Children’s
Literature, eg, Counting on Frank or the Doorbell
Rang
See Booker et al, pp.182-201 & pp.221-233
1. Groups of:
4 threes ... 3, 6, 9, 12
3 fours ... 4, 8, 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).
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
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!
4. ‘Area’ idea:
14
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
The ‘Area’ idea (extended):
33
24
Supports multiplication by place-value parts, eg, 2
tens by 3 tens is 6 hundreds...
Ultimately, that tenths by tenths are hundredths
and (2x+4)(3x+3) is 6x2+18x+12
5. Cartesian Product:
4 different
types of
filling
3 different
types of
bread
Eg, lunch options
2 different
types of
fruit
3 x 4 x 2 = 24
different options
Supports ‘for each’ idea and multiplication by 1
or more factors
6. Rate:
Eg, 5 sweets per bag. 13 bags of sweets. How many
sweets altogether?
These problems require thinking about the
‘unit’. In this case, 1 bag and 1 kg respectively
Eg, Jason bought 3.5 kg of potatoes at $2.95 per kg. How
much did he spend on potatoes?
Eg, Samantha’s snail travels 15 cm in 3 minutes.
Anna’s 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
MENTAL STRATEGIES FOR
MULTIPLICATION:
The traditional ‘multiplication tables’ are not really tables at
all but lists of equations which count groups, for
example:
1x3=3
2x3=6
3x3=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
1x4=4
2x4=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
This is grossly
inefficient
3 fours not seen to be the
same as 4 threes ...
10’s and beyond not
necessary
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
Eg, 6 threes? ...
THINK:
3 sixes ...
double 6, 12, and 1
more 6 ... 18
6
6
3
This involves a shift in focus:
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”.
From a
focus on
the
number
IN the
group
To a
focus on
the
number
OF
groups
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)
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
6 ones,
1 1 2 3 4 5 6 7 8 9
6 twos,
2 2 4 6 8 10 12 14 16 18
6 threes,
3 3 6 9 12 15 18 21 24 27
6 fours,
4 4 8 12 16 20 24 28 32 36
6 fives,
5 5 10 15 20 25 30 35 40 45
6 sixes,
6 6 12 18 24 30 36 42 48 54
6 sevens,
7 7 14 21 28 35 42 49 56 63
6 eights,
8 8 16 24 32 40 48 56 64 72
6 nines
9
9 18 27 36 45 54 63 72 81
The region model implicit in the alternative table
also supports the commutative idea:
Eg, 6 threes?
THINK:
….
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
The region model implicit in the alternative table
also supports the commutative idea:
Eg, 6 threes?
THINK:
3 sixes
This halves
the amount
of learning
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
Doubles Strategy (twos) :
2 ones, 2 twos, 2 threes, 2 fours, 2 fives ...
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 fours ...
THINK:
double 4 ... 8
2 sevens ...
THINK:
double 7 ... 14
7 twos ...
THINK:
double 7 ... 14
Doubles and 1 more group strategy (threes):
3 ones, 3 twos, 3 threes, 3 fours, 3 fives ...
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
3 eights
THINK:
double 8 and
1 more 8
16 , 20, 24
9 threes ...
THINK?
3 twenty-threes
THINK?
Doubles doubles strategy (fours):
4 ones, 4 twos, 4 threes, 4 fours, 4 fives ...
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
4 sixes
THINK:
double 4 ... 8
double
again, 16
8 fours ...
THINK?
4 forty-sevens
THINK?
‘Same as’ strategy (ones and zeros):
1 one, 1 two, 1 three, 1 four, 1 five, ...
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
1 of anything
is itself ... 8
ones, same
as 1 eight
Cannot show
zero facts on
table ...
0 of anything
is 0 ... 7 zeros,
same as 0
sevens
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
X 1 2 3 4 5 6 7 8 9
THINK: half of
1 1 2 3 4 5 6 7 8 9
10 sevens, 35
2 2 4 6 8 10 12 14 16 18
3
4
5
6
7
8
9
3
4
5
6
7
8
9
6
8
10
12
14
16
18
9
12
15
18
21
24
27
12
16
20
24
28
32
36
15
20
25
30
35
40
45
18
24
30
36
42
48
54
21
28
35
42
49
56
63
24
32
40
48
56
64
72
27
36
45
54
63
72
81
8 fives ...
THINK?
9 eights
THINK: less
than 10 eights,
1 eight less, 72
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
X 1 2 3 4 5 6 7 8 9
THINK: 3
1 1 2 3 4 5 6 7 8 9
sevens and 3
2 2 4 6 8 10 12 14 16 18
sevens, 42 ...
3 3 6 9 12 15 18 21 24 27
OR 5 sevens
and 1 more 7
4 4 8 12 16 20 24 28 32 36
5
6
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7
8
9
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
8 sevens
THINK: 7
sevens is 49,
and 1 more 7,
56
CONCEPTS FOR DIVISION:
1. How many groups in (quotition):
How many
fours in 12?
12 counters
1 four, 2 fours, 3 fours
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
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?
How many threes?
Sam has 48 marbles. He wants to give his
friends 6 marbles each. How many friends will
play marbles?
How many sixes?
Total and number in each group known –
Question relates to how many groups.
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 what’s are 18? ... 6 threes
Partition Action Stories:
42 tennis balls are shared equally among 7
friends. How many tennis balls each?
THINK: 7 what’s are 42?
Sam has 36 marbles. He packs them equally into
9 bags. How many marbles in each bag?
THINK: 9 what’s are 36?
Total and number of groups known –
Question relates to number in each group.
28 ÷ 7 = 4
groups of
7
7 groups
or parts
Q: 7 shares, how many
in each share?
Q: How many 7s in
28?
PARTITION
QUOTITION
This supports arrays, regions
and division more generally, in
particular, fractions and ratios
7 what’s are 28?
28
7
Meaning 28 divided by 7
What does 28 sevenths imply?
This suggests a count
of 7s, only practical for
small whole numbers
MENTAL STRATEGY FOR DIVISION:
• Think of multiplication
Does 7
represent the
number in each
group or
Eg, 56 divided by 7?
THINK: 7 what’s are 56?
the number of
… 7 sevens are 49, 7 eights are 56
groups?
So, 56 divided by 7 is 8
Work with fact families:
What do you know if you
know that 6 fours are 24?
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
FRACTIONS AND DECIMALS:
2
Traditional practices (eg, shade to show 5
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.
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”.
•
“You’ve got more than me, that’s 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
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
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’
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]
Explore paper folding, what do you notice as
the number of parts increases?
Fold a sheet of newspaper in
half. Repeat until it can’t 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
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
Equal parts
As the number of
parts increases, the
size of the part
decreases
The number of
parts names
the part
3. Recording common fractions and decimal
fractions – problems with recording, the
The numerator tells
fraction symbol, decimal numeration (to
‘how many’, the
tenths)
denominator tells
4. Consolidating fraction knowledge –
comparing, ordering/sequencing, counting,
and renaming.
‘how much’
Links to
multiplication and
division
Partitioning:
Counting and colouring parts of someone else’s
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.
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
The ‘halving’ strategy
For example,
Explore paper folding with coloured paper
squares, paper streamers and newspaper.
Both shapes
are 1 half
How are they different?
How are they the same?
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?
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?
It’s bigger than 2 and a half ... Smaller than 3 .... It’s 11
quarters ... It’s 5 halves and 1 quarter ... It could be 2
and 3 quarter slices of bread ...
Extend partitioning to diagrams:
Ask: What did the first
fold do?
It cut the top and
bottom edges in half
Estimate 1 half
Ask: What did the
second fold do?
It cut the top and
bottom edges in half
again
Ask: What did the
third fold do? It cut the
side edges in half.
How would you describe this strategy
using paper streamers?
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
Use to draw diagrams, for example,
Apply thirding strategy to
top and bottom edge,
halving strategy to side
edges to get sixths
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
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
Notice:
Halving
family
Thirding
family
Fifthing
family
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
Halving
and
Thirding
Halving
and Fifthing
Thirding
and Fifthing
As the number of parts increases, the size of the parts
gets smaller – the number of parts, names the part
Explore strategy combinations to recognise
that:
Thirds by quarters
give twelfths
quarters
thirds
thirds
Thirds by fifths
give fifteenths
fifths
What other fractions can
be generated by halving
and thirding or by
fifthing and thirding?
tenths
What other
fractions can be
generated by
fifthing and
tenths
halving?
Tenths by tenths
give hundredths
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
34
This tells
‘how many’
ones
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
Recording common fractions:
Introduce recording once key ideas have been
established through practical activities and
partitioning:

equal shares - equal parts Explore non-examples

fraction names are related to the total number
of parts (denominator idea – the more parts
there are, the smaller they are)
This tells how much

the number of parts required tells how many
(numerator idea – the only counting number)
This tells how many
Introduce the fraction symbol:
2 fifths
2
5
2
out of
5
2
5
This number tells how many
This number names the parts and tells
how much
Make and name mixed common fractions
Recognise:
third
3rd
• 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,
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”
Introducing Decimals:
Recognise decimals as fractions – use halving
and fifthing partitioning strategies to make and
represent tenths
Halves by fifths are tenths
fifths
7 out of ten parts, 7 tenths
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?
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?
Extend decimal place-value:
Recognise hundredths as a new place-value part:
1. Introduce the new unit: 1 tenth is 10 hundredths
5.3 5.4
6.0
hundredths
5.0
ones
2. Show, name and record ones, tenths &
hundredths
tenths
via partitioning
53 7
5.30
5.37
5.40
3. Consolidate: compare, order, count forwards
and backwards, and rename
Establish links between tenths and
hundredths, and hundredths and per cent:
0.7 is 7 tenths or 7
10
Recognise per cent ‘benchmarks’:
50% is a half, 25% is a quarter,
10% is a tenth, …
33 % is 1 third …
0.75 is 7 tenths, 5 hundredths
75 hundredths
75 per cent, 75%, or
75
100
Consolidating decimal place-value:
1. Compare decimals – which is larger, which is
smaller, why? Which is longer, 4.5 metres or 4.34 metres?
Which is heavier, 0.75 kg or 0.8 kg?
2. Order decimal fractions on a number line, eg,
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
3. Count forwards and backwards in place-value
parts, eg,
… 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, …
…5.23, 5.43, 5.63, 5.83, …
4. Rename in as many different ways as possible, eg,
4.23 is 4 ones, 2 tenths, 3 hundredths
4 ones, 23 hundredths
42 tenths, 3 hundredths
423 hundredths …
Extending Fraction & Decimal Ideas:
By the end of primary school, students are
expected to be able to:
Requires:
partitioning
• rename, compare and
strategies, fraction
order fractions with unlike
as division idea
denominators
and ‘region’ idea
• recognise decimal
for multiplication
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
Renaming Common Fractions:
1
3
3 parts
4 parts
3
4
Use paper
folding &
student
generated
diagrams
to arrive at the
generalisation:
2
6
9
12
9 parts
12 parts
If the total number of parts increase by a
certain factor, the number of parts required
increase by the same factor
Comparing common fractions:
Which is larger 3 fifths or 2 thirds?
But how do you know? ... Partition
fifths
thirds
THINK: thirds by fifths ... fifteenths
Comparing common fractions:
Which is larger 3 fifths or 2 thirds?
3
9
=
5
15
2
10
=
3
15
THINK: thirds by fifths ... fifteenths
Extend decimal place-value:
Recognise hundredths as a new place-value part:
1. Introduce the new unit: 1 hundredth is 10
thousandths
via partitioning
5.370
5.37 5.38
5.376
5.40
5.380
thousandths
6.0
tenths
5.30
5.3 5.4
ones
5.0
hundredths
2. Show, name and record ones, tenths, hundredths
and thousandths
53 7 6
3. Consolidate: compare, order, count forwards
and backwards, and rename
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)
Compare, order and rename decimal fractions:
a) Is 4.57 km longer/shorter than 4.075 km?
b) Order the the long-jump distances: 2.45m,
1.78m, 2.08m, 1.75m, 3.02m, 1.96m and 2.8m
c) 3780 grams, how many kilograms?
d) Express 7¾ % as a decimal
2
ones
9
tenths
0
hundredths
7
thousandths
1
Use Number Expanders to rename decimals
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
For example,
(Gillian Large, Year 5/6, 2002)
(Gillian Large, Year 5/6, 2002)
Games:
For example,
• Make a Whole
• Target Practice
• Fraction Concentration
(Make a Whole Game Board, Vicki Nally, 2002)
Make a Whole:
(Vicki Nally, 2002)
(Vicki Nally, 2002)
Make a Model, eg, a Think Board
(Gillian Large, 2002)