tasmanian department of education support for this project has been provided by the australian...
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TASMANIANDepartment of Education
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.
The key ideas and strategies that underpin Multiplicative Thinking
Presented by Dianne Siemon
• 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)
KEY IDEAS AND STRATEGIES:
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-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”
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 106 and 2 more4 and 4double 43 and 3 and 25 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.
* See WA Department of Education, First Steps in Mathematics
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.
Trusting the count is evident when children:
• 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)
• 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);
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: 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
For example:
Cover 5, count on
Cover 4, count on
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
For example:
Ten-frames
For example:
Bead Frame (to 20)
Double-decker bus scenario
Count: 6 and 6 is 12, and 1 more, 13
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)
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:
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
Rotate
and rename
3. Regions:
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.
Rotate
and rename
Note: For whole number multiplication continuous models are introduced after discrete – this is different for
fraction models!
4. ‘Area’ idea:
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
3 by 1 ten and 4 ones
3
14
3 by 1 ten ... 3 tens 3 by 4 ones ... 12 ones
Think: 30 ... 42
The ‘Area’ idea (extended):
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
24
33
5. Cartesian Product:
Supports ‘for each’ idea and multiplication by 1 or more factors
Eg, lunch options
3 different types of bread
4 different types of
filling2 different types of
fruit
3 x 4 x 2 = 24 different options
6. Rate:
Rate builds on the ‘for each’ idea and underpins proportional reasoning
Eg, Samantha’s snail travels 15 cm in 3 minutes. Anna’s snail travels 37 cm in 8 minutes. Which is the speedier snail?
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
This problem involves rate but actually asks for a comparison of ratios which requires 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:
1 x 3 = 32 x 3 = 63 x 3 = 94 x 3 = 125 x 3 = 156 x 3 = 187 x 3 = 218 x 3 = 249 x 3 = 2710 x 3 = 3011 x 3 = 3312 x 3 = 36
1 x 4 = 42 x 4 = 83 x 4 = 124 x 4 = 165 x 4 = 206 x 4 = 247 x 4 = 288 x 4 = 329 x 4 = 3610 x 4 = 4011 x 4 = 4412 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
3
6
6
3Eg, 6 threes? ...
THINK: 3 sixes ...
double 6, 12, and 1 more 6 ... 18
And the commutative principle:
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”.
This involves a shift in focus: 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)
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
An alternative ‘multiplication table’:
This actually represents the region idea and supports efficient, mental strategies (read across the row), eg,
6 ones, 6 twos, 6 threes, 6 fours, 6 fives, 6 sixes,
6 sevens, 6 eights, 6 nines
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: ….
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 ...
2 fours ... THINK:
double 4 ... 8
2 sevens ... THINK:
double 7 ... 14
7 twos ... THINK:
double 7 ... 14
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 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
3 twenty-threes THINK?
9 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
Doubles doubles strategy (fours):
4 ones, 4 twos, 4 threes, 4 fours, 4 fives ...
4 sixes THINK:
double 4 ... 8double
again, 16
4 forty-sevens THINK?
8 fours ... 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
‘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
Cannot show zero facts on
table ... 0 of anything
is 0 ... 7 zeros, same as 0
sevens
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
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
9 eights THINK: less
than 10 eights, 1 eight less, 72
8 fives ... 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
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
8 sevens THINK: 7
sevens is 49, and 1 more 7,
56
CONCEPTS FOR DIVISION:
1. How many groups in (quotition):
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
How many fours in 12?
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.
2. Sharing (partition):
18 counters
3 in each group
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
18 sweets shared among 6.
How many each?
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?
Total and number of groups known – Question relates to number in each group.
THINK: 7 what’s are 42?
THINK: 9 what’s are 36?
28 ÷ 7 = 47 groups or parts
groups of 7
Q: 7 shares, how many in each share?
PARTITIONPARTITION
Q: How many 7s in 28?
QUOTITIONQUOTITION
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 what’s are 28?
287
Meaning 28 divided by 7
What does 28 sevenths imply?
MENTAL STRATEGY FOR DIVISION:
• Think of multiplication
Work with fact families:
What do you know if you know that 6 fours are 24?
Eg, 56 divided by 7?
THINK: 7 what’s 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
Does 7 represent the number in each group or
the number of groups?
FRACTIONS AND DECIMALS:
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.
25
Introducing Fractions:
• “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
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”.
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:
Note: language only, no symbols
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
Use real-world examples AND non-examples to ensure students understand that EQUAL parts are required.
Share jelly-beans or smarties equally and unequally – discuss ‘fair shares’
Cut plasticene ‘rolls’ and ‘pies’ into equal and unequal parts – 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?
Halve paper strips of different lengths, compare halves – how are they the same? How are they different?
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
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
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
Partitioning:
• 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.
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:
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
rope and pegs
SmartiesKindergarten Squares
paper streamers
For example,
The ‘halving’ strategy
Explore paper folding with coloured paper squares, paper streamers and newspaper.
How are they different? How are they the same?
Both shapes are 1 half
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
Write down as many things as you can about your fraction. How many different ways can you find to name
your fraction?
2 and 3 quarters
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 ... estimateHalve the
remaining part
Fold kindergarten squares or paper streamers into 3 equal parts
Apply thirding strategy to top and bottom edge,
halving strategy to side edges to get sixths
Use to draw diagrams, for example,
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
Apply fifthing strategy to top and
bottom edge, halving strategy to side
edges to get tenths
Use to draw diagrams, for example,
4 5
Apply to number line
Notice: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
As the number of parts increases, the size of the parts gets smaller – the number of parts, names the part
Halving family
Thirding family
Fifthing family
Halving and Thirding
Halving and Fifthing
Thirding and Fifthing
Explore strategy combinations to recognise that:
Tenths by tenths give hundredths
Thirds by quarters give twelfths
tenths
tenths
thirds
quarters
What other fractions can be generated by fifthing and halving?
fifths
thirdsThirds by fifths give
fifteenths
What other fractions can be generated by halving and thirding or by fifthing and thirding?
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’
34This tells
‘how many’ tens
This tells ‘how many’
ones
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?
Informally describe and compare:
Construct fraction diagrams to compare more formally
Recording common fractions:
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)
Introduce recording once key ideas have been established through practical activities and partitioning:
This tells how much
This tells how many
Explore non-examples
2 fifths25
2out of
525
This number tells how many
This number names the parts and tells how much
Introduce the fraction symbol:
Make and name mixed common fractions
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,
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”
third 3rd
Introducing Decimals:
Recognise decimals as fractions – use halving and fifthing partitioning strategies to make and represent tenths
Name decimals in terms of their place-value parts, eg, “two and four tenths” NOT “two point four”
fifths
halves
Halves by fifths are tenths
7 out of ten parts, 7 tenths
2 2.4 3
Fifth then halve each part or halve then fifth each part, 2 and 4 tenths
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
3. Consolidate: compare, order, count forwards and backwards in ones and tenths, and rename
one and 3 tenthsones tenths
1 3
The decimal point shows where ones begin
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
2.Show, name and record ones, tenths & hundredths
3. Consolidate: compare, order, count forwards and backwards, and rename
5 3 7
via partitioning
ones
tent
hs
hund
redt
hs
5.0 5.3 5.4 6.0
5.30 5.37 5.40
Establish links between tenths and hundredths, and hundredths and per cent:
0.75 is 7 tenths, 5 hundredths75 hundredths75 per cent, 75%, or
0.7 is 7 tenths or
710
75100
Recognise per cent ‘benchmarks’: 50% is a half, 25% is a quarter, 10% is a tenth, …33 % is 1 third …
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,
4.23 is 4 ones, 2 tenths, 3 hundredths4 ones, 23 hundredths42 tenths, 3 hundredths423 hundredths …
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
Which is longer, 4.5 metres or 4.34 metres?Which is heavier, 0.75 kg or 0.8 kg?
… 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, …
…5.23, 5.43, 5.63, 5.83, …
Extending Fraction & Decimal Ideas:
By the end of primary school, students are expected to be able to:
• rename, compare and order fractions with unlike denominators
• recognise decimal fractions to thousandths
Requires: partitioning
strategies, fraction as division idea and ‘region’ idea for multiplication
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
Use paper folding & student
generated diagrams
to arrive at the generalisation:
If the total number of parts increase by a certain factor, the number of parts required
increase by the same factor
2 6
3 4
9 12
4 parts 12 parts
3 parts 9 parts
THINK: thirds by fifths ... fifteenths
Comparing common fractions:
Which is larger 3 fifths or 2 thirds?
But how do you know? ... Partition
thirds
fifths
THINK: thirds by fifths ... fifteenths
Comparing common fractions:
Which is larger 3 fifths or 2 thirds?
35
915=
23
=1015
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
3. Consolidate: compare, order, count forwards and backwards, and rename
5 3 7 6
via partitioning
ones
tent
hs
hund
redt
hs 5.0 5.3 5.4 6.0
5.30 5.37 5.38 5.40 thou
sand
ths
5.370 5.376 5.380
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
ones tenths hundredths thousandths
2 9 0 7 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)
(Gillian Large, 2002)
Make a Model, eg, a Think Board