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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

- 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)

“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’.

“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”

Close your eyes. What did you see?

What difference does this make?

What did you notice?

Would colour help? How? Why?

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

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-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)

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,

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

Ten-frames

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 8 and 6 …

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)

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

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).

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

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!

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

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

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

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, 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:

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 ...

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

6

Eg, 6 threes? ... THINK:

3 sixes ...

double 6, 12, and 1 more 6 ... 18

6

3

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

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,

6 ones,

6 twos,

6 threes, 6 fours,

6 fives,

6 sixes,

6 sevens, 6 eights, 6 nines

The region model implicit in the alternative table also supports the commutative idea:

Eg, 6 threes?

THINK:

….

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

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

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

9 threes ... THINK?

3 twenty-threes THINK?

Doubles doubles strategy (fours):

4 ones, 4 twos, 4 threes, 4 fours, 4 fives ...

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, ...

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 THINK: half of 10 sevens, 35

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 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

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

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.

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

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 what’s are 42?

THINK: 9 what’s are 36?

Total and number of groups known – Question relates to number in each group.

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 what’s are 28?

28

7

Meaning 28 divided by 7

What does 28 sevenths imply?

Does 7 represent the number ineach group or

the number ofgroups?

- Think of multiplication

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

Work with fact families:

What do you know if you know that 6 fours are 24?

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.

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

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

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:

- Prior knowledge and experience - informal experiences, fraction language, key ideas
- Partitioning – the missing link in building fraction knowledge and confidence, strategies for making, naming and representing fractions
- Recording common fractions and decimal fractions – problems with recording, the fraction symbol, decimal numeration (to tenths)
- 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

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

For example,

Explore paper folding with coloured paper squares, paper streamers and newspaper.

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?

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?

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

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

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

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

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

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 requiredtells how many (numerator idea – the only counting number)

Explore non-examples

This tells how much

This tells how many

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

- different meanings for ordinal number names, eg, ‘third’ can meanthird in line, the 3rd of Aprilor 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”

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?

Recognise tenths as a new place-value part:

- Introduce the new unit: 1 one is 10 tenths
- 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?

Recognise hundredths as a new place-value part:

- Introduce the new unit: 1 tenth is 10 hundredths
- 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

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

Consolidating decimal place-value:

- Compare decimals – which is larger, which is smaller, why?
- Order decimal fractions on a number line, eg,
- Count forwards and backwards in place-value parts, eg,
- 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 …

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

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

Which is larger 3 fifths or 2 thirds?

But how do you know? ... Partition

fifths

thirds

THINK: thirds by fifths ... fifteenths

Which is larger 3 fifths or 2 thirds?

3

5

9

15

=

2

3

10

15

=

THINK: thirds by fifths ... fifteenths

Recognise hundredths as a new place-value part:

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

via partitioning

5.0 5.3 5.4 6.0

thousandths

hundredths

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

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:

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

ones tenths hundredths thousandths

2 9 0 7 1

Use Number Expanders to rename decimals

Consolidating fraction knowledge:

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

Which is bigger? Why?

2/3 or 6 tenths ... 11/2 or 18/16

(Gillian Large, Year 5/6, 2002)

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

(Make a Whole Game Board, Vicki Nally, 2002)

(Vicki Nally, 2002)

Make a Model, eg, a Think Board

(Gillian Large, 2002)

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