Mathematics Developmental Continuum P-10

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Mathematics Developmental Continuum P-10 – Structure

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More about properties of operations: 2.75

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Commutative, associative and distributive properties: Spin, shuffle and split

The commutative, associative and distributive properties describe how the four number operations – addition, subtraction, multiplication and division – act on their own and with each other.

These properties provide a basis for computation, whether done mentally or using pencil and paper algorithms. We use these properties when we:

  • recognise that we can add 7 +36 + 3 by combining the 7 and 3 first to give 10, which is easy to add to 36 to give 46
  • quickly calculate 4 × 206 by working out 4 x 200 and 4 x 6 (as 800 and 24 respectively) and adding these results to get 824
  • divide 14 by 4 by first dividing 12 by 4 to give 3, and then dividing 2 by 4 to give ½, so the answer is 3½

The basic properties or rules, from which other number properties are derived, have technical names (commutative, associative and distributive) which are used in professional documents. It is NOT necessary to teach these names to young children but it is very important to support them to develop an understanding about what they mean and how they can use them to help with computation.

The properties and their implications are discussed below under the following headings:

 

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Commutativity

The commutative property applies to addition and multiplication. Written symbolically the commutative properties say that no matter what numbers a and b are used:

  • a + b = b + a (commutative property for addition)
  • a × b = b × a (commutative property for multiplication)

This means that for addition and multiplication, it doesn‘t matter which of the two numbers you start off with, and which number you use as the addend or the multiplicand (i.e. the number that gets added on or multiplied).

The commutative property is very useful for calculation. For example, if you need to work out 3 + 16, it is much easier to think about this as 16 + 3. This is particularly important for children at the “count on” stage of understanding about addition: starting at 3 and counting on 16 is much harder than starting at 16 and counting on 3. In student language, they need to learn to ‘spin’ around 3 + 16 to the easier 16 + 3.

Commutativity

Similarly the commutative property of multiplication is useful. For example, for a young student, working out 10 × 8 (10 groups of eight) may be easier as its ‘spin around’ of 8 × 10 (8 groups of ten), thinking that 8 tens are more obviously 80.

How to show addition and multiplication are commutative

The validity of these rules can be demonstrated with materials. For addition, if you have piles of objects representing each of the two numbers involved, the total that you have is not affected by whether you consider adding the first pile to the second or the other way around.

How to show addition and multiplication are commutative

For multiplication, commutativity is not at all obvious. A study of Brazilian children with very little education who worked as street vendors, showed that they could easily work out 3 groups of 50 cents, but they could not work out 50 groups of 3 cents. Children who had been to school could usually do both, because they had learned that 3 × 50 and 50 × 3 have the same answer.

Young students will mostly be convinced that multiplication is commutative just by looking at examples e.g. 3 × 4 = 12 and also 4 × 3 = 12. However, commutativity of multiplication is best demonstrated once the array model is understood. It is then just a matter of helping children to appreciate that the array representing 3 x 5 with 3 rows and 5 columns, for example, is the same size as the array representing 5 x 3 with 5 rows and 3 columns. All that is different in the arrays is the orientation, not the number of elements.

How to show addition and multiplication are commutative

What about subtraction and division?

There is no commutative rule for subtraction or for division. This is because it really does matter which way around the numbers are in a subtraction or a division calculation. For example, 6 − 4 = 2 whereas 4 − 6 = −2. Similarly, if we work out 100 ÷ 4 we get 25, whereas 4 ÷ 100 is 0.04.

One of the common errors in the written subtraction algorithm is to assume commutativity: students often subtract the numbers the wrong way around instead of renaming. They might think that if they can‘t subtract the bottom number from the top then they can subtract the top number from the bottom instead.

 

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The distributive property

The distributive property tells how multiplication works with addition. It is very useful for mental computation; it is the basis of all formal multiplication algorithms; and it is used extensively in algebra for factorisation.

Written in symbols, the distributive property says that for three numbers a , b, and c,

a × (b + c) = a × b + a × c
An example is 10 × (3 + 2) = 10 × 3 + 10 × 2

We say that “multiplication distributes over addition”. In other words, this means that if you multiply a number by the sum of two other numbers you get the same answer as if you multiply the first number by each of the other two numbers separately and add the results. For example, if you do 5 × (6 + 4) then this is just 5 × 10 (because we have to calculate the brackets first), and so the final answer is 50. On the other hand, if we do 5 × 6 + 5 × 4 we get 30 +20 which is also 50.

The distributive property is frequently used for mental multiplication. For example, if we want to multiply 7 by 130, we can calculate this by doing 7 × 100 and 7 × 30, to get 700 and 210 respectively, and then add the results to get 910. (Note that when we do 7 × 30, most of us will work out 7 × 3 and then multiply by 10, which is actually an application of the associative property as well – see below.)

Two ways to show the distributive property

The distributive property can be justified by thinking of multiplication as repeated addition.

An example of the general reasoning involved, can be shown by:

5 × 6 + 5 × 4 = 5 × (6 + 4) .
5 × 6 is 5 groups of 6, which is 6 + 6 + 6 + 6 + 6
5 × 4 is 5 groups of 4, which is 4 + 4 + 4 + 4 + 4
Add these two together (by rearranging in a convenient way)
6 + 6 + 6 + 6 + 6 + 4 + 4 + 4 + 4 + 4 = 10 + 10 + 10 + 10 + 10 + 10,
which is 5 groups of 10,
which is 5 × (6 + 4).

Another example:

Let‘s work out 7 × 5, with the assumption that we already know 6 × 5.
7 groups of 5 = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 6 groups of 5 + 1 group of 5
7 × 5 = 6 × 5 + 1 × 5

Two ways to show the distributive property

Array model: Once the array model of multiplication is understood the distributive property can be demonstrated with materials as shown in the diagram below. The array has 5 rows and 10 columns, so there are 50 = 5 × 10 counters in it. However, we can split the 10 columns into two sections (e.g. 6 and 4 as shown below), and calculate how many counters in each section separately (5 × 6 and 5 × 4).

Two ways to show the distributive property

Multiplication is also distributive over subtraction

The distributive property also works if subtraction is involved instead of addition: a × (b − c) = a × b − a × c

This version of the property can also make mental computation easier. For example, if we want to work out the price of 8 pairs of socks which cost $3.99 each, then instead of trying to figure out 8 × $3.99, we can work out 8 × $4 (which is $32) and then subtract 8 × 1 cent (which is 8 cents) to get the answer of $31.92.

Distributive property in long multiplication and algebra

Long multiplication algorithms use the distributive property extensively. As an example, 42 × 23 would be set out as follows:

Distributive property in long multiplication and algebra

The 6 on the first line of calculation is from 3 ones × 2 ones; the 12 on the same line is the result of 3 one × 4 tens which is 12 tens or 1 hundred and 2 tens. So, the first line of calculation really arises from 3 × (4 tens + 2 ones) and is the 12 tens and 6 ones arising from calculation 3 × 4 tens and 3 × 2 ones. Similarly the second line of calculation arises from using the 2 tens in the 23 to calculate 2 tens × (4 tens + 2 ones) which it does by calculating 2 tens × 2 ones, which is 4 tens, shown as a 4 on the second line in the tens column, and 2 tens × 4 tens, which is 8 hundreds, shown as an 8 on the second line in the hundreds column. Everything is then added together to get the result of 966

In algebra, the distributive property is used regularly in factorisation. For example, when we simplify the expression 2m − 4mn as 2m (1 − 2n) we are applying the distributive property.

Distributive property and division

Note that because division of a number is just the same as multiplying by the reciprocal of that number, there are also important consequences of the distributive property for division. These include

(a + b) ÷ c = (a ÷ c) + (b ÷ c)
and
(a − b) ÷ c = (a ÷ c) − (b ÷ c),

which can also be written in fraction form as follows.

Distributive property and division

However, division does not actually distribute over addition, because if the division sign is before the sum we can‘t distribute it over the sum. Symbolically,

a ÷ (b + c) `" (a ÷ b) + (a ÷ c)

This can be shown by means of an example:

24 ÷ (8 + 4) = 24 ÷ 12 = 2, but 24 ÷ 8 + 24 ÷ 4 = 3 + 6 = 9

It doesn‘t work with subtraction in place of addition, either. For example,

24 ÷ (8 - 2) = 24 ÷ 6 = 4, but 24 ÷ 8 - 24 ÷ 2 = 3 −  12 = −9

 

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Associativity

The four number operations are called binary operations, which means that they take two numbers as inputs and produce an answer as output. So, for example, to do subtraction you need two numbers to produce the answer. The thing to note about binary operations is that we never combine three numbers all at once; we only combine two numbers at a time. Associativity tells us how to add or multiply 3 or more numbers in one calculation: in fact it says that they can be grouped in any convenient way.

It is hard to explain exactly why the associative property is important, because it looks obvious. Written symbolically the associative property of addition says that if we have any three numbers a, b and c then

(a + b) + c = a + (b + c)

and the associative property of multiplication says that

(a × b) × c = a × (b × c).

For addition, this means that to add 3 numbers, you can either add the first pair of numbers and then add the third, or add the second pair of numbers and then add the result to the first. For example, to find 37 + 25 + 35, you can add the first two and then work out 62 + 35, or you can add the second two and then work out 37 + 60. The answer of 97 will be the same either way.

The same property applies for multiplication: for example, we can work out 2 × 3 × ð4 by working out 6 × 4 or by working out 2 × 12. Either way we get 24.

Using associativity for calculation: shuffling the numbers around

The associative property, too, is very useful for calculation. Taken together with the commutative property, the associative property means that for calculations that involve only addition or only multiplication, we can shuffle the numbers in any order we want and add or multiply whichever pairs we like, in such a way to make the calculations simpler to do in our head.

For example, if you need to work out 37 + 25 + 35, it is much easier to add the 25 and 35 first, rather than adding 37 and 25, but any shuffling will give the answer of 97.

Using associativity for calculation: shuffling the numbers around

Similarly it may be much easier to work out 10 × 8 × 8 by working out 8 x 8 first and leaving the easy multiplication by 10 until last, rather than working out 10 × 8, and then having to work out 80 × 8.

All that shuffling and combining of pairs is commutativity and associativity in action.

How to demonstrate associativity

The validity of associativity of addition and multiplication can also be demonstrated with materials, although many students will not see the point of this – they will think this rule is really obvious.

Addition: If you have piles of objects representing each of the three numbers involved, the total that you have is not affected by whether you add the first pile to the second and then add in the third, or whether you add the second to the third, and then add on the first.

Multiplication: For multiplication, associativity is harder to demonstrate, because with three numbers we no longer have the 2-dimensional array used to demonstrate commutativity, but a 3-dimensional array (a rectangular prism), with dimensions given by each of the three numbers. It doesn‘t matter what order of numbers you use to describe the prism, it will still be made up of the same number of cubes.

In the diagram below, both prisms have the same number of cubes, even though one might be thought of as (3 × 5) × 2, (i.e. with a base that is 3 × 5 and a height that is 2), and the other might be thought of as 3 × (5 × 2), which is a prism with base 5 × 2 and a height of 3.

Subtraction and division are not associative

Subtraction and division are not associative

As with commutativity, there is no associative rule for subtraction or for division. This is because it really does matter how the pairs of numbers are grouped for a subtraction calculation or a division calculation. For example, if we want to work out 9 − 3 − 2 should I do the first or second subtraction first?

If we do the first subtraction first we get (9 − 3) − 2 = 6 − 2 = 4. On the other hand, if we do the second subtraction first we get 9 − (3 − 2) = 9 − 1 = 8, which is different.

The two answers are not the same – how we group subtraction really matters, and the brackets are really important.

The same problem holds with division. For example, (100 ÷ 50) ÷ 2 = 2 ÷ 2 = 1, whereas 100 ÷ (50 ÷ 2) = 100 ÷ 25 = 4.

The end result is that brackets are really important for subtraction and division.

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References

Schliemann , A., Araujo, C., Cassunde, M., Macedo, S. & Niceas, L. (1998) Use of multiplicative commutativity by school children and street sellers. Journal for research in mathematics education. 29 (4) 422 – 435.