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1.07 Number operation properties

Lesson

Introduction

One of the hardest things to do in mathematics is to not use your calculator. It's just so much easier than working things out in your head. But what if you don't have a calculator or simply aren't allowed one? Is it even possible to solve something like 24\times13+78\times13 in your head? Yes. By using mental arithmetic strategies we can make questions easier by changing the way we approach them.

The image shows a numerical expression and answer using mental arithmetic strategies. Ask your teacher for more information.

But we don't have the tools to solve something like this yet. Let's start with some simple mental strategies to build up our tool kit.

The associative law

Try calculating 29+38+12 in your head. Which two numbers did you add together first? If you added the 38 and 12 together first, you were using the associative law. The associative law lets us evaluate the operations in any order, so long as the operations are all addition or all multiplication. Mathematically, this looks like we are adding or moving brackets to change which pair of numbers we add or multiply first.

An image showing the representation of associative law for addition. Ask your teacher for more information

The associative law for addition

This law doesn't apply to subtraction or division.

For example: 22-7-5 \neq 22-(7-5) and 24 \div 6 \div 2 \neq 24 \div (6 \div 2).

Examples

Example 1

Consider 27+48+13.

a

Which pair of numbers will be easiest to add together first?

A
48+13
B
27+13
C
27+48
Worked Solution
Create a strategy

Choose the pair of numbers where the units digits will sum up to 10 or less.

Apply the idea

Option A

8+3=11

Option B

7+3=10

Option C

7+8=15

Answer is option B: 27+13.

b

Find the missing value of the expression to complete the working out.

\displaystyle 27+48+13\displaystyle =\displaystyle 27+⬚+⬚Apply the associative law
\displaystyle =\displaystyle ⬚+⬚Evaluate the first addition
\displaystyle =\displaystyle ⬚
Worked Solution
Create a strategy

Use the associative property.

Apply the idea
\displaystyle 27+48+13\displaystyle =\displaystyle 27+13+48Apply the associative law
\displaystyle =\displaystyle 40+48Evaluate the first addition
\displaystyle =\displaystyle 88Evaluate
Idea summary

The associative law evaluates the operations in any order, if the operations are both addition or multiplication.

For example: 4+5+6=4+6+5.

The commutative law

Which of the following are true?

a) \,4+3=3+4 \quad \quadb) \,4-3=3-4 \quad \quadc) \,4\times 3=3\times 4 \quad \quadd) \,4\div 3=3\div 4

If you answered a) and c), you're correct. These two show applications of the commutative law which lets us swap the numbers on either side of the operation. Notice that since only a) and c) are true, the commutative law can be used for addition and multiplication, but not subtraction and division.

The image showing the representation of commutative law for multiplication. Ask your teacher for more information.

The commutative law for multiplication

Exploration

The commutative law for multiplication is demonstrated in the following applet.

Loading interactive...

The applet shows that the commutative law of multiplication holds, because even if we swap the numbers on either side of the multiplication the number of squares stays the same.

Examples

Example 2

Consider 4\times13\times5.

a

Which pair of numbers will be easiest to multiply together first?

A
4\times5
B
13\times5
C
4\times13
Worked Solution
Create a strategy

Choose the smaller pair of numbers and has a product that is multiple of 10.

Apply the idea

The smallest pair of numbers are 4 and 5, and its product is a multiple of 10.

Answer is option A: 4\times5.

b

Complete the working out below.

\displaystyle 4\times13\times5\displaystyle =\displaystyle 4\times⬚\times⬚Apply the commutative law
\displaystyle =\displaystyle ⬚\times⬚Evaluate the first multiplication
\displaystyle =\displaystyle ⬚
Worked Solution
Create a strategy

To apply the commutative law, rearrange the expression and write the pair of numbers that are easiest to multiply first. In this case, we want to use the pair found in part (a).

Apply the idea
\displaystyle 4\times13\times5\displaystyle =\displaystyle 4\times5\times13Apply the commutative law
\displaystyle =\displaystyle 20\times13Evaluate the first multiplication
\displaystyle =\displaystyle 260Evaluate
Idea summary

The commutative law swaps the numbers on either side of the operation, if the operations are both addition or multiplication.

For example: 7\times8=8\times7.

Reordering

This mental strategy is a similar to the commutative law but can be used when we have more than one of the same operation.

An image showing reordering of addition on a number line. Ask your teacher for more information.

When calculating 13+18+7 which number would it be easier to add first, 18 or 7? Unless you are a calculator it is usually easier to add 7 from 13 first and then add 18 to their sum. This is an example of using reordering.

Reordering lets us rearrange the numbers in the expression to make the calculations easier when solving from left to right. The rules for using reordering are that all the operations must be same and the first number cannot be moved (notice that 13 stays as the first number in the expression), unless the operation is also commutative (i.e. if it is either addition or multiplication).

Examples

Example 3

Consider 52-24-12.

a

Which of the following expressions is equal to 52-24-12?

A
(52-24)\times12
B
52-12-24
C
52-(24-12)
D
52\times24-52\times12
Worked Solution
Create a strategy

Evaluate each option.

Apply the idea

Original expression:

\displaystyle 52-24-12\displaystyle =\displaystyle 28-12Operate from left to right
\displaystyle =\displaystyle 16Evaluate

Option A:

\displaystyle (52-24)\times12\displaystyle =\displaystyle 28\times12Evaluate the parentheses
\displaystyle =\displaystyle 336Evaluate

Option B:

\displaystyle 52-12-24\displaystyle =\displaystyle 40-24Operate from left to right
\displaystyle =\displaystyle 16Evaluate

Option C:

\displaystyle 52-(24-12)\displaystyle =\displaystyle 52-12Evaluate the parentheses
\displaystyle =\displaystyle 40Evaluate

Option D:

\displaystyle 52\times24-52\times12\displaystyle =\displaystyle 1248-624Evaluate the multiplication
\displaystyle =\displaystyle 624Evaluate

Answer is option B: 52-12-24.

b

Which of the arithmetic rules can we apply to 52-24-12 to transform it into 52-12-24?

A
Associative Law
B
Reordering
C
Commutative Law
Worked Solution
Create a strategy

Recall what each of the rules does:

  • The associative law moves or introduces brackets.

  • The commutative law swaps the numbers on either side of an operation.

  • Reordering rearranges the order of the numbers in an expression.

Apply the idea

Notice that the order of the numbers changed, but there was not any distributing of parentheses.

The answer is option B: Reordering.

Idea summary

Reordering rearranges the order of the numbers in an expression.

For example: 13+18+7=13+7+18

The distributive law

The distributive law applies whenever there is multiplication outside some brackets and addition inside the brackets, such as the expression 3 \times (5+2). We apply the multiplication to each term inside the brackets individually, and add the results together:

This image shows the calculation of 3 times left parenthesis 5 plus 2 right parenthesis using dots. Ask your teacher for more information.

The distributive law for multiplication

This law will be very useful when we study algebra, but for now it is still a clever strategy for mental arithmetic.

Is there an easy way to solve 102 \times 13 There is. The trick is find an "easy to multiply by" number close to 102. In this case we can use 100. So instead of trying to calculate 102 \times 13 we can instead calculate (100+2) \times 13. We can then use the distributive law to expand the brackets to get (100 \times 13) + (2 \times 13), which is much easier to multiply and add.

The distributive law can be used whenever there are brackets with addition (or subtraction) inside, and multiplication on the outside (on either side). It can also be used when the brackets are being divided from the right side.

For example:

\begin{aligned} (24+6) \times 2 &= (24 \times 2)+ (6\times2) \\ 2 \times (24+6) &= (2\times 24) + (2 \times 6) \\ (24+6) \div 2 &= (24 \div 2) + (6 \div 2) \end{aligned}

We cannot apply the distributive law to division from the left side.

For example: 24 \div (6+2) \neq (24 \div 6) + (24 \div 2)

We can show the distributive law visually as well through area. We can either break up the total area into two simpler areas or subtract some excess area from an approximate total.

This image shows the multiplication of 102 times 13 and 99 times 13 using areas of rectangles. Ask your teacher for more information.

We can use a similar trick to make division questions like 168 \div 14 easier. What is a number close to 168 that is "easy to divide by 14"? One way we can break up 168 is into 140 and 28, splitting a difficult to divide number into two easy to divide numbers.

This image shows the division of 168 by 14 using blocks. Ask your teacher for more information.

Examples

Example 4

Consider 168\div7.

a

Complete the statement:

168\div7 is the same as (140+⬚)\div7

Worked Solution
Create a strategy

Find the number to be added to 140 that will give a sum of 168.

Apply the idea

140+28=168

168\div7 is the same as \left(140+28\right)\div7.

b

Complete the statement:

(140+28)\div7 is the same as 140\div 7 + ⬚\div 7

Worked Solution
Create a strategy

Consider the first expression. Find the other number that is being divided by 7.

Apply the idea

In \left(140+28\right)\div7 both 140 and 28 will be divided by 7.

\left(140+28\right)\div7 is the same as 140\div7+28\div7.

c

Which arithmetic rule explains the equality between 168\div7 and 140\div 7+28\div7?

Worked Solution
Apply the idea

From the previous parts we have seen:

\displaystyle 168\div7\displaystyle =\displaystyle (140+28)\div7168 was seperated into two smaller numbers
\displaystyle =\displaystyle 140\div7+28\div7The parentheses was distributed

Distributive Property is the arithmetic rule that explains the equality.

d

Find the missing value of the expression to complete the working out.

\displaystyle 168\div7\displaystyle =\displaystyle (140+28)\div7Separate into two manageable components
\displaystyle =\displaystyle 140\div7+28\div7Distribute the parentheses using the distributive property
\displaystyle =\displaystyle ⬚+⬚Evaluate the division
\displaystyle =\displaystyle ⬚Evaluate
Worked Solution
Create a strategy

Perform the divisions then add the results together.

Apply the idea
\displaystyle 168\div7\displaystyle =\displaystyle (140+28)\div7Separate into two manageable components
\displaystyle =\displaystyle 140\div7+28\div7Distribute the parentheses using the distributive property
\displaystyle =\displaystyle 20+4Evaluate the division
\displaystyle =\displaystyle 24Evaluate
Idea summary

The distributive law applies the operation outside the parentheses to the expression inside it in order to distribute the parentheses.

For example: 4\times(5+3)=(4\times5)+(4\times3).

Outcomes

MA4-4NA

compares, orders and calculates with integers, applying a range of strategies to aid computation

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