B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Properties of Operations with Rational Numbers

B2.1 A1About 45 minutes

Learning goals

I can use the commutative, associative and distributive properties to calculate with fractions and decimals in my head.
I can use the identity and zero properties.
I can use the relationships between operations, like "dividing by 0.5 is the same as multiplying by 2", to make a calculation easier and to check my work.
I can tell which properties work for subtraction and division and which do not.

Words to know

rational number
Any number that can be written as a fraction of two integers, like 3/4, −2, 0.6 or 2 1/2. You study them closely in the Number Sense unit.

commutative property
You can switch the order when you add or multiply: a + b = b + a and a × b = b × a.

associative property
You can change the grouping when you add or multiply: (a + b) + c = a + (b + c).

distributive property
Multiplying a sum (or difference) is the same as multiplying each part and then adding (or subtracting): a × (b + c) = a × b + a × c.

identity property
Adding 0 or multiplying by 1 leaves a number unchanged.

inverse operations
Operations that undo each other: addition and subtraction; multiplication and division.

reciprocal
The number you multiply by to get 1. The reciprocal of 2/3 is 3/2, and the reciprocal of 0.5 (which is 1/2) is 2.

Remember from Grade 7

In Grade 7 you used properties to do calculations like 25 × 48 in your head, you used the order of operations with exponents, you multiplied and divided fractions and decimals, and you added and subtracted integers. In Grade 8 you will use the same properties with all kinds of rational numbers: fractions, decimals, mixed numbers and (later in this unit) negative numbers. Properties are the tools that let you calculate quickly and explain why your shortcut works.

Let's start at the arena canteen

Your team buys 4 hot chocolates at $2.75 and 4 granola bars at $1.25.

One way: 4 × 2.75 = 11 and 4 × 1.25 = 5, so 11 + 5 = $16.

A faster way: each friend gets one of each, and 2.75 + 1.25 = 4. So the total is 4 × (2.75 + 1.25) = 4 × 4 = $16.

Both ways give the same answer because of the distributive property. Knowing the properties lets you choose the easiest path.

Each friend's hot chocolate and granola bar together cost $4, so 4 groups of $4 make $16. That is the distributive property: 4 × (2.75 + 1.25) = 4 × 2.75 + 4 × 1.25.

Property	What it says	Example with rational numbers
Commutative	a + b = b + a; a × b = b × a	0.25 × 7.3 × 4 = 0.25 × 4 × 7.3 = 1 × 7.3 = 7.3
Associative	(a + b) + c = a + (b + c); (a × b) × c = a × (b × c)	2/3 + (1/3 + 4/5) = (2/3 + 1/3) + 4/5 = 1 4/5
Distributive	a × (b + c) = a × b + a × c; a × (b − c) = a × b − a × c	8 × 2.99 = 8 × 3 − 8 × 0.01 = 24 − 0.08 = 23.92
Identity	a + 0 = a; a × 1 = a	3/4 × 5/5 = 15/20 (5/5 is just a form of 1)
Zero	a × 0 = 0	4.87 × 0 = 0

The five properties you will use most. Letters a, b and c stand for any rational numbers.

The distributive property as an area model (not to scale). A 6 by 2.5 rectangle splits into a 6 by 2 part (6 × 2 = 12) and a 6 by 0.5 part (6 × 0.5 = 3). Together: 12 + 3 = 15.

Relationships between operations

Operations are connected, and you can swap one for another:

Subtraction undoes addition and division undoes multiplication. Use this to check: if 15.6 ÷ 1.2 = 13, then 13 × 1.2 should be 15.6.
Dividing by a number is the same as multiplying by its reciprocal. 12 ÷ 0.5 = 12 × 2 = 24, and 9 ÷ 1/4 = 9 × 4 = 36.
Multiplying by a unit fraction is the same as dividing. × 0.5 means ÷ 2, × 0.25 means ÷ 4, and finding 20% means ÷ 5.
Subtracting is the same as adding the opposite. 7 − 2.5 = 7 + (−2.5). You will use this a lot with integers.

Watch out!

Subtraction and division are not commutative or associative.

10 − 4 = 6, but 4 − 10 = −6.
(24 ÷ 4) ÷ 2 = 6 ÷ 2 = 3, but 24 ÷ (4 ÷ 2) = 24 ÷ 2 = 12.

If you want to rearrange a subtraction, first rewrite it as adding the opposite.

Worked example

Example 1: Reorder to make a friendly product

Find 0.5 × 37 × 4 in your head.

Use the commutative property to move the friendly numbers together: 0.5 × 4 × 37.
0.5 × 4 = 2.
2 × 37 = 74.

Worked example

Example 2: Use the distributive property

Find 7 × 4.98 in your head.

4.98 is close to 5. Write it as 5 − 0.02.
7 × (5 − 0.02) = 7 × 5 − 7 × 0.02.
35 − 0.14 = 34.86.

34.86

Worked example

Example 3: Group to make whole numbers

Find 3/5 + 2.7 + 2/5 + 0.3.

Use the commutative and associative properties to group numbers that make wholes: (3/5 + 2/5) + (2.7 + 0.3).
3/5 + 2/5 = 5/5 = 1, and 2.7 + 0.3 = 3.
1 + 3 = 4.

Worked example

Example 4: Check with the inverse operation

Lena found that 15.6 ÷ 1.2 = 13. Check her answer.

Division is undone by multiplication, so check 13 × 1.2.
Use the distributive property: 13 × 1.2 = 13 × 1 + 13 × 0.2 = 13 + 2.6 = 15.6. ✓

Her answer is correct.

Think like a mathematician

Before you calculate, look for friendly pairs: numbers that make 1, 10 or 100 (like 0.25 and 4, or 2.7 and 0.3), or numbers that are close to a whole number (like 4.98). Then say which property you used. Explaining your strategy out loud is part of the Grade 8 math processes (A1).

Hands-on activity

Receipt mental math

You need: a grocery receipt or a store flyer, paper, a pencil, a calculator for checking.

Pick 3 or 4 items from the receipt or flyer.
Find the total in your head using a property. Look for prices that add to whole dollars, or prices just under a whole dollar.
Pick one item and imagine buying 3, 6 or 8 of it. Use the distributive property to find the cost.
Write down which property you used each time, then check with the calculator.

Model answer: 3 items at $4.99 → 3 × 5 − 3 × 0.01 = 15 − 0.03 = $14.97 (distributive property). Prices $2.35 and $1.65 → 2.35 + 1.65 = $4.00 (grouping friendly numbers).

Math talk: "Which item was easiest to multiply in your head? Why?"

Just for fun

Some things in real life are not commutative. Socks then shoes works great. Shoes then socks? Not so much.

Your turn

Find 25 × 3.6 × 4 in your head.

Find 9 × 6.99.

Find 1/4 + 5/8 + 3/4.

Find 12 × 3 1/4 using the distributive property.

Which property is shown? 4.2 × 1 = 4.2

Find 48 ÷ 0.5.

Is (36 ÷ 6) ÷ 2 equal to 36 ÷ (6 ÷ 2)?

More practice: warm-up

Find 0.2 × 17 × 5.

Find 2.6 + 3.8 + 1.4.

Find 5 × 102.

Find 3/7 × 0.

Find 8 × 0.25.

More practice: on your own

Find 15 × 2.4 using the distributive property.

Find 6 × 4.75 + 6 × 5.25.

Find 2/3 × 7 × 3/2.

Find 75 × 0.04.

A school orders 8 hoodies at $39.95 each. What is the total before tax?

Challenge

Find 4.5 × 18 − 4.5 × 8 in your head.

Maya says 1 ÷ 0.2 × 5 = 1 because 0.2 × 5 = 1. Is she right? Explain.

Find 99 × 3.7 + 3.7.

Check your understanding

Quiz

Which property is shown by 2.5 × (4 + 0.8) = 2.5 × 4 + 2.5 × 0.8?

What is 0.25 × 44?

Which statement is not always true?

Dividing by 1/3 is the same as:

What is 6 × 9.99?

To find 3/8 + 4.6 + 5/8 quickly, which two numbers should you add first?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.1 asks students to use the properties and order of operations, and the relationships between operations, to solve problems with rational numbers. This lesson focuses on the properties and relationships; the next lesson focuses on the order of operations.

Common difficulties:

Rearranging subtraction or division. Remind your child that only addition and multiplication can be reordered freely.
Using the distributive property on only one part (writing 7 × 4.98 as 7 × 5 − 0.02). Both parts must be multiplied by 7.
Mixing up "multiply by 0.5" and "divide by 0.5". × 0.5 halves a number; ÷ 0.5 doubles it.

How to help: At the store, ask "What would 3 of those cost?" for prices like $4.99, and ask your child to explain the shortcut. Praise the explanation, not just the answer. This builds the A1 process skills of reasoning and communicating.