B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Properties and the Order of Operations

B2.1 A1About 40 minutes

Learning goals

I can use the order of operations to work out expressions with more than one operation.
I can use brackets to show which part to do first.
I can use properties (any order, any grouping, splitting) to calculate with whole numbers, decimals and fractions.
I can check a calculation with the inverse operation.

Words to know

expression
A group of numbers and operation signs, like 6 + 4 × 5. It has a value, but no equal sign.

order of operations
The agreed order for doing operations: brackets first, then multiply and divide from left to right, then add and subtract from left to right.

brackets
The symbols ( ) that show which part of an expression to do first.

commutative property
The any order rule for adding and multiplying: 2.5 × 8 = 8 × 2.5.

associative property
The grouping rule for adding and multiplying: you can choose which two numbers to do first.

distributive property
The splitting rule: 6 × 4.5 = 6 × 4 + 6 × 0.5.

inverse operations
Operations that undo each other: adding and subtracting, multiplying and dividing.

Remember from Grade 5

In Grade 5 you used the properties of operations: you can add or multiply in any order, you can group numbers in a smart way, and you can split a number to multiply. You checked subtraction with addition and division with multiplication. In algebra you also saw that brackets ( ) tell you which part to do first.

In Grade 6 you will use these ideas with whole numbers, decimals and fractions, and learn the full order of operations, so that everyone gets the same answer to the same expression.

Let's start with a mystery

Priya and Jonah both work out 6 + 4 × 5.

Priya adds first: 6 + 4 = 10, then 10 × 5 = 50.
Jonah multiplies first: 4 × 5 = 20, then 6 + 20 = 26.

They can't both be right! Mathematicians all over the world agreed on one order, so an expression always has one value. Jonah followed that order. The answer is 26.

The order of operations

Brackets first. Do what is inside ( ).
Then multiply and divide, working from left to right.
Then add and subtract, working from left to right.

Multiplying does not always come before dividing. They are a team: do whichever comes first as you read from left to right. Adding and subtracting are a team too.

Many Canadian students remember this as BEDMAS. The E stands for exponents, which you will learn about in a later grade, so you can skip it for now.

Step	What to do	Example: 30 − (2 + 4) × 3
1. Brackets	Work inside the brackets	2 + 4 = 6, so 30 − 6 × 3
2. × and ÷, left to right	Multiply 6 × 3	30 − 18
3. + and −, left to right	Subtract	12

Follow the steps in order. The value of 30 − (2 + 4) × 3 is 12.

Worked example

Example 1: Brackets change the answer

Find the value of 6 + 4 × 5 and of (6 + 4) × 5.

6 + 4 × 5: no brackets, so multiply first. 4 × 5 = 20. Then 6 + 20 = 26.
(6 + 4) × 5: brackets first. 6 + 4 = 10. Then 10 × 5 = 50.

6 + 4 × 5 = 26 and (6 + 4) × 5 = 50. The brackets made a big difference!

Worked example

Example 2: A longer expression

Find the value of 48 ÷ (2 + 6) × 3 − 5.

Brackets first: 2 + 6 = 8. Now it is 48 ÷ 8 × 3 − 5.
Multiply and divide from left to right. Division comes first: 48 ÷ 8 = 6. Then 6 × 3 = 18. Now it is 18 − 5.
Subtract: 18 − 5 = 13.

48 ÷ (2 + 6) × 3 − 5 = 13

Rewrite the expression after each step. It gets shorter every time, and each step follows the order: brackets, then × and ÷ left to right, then + and −.

Properties still make math easier

The properties work with decimals and fractions too.

Any order and any grouping (adding and multiplying): 2.5 × 17 × 4 → (2.5 × 4) × 17 = 10 × 17 = 170.
Friendly pairs: 1/4 + 3.6 + 3/4 → (1/4 + 3/4) + 3.6 = 1 + 3.6 = 4.6.
Splitting (distributive property): 6 × 4.5 = 6 × 4 + 6 × 0.5 = 24 + 3 = 27.

Careful: any order and any grouping do not work for subtraction or division.

An area model for 6 × 4.5, drawn to scale. Split the 4.5 side into 4 and 0.5, find each smaller area, then add: 24 + 3 = 27.

Worked example

Example 3: Use properties to calculate in your head

Find 2.5 × 17 × 4 and 8 × 3.25.

2.5 × 17 × 4: change the order and group 2.5 × 4 first. 2.5 × 4 = 10. Then 10 × 17 = 170.
8 × 3.25: split 3.25 into 3 + 0.25. 8 × 3 = 24 and 8 × 0.25 = 2. Then 24 + 2 = 26.

2.5 × 17 × 4 = 170 and 8 × 3.25 = 26

Worked example

Example 4: Check with the inverse operation

Leo says 31.5 ÷ 7 = 4.5. Is he right?

Multiplication undoes division. Check: 4.5 × 7.
Split: 4 × 7 = 28 and 0.5 × 7 = 3.5. 28 + 3.5 = 31.5. ✓

Yes, 31.5 ÷ 7 = 4.5.

Think like a mathematician

Before you calculate, read the whole expression first. Circle the brackets. Underline the × and ÷ signs. Then work step by step and rewrite the shorter expression each time. Writing each step makes mistakes easy to find.

Just for fun

Why are brackets such good bodyguards?

They protect what's inside and make sure it always goes first!

Hands-on activity

Card target game

You need: a deck of playing cards with the 10s and face cards (jacks, queens, kings) removed (aces count as 1), paper and a pencil. Play with a grown-up or a sibling.

Choose a target number, like 24.
Deal 4 cards face up.
Each player tries to make the target using all 4 numbers once, with +, −, × and ÷ and brackets.
Write your expression. Your partner checks it by following the order of operations.
If nobody can make the target, get as close as you can. Closest wins the round.

Example: with 2, 3, 4 and 6 you can write 6 × 4 × (3 − 2) = 24 × 1 = 24.

Math talk: "Where did you need brackets? What would happen without them?"

Your turn

8 + 2 × 7 = ?

(8 + 2) × 7 = ?

20 − 12 ÷ 4 = ?

36 ÷ 4 × 3 = ?

5 × (9 − 3) + 2 = ?

0.25 + 5.6 + 0.75 = ?

4 × 3.5 = ? Split 3.5 into 3 + 0.5.

More practice: warm-up

10 − 2 × 3 = ?

(10 − 2) × 3 = ?

7 + 3 × 0 = ?

1/2 + 3 + 1/2 = ?

50 × 9 × 2 = ?

More practice: on your own

3 + 4 × (8 − 2) ÷ 3 = ?

100 − 64 ÷ 8 × 5 = ?

Put brackets in 2 + 3 × 4 + 1 so its value is 25.

8 × 6.25 = ? Use splitting.

Check with multiplication: is 23.4 ÷ 6 = 3.9 correct?

Challenge

Use 3, 4, 5 and 6 once each, with any operations and brackets, to make 24.

Maya says 24 − 6 + 4 = 14. What did she do wrong? What is the correct value?

At a bake sale you buy 3 muffins at $2.25 and 2 juices at $1.50. Write one expression for the total and find its value.

Check your understanding

Quiz

What is 5 + 3 × 4?

In 18 − (4 + 2) × 2, what do you do first?

What is 24 ÷ 4 × 2?

Which is equal to 7 × 4.3?

What is 0.5 × 46 × 2?

What is the best way to check 31.5 ÷ 7 = 4.5?

For parents and teachers

Big idea: The Ontario curriculum (B2.1) asks Grade 6 students to use the properties of operations and the relationships between operations to solve problems with whole numbers, decimals, fractions, ratios, rates and percents, including problems with multiple steps or multiple operations. To write and evaluate multi-operation expressions, students need the order of operations. Exponents are not part of Grade 6, so we teach brackets, then × and ÷, then + and −.

Common difficulties:

Always multiplying before dividing (or adding before subtracting). In 24 ÷ 4 × 2 the answer is 12, not 3. Say "partners: left to right".
Working left to right without looking for brackets or × first: 6 + 4 × 5 is 26, not 50.
Using "any order" with subtraction or division.

How to help: Play the card target game. Ask your child to rewrite the expression after every step instead of doing it all in their head. This lesson also builds A1 skills: being flexible with strategies, persevering when a target is hard to reach, and explaining thinking clearly to someone else.