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

The Order of Operations with Fractions, Decimals and Exponents

B2.1 A1About 45 minutes

Learning goals

I can use the order of operations (BEDMAS) with fractions, decimals and exponents.
I can treat a fraction bar as a grouping symbol.
I can work through nested brackets from the inside out.
I can write one expression for a multi-step problem and evaluate it.

Words to know

order of operations
The agreed order for doing operations: Brackets, Exponents, Division and Multiplication (left to right), Addition and Subtraction (left to right). Often remembered as BEDMAS.

expression
A combination of numbers and operation signs, like 3 + 0.5 × 4, with no equal sign.

evaluate
Find the value of an expression.

nested brackets
Brackets inside other brackets, like 2 × [5 − (1 + 0.5)]. Work from the inside out.

fraction bar
The line in a fraction. It means "divide" and also groups: everything on top is worked out before dividing by everything below.

Remember from Grade 7

In Grade 7 you used BEDMAS with whole numbers and exponents, like 4 + 3² × 2 = 22. In Grade 8 the numbers become fractions and decimals, the brackets can be nested, and you will write your own expressions for multi-step problems.

Let's start with a disagreement

Two friends evaluate 2 + 0.5 × 4².

Sam goes left to right: 2 + 0.5 = 2.5, then 2.5 × 4 = 10, then 10² = 100.
Ria uses the order of operations: 4² = 16, then 0.5 × 16 = 8, then 2 + 8 = 10.

Only one answer can be right, and it is Ria's: 10. Everyone in the world uses the same order of operations so that an expression has only one value.

Same numbers, different order, different answers. Following the order of operations (exponent, then multiply, then add) gives the one correct value: 10.

Step	What to do	Example: 3 + (1.2 + 0.8)² ÷ 0.5
B: Brackets	Work inside brackets first (innermost first). A fraction bar groups the top and the bottom too.	1.2 + 0.8 = 2 → 3 + 2² ÷ 0.5
E: Exponents	Evaluate powers.	2² = 4 → 3 + 4 ÷ 0.5
D M: Division and Multiplication	From left to right, whichever comes first.	4 ÷ 0.5 = 8 → 3 + 8
A S: Addition and Subtraction	From left to right, whichever comes first.	3 + 8 = 11

BEDMAS. Division and multiplication have equal rank, and so do addition and subtraction.

New in Grade 8: powers and fraction bars with rational numbers

Powers of fractions and decimals: (2/3)² = 2/3 × 2/3 = 4/9. 0.3² = 0.3 × 0.3 = 0.09 (not 0.9). (1/2)³ = 1/2 × 1/2 × 1/2 = 1/8.
The fraction bar is a grouping symbol. In (6 + 0.4)/0.8, add the top first: 6.4/0.8 = 6.4 ÷ 0.8 = 8. If you use a calculator, put brackets around the top and around the bottom.
Nested brackets: in 2 × [5.2 − (1.5 + 0.7)], start with the round brackets inside, then the square brackets.

Nested brackets work from the inside out: round brackets first, then square brackets, then multiply and divide from left to right. 2 × [5.2 − (1.5 + 0.7)] ÷ 0.4 = 15.

Worked example

Example 1: Fractions and an exponent

Evaluate 3/4 + 1/2 × (2/3)².

Exponent first: (2/3)² = 4/9.
Multiply: 1/2 × 4/9 = 4/18 = 2/9.
Add with a common denominator of 36: 3/4 + 2/9 = 27/36 + 8/36 = 35/36.

35/36

Worked example

Example 2: Nested brackets

Evaluate 2 × [5.2 − (1.5 + 0.7)] ÷ 0.4.

Innermost brackets: 1.5 + 0.7 = 2.2.
Square brackets: 5.2 − 2.2 = 3.
Now 2 × 3 ÷ 0.4. Left to right: 2 × 3 = 6, then 6 ÷ 0.4 = 15.

Worked example

Example 3: A fraction bar

Evaluate (4.8 + 1.2²) / (0.2 × 3).

Top: 1.2² = 1.44, and 4.8 + 1.44 = 6.24.
Bottom: 0.2 × 3 = 0.6.
Divide: 6.24 ÷ 0.6 = 62.4 ÷ 6 = 10.4.

10.4

Worked example

Example 4: Write one expression for a problem

Four friends go to a movie. Tickets are $12.50 each and they share 2 bags of popcorn at $7.00 each. They split the total equally. Write one expression and find each person's share.

Total cost: 4 × 12.50 + 2 × 7.00. Put it in brackets so it is worked out before sharing.
Expression: (4 × 12.50 + 2 × 7.00) ÷ 4.
Brackets: 4 × 12.50 = 50 and 2 × 7.00 = 14, so 50 + 14 = 64.
64 ÷ 4 = 16.

(4 × 12.50 + 2 × 7.00) ÷ 4 = $16 each

Watch out!

0.3² is 0.09, not 0.9. Count decimal places: 0.3 has one, so 0.3 × 0.3 has two.
Multiplication does not always come before division. 12 ÷ 4 × 3 = 3 × 3 = 9 (left to right), not 12 ÷ 12 = 1.
Write each step on a new line and copy the parts you have not used yet. Most mistakes happen when a number gets dropped.

Hands-on activity

Make the target

You need: a deck of cards with the 10s and face cards removed (Ace = 1), paper, a pencil. Play alone or with a partner.

Shuffle and turn over 4 cards.
Use all 4 numbers once each, with any of +, −, ×, ÷, brackets and exponents, to make 24. You may also make a fraction or a decimal from a card, like 1/2 or 0.5.
Write your expression. Your partner checks it with BEDMAS.
If 24 is impossible, get as close as you can. Closest expression wins the round.

Model answer: cards 3, 8, 2, 5 → (8 − 5) × 2³ = 3 × 8 = 24. Cards 6, 1, 2, 9 → (9 − 1) × 6 ÷ 2 = 8 × 6 ÷ 2 = 24.

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

Just for fun

In 2019, the expression 8 ÷ 2(2 + 2) went viral online, and people argued for days over whether the answer was 16 or 1. Math teachers everywhere had the same advice: use more brackets!

Your turn

6 + 0.5 × 8 = ?

(6 + 0.5) × 8 = ?

1/2 + 1/4 × 2 = ?

0.4² + 0.2 = ?

(3/5)² = ?

(9 + 0.6) / 1.2 = ?

10 − 2 × [3 − (1.2 + 0.3)] = ?

More practice: warm-up

4 + 3 × 0.5 = ?

(4 + 3) × 0.5 = ?

2³ × 0.1 = ?

12 ÷ 4 × 3 = ?

(1/2)² = ?

More practice: on your own

5 − 2/3 × 3/4 = ?

1.5² + 2 × 0.75 = ?

(2.4 + 3.6) / (0.5 × 4) = ?

3/4 ÷ 1/4 − 1/2 = ?

Add one pair of brackets to make this true: 2 + 0.5 × 4 = 10

Challenge

Ali buys 3 notebooks at $2.25 and 2 pens at $1.40, and pays with a $20 bill. Write one expression for his change and evaluate it.

Evaluate [(1/2 + 1/3) × 6]² ÷ 5.

Which is greater, 0.9² or 0.9³? Explain why.

Check your understanding

Quiz

What is 8 − 2 × 1.5?

What is 0.3²?

What is (5.4 + 0.6) ÷ 1.5?

What is (2/3)²?

What is 2 + 3 × 4²?

What is 18 ÷ 3 × 2?

For parents and teachers

Big idea: B2.1 includes using the order of operations in problems with rational numbers and multiple steps. The rules are the same as in Grade 7; the numbers are harder.

Common difficulties:

Doing multiplication before division (or addition before subtraction) every time. They have equal rank, so go left to right.
Squaring decimals: writing 0.3² = 0.9. Ask your child to count decimal places.
Ignoring the fraction bar as a grouping symbol, especially on a calculator. Teach them to type brackets around the top and the bottom.

How to help: Play the "Make the target" card game together. When your child makes an expression, ask "Where do you need brackets?" Checking each other's expressions builds the A1 skills of reasoning, proving and communicating.