Grade by Grade
C. Algebra · Patterns, Equations, Coding and Modelling

Equal or Not Equal? Comparing Expressions

C2.2 A1About 40 minutes
Learning goals
  • I can tell what an expression is.
  • I can decide if two expressions are equivalent (have the same value) using adding, subtracting, multiplying and dividing.
  • I can use the signs = and ≠ correctly.
  • I can explain my thinking with numbers, pictures or math rules.

Words to know

expression
Numbers and operation signs that show an amount, like 3 × 4 or 20 − 8. An expression has no equal sign.
value
How much an expression is worth. The value of 3 × 4 is 12.
equivalent
Having the same value. 3 × 4 and 2 × 6 are equivalent: both are 12.
=
The equal sign. It means has the same value as.
≠
The not equal sign. It means the two sides do not have the same value.
array
Objects arranged in equal rows and columns.

Remember from Grade 2

In Grade 2 you learned that = means the same as, and you balanced adding and subtracting sentences like 8 + 6 = 10 + 4. In Grade 3 you know how to multiply and divide too, so now we can compare all four kinds of expressions!

What is an expression?

An expression shows an amount using numbers and operations. Here are some expressions:

  • 7 + 5
  • 20 − 8
  • 3 × 4
  • 24 ÷ 2

The first three all have the value 12. They are equivalent. The last one, 24 ÷ 2, is also 12! (24 shared into 2 equal groups is 12 in each group.)

= or ≠?

  • When two expressions have the same value, we write =. Example: 3 × 4 = 2 × 6 (both are 12).
  • When they have different values, we write ≠, which means not equal. Example: 3 × 4 ≠ 3 + 4 (12 is not 7).
Think of = as a balance5 × 4= 2025 − 5= 205 × 4 = 25 − 54 × 5= 204 + 5= 94 × 5 ≠ 4 + 5
When both sides have the same value, they balance: =. When the values are different, one side is heavier: ≠.
=3 × 4 = 122 × 6 = 12
An array of 3 rows of 4 has 12 dots. An array of 2 rows of 6 also has 12 dots. So 3 × 4 = 2 × 6.

Three ways to check

  1. Find the value of each side. Work out both expressions, then compare.
  2. Use a picture. Arrays, counters or a number line can show if two amounts are the same.
  3. Use a math rule. Some rules tell us right away, without working anything out.
Math ruleExampleEquivalent?
You can add in any order8 + 5 and 5 + 8Yes: 8 + 5 = 5 + 8
You can multiply in any order6 × 3 and 3 × 6Yes: 6 × 3 = 3 × 6
Subtracting in a different order is not the same10 − 4 and 4 − 10No: 10 − 4 ≠ 4 − 10
Dividing in a different order is not the same8 ÷ 2 and 2 ÷ 8No: 8 ÷ 2 ≠ 2 ÷ 8
Switching the order works for adding and multiplying, but not for subtracting and dividing.
Worked example

Example 1: Find each value

Is 5 × 4 equivalent to 25 − 5? Write = or ≠.

  1. Left side: 5 × 4 is the same amount as 4 groups of 5. Count by 5s four times: 5, 10, 15, 20. The value is 20.
  2. Right side: 25 − 5 = 20.
  3. Both values are 20.

5 × 4 = 25 − 5. They are equivalent.

Worked example

Example 2: Multiplying and dividing

Compare 12 ÷ 2 and 3 × 2.

  1. 12 ÷ 2: share 12 into 2 equal groups. Each group gets 6.
  2. 3 × 2: 3 groups of 2 = 2 + 2 + 2 = 6.
  3. 6 and 6 are the same.

12 ÷ 2 = 3 × 2.

Worked example

Example 3: Not equal

Compare 16 ÷ 2 and 16 ÷ 4.

  1. 16 ÷ 2: 16 shared into 2 groups is 8 in each group.
  2. 16 ÷ 4: 16 shared into 4 groups is 4 in each group.
  3. Sharing among more groups gives fewer in each group. 8 is not the same as 4.

16 ÷ 2 ≠ 16 ÷ 4.

Worked example

Example 4: Use a rule instead of calculating

Is 9 × 7 = 7 × 9 true or false?

  1. We can multiply in any order. 9 groups of 7 is the same amount as 7 groups of 9.
  2. We don't even need to find the value (it is 63) to know they are equal!

True. 9 × 7 = 7 × 9.

Just for fun

Funny but true: 2 + 2 = 4 and 2 × 2 = 4. Of all the counting numbers, 2 is the only one where adding it to itself and multiplying it by itself give the same answer!

Worked example

Example 5: Which ones belong?

Which of these expressions are equivalent to 24? 4 × 6, 3 × 8, 30 − 6, 20 + 3, 12 + 12

  1. 4 × 6 = 24 ✔ (count by 6s: 6, 12, 18, 24)
  2. 3 × 8 = 24 ✔ (8, 16, 24)
  3. 30 − 6 = 24 ✔
  4. 20 + 3 = 23 ✘
  5. 12 + 12 = 24 ✔

4 × 6, 3 × 8, 30 − 6 and 12 + 12 are all equivalent. 20 + 3 is not: 20 + 3 ≠ 24.

Careful with the equal sign

The equal sign does not mean the answer comes next. It means both sides have the same value. So 12 = 3 × 4 is true, and so is 2 × 6 = 3 × 4.

Your turn

1

Write = or ≠: 6 + 6 □ 2 × 6

2

Write = or ≠: 4 × 5 □ 4 + 5

3

Write = or ≠: 18 − 9 □ 3 × 3

4

Write = or ≠: 10 ÷ 2 □ 2 × 2

5

True or false: 8 × 3 = 3 × 8

6

True or false: 9 − 3 = 3 − 9

7

Which is not equivalent to 10? 5 × 2, 20 ÷ 2, 15 − 5, 6 + 3

Hands-on activity

Expression match-up

You need: 12 small pieces of paper and a pencil.

  1. A grown-up writes these expressions, one on each paper: 2 × 5, 15 − 5, 20 ÷ 2, 3 × 4, 6 + 6, 24 ÷ 2, 4 × 4, 20 − 4, 8 + 8, 2 × 7, 10 + 4, 18 − 4.
  2. Mix them up and lay them face up.
  3. Find the value of each one and sort them into families that are equivalent.
  4. For each family, write a true number sentence, like 2 × 5 = 15 − 5.
  5. Pick two papers from different families and write a sentence with ≠.

(Answer key for grown-ups: the 10s are 2 × 5, 15 − 5, 20 ÷ 2; the 12s are 3 × 4, 6 + 6, 24 ÷ 2; the 16s are 4 × 4, 20 − 4, 8 + 8; the 14s are 2 × 7, 10 + 4, 18 − 4.)

More practice: warm-up

1

What is the value of 3 × 5?

2

Write = or ≠: 5 + 5 □ 10

3

Write = or ≠: 2 × 3 □ 2 + 3

4

True or false: 7 + 2 = 2 + 7

5

Write = or ≠: 8 ÷ 2 □ 2 × 2

More practice: on your own

1

Write = or ≠: 5 × 6 □ 40 − 10. Explain.

2

Write = or ≠: 20 ÷ 4 □ 20 ÷ 5. Explain.

3

Write 3 different expressions that are equivalent to 18. Use at least one × and one ÷.

4

Sara says 4 × 2 = 2 × 4 = 8 × 1. Is she right? Explain.

5

Without calculating, how do you know 6 × 9 = 9 × 6? Use a picture idea.

Challenge

1

Write = or ≠: (3 × 4) + 2 □ 2 × 7. (Do the part in brackets first.)

2

Find the missing number to make the expressions equivalent: 4 × 6 = 3 × □

3

Leo says 12 ÷ 3 = 3 ÷ 12 because you can switch the numbers. Explain why he is wrong.

Check your understanding

Quiz

1.

Which is an expression?

2.

Which is equivalent to 3 × 5?

3.

Which sign makes this true? 14 ÷ 2 □ 2 × 4

4.

Which is true?

5.

Which expression is not equivalent to 20?

6.

What does ≠ mean?

For parents and teachers

Big idea: Two expressions are equivalent when they have the same value (C2.2). In Grade 3, children compare expressions with all four operations, write = or ≠, and explain how they know. Using a rule (like you can multiply in any order) is a powerful shortcut.

Common difficulties:

  • Thinking = means "the answer is". Children may say 2 × 6 = 3 × 4 is "wrong" because there is no answer after the equal sign. Keep saying "is the same as".
  • Switching numbers in subtraction or division. 10 − 4 is not the same as 4 − 10.
  • Multiplication and division facts. If facts are still slow, let your child use arrays, counters or skip counting. Understanding comes first; speed comes with practice.

How to help: Use snacks to build arrays: 3 rows of 4 crackers, then rearrange them into 2 rows of 6. Ask, "Did the amount change?"