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Math · Expressions & Order of Operations

Expressions & Order of Operations: Parentheses, Exponents, Multiply, Divide, Add, Subtract

Read and evaluate numerical expressions reliably by grouping first, handling exponents, then multiplying/dividing and adding/subtracting from left to right, with estimation and inverse-operation checks.

Grades 5–6 · 40 practice questions · Original material

Learning objectiveTranslate words into numerical expressions, evaluate expressions using the standard order of operations, use parentheses to change grouping, interpret simple exponents, and check whether an answer is reasonable.

The idea

An expression shows a calculation without an equals sign. Order matters because the same numbers can produce different results when operations are grouped differently.

Use this sequence: parentheses or other grouping symbols first; then exponents; then multiplication and division from left to right; then addition and subtraction from left to right.

The rule is not 'multiplication always before division' or 'addition always before subtraction'. Operations at the same level are handled from left to right.

Worked examples

Example 1 · Multiplication before addition

Evaluate 8 + 3 × 4.

Multiply first: 3 × 4 = 12.
Then add: 8 + 12 = 20.

20.

Example 2 · Parentheses first

Evaluate (8 + 3) × 4.

Parentheses first: 8 + 3 = 11.
Then multiply: 11 × 4 = 44.

44.

Example 3 · Exponent

Evaluate 5 + 2³.

Exponent first: 2³ = 2 × 2 × 2 = 8.
Then add: 5 + 8 = 13.

13.

Example 4 · Same-level operations

Evaluate 24 ÷ 6 × 3.

Division and multiplication have the same priority.
Work left to right: 24 ÷ 6 = 4, then 4 × 3 = 12.

12.

Example 5 · Translate words

Write an expression for 'five more than three times 7'.

Three times 7 is 3 × 7.
Five more means add 5.

3 × 7 + 5.

Example 6 · Check reasonableness

Evaluate 50 − 6 × 7.

Multiply first: 6 × 7 = 42.
Then subtract: 50 − 42 = 8.
A quick estimate says the answer should be small and positive, so 8 is reasonable.

Common mistakes

These are tempting because they use familiar operations. Check what the question is actually measuring.

Working strictly left to right through every operation.

Why this fails: Different operation levels have different priority; grouping and exponents come first.

Doing multiplication before division even when division appears first.

Why this fails: Multiplication and division share one priority level and are handled left to right.

Doing addition before subtraction because A appears before S in a mnemonic.

Why this fails: Addition and subtraction share one priority level and are handled left to right.

Reading 3² as 3 × 2.

Why this fails: 3² means 3 × 3, so the value is 9.

Dropping parentheses when translating words.

Why this fails: Grouping can change the meaning and value of an expression.

Foundation

Foundation practice

Read symbols and apply the basic order of operations.

Evaluate

1. 6 + 2 × 5

Evaluate

2. 18 − 3 × 4

Evaluate

3. 20 ÷ 5 + 7

Evaluate

4. 7 + 12 ÷ 3

Parentheses

5. (6 + 2) × 5

Parentheses

6. 18 − (3 × 4)

Exponent

7. 2³ + 1

Exponent

8. 3² + 4

Vocabulary

9. Does 5 + 3 × 2 = 16?

Compare

10. Which is greater: 4 + 2 × 6 or (4 + 2) × 6?

Core

Core practice

Combine several operations and respect left-to-right rules within one level.

Evaluate

11. 30 ÷ 5 × 2

Evaluate

12. 24 ÷ 6 × 4

Evaluate

13. 20 − 7 + 3

Evaluate

14. 9 + 6 − 4

Evaluate

15. 4 × 3 + 18 ÷ 6

Evaluate

16. 36 ÷ 4 + 5 × 2

Parentheses

17. 48 ÷ (4 + 2)

Parentheses

18. 5 × (12 − 8) + 3

Exponent

19. 2 × 3² + 1

Exponent

20. 4² ÷ 8 + 6

Think

Think practice

Translate language, compare groupings and analyse errors.

Translate

21. Write an expression for 'four more than six times 5'.

Translate

22. Write an expression for 'three times the sum of 8 and 2'.

Translate

23. Write an expression for '40 divided by the difference of 9 and 4'.

Compare

24. Are 5 × (4 + 2) and 5 × 4 + 2 equal?

Error analysis

25. A student says 16 ÷ 4 × 2 = 2 because multiplication comes first. Correct it.

Error analysis

26. A student says 15 − 6 + 2 = 7 because addition comes first. Correct it.

Exponent meaning

27. What does 4³ mean?

Missing parentheses

28. Insert parentheses so 6 + 2 × 5 has value 40.

Missing parentheses

29. Insert parentheses so 24 ÷ 3 + 5 has value 3.

Reasoning

30. Why can two expressions with the same numbers and operations have different values?

Challenge

Challenge practice

Handle nested structure, missing values and multi-step contexts.

Evaluate

31. 3 × (2² + 5)

Evaluate

32. 64 ÷ (2³ × 2)

Evaluate

33. 5 + 2 × (9 − 3)² ÷ 6

Missing value

34. Find n if 4 × n + 3 = 27.

Missing value

35. Find n if 36 ÷ n + 2 = 8.

Context

36. A game awards 8 points for each of 4 wins and subtracts 3 points for each of 2 penalties. Write and evaluate the expression.

Context

37. Three friends each buy a $7 ticket and together pay a $6 booking fee. Write and evaluate the total.

Context

38. A box has 5 rows of 8 pencils. Six pencils are removed, then the rest are split equally among 2 classes. Evaluate the expression (5 × 8 − 6) ÷ 2.

Estimate and check

39. Without exact calculation, should 198 ÷ 6 + 4 × 9 be closer to 40 or 70?

Generalize

40. What is the safest way to evaluate a long expression?

Answer key

1. 16
Multiply 2 × 5 = 10, then add 6.
2. 6
Multiply 3 × 4 = 12, then subtract from 18.
3. 11
Divide 20 ÷ 5 = 4, then add 7.
4. 11
Divide 12 ÷ 3 = 4, then add 7.
5. 40
Parentheses first: 8 × 5 = 40.
6. 6
Evaluate the grouped multiplication, then subtract.
7. 9
2³ = 8; 8 + 1 = 9.
8. 13
3² = 9; 9 + 4 = 13.
9. No; it equals 11.
Multiply 3 × 2 first, then add 5.
10. (4 + 2) × 6
The values are 16 and 36.
11. 12
Work left to right: 30 ÷ 5 = 6, then 6 × 2 = 12.
12. 16
24 ÷ 6 = 4, then 4 × 4 = 16.
13. 16
Addition and subtraction share priority; work left to right: 20 − 7 = 13, then +3.
14. 11
Work left to right: 9 + 6 = 15, then 15 − 4 = 11.
15. 15
Do multiplication and division first: 12 + 3 = 15.
16. 19
36 ÷ 4 = 9 and 5 × 2 = 10; then add.
17. 8
4 + 2 = 6; 48 ÷ 6 = 8.
18. 23
12 − 8 = 4; 5 × 4 = 20; then +3.
19. 19
3² = 9; 2 × 9 = 18; then +1.
20. 8
4² = 16; 16 ÷ 8 = 2; then +6.
21. 6 × 5 + 4
Six times 5 is multiplied first; four more means add 4.
22. 3 × (8 + 2)
The words 'the sum of 8 and 2' must be grouped.
23. 40 ÷ (9 − 4)
The difference must be found before division.
24. No; they are 30 and 22.
Parentheses change which addition is included in the multiplication.
25. 8
Division and multiplication share priority; work left to right: 16 ÷ 4 = 4, then ×2.
26. 11
Subtraction and addition share priority; work left to right.
27. 4 × 4 × 4
The exponent 3 tells how many factors of 4 are multiplied.
28. (6 + 2) × 5
Grouping 6 + 2 first gives 8 × 5.
29. 24 ÷ (3 + 5)
Grouping 3 + 5 gives 24 ÷ 8 = 3.
30. Grouping and operation order can change which calculation happens first.
Order is part of the expression's meaning.
31. 27
2² = 4; 4 + 5 = 9; 3 × 9 = 27.
32. 4
2³ = 8; 8 × 2 = 16; 64 ÷ 16 = 4.
33. 17
9 − 3 = 6; 6² = 36; 2 × 36 ÷ 6 = 12; then +5.
34. 6
The expression value 27 means 4n = 24, so n = 6.
35. 6
36 ÷ n must equal 6, so n = 6.
36. 8 × 4 − 3 × 2 = 26
Score gains and penalties are separate products before subtraction.
37. 3 × 7 + 6 = 27
Three tickets cost 21, then add one shared fee.
38. 17
40 − 6 = 34; 34 ÷ 2 = 17.
39. 70
198 ÷ 6 is about 33 and 4 × 9 = 36; total is about 69.
40. Mark the grouping, evaluate exponents, then work multiplication/division and addition/subtraction left to right, checking each stage.
A fixed sequence prevents skipped or reordered operations.

Related practice

Integers & Coordinate Plane Percent & Proportional Reasoning Ratios & Unit Rates