Expressions and Patterns: practice solutions, Grade 5 – download the PDF

1 Operations in the right order ★★★

Multiplication and division come before addition and subtraction.

1. \( 14 + 6 \times 2 = 14 + 12 = 26 \)
2. \( 30 - 12 \div 4 = 30 - 3 = 27 \)
3. \( 5 \times 4 - 3 = 20 - 3 = 17 \)
4. \( 48 \div 6 + 9 = 8 + 9 = 17 \)

2 Parentheses first ★★★

Work inside the parentheses first.

1. \( 20 \times 2 = 40 \)
2. \( 18 \div 3 = 6 \)
3. \( 5 \times 5 = 25 \)
4. \( 60 \div 10 = 6 \)

3 Left to right ★★★

1. \( 40 - 15 = 25 \), then \( 25 + 5 = 30 \).
2. \( 24 \div 4 = 6 \), then \( 6 \times 3 = 18 \).
3. \( 9 \times 2 = 18 \), then \( 18 \div 6 = 3 \).
4. \( 50 - 20 = 30 \), then \( 30 - 10 = 20 \).

4 From words to symbols ★★★

1. \( (6 + 9) \times 4 = 15 \times 4 = 60 \)
2. \( 6 \times 4 + 9 = 24 + 9 = 33 \)
3. \( (25 - 7) \div 3 = 18 \div 3 = 6 \)
4. \( 36 \div 4 - 2 = 9 - 2 = 7 \)

Parentheses were needed in (a) and (c) because the first operation named is addition or subtraction.

5 Continue the pattern ★★★

Add 4 each time: \( 13 + 4 = 17 \), \( 17 + 4 = 21 \), \( 21 + 4 = 25 \), \( 25 + 4 = 29 \).

The next four terms are \( 17, 21, 25, 29 \).

6 Ordered pairs from two patterns ★★★

Pattern A: \( 0, 2, 4, 6, 8 \). Pattern B: \( 0, 10, 20, 30, 40 \).

The ordered pairs are \( (0, 0),\ (2, 10),\ (4, 20),\ (6, 30),\ (8, 40) \).

7 Reading points ★★★

Read the move to the right first, then the move up.

\( A(2, 3) \), \( B(5, 1) \), \( C(4, 6) \), \( D(0, 4) \). Point D is on the y-axis because its x-coordinate is 0.

8 Two kinds of operations ★★★

Parentheses: \( 8 + 5 = 13 \). Expression: \( 3 \times 13 - 12 \div 4 \).

Multiply and divide: \( 3 \times 13 = 39 \) and \( 12 \div 4 = 3 \). Then \( 39 - 3 = 36 \).

The value is \( 36 \).

9 Nested groups ★★★

1. \( 3 + 5 = 8 \); \( 20 - 8 = 12 \); \( 12 \times 4 = 48 \).
2. \( 2 + 4 = 6 \); \( 36 \div 6 = 6 \); \( 2 \times 6 = 12 \); \( 12 + 1 = 13 \).
3. \( 9 - 3 = 6 \); \( 2 + 6 = 8 \); \( 5 \times 8 = 40 \); \( 40 \div 4 = 10 \).

10 Find the mistake ★★★

Leo added \( 8 + 2 = 10 \) first, then multiplied by 5. But multiplication comes before addition.

Correct: \( 2 \times 5 = 10 \), then \( 8 + 10 = 18 \).

With parentheses, \( (8 + 2) \times 5 = 10 \times 5 = 50 \).

11 Expressions for stories ★★★

1. \( 3 \times 8 - 5 = 24 - 5 = 19 \). Ana has 19 stickers left.
2. \( 5 \times 12 - 8 = 60 - 8 = 52 \). The team pays 52 dollars.
3. \( 4 \times (7 + 3) = 4 \times 10 = 40 \). The total is 40 dollars.

12 Which is larger? ★★★

1. \( 6 \times (23 + 19) \) is larger: it is 6 times as large as \( 23 + 19 \).
2. \( 62 - 18 \) is larger: multiplying by 0.5 gives half of it.
3. \( 45 + 30 \) is larger: dividing by 5 makes it one fifth as large.

13 Say it in words ★★★

1. Twice the sum of 15 and 4.
2. The sum of 15 and 4, divided by 2 (half of the sum).
3. Three times 10, plus 4. Without parentheses the 4 is added after the multiplication.

14 Comparing two patterns ★★★

A: \( 0, 4, 8, 12, 16, 20 \). B: \( 0, 8, 16, 24, 32, 40 \).

Pairs: \( (0, 0),\ (4, 8),\ (8, 16),\ (12, 24),\ (16, 32),\ (20, 40) \).

Each term of B is twice the matching term of A. The reason: 8 is \( 2 \times 4 \), so B grows twice as fast.

15 Graph two patterns ★★★

A: \( 0, 5, 10, 15, 20 \). B: \( 0, 10, 20, 30, 40 \).

Pairs: \( (0, 0),\ (5, 10),\ (10, 20),\ (15, 30),\ (20, 40) \).

The points are on a straight line through the origin. Each y-value is twice its x-value.

16 Different starting numbers ★★★

1. A: \( 1, 4, 7, 10, 13 \). B: \( 2, 8, 14, 20, 26 \). Pairs: \( (1, 2),\ (4, 8),\ (7, 14),\ (10, 20),\ (13, 26) \).
2. B is always twice A, so the next term of B is \( 2 \times 16 = 32 \) (check: \( 26 + 6 = 32 \)).
3. The 10th term adds the rule 9 times: A: \( 1 + 3 \times 9 = 28 \). B: \( 2 + 6 \times 9 = 56 \), which is \( 2 \times 28 \).

17 Four times as large ★★★

1. P: \( 0, 3, 6, 9 \). Q: \( 0, 12, 24, 36 \). Pairs: \( (0, 0),\ (3, 12),\ (6, 24),\ (9, 36) \).
2. Q is 4 times P. For \( x = 15 \): \( y = 4 \times 15 = 60 \).
3. \( 4 \times 21 = 84 \) so \( (21, 84) \) is on the graph. \( 4 \times 12 = 48 \), not 40, so \( (12, 40) \) is not. \( 4 \times 30 = 120 \) so \( (30, 120) \) is.

18 Use all four numbers ★★★

One answer is \( 8 + 5 \times 3 - 2 \). Multiply first: \( 5 \times 3 = 15 \). Then \( 8 + 15 = 23 \) and \( 23 - 2 = 21 \).

Other answers may exist. What matters is that each number is used once and the order of operations is respected.

19 Saving money ★★★

1. Sam: \( 0, 6, 12, 18, 24, 30 \). Ava: \( 0, 12, 24, 36, 48, 60 \).
2. \( (0, 0),\ (6, 12),\ (12, 24),\ (18, 36),\ (24, 48),\ (30, 60) \).
3. Ava always has twice as much as Sam, so \( 2 \times 42 = 84 \). Ava has 84 dollars.

20 Is it true? ★★★

Left side: \( 4 \times 22 = 88 \). Right side: \( 4 \times 13 + 9 = 52 + 9 = 61 \). They are not equal, so Marco is wrong.

The 4 must multiply both numbers: \( 4 \times (13 + 9) = 4 \times 13 + 4 \times 9 = 52 + 36 = 88 \).

21 Check the points ★★★

\( 3 \times 2 = 6 \), \( 3 \times 4 = 12 \), \( 3 \times 6 = 18 \). Mia is right.

\( 3 \times 5 = 15 \): \( (5, 15) \) is on the graph. \( 3 \times 8 = 24 \): \( (8, 24) \) is on it. \( 3 \times 10 = 30 \), not 20, so \( (10, 20) \) is not.