Integers and Rational Numbers: practice solutions, Grade 6 – download the PDF

1 Writing situations with signed numbers ★★★

Below, owing and losses are negative; above and gains are positive.

1. \( -12 \) feet.
2. \( +8 \) yards, or simply \( 8 \).
3. \( -15 \) dollars.
4. \( +6 \) degrees, or \( 6 \).

2 Finding opposites ★★★

The opposite changes the sign: \( -9 \), \( 14 \), \( 0 \), \( 3.5 \), \( -\dfrac{7}{8} \). Zero is its own opposite.

3 Comparing integers ★★★

1. \( -5 \lt 3 \): a negative number is less than a positive number.
2. \( -8 \lt -2 \): \( -8 \) is farther left.
3. \( 0 \gt -1 \): zero is greater than any negative number.
4. \( 4 \gt -10 \).

4 Absolute values ★★★

The absolute value is the distance from zero: \( |-9| = 9 \), \( |12| = 12 \), \( |0| = 0 \), \( |-2.5| = 2.5 \), \( |-\dfrac{3}{4}| = \dfrac{3}{4} \).

5 Reading a number line ★★★

Each tick is one unit. A is at \( -6 \), B is at \( -2 \), C is at \( 3 \) and D is at \( 7 \).

6 Naming quadrants ★★★

Look at the signs. \( (4, 7) \): both positive, quadrant I. \( (-3, 5) \): quadrant II. \( (-6, -1) \): quadrant III. \( (2, -8) \): quadrant IV. \( (0, 5) \): the x-coordinate is 0, so it is on the y-axis, in no quadrant.

7 True or false? ★★★

1. True. An integer \( n \) can be written \( \dfrac{n}{1} \).
2. False. Zero is neither positive nor negative.
3. True: \( -(-7) = 7 \).
4. False. An absolute value is a distance, so \( |-5| = 5 \).

8 Ordering integers and decimals ★★★

1. Negatives first, with the largest absolute value leftmost: \( -7 \lt -3 \lt -1 \lt 0 \lt 2 \).
2. \( -1.2 \lt -0.8 \lt 0.08 \lt 0.5 \).

9 Fractions on the number line ★★★

1. \( -0.75 \), \( 0.5 \) and \( -1.25 \). Order: \( -\dfrac{5}{4} \lt -\dfrac{3}{4} \lt \dfrac{1}{2} \).
2. With denominator 12: \( -\dfrac{2}{3} = -\dfrac{8}{12} \) and \( -\dfrac{3}{4} = -\dfrac{9}{12} \). Since \( -9 \lt -8 \), \( -\dfrac{3}{4} \) is less.

10 A week of temperatures ★★★

1. Coldest first: Tuesday \( (-9) \), Friday \( (-6) \), Monday \( (-4) \), Thursday \( (-1) \), Wednesday \( (2) \).
2. Tuesday and Friday, since \( -9 \lt -5 \) and \( -6 \lt -5 \).
3. \( 2 - (-9) = 11 \). The temperature rose 11 degrees.

11 Money and debts ★★★

1. She owes 45 dollars.
2. \( -45 + 30 = -15 \). The balance is \( -15 \) dollars: she still owes 15 dollars.
3. Owing 60 dollars is \( -60 \) and owing 35 dollars is \( -35 \). Since \( -60 \lt -35 \), the first friend owes more.

12 Absolute value versus order ★★★

1. \( |-8| = 8 \gt 5 = |5| \), but \( -8 \lt 5 \).
2. False. A larger absolute value only means farther from zero. \( -12 \) lies left of 7, so \( -12 \lt 7 \).
3. \( |-0.3| = 0.3 \) and \( |0.25| = 0.25 \). Since \( 0.25 \lt 0.3 \), the number \( 0.25 \) has the smaller absolute value.

13 Reading coordinates ★★★

A\( (-4, 3) \) is in quadrant II. B\( (2, 5) \) is in quadrant I. C\( (-5, -2) \) is in quadrant III. D\( (3, -4) \) is in quadrant IV.

14 Reflecting a point ★★★

Across the x-axis, change the sign of y: \( (5, 2) \). Across the y-axis, change the sign of x: \( (-5, -2) \). Across both: \( (-5, 2) \).

15 Submarine and seabird ★★★

1. They are on opposite sides of sea level: \( 120 + 240 = 360 \) feet.
2. \( -240 + 85 = -155 \), so the submarine is at \( -155 \) feet. The distance is \( 120 + 155 = 275 \) feet.
3. \( |-240| = 240 \), so it must rise 240 feet.

16 Rectangle on the grid ★★★

1. A and B share \( x = -6 \): \( AB = 4 + 3 = 7 \). B and C share \( y = -3 \): \( BC = 6 + 5 = 11 \).
2. Perimeter: \( 2(7 + 11) = 36 \) units. Area: \( 7 \times 11 = 77 \) square units.

17 Walking across the map ★★★

1. Library: \( (3, 2) \). School: \( (3, -2) \).
2. Same y-coordinate: \( 3 + 3 = 6 \) units, which is \( 6 \times \dfrac{1}{2} = 3 \) miles.
3. Library to school: same x-coordinate, \( 2 + 2 = 4 \) units, so 2 miles. Total: \( 3 + 2 = 5 \) miles.

18 Two reflections in a row ★★★

1. First image: \( (-7, -3) \). Second image: \( (7, -3) \). The start is in quadrant II, the first image in quadrant III, the final image in quadrant IV.
2. Both coordinates were changed to their opposites.
3. \( (-2.5, 4) \to (-2.5, -4) \to (2.5, -4) \).

19 Changing lake level ★★★

In quarters: \( -3,\ +6,\ -9,\ +1 \).

1. \( -2\dfrac{1}{4} \lt -\dfrac{3}{4} \lt \dfrac{1}{4} \lt 1\dfrac{1}{2} \).
2. Week 3, with a drop of \( 2\dfrac{1}{4} \) inches.
3. \( \dfrac{-3 + 6 - 9 + 1}{4} = -\dfrac{5}{4} \), so the lake fell \( 1\dfrac{1}{4} \) inches in all.

20 Reasoning with absolute value ★★★

1. \( -3, -2, -1, 0, 1, 2, 3 \): seven integers.
2. Those greater than \( -2 \): \( -1, 0, 1, 2, 3 \), five integers.
3. \( 6.5 \) and \( -6.5 \).
4. Being farther from zero on the negative side means being farther left, so \( -7 \) is less than \( -2 \).

21 Missing vertex ★★★

1. The fourth vertex shares x with the first and y with the third: \( (-2, 3) \).
2. Horizontal side: \( 2 + 4 = 6 \). Vertical side: \( 1 + 3 = 4 \). Perimeter \( 2(6 + 4) = 20 \) units; area \( 6 \times 4 = 24 \) square units.