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Math · Integers & Coordinate Reasoning

Integers: Positive & Negative Numbers, Opposites, Absolute Value, Compare & Coordinate Plane

Extend number sense below zero: place integers on a number line, identify opposites, use absolute value, compare signed numbers and reason with coordinates and real-world contexts.

Grades 5–6 · 40 practice questions · Original material

Learning objectiveInterpret positive and negative integers in context; locate, compare and order integers; identify opposites and absolute value; and use signed coordinates with estimation and reasoning checks.

The idea

Positive and negative integers describe direction from a reference point: above or below zero, gain or loss, and forward or backward.

On a number line, numbers farther right are greater. This remains true even when both numbers are negative.

Absolute value measures distance from zero, so it is never negative. Opposites have the same absolute value but different signs.

Worked examples

Example 1 · Temperature

A temperature changes from 3°C to −4°C. Which temperature is colder?

Place both values on a number line.
−4 is left of 3, so −4 is less than 3.

−4°C is colder.

Example 2 · Opposites

What is the opposite of −7?

Opposites are the same distance from zero on different sides.
−7 is 7 units left of zero, so its opposite is 7.

Example 3 · Absolute value

Find |−12|.

Absolute value is distance from zero.
−12 is 12 units from zero.

12.

Example 4 · Compare negatives

Which is greater: −3 or −8?

Locate both on a number line.
−3 is farther right, so −3 is greater.

−3 > −8.

Example 5 · Order

Order 4, −2, 0, −7, 3 from least to greatest.

Start with the farthest-left negative value.
Move right across the number line.

−7, −2, 0, 3, 4.

Example 6 · Coordinate point

Point A is (−3, 4). What do the signs tell you?

The x-coordinate −3 means move 3 units left of the origin.
The y-coordinate 4 means move 4 units up.

Left 3, up 4.

Common mistakes

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

Thinking −8 is greater than −3 because 8 is greater than 3.

Why this fails: On the number line, −8 is farther left, so it is smaller.

Saying |−6| = −6.

Why this fails: Absolute value is a distance from zero, so |−6| = 6.

Calling 0 positive or negative.

Why this fails: Zero is neither positive nor negative.

Changing only the size when finding an opposite.

Why this fails: The opposite keeps the same distance from zero and changes side/sign.

Reading (−2, 5) as down 2 and right 5.

Why this fails: Ordered pairs are (x, y): horizontal movement first, vertical movement second.

Foundation

Foundation practice

Read, locate and name integers.

Integer meaning

1. Which integer represents 5 degrees below zero?

Integer meaning

2. Which integer represents an elevation 12 m above sea level?

Number line

3. Which is farther right: −4 or 2?

Compare

4. Fill in: −3 ___ 1.

Compare

5. Fill in: −2 ___ −9.

Opposite

6. What is the opposite of 8?

Opposite

7. What is the opposite of −11?

Absolute value

8. Find |−7|.

Absolute value

9. Find |5|.

Zero

10. Is 0 positive, negative, or neither?

Core

Core practice

Compare, order and connect signed values to contexts.

Order

11. Order −6, 3, −1 from least to greatest.

Order

12. Order 0, −4, 5, −2, 1 from greatest to least.

Temperature

13. Morning is −8°C and afternoon is −2°C. Which is warmer?

Money

14. A bank change of −$15 means what?

Elevation

15. Which is lower: −20 m or −5 m relative to sea level?

Distance from zero

16. Which is farther from zero: −9 or 6?

Absolute value

17. Compare |−4| and |3|.

Opposites

18. What integer is 13 units from zero on the negative side?

Missing value

19. If x is the opposite of −14, find x.

Reasoning

20. Which is greater: −1 or −100?

Think

Think practice

Use signs, absolute value and coordinates to reason rather than guess.

Error analysis

21. A student says −12 > −5 because 12 > 5. Correct the claim.

Error analysis

22. A student says |−10| = −10. Correct it.

Coordinate

23. For point (−4, 3), which direction is the x-move?

Coordinate

24. For point (2, −5), which direction is the y-move?

Coordinate quadrant

25. Which point lies left of and above the origin: (3,4), (−3,4), (−3,−4), or (3,−4)?

Symmetry

26. Point A is (5, −2). Reflect it across the y-axis.

Symmetry

27. Point B is (−4, 6). Reflect it across the x-axis.

Distance

28. How far apart are −3 and 4 on a number line?

Distance

29. How far apart are −8 and −2?

Reasoning

30. Can two opposite integers have different absolute values?

Challenge

Challenge practice

Combine integer reasoning with multi-step contexts and coordinates.

Temperature change

31. Temperature rises from −6°C to 3°C. By how many degrees?

Temperature change

32. Temperature falls from 4°C to −7°C. By how many degrees?

Elevation change

33. An elevator moves from floor −2 to floor 6. How many floors?

Net change

34. A game score changes +8, then −12. What is the net change?

Net change

35. A diver changes depth by −5 m, then +3 m relative to a starting level. Net change?

Coordinate distance

36. Points A(−2, 4) and B(5, 4) share the same y-coordinate. Horizontal distance?

Coordinate distance

37. Points C(3, −6) and D(3, 2) share the same x-coordinate. Vertical distance?

Equation

38. x and −9 are opposites. Find x.

Absolute-value equation

39. Name both integers x such that |x| = 6.

Generalize

40. Why is a number line more reliable than comparing the digits in two negative integers?

Answer key

1. −5
Below zero is represented by a negative integer.
2. 12
Above the reference level is positive.
3. 2
Greater numbers lie farther right on a number line.
4. −3 < 1
Every negative integer is less than every positive integer.
5. −2 > −9
−2 lies to the right of −9.
6. −8
Opposites are the same distance from zero on opposite sides.
7. 11
Changing side of zero changes the sign.
8. 7
−7 is 7 units from zero.
9. 5
5 is 5 units from zero.
10. Neither.
Zero is the reference point and is neither positive nor negative.
11. −6, −1, 3
Read from left to right on the number line.
12. 5, 1, 0, −2, −4
Greatest-to-least moves from right to left.
13. −2°C
−2 is greater than −8.
14. A decrease or withdrawal of $15.
The negative sign represents a change below the reference balance.
15. −20 m
−20 is farther below zero.
16. −9
Compare absolute values: 9 > 6.
17. |−4| > |3|
The distances are 4 and 3.
18. −13
Negative side means left of zero.
19. 14
The opposite changes the sign.
20. −1
−1 is much closer to zero and farther right.
21. −12 < −5.
Negative numbers reverse that size intuition because −12 is farther left.
22. |−10| = 10.
Absolute value is distance and cannot be negative.
23. 4 units left.
A negative x-coordinate is left of the origin.
24. 5 units down.
A negative y-coordinate is below the origin.
25. (−3, 4)
Left requires negative x; above requires positive y.
26. (−5, −2)
Reflection across the y-axis changes the sign of x only.
27. (−4, −6)
Reflection across the x-axis changes the sign of y only.
28. 7 units
From −3 to 0 is 3; from 0 to 4 is 4; total 7.
29. 6 units
The difference in positions is 6 units.
30. No.
Opposites are equally far from zero.
31. 9°C
Move 6 degrees to zero and 3 more to 3.
32. 11°C
The total downward change crosses zero: 4 + 7 = 11.
33. 8 floors
From −2 to 0 is 2, then 6 more.
34. −4
8 − 12 = −4.
35. −2 m
−5 + 3 = −2.
36. 7 units
From −2 to 5 is 7.
37. 8 units
From −6 to 2 is 8.
38. 9
Opposite integers sum to zero.
39. −6 and 6
Both are 6 units from zero.
40. It shows actual order: farther right always means greater.
The sign changes how magnitude relates to order, while the number line preserves the rule.

Related practice

Percent & Proportional Reasoning Ratios & Unit Rates Decimals