099r. Reviewing Integers and Integer Operations

Learning Intentions

• To understand that integers can be negative, zero or positive
• To understand how to use a number line to add or subtract positive integers
• add a positive integer to a negative integer
• subtract a positive integer from a positive or negative integer

Pre-requisite Summary

• Know that whole numbers include $0,1,2,3,\dots$
• Be able to read and mark values on a number line
• Understand that moving right on a number line increases value
• Understand that moving left on a number line decreases value
• Recall basic addition and subtraction facts with positive whole numbers
• Know that subtraction can be interpreted as moving back a number of steps
• Be familiar with comparing numbers as greater than or less than
• Understand that zero separates positive and negative numbers on a number line

Worked Examples

Worked Example 1

Classify each integer as negative, zero or positive:
a) $-6$
b) $0$
c) $9$

Worked Example 2

Use a number line to find each answer:
a) $4+3$
b) $7-5$

Worked Example 3

Add a positive integer to a negative integer:
a) $-5+2$
b) $-8+6$

Worked Example 4

Subtract a positive integer from a positive integer:
a) $9-12$
b) $6-8$

Worked Example 5

Subtract a positive integer from a negative integer:
a) $-3-4$
b) $-7-5$

Worked Example 6

Use a number line to solve:
a) $-4+7$
b) $2-6$
c) $-2-3$

Problems

Problem 1

Classify each integer as negative, zero or positive:
a) $-11$
b) $0$
c) $14$

Problem 2

Use a number line to find each answer:
a) $5+4$
b) $8-6$

Problem 3

Add a positive integer to a negative integer:
a) $-6+4$
b) $-9+3$

Problem 4

Subtract a positive integer from a positive integer:
a) $7-10$
b) $5-9$

Problem 5

Subtract a positive integer from a negative integer:
a) $-2-6$
b) $-10-3$

Problem 6

Use a number line to solve:
a) $-5+8$
b) $3-7$
c) $-1-4$

Exercises

Understanding and Fluency

1. State whether each integer is negative, zero or positive:
   a) $-4$
   b) $12$
   c) $0$
   d) $-15$

2. Place these integers in order from least to greatest:
   a) $-3,2,0,-1$
   b) $5,-2,1,-6$
   c) $-8,-5,0,4$

3. Use a number line to evaluate:
   a) $3+4$
   b) $6-2$
   c) $1-5$

4. Add a positive integer to a negative integer:
   a) $-7+5$
   b) $-2+6$
   c) $-9+4$

5. Subtract a positive integer from a positive integer:
   a) $8-3$
   b) $4-9$
   c) $12-15$

6. Subtract a positive integer from a negative integer:
   a) $-4-3$
   b) $-1-7$
   c) $-10-2$

7. Use a number line to solve:
   a) $-3+5$
   b) $2-7$
   c) $-6-4$

8. Complete each statement:
   a) $0$ is neither ______ nor ______
   b) Integers to the right of $0$ are ______
   c) Integers to the left of $0$ are ______
   d) Subtracting a positive integer means moving ______ on the number line

Reasoning

9. Explain why $-2$ is less than $1$ using a number line.

10. A student says that $-5$ is greater than $-2$ because $5>2$. Explain the mistake.

11. Noah says that $-4+7=-11$ because he added $4$ and $7$ and kept the negative sign. Explain why this is incorrect.

12. Explain why subtracting a positive integer from a negative integer always gives a smaller number.

13. A student says $3-8=5$. Describe what the student has forgotten about the number line.

Problem-solving

14. The temperature is $-3^{\circ }\text{C}$ in the morning. It rises by $5^{\circ }\text{C}$. What is the new temperature?

15. A diver is at $-6$ m relative to sea level. She moves up $4$ m. What is her new position?

16. A lift starts on floor $2$ and goes down $7$ floors. What floor does it reach?

17. A submarine is at $-4$ m and then descends $3$ m more. What is its new position?

18. A game score starts at $0$. A player loses $5$ points and then gains $3$ points. What is the final score?

19. At night the temperature is $2^{\circ }\text{C}$. It drops by $6^{\circ }\text{C}$. What is the new temperature?

Potential Misunderstandings

• Students may think integers are only negative numbers
• Students may forget that $0$ is an integer
• Students may think $0$ is positive or negative
• Students may confuse the size of a negative number with the size of its digits, for example thinking $-8>-3$
• Students may move the wrong direction on a number line when subtracting
• Students may think subtracting always makes a number stay positive
• Students may add the digits correctly but choose the wrong sign for the answer
• Students may think subtracting a positive integer from a negative integer makes the number closer to zero
• Students may ignore the starting point on the number line and only count the jumps
• Students may confuse moving left and right with increasing and decreasing values