051. Adding and Subtracting Negative Integers

Learning Intentions

• To understand that adding a negative integer is the same as subtracting a positive integer
• To understand that subtracting a negative integer is the same as adding a positive integer
• use a number line to add and subtract negative integers

Pre-requisite Summary

• Understand that integers can be positive, negative or zero
• Be able to locate integers on a number line
• Know that moving right on a number line increases value
• Know that moving left on a number line decreases value
• Understand addition and subtraction of positive integers on a number line
• Recognise that opposite integers are the same distance from zero in opposite directions
• Be able to compare integers and decide which is greater
• Understand that subtraction can be interpreted as a movement or change on a number line

Worked Examples

Worked Example 1

Use a number line to show that adding a negative integer is the same as subtracting a positive integer:
a) $3+(-4)$
b) $3-4$
c) Compare the two results.

Worked Example 2

Use a number line to show that subtracting a negative integer is the same as adding a positive integer:
a) $2-(-5)$
b) $2+5$
c) Compare the two results.

Worked Example 3

Use a number line to add negative integers:
a) $-1+(-3)$
b) $4+(-6)$
c) $0+(-5)$

Worked Example 4

Use a number line to subtract negative integers:
a) $-2-(-4)$
b) $3-(-2)$
c) $-5-(-1)$

Worked Example 5

Describe the movement on the number line and find the result:
a) Start at $4$, then add $-3$
b) Start at $-2$, then subtract $-5$
c) Start at $1$, then add $-6$

Worked Example 6

Rewrite each calculation and then evaluate using a number line:
a) $5+(-2)$
b) $-3-(-4)$
c) $-6+(-1)$

Problems

Problem 1

Use a number line to show that adding a negative integer is the same as subtracting a positive integer:
a) $6+(-3)$
b) $6-3$
c) Compare the two results.

Problem 2

Use a number line to show that subtracting a negative integer is the same as adding a positive integer:
a) $1-(-4)$
b) $1+4$
c) Compare the two results.

Problem 3

Use a number line to add negative integers:
a) $-2+(-4)$
b) $5+(-7)$
c) $0+(-6)$

Problem 4

Use a number line to subtract negative integers:
a) $-3-(-5)$
b) $4-(-3)$
c) $-6-(-2)$

Problem 5

Describe the movement on the number line and find the result:
a) Start at $5$, then add $-4$
b) Start at $-1$, then subtract $-6$
c) Start at $2$, then add $-7$

Problem 6

Rewrite each calculation and then evaluate using a number line:
a) $7+(-3)$
b) $-4-(-5)$
c) $-8+(-2)$

Exercises

Understanding and Fluency

1. Rewrite each addition of a negative integer as subtraction, then evaluate:
   a) $4+(-3)$
   b) $7+(-2)$
   c) $-1+(-5)$

2. Rewrite each subtraction of a negative integer as addition, then evaluate:
   a) $3-(-4)$
   b) $-2-(-6)$
   c) $5-(-3)$

3. Use a number line to add negative integers:
   a) $2+(-5)$
   b) $-3+(-2)$
   c) $6+(-8)$

4. Use a number line to subtract negative integers:
   a) $1-(-3)$
   b) $-4-(-2)$
   c) $-7-(-5)$

5. Evaluate:
   a) $-2+(-4)$
   b) $3+(-7)$
   c) $-5+(-1)$

6. Evaluate:
   a) $4-(-6)$
   b) $-3-(-4)$
   c) $2-(-8)$

7. Describe the movement on the number line and find the result:
   a) start at $3$, move left $5$
   b) start at $-2$, move right $4$
   c) start at $1$, move left $6$

8. Describe the movement on the number line and find the result:
   a) start at $-4$, move right $7$
   b) start at $5$, move left $3$
   c) start at $-1$, move left $4$

Reasoning

9. Explain why $5+(-3)$ gives the same result as $5-3$.

10. A student says $4-(-2)=2$. Explain the mistake.

11. Explain why subtracting a negative integer moves right on a number line.

12. A student says $-3+(-4)=1$ because “two negatives make a positive”. Explain why this is incorrect.

Problem-solving

13. A temperature is $2^{\circ }\text{C}$ and then changes by $-5^{\circ }\text{C}$. What is the new temperature?

14. A lift is on floor $-3$ and then goes up $4$ floors. Write this as subtracting a negative integer or adding a positive integer, and find the new floor.

15. A bank balance is represented by $5$, then a debt of $7$ is added. Write this as an integer calculation and find the new balance.

16. A diver is at $-6$ m and rises $5$ m. Write this as subtracting a negative integer or adding a positive integer, and find the new position.

17. A game score is $4$, then changes by $-9$. What is the new score?

18. A submarine is at $-2$ m and then moves up $6$ m. Represent the change using integers and find the new position.

Potential Misunderstandings

• Students may think adding a negative integer means moving right instead of left on the number line
• Students may think subtracting a negative integer means moving left instead of right
• Students may incorrectly apply the rule “two negatives make a positive” to addition, where it does not apply in that way
• Students may ignore brackets and confuse $-3-(-4)$ with $-3-4$
• Students may start at zero instead of starting at the given integer on the number line
• Students may confuse the sign of a number with the operation sign
• Students may think $a+(-b)$ and $a-(-b)$ mean the same thing
• Students may count jumps incorrectly on the number line when moving across zero

Next: 052e. Multiplying and Dividing Integers