050. Adding and Subtracting Integers on a Number Line

Learning Intentions

• To understand the number line model for adding and subtracting positive integers
• use a number line to add a positive integer to any integer
• use a number line to subtract a positive integer from any integer

Pre-requisite Summary

• Understand that integers can be negative, positive or zero
• Be able to locate integers on a number line
• Know that moving right on a number line means the value increases
• Know that moving left on a number line means the value decreases
• Be able to compare integers and identify which is greater
• Understand that addition can be modelled as movement on a number line
• Understand that subtraction can be modelled as movement on a number line

Worked Examples

Worked Example 1

Use a number line to add a positive integer:
a) Start at $3$ and add $4$
b) Start at $-2$ and add $5$

Worked Example 2

Use a number line to subtract a positive integer:
a) Start at $6$ and subtract $3$
b) Start at $1$ and subtract $4$

Worked Example 3

Use a number line to add a positive integer to any integer:
a) $-5+2$
b) $0+6$
c) $4+3$

Worked Example 4

Use a number line to subtract a positive integer from any integer:
a) $-1-3$
b) $2-5$
c) $7-2$

Worked Example 5

Describe the movement on the number line and find the result:
a) Start at $-4$, move right $7$ places
b) Start at $5$, move left $8$ places

Worked Example 6

Explain whether each calculation should move left or right on a number line, then find the result:
a) $-3+4$
b) $-3-4$
c) $2+5$

Problems

Problem 1

Use a number line to add a positive integer:
a) Start at $2$ and add $5$
b) Start at $-3$ and add $4$

Problem 2

Use a number line to subtract a positive integer:
a) Start at $7$ and subtract $2$
b) Start at $0$ and subtract $3$

Problem 3

Use a number line to add a positive integer to any integer:
a) $-6+3$
b) $1+5$
c) $3+4$

Problem 4

Use a number line to subtract a positive integer from any integer:
a) $-2-4$
b) $4-6$
c) $8-3$

Problem 5

Describe the movement on the number line and find the result:
a) Start at $-5$, move right $6$ places
b) Start at $4$, move left $7$ places

Problem 6

Explain whether each calculation should move left or right on a number line, then find the result:
a) $-4+3$
b) $-4-3$
c) $1+6$

Exercises

Understanding and Fluency

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

2. Use a number line to add:
   a) $0+6$
   b) $3+2$
   c) $-7+3$

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

4. Use a number line to subtract:
   a) $7-5$
   b) $0-2$
   c) $-5-1$

5. Describe the movement and find the result:
   a) start at $-3$, move right $4$
   b) start at $2$, move left $5$
   c) start at $0$, move right $3$

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

7. Calculate using a number line:
   a) $-8+6$
   b) $3-7$
   c) $-1+2$

8. Calculate using a number line:
   a) $-3-2$
   b) $5+4$
   c) $-6+7$

Reasoning

9. Explain why adding a positive integer means moving to the right on a number line.

10. A student says that $-2+5$ means move left $5$ places. Explain the mistake.

11. Explain why $3-6$ ends at a negative integer.

12. A student starts at $-4$ and solves $-4-3$ by moving right $3$ places. Explain why this is incorrect.

Problem-solving

13. A lift is on floor $-2$ and goes up $5$ floors. What floor does it reach?

14. A temperature is $3^{\circ }\text{C}$ and then drops by $7^{\circ }\text{C}$. What is the new temperature?

15. A diver is at $-6$ m and rises $4$ m. What is the diver’s new position?

16. A game score starts at $1$ and decreases by $5$. What is the new score?

17. A submarine is at $-3$ m and rises $8$ m. What integer represents its new position?

18. A bank account balance is represented by $-4$ and then decreases by $3$. What is the new integer balance?

Potential Misunderstandings

• Students may confuse adding with moving left instead of right on the number line
• Students may confuse subtracting with moving right instead of left on the number line
• Students may start from zero instead of from the given integer
• Students may count the starting point as the first jump incorrectly
• Students may think a negative starting number changes the direction of movement for adding a positive integer
• Students may not recognise that subtracting a positive integer can move the result further into the negative numbers
• Students may make counting errors in the number of jumps on the number line
• Students may confuse the sign of the starting integer with the operation being performed

Next: 051. Adding and Subtracting Negative Integers