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)

b)

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)

b)

c) Compare the two results.

Worked Example 3

Use a number line to add negative integers:

a)

b)

c)

Worked Example 4

Use a number line to subtract negative integers:

a)

b)

c)

Worked Example 5

Describe the movement on the number line and Solve the result:

a) Start at , then add

b) Start at , then subtract

c) Start at , then add

Worked Example 6

Rewrite each calculation and then Evaluate Use a number line:

a)

b)

c)

Problems

Problem 1

Use a number line to show that adding a negative integer is the same as subtracting a positive integer:

a)

b)

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)

b)

c) Compare the two results.

Problem 3

Use a number line to add negative integers:

a)

b)

c)

Problem 4

Use a number line to subtract negative integers:

a)

b)

c)

Problem 5

Describe the movement on the number line and find the result:

a) Start at , then add

b) Start at , then subtract

c) Start at , then add

Problem 6

Rewrite each calculation and then evaluate using a number line:

a)

b)

c)

Exercises

Understanding and Fluency

Exercise 1.

Rewrite each addition of a negative integer as subtraction, then evaluate:

a)

b)

c)

Exercise 2.

Rewrite each subtraction of a negative integer as addition, then evaluate:

a)

b)

c)

Exercise 3.

Use a number line to add negative integers:

a)

b)

c)

Exercise 4.

Use a number line to subtract negative integers:

a)

b)

c)

Exercise 5.

Evaluate:

a)

b)

c)

Exercise 6.

Evaluate:

a)

b)

c)

Exercise 7.

Describe the movement on the number line and find the result:

a) start at , move left

b) start at , move right

c) start at , move left

Exercise 8.

Describe the movement on the number line and find the result:

a) start at , move right

b) start at , move left

c) start at , move left

Reasoning

Exercise 9.

Explain why gives the same result as .

Exercise 10.

A student says . Explain the mistake.

Exercise 11.

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

Exercise 12.

A student says because “two negatives make a positive”. Explain why this is incorrect.

Problem-solving

Exercise 13.

A temperature is and then changes by . What is the new temperature?

Exercise 14.

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

Exercise 15.

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

Exercise 16.

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

Exercise 17.

A game score is , then changes by . What is the new score?

Exercise 18.

A submarine is at m and then moves up 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 with
• 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 and mean the same thing
• Students may count jumps incorrectly on the number line when moving across zero

Next: 052e. Multiplying and Dividing Integers