# Lesson 5Representing Subtraction

Let's subtract signed numbers.

### Learning Targets:

• I can explain the relationship between addition and subtraction of rational numbers.
• I can use a number line to subtract positive and negative numbers.

## 5.1Equivalent Equations

For the equations in the second and third columns, write two more equations using the same numbers that express the same relationship in a different way. If you get stuck, consider looking at the examples in the first column.

## 5.2Subtraction with Number Lines

1. Here is an unfinished number line diagram that represents a sum of 8.

1. How long should the other arrow be?
2. For an equation that goes with this diagram, Mai writes .
Tyler writes . Do you agree with either of them?
3. What is the unknown number? How do you know?
2. Here are two more unfinished diagrams that represent sums.

For each diagram:

1. What equation would Mai write if she used the same reasoning as before?
2. What equation would Tyler write if he used the same reasoning as before?
3. How long should the other arrow be?
4. What number would complete each equation? Be prepared to explain your reasoning.
3. Draw a number line diagram for What is the unknown number? How do you know?

1. Match each diagram to one of these expressions:

2. Which expressions in the first question have the same value? What do you notice?
3. Complete each of these tables. What do you notice?

expression value
expression value

### Are you ready for more?

It is possible to make a new number system using only the numbers 0, 1, 2, and 3. We will write the symbols for adding and subtracting in this system like this: and . The table shows some of the sums.

0 1 2 3
0 0 1 2 3
1 1 2 3 0
2 2 3 0 1
3
1. In this system, and . How can you see that in the table?
2. What do you think should be?
4. What do you think should be?
6. Can you think of any uses for this number system?

## Lesson 5 Summary

The equation is equivalent to . The diagram illustrates the second equation.

Notice that the value of is 2.

We can solve the equation by adding -5 to both sides. This shows that

Likewise, is equivalent to .

Notice that the value of is -2.

We can solve the equation  by adding -5 to both sides. This shows that

In general:

If , then . We can add to both sides of this second equation to get that

## Lesson 5 Practice Problems

1. Write each subtraction equation as an addition equation.

2. Find each difference. If you get stuck, consider drawing a number line diagram.

3. A restaurant bill is $59 and you pay$72. What percentage gratuity did you pay?
4. Find the solution to each equation mentally.

5. One kilogram is 2.2 pounds. Complete the tables. What is the interpretation of the constant of proportionality in each case?

pounds kilograms
2.2 1
11
5.5
1

______ kilogram per pound

kilograms pounds
1 2.2
7
30
0.5

______ pounds per kilogram