# Lesson 5: Representing Subtraction

Let's subtract signed numbers.

## 5.1: Equivalent 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.

$2+ 3= 5$

$3 + 2 = 5$

$5 - 3 = 2$

$5 - 2 = 3$

$9+ (\text- 1)= 8$

$\text- 11+ x= 7$

## 5.2: Subtraction 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 $3 + {?} = 8$.
Tyler writes $8 - 3 = {?}$. 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 $(\text-8) - (\text-3) = {?}$ What is the unknown number? How do you know?

1. Match each diagram to one of these expressions:

$3 + 7$

$3 - 7$

$3 + (\text- 7)$

$3 - (\text- 7)$

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
row 1 $8 + (\text- 8)$
row 2 $8 - 8$
row 3 $8 + (\text-5)$
row 4 $8 - 5$
row 5 $8 + (\text-12)$
row 6 $8 - 12$
expression value
row 1 $\text-5 + 5$
row 2 $\text-5 - (\text-5)$
row 3 $\text-5 + 9$
row 4 $\text-5 - (\text-9)$
row 5 $\text-5 + 2$
row 6 $\text-5 - (\text-2)$

## Summary

The equation $7 - 5 = {?}$ is equivalent to ${?} + 5= 7$. The diagram illustrates the second equation.

Notice that the value of $7 + (\text-5)$ is 2.

We can solve the equation ${?} + 5= 7$ by adding -5 to both sides. This shows that $7 - 5= 7 + (\text- 5)$

Likewise, $3 - 5 = {?}$ is equivalent to ${?} + 5= 3$.

Notice that the value of $3 + (\text-5)$ is -2.

We can solve the equation ${?} + 5= 3$ by adding -5 to both sides. This shows that $3 - 5 = 3 + (\text- 5)$

In general:

$$a - b = a + (\text- b)$$

If $a - b = x$, then $x + b = a$. We can add $\text- b$ to both sides of this second equation to get that $x = a + (\text- b)$