Let's subtract signed numbers.
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.
$2+ 3= 5$
$3 + 2 = 5$
$5 - 3 = 2$
$5 - 2 = 3$
$9+ (\text- 1)= 8$
$\text- 11+ x= 7$
Here is an unfinished number line diagram that represents a sum of 8.
Here are two more unfinished diagrams that represent sums.
For each diagram:
Match each diagram to one of these expressions:
$3 + 7$
$3 - 7$
$3 + (\text- 7)$
$3 - (\text- 7)$
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 | $\text-5 - (\text-2)$ |
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: $2 \oplus 1 = 3$ and $2\ominus 1 = 1$. The table shows some of the sums.
| $\oplus$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 |
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.
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 $x = a + (\text- b)$
