Connecting addition and subtraction
In this topic you will
- Use the connection between addition and subtraction to find an inverse calculation.
- Find the addition and subtraction fact family for a set of $3$ numbers.
- Use rounding to estimate a solution and use the inverse calculation to check a solution.
Key Words
- fact family
- inverse
Show Definitions
- fact family: A group of number facts that use the same $3$ numbers, for example two additions and two subtractions.
- inverse: An operation that undoes another operation; for example, addition and subtraction are inverses.
EXERCISES
1. Find the inverse calculations.
| Calculation | Inverse |
|---|---|
| $4 + 6 = 10$ | |
| $12 + 7 = 19$ | |
| $21 + 26 = 47$ | |
| $10 - 8 = 2$ | |
| $17 - 6 = 11$ | |
| $39 - 16 = 23$ |
👀 Show answer
Inverse calculations:
$4 + 6 = 10$ → $10 - 4 = 6$ and $10 - 6 = 4$
$12 + 7 = 19$ → $19 - 12 = 7$ and $19 - 7 = 12$
$21 + 26 = 47$ → $47 - 21 = 26$ and $47 - 26 = 21$
$10 - 8 = 2$ → $8 + 2 = 10$ and $2 + 8 = 10$
$17 - 6 = 11$ → $6 + 11 = 17$ and $11 + 6 = 17$
$39 - 16 = 23$ → $16 + 23 = 39$ and $23 + 16 = 39$
2. Find the inverse calculation to check each calculation.
$19 - 4 = 15$
$28 - 15 = 13$
$37 + 22 = 59$
$25 + 23 = 48$
👀 Show answer
$19 - 4 = 15$ → $15 + 4 = 19$
$28 - 15 = 13$ → $13 + 15 = 28$
$37 + 22 = 59$ → $59 - 37 = 22$ and $59 - 22 = 37$
$25 + 23 = 48$ → $48 - 25 = 23$ and $48 - 23 = 25$
3. Write the $6$ number bonds for $10$. Write the inverse calculation for each number bond for $10$.
👀 Show answer
Number bonds for $10$ and their inverses:
$0 + 10 = 10$ → $10 - 0 = 10$ and $10 - 10 = 0$
$1 + 9 = 10$ → $10 - 1 = 9$ and $10 - 9 = 1$
$2 + 8 = 10$ → $10 - 2 = 8$ and $10 - 8 = 2$
$3 + 7 = 10$ → $10 - 3 = 7$ and $10 - 7 = 3$
$4 + 6 = 10$ → $10 - 4 = 6$ and $10 - 6 = 4$
$5 + 5 = 10$ → $10 - 5 = 5$
4. Find the missing digit in each calculation.
a. $7\Box - 35 = 41$
b. $52 + \Box7 = 99$
c. $\Box3 - 42 = 21$
d. $45 + 3\Box = 78$
Talk to your partner about how you found the missing digit. Did you use the inverse calculation or something else?
👀 Show answer
a. $7\Box - 35 = 41$. Use inverse: $41 + 35 = 76$, so the missing digit is $6$.
b. $52 + \Box7 = 99$. Find the missing addend: $99 - 52 = 47$, so the missing digit is $4$.
c. $\Box3 - 42 = 21$. Use inverse: $21 + 42 = 63$, so the missing digit is $6$.
d. $45 + 3\Box = 78$. Find the missing addend: $78 - 45 = 33$, so the missing digit is $3$.
5. Write the fact family for this part whole representation.
👀 Show answer
Use the two parts and the whole to write the fact family: if the parts are $a$ and $b$ and the whole is $c$, then $a + b = c$, $b + a = c$, $c - a = b$, and $c - b = a$.
6. Complete each fact family house.
👀 Show answer
House with $3$, $6$, $9$:
$6 + 3 = 9$
$3 + 6 = 9$
$9 = 6 + 3$
$9 = 3 + 6$
$9 - 6 = 3$
$9 - 3 = 6$
$3 = 9 - 6$
$6 = 9 - 3$
House with $17$, $8$, $9$:
$8 + 9 = 17$
$9 + 8 = 17$
$17 = 8 + 9$
$17 = 9 + 8$
$17 - 8 = 9$
$17 - 9 = 8$
$9 = 17 - 8$
$8 = 17 - 9$
Think like a Mathematician
The wind has blown the roof away from a fact family house! Here is part of the roof.
What do you think?
Show Answers
- 1: A fact family uses $3$ numbers. If the largest number is $15$ (the whole) and one part is $6$, then the missing part is $15 - 6 = 9$. The fact family is: $6 + 9 = 15$, $9 + 6 = 15$, $15 - 6 = 9$, $15 - 9 = 6$.
- 2: If $15$ is the whole and $6$ is a part, then the missing part must be $9$, so the fact family is fixed. The only “more than one” possibility is swapping the two parts (writing $6 + 9$ or $9 + 6$), but it is the same fact family.
EXERCISES
7. The number sentence $40 + 30 = 70$ shows two complements of $70$. Write the $4$ subtraction calculations using the same set of three numbers.
👀 Show answer
$70 - 40 = 30$
$70 - 30 = 40$
$30 = 70 - 40$
$40 = 70 - 30$
8. Estimate, solve and use the inverse calculation to check.
Quick Math Tip
Estimate by rounding to tens: Round each number to the nearest $10$ first to get a quick estimate of the answer, then do the exact calculation to check.
a. $42 + 37$
b. $89 - 62$
c. $77 - 34$
d. $54 + 33$
👀 Show answer
a. Estimate: $42 \approx 40$, $37 \approx 40$, so $40 + 40 = 80$.
Exact: $42 + 37 = 79$.
Inverse check: $79 - 42 = 37$ (or $79 - 37 = 42$).
b. Estimate: $89 \approx 90$, $62 \approx 60$, so $90 - 60 = 30$.
Exact: $89 - 62 = 27$.
Inverse check: $62 + 27 = 89$.
c. Estimate: $77 \approx 80$, $34 \approx 30$, so $80 - 30 = 50$.
Exact: $77 - 34 = 43$.
Inverse check: $34 + 43 = 77$.
d. Estimate: $54 \approx 50$, $33 \approx 30$, so $50 + 30 = 80$.
Exact: $54 + 33 = 87$.
Inverse check: $87 - 54 = 33$ (or $87 - 33 = 54$).