Inverse operations & fact families
Theory
Three numbers such as \(4\), \(6\) and \(24\) make a fact family: \(4 \times 6 = 24\), \(6 \times 4 = 24\), \(24 \div 4 = 6\) and \(24 \div 6 = 4\). Knowing one fact gives you all four, because multiplication and division are inverse operations. This Year 5 maths page shows fact families with arrays and triangles, for multiplication and division and for addition and subtraction.
An array shows multiplication as rows and columns. \(4\) rows of \(6\) is \(4 \times 6 = 24\); turned on its side it is \(6 \times 4 = 24\). The order of the factors does not change the product.
The same array shows division: \(24\) shared into \(4\) rows gives \(6\) in each row, so \(24 \div 4 = 6\); shared into \(6\) columns gives \(4\), so \(24 \div 6 = 4\).
A fact family is the set of related facts made from three numbers. For multiplication and division there are four facts; addition and subtraction work the same way: \(8 + 15 = 23\), \(15 + 8 = 23\), \(23 - 8 = 15\), \(23 - 15 = 8\).
Because division undoes multiplication, you can check a division by multiplying: \(72 \div 6 = 12\) is right because \(12 \times 6 = 72\).
One multiplication fact gives four facts:
How to write a fact family
- Multiply the two smaller numbers both ways round.
- Divide the product by each of the smaller numbers.
How to find a missing number
- Find the fact family it belongs to.
- Use the related fact: \(\;\rule{1.2cm}{0.4pt}\; \times 7 = 77\) is the same family as \(77 \div 7\).
| \(4 \times 6\) | \(=\) | \(24\) |
| \(6 \times 4\) | \(=\) | \(24\) |
\(24\) counters; \(4 \times 6 = 24\) and \(6 \times 4 = 24\).
| \(8 \times 11\) | \(=\) | \(88\) |
| \(11 \times 8\) | \(=\) | \(88\) |
| \(88 \div 8\) | \(=\) | \(11\) |
| \(88 \div 11\) | \(=\) | \(8\) |
All four facts use the same three numbers.
| \(23 - 8\) | \(=\) | \(15\) |
| \(23 - 15\) | \(=\) | \(8\) |
\(23 - 8 = 15\) and \(23 - 15 = 8\).
| \(12 \times 6\) | \(=\) | \(72\) |
The check gives 72, so Kim is right.
(i) \(\Box \times 7 = 77\)
(ii) \(78 \div \Box = 6\)
(iii) \(\Box \div 5 = 14\)
| \(77 \div 7\) | \(=\) | \(11\) |
| \(78 \div 6\) | \(=\) | \(13\) |
| \(14 \times 5\) | \(=\) | \(70\) |
(i) \(11\)
(ii) \(13\)
(iii) \(70\)
| \(40 \div 8\) | \(=\) | \(5\) |
\(40 \div 8 = 5\); the others use different numbers or operations.
(i) How many are in each row?
(ii) If they are put into \(6\) equal columns instead, how many are in each column?
| \(24 \div 4\) | \(=\) | \(6\) |
| \(24 \div 6\) | \(=\) | \(4\) |
(i) \(6\)
(ii) \(4\)
Common pitfalls
Frequently asked questions
What is a fact family?
The related facts made from three numbers. For \(8\), \(11\) and \(88\): \(8 \times 11 = 88\), \(11 \times 8 = 88\), \(88 \div 8 = 11\), \(88 \div 11 = 8\).
How does an array show a fact family?
Rows times columns gives the total; the total divided by the rows gives the columns, and divided by the columns gives the rows.
How do you check a division?
Multiply the answer by the number you divided by: \(12 \times 6 = 72\) checks \(72 \div 6 = 12\).
Do addition and subtraction have fact families?
Yes: \(8 + 15 = 23\), \(15 + 8 = 23\), \(23 - 8 = 15\) and \(23 - 15 = 8\).
How does a fact family help find a missing number?
\(\Box \times 7 = 77\) belongs with \(77 \div 7 = 11\), so the missing number is \(11\).