Lesson 7 of 10 · Grades 3 to 4
Division facts and fact families
Three numbers can make a whole family of facts. If you know 7, 8 and 56, you can build 7 × 8 = 56, 8 × 7 = 56, 56 ÷ 7 = 8 and 56 ÷ 8 = 7. One family, four facts, and every one of them is true.
One fact becomes three
A division fact is a multiplication fact seen from the other side. Start with 7 × 8 = 56. You can read the same three numbers backwards: 56 ÷ 7 = 8 and 56 ÷ 8 = 7. The product 56 sits at the top of the family, and the two factors sit below.
Three numbers that fit together this way make a fact family. A complete family has four facts: two multiplications and two divisions. The two multiplications are the turn-around pair, and each division undoes one of them.
Building a family
To build a fact family, start with a multiplication you know. Write it down with its two factors and its product. Then write the turn-around multiplication, and finish with the two divisions. For 4, 9 and 36, the family is 4 × 9 = 36, 9 × 4 = 36, 36 ÷ 4 = 9 and 36 ÷ 9 = 4.
Every whole multiplication fact gives a family like this, as long as none of the factors is zero. Multiplication by zero is a special case; it gets its own rule later in your studies.
Using the family to remember
Fact families shrink the amount of remembering you have to do. If a division fact slips your mind, rebuild it from the multiplication fact that stands beside it. Stuck on 48 ÷ 6? Ask which number times 6 makes 48. The answer is 8, so 48 ÷ 6 = 8 and the whole family is back.
The family also gives you a checking tool. When you work out a division, multiply the quotient by the divisor. If you get the dividend, the family fits together and the answer is right. That check is the whole subject of the final lesson in this course.
Useful tip
Write a fact family as a triangle of numbers rather than four sentences. The three numbers on the corners remind you that only three numbers are involved. If your family tries to use a fourth number, something has gone wrong.
Worked example
Build the whole fact family for the numbers 7, 8 and 56.
Step 1: write the multiplication with the two smaller numbers: 7 × 8 = 56.
Step 2: swap the factors: 8 × 7 = 56.
Step 3: divide the product by each factor: 56 ÷ 7 = 8 and 56 ÷ 8 = 7.
Step 4: check the family. Every fact uses the same three numbers, and no other whole number belongs to the family.
Answer: 7 × 8 = 56, 8 × 7 = 56, 56 ÷ 7 = 8 and 56 ÷ 8 = 7.
Remember
- A fact family uses three numbers: two factors and a product.
- It holds two multiplications and two divisions.
- Each division undoes one multiplication.
- Rebuild a forgotten division fact from its multiplication partner.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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Write the whole fact family for the numbers 3, 4 and 12.
Show answer
3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4 and 12 ÷ 4 = 3.
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Complete this fact: 56 ÷ 8 = ?
Show answer
7. Eight groups of 7 make 56.
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Complete this fact: 56 ÷ 7 = ?
Show answer
8. Seven groups of 8 make 56.
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One fact from the family of 5, 6 and 30 is 6 × 5 = ?
Show answer
30.
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Use 9 × 7 = 63 to write a division fact.
Show answer
63 ÷ 9 = 7, or 63 ÷ 7 = 9. Both come from the same family.
Puzzle · Fact family
Puzzle time
The triangle shows the fact family for 7, 8 and 56. Which equation does NOT belong to the family?
Show a hint
Check each equation against the three numbers on the triangle. One of the equations gives the wrong quotient.
Show the answer
Answer: 56 ÷ 7 = 9. 56 ÷ 7 = 8, not 9, so the last equation is false. The other three facts all use 7, 8 and 56 in ways that are true.