Division in the times tables · البرمجة مع مارتن

Families: three numbers, four questions

Last time you found out that division is multiplying backwards. Today we will get the most out of that.

Take three numbers: 4, 6 and 24. 4 × 6 = 24 holds. But from those same three numbers you get four questions at once:

4 × 6 = 24, 6 × 4 = 24, 24 ÷ 4 = 6, 24 ÷ 6 = 4.

That is called a fact family. Three numbers that belong together — like mum, dad and child. When you know one member of the family, you know them all.

What does that mean for you? That you have already learned division in the times tables — last week, when you practised times. Every fact you know (7 × 8 = 56) gives you two divisions for free (56 ÷ 7 = 8 and 56 ÷ 8 = 7).

Today we will build families and take them apart until you see them everywhere.

Ema plays a family game with dad. Dad says three numbers: “3, 7, 21.” Ema has to pour out all four questions as fast as she can: 3 × 7 = 21, 7 × 3 = 21, 21 ÷ 7 = 3, 21 ÷ 3 = 7. Then they swap roles. Ema tries to catch dad out: “2, 5, 11!” Dad frowns: “That is not a family. 2 × 5 is 10, not 11.” Ema laughs — you cannot catch him, but at least she found out that you know a family by multiplying: the middle numbers must give the big one.

💡

In a family the biggest number is always the times result — and in division it always stands at the start. 24 ÷ 6, never 6 ÷ 24.

How to make four from one fact

How to build a family:

Take a times fact you know: say 7 × 9 = 63.
Swap the factors: 9 × 7 = 63. (You already know that — order in multiplying does not matter.)
Flip to division: biggest number first, divide by one of the pair: 63 ÷ 7 = 9.
And by the other: 63 ÷ 9 = 7.

Four questions from one fact.

How to use it when calculating: when you get 63 ÷ 9, do not look for something new. Ask: “which family do 63 and 9 belong to?” You remember 7 × 9 = 63 — and the answer is 7.

Special families:

Dividing by one: 8 ÷ 1 = 8. Sharing eight things into one pile — the pile gets everything.
Dividing by itself: 8 ÷ 8 = 1. Eight things into eight piles — each gets one.
You never divide by zero. Sharing 8 sweets among zero children makes no sense — the question “how many does each get” has nobody. The write-up 8 ÷ 0 is forbidden and stays forbidden even at secondary school.

Building a family

Task: build a family from the numbers 6, 8, 48.

Check that it is a family: 6 × 8 = 48. Yes (six through five: 40 + 8).

Four questions:

6 × 8 = 48
8 × 6 = 48
48 ÷ 6 = 8
48 ÷ 8 = 6

Notice: 48 is the biggest and in the divisions it always stands first. The smaller numbers take turns in the divisions.

Division through a family

Example: 56 ÷ 8.

Step 1: I look for a family: 8 × what = 56?

Step 2: I remember the helper 5, 6, 7, 8 → 7 × 8 = 56.

Step 3: so 56 ÷ 8 = 7.

Check: 7 × 8 = 56. It fits.

One more: 42 ÷ 6. Family: 6 × 7 = 42. So 42 ÷ 6 = 7.

Spot a fake family

Which threes are families?

a) 4, 9, 36 — check: 4 × 9 = 36. Yes, a family. (9 × 4, 36 ÷ 4 = 9, 36 ÷ 9 = 4.)

b) 5, 6, 35 — check: 5 × 6 = 30, not 35. Fake! Into a family with 5 and 35 belongs 7, because 5 × 7 = 35.

c) 7, 7, 49 — check: 7 × 7 = 49. A family, but a special one: mum and dad are the same, so it only has two different questions: 7 × 7 = 49 and 49 ÷ 7 = 7.

⚠️

Always do the check by multiplying, not by another division. You work out 54 ÷ 6 = 9? Check: 9 × 6 = 54. If 9 × 6 came out 56, you know at once that the quotient 9 is wrong… or that you mixed up the times tables. You want to know both right away.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. البداية with ten. When it goes well, add more.

Divide through families. For each question say the whole family in your head.

Practise the questions

On paper

Families are best learned by writing. Four lines per family.

Write whole families (all four questions) for the threes: 3, 8, 24 — 7, 9, 63 — 6, 6, 36.
Fill the missing family member and write the whole family: 4, ☐, 32 — ☐, 9, 45 — 8, ☐, 72.
Work out through a family and write the check: 27 ÷ 3, 64 ÷ 8, 35 ÷ 7.
Find the cheat: 6, 7, 44 — why is that not a family? Which number would you swap, and for what?

Now you 💪

From the fact 7 × 9 = 63 you pour out the other three questions of the family.
You can work out 56 ÷ 8 through a family and back it with the check 7 × 8 = 56.
You know why you never divide by zero, and you can say it in your own words.

✅

Done when: You turn every division in the times tables into a family and check the result by multiplying.

What to take from this lesson

Three family numbers give four questions: two times and two divisions.
You do not solve division as a new thing — you look for a family you already know from the times tables.
The biggest number in the family always stands first in a division.
Check: quotient × divisor = dividend.
Dividing by one leaves the number as it is, dividing by itself gives 1, you never divide by zero.