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Making ten

Numbers · level 1, lesson 5 of 6

A question to start

Try to work out 8 + 5 in your head. Not with fingers. Just think.

Was it hard? Most people find 8 + 5 harder than 10 + 3. But here is a surprise: 8 + 5 and 10 + 3 are the same sum. They both make 13. If you can turn the hard sum into the easy one, you never have to count on 5 again.

The trick is the number ten. Ten is the friendliest number there is. In this lesson you will learn all the ways to make ten, and then use them to make hard sums easy.

The idea

Why ten is friendly

Adding to ten is easy: 10 + 3 is "ten and three", which we say thirteen. 10 + 6 is "ten and six", sixteen. You do not need to count at all. The answer is in the name. (You will see why in the next lesson.)

So if we can turn any sum into "ten and some more", adding becomes easy. To do that, we need to know, by heart, how to make ten.

All the ways to make ten

Two numbers that make ten together are called partners of ten, or a number bond to ten. Here they are, in a staircase. Each row has 10 squares: some orange and the rest blue.

1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 6 + 4 7 + 3 8 + 2 9 + 1

Read the rows: 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5, 6 + 4, 7 + 3, 8 + 2, 9 + 1. Every row makes 10, because every row has 10 squares. There are also 0 + 10 and 10 + 0: no orange, or no blue.

Look at the staircase. As you go down one row, one square turns from blue to orange. The orange part gets one bigger, and the blue part gets one smaller. The total stays 10. That is why the orange numbers go up 1, 2, 3, ... while the blue numbers go down 9, 8, 7, ...

Look again. The bottom half is the top half turned around: 6 + 4 is 4 + 6, and 9 + 1 is 1 + 9. So there are really only these five pairs to learn, plus 0 and 10:

1 and 9 · 2 and 8 · 3 and 7 · 4 and 6 · 5 and 5

Seeing the partners on a ten frame

The ten frame from lesson 2 shows partners of ten without any work. The counters are one partner. The empty boxes are the other partner. Together they fill the frame, which is ten.

So if a ten frame shows 7 counters, you can see straight away that there are 3 empty boxes. 7 and 3 make 10. Your fingers do the same thing: hold up 7 fingers, and 3 are down.

A number bond picture

Mathematicians often draw a number bond like this: the whole on top, and its two parts below, joined by lines.

1073

One picture holds four facts: 7 + 3 = 10, 3 + 7 = 10, 10 − 3 = 7 and 10 − 7 = 3. If you know the parts, you know the whole. If you know the whole and one part, you know the other part.

The big trick: make a ten first

Now back to 8 + 5. Put 8 on one ten frame and 5 on another.

8 + 5: the first frame has 2 empty boxes

The first frame has 2 empty boxes, because 8 and 2 make 10. So move 2 blue counters across to fill it.

a full ten and 3 more: 10 + 3 = 13

Now one frame is full (that is ten), and the other has 3. Ten and three is thirteen. Nothing was added or taken away; we just moved 2 counters. So 8 + 5 = 13.

In words: 8 + 5 is 8 + 2 + 3, because 5 is made of 2 and 3. And 8 + 2 is 10. So it is 10 + 3, which is 13.

Why does it work? Splitting the 5 into 2 and 3 does not change how many there are. Adding them in two goes, first 2 and then 3, gives the same total as adding all 5 at once. The 2 was chosen on purpose: it is the partner of 8.

The recipe is:

  1. Look at the bigger number. What is its partner of ten? (For 8, it is 2.)
  2. Split the other number into that partner and the rest. (5 is 2 and 3.)
  3. Make the ten, then add the rest. (10 and 3 is 13.)

Making ten when you take away

The trick also works backwards. Find 13 − 5.

13 is ten and 3. First take away the 3, to get down to 10. You still have to take away 2 more (because 5 is 3 and 2). 10 − 2 is 8, which you know from the partners of ten. So 13 − 5 = 8.

Going down to ten first, then past it, is just like going up to ten first, then past it. Ten is a stepping stone in the middle of the river.

Worked examples

1. A partner of ten. What goes with 6 to make 10?

Think of the staircase, or a ten frame with 6 counters: there are 4 empty boxes. 6 + 4 = 10.

2. Make ten to add. Find 9 + 4.

The partner of 9 is 1. Split 4 into 1 and 3. Then 9 + 1 = 10, and 10 + 3 = 13. So 9 + 4 = 13.

Adding 9 is always like this: take 1 to make ten, and the rest goes on top. 9 + 6 is 10 + 5 = 15. 9 + 7 is 10 + 6 = 16.

3. The other way round. Find 3 + 8.

Start from the bigger number, 8. Its partner is 2. Split 3 into 2 and 1. Then 8 + 2 = 10, and 10 + 1 = 11. So 3 + 8 = 11.

4. Make ten with 7. Find 7 + 6.

The partner of 7 is 3. Split 6 into 3 and 3. Then 7 + 3 = 10, and 10 + 3 = 13. So 7 + 6 = 13.

5. Going down through ten. Find 15 − 7.

15 is ten and 5. Take away the 5 first: 15 − 5 = 10. 7 is 5 and 2, so 2 more still have to go: 10 − 2 = 8. So 15 − 7 = 8. Check by adding: 8 + 7 is 8 + 2 + 5 = 10 + 5 = 15. ✓

Try it

  1. Fill in the partners of ten: 4 + ___ = 10, 7 + ___ = 10, 1 + ___ = 10, 5 + ___ = 10.
  2. A ten frame shows 6 counters. How many empty boxes?
  3. Use "make ten" to find 9 + 5. Show the split.
  4. Use "make ten" to find 8 + 6.
  5. Use "make ten" to find 6 + 7.
  6. Find 12 − 4 by going down to ten first.
  7. A bag holds 10 marbles. Some are red and some are blue. There are 3 more red than blue. How many of each? (Hint: look along the staircase.)
  8. Is 8 + 5 the same as 10 + 3? Explain with counters why it is.
Answers
  1. 4 + 6 = 10, 7 + 3 = 10, 1 + 9 = 10, 5 + 5 = 10.
  2. 4 empty boxes, because 6 and 4 make 10.
  3. The partner of 9 is 1, so split 5 into 1 and 4: 9 + 1 = 10, then 10 + 4 = 14.
  4. The partner of 8 is 2, so split 6 into 2 and 4: 8 + 2 = 10, then 10 + 4 = 14.
  5. Start with the bigger number, 7. Its partner is 3, so split 6 into 3 and 3: 7 + 3 = 10, then 10 + 3 = 13. (Or start with 6: its partner is 4, so split 7 into 4 and 3: 6 + 4 = 10, then 10 + 3 = 13. Same answer.)
  6. 12 is ten and 2. Take away 2 to reach 10. 4 is 2 and 2, so take away 2 more: 10 − 2 = 8. Check: 8 + 4 = 12. ✓
  7. Look for a row of the staircase where the two numbers are 3 apart. 4 and 6 are 2 apart. 3 and 7 are 4 apart. There is no row that is exactly 3 apart! So this bag cannot exist. (Why? When one colour goes up by 1, the other goes down by 1, so the gap always changes by 2: 0, 2, 4, 6, 8, 10. The gap is never 3.)
  8. Yes. Put 8 counters and 5 counters on two ten frames. Move 2 of the 5 across to fill the first frame. Now there is a full ten and 3 more. No counters were added or lost, only moved, so 8 + 5 = 10 + 3 = 13.

Common mistake

Splitting the wrong number. For 8 + 5, you split the 5, so that the 8 can become 10. If you split the 8 instead, you do not get a ten. Always ask: "which number is close to ten?", and split the other one.

Forgetting the rest. For 9 + 4, some people say "9 + 1 = 10, so the answer is 10". But only 1 of the 4 has been added. The other 3 still have to go on: 10 + 3 = 13.

Where it leads