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Putting together and taking away

Numbers · level 1, lesson 4 of 6

A question to start

Jack has 3 marbles. Then he wins 9 more. How many does he have now?

Sara works it out by counting every marble from the start: "one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve". That is twelve counting words.

Jack does it differently. He says "nine... ten, eleven, twelve". Just three words after the nine. He gets twelve too.

But wait. Jack had the 3 marbles first, and won the 9 after. Is he allowed to start at 9? Is that cheating? By the end of this lesson you will know why Jack's way is allowed, and why it is so much faster.

The idea

Adding means putting together

When we add, we put two groups together and find how many there are all together.

+ = 3 + 4 = 7

We write this as 3 + 4 = 7, and we read it "three plus four equals seven". The two new signs are:

  • + means "put together with" (we say "plus" or "add").
  • = means "is the same amount as" (we say "equals" or "makes").

The answer to an adding question has a name: the total (or the sum).

The sign = is like a balance. On one side is 3 + 4, on the other side is 7. They are the same amount, so the balance stays level. That is why we can also write it the other way round: 7 = 3 + 4. It is still true.

Adding in either order gives the same total

Look at the picture of 3 + 4 again, and walk around to the other side of the table. Now you see the 4 blue dots first and the 3 orange dots second. That is 4 + 3. But it is the same dots! Nothing was added or taken away.

So 3 + 4 = 4 + 3. This is true for every pair of numbers. When you put two groups together, it does not matter which group you pour in first; the total is the same.

Counting on

To find 6 + 3, you could count all the things from one. But you already know the first group has 6. So start at 6 and count on 3 more:

"six... seven, eight, nine." Hold up a finger for each number you count on: one finger for seven, one for eight, one for nine. When 3 fingers are up, stop. The answer is 9.

On the number line, counting on is hopping to the right, one hop for each one you add. Here is 9 + 3:

0123456789101112 start at 9, hop 3 to the right: 9 + 3 = 12

Count on from the bigger number

Now Jack's puzzle. 3 + 9 = 9 + 3, because the order does not change the total. So Jack is allowed to start with the 9 and count on 3: "nine... ten, eleven, twelve". That is only 3 counting words instead of 12.

Starting with the bigger number is always quicker, because then you count on the smaller number. Try 2 + 8 both ways. Starting at 2 you have to count on 8: "three, four, five, six, seven, eight, nine, ten". Starting at 8 you only count on 2: "nine, ten". Same answer, much less work. That is not cheating. It is being clever.

Subtracting means taking away

When we subtract, we start with a group and take some away. Then we find how many are left.

8 − 3 = 5

There were 8 apples. 3 were eaten (crossed out). 5 are left. We write 8 − 3 = 5 and read it "eight minus three equals five" or "eight take away three is five". The sign − means "take away".

On the number line, taking away is hopping to the left. Here is 12 − 5: start at 12, hop 5 to the left, land on 7. This is called counting back: "twelve... eleven, ten, nine, eight, seven".

0123456789101112 start at 12, hop 5 to the left: 12 − 5 = 7

Order does matter for taking away. 8 − 3 is 5, but 3 − 8 does not make sense yet: you cannot take 8 apples from a bowl of 3. (Much later, in level 4, you will meet numbers below zero, and then it will make sense.)

Subtracting also means "how many more?"

Here is a different kind of question. Tom has 9 stickers. Ana has 6 stickers. How many more does Tom have?

Nobody takes anything away here. But match them up: Ana's 6 stickers pair with 6 of Tom's, and Tom has 3 left over. So Tom has 3 more. And this is the same as 9 − 6 = 3.

Why is it the same? Think of Tom's 9 stickers. Cover up the 6 that have partners. What is left uncovered is the extra. Covering up 6 out of 9 is just like taking 6 away from 9. So "how many more?" is also a subtraction.

You can also find it by counting up: start at the smaller number, 6, and count up to 9: "seven, eight, nine". That is 3 numbers, so the difference is 3. Counting up is often easier than counting back, especially when the two numbers are close together, like 15 − 13 (count up "fourteen, fifteen": 2).

Adding and taking away undo each other

If you have 5 apples, get 3 more, and then 3 are eaten, you are back to 5. Adding 3 and then taking 3 away gets you back where you started. That is why every adding fact gives you taking-away facts for free: if 5 + 3 = 8, then 8 − 3 = 5. You will use this a lot in level 2.

Zero

Adding 0 changes nothing: 7 + 0 = 7 (you put together 7 things with no things). Taking away 0 changes nothing: 7 − 0 = 7. Taking away everything leaves nothing: 7 − 7 = 0.

Worked examples

1. Counting on. 8 birds sit on a fence. 4 more land. How many birds now?

Start at the bigger number, 8, and count on 4: "nine, ten, eleven, twelve". 8 + 4 = 12 birds.

2. Turning it round. Find 2 + 9.

2 + 9 = 9 + 2. Start at 9 and count on 2: "ten, eleven". 2 + 9 = 11.

3. Taking away. There are 10 balloons. 4 pop. How many are left?

Count back 4 from 10: "nine, eight, seven, six". 10 − 4 = 6 balloons are left. Check with fingers: hold up 10, put 4 down, 6 are still up.

4. How many more? A rover on Mars drives 7 metres on Monday and 11 metres on Tuesday. How many more metres on Tuesday?

This is a "how many more" question, so it is 11 − 7. Count up from 7 to 11: "eight, nine, ten, eleven". That is 4. The rover drove 4 more metres on Tuesday.

5. Undoing. You know 6 + 7 = 13. What is 13 − 7?

Taking away the 7 undoes adding the 7, so you are back at 6. 13 − 7 = 6.

Try it

  1. Find 5 + 2 and 2 + 5. What do you notice?
  2. Count on from the bigger number: 3 + 8.
  3. Count back: 11 − 3.
  4. There are 9 cookies. You eat 2. How many are left?
  5. Mia has 12 crayons and Leo has 8. How many more does Mia have? (Try counting up.)
  6. Fill in the gap: 7 + ___ = 10.
  7. You know 4 + 9 = 13. Write two taking-away facts that you now know for free.
  8. A frog sits on 5 on the number line. It hops 4 to the right and then 6 to the left. Where does it land?
Answers
  1. Both are 7. Adding in either order gives the same total.
  2. Start at 8, count on 3: "nine, ten, eleven". 11.
  3. Start at 11, count back 3: "ten, nine, eight". 8.
  4. 9 − 2: count back 2 from 9, "eight, seven". 7 cookies are left.
  5. 12 − 8. Count up from 8 to 12: "nine, ten, eleven, twelve" is 4 numbers. Mia has 4 more.
  6. Count up from 7 to 10: "eight, nine, ten" is 3 numbers. 7 + 3 = 10.
  7. 13 − 9 = 4 and 13 − 4 = 9. Taking away undoes adding.
  8. 4 to the right: 5 → 9. Then 6 to the left: 9 → 3. The frog lands on 3. (In one go: it went 2 further left than right, so it ends 2 to the left of 5.)

Common mistake

Counting the starting number again. For 8 + 4, some people say "eight, nine, ten, eleven" and stop at eleven, because they counted 4 words. But "eight" is where you start, not one of the 4 you add. Say "eight" with no finger up, then put up one finger for each number after it: "nine (1), ten (2), eleven (3), twelve (4)". The answer is 12.

Taking away the wrong way round. 9 − 6 and 6 − 9 are not the same. Always start with the group you have, and take away from it.

Where it leads