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Fact families and mental tricks

Numbers · level 2, lesson 3 of 7

A question to start

How many adding facts are there from 0 + 0 up to 10 + 10? Each of the two numbers can be anything from 0 to 10, which is 11 choices each, so there are 11 rows of 11 facts: 121 facts. And there are just as many taking-away facts. That is a lot to remember!

Here is the good news. You do not need to remember them one by one. If you know one fact, you get three more for free. And with a handful of tricks, most facts can be worked out in a second from facts you already know. This lesson shows you those tricks, and why each one works, so you can trust them.

The idea

Fact families: one fact gives four

Take the fact 7 + 8 = 15. Think of it as a whole (15) made of two parts (7 and 8). We can draw it as a fact triangle: the whole at the top, the parts at the bottom.

1578 7 + 8 = 158 + 7 = 15 15 − 8 = 715 − 7 = 8

From this one triangle come four facts:

  • 7 + 8 = 15 and 8 + 7 = 15 (the two parts, in either order, make the whole);
  • 15 − 8 = 7 and 15 − 7 = 8 (the whole, take away one part, leaves the other part).

These four facts are called a fact family, because they all use the same three numbers.

Why do they belong together? Picture 15 counters: 7 red and 8 blue. Count all of them: that is 7 + 8 = 15. Take away the blue ones: 7 red are left, so 15 − 8 = 7. Take away the red ones: 8 blue are left, so 15 − 7 = 8. It is the same picture every time. Adding joins the parts; subtracting splits them apart again.

Think adding to subtract

So every subtraction can be turned into a missing-part question. To find 15 − 8, ask: "8 plus what makes 15?" You know 8 + 7 = 15, so 15 − 8 = 7.

This is often easier than counting back eight steps. It also means that if you know your adding facts, you already know your subtracting facts.

Doubles

A double is a number added to itself. Many people find doubles easy to remember, perhaps because they come in pairs, like two hands or two feet.

Double 1 + 1 2 + 2 3 + 3 4 + 4 5 + 5 6 + 6 7 + 7 8 + 8 9 + 9 10 + 10
Total 2 4 6 8 10 12 14 16 18 20

Notice that every double is an even number: you can always share it into two equal groups, because that is how it was made.

Near doubles

What is 6 + 7? It is almost a double. 7 is one more than 6, so

6 + 7 = 6 + 6 + 1 = 12 + 1 = 13.

6 + 6 = 12 and 1 more is 13

The picture shows why: 6 dots on top, 7 below. Pair them up and you have a double, 6 + 6, with one dot left over.

You can also go down instead of up: 6 + 7 = 7 + 7 − 1 = 14 − 1 = 13. Both ways give 13.

Near doubles work whenever two numbers are next to each other: 4 + 5, 7 + 8, 8 + 9. If they are two apart, like 6 + 8, take one from the bigger and give it to the smaller: 6 + 8 = 7 + 7 = 14. (Nothing was lost, it just moved.)

Adding 9: add 10, then take 1 away

Adding 10 is easy: only the tens digit changes (46 + 10 = 56). Adding 9 is nearly the same, because 9 is one less than 10.

So to add 9, add 10, then take 1 away: 46 + 9 = 56 − 1 = 55.

465556 + 10− 1

Why does it work? Adding 10 adds one too many, since we only wanted 9. Taking 1 away fixes the extra one. The same trick works for 19 (add 20, take 1), 29 (add 30, take 1), and even 99 (add 100, take 1): 245 + 99 = 345 − 1 = 344.

Subtracting 9: take 10 away, then add 1 back

To find 53 − 9: take away 10 to get 43. But we only had to take away 9, so we took away one too many. Give it back: 43 + 1 = 44.

Be careful with the direction. When adding 9, you add 10 and take 1 away. When subtracting 9, you take 10 away and add 1 back. Always ask: "Did I do too much, or too little? Then fix it."

Make a friendly number

The 9 trick is one example of a bigger idea: change a number into a friendly one, then fix it. Friendly numbers are the ones that end in 0, like 20, 50 or 300.

  • 38 + 25: move 2 from the 25 over to the 38. Now it is 40 + 23 = 63. (The total did not change: one number went up by 2, the other went down by 2.)
  • 64 − 28: 28 is 2 less than 30. 64 − 30 = 34, but we took away 2 too many, so add 2 back: 36.

Estimating to check

An estimate is a quick, rough answer: not exact, but close. It is a great way to catch mistakes. The usual way is to round each number to the nearest ten first.

To round 38 to the nearest ten: 38 is between 30 and 40, and it is closer to 40. So 38 rounds to 40. A number ending in 5, like 45, is exactly halfway; the usual rule is to round it up, to 50.

Then: 38 + 45 is about 40 + 50 = 90. The exact answer is 83. That is fairly close, so 83 makes sense. If you had written 183 or 73, the estimate would warn you that something went wrong.

Worked examples

1. A fact family. Write the fact family for 6, 9 and 15.

6 + 9 = 15, 9 + 6 = 15, 15 − 9 = 6, 15 − 6 = 9.

2. Think adding. Find 17 − 9.

"9 plus what makes 17?" 9 + 8 = 17 (9 + 1 = 10, then 7 more is 17, so 1 + 7 = 8). So 17 − 9 = 8.

3. Near double. Find 8 + 9.

8 + 8 = 16, and 9 is one more than 8, so 8 + 9 = 16 + 1 = 17.

4. Adding 9. Find 67 + 9.

67 + 10 = 77, then take 1 away: 76.

5. Subtracting 19. Find 82 − 19.

19 is one less than 20. 82 − 20 = 62. We took away one too many, so add 1 back: 63. Check: 63 + 19 = 63 + 20 − 1 = 83 − 1 = 82. ✓

6. Estimating. Ella adds 47 + 36 and gets 73. Is that sensible?

Round: 47 is about 50, 36 is about 40. 50 + 40 = 90. Ella's 73 is far from 90, so something is probably wrong. The real answer: 40 + 30 = 70, 7 + 6 = 13, total 83. (She forgot to carry the ten.)

Try it

  1. Write the whole fact family for 5, 8 and 13.
  2. Find 14 − 6 by thinking "6 plus what makes 14?".
  3. Use a double to find 7 + 8 and 9 + 10.
  4. Use "add 10, take 1" to find 35 + 9 and 128 + 9.
  5. Use "take 10, add 1" to find 71 − 9.
  6. Find 56 + 29 using a friendly number.
  7. Round each number to the nearest ten, then estimate: 63 + 28. Then work out the exact answer.
  8. Sam says: "To add 9, I add 10 and add 1." Try it on 20 + 9. What goes wrong, and why?
Answers
  1. 5 + 8 = 13, 8 + 5 = 13, 13 − 8 = 5, 13 − 5 = 8.
  2. 6 + 8 = 14 (6 + 4 = 10, then 4 more), so 14 − 6 = 8.
  3. 7 + 8: 7 + 7 = 14, plus 1 is 15. 9 + 10: 9 + 9 = 18, plus 1 is 19 (or 10 + 10 = 20, minus 1).
  4. 35 + 10 = 45, take 1: 44. 128 + 10 = 138, take 1: 137.
  5. 71 − 10 = 61, add 1 back: 62. Check: 62 + 9 = 71. ✓
  6. 29 is one less than 30. 56 + 30 = 86, take 1 away: 85.
  7. 63 is about 60, 28 is about 30. Estimate 60 + 30 = 90. Exact: 63 + 28 = 63 + 30 − 2 = 93 − 2 = 91. The estimate is close. ✓
  8. Sam gets 20 + 10 + 1 = 31, but 20 + 9 = 29. Adding 10 already adds one too many, so he must take 1 away, not add another 1. His way adds 11, not 9.

Common mistake

Fixing in the wrong direction. With "add 10 and fix", it is easy to fix the wrong way, like Sam in question 8. Before you fix, ask: "Did I add too much or too little?" If you added 10 when you only wanted 9, you added too much, so take 1 away. If you took 10 away when you only wanted to take 9, you took too much, so add 1 back.

Mixing up the fact family. 8 − 15 is not in the family of 7, 8 and 15. In a taking-away fact, the biggest number, the whole, always comes first.

Where it leads