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Missing numbers with bigger numbers

Algebra · level 2, lesson 1 of 6

A question to start

Here are three puzzles. In each one, a number is hiding in the box.

  • 8 + ? = 15
  • 20 − ? = 12
  • ? − 6 = 9

They look alike, but they are three different kinds of puzzle. In one of them you must add to find the box, even though the sentence has a take-away sign in it. Which one? By the end of this lesson you will have one simple question that cracks all three, and many more with bigger numbers. (The answers are in the worked examples.)

The idea

The box is a mystery number

In level 1 the mystery number hid under a cup. Now it hides in a box, written ? (some books draw an empty square). The box stands for one number. Your job is to find the number that makes the sentence true: both sides of the = the same amount.

Every adding fact has a whole and two parts

Think of the fact 8 + 7 = 15. There is a whole, 15, made of two parts, 8 and 7. We can draw this as a fact triangle: the whole at the top, the two parts at the bottom.

158? whole on top, parts below 15 (whole) 8 ? the two parts are as long as the whole

The three numbers make a fact family of four facts (see Fact families in Numbers):

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

The one question that cracks every puzzle

Every missing-number puzzle with + or − is about a whole and two parts. So ask:

Is the box the whole, or is it a part?

  • If the box is a part, find it by taking the part you know away from the whole (or by counting up from the part to the whole).
  • If the box is the whole, find it by adding the two parts.

How do you know which number is the whole? In an adding sentence like 8 + ? = 15, the two numbers being added are the parts, and the number on the other side is the whole. In a take-away sentence like 20 − ? = 12, the number you start with (before the −) is the whole, and the other two are the parts.

Puzzle 1: 8 + ? = 15 (the box is a part)

The whole is 15. One part is 8. The box is the other part. So the box is 15 − 8.

A good way to find it is to count up, jumping through 10. From 8, jump 2 to reach 10. From 10, jump 5 to reach 15. Altogether you jumped 2 and 5, which is 7.

8 10 15 jump 2 jump 5 from 8 to 15 is 2 and 5 more: 7

The box is 7. Check: 8 + 7 = 15 ✓.

Why jump to 10 first? Because 10 is a friendly number. You know straight away that 8 needs 2 more to make 10 (they are partners of 10), and that 10 needs 5 more to make 15 (fifteen is "ten and five"). Two easy jumps instead of seven small counts.

Puzzle 2: 20 − ? = 12 (the box is a part)

The whole is 20, because that is what we start with. Some is taken away (the box), and 12 is left. The part taken away and the part left make up the whole. So the box and 12 make 20: this is the same as 12 + ? = 20.

Count up from 12 to 20. From 12, jump 8 to reach 20 (2 and 8 make 10, so 12 and 8 make 20). The box is 8. Check: 20 − 8 = 12 ✓.

Puzzle 3: ? − 6 = 9 (the box is the whole)

Here the box is before the take-away sign. So the box is what we start with: the whole. 6 is taken away and 9 is left. The two parts are 6 and 9, and the whole is the two parts put together. So the box is 9 + 6.

9 and 6: jump 1 from 9 to 10, then 5 more to 15. The box is 15. Check: 15 − 6 = 9 ✓.

This is the puzzle that needs adding, even though it has a take-away sign in it. If you just took 6 from 9 you would get 3, and 3 − 6 does not make 9: you cannot even take 6 away from 3. Asking "is the box the whole?" saves you.

Bigger numbers, same idea

The question works for any numbers. 25 + ? = 40: the whole is 40, one part is 25. Count up from 25: jump 5 to 30, then 10 to 40. The box is 5 and 10, which is 15. Check: 25 + 15 = 40 ✓.

25 30 40 jump 5 jump 10 from 25 to 40 is 5 and 10 more: 15

50 − ? = 32: the whole is 50 and the part left is 32. Count up from 32: jump 8 to 40, then 10 to 50. The box is 8 and 10, which is 18. Check: 50 − 18 = 32 ✓ (50 take away 10 is 40, take away 8 more is 32).

Why counting up gives the same answer as taking away

Look at the number line for 8 and 15. The jumps measure the gap between 8 and 15. Taking 8 away from 15 also finds the gap: it tells you how much of 15 is left once the first 8 is gone. The gap is the same whether you walk it from left to right (counting up) or from right to left (counting back). So 15 − 8 and "8 plus what makes 15" are always the same number.

Worked examples

1. The three starting puzzles.

  • 8 + ? = 15: the box is a part. 15 − 8 = 7. Check: 8 + 7 = 15 ✓.
  • 20 − ? = 12: the box is a part. Count up from 12 to 20: 8. Check: 20 − 8 = 12 ✓.
  • ? − 6 = 9: the box is the whole. Add the parts: 9 + 6 = 15. Check: 15 − 6 = 9 ✓. This is the one that needs adding.

2. The box first. ? + 9 = 14. Adding in either order gives the same total, so this is 9 + ? = 14. The box is a part. From 9, jump 1 to 10, then 4 to 14: 1 and 4 make 5. Check: 5 + 9 = 14 ✓.

3. Using a double. 7 + ? = 14. You may know the double 7 + 7 = 14. So the box is 7. Check: 7 + 7 = 14 ✓. Knowing doubles makes puzzles near them quick too: 7 + ? = 15 must be one more, 8, because 15 is one more than 14.

4. Tens. 30 + ? = 70. Think in tens: 3 tens and how many tens make 7 tens? 4 tens. The box is 40. Check: 30 + 40 = 70 ✓.

5. A two-digit take-away. ? − 14 = 21. The box is before the −, so it is the whole. Add the parts: 21 + 14. 21 and 10 is 31, and 4 more is 35. The box is 35. Check: 35 − 14 = 21 ✓ (35 take away 10 is 25, take away 4 more is 21).

Try it

  1. 6 + ? = 13
  2. ? + 8 = 17
  3. 16 − ? = 9
  4. ? − 7 = 8
  5. 40 + ? = 100
  6. 35 − ? = 20
  7. ? − 25 = 25
  8. Ravi says the answer to ? − 5 = 12 is 7. Check his answer, explain his mistake, and find the right answer.
Answers
  1. 7. The box is a part. From 6, jump 4 to 10, then 3 to 13: 4 and 3 make 7. Check: 6 + 7 = 13 ✓.
  2. 9. The same as 8 + ? = 17. From 8, jump 2 to 10, then 7 to 17: 2 and 7 make 9. Check: 9 + 8 = 17 ✓.
  3. 7. The whole is 16, 9 is left, so the box and 9 make 16. From 9, jump 1 to 10, then 6 to 16: 1 and 6 make 7. Check: 16 − 7 = 9 ✓.
  4. 15. The box is the whole. Add the parts: 8 + 7. That is a near double: 7 + 7 is 14, and one more is 15. Check: 15 − 7 = 8 ✓.
  5. 60. In tens: 4 tens and 6 tens make 10 tens, which is 100. Check: 40 + 60 = 100 ✓.
  6. 15. The whole is 35 and 20 is left. From 20, jump 10 to 30, then 5 to 35: 15. Check: 35 − 15 = 20 ✓.
  7. 50. The box is the whole. Add the parts: 25 + 25 = 50 (a double: two quarters of a dollar make half a dollar). Check: 50 − 25 = 25 ✓.
  8. Check: 7 − 5 is 2, not 12. So 7 is wrong. Ravi took 5 from 12. But the box is before the − sign, so it is the whole, and 5 and 12 are the parts. Add them: 12 + 5 = 17. Check: 17 − 5 = 12 ✓.

Common mistake

Always taking away the two numbers you see. In ? − 5 = 12, taking 5 from 12 gives 7, but 7 − 5 is 2, not 12. The box is the whole here, so you must add. Always ask "is the box the whole or a part?" first.

Not checking. Put your answer in the box and do the sentence. A check costs a few seconds and catches the mistake above every time.

Where it leads