Chapter 4 of 7

All 7 chapters
  1. 1Number sequences, readunderstood
  2. 2Growing shape patterns, readunderstood
  3. 3Number machines, readunderstood
  4. 4Missing numbers, readunderstood
  5. 5Odd and even rules, readunderstood
  6. 6Square and triangle numbers, readunderstood
  7. 7Jumping far ahead, readunderstood

M2.8.4 Chapter 4

Missing numbers

What is the missing number in 36 + ? = 50?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the missing number in 36 + ? = 50
  • Find the missing number in ? × 6 = 42
  • Use the opposite operation to check a missing number

Here is a sum with a hidden number: 36 + ? = 50.

The ? stands for a missing number. Think: 36 plus what makes 50?

In the pictures, a hatched box with a ? marks it.

Draw it first

See the missing part

  1. A long box labelled 50. Under it, a box labelled 36 fills most of the width. A hatched box labelled ? fills the rest. In step 4 the ? becomes 14, and a label says 50 − 36 = 14.50

    1 Draw the whole, 50, as one long box.

  2. A long box labelled 50. Under it, a box labelled 36 fills most of the width. A hatched box labelled ? fills the rest. In step 4 the ? becomes 14, and a label says 50 − 36 = 14.5036

    2 Under it, draw the part you know. It is 36.

  3. A long box labelled 50. Under it, a box labelled 36 fills most of the width. A hatched box labelled ? fills the rest. In step 4 the ? becomes 14, and a label says 50 − 36 = 14.5036?

    3 The rest of the box is the missing part. Mark it ?.

  4. A long box labelled 50. Under it, a box labelled 36 fills most of the width. A hatched box labelled ? fills the rest. In step 4 the ? becomes 14, and a label says 50 − 36 = 14.50361450 − 36 = 14

    4 Take the part you know from the whole. 50 − 36 = 14.

The whole is 50. Take away the part you know. That leaves the missing part.

Worked example

Find the missing number. 36 + ? = 50.

  1. The whole is 50. The part you know is 36.
  2. Take the known part away. 50 − 36 = 14.
  3. Check. Put 14 where the ? is. 36 + 14 = 50.

Answer: 14

Times works the same way. Look at ? × 6 = 42. What times 6 makes 42?

Use the opposite operation. It is the job that undoes another job.

A table with the heads Job and Opposite. The job + has the opposite −. The job − has the opposite +. The job × has the opposite ÷. The job ÷ has the opposite ×.JobOpposite+−−+×÷÷×
Each job and the job that undoes it.
A long box labelled 42. Under it, seven boxes in a row, each labelled 6. Seven sixes fill the long box.426666666
How many sixes fit in 42? Divide 42 by 6.

Worked example

Find the missing number in ? × 6 = 42.

  1. The job is × 6. Its opposite is ÷ 6.
  2. Divide 42 by 6. 42 ÷ 6 = 7.
  3. Check. Put 7 where the ? is. 7 × 6 = 42.

Answer: 7

See it three ways

With things

Put out 9 stones. You need 15 in all. Add stones until you have 15. You add 6.

As a picture

Fifteen stones in two rows. The first 9 stones are filled in. The other 6 are open.

In symbols

9 + ? = 15. The opposite of + is −. 15 − 9 = 6.

Try it

Find the hidden stones

You need: 15 stones, beans or seeds, a cup or a bowl, a pencil and paper.

  1. Count out 15 stones. Write 15.
  2. Cover some stones with the cup. Do not count them.
  3. Count the stones you can see. Write that number, then + ? = 15.
  4. Take the number you can see away from 15. That is your answer.
  5. Lift the cup and count the hidden stones.

What you will see: Your answer matches the hidden stones. Take away finds the missing number in a sum with +.

If you cannot try it, this is what happens.
Nine stones in view beside a hatched box labelled ?, which hides the rest. Under them, a label says 9 + ? = 15.?9 + ? = 15

A wrong idea many people have

In 36 + ? = 50, you see a +. So the ? is 36 + 50 = 86.

Why it feels right: You see a plus sign, so your head says add.

The test that breaks it: Put 86 where the ? is. 36 + 86 = 122, not 50. The missing number must be less than 50.

What is true: The total, 50, is on the other side of the =. Take the part you know away: 50 − 36 = 14.

Key words

missing number
a hidden number, which you find.
opposite operation
the job that undoes another job.

Back to your guess

The answer is B: 14. Take 36 away from 50.

36 + 14 = 50. The whole is 50 and you know one part, 36. Take the part away from the whole. 50 − 36 = 14.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Find the missing number. 29 + ? = 71

    See the answer

    42

    Take the known part from the whole. 71 − 29 = 42. Check: 29 + 42 = 71.

  2. Find the missing number in ? × 8 = 56.

    See the answer

    7

    The opposite of × is ÷. 56 ÷ 8 = 7. Check: 7 × 8 = 56.

  3. Amina says the ? in 25 + ? = 60 is 85. Is she right?

    Tap your answer.

    1. Yes, that is right. The whole is 60. 60 − 25 = 35. Check: 25 + 35 = 60.

    2. Not this one. You added the two numbers you can see. Put 85 back to check: 25 + 85 is far more than 60. Try another.

    3. Not this one. You took the smaller digit from the bigger in each place. Put 45 back: 25 + 45 = 70, not 60. Try another.

    See the answer

    No. 25 + 85 = 110. The missing number is 35.

    The whole is 60. 60 − 25 = 35. Check: 25 + 35 = 60.

    • If you chose B: You added the two numbers you can see. Put 85 back to check: 25 + 85 is far more than 60.
    • If you chose C: You took the smaller digit from the bigger in each place. Put 45 back: 25 + 45 = 70, not 60.

  4. Tariq has 48 beans. He gets some more and has 72. How many more did he get?

    See the answer

    24 beans

    48 + ? = 72. The opposite of + is −. 72 − 48 = 24. Check: 48 + 24 = 72.

  5. From an earlier chapter

    A machine does × 4, then + 3. The output is 31. What went in?

    See the answer

    7

    Undo + 3 first. 31 − 3 = 28. Then undo × 4. 28 ÷ 4 = 7. Check: 7 × 4 + 3 = 31.

  6. Kofi finds the missing part. What is the hatched part?

    A long box labelled 80. Under it, a box labelled 53, then a hatched box labelled ? for the rest.8053?
    Tap your answer.

    1. Not this one. You took the smaller digit from the bigger in each place. Put 33 back: 53 + 33 = 86, not 80. Try another.

    2. Not this one. You added the two numbers you can see. The part must be less than the whole, 80. Try another.

    3. Yes, that is right. The whole is 80. Take the part you know away. 80 − 53 = 27. Check: 53 + 27 = 80.

    See the answer

    27

    The whole is 80. Take the part you know away. 80 − 53 = 27. Check: 53 + 27 = 80.

    • If you chose A: You took the smaller digit from the bigger in each place. Put 33 back: 53 + 33 = 86, not 80.
    • If you chose B: You added the two numbers you can see. The part must be less than the whole, 80.

  7. Find the missing number in ? × 9 = 63.

    Tap your answer.

    1. Not this one. You took 9 away from 63. The opposite of × is ÷, not −. Try another.

    2. Yes, that is right. The opposite of × is ÷. 63 ÷ 9 = 7. Check: 7 × 9 = 63.

    3. Not this one. 8 × 9 = 72. That is one 9 too many. 7 × 9 = 63. Try another.

    See the answer

    7

    The opposite of × is ÷. 63 ÷ 9 = 7. Check: 7 × 9 = 63.

    • If you chose A: You took 9 away from 63. The opposite of × is ÷, not −.
    • If you chose C: 8 × 9 = 72. That is one 9 too many. 7 × 9 = 63.

  8. Mei finds that 50 − 36 = 14. Which sum checks her answer?

    Tap your answer.

    1. Not this one. You added the whole and the answer. Add the answer to the part you took away. Try another.

    2. Yes, that is right. The opposite of − is +. Put the answer back with the part you know. 36 + 14 = 50, the whole.

    3. Not this one. You took the answer from the part. Add them instead, because + undoes −. Try another.

    See the answer

    36 + 14 = 50

    The opposite of − is +. Put the answer back with the part you know. 36 + 14 = 50, the whole.

    • If you chose A: You added the whole and the answer. Add the answer to the part you took away.
    • If you chose C: You took the answer from the part. Add them instead, because + undoes −.

  9. Sipho puts 54 stones in 6 equal bags. How many stones are in each bag?

    A label says 54 stones in all. Under it, six hatched boxes in a row, each labelled ?.54 stones in all??????
    Tap your answer.

    1. Yes, that is right. Share 54 into 6 equal bags. 54 ÷ 6 = 9. Check: 6 × 9 = 54.

    2. Not this one. You took 6 away from 54. The bags are equal, so share 54 into 6 groups with ÷. Try another.

    3. Not this one. 6 × 8 = 48. That is 6 short of 54. 6 × 9 = 54. Try another.

    See the answer

    9

    Share 54 into 6 equal bags. 54 ÷ 6 = 9. Check: 6 × 9 = 54.

    • If you chose B: You took 6 away from 54. The bags are equal, so share 54 into 6 groups with ÷.
    • If you chose C: 6 × 8 = 48. That is 6 short of 54. 6 × 9 = 54.

Keep it

Copy this into your exercise book.

To find a missing number, use the opposite operation. For 36 + ? = 50, do 50 − 36 = 14. For ? × 6 = 42, do 42 ÷ 6 = 7. Then check.

Draw: A long box 50. Under it, a box 36 and a hatched box with a ?.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (603 KB). It opens on any phone.