Chapter 3 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

M2.8.3 Chapter 3

Number machines

A machine does × 3, then + 1. What comes out if I put in 4?

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 output of a machine like 'times 3, then add 1'
  • Find the rule from pairs of inputs and outputs
  • Work backwards from an output to find the input

A number machine changes numbers by a rule. Our machine does two jobs, in order.

The number you put in is the input. The number that comes out is the output.

Draw it first

Run a number through the machine

  1. A number machine drawn as boxes in a row. Step 1 shows two boxes, × 3 and + 1. Step 2 adds an input box 4, with the caption input. Step 3 shows 12 over the arrow between the two jobs. Step 4 adds an output box 13, with the caption output.× 3+ 1

    1 The machine does two jobs. First × 3. Then + 1.

  2. A number machine drawn as boxes in a row. Step 1 shows two boxes, × 3 and + 1. Step 2 adds an input box 4, with the caption input. Step 3 shows 12 over the arrow between the two jobs. Step 4 adds an output box 13, with the caption output.4× 3+ 1input

    2 Put in 4. This is the input.

  3. A number machine drawn as boxes in a row. Step 1 shows two boxes, × 3 and + 1. Step 2 adds an input box 4, with the caption input. Step 3 shows 12 over the arrow between the two jobs. Step 4 adds an output box 13, with the caption output.4× 3+ 112input

    3 The first job gives 4 × 3 = 12.

  4. A number machine drawn as boxes in a row. Step 1 shows two boxes, × 3 and + 1. Step 2 adds an input box 4, with the caption input. Step 3 shows 12 over the arrow between the two jobs. Step 4 adds an output box 13, with the caption output.4× 3+ 11312inputoutput

    4 The second job gives 12 + 1 = 13. This is the output.

The order matters. Now do + 1 first, then × 3. Put in 4 again.

Two machines. The top machine does × 3 then + 1. Put in 4, get 12 in the middle, and 13 comes out. The bottom machine does + 1 then × 3. Put in 4, get 5 in the middle, and 15 comes out.4× 3+ 113124+ 1× 3155
The same jobs in a different order give a different output.

Worked example

Put 7 in the machine. It does × 3, then + 1.

  1. First job: 7 × 3 = 21.
  2. Second job: 21 + 1 = 22.
  3. The output is 22.

Answer: 22

Squares of sticks use this machine too. Put in 4 squares. 13 sticks come out.

It is the stick rule from the last chapter.

Sometimes you see the inputs and the outputs, but not the rule.

A table with the heads Input and Output. Input 1 gives output 5. Input 2 gives output 7. Input 3 gives output 9.InputOutput152739
What does the machine do?

Worked example

Find the rule. 1 gives 5, 2 gives 7, 3 gives 9.

  1. The inputs go up by 1. The outputs go up by 2. So the rule starts with × 2.
  2. Try × 2 on the input 1. 2 × 1 = 2. The output is 5.
  3. It needs 3 more. 5 − 2 = 3. So add 3.
  4. Test every pair. 2 × 2 + 3 = 7. 2 × 3 + 3 = 9.

Answer: × 2, then + 3

Now go backwards. The output is 22. What did you put in?

Undo each job with its opposite. Start with the last job.

The opposite of + is −. The opposite of × is ÷.

Going backwards. The output 22 goes in the first box, with the caption output. The jobs are − 1 and ÷ 3. The number between them is 21. The last box is 7, with the caption input.22− 1÷ 3721outputinput
Start at the output. Undo the last job first.

Worked example

The machine does × 3, then + 1. The output is 22. Find the input.

  1. The last job was + 1. Undo it first. Take 1 away. 22 − 1 = 21.
  2. The first job was × 3. Undo it. 21 ÷ 3 = 7.
  3. Check. 7 × 3 + 1 = 22.

Answer: 7

See it three ways

With things

Put in 2. Make 2 groups of 3 beans. Then add 1 more bean. Count them all.

As a picture

Two groups of three beans, then one more bean filled in on the right.

In symbols

2 × 3 + 1 = 7. The input is 2. The output is 7.

Try it

Be the number machine

You need: a pencil and paper, a friend, if you have one.

  1. Write a rule: × 2, then + 1. Keep it hidden from your friend.
  2. Ask your friend for the inputs 1, 2, 3 and 4. Give each output.
  3. Write the inputs and outputs in a table.
  4. Ask your friend to find the rule. On your own, cover the rule and find it.

What you will see: The outputs 3, 5, 7, 9 go up by 2. That shows the × 2. The first output needs 1 more.

If you cannot try it, this is what happens.
A table with the heads Input and Output. Input 1 gives output 3. Input 2 gives output 5. Input 3 gives output 7. Input 4 gives output 9.InputOutput13253749

A wrong idea many people have

To go backwards from 22, divide by 3 first. Then take 1 away.

Why it feels right: The machine does × 3 first. So it seems you undo × 3 first.

The test that breaks it: 22 ÷ 3 leaves a bit over. Say you use 7, then take 1 away to get 6. Put 6 in the machine. 6 × 3 + 1 = 19, not 22.

What is true: Undo the last job first. You tie string round a parcel last. To open it, you untie the string first.

Key words

number machine
a rule that changes a number, one job at a time.
input
the number you put in.
output
the number that comes out.
opposite
the job that undoes another. + and − are opposites.

Back to your guess

The answer is B: 13. Times 3 gives 12, then add 1.

The machine does its jobs in order. 4 × 3 = 12, then 12 + 1 = 13. Add 1 first and you get 15, which is a different machine.

Check yourself

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

  1. A machine does × 5, then − 2. Put in 6. What comes out?

    See the answer

    28

    First job: 6 × 5 = 30. Second job: 30 − 2 = 28.

  2. A machine does + 2, then × 4. Put in 3. What comes out?

    Tap your answer.

    1. Yes, that is right. First job: 3 + 2 = 5. Second job: 5 × 4 = 20.

    2. Not this one. You did × 4 first. 3 × 4 + 2 = 14. This machine adds 2 first. Try another.

    3. Not this one. You added 2 and then added 4. The second job is × 4, not + 4. Try another.

    See the answer

    20

    First job: 3 + 2 = 5. Second job: 5 × 4 = 20.

    • If you chose B: You did × 4 first. 3 × 4 + 2 = 14. This machine adds 2 first.
    • If you chose C: You added 2 and then added 4. The second job is × 4, not + 4.

  3. Find the rule. 2 gives 9, 3 gives 13, 4 gives 17.

    A table with the heads Input and Output. Input 2 gives output 9. Input 3 gives output 13. Input 4 gives output 17.InputOutput29313417
    See the answer

    × 4, then + 1

    The inputs go up by 1 and the outputs by 4. So start with × 4. 2 × 4 = 8, and the output is 9, so add 1. Test: 3 × 4 + 1 = 13 and 4 × 4 + 1 = 17.

  4. A machine does × 2, then + 5. The output is 19. What went in?

    See the answer

    7

    Undo + 5 first. 19 − 5 = 14. Then undo × 2. 14 ÷ 2 = 7. Check: 7 × 2 + 5 = 19.

  5. From an earlier chapter

    How many sticks make a row of 7 squares?

    See the answer

    22 sticks

    4 sticks for the first square. Then 6 more squares add 3 each. 6 × 3 = 18, and 4 + 18 = 22. Or 3 × 7 + 1 = 22.

  6. A machine does × 2, then + 4. The output is 20. What went in?

    A number machine drawn as boxes in a row. The input box is ?, the jobs are × 2 and + 4, and the output box is 20. The number between the jobs is ?. The captions are input and output.?× 2+ 420?inputoutput
    Tap your answer.

    1. Not this one. You undid × 2 first: 20 ÷ 2 = 10, then 10 − 4 = 6. Undo the last job first, which is + 4. Try another.

    2. Yes, that is right. Undo the last job first. 20 − 4 = 16. Then undo × 2. 16 ÷ 2 = 8. Check: 8 × 2 + 4 = 20.

    3. Not this one. You did + 4 and × 2 again. To go back, use the opposite of each job: − 4 and ÷ 2. Try another.

    See the answer

    8

    Undo the last job first. 20 − 4 = 16. Then undo × 2. 16 ÷ 2 = 8. Check: 8 × 2 + 4 = 20.

    • If you chose A: You undid × 2 first: 20 ÷ 2 = 10, then 10 − 4 = 6. Undo the last job first, which is + 4.
    • If you chose C: You did + 4 and × 2 again. To go back, use the opposite of each job: − 4 and ÷ 2.

  7. Find the rule. 1 gives 5, 2 gives 8, 3 gives 11.

    A table with the heads Input and Output. Input 1 gives output 5. Input 2 gives output 8. Input 3 gives output 11.InputOutput1528311
    Tap your answer.

    1. Not this one. The outputs go up by 3, but 1 + 3 = 4, not 5. A rule must fit every pair. Try another.

    2. Not this one. You used the first output, 5. 1 × 5 = 5 fits, but 2 × 5 = 10, not 8. Try another.

    3. Yes, that is right. The outputs go up by 3, so start with × 3. 1 × 3 = 3, and the output is 5, so add 2. Test: 2 × 3 + 2 = 8 and 3 × 3 + 2 = 11.

    See the answer

    × 3, then + 2

    The outputs go up by 3, so start with × 3. 1 × 3 = 3, and the output is 5, so add 2. Test: 2 × 3 + 2 = 8 and 3 × 3 + 2 = 11.

    • If you chose A: The outputs go up by 3, but 1 + 3 = 4, not 5. A rule must fit every pair.
    • If you chose B: You used the first output, 5. 1 × 5 = 5 fits, but 2 × 5 = 10, not 8.

  8. A machine does × 4, then − 3. Put in 5. What comes out?

    A number machine drawn as boxes in a row. The input box is 5, the jobs are × 4 and − 3, and the output box is ?. The number between the jobs is ?. The captions are input and output.5× 4− 3??inputoutput
    Tap your answer.

    1. Not this one. You added 3. The second job is − 3, so take 3 away. Try another.

    2. Yes, that is right. First job: 5 × 4 = 20. Second job: 20 − 3 = 17.

    3. Not this one. You took 3 away first, then did × 4. The jobs go in order: × 4 comes first. Try another.

    See the answer

    17

    First job: 5 × 4 = 20. Second job: 20 − 3 = 17.

    • If you chose A: You added 3. The second job is − 3, so take 3 away.
    • If you chose C: You took 3 away first, then did × 4. The jobs go in order: × 4 comes first.

  9. Leila's machine does ÷ 2, then + 6. The output is 15. What went in?

    A number machine drawn as boxes in a row. The input box is ?, the jobs are ÷ 2 and + 6, and the output box is 15. The number between the jobs is ?. The captions are input and output.?÷ 2+ 615?inputoutput
    Tap your answer.

    1. Not this one. You undid ÷ 2 first: 15 × 2 = 30, then 30 − 6 = 24. Undo the last job first, which is + 6. Try another.

    2. Not this one. You undid + 6 but not ÷ 2. The opposite of ÷ 2 is × 2, so 9 × 2 = 18. Try another.

    3. Yes, that is right. Undo the last job first. 15 − 6 = 9. Then undo ÷ 2. 9 × 2 = 18. Check: 18 ÷ 2 + 6 = 15.

    See the answer

    18

    Undo the last job first. 15 − 6 = 9. Then undo ÷ 2. 9 × 2 = 18. Check: 18 ÷ 2 + 6 = 15.

    • If you chose A: You undid ÷ 2 first: 15 × 2 = 30, then 30 − 6 = 24. Undo the last job first, which is + 6.
    • If you chose B: You undid + 6 but not ÷ 2. The opposite of ÷ 2 is × 2, so 9 × 2 = 18.

Keep it

Copy this into your exercise book.

A number machine does jobs in order. The input goes in. The output comes out. To go backwards, undo the last job first, with its opposite.

Draw: Four boxes in a row: input 4, × 3, + 1, output 13. Write 12 above the middle arrow.

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.