Chapter 1 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

M2.8.1 Chapter 1

Number sequences

What number comes next in 3, 7, 11, 15?

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

  • Continue a sequence that goes up or down in equal steps
  • Find the step and the missing numbers in a sequence

Look at this list of numbers: 3, 7, 11, 15. What comes next?

A sequence is a list of numbers. The numbers follow a rule.

Here, each number is 4 more than the one before. The jump of 4 is the step.

Draw it first

Find the step, then the next number

  1. Boxes in a row holding 3, 7, 11 and 15. Step 2 labels each jump +4. Step 3 adds a hatched box 19 after 15, with a last jump +4.371115

    1 Write the numbers in boxes: 3, 7, 11, 15.

  2. Boxes in a row holding 3, 7, 11 and 15. Step 2 labels each jump +4. Step 3 adds a hatched box 19 after 15, with a last jump +4.371115+4+4+4

    2 Look at each jump. 3 to 7 is +4. 7 to 11 is +4. 11 to 15 is +4.

  3. Boxes in a row holding 3, 7, 11 and 15. Step 2 labels each jump +4. Step 3 adds a hatched box 19 after 15, with a last jump +4.37111519+4+4+4+4

    3 Jump 4 more. 15 + 4 = 19. The next number is 19.

To find the step, take the first number from the second. 7 − 3 = 4.

Then check every pair. 11 − 7 = 4 and 15 − 11 = 4. The step is the same each time.

A sequence can go down too. This one goes down by 5 each time.

Boxes in a row holding 50, 45, 40 and 35. Each jump from one box to the next is labelled −5.50454035−5−5−5
Each number is 5 less than the one before. 50 − 45 = 5.

Worked example

Continue 2, 5, 8, 11 with the next two numbers.

  1. Find the step. 5 − 2 = 3.
  2. Check the other pairs. 8 − 5 = 3 and 11 − 8 = 3.
  3. Add 3 to the last number. 11 + 3 = 14.
  4. Add 3 again. 14 + 3 = 17.

Answer: 14, 17

You can show a sequence on a number line. Each hop is one step.

A number line with marks at 6, 12, 18, 24 and 30. Each hop from one mark to the next is labelled +6.612182430+6+6+6+6
The hops are all the same length. The step is 6.

Sometimes a number is missing. The last two numbers tell you the step.

Boxes in a row holding 12, a hatched box labelled ?, then 24 and 30. The jump from 24 to 30 is labelled +6.12?2430+6
Only the last jump is known. It is +6.

Worked example

Find the missing number in 12, ?, 24, 30.

  1. Use the last two numbers. 30 − 24 = 6. The step is 6.
  2. The missing number is one step after 12. 12 + 6 = 18.
  3. Check. 18 + 6 = 24. It fits.

Answer: 18

See it three ways

With things

Make a pile of 3 stones. Make a pile of 7. Make a pile of 11. Each pile is 4 stones bigger.

As a picture

Three rows of dots. The first row has 3 dots. The second row has 7 dots, and the first 3 are filled in. The third row has 11 dots, and the first 7 are filled in.

In symbols

3, 7, 11, 15. The step is 4. 7 − 3 = 4. 11 − 7 = 4. 15 − 11 = 4.

Try it

Grow a sequence with stones

You need: about 26 stones, beans or seeds, a pencil and paper.

  1. Put 2 stones in a row. Write 2.
  2. Make a new row with 3 more stones than the first. It has 5. Write 5.
  3. Add 3 more again. Write the new number.
  4. Do it once more. Write the number.

What you will see: Each row is 3 stones longer than the row above. Your numbers are 2, 5, 8, 11. The step is 3 every time.

If you cannot try it, this is what happens.
Four rows of stones. The rows have 2, 5, 8 and 11 stones. Each row is 3 stones longer than the one above.

A wrong idea many people have

The numbers 4, 8, 12 double each time. The next number is 24.

Why it feels right: 4 to 8 is a double. So it looks like doubling.

The test that breaks it: Check the next pair. Double 8 is 16, but the list says 12. The rule fails on the second pair.

What is true: Check the step on every pair. 8 − 4 = 4 and 12 − 8 = 4. The rule is add 4, so the next number is 16.

Key words

sequence
a list of numbers that follows a rule.
step
how much the numbers go up or down each time.

Back to your guess

The answer is A: It goes up by 4 each time, so 19.

The numbers go up by 4 each time. 15 + 4 = 19. The list is a sequence. The jump is its step.

Check yourself

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

  1. What comes next? 6, 11, 16, 21, ...

    Tap your answer.

    1. Yes, that is right. The step is 5, because 11 − 6 = 5. 16 − 11 = 5 and 21 − 16 = 5. So 21 + 5 = 26.

    2. Not this one. You counted on by 1. Find the step first: 11 − 6 = 5. Try another.

    3. Not this one. You doubled 21. Doubling fails on the first pair: 6 doubled is 12, not 11. The step is 5. Try another.

    See the answer

    26

    The step is 5, because 11 − 6 = 5. 16 − 11 = 5 and 21 − 16 = 5. So 21 + 5 = 26.

    • If you chose B: You counted on by 1. Find the step first: 11 − 6 = 5.
    • If you chose C: You doubled 21. Doubling fails on the first pair: 6 doubled is 12, not 11. The step is 5.

  2. Write the next two numbers: 90, 80, 70, 60, ...

    See the answer

    50, 40

    The numbers go down by 10 each time. 60 − 10 = 50 and 50 − 10 = 40.

  3. Find the missing number: 8, 14, ?, 26, 32

    See the answer

    20

    The last two numbers give the step. 32 − 26 = 6. Then 14 + 6 = 20, and 20 + 6 = 26.

  4. Mei says 2, 4, 7 goes up by 2 each time. Is she right?

    Tap your answer.

    1. Yes, that is right. 4 − 2 = 2 but 7 − 4 = 3. The steps are not the same. So 2, 4, 7 has no equal step.

    2. Not this one. You checked only the first pair. Check every pair: 7 − 4 = 3, not 2. Try another.

    See the answer

    No. The first jump is 2 and the second is 3.

    4 − 2 = 2 but 7 − 4 = 3. The steps are not the same. So 2, 4, 7 has no equal step.

    • If you chose B: You checked only the first pair. Check every pair: 7 − 4 = 3, not 2.

  5. Write the first five numbers of a sequence. It starts at 100 and goes down by 20 each time.

    See the answer

    100, 80, 60, 40, 20

    Take 20 away each time. 100 − 20 = 80, 80 − 20 = 60, 60 − 20 = 40, 40 − 20 = 20.

  6. What comes next? 47, 41, 35, 29, ...

    Boxes in a row holding 47, 41, 35 and 29, then a hatched box labelled ?.47413529?
    Tap your answer.

    1. Not this one. You went up by 6. These numbers go down, so take 6 away. Try another.

    2. Not this one. You took away 12. That is two jumps in one: 47 − 35 = 12. One jump is 47 − 41 = 6. Try another.

    3. Yes, that is right. The numbers go down by 6 each time. 47 − 41 = 6 and 41 − 35 = 6. So 29 − 6 = 23.

    See the answer

    23

    The numbers go down by 6 each time. 47 − 41 = 6 and 41 − 35 = 6. So 29 − 6 = 23.

    • If you chose A: You went up by 6. These numbers go down, so take 6 away.
    • If you chose B: You took away 12. That is two jumps in one: 47 − 35 = 12. One jump is 47 − 41 = 6.

  7. Ravi writes 5, 9, 13, 17. What number comes next?

    Tap your answer.

    1. Not this one. You counted 5, 6, 7, 8, 9. That is 5 numbers but only 4 jumps. The step is 9 − 5 = 4. Try another.

    2. Yes, that is right. The step is 4, because 9 − 5 = 4. 13 − 9 = 4 and 17 − 13 = 4. So 17 + 4 = 21.

    3. Not this one. You doubled 17. Doubling fails on the first pair: 5 doubled is 10, not 9. Try another.

    See the answer

    21

    The step is 4, because 9 − 5 = 4. 13 − 9 = 4 and 17 − 13 = 4. So 17 + 4 = 21.

    • If you chose A: You counted 5, 6, 7, 8, 9. That is 5 numbers but only 4 jumps. The step is 9 − 5 = 4.
    • If you chose C: You doubled 17. Doubling fails on the first pair: 5 doubled is 10, not 9.

  8. What is the step in 14, 21, 28, 35?

    A number line with marks at 14, 21, 28 and 35, equal hops apart.14212835
    Tap your answer.

    1. Not this one. You took 14 from 35. That is three steps together. A step is one jump, from one mark to the next. Try another.

    2. Not this one. You used the first number. The step is the jump from one number to the next. Try another.

    3. Yes, that is right. Take one number from the next: 21 − 14 = 7. Check: 28 − 21 = 7 and 35 − 28 = 7.

    See the answer

    7

    Take one number from the next: 21 − 14 = 7. Check: 28 − 21 = 7 and 35 − 28 = 7.

    • If you chose A: You took 14 from 35. That is three steps together. A step is one jump, from one mark to the next.
    • If you chose B: You used the first number. The step is the jump from one number to the next.

  9. Hana marks 5, a gap, 17 and 23 in equal hops. What number fills the gap?

    A number line with marks at 5, then a hatched box labelled ?, then 17 and 23. An arrow joins each mark to the next.5?1723
    Tap your answer.

    1. Yes, that is right. The last two marks give the step: 23 − 17 = 6. The gap is one step after 5: 5 + 6 = 11. Check: 11 + 6 = 17.

    2. Not this one. A step of 5 does not fit: 10 + 5 = 15, not 17. Use the last two marks: 23 − 17 = 6. Try another.

    3. Not this one. You took 5 from 17. That is two hops, the gap and the one after it. One hop is 23 − 17 = 6. Try another.

    See the answer

    11

    The last two marks give the step: 23 − 17 = 6. The gap is one step after 5: 5 + 6 = 11. Check: 11 + 6 = 17.

    • If you chose B: A step of 5 does not fit: 10 + 5 = 15, not 17. Use the last two marks: 23 − 17 = 6.
    • If you chose C: You took 5 from 17. That is two hops, the gap and the one after it. One hop is 23 − 17 = 6.

Keep it

Copy this into your exercise book.

A sequence is a list of numbers that follows a rule. Its step is the same each time. For 3, 7, 11 the step is 7 − 3 = 4. Check every pair.

Draw: Four boxes with 3, 7, 11 and 15. Write +4 on each jump between them.

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.