Chapter 7 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

M2.8.7 Chapter 7

Jumping far ahead

How many sticks do 10 squares need? Must I draw all ten?

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 10th shape or number without drawing them all
  • Write a pattern rule in words and test it

You made squares from sticks in a row. 1 square needs 4 sticks. 2 squares need 7.

Drawing 10 squares is slow. There is a quicker way. Look for a rule. A rule gets you the answer from the position. That is the place in the pattern, such as 1st or 10th.

Draw it first

Find the rule for stick squares

  1. Squares made of sticks in a row. Step 1: one square, with a dashed start stick labelled 1 and the other 3 sticks labelled +3, and the sum 3 × 1 + 1 = 4. Step 2: a second square labelled +3, and the sum 3 × 2 + 1 = 7. Step 3: a third square labelled +3, and the sum 3 × 3 + 1 = 10. Step 4: the label 10 squares: 3 × 10 + 1 = 31. The row still shows 3 squares.3 × 1 + 1 = 41+3

    1 One square is 4 sticks. Think of it as 1 start stick, then 3 more.

  2. Squares made of sticks in a row. Step 1: one square, with a dashed start stick labelled 1 and the other 3 sticks labelled +3, and the sum 3 × 1 + 1 = 4. Step 2: a second square labelled +3, and the sum 3 × 2 + 1 = 7. Step 3: a third square labelled +3, and the sum 3 × 3 + 1 = 10. Step 4: the label 10 squares: 3 × 10 + 1 = 31. The row still shows 3 squares.3 × 2 + 1 = 71+3+3

    2 A second square adds only 3. It shares a stick with the first.

  3. Squares made of sticks in a row. Step 1: one square, with a dashed start stick labelled 1 and the other 3 sticks labelled +3, and the sum 3 × 1 + 1 = 4. Step 2: a second square labelled +3, and the sum 3 × 2 + 1 = 7. Step 3: a third square labelled +3, and the sum 3 × 3 + 1 = 10. Step 4: the label 10 squares: 3 × 10 + 1 = 31. The row still shows 3 squares.3 × 3 + 1 = 101+3+3+3

    3 A third square adds 3 more. 3 × 3 + 1 = 10 sticks.

  4. Squares made of sticks in a row. Step 1: one square, with a dashed start stick labelled 1 and the other 3 sticks labelled +3, and the sum 3 × 1 + 1 = 4. Step 2: a second square labelled +3, and the sum 3 × 2 + 1 = 7. Step 3: a third square labelled +3, and the sum 3 × 3 + 1 = 10. Step 4: the label 10 squares: 3 × 10 + 1 = 31. The row still shows 3 squares.10 squares: 3 × 10 + 1 = 311+3+3+3

    4 Every square adds 3. So 10 squares need 3 × 10 + 1 = 31 sticks.

The rule in words: take the number of squares. Times it by 3. Then add 1.

A table with the heads Squares and Sticks. Row 1: 1, 4. Row 2: 2, 7. Row 3: 3, 10. Row 4: 4, 13. Row 5: 5, 16. Row 6: 10, 31.SquaresSticks14273104135161031
Here the position is the number of squares. The last row used the rule.

Test a rule on squares you already know. 3 × 1 + 1 = 4. 3 × 2 + 1 = 7. 3 × 4 + 1 = 13. All three match the table.

The picture shows why. There is 1 start stick, then 3 for each square. So you can trust the rule.

Worked example

How many sticks do 10 squares need?

  1. Each square adds 3 sticks. 10 squares give 3 × 10 = 30 sticks.
  2. Add the 1 start stick. 30 + 1 = 31.
  3. Check by adding 3 nine times to 4. That also gives 31.

Answer: 31 sticks

Rules work for number patterns too. Take 5, 8, 11, 14. The 1st number is 5. The step is 3.

Worked example

Find the 10th number of 5, 8, 11, 14.

  1. The 1st number is 5. The step is 3.
  2. From the 1st to the 10th there are 9 steps, not 10.
  3. 9 steps of 3 is 9 × 3 = 27.
  4. 5 + 27 = 32. The 10th number is 32.

Answer: 32

Here is a second rule for the same numbers. Times the position by 3, then add 2.

Test it. 3 × 1 + 2 = 5. 3 × 2 + 2 = 8. 3 × 3 + 2 = 11. For the 10th: 3 × 10 + 2 = 32. Both rules agree.

See it three ways

With things

Make 4 squares in a row with sticks. Count 1 start stick. Then count 3 for each square.

As a picture

Four squares made of sticks in a row. The dashed start stick is labelled 1. Under each square is the label +3. Below the squares is the sum 3 × 4 + 1 = 13.1+3+3+3+33 × 4 + 1 = 13

In symbols

3 × 4 + 1 = 13. So 4 squares need 13 sticks.

Try it

Test the rule with sticks

You need: 16 sticks, straws or strips of paper, a pencil and paper.

  1. Make 1 square. Count the sticks. You should count 4.
  2. Add 3 sticks to make 2 squares. Count 7.
  3. The rule says 5 squares need 3 × 5 + 1 = 16 sticks. Write it down.
  4. Now make 5 squares in a row. Count all the sticks.

What you will see: You count 16 sticks. The rule gave the same number without any sticks.

If you cannot try it, this is what happens.
Five squares made of sticks in a row. The dashed start stick is labelled 1. Under each square is the label +3. Below the squares is the sum 3 × 5 + 1 = 16.1+3+3+3+3+33 × 5 + 1 = 16

A wrong idea many people have

5 squares need 16 sticks, so 10 squares need double: 32.

Why it feels right: Twice as many squares seems to need twice as many sticks.

The test that breaks it: Make 1 square. It needs 4 sticks. Make 2 squares. They need 7 sticks, not 8.

What is true: Doubling counts the shared stick twice. Use the rule: 3 × 10 + 1 = 31.

Key words

position
the place in the pattern, such as 1st, 2nd or 10th.

Back to your guess

The answer is B: No. A rule gives the answer: 3 × 10 + 1 = 31.

A rule jumps ahead. Each square adds 3 sticks, plus 1 to start. 3 × 10 + 1 = 31. Doubling 16 gives 32, which is too many.

Check yourself

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

  1. Stick squares need 4 sticks, then 7, then 10. How many sticks do 12 squares need?

    See the answer

    37 sticks

    Each square adds 3, plus 1 to start. 3 × 12 + 1 = 37.

  2. Find the 10th number of 2, 6, 10, 14.

    Tap your answer.

    1. Yes, that is right. The 1st number is 2. There are 9 steps of 4. 9 × 4 = 36. 2 + 36 = 38.

    2. Not this one. You may have done 10 × 4. But the 1st number is 2, not 4. Test your rule on the 1st number. Try another.

    3. Not this one. You added 4 ten times. From the 1st to the 10th there are 9 steps, not 10. Try another.

    See the answer

    38

    The 1st number is 2. There are 9 steps of 4. 9 × 4 = 36. 2 + 36 = 38.

    • If you chose B: You may have done 10 × 4. But the 1st number is 2, not 4. Test your rule on the 1st number.
    • If you chose C: You added 4 ten times. From the 1st to the 10th there are 9 steps, not 10.

  3. Mei tests the rule "times by 2, then add 1". Does it work for 3 squares of sticks?

    See the answer

    No. It gives 7, but 3 squares need 10 sticks.

    2 × 3 + 1 = 7. The table says 3 squares need 10. The rule that works is 3 × 3 + 1 = 10.

  4. Ana knows 10 squares need 31 sticks. How many sticks do 20 squares need?

    Tap your answer.

    1. Not this one. You did 3 × 20 and forgot the start stick. Add 1. Try another.

    2. Yes, that is right. Each square adds 3. 3 × 20 = 60. Add the start stick: 61.

    3. Not this one. You doubled 31. That counts the shared stick twice. Use 3 × 20 + 1. Try another.

    See the answer

    61

    Each square adds 3. 3 × 20 = 60. Add the start stick: 61.

    • If you chose A: You did 3 × 20 and forgot the start stick. Add 1.
    • If you chose C: You doubled 31. That counts the shared stick twice. Use 3 × 20 + 1.

  5. A pattern is 3, 8, 13, 18. Find the 8th number.

    See the answer

    38

    The 1st number is 3. There are 7 steps of 5. 7 × 5 = 35. 3 + 35 = 38. Rule check: 5 × the position, take away 2. 5 × 8 − 2 = 38.

  6. From an earlier chapter

    A number machine does "times 3, then add 1". What comes out when 8 goes in?

    Tap your answer.

    1. Not this one. You did times 3 but forgot to add 1. Go to M2.8.3. Try another.

    2. Yes, that is right. 8 × 3 = 24. Then 24 + 1 = 25.

    3. Not this one. You added 1 first, then times 3. The order matters. Times 3 comes first. Go to M2.8.3. Try another.

    See the answer

    25

    8 × 3 = 24. Then 24 + 1 = 25.

    • If you chose A: You did times 3 but forgot to add 1. Go to M2.8.3.
    • If you chose C: You added 1 first, then times 3. The order matters. Times 3 comes first. Go to M2.8.3.

  7. Find the 8th number of 4, 7, 10, 13.

    Boxes holding 4, 7, 10, 13, with the labels 1st, 2nd, 3rd and 4th under them. Then three dots (...) and a hatched box labelled ?, with the label 8th under it.41st72nd103rd134th...?8th
    Tap your answer.

    1. Not this one. You added 3 eight times. From the 1st to the 8th there are 7 steps, not 8. Try another.

    2. Yes, that is right. The 1st number is 4. There are 7 steps of 3. 7 × 3 = 21. 4 + 21 = 25.

    3. Not this one. You did 8 × 3, but the 1st number is 4, not 3. Test a rule on the 1st number. Try another.

    See the answer

    25

    The 1st number is 4. There are 7 steps of 3. 7 × 3 = 21. 4 + 21 = 25.

    • If you chose A: You added 3 eight times. From the 1st to the 8th there are 7 steps, not 8.
    • If you chose C: You did 8 × 3, but the 1st number is 4, not 3. Test a rule on the 1st number.

  8. Positions 1 to 4 hold 4, 7, 10 and 13. Which rule gives them?

    A table with the heads Position and Number. Position 1 has number 4. Position 2 has number 7. Position 3 has number 10. Position 4 has number 13.PositionNumber1427310413
    Tap your answer.

    1. Not this one. 1 × 3 = 3, but the 1st number is 4. Test a rule on the 1st number. Try another.

    2. Not this one. 1 + 3 = 4 fits, but 2 + 3 = 5, not 7. A rule must fit every position. Try another.

    3. Yes, that is right. The numbers go up by 3, so start with × 3. 1 × 3 = 3, and the 1st number is 4, so add 1. Test: 2 × 3 + 1 = 7 and 4 × 3 + 1 = 13.

    See the answer

    Times the position by 3, then add 1.

    The numbers go up by 3, so start with × 3. 1 × 3 = 3, and the 1st number is 4, so add 1. Test: 2 × 3 + 1 = 7 and 4 × 3 + 1 = 13.

    • If you chose A: 1 × 3 = 3, but the 1st number is 4. Test a rule on the 1st number.
    • If you chose B: 1 + 3 = 4 fits, but 2 + 3 = 5, not 7. A rule must fit every position.

  9. A pattern goes 50, 46, 42, 38. Find the 10th number.

    Boxes holding 50, 46, 42, 38, with the labels 1st, 2nd, 3rd and 4th under them. Then three dots (...) and a hatched box labelled ?, with the label 10th under it.501st462nd423rd384th...?10th
    Tap your answer.

    1. Yes, that is right. The 1st number is 50. There are 9 steps of 4 down. 9 × 4 = 36. 50 − 36 = 14.

    2. Not this one. You took 4 away ten times. From the 1st to the 10th there are 9 steps, not 10. Try another.

    3. Not this one. You added 4 nine times. This pattern goes down, so take 4 away each time. Try another.

    See the answer

    14

    The 1st number is 50. There are 9 steps of 4 down. 9 × 4 = 36. 50 − 36 = 14.

    • If you chose B: You took 4 away ten times. From the 1st to the 10th there are 9 steps, not 10.
    • If you chose C: You added 4 nine times. This pattern goes down, so take 4 away each time.

Keep it

Copy this into your exercise book.

A rule lets you jump ahead. Stick squares: times the number of squares by 3, then add 1. 10 squares need 31 sticks. Test a rule on squares you know, and see why it works.

Draw: Draw 3 squares of sticks in a row. Mark the start stick. Write 3 × 3 + 1 = 10 under 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.