Chapter 2 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

M2.8.2 Chapter 2

Growing shape patterns

How many sticks do I need for a row of 5 squares?

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

  • Draw the next shape of a growing pattern of squares
  • Make a table of the number of squares and sticks

You can make a row of squares from sticks. Let us count the sticks.

Each new shape has one more square than the shape before. This is a growing pattern.

Draw it first

Grow a row of squares

  1. Sticks making a row of squares. Step 1 is one square, labelled 4 sticks. Step 2 adds a square and is labelled 4 + 3 = 7 sticks. Step 3 adds another and is labelled 7 + 3 = 10 sticks. Step 4 adds one more and is labelled 10 + 3 = 13 sticks.4 sticks

    1 Make one square. It needs 4 sticks.

  2. Sticks making a row of squares. Step 1 is one square, labelled 4 sticks. Step 2 adds a square and is labelled 4 + 3 = 7 sticks. Step 3 adds another and is labelled 7 + 3 = 10 sticks. Step 4 adds one more and is labelled 10 + 3 = 13 sticks.4 + 3 = 7 sticks

    2 Add a square at the end. It shares a stick, so add only 3. 4 + 3 = 7.

  3. Sticks making a row of squares. Step 1 is one square, labelled 4 sticks. Step 2 adds a square and is labelled 4 + 3 = 7 sticks. Step 3 adds another and is labelled 7 + 3 = 10 sticks. Step 4 adds one more and is labelled 10 + 3 = 13 sticks.7 + 3 = 10 sticks

    3 Add another square. 7 + 3 = 10.

  4. Sticks making a row of squares. Step 1 is one square, labelled 4 sticks. Step 2 adds a square and is labelled 4 + 3 = 7 sticks. Step 3 adds another and is labelled 7 + 3 = 10 sticks. Step 4 adds one more and is labelled 10 + 3 = 13 sticks.10 + 3 = 13 sticks

    4 Add one more. 10 + 3 = 13.

A new square uses the end stick of the last square. That stick is counted once.

Two pictures of two squares. The top pair of squares do not touch, and the label says apart: 8 sticks. The bottom pair are joined and share a stick, and the label says joined: 7 sticks.apart: 8 sticksjoined: 7 sticks
Squares that touch share a stick. Squares apart do not.

A table puts the squares and the sticks side by side.

A table with the heads Squares and Sticks. 1 square needs 4 sticks. 2 squares need 7. 3 squares need 10. 4 squares need 13. 5 squares need 16, and that box is shaded.SquaresSticks1427310413516
The sticks go up by 3 each row. The shaded box answers our question.

Worked example

How many sticks make a row of 5 squares?

  1. Fill a table. 1 square needs 4 sticks.
  2. Add 3 for each new square. 4, 7, 10, 13, 16.
  3. 5 squares need 16 sticks.

Answer: 16 sticks

There is a quick way too. Take 3 times the squares. Then add 1.

Why? Each square has 3 sticks of its own. One more stick closes the first square.

For 5 squares: 3 × 5 + 1 = 16. It gives the same answer as the table.

See it three ways

With things

Use sticks. Make 1 square. Then join a second square, then a third. Count the sticks each time.

As a picture

Three rows of squares made from sticks. One square is labelled 4. Two joined squares are labelled 7. Three joined squares are labelled 10.4710

In symbols

4, 7, 10. The step is 3. 7 − 4 = 3. 10 − 7 = 3.

Try it

Build squares with sticks

You need: about 16 sticks of the same length, such as twigs or straws, a pencil and paper.

Be safe: Ask a grown up to cut the sticks. Never use matches.

  1. Make one square with 4 sticks. Write 4.
  2. Add a square at the end. Use the end stick of the first square.
  3. Count the new sticks you used. Write the total.
  4. Keep adding squares up to 4. Write each total.

What you will see: Each new square needs only 3 sticks. Your totals are 4, 7, 10, 13.

If you cannot try it, this is what happens.
A row of 4 squares made from sticks, labelled 4 squares, 13 sticks.4 squares, 13 sticks

A wrong idea many people have

Two squares in a row need 8 sticks. Each square needs 4.

Why it feels right: One square has 4 sticks. So two squares seem to need 4 + 4.

The test that breaks it: Build two squares that touch, with real sticks. Count them. You use 7. The middle stick is a side of both squares.

What is true: Joined squares share sticks. Each new square adds 3, not 4. Two squares need 7 sticks.

Key words

growing pattern
a pattern where each shape is bigger than the shape before.
table
numbers in rows and columns, such as squares and sticks.

Back to your guess

The answer is B: 16 sticks. The first square needs 4, and each new one adds 3.

Squares in a row share sticks. The first square needs 4. Each new square adds 3. So 5 squares need 16 sticks.

Check yourself

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

  1. How many sticks make a row of 6 squares?

    See the answer

    19 sticks

    4, 7, 10, 13, 16, 19. Or take 3 times 6, then add 1. 3 × 6 + 1 = 19.

  2. Kofi says 3 squares in a row need 12 sticks. Is he right?

    Tap your answer.

    1. Yes, that is right. The first square needs 4. The next two add 3 each. 4 + 3 + 3 = 10.

    2. Not this one. You gave every square 4 sticks. Joined squares share sticks, so each new one adds only 3. Try another.

    3. Not this one. You used 3 sticks for every square. The first square needs 4 sticks. Try another.

    See the answer

    No. They need 10 sticks.

    The first square needs 4. The next two add 3 each. 4 + 3 + 3 = 10.

    • If you chose B: You gave every square 4 sticks. Joined squares share sticks, so each new one adds only 3.
    • If you chose C: You used 3 sticks for every square. The first square needs 4 sticks.

  3. Fill the empty boxes. Find the sticks for 3, 4 and 5 squares.

    A table with the heads Squares and Sticks. 1 square needs 4 sticks. 2 squares need 7 sticks. The boxes for 3, 4 and 5 squares are empty.SquaresSticks1427345
    See the answer

    10, 13, 16

    Add 3 each time. 7 + 3 = 10, 10 + 3 = 13, 13 + 3 = 16.

  4. A pattern grows from 1 square, to 2, to 3. Here is the shape with 3. Draw the next shape. How many sticks does it need?

    Three squares joined in a row, made from sticks.
    See the answer

    4 squares in a row, with 13 sticks

    Three squares use 10 sticks. The fourth square adds 3. 10 + 3 = 13.

  5. From an earlier chapter

    Find the missing number: 5, 9, ?, 17, 21

    See the answer

    13

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

  6. A row grows 1 square, 2 squares, 3 squares. Which shape comes next?

    Three shapes made of sticks, labelled A, B and C. A is 4 squares in a row. B is 4 squares in a 2 by 2 block. C is 3 squares in a row with one more square on top.ABC
    Tap your answer.

    1. Yes, that is right. Each shape adds 1 square to the end of the row. After 3 comes 4. Shape A has 4 squares in a row.

    2. Not this one. The pattern adds one square to the end of the row. It does not start a new row. Try another.

    3. Not this one. Each new square joins the end of the row, not the top. Try another.

    See the answer

    Shape A

    Each shape adds 1 square to the end of the row. After 3 comes 4. Shape A has 4 squares in a row.

    • If you chose B: The pattern adds one square to the end of the row. It does not start a new row.
    • If you chose C: Each new square joins the end of the row, not the top.

  7. Amina made this table. One row is wrong. Which row?

    A table with the heads Squares and Sticks. 1 square, 4 sticks. 2 squares, 7 sticks. 3 squares, 10 sticks. 4 squares, 12 sticks. 5 squares, 16 sticks.SquaresSticks1427310412516
    Tap your answer.

    1. Not this one. This row is right: 13 + 3 = 16. The row before it is the one that breaks the step. Try another.

    2. Not this one. 3 × 4 = 12 counts the shared sticks twice. Joined squares share, so 7 + 3 = 10 is right. Try another.

    3. Yes, that is right. The sticks go up by 3 each row. 10 + 3 = 13, not 12. The 4 squares row should say 13.

    See the answer

    4 squares: 12

    The sticks go up by 3 each row. 10 + 3 = 13, not 12. The 4 squares row should say 13.

    • If you chose A: This row is right: 13 + 3 = 16. The row before it is the one that breaks the step.
    • If you chose B: 3 × 4 = 12 counts the shared sticks twice. Joined squares share, so 7 + 3 = 10 is right.

  8. Sipho joins 10 squares in a row. How many sticks does he need?

    Tap your answer.

    1. Not this one. You did 3 times 10 and forgot the 1 start stick. Add 1. Try another.

    2. Not this one. You gave every square 4 sticks. Joined squares share sticks. Try another.

    3. Not this one. You added 3 ten times. The first square is 4 sticks. Only 9 squares add 3. Try another.

    4. Yes, that is right. Take 3 times the squares, then add 1. 3 × 10 + 1 = 31. Check: 4 + 3 × 9 = 31.

    See the answer

    31

    Take 3 times the squares, then add 1. 3 × 10 + 1 = 31. Check: 4 + 3 × 9 = 31.

    • If you chose A: You did 3 times 10 and forgot the 1 start stick. Add 1.
    • If you chose B: You gave every square 4 sticks. Joined squares share sticks.
    • If you chose C: You added 3 ten times. The first square is 4 sticks. Only 9 squares add 3.

  9. Ana used 19 sticks to make a row of joined squares. How many squares is that?

    Tap your answer.

    1. Yes, that is right. Count up the table. 4, 7, 10, 13, 16, 19. That is 6 squares. Check: 3 × 6 + 1 = 19.

    2. Not this one. You counted the steps of 3. 4 + 5 × 3 = 19. The first square is not a step, so count it too. Try another.

    3. Not this one. You shared 19 sticks by 4. Joined squares share sticks, so 4 squares need only 13. Try another.

    See the answer

    6

    Count up the table. 4, 7, 10, 13, 16, 19. That is 6 squares. Check: 3 × 6 + 1 = 19.

    • If you chose B: You counted the steps of 3. 4 + 5 × 3 = 19. The first square is not a step, so count it too.
    • If you chose C: You shared 19 sticks by 4. Joined squares share sticks, so 4 squares need only 13.

Keep it

Copy this into your exercise book.

Squares in a row share sticks. One square needs 4 sticks. Each new square adds 3. So 2 squares need 7, 3 need 10, and 4 need 13.

Draw: A row of 3 squares made from sticks. Beside it, a table of squares and sticks.

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.