Skip to content

Growing shapes

Patterns and sequences · level 2, lesson 5 of 6

A question to start

Maya makes squares from toothpicks, in a row, each one joined to the last.

shape 1 shape 2 shape 3 shape 4

One square takes 4 toothpicks. Two squares take 7. Three squares take 10.

How many toothpicks does she need for a row of 10 squares? Maya says "4 for each square, so 40". Is she right?

The idea

Count them, and look for the rule

Count the toothpicks in each shape:

  • Shape 1 (one square): 4 toothpicks.
  • Shape 2 (two squares): 7 toothpicks.
  • Shape 3 (three squares): 10 toothpicks.
  • Shape 4 (four squares): 13 toothpicks.

So the numbers are 4, 7, 10, 13. The gaps are +3, +3, +3. The rule in words is: start at 4 and add 3 each time.

That already shows Maya is wrong. If each square took 4, the gaps would be +4. Each new square only adds 3. Why?

Way one: each new square adds 3

shape 1 shape 2 shape 3 shape 4

Look at the colours. The first square (green) uses 4 toothpicks. Every square after that is built onto the side of the one before. The side it shares is already there. So each new square needs only a top, a bottom and one new side: 3 toothpicks.

So a row of 10 squares is:

  • 4 toothpicks for the first square,
  • and then 9 more squares (the 2nd to the 10th), each adding 3.

9 threes is 27. Then 4 + 27 = 31 toothpicks.

Maya said 40 because she forgot that the squares share sides. 10 squares do have 40 sides between them, but 9 of those are shared (one between each pair of neighbours), and a shared toothpick counts once, not twice. 40 − 9 = 31. ✓

Way two: tops, bottoms and uprights

Here is a completely different way to count the same shapes.

shape 1 shape 2 shape 3 shape 4

Sort the toothpicks by where they are:

  • Tops (orange): one for each square.
  • Bottoms (blue): one for each square.
  • Uprights (green): look closely. For 1 square there are 2 uprights. For 2 squares there are 3. For 3 squares there are 4. There is always one more upright than squares, like posts in a fence.

For 10 squares: 10 tops, 10 bottoms and 11 uprights. 10 + 10 + 11 = 31.

Why two ways are better than one

The two ways look completely different, but they gave the same answer: 31. That is not luck. Both ways count the same toothpicks, just sorted into different piles. If you count a pile of toothpicks carefully, it does not matter how you group them; you must get the same total.

So when two different ways agree, you can be very sure you are right. When they disagree, one of them has a mistake, and you know to look again. Mathematicians love counting things in two ways for exactly this reason.

A table

A good way to keep track of a growing shape is a table. The top row is the shape number. The bottom row is how many toothpicks it takes.

Shape number 1 2 3 4 5 6
Toothpicks 4 7 10 13 16 19

Reading across the bottom row, each number is 3 more than the one before. That is the "start at 4 and add 3" rule.

But the table also lets you go down, from a shape number straight to its toothpicks, using way two: shape 6 is 6 tops, 6 bottoms and 7 uprights, so 6 + 6 + 7 = 19. ✓ You do not have to work through shapes 1 to 5 first. That is a big help for faraway shapes like shape 50 (50 + 50 + 51 = 151).

Working backwards

Can a row of squares use exactly 25 toothpicks? How many squares?

Use way one. The first square takes 4, leaving 25 − 4 = 21 toothpicks. Each new square takes 3, and 21 is seven 3s. So there are 7 new squares after the first one: 8 squares in all.

Check with way two: 8 tops, 8 bottoms, 9 uprights. 8 + 8 + 9 = 25. ✓

What about 20 toothpicks? 20 − 4 = 16, and 16 is not a whole number of 3s (five 3s is 15, six 3s is 18). So there is no shape with exactly 20 toothpicks: you would have 1 left over. The numbers in this pattern are only 4, 7, 10, 13, 16, 19, 22, 25, ...

Another growing shape: triangles

shape 1 shape 2 shape 3 shape 4

Now triangles, pointing up and down in turn. The first triangle takes 3 toothpicks (one green, then an orange and a blue). Each new triangle shares a side with the one before, so it needs only 2 more: one orange and one blue.

Shape number 1 2 3 4 5
Toothpicks 3 5 7 9 11

Rule: start at 3 and add 2 each time. Look at the colours another way: each triangle has one orange and one blue toothpick, and there is one extra green one at the start. So shape 10 uses 10 orange, 10 blue and 1 green: 21.

Worked examples

Example 1. Shape 7 of the squares, two ways.

  1. Way one: 4 for the first square, then 6 more squares at 3 each. 6 threes is 18. 4 + 18 = 22.
  2. Way two: 7 tops, 7 bottoms, 8 uprights. 7 + 7 + 8 = 22.
  3. They agree. ✓

Example 2. Fill in the table. Complete the table for the triangles.

Shape number 1 2 3 6 8
Toothpicks 3 5 7 ? ?
  1. Each triangle has 1 orange and 1 blue, plus 1 green at the start.
  2. Shape 6: 6 + 6 + 1 = 13.
  3. Shape 8: 8 + 8 + 1 = 17.
  4. Check by adding 2s from shape 3: 7, 9 (shape 4), 11 (shape 5), 13 (shape 6) ✓, 15 (shape 7), 17 (shape 8) ✓.

Example 3. Which shape has 40 toothpicks? (Squares in a row.)

  1. Take away the first square: 40 − 4 = 36.
  2. 36 is twelve 3s. So 12 more squares.
  3. 13 squares. Check: 13 + 13 + 14 = 40. ✓

Example 4. A two-row block. Squares stacked in two rows, two squares tall, growing to the right. Shape 1 is a column of 2 squares, which takes 7 toothpicks (3 across: top, middle, bottom; and 2 on the left and 2 on the right). Each new column adds 3 across and 2 up: 5 more. So: 7, 12, 17, 22, ... Shape 4 has 22 toothpicks.

Try it

  1. How many toothpicks for a row of 5 squares? Show it two ways.
  2. How many toothpicks for a row of 12 squares?
  3. Copy and complete the table for squares in a row: shape 1, 2, 3, 8, 10.
  4. Can a row of squares use exactly 31 toothpicks? 33 toothpicks? Explain.
  5. How many toothpicks for 7 triangles in a row?
  6. Which triangle shape uses 31 toothpicks?
  7. A row of pentagons (five-sided shapes) shares one side between neighbours. The first pentagon takes 5 toothpicks. How many does each new one add? How many for 4 pentagons?
  8. Maya now says: "Each square has 4 sides, and 10 squares share 9 sides, so it is 40 − 9." Explain why this is right.
Answers
  1. Way one: 4 + 3 + 3 + 3 + 3 = 16. Way two: 5 tops, 5 bottoms, 6 uprights: 5 + 5 + 6 = 16.
  2. 12 tops, 12 bottoms, 13 uprights: 12 + 12 + 13 = 37. (Way one: 4 + 11 threes = 4 + 33 = 37.)
  3. Shape 1: 4. Shape 2: 7. Shape 3: 10. Shape 8: 8 + 8 + 9 = 25. Shape 10: 10 + 10 + 11 = 31.
  4. 31: yes, it is shape 10 (31 − 4 = 27 = nine 3s, so 1 + 9 = 10 squares). 33: no. 33 − 4 = 29, and 29 is not a whole number of 3s (nine 3s is 27, ten 3s is 30).
  5. 7 orange, 7 blue, 1 green: 15. (Or 3, 5, 7, 9, 11, 13, 15.)
  6. Take away the 1 green: 30. Half of 30 is 15 orange and 15 blue. So shape 15.
  7. Each new pentagon shares one side, so it adds 5 − 1 = 4. 4 pentagons: 5 + 4 + 4 + 4 = 17.
  8. 10 squares have 10 × 4 = 40 sides if they are all separate. But each pair of neighbours shares one toothpick, and 10 squares in a row have 9 pairs of neighbours. Each shared toothpick was counted twice, so take 9 away:
  9. This is a third way to count, and it agrees with the other two!

Common mistake

Counting shared sides twice. "4 toothpicks per square" forgets that neighbours share a side. Build the shape, or draw it, and check your rule on shape 2 before you trust it.

Off by one in the uprights. 10 squares need 11 uprights, not 10. There is always one more post than gaps.

Where it leads