Problem
Ripeka and Jan were sitting around playing with toothpicks when Ripeka started to make a pattern of squares.

How many toothpicks would she need to make a pattern like this that had 9 squares?

Variation: How many squares could Ripeka make with 23 toothpicks?

What is this problem about?
In this problem children need to find a pattern and then apply it to a practical situation. There are two ways to apply the pattern. In the original problem Ripeka has to find how many toothpicks she needs to make 9 squares. But the problem can be looked at another way (see Variation). Given the number of toothpicks, how many squares can she make?

In either direction the problem can build a foundation for algebra by enabling the children to see a link between variables. The variables here are the numbers of toothpicks and the numbers of squares. To be of value the children do not necessarily have to write this link formally as we have done in the solution. For instance, it can be done using a table.

The Extension takes a different perspective. Here the way is open for children to come up with their own arrangement in an attempt to minimise the number of toothpicks needed to make 9 squares. (This can also be turned around and the maximum number of squares can be sought using a given number of toothpicks.) Hopefully this will lead to creative use of their imagination. It is a useful problem to use for the development band.

Achievement Objectives
Algebra (Level 3)
- describe in words, rules for continuing number and spatial sequence patterns
- state the general rule for a set of practical problems

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a drawing, use equipment, make a table).

Resources
Toothpicks
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- state the general rule for a practical situation.
- describe a continuing pattern

1. Using toothpicks ask the children to form a single square. Next ask them to make 2 squares. Discuss – how many toothpicks do you need? What is the smallest number? (7)
2. Pose the problem.
3. As the children work on the problem ask them questions that focus their thinking on the number pattern that emerges.
   How many toothpicks do you need to make 3 squares? 4 squares?
   Can you predict how many you will need for 5? Why do you think that?
   Can you see a pattern in the number of toothpicks you need? Describe this?
4. Encourage the children to write down using their own words the rule for finding the number of toothpicks.
5. Share findings.

Extension to the problem
If Bob tried another pattern with the toothpicks what is the smallest number of toothpicks he would need to make 9 squares all the same size?

Solution
Ripeka’s pattern gives a pattern in the number of toothpicks she uses. To make 1 square she uses 4 toothpicks; to make 2 squares she uses 7 toothpicks; to make 3 squares she uses 10 toothpicks. For each new square she needs a further 3 toothpicks. If she wants to make # squares she will need 3# + 1 toothpicks. So 9 squares needs 3 x 9 + 1 = 28 toothpicks.

This problem can, of course, be done without relying so much on formal algebra. A table can be used, all the squares can be made and the toothpicks counted, and so on. However, you should point out the relation between the squares and the toothpicks and get them to look out for similar situations elsewhere. They could be encouraged to make up their own matchstick pattern.

Variation: Since 3# + 1 = 25, then # = 8. Ripeka can make 8 squares with 25 toothpicks.

Extension: This could be left as a puzzle to see who can use the least toothpicks. We can get 9 squares using 24 toothpicks.