In this problem students 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 students to see a link between variables. The variables here are the numbers of toothpicks and the numbers of squares. To be of value the students 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 students 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.
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?
How many squares could Ripeka make with 25 toothpicks?
- Using toothpicks ask the students 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)
- Pose the problem.
- As the students 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?
- Encourage the students to write down using their own words the rule for finding the number of toothpicks.
- Share findings.
Extension to the problem
If Nikau tried another pattern with the toothpicks what is the smallest number of toothpicks he would need to make 9 squares all the same size?
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 be done without relying on formal algebra. A table can be used, all the squares can be made and the toothpicks counted. However, it is important that students see the relationship between the squares and the toothpicks, and that they are able to recognize similar situations in other patterns. Encourage them to make up their own matchstick pattern.
Since 3# + 1 = 25, then # = 8. Ripeka can make 8 squares with 25 toothpicks.
Solution to the extension:
This could be left as a puzzle to see who can use the fewest toothpicks. Here the 9 squares use 24 toothpicks.