The unit investigates patterns made using matchsticks and tiles. The relation between the number of the term of a pattern and the number of matchsticks (tiles) that that term has, is explored with a view to finding a general rule that can be expressed is several ways.
- Predict the next term of a spatial pattern.
- Find a rule to give the number of matchsticks (tiles) in a given member of the pattern.
- Find the member of the pattern that has a given number of matchsticks (tiles).
In many textbooks, a common approach to developing rules from patterns is to use matchstick pictures. This unit develops these ideas further in terms of techniques that can be used to get children ‘inside the patterns’ and in so doing develop some skills that are part of a tool box for the development of algebraic thinking. Specifically, the approach adopted by this unit focuses on:
- constructing and manipulating representations (figures/shapes) that lead to the expression of a generalisation
- visualisation as a way of seeing how patterns are constructed
- identifying the recursive nature of some sequences
- communicating generalisations to others
Links to Numeracy
This unit provides an opportunity to focus on the strategies students are using to solve number problems, in particular their strategies for solving multiplication and division problems. The matchstick patterns are all based on multiplicative sequences.
Encourage students to think about these patterns by focusing on the different strategies that can be used to calculate successive numbers in the pattern. For example, the pattern for the triangle path made from 9 matchsticks can be seen as in a variety of ways:
3 + 2 + 2 + 2
1 + 2 + 2 + 2 + 2
3 + (3 X 2)
1 + (4 X 2)
Questions to develop strategic thinking
What numbers could you use to describe the way the pattern is made?
How did you work that out?
Can you think of any other numbers you could use?
Encourage students to explain their thinking and to see the patterns in a variety of ways.
It is this focus that sets this unit aside from the more traditional approaches using matchstick patterns. It is envisaged that these skills will be part of the support required that will allow children to engage in the more traditional forms of algebra at higher levels.
- Matchsticks with the heads burnt, or toothpicks as an alternative
- Dot paper as an alternative to using matchsticks
Here we look at a simple pattern created by putting matchsticks together to form a connected path of triangles.
- Introduce the session by telling the students that Kiri made the following matchstick paths using 1, 2, and 3 triangles – she called them a 1-triangle path, a 2-triangle path and a 3-triangle path.
Ask the students to use Kiri’s method to make a 4- and then a 5-triangle path.
How many extra matchsticks would be needed to make a 6-triangle path? A 7-triangle path?
How many matchsticks would Kiri need to make a 20-triangle path?
- Kiri noticed that if she rearranged the matchsticks, she could count them quite quickly. The following picture shows how she rearranged them.
- How does Kiri’s method work?.
How would Kiri rearrange a 7-triangle path?
- Tell the class that Kiri says that using her method, she can see a short cut way of counting the number of matchsticks needed to make a 10-triangle path. Get them to write down, using pictures to help them explain, what Kiri’s short cut method might be.
- Let’s call Kiri’s method, Kiri’s Rule.
Using Kiri’s Rule, how many matchsticks will be needed to make a 20-triangle path?
How big a path can Kiri make with 201 matchsticks?
- Kiri’s friend Jamie arranged his matchsticks differently. His pictures looked like this:
What is Jamie’s Rule?
What is Jamie’s picture for a 12-triangle path?
- Jamie says that using his method, he can see another short cut way of counting the number of matchsticks needed to make a 10-triangle path. Get the class to write down, using pictures to help them explain, what Jamie’s Rule is.
How many matchsticks will be needed to make a 20-triangle path?
How big a path can Jamie make with 201 matchsticks?
- Get the children to explain how Kiri’s Rule is different from Jamie’s Rule.
- Ask the class: How would Jamie explain to someone else how he could find the number of matches needed to make a path consisting of any number, say 1000, of triangles?
Here we look at a simple pattern created by putting matchsticks together to form a connected path of squares.
- Following the same general procedure as above, allow the students to explore ways of counting the number of matchsticks that are needed to make square paths. Present the children with the following picture.
- Have them make a 4-square and 5-square path. Focus questions on how many extra matchsticks were added each time.
- Ask them how they could develop a quick and easy way of finding the number of matchsticks needed to make a 20-square path.
- Let the students work in groups of 2. Ask the groups to make a picture showing how each path is made. They can experiment with the matchsticks, and record their pictures.
Is there only one possible picture?
What might some of the others look like?
- Some pictures will be very helpful in counting the number of matchsticks needed to make a 8-square path – some will not. Have the children choose the picture that they think best explains how successive square paths are made up AND gives a quick and easy method for counting the matchsticks needed for an 8 -square path.
- Have the children use their ‘best method’ to verify that there are 76 matchsticks needed to make a 25-square path.
- They can use this method to predict the number of matchsticks needed to make 20-, 36- and 100-square paths.
- Have them write down how they would use their method to count the number of matchsticks needed to make a square path consisting of any number of squares, say 1000 squares.
- How many squares are in a square path with 31, 304 and 457 matchsticks?
How many matchsticks will be left over if you make the biggest square path that you can with 38, 100 and 1000 matchsticks?
The ideas learnt in the last two sessions are reinforced here using ‘house paths’.
- Use the techniques developed in the last two sessions to explore the following problem:
A new matchstick path is being designed. It is called a house path. Some of them are shown below. Develop a counting rule, that is, a short-cut way of counting the number of matchsticks needed to make a 1000-house path.
- Have the children illustrate how they developed their counting rule. They could do this, for example, by using pictures, words or numbers (or some combination of these).
- Get the class to discuss the various approaches that were used and methods that were obtained.
- Allow time for the class to write up its conclusions.
What’s My Path?
Once again the ideas of the first two sessions are used and therefore reinforced in another context. This time the problem gives a rule and the students have to find the pattern.
- Give students the following problem:
My friend made a picture that showed how a matchstick path was made. She named it:
5 lots of 4 and add 2 (this was the counting rule used to make the path)
She sent it to me via email. However, I was only able to read the name of the path and not see the picture!
Make some possible pictures that she could have sent.
- It is worth noting that there are many answers to this. So even if two groups get a different answer, they may still both be correct.
- Allow the class time both to report back and discuss their solutions, and to write up what they have discovered.
Odd Ways of Seeing Things
The usual basic ideas continue to be explored but this time with a more complicated spatial pattern.
- Show the class the pattern below that is made up of square tiles. The 1st, 2nd, 3rd and 4th terms of the sequence are shown.
- Two students, Paul and Penina, discover two different ways that they think the shapes are related. Penina's way of seeing how each shape is made is given below.
- Explain what Penina has done, and what thinking might be behind her method.
- This is Paul’s way of seeing how each shape is made:
- Both of these students have written down a way of finding the number of tiles (not matchsticks!) needed to make the 10th shape in the pattern.
Which way belongs to whom?
(1) 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 – 1
(2) 10 + 9
- Penina notices that adding 2 all the time is the same as multiplication by 2. She refines her method further into a short cut. Use her short cut to find the 100th shape in the sequence?
- Check Penina’s result using Paul’s method.
- What term of the sequence has 45 tiles?