The unit investigates patterns made using matches and tiles. The relation between the number of the term of a pattern and the number of matches that that term has, is explored with a view to finding a general rule that can be expressed in several ways.
- Predict the next term of a spatial pattern.
- Find a rule to give the number of matches in a given term of the pattern.
- Find the member of the pattern that has a given number of matches.
This unit builds the concept of a relation using growing patterns made with matches. A relation is a connection between the value of one variable (changeable quantity) and another. In the case of matchstick patterns, the first variable is the term, that is the step number of the figure, e.g. Term 5 is the fifth figure in the growing pattern. The second variable is the number of matches needed to create the figure.
Relations can be represented in many ways. The purpose of representations is to enable prediction of further terms, and the corresponding value of the other variable, in a growing pattern. For example, representations might be used to find the number of matches needed to build the tenth term in the pattern. Important representations include:
- Tables of values
- Word rules for the nth term
- Equations that symbolise word rules
- Graphs on a number plane
Further detail about the development of representations for growth patterns can be found on pages 34-38 of Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics.
Links to Numeracy
This unit provides an opportunity to focus on the strategies students use to solve number problems.The matchstick patterns are all based on linear relations. This means that the increase in number of matches needed for the ‘next’ term is a constant number added to the previous term.
Encourage students to think about linear 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 matches 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 and how it grows?
- What do the numbers and operations tell you about the pattern?
- In what order do we perform the calculations like 3 + 3 x 2? (Note order of operations)
- Are the expressions the same in some way? For example, How is 3 + 2 + 2 + 2 the same as 3 + 3 x 2?
- Which expressions are the most efficient ways to calculate the number of matches?
Strategies for representation and prediction will support students to engage in the more traditional forms of algebra at higher levels.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:
- providing matchsticks so students can build the growth patterns
- using colour to highlight repeating elements in diagrams of the growth patterns
- easing the calculation demands by providing calculators
- modelling creating tables and other ways for students to record their working and ease demands on their working memory.
Tasks can be varied in many ways including:
- reducing the ‘distance’ of the terms involved, particularly predicting the number of matches for terms that are easy to build and check
- reducing the complexity of the patterns, e.g. increasing in twos, threes, and fives rather than sixes, twelves, etc
- collaborative grouping so students can support others
- reducing the demands for a product, e.g. oral presentation rather than a lot of calculations and words.
The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. Matches are a cheap and accessible resource but may not be of interest to your students. They might be more interested in other thin objects such as leaves or lines on tapa (kapa) cloth. You might find growth patterns in friezes on buildings in the community. Be aware of opportunities to learn that connect to the everyday experiences of your students.
- Matches with the heads burnt, or toothpicks, iceblock sticks, nursery sticks, trimmed bamboo skewers, etc.
- Dot paper as an alternative to using matches
- PowerPoint One
Note: All of the patterns used in this unit are available in PowerPoint 1 to allow easy sharing with data projector or similar.
Session 1: Triangle Paths
In this session we look at a simple pattern created by putting matches 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. Note that 1, 2, and 3 are the term numbers in Kiri’s pattern.
- Ask the students to use Kiri’s method to make a 4- and then a 5-triangle path.
How many extra matches would be needed to make a 6-triangle path? A 7-triangle path?
How many matches would Kiri need to make a 20-triangle path?
- Let students work out the number of matches needed for the 20th term. Use think, pair, share to allow students to compare their strategies.
- Kiri noticed that if she rearranged the matches, 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?
What expression would she write to show her calculation? (1 + 7 x 2 or 1 + 2 + 2 + 2 + 2 + 2 + 2 + 2)
- Tell the class that Kiri says that using her method, she can see a short cut way of counting the number of matches needed to make a 10-triangle path. Get them to write down, using pictures support their explanation, what Kiri’s short cut method might be.
- Let’s call Kiri’s method, Kiri’s Rule. Ask:
Using Kiri’s Rule, how many matches will be needed to make a 20-triangle path?
Reverse the problem by asking: How big a path can Kiri make with 201 matches?
- Allow students time to develop an answer and compare their strategies.
- Do students that relied on repeated addition change to multiplicative strategies with increased demand?
- Are students able to recognise that the term number is required, not the number of matches?
- Can students ‘undo’ their previous rules to find the term number?
- Kiri’s friend Jamie arranged his matches differently. His pictures looked like this:
What is Jamie’s Rule?
What is Jamie’s picture for a 12-triangle path?
What expression could Jamie write for the 12-triangle path (Term 12)
How are 3 + 2 + 2 + …+ 2 and 3 + 11 x 2 the same?
- Jamie says that using his method, he can see another short cut way of counting the number of matches needed to make a 10-triangle path. Get the class to write down, using pictures to support their explanations, what Jamie’s strategy is.
How many matches will be needed to make a 20-triangle path?
How big a path can Jamie make with 201 matches?
- Get the students to explain how Kiri’s Rule is different and the same compared to Jamie’s Rule.
- Ask the class: How would Kiri and Jamie explain to someone else how they could find the number of matches needed to make a path consisting of any number, say 1000, of triangles?
Vey-un has another way to work out the number of matches for a 10-triangle pattern. He writes 10 x 3 – 9 and gets the same number of matches as Kiri and Jamie, 21.
Ask students to explain how Vey-un’s strategy works. What do the numbers in his calculation refer to?
[Vey-un imagines ten complete triangles that require 10 x 3 = 30 matches to build. He imagines that the ten triangles join and that creates nine overlaps. He subtracts nine from 30 to allow for the overlapping matches.]
Session 2: Square Paths
Here we look at a simple pattern created by putting matches 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 matches that are needed to make square paths. Present the students with the following picture.
- Have your students construct a 4-square and 5-square path with matches or by drawing. Focus on how many extra matches were added each time. Where are the additional matches located?
- Ask your students how they could develop a quick and easy way of finding the number of matches needed to make a 20-square path.
What would Kiri, Jamie and Vey-un do for this square pattern?
- Let the students work in groups of two or three. Ask the groups to make a picture showing how the 20-square path is made. They can experiment with the matches, and record their pictures. Do you need to draw every square?
Is there only one possible way to look at the pattern?
What might some of the others ways look like?
- Some pictures will be very helpful in counting the number of matches needed to make a 20-square path – some will not. Have the students 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 matches needed for an 20 -square path. Note the cumbersome nature of repeatedly drawing squares and repeatedly adding three matches.
What is a more efficient way to draw or calculate the total number of matches?
- Have the students use their ‘best method’ to verify that there are 61 matches are needed to make a 20-square path.
- Compare the way the rules might be written:
Kiri [1 + 20 x 3] Jamie [4 + 19 x 3 Vey-un [20 x 4 – 19]
- Students can use these methods or their own ways to predict the number of matches needed to make 14-, 36- and 100-square paths.
- Ask them to write down how they would use their method to count the number of matches needed to make a square path consisting of any number of squares, say 1000 squares. Depending on the comfort of students with their rules you might use algebraic notation to represent the word rules:
Kiri [1 + 3n] Jamie [4 + 3 (n-1] Vey-un [4n – (n-1)]
- Reverse the problems so students must work out the term number for a given number of matches.
How many squares are in a square path with 31, 304 and 457 matches?
How many matches will be left over if you make the biggest square path that you can with 38, 100 and 1000 matches?
- Are students able to ‘undo’ their rules of find missing terms?
Kiri calculates “One plus three times the term number” to find the number of matches.
If Kiri knows the number of matches, how should she undo her rule to find the term number? [Note that order of undoing is important, subtract one then divide by three.]
Session 3: House Paths
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. The first three terms are shown below. Develop a counting rule, that is, a short-cut way of counting the number of matches needed to make a 1000-house path.
- Have the students illustrate how they developed their counting rule. They could do this, by using pictures, words or numbers (or some combination of these).
Do you need to draw every house?
Do you need to add on 999 times?
What do you think Kiri, Jamie and Vey-un might do with this pattern?
- 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 about the most efficient strategies.
- Latitia has 503 matches. How many houses are in her path if she uses all the matches? Will she have any matches left over?
Session 4: What’s My Path?
The ideas of the first three sessions are extended and reinforced in another context. This time the problem gives a rule and the students find the pattern.
- Give students the following problem:
My friend made a picture that showed how her fifth 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.
We have many different pictures that match the word rule. How are they different and how are they the same?
[The common property is that the pattern starts with two matches and build on using four matches for each additional shape]
- Examples might be:
- Allow the class time both to report back and discuss their solutions, and to write up what they have discovered.
- Ask: If my friend wanted to build a n-path how many matched would she need? N means any number you give her, say 1000, 53 or 214.
- Another friend sent this n-rule. Can you draw a pattern that matches his rule?
“n minus one then multiplied by five then add six”
What might the pattern look like?
One possible answer is:
Session 5: Other Ways of Seeing Things
In this session, the concept of a relation is explored with a more complicated spatial pattern.
- Show the class the pattern below that is made up of matches. The 1st, 2nd, and 3rd terms of the sequence are shown.
- Challenge the students with this problem:
Find many different ways to work out the total number of matches in Term 10.
- Remind students about the ways that Kiri, Jamie and Vey-un represented their patterns, including rules that work for any term.
- Let students work in pairs or threes. Ensure they record their thinking using diagrams and expressions. Do your students:
- Look for the growth between terms, i.e. 12 matches.
- Create tables of values to represent the number of matches for each term
- Use multiplicative strategies to predict the number of matches for term 10
- Gather the class to process the ideas. Highlight the efficiency of multiplicative strategies such as 10 x 12 – 8 and 4 + 9 x 12 compared to additive strategies like 4 = 12 + 12 + …
- Ask students to connect the numbers and operations in their expressions to the figural pattern of matches.
Why is the number of matches increasing by 12 each term?
How many groups of 12 matches will be in the 10th term?
Why does Kiri subtract 4 at the end?
- How could our rules be used to predict the number of matches needed for Term 23? Term 101? Term n?
- If Taylor uses 604 matches to build a figure in this pattern, what Term does she make?
- To assess the ability of students to personally make predictions and create general rules pose this assessment task.
Here is a pattern of growing stars made with matches.
How many matches are needed to make Term 15, that has 15 stars?
Can you write a rule for the number of matches needed to make Term n, any term?
If you have 244 matches, what is biggest number of stars you can make in this pattern?