Match Triangles

Task 178 ... Years 4 - 8

Summary

This is one of several tasks that begin with an easily accessed concrete/visual pattern, but lead on to considerable algebraic possibilities. Students are asked to make triangle chains, similar to those used in engineering for strengthening structures. The main challenge in the problem is:
  • If I tell you the number of triangle sections, can you tell me the number of 'matches' I need to make it?
This task is a partner for Task 154, 4 Arm Shapes and others. Using a suite of tasks like this means that algebra becomes concrete and visual, and it makes sense.

This cameo has a From The Classroom section which has a PowerPoint report from a primary school class and the cameo also includes an Investigation Guide.

Match Triangles also appears on the Picture Puzzles Pattern & Algebra B menu where the problem is presented using one screen, two learners, concrete materials and a challenge.

 

Materials

Content

  • basic number skills
  • seeking & seeing patterns
  • generalisation
  • equivalent algebraic expressions
  • symbolic representation
  • substituting into equations
  • solving equations
  • graphing ordered pairs
  • relationship to gradient and y intercept
Match Triangles

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

   

Answers are:

No. of Triangles 1 2 3 4 5 ... 10 ... 100
Number of Matches 3 5 7 9 11 ... 21 ... 201

and students should keep this record in their journal.

The iceberg begins with the last challenge: Explain your answer.
There are at least four ways to do this and it is important to realise that the way the student 'sees' is the one that makes sense to them. The way we 'see' the generalisation may be different, but it is not more correct.

  • Generalisation A
    To find the number of matches count two for each triangle, then and add one to close off the end.
  • Generalisation B
    To find the number of matches start with one, then add two for every triangle.
  • Generalisation C
    To find the number of matches multiply the number of triangles by three (3 matches to make a triangle), then subtract one less than the number of triangles (because there will be double matches at all the joins).
  • Generalisation D
    To find the number of matches start with 3 (the first triangle), then add two for each of the remaining triangles.
Encourage students to record their view and at least one other. Once the generalisation has been made orally, record in words as here. The written words are the genesis of symbolic representation as an equation:
  • Generalisation A
    To find the number of matches (M =) count two for each triangle (2T), then and add one to close off the end (+ 1).
  • Generalisation B
    To find the number of Matches (M =) start with one (1), then add two for every triangle (+ 2T).
  • Generalisation C
    To find the number of Matches (M =) multiply the number of triangles by three (3T), then subtract one less than the number of triangles (- [T - 1]).
  • Generalisation D
    To find the number of Matches (M =) start with 3 (3), then add two for each of the remaining triangles (+ 2[T - 1]).
which become:
  • Generalisation A ... M = 2T + 1
  • Generalisation B ... M = 1 + 2T
  • Generalisation C ... M = 3T - (T - 1)
  • Generalisation D ... M = 3 + 2(T - 1)
These are all equivalent algebraic expressions and, by reference to the match triangle pattern, students will be able to tell you what each symbol means and why particular operations and numbers are there.

Extend further with questions such as:

  • Do these different ways of seeing the pattern give the same answers for 5, 17, 26 triangles?
  • Suppose I tell you the number of matches I have. Can you tell me number of triangles in the chain I could build?
  • Can I tell you any number for the number of matches? Discuss.
  • Choose any five numbers for the triangles. Work out the number of matches in each case and make pairs of numbers like this (triangles, matches). If these pairs were plotted on a graph what would you expect to see? Plot them to check your hypothesis.
  • If you joined up these dots with a pencil line, how could you measure the slope (gradient) of the line? Which number does it go through on the vertical axis?
  • What happens if we change the match pattern?
All tasks have three lives. The task form (above) is an invitation to work like a mathematician. To this can be added an Investigation Guide to lead students deeper into the iceberg of a task. This is the second life. The third life is as a whole class investigation to model the work of a mathematician. This third life is described below.

Usually teachers prepare their own Investigation Guides but in this case two teachers have gone further. Jodi Wilson and Maria Antoniou, Mt. Eliza Secondary College, have prepared their own Guide and then offered it to colleagues through this cameo. We very much appreciate this form of sharing and encourage others to do the same.

There are also Guides like this included for the 20 tasks in the Maths With Attitude Pattern & Algebra Years 7 & 8 kit and for 10 tasks in the Maths With Attitude Pattern & Algebra Years 9 & 10 kit. Guides such as these can lead to students publishing a report of their investigation. See Recording & Publishing for examples of student reports in various forms and see Assessment for a rubric for assessing such reports that has also been submitted by Jodi & Maria.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

   

To convert this task to a whole class lesson with the purpose of addressing all the mathematics above while modelling what it means to work like a mathematician, you need plenty of pop sticks or an alternative such as straws cut to size. However, for a greater level of involvement, which is sensible management because it adds purpose to using the sticks before they are distributed, begin with each student quickly preparing a newspaper roll. Students bring this to a central floor space and a whole class model of the triangle chain is quickly constructed. Discussion and challenge begins here and the table top models are then used to explore and confirm hypotheses before returning to the public floorboard model for discussion and extension. (See From The Classroom below.)

Many teachers report that an added advantage of lessons like these is that when the next lesson is a related 'toolbox lesson', perhaps from a text book, students often comment that the ...stuff in the book is simple.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 164, Match Triangles.

    

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The Match Triangles task is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Pattern & Algebra Years 7 & 8

The Match Triangles lesson is an integral part of:

  • MWA Pattern & Algebra Years 7 & 8

Match Triangles task is also included in the Task Centre Kit for Aboriginal Students.

From The Classroom

St. Monica's Primary School
Evatt

Sarah Collis
Year 5
Miss Collis called this an 'aha' moment and couldn't stop smiling and we felt successful.

So say the Year 5 students of St. Monica's Primary School. Sarah writes:

Match Triangles was our first whole class investigation and the other teachers were wondering how they could implement them into their maths program. So the students developed this short PowerPoint explaining our steps and sharing our experience to demonstrate how engaged the children were and how much they enjoyed the challenge of going beyond what was on a piece of paper.

Green Line
Follow this link to Task Centre Home page.