# Unseen Triangles

### Task 179 ... Years 4 - 8

#### Summary

This is another of several tasks which begin with an easily accessed concrete/visual pattern, but lead on to considerable algebraic possibilities. Students are asked to make a range of 'mountain peaks' which look like each mountain partially obscures the one behind. The main challenge in the problem is:
• If I tell you the number of mountain peaks, can you tell me the number of 'matches' I need to make it?
This task is a partner for Task 178, Match Triangles, Task 154, 4 Arm Shapes, Task 147, Garden Beds and others. Using a suite of tasks like this means that algebra becomes concrete and visual, and it makes sense.

Unseen 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.

#### Content

• algebra, concept of a variable / function
• algebra, equivalent expressions
• algebra, factorisation
• algebra, generalisation in words & symbols
• algebra, linear
• equations, creating
• equations, substitution & solution
• graphical representation
• mental arithmetic
• patterns, number
• patterns, visual
• recording mathematics #### Iceberg

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

 No. of Peaks 1 2 3 4 5 ... 10 ... 100 Number of Matches 6 10 14 18 22 ... 42 ... 402

and students should keep this record in their journal.

The iceberg begins with the last challenge: Find at least one other way to explain it to me. Why? Because when a mathematician is working on a problem there is no one else who they can ask about whether they are correct. Therefore they must ask:

• Can I check it another way?
There are at least three 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 four for each mountain that is partly seen, then and add six for the one in full view.
• Generalisation B
To find the number of matches count six for every mountain, then subtract two for each of the mountains except the one in full view.
• Generalisation C
To find the number of matches count four for every mountain, then add two to complete the one mountain peak in full view.
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 four for each mountain that is partly seen [4(P - 1)], then and add six (+ 6) for the one in full view.
• Generalisation B
To find the number of matches (M =) count six for every mountain (6P), then subtract two for each of the mountains except the one in full view [-2(P - 1)].
• Generalisation C
To find the number of matches (M =) count four for every mountain (4P), then add two to complete the one mountain peak in full view (+ 2).
which become:
• Generalisation A ... M = 4(P - 1) + 6
• Generalisation B ... M = 6P - 2(P - 1)
• Generalisation C ... M = 4P + 2
These are all equivalent algebraic expressions and, by reference to the physical pattern, students will be able to tell you what each symbol means and why particular operations and numbers are there. Interestingly also, the simplest algebraic form is not the one students necessarily think of first. Perhaps that is because the natural way for some to make the range of peaks is to start with the fully seen one, but for whatever reason, the simplest form is not necessarily the most logical.
• If given a table of values like the one above in a textbook and asked to find the rule, which form would you expect to find in the back of the book?

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 peaks in the mountain range I could build?
• Can I tell you any number for the number of matches? Discuss.
• Choose any five numbers for the peaks. Work out the number of matches in each case and make pairs of numbers like this (peaks, 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? Are these girls on the way to developing a pattern of diamonds (rhombii)?

All tasks have three lives. The task life (above) is an invitation to work like a mathematician. To this can be added an Investigation Guide which leads students deeper into the iceberg of a task. Staff construct the Guide in advance based on the extension questions. 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.

Generally, the use of an Investigation Guide follows the completion of the card and a discussion with the teacher which helps to reveal something of the extension questions. Then the teacher asks:

• Okay, would you like to tackle some of these questions now?
If yes, the teacher tells the students where to find the Investigation Guide and they begin. If no, the students complete their journal entry for the task by recording one or two questions they can start with next time they come back to this task.

Guides like this are 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.

Investigation Guides 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 been submitted by Jodi Wilson and Maria Antoniou, Mt. Eliza Secondary College, and another from Damian Howison and the staff from Mackillop College, Swan Hill.

#### 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 mountain peaks is quickly constructed. You can see photos of this approach in the Match Triangles cameo. Discussion and challenge begins with the floorboard model and the table top models are then used to explore and confirm hypotheses before returning to the public model for discussion and extension.

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 20, Unseen 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 Unseen Triangles task is an integral part of:

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

The Unseen Triangles lesson is an integral part of:

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

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