# Equilateral Triangles

### Task 170 ... Years 2 - 8

#### Summary

Designed to invite younger students to think 'outside the square' this task hints at both algebraic pattern and the idea that three dimensional objects are made from flat surfaces which have familiar names. The doors it opens are extensive. The task is a partner to Six Square Puzzle which offers related challenges in the context of squares.

#### Materials

• 9 equal rods and some blu-tac

#### Content

• fractions, value relations
• patterns, number
• patterns, visual
• shapes, recognition
• shapes, properties
• spatial perception, 2D or 3D #### Iceberg

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

The solutions to Questions 1 and 2 are:  The answers to the two challenges are:  The first diagram is a 2D representation of a 3D object.

At this point the students might consider they have finished the task, but no. It is important with every task that students are encouraged to see that there can be more. For example, in this case we could suggest:

• Challenge 1 and Challenge 2 are connected. Explain how.
and if they can't see that Challenge 2 is the net of Challenge 1, then the statement is recorded in their journal as something to consider when they revisit this task. Another development from the challenges is to explore the language 'faces, edges and vertices' in context and introduce the name 'tetrahedron' for the object they have made. This can be extended further with Challenges such as:
• The ancient Greek mathematician Plato showed there are two other objects that only have equilateral triangle faces. Find the names of these objects.
• How many faces, edges and vertices do these other objects have?
Work with naming objects and their parts can be extended with Task 226, Playing With Objects.

Questions 1 and 2 can also be extended with, in either case, questions like this: There are two triangles in this 'chain'. Extend the same chain so that it has 3 triangles, ... 4 triangles, ...10 triangles.

• If I tell you any number of triangles can you tell me the number of sticks you need to make the chain?
• If I tell you any number of sticks I have can you tell me the number of triangles in the longest chain I can make?  Number pattern work with triangles is extended further with Task 178, Match Triangles, and Task 179, Unseen Triangles.

Challenge 2 suggests a number pattern too. It is a template for making a new equilateral triangle from 4 unit triangles. That means that four copies of the Challenge 2 result would make the next size equilateral triangle and it would include 16 unit triangles. But how many triangles altogether in this larger diagram? This mathematics is extended in Task 42, Triangles Around Triangles and leads to a different algebraic link in Task 186, Tetrahedron Triangles.

In a different direction, Size 1 is the original unit triangle. Size 2 is the Challenge 2 result. Size 4 is this new triangle (why Size 4?).

• If I tell you any Size triangle created in this way, can you tell me the number of unit triangles inside it?
• If I tell you any Size triangle created in this way, can you tell me the number of sticks you need to make it? (more difficult)
And the backwards questions?

In this interpretation of the Challenge 2 result, the equilateral triangle grows as the construction pattern repeats. This type of tile - one that self-replicates - is called a Rep-tile. A 'sphinx', is another Rep-tile and Task 166, Sphinx, is explored in huge detail as a model for exploring the mathematics of Rep-tiles. Trisquares are Rep-tiles too and Tricubes carry the same mathematical questions into three dimensions.

Lastly (or is it?) the Challenge 2 result opens the door to Value Relations questions such as:

• If the unit triangle is worth 1 (... 3, 1.5, \$2.70, 1/3, ...) what is the larger triangle worth? ... what is the rhombus worth?
And the backwards questions?

#### 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.

Straws or popsticks or similar material and a little blu-tac per pair is all you need to turn this task into a whole class investigation that will allow you to take any of the directions above. The investigation could also be supported by writing Investigation Guides that provide a set of further questions to lead students into any of these directions. When you have created and trialed Guides for this activity we would be happy to share them through this cameo.

At this stage, Equilateral Triangles does not have a matching lesson on Maths300, however Lesson 107, Newspaper Shapes, explores making 2D patterns and 3D objects with newspaper tubes and Lesson 164, Match Triangles, uses newspaper tubes and popsticks to explore more deeply the algebra opportunities which arise from the Question 2 solution above.

#### 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.

Equilateral Triangles is not in any MWA kit. However it can be used to enrich the Space & Logic kit at Years 3/4 and the Pattern & Algebra kit at 7/8. This task is included in the Task Centre Kit for Aboriginal Students with the title Triangles Galore. 