# Growing Trisquares

### Task 238 ... Years 4 - 10

#### Summary

The crux of this problem is that 4 Trisquares can be joined to make a new, scaled up, Trisquare. This provides a 'template' for constructing the next size, and the next, and the next... The visual pattern can also be represented as a number pattern and this leads to graphing, equation work and scale factors.

This task is a partner to Task 237, Trisquares. Between them present the mathematics which also appears in Task 166, Sphinx and its Sphinx Album. Growing Trisquares is also related to Task 115, Dividing Shapes.

This cameo has a From The Classroom which shows magnificent work from a Year 9 class at Wade High School who were so enthusiastic about the challenge that they cut out masses of paper Trisquares to complete an incredible set of posters.

#### Materials

• 8 trisquare shapes or cut your own from this Trisquare Board
(Suggestion: Print on thin card and laminate before cutting)
• Graph paper

#### Content

• spatial perception
• perimeter & area
• square numbers
• powers of 2
• proportion
• similar shapes
• value relations and fractions
• prime and composite numbers

#### Iceberg

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

The starting point for this investigation is the solution for Question 1. Everything else grows from this.

This solution is a template for showing how 4 Trisquares must be arranged to form a larger, proportional Trisquare - a similar shape. However, the solution itself is a Trisquare, so 4 of this size can be made into the next size Trisquare ...and 4 of those into the next size and so on to approach an area of infinity. (You might also like to explore the mind experiment of imagining the four smaller trisquares that make the original ...and the four smaller trisquares from which each of those is made ... and so on to trisquares with an area approaching zero.

At a mathematics workshop for students in Wagga Wagga, these children were very excited about combining their Trisquares to make a Size 4.

As the size of the Trisquare grows, a mathematical relationship develops between that number and the number of Trisquares units making it up.

 Size Trisquares 1 1 2 4 4 16 8 64 ... ...

Perhaps Jenny saw this pattern, thought about Size 3 then guessed she might need 9 unit Trisquares and that's how her challenge on the card came about. Of course, if Size 3 can be made, then we must ask about Sizes 5, 6 and 7 and so on. Size 6 could be interesting. Perhaps it could be made from either four Size 3s or nine Size 2s?

To round off the investigation, one might ask what would happen if a Trisquare became a Tricube.

#### 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 lesson to a whole class investigation you will need lots of Trisquares. You could save this drawing, import it into a word processor and make a master from which students cut their pieces...

...or you could use the class set of Trisquares available from Mathematics Centre.

The photos in the From The Classroom section below probably say all that needs to be said about the whole class lesson possibilities of this task.

At this stage, Growing Trisquares does not have a matching lesson on Maths300. However Lesson 25, Sphinx, is built around the same mathematics and includes a broad suite of examples from classrooms around the world.

#### 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.

Growing Trisquares is not in any MWA kit. However it can be used to enrich the Pattern & Algebra kit at Years 5/6 and the Pattern & Algebra kit at Years 9/10.

## From The Classroom

#### Wade High School Griffith, NSW

Year 9
This group of students were not known for their efforts in mathematics, but Task 115, Dividing Shapes, caught their imagination and led to this amazing work with 'home made' Trisquares. As the Trisquares grew so did the students understanding of number patterns, number pairs, graphing, perimeter and area.