Trisquares

Task 237 ... Years 2 - 10

Summary

Trisquares, as their name implies, are made from three squares with matching edges. They are much more interesting than squares and in this task students have an opportunity to explore the wide variety of shapes which can be made from them and to link the exploration to perimeter and area.

This task is a partner to Task 238, Growing Trisquares. Between them present the mathematics which also appears in Task 166, Sphinx and its Sphinx Album. Trisquares is also related to Task 115, Dividing Shapes.

This cameo has a From The Classroom section with snippets from a Year 5 class using Tricubes (they didn't have Trisquares) to develop the differences and connections between perimeter and area.

Trisquares also appears on the Picture Puzzles Shape & Measurement A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. It is the basis of two puzzles - one related to building shapes with trisquares and tessellating, and the other related to measuring area and perimeter using a square as the unit.

 

Materials

Content

  • spatial perception
  • logical strategies
  • perimeter & area
  • value relations and fractions

Trisquares

Iceberg

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

There are many possible shapes students could draw to answer Question 1. For example:



How many 2-Trisquare shapes are there? We don't know ... but the card does suggest a way to find out and the five examples above are a starting point. There is one Trisquare in each of these drawings which is oriented in the same way, in other words, it is stationary and the other one is in various positions around it. Some students might like to take on this challenge and organise and record their search. We would be happy to publish student work.

Another approach would be to keep a class display of 2-Trisquare shapes which grows as each pair tries the task. We would also be happy to publish photos of your display.

Questions 4 and 5 on the card are designed to encourage students to realise that when two Trisquares touch under the rules, a section of the perimeter of each becomes an internal line. (Perimeter is measured by the side of a unit square.) So the shape with:

  • the shortest perimeter will turn the maximum number of perimeter units into internal lines.
  • the longest perimeter will turn the minimum number of perimeter units into internal lines.
The perimeter of one Trisquare is eight units, so the perimeter of two which don't touch is 16 units. The minimum amount of touch would be one unit to one unit, so the longest perimeter of 2-Trisquare shape would be 14 units. The maximum amount of touch would be 3 units to 3 units, so the shortest perimeter would be 10 units. There are several 14 perimeter shapes, but only one 10 perimeter shape.

This type of reasoning also applies to the challenge question, but the one 4-Trisquare shape students sometimes don't find is:

In fact, you may have to challenge students to find this new 4-Trisquare Trisquare. Once found it can become the focus of a different challenge - Master & Robot:
  1. The pieces are spread randomly on the table to start.
  2. One person is the Master and must keep their hands behind their back.
  3. One person is the Robot and must do exactly what the Master says. The Robot does not think for itself.
  4. The aim is for the Master to find the language to instruct the Robot to create the 4-Trisquare Trisquare.
  5. Reverse roles, then discuss and record the words which made it easier for the Robot.

Extensions

  1. Students draw their favourite 4-Trisquare shape in outline only on black paper and paste it to a white card. This becomes a challenge for card for others to reconstruct the way the shapes made the outline. As each pair creates their card, a class set of spatial puzzles builds up.

Teachers from Ístersund, Sweden, drawing and recreating 4-Trisquare shapes.

Can you make our shape?

Ja easy! ... Huh? ... Nej?

Ja!

 

 
  1. The cards also provide data for a discussion about creating 'families' of 4-Trisquare shapes using criteria such as symmetry, rotational symmetry, 'holes', polygons, concave, convex, obtuse angles, acute angles...
  2. At first look a Trisquare is made from 3 identical squares. But it could also be seen as constructed from 6 identical triangles, ...or 12 identical triangles, ...or 12 identical squares, ... or 2 identical trapeziums ...4 identical (smaller) trisquares.
  3. Value Relations questions are appropriate when a whole shape is divided into equal parts. A value is given to one piece (which may be a part or the whole) and the value of the other part(s) has to be calculated. For example:
    • The trisquare is worth 9, what is the value of the square ...the triangle ...the trapezium ...the smaller square ...the smaller triangle ...the smaller trisquare?
    • The trapezium is worth 3, what is the value of the trisquare ...the square ...the triangle ...the smaller square ...the smaller triangle ...the smaller trisquare?
    • The trapezium is worth 4, what is the value of the trisquare ...the square ...the triangle ...the smaller square ...the smaller triangle ...the smaller trisquare?
    • These questions can be explored using an Investigation Guide to extend the task. (When you create one we would be happy to share it through Mathematics Centre.)

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 ideas above easily develop into the lesson. Using square dot paper instead of line paper, as above, can lead to an interesting question of proportion (and the occasional error). For example do each of the 4-Trisquare shapes recorded in this student's work have an area of 12 squares?

At this stage, Trisquares does not have a matching lesson on Maths300. However, the Value Relations extension above is the focus of Lesson 99, What's It Worth?, which includes a Classroom Contribution with an Investigation Guide for Year 7, an extensive set of photos showing student work and companion software.

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.
   

Trisquares is not in any MWA kit. However it can be used to enrich the Chance & Measurement kit at Years 3/4 and the Number & Computation kit at Years 7/8.

From The Classroom

Engelska Skolan
Huskvarna, Sweden

Year 5
I was asked to run a Discussion Lesson about perimeter and area. I only had my Tricubes with me, Trisquares are designed for the purpose, so I had to make adjustments. None-the-less teachers thought the lesson worked quite well anyway (see the Features Daisy below) and it did have the advantage of opening the door to introducing other measurements such as volume and surface area.
Doug. Williams

Not surprisingly, the first few minutes of the lesson
had to be turned over to free play building.

Then we were all happy to concentrate for a while on
using just one Tricube.

 

How many ways can one Trisquare be placed on the table?
We found 4 ways and noticed that each way had a different base area

(0, 1, 2 or 3 squares) and only one of these looked like
a single storey building.

 

We drew a top view of this building and recorded
its base area and perimeter.
Then we explored the perimeter of single storey buildings
made from just two Tricubes.

 

There are many single storey 2 Tricube buildings. Of course they all have
the same base area, but what can we discover about their perimeters.
We chose one each that was different from everyone else on the table
and then made a (rather cramped) human graph of the perimeters.

 

  • What is the minimum perimeter?
  • What is the maximum perimeter?
  • How many 2 Tricube single storey buildings can we find for each perimeter between the maximum and minimum?
  • Why are there no buildings with odd numbered perimeters?

In discussion afterwards the first thing teachers did was identify features of the lesson which they felt had encouraged involvement and learning. The Features Daisy summarises that part of the discussion.

Then we explored what the next lesson might be. Perhaps buildings from 3 and 4 tricubes. Perhaps looking at any of the 2, 3 or 4 that make rectangles to notice the easy way to count the base area. Perhaps exploring the multiplicative growth in area and perimeter for the special case explored in the Task 238, Growing Trisquares.

One thing was clear though. With at least one more lesson like this, text book problems about area and perimeter were going to be no issue.

Green Line
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