Task 237 ... Years 2 - 10
SummaryTrisquares, 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 they 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 Picture 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.
IcebergA 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:
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:
This type of reasoning also applies to the challenge question, but the one 4-Trisquare shape students sometimes don't find is:
Can you make our shape?
Ja easy! ... Huh? ... Nej?
Whole Class InvestigationTasks 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. A print master is supplied above in materials. Or you could make, or have made, a class set of Trisquares. 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
|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.
|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.
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.