Diamonds & Rectangles

Task 40 ... Years 4 - 10

Summary

Shapes are sorted into categories, then placed in bags which are all incorrectly labelled. Is it possible to make one selection from one bag and be able to work out what is in all the bags? That's the challenge, and once solved the next challenge is to be able to explain your reasoning to someone else.

Materials

3 containers and 3 label cards
6 diamonds and 6 rectangles

Content

problem solving strategies such as:
- trying every possible case
- if-then reasoning

Iceberg

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

Note: Label cards may have to be cut out as separate cards before the task is used.
Some teachers may prefer to rename the task 'Rectangles & Rhombi' on the grounds that a diamond is really a three dimensional object.
Some teachers may prefer to read the answer below before trying the problem themselves. That would be a pity. Please try it before you read the solution process below. Use any objects you have nearby - $1 and $2 coins for example.

The critical steps in the solution are:

Choose from the Mixed bag first. Whatever you choose tells you what is in that bag, because the wrong labelling implies it is not a mixture. Therefore the only possibility is a bag of what you bring out.
There are two bags remaining. One is incorrectly labelled with the name of the shape you just discovered. Forget this bag for a minute and consider the other one.
This other bag can't be what it is labelled; and it can't be what you just discovered in the Mixed bag. So it must be the only remaining choice.
Now you know what two bags must be so you also know the third.

The problem is designed to give a unique solution, but what happens if only two bags have been mislabelled? Explore the possibilities. Suppose we are told only one bag has been labelled correctly. Explore the possibilities?

There is another problem lurking here which would be better explored using the same shape (or object) in two colours, such as the counting bears suggested for the class lesson below.

Put all 12 shapes in a bag. Select two at random. What is the chance your selection will be a pair? What happens if you change the mix in the bag?

(This problem is a bit difficult to address satisfactorily with the equipment in the task because the students can feel the sharper points of the 'diamonds'.)

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.

Of course it is not necessary to use diamonds and rectangles for this task. Any objects or shapes will do as long as they can't be seen inside the container. You might have toy farm animals (6 sheep and 6 horses for example) or counting bears or cubes in two colours (Red Bear Bag, Blue Bear Bag, Mixed Bear Bag). Containers might be margarine containers.

If you do want to use shapes then use a computer to design them, print them on card, and laminate so you have a permanent set.

The focus of a whole class investigation would be:

strategies for solving the problem,
what happens if...?
ways the solution could be explained to others.

Communicating mathematics is a very important part of a mathematician's work. Would students create a poster, a Power Point, a sequence of digital photos, a comic strip...?

At this stage Diamonds & Rectangles does not have a matching lesson on Maths300.

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 Diamonds & Rectangles task is an integral part of:

MWA Space & Logic Years 5 & 6

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

Follow this link to Task Centre Home page.