Cuboids: a context for learning

I recently began teaching my S1 class (12-year-olds) about volume. This is a topic most of them had met at primary school – they were keen to tell me they knew that

Volume = length x breadth x height.

In addition, since joining my class, they have learned to find the areas of rectangles and right-angled triangles using formulae, and can find the area of composite 2-D shapes involving combinations of these. While teaching them about area I had used both enactive and iconic representations to scaffold their learning. For example, we drew some rectangles and triangles on paper and card, cut them out and combined them in a range of ways to create composite figures. We then drew these, before calculating their areas.

I was keen to build on this multiple-representational approach and decided to break out the recently-delivered Cuisenaire rods for the first time with a class. I took some photographs during the first lesson and posted one on Twitter which generated a few likes, retweets and helpful observations: clearly some other teachers were also interested in this approach. Having completed three lessons so far, I wanted to reflect on and try to make sense of this experience so far.

What follows here is not so much a record of the learning that has taken place with my class to date. It’s more about exploring how this approach can help me improve my teaching of this and other topics going forward. Hopefully, readers may also benefit in terms of the learning and teaching in their classes too. For the most part, it consists of questions and images. These have been designed with my own class in mind. The question-set is not meant to be either definitive or exhaustive. It will need to be adapted, extended, supplemented, pared back or, like the physical models that the pupils and teacher build with the rods, deconstructed and reconfigured to suit different learning contexts. As Caleb Gattegno – who did most to popularise these rods devised by Georges Cuisenaire – asserted, the act of teaching should be subordinated to the process of learning.

What do you notice about the image below?

What questions do you have?

Could you build a copy of this shape using white rods only?

How many white rods would you need?

How do you know?

Could you build another using red rods only? How many?

Could you repeat this single-colour task with any other colours? Which ones? How do you know?

The number of white rods used to make a larger shape can help us describe its volume. Can you explain why?

How many faces does a white rod have? Can you see them all at the same time?

If two white rods are placed side-by side, how many faces can you see? (You are allowed to pick up and turn the shape around – but you must keep the two rods together, as if stuck with glue.)

What other colour of rod would this most resemble?

What about three white rods stuck together? Or four? Or more?

The number of white faces that can be seen if we are allowed to turn a shape around can help us determine the surface area of the shape.

What questions do you have?

What are the volume and surface area of the large shape in the first picture? (Remember to include the faces that could be seen if you were able pick it up and turn it around.) How did you arrive at your answers?

In mathematics, we say shapes that are exactly the same as each other are “congruent”. One way to check this is to put one on top of the other. If they can be made to fit exactly we know the two shapes are congruent.

Imagine we made two congruent copies of the shape in the picture and put them one on top of the other. Can you describe how it would look? Does the new shape have a name?

Here is a much smaller cuboid made of different rods.

What are its dimensions? What is the volume of this shape? What is its surface area?

How do you know?

Here are some cuboids made of discrete layers. (What do you think this means?)

In what ways are these three cuboids the same? In what ways are they different?

Here’s another.

Can you make a copy of this cuboid using rods of just one colour? And another using a different colour? And another …?

Imagine we added another congruent layer on top. What would the volume and surface area of the new cuboid be?

What if we added a new layer on the front, or on the side? What are the volumes and surface areas of the resulting cuboids?

Make a cuboid with volume 48 cubic units.

What is it’s surface area?

Make another with the same volume but different dimensions. What is the surface area of your new cuboid? Can you make another? How many different ones could you make? How do you know?

Chose a volume and make as many cuboids as you can with that volume. Work out and note down the surface area of each one.

What questions do you have?

Here is a composite shape made from groups of red, blue, green, black and yellow rods. What is the volume of the composite shape? What is its surface area?

Here is a photograph of a cuboid taken from directly above so that we can only see its top surface. What two dimensions do we know and what are their values? If I tell you its volume is 150 cubic units, can you work out the value of its other dimension?

Here’s another viewed from one side. What dimensions do and don’t we know from this picture? If it has a volume of 168 cubic units, what is its surface area?

Here is a model of a staircase seen from one side. What do you notice about it? What do you wonder? Can you create some questions for others to work out?

Thank you, as ever, for reading my thoughts about learning teaching and doing mathematics. I hope your time spent here proves worthwhile.

2 thoughts on “Cuboids: a context for learning

    1. Thanks Peter. I didn’t actually pose that question to the class – I haven’t completed the teaching sequence. If/when I do, I would be looking for a range of values for all three dimensions.

      Like

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