Minds On

Making shapes

How many shapes can you create that are made up of 4 squares?

4 identical squares.

You could join the 4 squares together to make a bigger square. The following image is an example:

4 squares two by two to make a big square.

Now it’s your turn! Create as many different shapes as you can using 4 squares.

Record your shapes and ideas in your notebook or another method of your choice.

Once you have tried to create different shapes, consider the geoboards in the following Sample Answers button.

Press the ‘Sample Answers’ button to reveal some possible shapes.

3 squares in a row with 1 square below the square on the right. 2 squares stacked on each other. Another 2 squares are joined to the right of the bottom square. 2 squares stacked vertically with 2 squares to the right of the bottom square. 3 squares side by side in a row. There is a square below the square in the middle. 3 squares side by side in a row. There is a square below the square in the middle.

Are the shapes from the ‘Sample Answers’ similar to the shapes you created?

What is the same about all the shapes? What is different?

When we combine shapes or put them together to make larger shapes, this is called composing. In this task, we combined 4 squares to make larger shapes.

Action

It’s all about the area!

Area tells us about the amount of space in a shape.

All the shapes from the Minds On section had the same area. Each shape was made with 4 squares.

The size of the square and the number of squares used stayed the same.

This means the area stayed the same too.

3 squares in a row with 1 square below the square on the right. 2 squares stacked on each other. Another 2 squares are joined to the right of the bottom square. 2 squares stacked vertically with 2 squares to the right of the bottom square. 4 squares in a row. 3 squares side by side in a row. There is a square below the square in the middle.

Decomposing shapes

In the Minds On activity, we combined squares to create new shapes.

We can also break larger shapes into smaller shapes. This is called decomposing shapes.

Let’s think about decomposing a shape into smaller shapes.

Examine the following shape. What is the shape? You can record your answer digitally, orally, or in print.

A geoboard with a square outlined. The square is 4 units across and 4 units deep.

Press the ‘What is the Shape?’ button to reveal the original shape.

The original shape is a square.

How can the shape be broken (decomposed) into smaller shapes? You can record your ideas digitally, orally, or in print.

Did you come up with some ideas of how the shape can be decomposed? When you’re ready, press the ‘Decomposing the Shape’ button to reveal some examples.

The following images are some ways the square can be decomposed. A square on a geoboard split into 2 rectangles. A square on a geoboard split into 2 triangles. A square on a geoboard split into three rectangles and a square. A square on a geoboard split into 16 smaller squares.

What shapes can you identify from the decomposed shapes?

Press the ‘Shapes in the Square’ button to reveal some shapes we can identify.

The shapes in the square are triangles, rectangles, and smaller squares.

Is the area the same?

Examine the following pairs.

Do both images have the same area? Select “yes” or “no” for each pair of shapes.

Composing shapes

We can decompose a shape and move the pieces around to compose another shape.

If we use all the same pieces, the area will stay the same.

Student Tips

Review

Before we move on, let’s review some important words!

  • Compose is to put together or to combine.
  • Decompose is to take apart or break apart.

Is the area the same? (part 2)

Examine the following pairs.

Do both images have the same area? Select “yes” or “no” for each pair of shapes.

We can decompose a shape into smaller shapes and move the pieces around to compose another shape. The area will stay the same.

Brainstorm

Pause and reflect

What did you notice about the area of the shape when you moved around the squares?

How did you know the area of two shapes was the same?

Record your ideas in a method of your choice.

Consolidation

Making and breaking

In the following tasks, you will be decomposing and composing shapes! You can use an online geoboard or another method of your choice for these tasks.

Task 1: Decomposing

A square being split diagonally into 2 equal right-angle triangles.

Create a large shape using a method of your choice.

After you have made your shape, decompose the shape into:

  • two smaller shapes
  • three smaller shapes
  • four smaller shapes

What are the smaller shapes?

How does the area of all the smaller shapes compare to the area of the large shape?

Record your answers digitally, orally, or in print.

Task 2: Decomposing and composing

Two squares stacked on top of each other next to a square being split diagonally into 2 equal right-angle triangles.

For this task, you will create a large shape using a method of your choice.

Then, you will decompose! Break the shape down into 4 smaller shapes.

After that, you will compose! How many different larger shapes can you make by combining the small shapes?

Let’s explore an example! Press the following tabs to access steps you may take to create, decompose, and compose a shape.

First, I create the following shape on a geoboard.

An irregular shape on a geoboard.

Next, I broke the shape down into smaller shapes. The following image displays the broken down shapes.

A shape on a geoboard. It is made of a square, rectangle, big triangle, and small triangle.

I chose a square, a rectangle, and two different triangles.

A square, rectangle, big triangle, and small triangle.

Then I rearranged my smaller shapes to make new ones. I used the shapes to compose new shapes. The following images display the new shapes.

A shape made of a square with a rectangle on 1 side and 2 triangles on the other side. A square and rectangle beside each other. There are 2 triangles on the other side of the square. A square, rectangle and 2 triangles combined. The shape is similar to a house with a roof.

Now it’s your turn. Create, decompose, and then compose!

Record your answer using a method of your choice.

Composing some final thoughts

In the Action section, we learned that two-dimensional shapes can be combined to make larger shapes or broken up to make smaller shapes. We called this composing and decomposing shapes.

We also learned that if we break a shape in to smaller parts and then put those parts back together in a different way, the area of the shape (how much space it covers) stays the same, even though the shape has changed.

Examine the new shapes you created with the smaller shapes.

How can you prove that the area (the space the shapes cover) is the same as your original shape?

Reflection

How do you feel about what you have learned in this activity? Which of the next four sentences best matches how you are feeling about your learning? Press the button that is beside this sentence.

I feel...

Now, record your ideas about your feelings using a voice recorder, speech-to-text, or writing tool.