Lesson outline

This task uses Polypad’s tetrominoes as a context for exploring enlargements (dilations) and scale factors. Tetrominoes are shapes which can be made from 4 individual squares, and will be instantly recognisable to any student who has played the classic game Tetris.

Lesson objective: Understand how the area of a shape changes when it is enlarged.

Lesson activity

Warm-up

Show students this Polypad canvas. Explain that two of the tetrominoes have been enlarged (dilated) correctly, but two of them have mistakes. Invite students to work out the scale factor by which the two correct enlargements have been drawn, and to use the pen tool (or copy the diagrams onto squared paper) to correct the mistakes in the incorrect enlargements.

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/904d8bfd-5cc7-4fb6-a2f1-0f46d841e473/Screen_Shot_2021-02-21_at_3.22.47_PM.png

https://mathigon.org/polypad/Ri6FFre4bs0x1w

Solution: The square has been correctly enlarged by a scale factor of 3, and the L tetromino has been correctly enlarged by a scale factor of 5. The T tetromino and the Z tetromino have not been enlarged correctly.

If students haven’t seen tetrominoes before, and you think they could benefit from additional scaffolding before the warm up above, you could challenge them to find all the different arrangements of four 1-square tiles. They can use this canvas as a starting point.

Main Activity

Some shapes can be used to tile an enlargement of themselves – this is sometimes called a rep-tile. In this activity, students will explore how the L tetromino can be tiled to make enlargements of itself.

First, invite students to draw enlargements of the L tetromino by a scale factor of 2, 3, 4 and 5, using the pen tool. Students should enable the grid background using the grid button on the right.

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/cb0392d8-99e6-4dc1-9011-b4299667428d/image2.png

Next, challenge your students to fill each of these outlines using copies of the green L, without any gaps or overlaps!

If students are struggling, you can relax the constraints and allow them to use a mixture of different tetrominoes. Since all tetrominoes have the same area, this won’t affect the following questions.

There are many different possible solutions, but here are some examples:

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/0fede667-4a99-48c6-a71e-8072c8ac74f1/image4.png

Once students have completed some tilings, ask them to record how the area changed for each shape they have tiled. Students could record their results in a table, like the example below. Consider whether your students could benefit from receiving a blank copy of the table to fill in.

https://s3-us-west-2.amazonaws.com/secure.notion-static.com/09c5f821-904f-4326-8d48-3fef73f4ef8c/Screenshot_2021-02-14_at_13.06.26.png

It may not be immediately obvious to students that the area of each shape is four times the number of tetrominoes used, since each tetromino has an area of four. Make sure this emerges during any class discussion.