Developing the Concept of similarity

Simple practical tests for similarity

The One Eye Test

Once created these shapes can be used to develop the concept of similarity using simple practical tests. The first is the “One eye test”. Hold two of the rectangles, one in each hand. Now close one eye. Adjust the shapes so that the image of one overlaps the other. If the shapes are similar then the nearer shape can be made to perfectly cover the farther one.

Nesting Shapes

Another property that emerges from the family of folded shapes is that the can be nested to create a growing pattern.

Make two families of rectangles from two A4 sheets of different colour.

Nest the rectangles inside each other. Here are a couple of examples of the shapes you can make.

When this is done then pupils will discover that corresponding corners lie on straight lines. Starting with families of rectangles can create their own shapes.

Click here to see some of the patterns you can make with these rectangles

Note how the corners of the similar rectangles lie along the diagonal of the rectangle.

Similar shapes of many kinds can be made from the family of A rectangles. For example if you fold corner to opposite corner you can make a pentagon.

Do this with rectangles of different sizes and try out the One Eye Test with them. Or build them into nesting patterns

Similarity in 3 dimensional shapes

The property of similarity is of course maintained if the “A” size rectangles are made into 3 dimensional shapes.

Here tetrahedrons are folded from A4 and A6 rectangles.

First find the mid points of each side by folding. Crease along all the dotted lines. Then fold up along these creases. Use tape to join sides.

Bring the two halves together to create the tetrahedron.

The tetrahedrons made in this w32ay from any A rectangles will be similar.

One fascinating property of the Tetrahedron formed in this way is that it will tessellate in 3 dimensions.

Make a number of A6 tetrahedrons and build them into a larger tetrahedron and see how many it takes. This is a very good way to discover the relationship in volume between similar solids.

For more 3D shapes that can be folded from A paper click here.