Reptiles are shapes that tile to make a larger version of themselves. Here is an example, a trapezium. Notice how the whole is made up of four identical parts.

What transformations (translations, reflections, rotations) map the top trapezium onto the other three?

How might 16 copies of the shape tile to form an even larger copy of itself?

Here are some other reptiles. Four copies of each shape will tile to form a larger copy of itself. Figure out which transformations are needed.

Notice that these shapes are Rep-4 because you need four copies to make a larger whole of the same shape.

Can you find some other Rep-4 shapes?

The examples of Rep-4 shapes are simple enough that the tiling can be found by trial and error. However, the time a student takes to find the arrangement is greatly reduced if they attend to congruent angles. Students may produce cut-outs of the shapes and manipulate them to find the arrangement. Alternatively, they may draw large copies of the shape and subdivide it into four identical shapes. Square and triangular dot paper is useful.

Below the student looks for congruent angles and combines that approach with trial and error.

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The mountain reptile is harder to solve as there are two possible orientations for the starting shape.

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Students using a trial and error approach are likely to find some simple Rep-4 shapes, such as the square, equilateral triangle and parallelogram.

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The first important principle to notice about Rep-4 shapes is that the side length of the larger shape is twice that of the smaller shape. If side length is doubled then the area of the resulting shape is 2 x 2 = 4 of the initial shape.

Students who attend to this property of enlargement, and look for congruent angles, are likely to solve the puzzles more easily. They will also notice the transformations that map one shape onto the adjacent one.

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Creation of ‘new’ reptiles is also easier if students look at the transformations that occur with the given examples.

To test a shape for reptilian tendencies it seems that translation is the easiest transformation to attempt first. If that does not work, then try other possible transformations like reflection and rotation. Even so, non-reptiles are much easier to find than reptiles. Suggest to students that they start with shapes composed of squares and equilateral triangles. Simple shapes are more readily tiled than complex shapes. Here are some examples:

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