The purpose of this activity is to engage students in applying transformations to create repeating tiling patterns.
The background knowledge presumed for this task is outlined in the diagram below:
This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.
The arithmetic approach (show more)
- The student uses trial and error approaches with some acknowledgement of angles and side lengths to tile the 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.Click on the image to enlarge it. Click again to close.
The mountain reptile is harder to solve as there are two possible orientations for the starting shape.Click on the image to enlarge it. Click again to close.
Students using a trial and error approach are likely to find some simple Rep-4 shapes, such as the square, equilateral triangle and parallelogram.Click on the image to enlarge it. Click again to close.
The procedural approach (show more)
- The student applies the properties of transformations, including enlargement, to find how four copies of the shape form a larger version of itself.
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.Click on the image to enlarge it. Click again to close.
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:
Click on the image to enlarge it. Click again to close.