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Home , Disphenoid, Honeycomb (geometry), Pyramid (geometry)

This is a set of activities using both Isosceles and Equilateral .

3D tasks

Technical information This pack includes 25 Equilateral Triangles and 25 Isosceles Triangles. You will also need a small tube of Copydex. If you are new to making models with Copydex we suggest you start by watching www.atm.org.uk/Using-ATM-MATs

Different polyhedra to make

Kit A: 4 Equilateral MATs. Place one triangle on your desk and join the other three triangles to its three sides, symmetrically. Now add glue to one of each of the three added triangles so that all can be joined into a

This is your first polyhedron.

Kit B: 3 Isosceles triangles and 1

Kit C: 4 Isosceles triangles

Kit D: 6 Equilateral Triangles

Kit E: 6 Isosceles Triangles

Kit F: 3 Equilateral and 3 Isosceles Triangles. (are there different possible polyhedra this time?)

Kit G: 8 Equilateral Triangles. One possibility is a Regular , but is there more than one Octahedron?

Kit H: 4 Equilateral and 4 Isosceles Triangles (are there different possible polyhedra this time?)

Kit I: 2 Equilateral Triangles and 6 Isosceles Triangles

Kit J: Go back to the Regular Octahedron (from Kit G) and add a Regular to one of its faces, then add another … until four have been added. Keep track (in a table) of how many faces and vertices the Regular Octahedron and the next four polyhedra have. Hint: if you spaced the added Tetrahedra equally around the Octahedron you will find a bigger version of a previously made solid.

If you have made a larger Tetrahedron, look at its edge lengths, and its faces. What ratios are to be found here? What ratio will there be between the ?

Use this knowledge to work out the ratio of the between a Regular Tetrahedron and a regular octahedron made from the same sized Equilateral Triangles. Knowing that the Regular Octahedron is a Square-based doubled, what can you find out about the ratio of volumes between the Square-based pyramid and Regular Tetrahedron?

Information: Regular Tetrahedra and Octahedra fit together into a that fills all of the space – like a 3D .

Knowing that, imagine you have a supply of (T) Regular Tetrahedra, (O) Regular Octahedra, and (S) Square that are half a Octahedron. How many of each polyhedron will be needed to make larger versions of these?

Notes and names for the polyhedra

Kit A Regular Tetrahedron. 4 faces, 4 vertices, 6 edges Kit B Isosceles Tetrahedron. 4 faces, 4 vertices, 6 edges. Kit C Tetrahedron. 4 faces, 4 vertices, 6 edges. Kit D Triangular Di-Pyramid Octahedron. 6 faces, 5 vertices, 9 edges. Kit E Isosceles Triangular Di-Pyramid. 6 faces, 5 vertices, 9 edges. Kit F 2 possible Triangular Di-Pyramids. 6 faces, 5 vertices, 9 edges. Composed of Kits A and B combined by adding or subtracting Kit A from Kit B. Kit G Regular Octahedron or Bi-augmented Tetrahedron. 6 faces, 5 vertices, 9 edges. Kit H Elongated Triangular Anti-. 6 faces, 5 vertices, 9 edges. The Isosceles Triangles stretch the central band of the Regular Octahedron so it is elongated. Kit I This is a similar problem to Kit F. There will be two polyhedron, a convex di-prism and a concave model. All of these di-pyramids are the structure of many crystals Kit J Enlarged Regular Tetrahedron. 4 faces, 4 vertices, 6 edges. Twice the edge lengths, four times the surface , eight times the volume. Several other polyhedron are possible! 2D tasks

Explore what you can make using just the Equilateral Triangles leaving regular gaps.

What fraction of the Equilateral triangle tile is yellow?

Next do the same task using just isosceles triangles. Again use regular gaps.

Look at the tile. What is the ratio of the two side lengths in this triangle? This tile is designed to have exactly half its area orange. What does this tell you about lengths in this triangle?

If you decide to use some, or all of these tasks with a class of pupils you will need to purchase 200 equilateral triangles and 200 isosceles triangles from ATM. MATs are available in packs of 100's (50's for Decagons and Dodecagons). See the ATM website for price details www.atm.org.uk/MATs

Available MATs Tiles

Equilateral triangles (x 100 MATs) Squares (x 100 MATs) Regular Pentagons (x 100 MATs) Regular (x 100 MATs) Regular Octagons (x 100 MATs) Isosceles Triangles (x 100 MATs) (x 100 MATs) Regular Decagons (x 50 MATs) Regular Dodecagons (x 50 MATs)

A short video is available showing how to make models using ATM MATs using copydex glue www.atm.org.uk/Using-ATM-MATs Dry copydex is easy to remove from school tables and MATs once it has dried but can be much harder to remove from clothing so an additional piece of advice is to ensure that the class have rolled up their sleeves or put on painting overalls before you open the copydex.

Association of Teachers of Mathematics, 2A Vernon Street, Vernon House, Derby DE1 1FR www.atm.org.uk