12 equilateral triangles

The prompt

Mathematical inquiry processes: Interpret; explore; find rules; generalise. Conceptual field of inquiry: Angle properties of equilateral triangles; tessellation.

The prompt was devised by David Aaron (a primary teacher in Blackpool, UK) for his year 6 class. The inquiries that developed from the prompt covered, amongst other things:

  • Angles in triangles

  • Angles on a straight line

  • Angles around a point

  • Symmetry

  • Angle properties of parallel lines

  • Exterior angles

  • 2-d shapes composed of equilateral triangles

  • The number of equilateral triangles of different sizes in the prompt.

David said: "All this in one lesson! I had to rein things in, but it personalised the learning brilliantly."

An alternative prompt

Billy Adamson, Head of Mathematics at Thurston Community College (Bury St Edmunds, UK), used this prompt (originally from the SMILE Wealth of Worksheets) with year 8 classes as an introduction to shapes and angles. He reported that the prompt led to "great exploration" (see an example below) and he "loved the energy and excitement the students displayed."

A year 8 student annotated the diagram.

Exploring the prompts

The pictures come from pupils in a Year 5 class at Living Waters Lutheran School (Alice Springs, Australia). Rebekah Clark, their teacher, describes the inquiry: "I gave the pupils the choice of the two prompts at the beginning of the unit on angles. Most of them chose to label and classify angles first, then started to find the degrees by using other angles in the image." Rebekah concluded that it was an "awesome start to our unit on angles that involved fascinating discussions and misconceptions."

Tessellation prompt

All triangles and quadrilaterals tessellate.

The prompt is true. In the tessellation of a triangle, each vertex (the point of intersection of three or more tiles) is made up of two sets of each of the three angles. For the tessellation of a quadrilateral, each vertex is made up of one each of the four angles.

Students might decide the prompt is true by designing triangles and quadrilaterals and testing whether they tessellate.

The prompt is designed to encourage students to make conjectures about pentagons and other polygons.

The only regular polygons that tessellate are equilateral triangles, squares and hexagons because the size of their interior angles are divisors of 360o. However, students might design pentagonal tiles that tessellate or explore the eight semi-regular tessellations made up of two or more regular polygons.

The kites (left, produced using Mathigon polypad) tessellate because the angles at the vertices sum to 360o. Each of the four angles in the kite meet at a vertex. There are two types of vertex in the tessellation of the rhombuses (centre). Either three or six equal angles form a vertex.

The third pattern (right) is an example of an irregular pentagon that tessellates. Currently, there are 15 types of convex pentagons that are known to tile the plane using the same shape. Unlike the other two diagrams, the third is not a side-to-side tessellation.

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