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 an alternative design (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" and he "loved the energy and excitement the students displayed."

A year 8 student's annotation of the diagram.

Exploring the prompts

Inquiry as the start of a unit

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."

Inquiry is a highly effective way to start a unit of work. In their response to the prompt, students draw on their existing knowledge and can, through their comments and questions, suggest lines of proximal development. The teacher can assess students' current understanding and identify the misconceptions they hold. Moreover, the teacher can begin to plan the most effective ways for students to encounter the new concepts they will require during the inquiry.

Question and notice

Chris McGrane, Principal Teacher of Mathematics at Holyrood Secondary School (Glasgow, Scotland), posted pictures of the inquiry on twitter. They show the responses of his S1 (grade 6, year 7) class to the prompt. Chris reports that students worked individually before discussing in pairs and then in groups. They noticed properties of shapes involving angles, symmetry and similarity and generated questions to investigate. The inquiry led to "some really great mathematical discussion."

March 2022

Chris McGrane is author of Mathematical Tasks: The Bridge Between Teaching and Learning. He discusses the Inquiry Maths model between pages 202 and 206.