Copyright © University of Cambridge. All rights reserved.

Teachers may find the article Angle Measurement: an Opportunity for Equity of interest.

All students will need a sheet of nine-peg
circles to jot down ideas during discussion.

Demonstrate how the geoboard works - clicking on a coloured rubber band, dragging it onto a peg and then "stretching" it out onto two more pegs to make a triangle.

"How many different triangles can you make which consist of the centre point and two of the points on the edge? "

"Can you find all the angles of the triangles?"

At some stage students will want to know what makes triangles "different". For the purposes of this exercise, triangles that are congruent are considered to be the same.

After a short time bring the class together to discuss findings. Issues that you could touch upon:

- caclulating angles at the centre for different numbers of pegs
- rules for finding the base angles of the isosceles triangles

Draw a triangle on the geoboard with three vertices on the
edge of the circle which encloses the centre point.

"Can you find all the angles
of this triangle?"

How this builds on their previous work may not be immediately
obvious to most students. After a few minutes, if no progress is
being made that can be shared, add one isosceles triangle inside
the original triangle (see the
solution ). A further isosceles triangle can be added if
needed.

Discuss efficient methods.

"Now draw all the possible
triangles with three vertices on the edge of the circle and find
their angles."

At some stage discuss how to cope with triangles that do not
enclose the centre.

This handout
presents the whole problem on one sheet.

The
virtual geoboard environment allows you to create triangles in
circles with a variable number of pegs. The virtual geoboard
resource also has links to sheets with printed circles with
different numbers of pegs, a full set of these sheets can also be
found
here.

Ask students to work out the angles of triangles in 10-peg,
12-peg, 15-peg ... circles.

Lens Angle
provides an interesting challenge that requires students to apply
the properties of triangles in circles.

Students can start by finding the angles of triangles in 9-peg and 12-peg circles which contain the centre of the circle. They can then move on to triangles that do not contain the centre of the circles.

Printable sheets with all the possible triangles in 9-peg and 12-peg circles can be found here and here .

Sethe extension section of these notes for other useful links.