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People forget how to measure angles, probably because it is not something that they do every day (or even *once*) after leaving school. I'm writing about a couple of short activities I use to remind people how to measure and describe angles, and to develop knowledge about polygons and circle theorems.

The shape and space part of the GCSE syllabus has vocabulary to learn and a lot of facts that you have to know. The exam tests knowledge of the facts using puzzles; so you have to 'decode' a situation to work out which facts are needed. At Foundation level the puzzles are one or two steps, and part b of a question is usually independent of part a. At higher level, the range of facts to learn increases (circle theorems as well as parallel lines and basic angle facts) *and* the puzzles become more complex. Part b of a question will depend on the result of part a of a question; and part a may not be 'easy', so you can only gain access to the marks for part b of the question if you *really* know and can apply the facts.

- Download the polygons: measure the internal angles investigation
- Download a circle with centre template for the circle theorem investigation
- Download a vocabulary sheet that is used at the end of the lesson as a summary
- Download a sheet of 'emergency protractors' that can be printed onto acetate or printed onto paper and laminated
- Tim Devereux's Java animations of the circle theorems

I have an acute, right, obtuse, 180 degree and a reflex angle drawn on a whiteboard slide or flipchart ready for the start of the lesson. The angles are unlabelled but shown with small arcs. Whole class; we name each of the angles. 'Acute' seems to be remembered by *someone* in each class, 'obtuse' is less known and 'reflex' is news to most people. I write the range of sizes for each named angle as an inequality, linking back to the grouped frequency distributions in data handling.

Then the students registered at foundation level try the polygon investigation and the students registered at higher level try the circle theorem investigation.

As an introductory investigation I use a sheet with a triangle, a quadrilateral, a pentagon, hexagon and octagon. To save time with small groups each student takes a shape and measures each angle and finds the total while I get round everyone to check for accurate and sensible protractor use. I put the table on the whiteboard, with columns for the name of the shape, the number of sides, and the total internal angle. The students with the triangle and the quadrilateral have time to draw a second different example of each shape and verify that the internal angles do add up to the same total.

As students measure and add up the angles, the error tells me how accurate the measurements are. I mention the tolerance on angles allowed for measurements. The pentagon is scalene with 'difficult' angles and a good test of accuracy.

Once the table has been written up on the whiteboard we can try spotting the pattern. The term to term rule is usually identified as 'add 180' fairly quickly. Students can deal with the need to cope with the approximations in the measurements. We get the nth term so linking this work with sequences as an extension task. Students can use the pattern to predict the number of sides and the total internal angle for the 7 sided figure. The name usually needs a google search as names are conventional not logically determined!

I have a few dozen of the circle templates photocopied just in case the students don't have compasses with them, or in case of 'sloppy compass syndrome'.

Each student draws a diameter and then draws a *different* triangle on that diameter. Measures the angle at the vertex that touches the circumference. Compare notes. Label vertices A, B and C where B is the vertex on the circumference, and then we refer to the angle ABC to give practice in that convention.

Using a fresh template, each student draws a quadrilateral with all four vertices (A,B,C,D going clockwise) on the circumference. Add up their ABC and CDA angles and compare results. Then add the BCD and DAB angles and compare results.

By this time, the foundation students are ready to summarise the results on the table on the whiteboard; they 'present' to the higher level students just to make sure that the polygon total internal angle sequence is covered. I usually use directed questionning to pull out the idea of breaking up the quadrilateral, pentagon &c into the smallest possible number of triangles. Each triangle contributes 180 degrees to the total internal angle for each polygon.

The students registered for higher level need coaching as much as those registered at foundation level.

- Accurate positioning of the protractor centre over the vertex of the angle. A displacement of a few millimetres can result in an angular error of several degrees. Students seem to like to fiddle the positioning so the angle is a multiple of five degrees!
- Alignment of the 'zero' line on the protractor with one of the lines forming the angle to be measured.
- Using the correct scale on the protractor. I suggest that students 'name' the angle as acute or obtuse before measuring it so they
*know*beforehand if the angle is, for example, 150 or 30 degrees - A few students use the protractor like a ruler; they
*move*the centre along the arms of the angle. I appeal to the classroom clock and describe how an angle is formed from two arms

Once the investigations are complete I go through the basic angle facts; angles at a point; on a line; angles in a triangle; interior + exterior angle; leading onto parallel lines. Then I have worksheets that build up in complexity of puzzle.

I'm writing examples that introduce the circle theorems one by one with simple illustrations; and gotchas like remembering that a triangle with one vertex on the centre and the other two on the circumference will be isosceles.

Keith Burnett, Last update: Tue Mar 12 2013