# Angle Measurement: An Opportunity for Equity

I was observing a lesson recently in which the teacher was showing her year seven class how to use a protractor. Admittedly she was following the scheme of work specified by her head of department, but it saddened me to see so many children doing something which they could clearly already do. The teacher was aware of this but felt she hadn't the resources or the authority to do something
different. During the lesson, as pupils were working their way through one of the most turgid exercises imaginable, I began thinking about how such work might be made more relevant to such classes - that is classes where the evidence shows that a substantial number of children can already manage, with competence, the skill being taught. Also, I wanted a means by which practice could be embedded
within a more meaningful and mathematically coherent activity.

In this particular lesson pupils were working from a text which contained, on one page, a picture similar to that in figure 1, albeit with a greater number of labels. The first question invited them to measure, say, angle $\angle AOB$. This was followed by almost fifty requests for them to repeat the process. The exercise itself was repeated twice more over the page. That is, pupils were given
almost one hundred and fifty opportunities to repeat the same task. Ironically, the way in which the task was presented meant that no measurement was needed - pupils could simply read off the required angle from the scale.

*Aside: for reasons of classroom management, teachers may feel anxious about pupils' use of pairs of compasses. In such cases there is no reason why they ought not to provide them with pre-drawn copies.*

- What sorts of triangle can be found? What are their properties?
- How many different triangles are there?
- What are their angles?
- What might be inferred about triangles with right-angles in them?
- What is the largest angle that can be found in such triangles?
- What is the smallest angle that can be found in such triangles?
- What difference would it make if the centre of the circle were allowed as a vertex?
- What can be said about all such triangles?

*Aside: I would not use a protractor like the one above for work like this. Angle is dynamic, not static. It is a measure of turn and a protractor - either half circle or full circle - fails to acknowledge this. I want my pupils to have access to a dynamic angle measurer - one with a rotating arm that allows the learner to see clearly the angle turned.*

- Under what conditions are your triangles obtuse-angled?
- Under what conditions are your triangles acute-angled?
- When are they right-angled?
- How many different triangles can you get on an $n$-point circle?
- What is the largest angle that can be found in a triangle drawn in an $n$-point-circle?
- Can you prove why your triangles yield the angles they do?

Of course, triangles are but one line of enquiry. We could, as shown in Figure 5, explore quadrilaterals drawn in circles.

How many quadrilaterals can you get on different point circles? What can we infer about the sum of opposite angles of a quadrilateral drawn in a circle? Is this true for all quadrilaterals? How might one prove that the sum of opposite angles of a cyclic quadrilateral sum to $180^{\circ}$?

Another avenue might be to explore some elementary circle theorems. For example, in Figure 6, we can see an allusion to the angle at the centre being twice the angle at the circumference when drawn from the same chord.

Thus, in a lesson intended to consolidate the skills of measuring angles, pupils may, additionally:

- do some constructions
- investigate properties of triangles drawn in circles
- investigate properties of quadrilaterals drawn in circles
- investigate circle theorems
- investigate angles in similar shapes
- engage with proof and justification

- mathematics is presented to learners as a coherent body of knowledge
- less time need be spent on subsequent topics because the ground work has already been covered
- mathematics becomes a problem solving activity in which learners construct knowledge inductively and then prove their findings deductively
- mixed ability classes can work meaningfully on a task and, therefore, fulfil the equal opportunities aims espoused by schools but rarely realised.

#### References

Paul Andrews, University of Cambridge Faculty of Education. (published in Mathematics in School (2002), 31 (5), by The Mathematical Association )

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