Lesson 28 Graph of the rotating arm

Episode 1

Reasoning Resources
Whole class preparation
Understanding angle as a measure of turn and that when the turn is frozen’ we can read off the angle at that moment and find the sine ratio.

Distinguishing between the dynamic and static aspects of angle.
Today we're going to look at the sine ratio, which is a property of an angle and which changes as the angle changes. Explain the apparatus, pointing out that both arms are the same length and graduated in 10 sections. The centrally fixed arm rotates to show different angles and the hanging arm crosses the horizontal diameter at a point which can be read from the marks on the arm. Use the apparatus to show a few angles, Remind pupils that angle is actually a turn and is not related to the length of the arms. When we freeze the turn we lose the sense of turning and only have a picture of the arms. For each angle shown, read off the value of the hanging arm to 1 d.p.,e.g. 4.8 squares and divide it by the length of the radius (10). Explain that this gives the sine value rounded to one decimal place.

Display Worksheet 1 and show how to work out the sines of the angles up to 90° by drawing the rotating arm and finding where the hanging arm crosses the diameter.
Pair and small group work
Using Worksheet 1, pupils begin to find the values of sines to 90°. Bring pupils back together to show how to plot the sine ratios on the graph on Worksheet 2. What happens when we plot the sine values on a graph?

Pupils with calculators can find the more accurate values for the sines of angles. This would open up a useful discussion on the high degree of accuracy needed for calculations in real engineering work as opposed to the approximate values needed when we want to look at the overall pattern of change.
Seeing that the values of the sines produce a curved graph. Whole class sharing and discussion
Degrees of accuracy for sines depend on use. Approximation to 1 or 2 decimal places is adequate for looking at the pattern of change of function. Asking how we can describe the graph will steer the pupils’ descriptions to go beyond ‘increases from 0 up to 1’ to give a flavour of the curve levelling off, highlighting that the increase is greater nearer 0° and less nearer 90°.
Interpreting the graph in terms of steepness of the graph at different points. Pair and group work
Beginning to understand what the symmetry of the graph means in terms of the sines of different angles — that different angles have the same value of sine. Ask pairs or groups to continue with question 3, this time finding the sines of angles up to 180°. Give out tracing paper so that they can explore the symmetry of the two halves of the graph. Help pupils to trace parts of the graph, then turn or reverse the tracing paper to check symmetry, and challenge them to put in words what they find.
Whole class sharing/discussion
Preparing to think of what happens to the vertical projection when the angle is more than 180°. How would you describe the graph now? ‘the function is symmetrical’; ‘it decreases in the same manner as it increases’;‘at the same rate’ Why is the change in the function so uneven while the change in angle is even?

Pupils should attempt to explain that we are looking at the change in one particular aspect of change, which is the projection above the horizontal line, reading that from the vertical scale. It can be called ‘shadow of the hanging arm to the vertical central line’


Thinking Mathematics Lessons Copyright © by Michael Shayer and Mundher Adhami. All Rights Reserved.

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