Teaching, questioning but not telling. Angle at circumference of a semi-circle.

I wrote an earlier blog on this subject but thought it might be useful to break it down into a series of teacher to student questions and instructions to get a more active learning experience in the classroom. This can be done with geoboards, printed sheets or by using software like Geogebra. At the end are notes on the numbered graphics that follow.

[1]  What is the angle at the circumference of this semi-circle?

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[2] What do we know about the sum of the internal angles of any triangle?

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[3] What do we know about lines OP, OQ and OR?

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[4] How many triangles can you see now and what type are they?

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[5] Look at the isosceles triangles and label known angles.

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[6] If internal angles of any triangle sum to 180˚ how can we express the known angles in this triangle as an equation?

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[7] If we divide both sides of the last equation by 2, what can be said about the angle x + y and consequently the angle at the circumference of a semi-circle?

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The opening question [1] asks the question that we are aiming to answer as a water-tight proof. Some pupils will know the answer and they can be challenged to prove the the answer (90˚) is true for any angle made at any point on the circumference. By asking the question about the sum of all interior angles of a triangle [2] we are setting up a pathway to the proof of this theorem. By establishing and making it clear to students that point O is the centre of the diameter of the semi-circle [3] we are able to guide pupils in the first instant into verifying that lines OP and OP are equal in length and as such represent a radius of the semi-circle. The question to follow should be aimed at getting pupils to reason that the line OQ is also a radius and as such is equal to length of each of the other two lines.

In [4] we challenge students to name the three triangles that now exist. The lines OP, OQ and OR are all shown to be equal by the single dash on each of them and because of this we want pupils to acknowledge that the two smaller triangles are isosceles. The naming of the original larger triangle can be a source of discussion that will be well worth five minutes of class time as at this stage all that can be confidently stated is that it is probably some sort of scalene triangle or the extreme possibility that it too is another isosceles. This would be a worthy mini session within the teaching of this subject matter to give students a chance to express their reasons and ideas. When the students are clear about the existence of the two internal isosceles triangles they can replace the originally named angle (z) [2] at the circumference [5] for angles x and y. Making links between the different subject matter in mathematics is to be encouraged and at [6] we do just that by using algebra to define the the angles labelled as an equation. They should be able to simplify and factorise expressions in order to make sense of this. if after factorisation they arrive at the equation 2(x + y) = 180˚ then division by 2 on both sides [7] will reveal that x + y = 90˚ and as the point Q on the circumference was randomly placed we can conclude that this angle will always be a right-angle.

By using mathematical reasoning based on this lesson how would you as a teacher feel about the following question type?

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