Today was the first day of my first block of professional experience at a high-school in Tasmania. I’m teaching a unit on Pythagoras’ Theorem (which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides usually written: \(a^{2}+b^{2}=c^{2}\)) to two classes.
I introduced the topic by drawing a right triangle on the board, labelling its angles A, B, and C in ascending order, then labelling the sides a, b, and c after their opposite angle. C is obviously the right angle, and c is the hypotenuse. I then stated the proposition that: \(a^{2}+b^{2}=c^{2}\) illustrated that proposition by way of drawing a triangle \(abc\) with a square of the appropriate size on each side and proposed that we would, as a group, be proving that it is true (that it is a theorem).
I introduced the students to two different dissection proofs of Pythagoras’ Theorem:
The first dissects two squares \((a+b)^2\), one into \(a^2\), \(b^2\) and four triangles \(0.5ab\); and the other into \(c^2\) and four triangles \(0.5ab\). We can describe the relationship between the areas of the two squares as \(a^{2}+b^{2}+2ab = c^{2}+2ab\). The \(2ab\) on each side cancel each other out (being trivially equal) and we are left with \(a^{2}+b^{2}=c^{2}\). Q.E.D.
The second proof beings with a diagram of the square \(c^{2}\) (drawn using four right triangles arranged within a square \((a+b)^{2}\)). If we draw within the \(c^{2}\) four right triangles arranged to leave a gap in the centre (a square of \((b-a)^{2}\)). These five pieces taken from \(c^{2}\) can be rearranged to form \(a^{2}+b^{2}\). Q.E.D.
These two proofs can be seen at the Digital Mathematics Archive’s page of dissection proofs of Pythagoras’ Theorem. The first proof can be seen in the second diagram on that page, and the second proof is illustrated (in the same way I explained it) in the sixth diagram. Both classes managed to complete both of these proofs in just over 40 minutes, though I think that the first group may have suffered due to my nervousness.
My colleague teacher (who usually teaches these classes) gave me a lot of feedback on my performance. I managed to miss many, many, many of the little bits and pieces that keep things together:
introducing myself;
beginning with a question (to find out what they already know and get them thinking about the topic);
writing pertinent terms on the board (like “Pythagoras’ Theorem” for example);
structuring the lesson into ‘direction’ and ‘doing’ blocks effectively; and
three pages of other bits and pieces.
Hopefully, I’ll be able to resolve most of them by the end of this two week placement.