Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Mind Your Ps and Qs

Why do this problem

This problem will help to train students in the art of careful, logical, pure thinking which will help to develop their general mathematical skill. It will require students to address issues surrounding integration, use of functions, and inequalities, without needing to go into any particular detail with calculation of integrals.

Possible approach

Note: This problem might work best once students have tried Iffy logic as a starter at a previous time.

Give the students Mind Your Ps and Qs and let them read it carefully and think about what it is asking.

Suggest that students discuss in pairs what they think that the arrow symbols mean. Then, as a group discuss, for example, why

$$x> 4 \Rightarrow x> 3\mbox{ and } x=-2 \Leftrightarrow x^3=-8$$
are correct but
$$|x|> 2 \Rightarrow x> 1 \mbox{ and }x^2=4\Leftrightarrow x=2$$
are incorrect.

The next step is to ensure that everyone can construct their own individual examples of correct mathematical statements using propositions from the list. Once students have a couple of examples of such statements they should share them with the class and explain their reasoning. Do others agree or disagree? TALKING about such results will quickly highlight woolly or fallacious thinking and is an important part of the mathematical process.

Once the group has a feel for constructing the implications, they need to concentrate on using all of the statements to construct a complete set of 8 statements. Encourage students to consider their reasoning clearly in each case. Can the class complete the task with a clear explanation in each case?

Key questions

• What do the arrow symbols mean?
• Have you read the question carefully?
• Are there certain statementswhich look likely to go together in a pair?
• If an integral is positive or zero, what can we say about the area enclosed?
• Do you remember what the graphs of $\cos$ and $\sin$ look like?

Possible support

It is rather helpful to draw diagrams and number lines when thinking about inequalities. Shade the parts of the number line which apply to a particular inequality to help see which way round the logic flows.

If possible, start off with Iffy logic and the support materal suggested there.

Possible extension

Are there multiple solutions? Can students make up a similar set of questions to give to each other to try? Can they write down really clear explanations of why, for example, $x> 4\Rightarrow x> 2$?