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Challenge Problems (#10)

Saturday 9 April 2016

To me - and to most people who enjoy mathematics - solving a problem that requires insight, perseverance and creative use of mathematical knowledge and techniques can be a very enjoyable and rewarding activity.  Most of what students do in mathematics classes around the world is solve exercises, not problems.  An 'exercise' is a question for which a student usually knows beforehand what strategy and technique(s) to apply.  An exercise set for homework will contain questions that will have students practice some facts and/or techniques they were recently taught. For example, use the sine rule to solve for a missing side or angle in a triangle; or set up a definite integral to find the area between two curves. This is all necessary and valuable - but students also need (I believe) regular experiences in solving a true problem - that is, a challenging problem for which it is not obvious what mathematical knowledge or technique(s) to apply.

There are many good books and websites that contain mathematical problems and puzzles. The tough task for a teacher interested in having their students tackle challenging problems is to find ones that are at a sufficient level of difficulty and involve mathematics that is appropriate and useful for the particular content that is being taught in their course.

I've worked hard to come up with such problems suitable for IB Maths HL and SL students. Of course, generally speaking, Maths HL students are going to be more receptive and capable in tackling difficult problems - but, I believe, Maths SL students also benefit from practicing problem-solving skills in a genuine way. Today, I am adding my 10th Challenge Problem - called Half the Area. I strive to come up with new problems or a modification of a known problem. Devising a new problem is not easy, but I'm always on the outlook for interesting questions that can lead to a problem suitably challlenging for Maths HL & SL students.

Recently in a Maths SL class, I asked students to quickly determine the exact area (so do not use a GDC) bounded by the x-axis and the graph of $y=\mathrm{sin}x$ on the interval $0\le x\le \text{π}$ . It's good practice in setting up an appropriate definite integral, carrying out some simple anti-differentiation, and then evaluating the definite integral. The result of exactly 2 square units is interesting; most students do not expect to get an exact integer value for this area. But then the question came up in class about how we could divide this region in half. After a brief discussion, we thought it best to divide it with a horizontal line, $y=k$ , so that the area of the region under $y=\mathrm{sin}x$ is divided into two regions each with an area of exactly 1 square unit.

Often a useful first step in a problem like this is to make a good guess at the answer.  This is something students often do not consider doing. Simple logic informs any thoughtful student that the value of k must be less than $1}{2}$ .

It's also fun to use technology to 'play' around to refine a good guess even better.  See the video below showing an animation on my TI-Nspire CX that tries to answer this problem.
Is it possible that the value of k is exactly 0.36 ?

Of course, we know that GDCs can give inaccurate or false results in certain situations. So, it seems reasonable (and correct, in this case) to be suspicious that the equation of the horizontal line that divides the region in half is exactly $y=0.36$ .

Is there an exact value for k ?  I don't know, although I believe there is none.  I've calculated k (approximate to 12 significant figures) to be 0.360034982809

As a result of my attempt to express k exactly, I did determine the following interesting result.

The desired value of k satisfies the following equation:

$\mathrm{arcsin}k=\frac{\text{π}\text{\hspace{0.17em}}k\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1-2\sqrt{1-{k}^{2}}}{2k}$ This interesting result - which makes use of a healthy dose of integral calculus, trigonometry & problem solving - led me to devise my 10th Challenge Problem.  Take a look at this Half the Area Challenge Problem - and the other Challenge Problems that I've posted thus far. There are more to come.