Mathematical Exercises

Mathematical Exercises (MEs) are not the typical exercises found in textbooks. They are problems which are related to the topic to be taught, but which are not necessary for the understanding of the topic itself. They are more for stimulating mathematical thoughts and making the lesson interesting. They need not be fixed questions detailed to students by the tutor, but could be anything the tutor (or even the students) think of that relates to the topic at hand.

Example 1

For example, after studying the topic on 'Hyperbolic Functions', the tutor can ask the students to create a 'Generalized Hyperbolic Functions' and ask questions like 'What's the consequence of doing so?' or 'Is the generalization helpful?'. You'll be surprised what the students can come up with. A possible generalization would be to replace the e in coshx = (ex + e-x)/2 by any constant real number, a. We denote this by cospx (the tutor can ask the students to come up with their own notations). A consequence of this is that on differentiation, we have D(cospx) = sinhxlna. This results in the extra term, lna. So it does not seem helpful at all to generalized the function in this way. What about the identity cosh2x - sinh2x = 1? Is it affected? A short exercise would show that the identity is preserved.

What the students gain from this exercise is that they learned about generalizing problems in Mathematics, which is one of the ways Mathematics develop, and in some case we get to know more about the specific problem at hand by looking at a more general form. However, in our 'pyperbolic' case, generalization does not really help at all, but this does not defeat the purpose of the exercise. It just goes to show that not all generalizations are useful. Furthermore, students may think of other more interesting ways of generalization, which certainly serve the purpose of arousing their thoughts. Another interesting aspect of this exercise is that students get to understand more of Hyperbolic Functions, because in working out D(cospx) and verifying the identity still holds, they need to work through the very proofs of D(coshx) and the original identity.

Example 2

If you are teaching a topic on graphs and special functions (exponential, trigonometric, logarithmic, etc), you could ask students to think of new functions and try to study these new functions in depth. One possibly interesting function to work at would be y = xx. How does the graph look like? Is it continuous? Does it have any special features? Is it a function at all? Students can use a brute force method, by plotting many point on a piece of graph paper to visualize the curve, or they could use standard softwares to generate the graph, or they could employ Analysis to understand it more thoroughly. You'll be surprised by the multitude of ways they can think of to approach the problem.

It is a function alright and it has many interesting features. It is not continuous over the whole domain of x. In fact, it is only continuous for x > 0. It is not defined at x = 0 and also for many negative x values. One feature that is not obvious from a plot of the curve (if you use a large domain of x) is that there is a minimum point at x = 1/e = 0.3678... The reason being that the curve around this point is quite gentle. The exact value of 1/e can only be obtain via Analysis and not through curve-plotting. All these subtleties should be raised to the students doing the exercise.

Here's the Analysis:
y = xx => lny = xlnx
=> y'/y = 1 + lnx
=> y' = xx(1 + lnx)

Now y' = 0 => 1 + lnx = 0 => x = 1/e = 0.3678...
It is easy to verify that y"(1/e) > 0, showing that it is indeed a minimum point.
Also from the graph, it seems that y is 0 at all negative values that the function is defined, but we know that it is just that the y values are very small (particularly for the more negative x).

Example 3

Not all MEs need to be about new stuffs. Another useful ME is to ask students to generate their own proofs of known theorems. One top contender would be Pythagoras Theorem. There are virtually hundreds of proofs of this famous theorem out there. It does not matter that the proofs the students thought of are already in the literature (which is most likely the case), what matters more is that they really think of a proof. The proof need not be one they have never seen before. If it is one they read in a text, it serves to consolidate their understanding of the theorem. Another important aspect of this ME is that the idea of a proof is instilled in the minds of the students. The importance of proofs is in the mind of all mathematicians and it is necessary that amateur learners of Mathematics also have the same mentality.

Example 4

Also suitable as MEs are unsolved problems in Mathematics. Of course the aim is not to solve these problems at all (if that happens it would really be a bonus!). Many times new branches of Mathematics blossom from studies on unsolved problems. An example is Henri Poincar�s discovery of Chaos Theory while working on the Three-Body Problem. A great contender for this category of MEs is Fermat's Last Theorem (although FLT is no longer an unsolved problem). Many great mathematicians have worked on it through the 350 years since it's formulation by Pierre de Fermat. What's interesting about it is not just that many new Mathematics is created but also that many fallacious 'proofs' have been submitted through the ages. The students could be taught to recognize mistakes in 'proofs', by presenting them with fallacious 'proofs' (some of which may come from the students themselves) and asking them what's wrong with the logic of the 'proofs'.

My favourite example is a fallacious proof of FLT, which goes like this:
Assume an + bn = cn and r is a natural number.
Let a>b.
Now an+r + bn+r = aran + brbn
= ar(cn - bn) + brbn
= arcn + (br - ar)bn
= cn+r + (ar - cr)cn + (br - ar)bn

Since c>a, we have cr>ar or ar - cr<0.
Also b<a, we have br<ar or br - ar<0.
Thus an+r + bn+r != cn+r.

Since when n=2, we have a2 + b2 = c2, this means for higher exponents (m>2), am + bm != cm (QED)

The error here is not in the logic of the statements, but rather the wrong interpretation of the proof. What is being proven here is simply that Pythagorean Triples cannot be Fermat Triples, and not that FLT is proven true.

Example 5

Of course we should not forget puzzles as the traditional means to get students interested in Mathematics. They work most of the time. Since old days, children have indulged themselves in tic-tac-toe, Sam Lyod's 15-puzzle, etc. And presently, the computer game available on all PCs, Minesweeper, has been adopted by certain institutions to stimulate logical thinking. All these games involved a certain set of simple rules and players need to make logical deductions to achieve the goal, while adhering to the rules. Instead of just playing the games, students can be asked to analyze the strategy of the games. The simplest to start off is tic-tac-toe. Ask them to come up with a sure-win strategy, if it exists. Ask them whether it is possible to ensure that the opponent never wins.

Such an exercise introduces students to topics they will most probably not encounter until they reach University level and choose to major in these topics. Analysis of strategies in games and puzzles belong to Game Theory, and students are not likely to learn it unless they major in Mathematics or Economics. Exposing students to a wide range of mathematical concepts is essential for all-round education, as all branch of Mathematics find applications in the real world. The answer to the questions posed for tic-tac-toe is that there is no sure-win strategy, but you can always prevent the opponent from winning (by forcing a draw).

Example 6

Another form of ME is speed calculation. How do we make calculations without pen and paper or a calculator? There are a few tricks. We can do addition, simple subtraction, doubling and halving very quickly mentally. By breaking down a sum into these simple steps, we can make seemingly tough calculations without any tool. Take for example the conversion from Fahrenheit to Celsius. The formula is C = 5/9 (F-32). How can we make this conversion mentally? Observe that we can rearrange the formula to C = 10/9 * 1/2 * (F-32) = 1.111... * 1/2 * (F-32). Let x = 1/2 * (F-32), hence C = 1.111... * x = x + 0.1*x + 0.01*x + ... This might look like we're getting nowhere. But this is actually easier to see with an example: e.g. F=60.
step1: 60-32=28
step2: 28/2=14
step3: 14
+1.4
+0.14
+...
=15.55555...

It might not appear easy at first glance but after a few examples, the algorithm become second nature to apply. Now, what's the algorithm for the conversion from C to F? Try it.

Example 7

The examples of MEs could go on and on, they are just limited by your imagination, so I'll just present another one here. This one arise from exercises in a textbook on Elementary Geometry. As we all know there are many Circle Theorems in Geometry. The number of such theorems need not be limited to those presented in the book, you can create your own ones. Here is it:

Sum of Side Angles
I am going to show that ^OBA + ^OCA = ^BAC and that ^OBD - ^OCD = ^BDC. These can be combined together as a single theorem if we adopt a suitable convention, so that the second equation becomes ^OBD + (-^OCD) = ^BDC. I call the result "Sum of Side Angles".

The first equation follows from the fact that triangles OAC and OAB are both isosceles. Similarly, the second equation depends on triangle OCD and OBD being isosceles.

So much has been said, but what really matters is action. Just let our imagination run wild and think of interesting MEs to uncover mathematical talents!

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