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P.o.t.D.- 250 Problems

Sunday 6 May 2018

I started writing problems for my Problem of the Day (P.o.t.D.) section on this site 15 months ago - and there are now 250 problems available - equally distributed between HL and SL. The distinction between a problem that I post on the HL problems page and one that I post on the SL problems page is certainly not well-defined. I try to compose problems that require some application of problem solving skills. Problems that I think are suitable for HL students usually involve more sophistication with regard to figuring out a successful strategy (not always just one method that works), and often a higher degree of rigour with regard to mathematical techniques. In a nutshell, the key differences between the Maths HL and Maths SL course is the much higher level of sophistication and rigour that exists in HL.

I think that the 250th problem I just added to the P.o.t.D. section (126th HL problem) is a good HL problem from the standpoint of demanding some thinking from the student (or teacher). The problem evolved from the 125th HL problem that I wrote. My students were working on some differential calculus exercises (not necessarily 'problems') - and one of them involved finding maximum and minimum points for the rational function $$y = \frac{{{x^2} + 3}}{{x + 1}}$$. Not a very demanding question. But then I wondered about the distance between the two branches of the graph of the rational function. My students had done some optimization questions (points on a parabola nearest a given point, max volume of open top box, etc) and I thought that it would be interesting to find the minimum distance between the two branches of graph. It's not the distance between the maximum and minimum points (just a little less).

This led me to consider finding the minimum distance between the graphs of two differet equations ... how about a circle and a parabola. And I thought it would be a nice problem in which a student would need to consider appropriate use of technology to help them solve it. Not only using technology in the solution, but also in assisting them in coming up with a strategy.  It's not a super difficult question (with a GDC) but there is an important insight required (at least in the solution method I used) that many students may not come up with unless they see the conditions of the problem illustrated dynamically. For my Problems of the Day for which a GDC is allowed (always stipulated in the instructions for each problem wheter a GDC is allowed or not), I encourage my students to consider using technology other than their GDC - for example, Geogebra. I like my students to gain some experience with Geogebra during  the course, so that they feel confident enough with it so they can consider using it with their Exploration (IA). Below is a Geogebra applet that dynamically illustrates HL P.o.t.D. #126 - and can help a student gain insight into devising a solution strategy. Can you?  (solution is given on 2nd page of problem)