# 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)

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