[Random Musings] Optimisation

G'day folks.

Today's post is gonna be a bit random (as the title suggests) and a bit anecdotal simply based on my experience teaching this topic.

So last Monday I had the chance to teach optimisation to a lower-attaining student. Her understanding of calculus is frankly quite shaky, but I wasn't prepared to know that her geometry skills are also severely lacking. She didn't know how to find the areas of shapes as well as the volumes of prisms; initially I planned to use a short activity on expressing perimeter/volume in algebra but ended up reteaching the formulae (no time to explain why the formulae work or do more examples, because we only meet once a week for an hour).

From my experience reading the examiners' reports, optimisation is usually the worst-performing question and the problem usually lies in setting up the models properly. I suspect that my student was far from alone - many people I've talked to struggled with optimisation for the same reason.

Solution?

Make sure you know your geometry stuff before attempting optimisation.

For students, this means going over your geometry notes and making sure you know the most basic formulae for

• area and perimeter of a square, rectangle, circle, triangles, polygons and any combination of the five (say, find the area of the shaded region in a question)
• surface area and volume of a cube, rectangular prism (cuboid, for British folks), cone, pyramid and sphere (and any combiantion of the five)
• Thales' theorem and similar triangles (these are more for a grade 7 question)

For teachers, I think it's a good idea to spend an hour or two just recapping all of this stuff unless you have a very solid class.

How is an optimisation problem related to curve sketching?

I don't know if it's just me but I couldn't connect the two when I was introduced to optimisation. They're legit the same. Let me explain.

To solve an optimisation you

• Make a sketch, define your variables and set up your model
• Find the derivative, set it equal to 0 and solve for your variable
• Answer the question

The second step is precisely what you do when you're finding the extrema!

Massive difference: function isn't given and you gotta read the question very carefully to make sure you know what you're being asked to do.

 

The challenge from these questions is in the first part (and if you do read the examiners' reports they almost always mention students perform poorly on the first part(s), aka setting up the model).

Optimisation in IB/A-level

Most IB optimisation questions are split into several parts - setting up the model, finding the max/min and checking if your answer is indeed a max/min (1st or 2nd derivative test).

Usually making their way to section B or the very end of section A, this is usually a more time-consuming question than your friendly section A questions. (For AI, translate "section B" to Paper 2 and "section A" to Paper 1.)

Because of the way the question is laid out, if you don't know how to set up the model, TAKE THEIR MODEL AND USE IT. The setup is usually worth around 50% of the marks (in the trickiest cases), usually less, so don't throw your hands too early.

The nature of the question makes it intrinsically difficult, so if you know how to answer it properly, you're already ahead of the curve. I'd say, if you're aiming for a 7, you must know how to answer most optimisation questions thrown at you. You really don't have that much buffer to throw away a question like this. Besides it's pretty predictable - if you know how to find the max volume of a cone inscribed a sphere, you're gucci.

Common misconceptions

This section is more for myself than anyone else. If you find it useful, please let me know. c;

(1) There are types of optimisation questions.

Yes, if you classify it by the surface structure. No, if you actually understand what you're doing. End of the day, you want to know where the extrema lies on the function you've just found.

(2) If a student doesn't know how to solve an optimisation question, they should simply solve more optimisation questions.

This is true, but just too time-consuming.

The hard part is setting up the model, so I'd rather spend time on building the correct model before getting too much into the mechanic-of-getting-the-answer. Furthermore, it's potentially misleading - kids think that they can't solve these questions because they are weak at calculus but really geometry is blocking their way.

(3) Finding the derivative when the function to be optimised has several variables

In calculus 1, you can't do that. Of course, when you're in multivariable calculus, it's WAY simpler to solve some calc 1 questions. But with the tools you currently have, changing everything to a single variable is the most feasible option.

Teaching sequence

If I'm allowed to teach this topic given sufficient time (say, 2-3 hours) to a middle-attaining student, here are the things that I plan to do. I haven't had the chance to do it yet, but I shall when the time comes.

This section should take place after a good discussion of curve sketching and the first/second derivative tests.

- algebra review on solving for one variable and substituting it back into the function to make the two-variable function multivariable

- review on geometry with algebra to get them used to setting up the model

- teach how to solve an optimisation question using calculus (then use a question that they could simply find the vertex/turning point of a quadratic function to marry the two together), making explicit connections to finding the extrema of a function

- let them try some questions on their own, potentially including some trickier questions (algebraic/geometric or bounded by a domain)

That's all I have to say re: optimisation. Probably a break from a LOOOOONG list of tests for the last couple months; finally found myself in the mood to write something potentially meaningful to others.

I'll see you on Tuesday. c:

Andrew