How to make a kid understand geometry and help him solve problems?

I'm tutoring a 13 year-old boy, a middle school student.

He has almost no problem with elementary algebra: he just applies the rules and everything falls into place. However he does struggle with geometry, since straight applying rules isn't always enough. The more complex the problem, the less he's able to find a path to the solution.

I'll give an example. We encountered a problem like this one:

Where the goal is to find the area of the shaded region. He was able to find the area of circular sector and triangle, but he could not put the pieces together without my help.

I don't want to suggest a solving path every time, because I'd like to improve his ability of finding one on his own. So, here's my question: what should I do in order to improve his problem-solving skills, especially when he needs to use some creativity to reach the solution?

• "What should I do in order to improve his problem-solving skills?" well you should suggest a solving path every time. I know you don't want to but if you make sure that he understand the idea then it's worth it. In this specific example the student needs to understand the "Technic" that in order to find the area of a certain region it sometimes useful to find the area of other regions (and subtract in this case or add in other cases). If from now on he is able to do this on his own then congratulation you improved his problem-solving skills. – Yanko Nov 22 '18 at 18:35
• @Yanko, thank you for your point of view, I understand what you're saying. Let me clear that when I say "I don't want to suggest a solving path", it's not that I don't feel like to, but I fear that it will make him lazy and that he will not even try in the future. However I do see your point and the need to show a technique in order to teach how it's done. – francescop21 Nov 22 '18 at 18:43

1 Answer

Since he's good at algebra, you could suggest he try to apply it in geometry when he gets stuck. I don't mean it'll be a feasible method on its own for all geometry; traditional Euclidean methods are often more productive than writing everything in Cartesian coordinates. However, consider your problem example. We want to compute a sum of two grey areas $$g_1+g_2$$, given that the white areas $$w_1,\,w_2$$ satisfy $$g_1+w_1=g_2+w_2=32$$ and $$g_1+w_1+g_2=16\pi$$.

I realise people at our level effectively think through the problem that way anyway, but your student might not have realised he can do that. Some students are excellent at juggling facts about letters' unknown values, but not so good (until better instructed) at applying this to specific meanings for those letters. In other words, they could solve the equations I just wrote, but wouldn't spot they're lurking in the diagram. I once knew a maths teacher who told me he'd encountered students who knew how to solve $$ax^2+bx+c=0$$, but not $$at^2+bt+c=0$$. I kid you not!

• Thank you for your contribution, unfortunately he doesn't know Cartesian coordinates yet, so this approach, while interesting, may be off-limits. – francescop21 Nov 22 '18 at 19:13
• @francescop21 No, it isn't. I didn't use Cartesian coordinates in that discussion; I just mention them because, at a certain level in geometry problems, students sometimes try to overuse that strategy. (As someone who uses to train for Olympiads, I know from first-hand experience how tempting it is.) But here all we're using is simultaneous equations in a few areas. – J.G. Nov 22 '18 at 19:17
• Oh, now I see what you're saying! – francescop21 Nov 22 '18 at 19:21