## Finding a Locus: Algebra and Geometry

Last time we looked at the meaning of the concept of locus. This time, we’ll explore seven examples, from two students. We’ll look at both algebraic (equation) and geometric (description) perspectives.

Last time we looked at the meaning of the concept of locus. This time, we’ll explore seven examples, from two students. We’ll look at both algebraic (equation) and geometric (description) perspectives.

(A new question of the week) Definite integrals can sometimes be solved by finding an antiderivative; but when that is either difficult or impossible, there may be special tricks available. Here we’ll lead a student gradually to a solution using symmetry; and then we’ll look at an earlier problem that used essentially the same trick …

(A new question of the week) Suppose we have a question that can be answered with Yes, No, or Maybe, and that whenever two people with different opinions meet, their discussion convinces each of them that neither can be right, so they both change to the other opinion. Given initial numbers of people with each …

A recent question about lottery numbers reveals that a seemingly special event is in fact surprisingly common: namely, the presence of consecutive numbers in a lottery drawing. The calculation is an interesting one, and we’ll also see a way to check our answer, then compare it to reality.

Last time, we considered the Chain Rule for derivatives. This time, we’ll look at the product and quotient rules, focusing on how to keep the formulas straight, and make them easier to apply. We’ll look primarily at the quotient rule to start with, and then examine the product rule at the end.

Having recently helped some students (in person) with the rules of differentiation, I’m reminded to do so here, starting with the chain rule. It is easy to make this topic look harder than it really is; the two main ways to state the rule are often confusing, and different approaches fit different problems. We’ll try …

I am always interested in problems that can be solved in different ways, particularly because this can give a student a chance to be creative, as well as learning from experience that you don’t have to do it “the teacher’s way”. Here we’ll use trigonometry, and two different ways to add lines to a figure …

Last week’s discussion reminded me of another question, from July, about a probability problem that was solved in a hard (but educational) way and an easy way. This instance is more extreme, and, due to its length, requires extreme editing in order to fit here.

Sometimes there are several ways a problem could be solved, and we encourage students to pursue the path they’ve started on, rather than give them a method that may not be natural to them. (And we encourage them to start, if they haven’t yet, in part so we can see what might be most helpful …

I’ve had several occasions in face-to-face tutoring lately to refer to a past post on mixing (that is, composition) of trig and inverse trig functions. Several recent questions have touched directly or indirectly on this same general idea and extended it, so I thought I’d post them.