Since we just looked at a complicated rational inequality, let’s look at some simpler rational equations, first a linear equation with fractions, and then truly rational equations, in which the variable(s) appear in the denominator. This discussion dealt with a common confusion I’ve seen in students.
When a challenging type of problem is written with unexpectedly large numbers, it can look impossible. Today’s discussion illustrates how to get past the hurdles.
(An archive question of the week) Students often struggle with solving an equation with several variables, for one of those variables. This is also called “solving a formula”, or a “literal equation”; or “making one variable the subject”. Learning to use variables instead of just numbers (as we looked at last week) is the first …
(A new problem of the week) Last week I mentioned “non-routine problems” in connection with the idea of “guessing” at a method. Let’s look at a recent discussion in which the same issues came up. How do you approach a problem when you have no idea where to start? We’ll consider some interesting implications for …
(An archive question of the week) We’re looking at extended discussions of a single topic, which illustrate how we try to guide a student to a deeper understanding. Here, a student asks how to solve an equation, and Doctor Ian takes him through the whole process, clarifying what it means to solve an equation, and …
Today we’ll look at a question from a student who was troubled by the amount of guessing needed to solve certain problems. This leads to an interesting survey of different kinds of guessing, and ways to develop that skill.
(An archive question of the week) I’m looking for past questions that led to deep discussions. This week, we have a case where a student realized he was doing algebra by rote, not thinking about what variables really mean. This realization was triggered by a step that many students stumble over, where parameters change their …
(An archive question of the week) There are several ways to restrict the range of values you need to test when you are searching for zeros of a polynomial (using the Rational Zero Test or the Intermediate Value Theorem, for example). One of them can be quite useful for difficult problems, but can be hard …
Last time we looked at some false proofs, which are often used to help students understand what does and does not constitute a valid proof, and in particular, to remind them to be careful in algebraic proofs, looking for issues like division by zero and taking square roots. This time, we’ll look at two similar …
One of the ways to show the need for careful proof, which I discussed last time, is the existence of apparent proofs of false “facts”. These are also useful for training us to think carefully. In this and the next couple posts, I will look at a few categories of false proofs, focusing on what …