Finding the Area of a Circle
Students often wonder where the formula for the area of a circle comes from; and knowing something about that can help make it more memorable, as I discussed previously about other basic area formulas.
A Fraction Word Problem – Algebra or Not?
Sometime soon I will do a series of posts on word problems, which are a common point of difficulty with students. But here is one recent example from a high school student, where language was the main difficulty, but the algebra is worth discussing as well. We’ll look a little more deeply into the problem …
Principles for Solving a Formula
(An archive question of the week) Last time I discussed issues that arise in solving a simple algebraic equation. In researching that, I found a discussion of solving a formula for a variable (which in some countries is called “making x the subject”, that is, changing an equation involving x into the form “x = …
Principles for Solving an Equation
Questions about solving algebraic equations are common. Here I will bring together several answers where we discussed the basic principles for solving relatively simple equations, which are important to learn well before moving on to quadratic equations and beyond.
What is Cosine – With a Twist
(A new question of the week) Having written last week about the definitions of trigonometric functions, I want to look at a question from a few months ago that illustrates a rather common mistake students make in applying those definitions. It also demonstrates the patience required to find out what is in a student’s mind, …
What is a Fraction, Really?
(An archive question of the week) One of the things I have learned as a Math Doctor is that it can be dangerous looking up a definition online. Sources vary — not because they are wrong, but because definitions depend on context, so you can easily find what appear to be contradictions because they refer …
Dividing Fractions: How and Why
Fractions have always given students trouble, and we have had many questions about working with them. Even looking only at division of fractions, I have had to restrict my attention to a few sample answers. These show the reasons for the standard method, presented in a variety of ways, together with some alternative methods.
Proving an Identity in Different Ways
(A new question of the week) Having discussed trigonometric identities on Monday, let’s make this Trig Week, by looking at a discussion from two months ago in which we were asked about alternative routes to a proof.
What Are Trig Functions, Really?
(An archive question of the week) Trigonometric functions are sometimes introduced without a deep explanation of their meaning; they are just buttons to push on a calculator, or names to write in an equation. Even when a textbook gives a careful presentation, there are so many facets to the concept that it can be easy …
Different Ways to Prove a Trigonometric Identity
Proving trigonometric identities can be a major challenge for students, as it is often very different from anything they have previously done. Often they confuse this concept with solving an equation. But also, they may be give overly rigorous standards to comply with. Here, I will look at several discussions we have had about different …
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