A System of Exponential Equations
What do you do when you are given a problem that starts with a “lie” and ends with a wrong answer? We’ll go in several directions with this problem, a system of two exponential equations in two variables.
Textbook errors
How to Think Through Probability Problems
It’s been a while since we’ve looked at probability. Here, we’ll look at three questions that we received last year. In each case, we have to detect an error! They’re good examples of what can go wrong, and what to do when your answer appears to be wrong.
Inverse Trig Problems With Unstated Restrictions
Here are a pair of problems related to the last of the three we looked at last time, involving inverse trigonometric function identities with subtle issues. They were probably intended to be rather simple problems (though going beyond what I typically see in American classes) by ignoring the subtleties, but if you approach them as …
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Pitfalls of Inverse Trig Functions
A couple recent questions involved errors made both by students and by the authors of their textbooks, involving trigonometric or inverse trigonometric functions. These offer some good lessons in pitfalls to be aware of.
Algebra Word Problems: Learning from Mistakes
A recent series of questions from an insightful high school student about word problems, provided a number of opportunities to discuss how to find and correct your mistakes – or the book’s! We’ll look at five.
A Challenging Triangle Trigonometry Problem
(A new question of the week) Trigonometry identities can be hard to prove, and more so when they are specifically about a triangle.
Adjoints and Inconsistency: A Questionable Test
(A new question of the week) Last week we looked at how the adjugate matrix can be used to find an inverse. (This was formerly called the [classical] adjoint, a term that is avoided because it conflicts with another use of the word, but is still used in many sources.) I posted that as background …
Proving Two Groups are Isomorphic
(A new question of the week) Two weeks ago, in Proving Certain Polynomials Form a Group, we joined a beginner in learning about groups. Here we will pick up where that left off, learning how to prove that the group we saw there, a subset of polynomials, is isomorphic to a group of matrices. As …
A Finite Series Workout
(A new question of the week) A question from the end of August led a student and a Math Doctor to an extra challenge, by way of an apparent typo in the problem. We particularly enjoy working with students who are willing to take on extra work in order to learn more than they need …
A Bad Geometry Proof Problem: SSA
(A new question of the week) It’s surprising how many questions we get that end up being about problems that are poorly worded or simply wrong. But these can be as illuminating as good problems, by showing ways to catch the error. This is one simple in itself, but will lead us into a common …