A Rational Inequality with Huge Exponents
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.
Chinese Remainders With and Without the Theorem
(An archive question of the week) My title is tongue-in-cheek, as we’ll be looking at the Chinese Remainder Theorem, which is really a Chinese theorem about remainders, not a theorem about “Chinese remainders”. But we’ll work on a problem that can be solved with or without knowledge of the theorem, and with various doses of …
Ranking a Word Among Its Permutations
(A new question of the week) There are some topics that appear to be standard in certain parts of the world, but far less familiar in our own. Sometimes it takes two of us to recognize what a student is asking, due to language issues and different past experience with such questions. This is an …
Interpreting and Solving a Counting Problem
(An archive question of the week) Combinatorics can be inherently tricky; making up your own problem is doubly so. Here we have a problem created by a teacher, who then is not entirely sure what it means. How can we figure out what meaning to give it? Combine that with working out how to solve …
Arranging Letters with Duplicates
(A new question of the week) Here is a recent discussion with a frequent user of our service, Kurisada, about combinatorics. He is new to the subject, so this involved several introductions to new ideas.
How Many Different Pizzas?
(An archive question of the week) We’ve been looking at examples of extended discussions with students about various kinds of problems. Here, we have one (not from a student) that led to some good thinking about combinatorics – the techniques of counting the ways something can happen.
Circumcircles and the Law of Sines
(A new question of the week) Today I want to look at a recent question that led into both geometrical and trigonometrical solutions, and particularly a useful perspective on the Law of Sines.
Too Many Variables?
(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 …
Non-routine Algebra Problems
(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 …
How Variables and Equations Work
(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 …