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 …
Triangles in a Circle: Two Methods
(A new question of the week) Last week we looked at a question about a triangle inscribed in a semicircle. Not long after that question, the same student, Kurisada, asked a question about triangle inscribed in a circle, which had some connections to the other. As we enjoy doing, we led the student through several …
Too Much Guessing?
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.
A Triangle in a Semicircle
(A new question of the week) Like many questions we get, this one can be solved in many ways. We like to guide a student to whatever solution will fit what they have learned; along the way, we may find various additional methods, and side trips into other topics of interest.
When Parameters Become Variables
(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 …