# Month: July 2019

## 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. Arranging 8 letters, 2 identical There are 8 letters : A, B, H, N, O, S, U, U I was …

## 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. The problem: Triple toppings Here’s the question, from …

## 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. The problem Here is the question, from March: A quadrilateral ABCD is put inside a circle with radius 1. AB = √3, …

## 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. When do we need to guess in math? The question is from 2017: More Methodical Than …