# Strategies

## Finding a Function Value Recursively

(A new question of the week) May was a particularly good month for interesting questions! Here is one requiring us to find one value of a function, based on an unusual property: If $$a+b=2^x$$, then $$f(a)+f(b)=x^2$$. The problem turned out to be not as hard as it looked, yet the function itself is quite interesting …

## A Random Walk on a Graph

(A new question of the week) It seems that most of the interesting questions recently have been about relatively advanced topics, though commonly in introductory classes. Here, we’ll help a student think through a problem introducing the idea of a random walk on a graph. (“Graph” here doesn’t mean the graph of an equation, which …

## Filling a Cistern: Three Pipes, No Numbers

(A new question of the week) Today we’ll look at a classic algebra word problem: Finding how long it takes to fill a cistern through two pipes, with a drain open. But usually these problems are given with specific numbers, as a simple exercise in algebra. What if it’s all variables? the discussion provides some …

## Arithmetic Series, Backward

Here is a recent question about arithmetic sequences and series (specifically, reversing the process to find the number of terms given the sum), that nicely illustrates a common type of interaction with a student: gathering information about both problem and student, then guiding them to use what they know, or giving new information as needed. …

## Fractions and Felonies

(A new question of the week) A recent question involved a word problem about fractions, which will fit in nicely with the current series on fractions. We’ll explore several ways to solve a rather tricky fraction word problem, some avoiding fractions as much as possible, some focusing on the meaning of the fractions, and others …

## One-sided limits of a composite function

(A new question of the week) A good way to develop a sense of what limits are and how they work comes from working with visual representations of them, in the form of graphs. In particular when the functions are defined by graphs rather than by equations, we have a lot more flexibility in creating …

## Cutting and Rearranging a Rectangle

Last week we looked at a puzzle about cutting a square cake into equal pieces. Here we will be trying to cut a rectangle into two pieces and rearranging them to make a different rectangle. Three of the questions we’ll look at came within two weeks in 2001, but we’ll take them in a logical …

## Finding an Angle With and Without Trig

(A new question of the week) Usually when we have a figure labeled with some lengths and angles, we can expect to find unknown angles using trigonometry. When we are expected to do this using geometry alone, we can expect that there is something special about the figure that makes it possible. But how to …

## A Mind-Stretching Exercise with a Stretched Cosine

(A new question of the week) A question in September, about graphing a horizontally-stretched cosine function, led to a long conversation. Between a typo in the problem and some inside-out thinking, this surprisingly non-routine problem led to some good mind-stretching! I have edited this down considerably by removing distractions from the main ideas, but it …

## Overlapping Square Tiles

(A new question of the week) An interesting geometry question came to us in July, about the area of overlap between two squares. The discussion was not long, but leads to some interesting ideas. Here is the initial question, from Vignesh: This is the problem I need help with: Two square tiles of area 9 …