(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.
For the next couple weeks, we’ll look at a few “dissection problems”, which involve cutting a shape in various ways. Here, we look at what turns out to be a simple problem, though it seems at first that it would be complicated: dividing a cake into pieces with the same amount of cake and the …
(A new question of the week) A question from last month provides an opportunity to show how to develop an algebraic proof of a combinatorial identity involving factorials. We’ll be looking over Doctor Rick’s shoulder as he guides a student through the maze. I’ll also add in a previously published version of the same proof …
Last week, we looked at two solutions to the problem of finding the probability that you can make a triangle using three pieces of a stick, if we cut it at two independently chosen, random locations. This time, we look another solution to that problem, and a similar solution to the version in which we …
(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 …
This week we look at questions about how likely it is that you can make a triangle out of three random pieces of a stick. As always in probability, the first issue comes in deciding how the process is to be done (that is, what does it mean to break a stick randomly?); we’ll also …
(A new question of the week) A good way to check (and hopefully build) a student’s depth of understanding is to assign non-routine problems, in which familiar ideas are twisted around so you have to come at them from a different direction. Here we’ll look at a question about a graph that can be solved …
I’ll close this series on averages with a quick look at the mode. Unlike the other “averages”, this doesn’t always exist, and when it is, it is not always unique. In fact, as we’ll see, sometimes we can’t be sure whether there is no mode, or many modes. How do we handle these odd cases?
(A new question of the week) Here is an intriguing question we got at the end of September from an adult whose name I’ll shorten to Arun.
In our series on averages, last week we introduced the idea of the weighted average (or weighted mean), where each item has a weight attached. The classic examples all involve grade averages in various ways. This time, we’ll look at how weighted averages arise when you need to average several averages together, something we touched …