(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) 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) Here is an intriguing question we got at the end of September from an adult whose name I’ll shorten to Arun.
One thing we enjoy doing is guiding a student through the process of problem-solving. Here is a problem from August that illustrates how to think through a complicated geometrical proof. In particular, this uses some circle theorems involving chords, secants, and tangents, together with a bit of algebra.
As the capstone of this series on counting, lets look at something that’s a little harder to count by drawing: What is the maximum number of regions into which all of 3-dimensional space can be divided by n planes? We’ll look at two significantly different perspectives.
Last week we looked at counting the maximum number of pieces into which a circle can be cut by n chords (straight lines). Here we will look at a similar-sounding problem where we use all the chords formed by n points on the circle. We’ll also see an important example of why we shouldn’t jump …
We’ve looked at how to count diagonals in a polygon; this week and next, I want to consider two different problems (though they look similar at first) dealing with chords of a circle (which are practically the same thing as diagonals of a polygon). In each, what we count will be the regions into which …
(An archive question of the week) In gathering information on how to count the diagonals of a polygon, I found this long discussion about a similar-sounding issue, which is hardly more difficult, yet far more complex. It was interesting to explore what the question means, and take it in different directions, on the way to …