In this post, I discuss the Sleeping Beauty problem, which explores how perception can affect assessment of probabilities. First, the video that introduced me to the concept: If you’re not in a position to or would rather not watch the video, here’s the gist of the puzzle: You have devised a means of administering an…
Author: Clio
Nested Isosceles Triangles
Today, I’m going to write up a quick geometric proof. Here’s the original puzzle that inspired it. Given that \(AC = AB\) and that \(AE = DE = CD = BC\), what is the measurement of \(\angle A\)? In other words, the diagram has four isosceles triangles: \(\Delta AED\), \(\Delta EDC\), \(\Delta CBD\), and \(\Delta ABC\). My strategy…
Types of Numbers
Elementary school students spend most of their time working with counting numbers, that is, the non-negative integers. As students progress through secondary school, they work increasingly with non-integers, eventually entering the complex number plane. However, many of them maintain the desire to tie these back to their comfort zone in \(\mathbb{N}^0\). In this post, I’ll…
Improper vs Mixed Fractions: A Six-Year-Old’s Perspective
I often discuss mathematics with my six-year-old son. As a teacher, my goal is to try to pinpoint where it is that student understandings go astray. As a parent, my goal is to teach my son some mathematics. We’ve discussed division before, and I was inspired to explore it again because of some multiplication he’d…
Misadding Fractions
A common mistake students make when adding fractions is to add both the numerators and the denominators (I’ll use a special symbol to reinforce that this is not proper addition): \[ \frac{2}{5} \heartsuit \frac{3}{7} = \frac{2+3}{5+7} = \frac{5}{12} \] The general approach is to tell students that that doesn’t normally work. However, while errors in process sometimes…
Milne: “Are” vs “Is”
I noted in an earlier post that in 1893, Milne used “are” as a casual speech reading of the equality sign, rather than the “is” that I’m used to. Adam Liss notes that “are” is also used in Danny Kaye’s 1952 movie Hans Christian Andersen. On the other hand, by the early 1960s, the Beatles were using…
The Haves and the Have-Gots
This is a post on language and linguistics, one of my other passions. This is also largely an educated riff; there has been a lot written on this subject, and I do not profess to higher mastery than what comes before. While reading Milne’s Elements of Arithmetic (1893), I noticed questions like these: “How many boys…
Milne on Using “And”
At a used bookstore today, I picked up the 1893 text Elements of Arithmetic: For Primary and Intermediate Classes in Public and Private Schools by Dr. William J. Milne. One thing that I noticed was that he is adamant that “and” is never to be used when naming integers: “In reading numbers expressed by three figures, the…
Volume of a tetrahedron
(Edited 6/20/23: I lost the images for this post, and they’re in 3D. I haven’t reconstructed them, so there’s an additional challenge for you!) This is a challenging one: Given all the information at one corner of a tetrahedron (all three surface angles and all three edge lengths), what is the volume of the tetrahedron?…
2048: How Many Fours?
Problem You have just completed a game of 2048, and you want to know what percentage of initial tiles were fours. How can you do so? Rules First, the rules of 2048. In its basic form, this app consists of a 4×4 grid containing some tiles. On a turn, the user slides one of the…