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…
Author: Clio
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…
Angles in the Pentagon
(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!) [Image lost, not reconstructed.]Here’s a geometry challenge. A plane intersects a cube in such a way as to form a pentagon. If AL, FJ, and CM are all one-fourth of the…
Consumer Math
Consumer math represents the most immediate and practical response to the student mantra, “When am I ever going to use this?” I was thinking about this yesterday during a late night run to Meijer to get some paper. They had two options: A ream of 500 sheets for $4, or a ream of 750 sheets…
Addition and Multiplication: Units
It is the habit among mathematics teachers, particularly at the elementary level, to present multiplication as repeated addition. The inimitable Keith Devlin, among others, has ranted about this, but it’s easy enough to see the temptation. When dealing with integers, multiplication and iterated addition will return the same numbers. Historically, it may be the case…
Ratios vs Fractions
Several middle school math teachers have told me that there’s an important distinction between fractions and ratios that students don’t get. When I ask them what it is, the teachers can’t tell me; “it’s complicated”, they say. I’ve been troubled by that response. For me, ratios and fractions both involve division, and they certainly look…