(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…
Author: Clio
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…
White and Blue Elephants
First, a riddle… Q. How do you shoot a white elephant? A. With a white elephant gun. Q. How do you shoot a blue elephant? A. Paint him white, then shoot him with a white elephant gun. Back to Units In this TEDx Talk, Randy Palisoc argues that mathematics should be taught as a language. While…
Zip and Abby
There are a lot of trite websites and apps available for teaching elementary education concepts. And then there are the occasional gems. Zip and Abby, from The Learning Chest, is one of the true gems. The goal of Zip and Abby is not to teach simple “math facts” or to drill on numbers as abstract…
Pizza Math
This gem is timely to my thinking about ratios and units: It seems to have situated itself broadly enough across the Internet that I don’t know if it’s real or a fabrication, but it seems plausible enough. There are, at least, lots of non-teachers who are equally convinced that the question is a trick because,…
Division vs. Ratio
I’ve noticed that the teachers of fractions tend to make a strong distinction between division and ratios, but I haven’t entirely understood why. In my mind, ratios and division are intimately related, even inextricably so. However, my reflections on the abstract unit has brought me to a realization that there is one significant difference between…
Philosophical Natterings: Abstraction
Background One of the thoughts I find myself returning to frequently is this: There is the belief shared among high school mathematics teachers that the struggle students have with algebra is that it’s the first time they’re exposed to abstraction. This isn’t true. The first abstraction in mathematics comes at such an early age, and…
Tangents and the Pythagorean Theorem
A common exercise that’s used to reinforce the concept that the tangent of a circle is perpendicular to its radius involves finding the radius of a circle given two measurements which are related to the tangent and the diameter secant. For example, students might be asked to find the radius of this circle: In the…