Dan Meyer offered this puzzle. The essence of it is, given an arbitrary number for volume, can we build an algorithm that will always generate the integer side lengths that give us the least surface area? He put it in more “real world” terms than that, but that’s the gist. If we did it in two…
Author: Clio
Choosing a Strategy
(Reposted from my blog for my students) I will often tell my students to select strategies that work best for them to solve a problem, rather than focusing on a single specific strategy. What do I mean by this? I don’t mean that students should use inefficient strategies. One of the things that successful students…
Filling in Blanks: The General Case
Discussion about the puzzle behind the previous post led to this question: Is it possible to construct a similar question, where there are three of five values and a mean given, and where the median is half one of the missing values, with has exactly two solutions? That is, given the values {a, b, c,…
Filling in Blanks
Here’s an interesting mean/median question. Assume your lab assistant took five readings, but only recorded three of them: -3, 5, and 7. The assistant also recorded that the mean was 6, and (for reasons lost on you) that the median was half of one of the missing readings. What were the readings? Actually, the problem…
It’s Always Monday
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…
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…