## 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…

Algebra

## 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…

## 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…

Uncategorized

## 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…

Mathematics

## 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…

## 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…

## 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…

## 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…

## 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…

## 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”…

## Volume of a tetrahedron

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? The volume of any pyramid is equal to the…

Mathematics

## 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…

Mathematics