# Mathematics

## Transformation Rules

We’re working on rigid transformations in my Geometry classes. The basic transformation rules for translation and reflection over a vertical or horizontal line are straightforward; here, they’re written as functions, rather than the briefer vector notation. Translation of $$h$$ horizontally…

Mathematics

## Dan Meyer’s Really Really Really Difficult Puzzle

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…

## 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…

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…

## 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

## Angles in the Pentagon

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 side length of the cube, what are the angles in the pentagon?…

Mathematics

## 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…

Mathematics