Exploring Fundamental Concepts In Elementary Mathematics

 

This series of essays represents the bulk of the course I've offered at mathematics camps in the United States (Math Zoom, Awesome Math, Epsilon Camp, MathWorks), at AMSAT in Bangalore India and Teacher Circle workshops sponsored by the American Institute of Mathematics. The idea is that we can motivate students to develop the mathematics needed to solve some interesting problems by starting with the problems themselves. Several of the essays below have been incorporated in the course Introduction to Mathematical Reasoning, Math 111, offered by the Saylor Foundation.

 

Place Value Problems is an essay on the representation of rational and irrational numbers in our standard powers of 10 notation. The sister essay Fusing Dots should be read soon after this one. In PVP we explore problems with digits. For example why do the two-digit numbers ab  and ba always differ by a multiple of 9.

Fusing Dots, also known as Exploding Dots, is the beautiful idea of James Tanton. We use it here to create exotic number systems: Fibonacci, base phi, Cantor (aka factorial), and negative base.

Problems with Digits is a set of problems designed to reinforce the place-value ideas of the first two essays.

Counting Problems is an essay on counting the number of samples taken from a population when we can answer the two questions, Does order matter? and Are we allowed to repeat an item? This is a terrific topic for students interested in competitions (very popular in the United States).

Bug Problems is a collection of problems involving a bug that crawls around the plane with a fixed itinerary. Questions like where is the bug after 2012 minutes and when will the bug get to a certain point are typical. The level of mathematical maturity needed is slightly higher for this topic.

Boxes and Cubes is an essay on visualizing geometric configurations. For example suppose all six faces on an n x n x n cube composed of n cubed unit cubes are painted red. How many of the cubes have some red paint on them?

Divisor Problems explains how to examine the divisor structure of a composite number geometrically using a device known as the Hasse diagram of the number. I use ZomeTools to build the Hasse diagrams for numbers like 30, 300, and 63000.

Fractions is a collection of problems about unit fractions, Simpson's paradox, and mediants.

Decanting problems can be solved using the Euclidean algorithm.

KenKen is a puzzle I love. Here's an essay on how to play it.

Epsilon problems

Other essays I'm writing now are on Fibonacci numbers, Toothpick problems, counting snakes and descents, and magic geograms.