I was recently reading a book on Greenfoot, a Java-based GUI intended for teaching programming to high schoolers and college underclassman. In the “Newton’s Lab” project, the writer assuaged the reader who might be leery of the mathematics in that particular project. Remember, the reader was told: Programming can do a variety of things, including…
Category: Mathematics
10 vs Ten
What does “ten” mean? Here are some dictionary definitions: The number 10. (MacMillan) The cardinal number equal to 9 + 1. (American Heritage) Equivalent to the product of five and two; one more than nine; 10. (Oxford) Superficially, these seem like comparably valid definitions: Ten is the number that comes after nine, that is, 10….
Intersecting Secants
In this entry, I’m going to be discussing how mathematicians tend to approach the world, and why we need better PR. I’m currently teaching High School Geometry. Here is what the book has to say about the “Segment of Chords Theorem”: “If two chords intersect in a circle, then the products of the lengths of…
Finding Quadratic Solutions
The topic in my geometry class today involved finding solutions to quadratic equations. The actual topic was on the ratio of lengths of secants in a circle, but that’s for another post. For a specific example, consider this problem. Two secants intersect outside a circle. The first secant line has segments of 15 (between the…
Pascal’s Triangle and Dice Rolls
Pascal’s Triangle Pascal’s Triangle represents the coefficients of a binomial such as \(x + 1\) raised to a power. Row n of the triangle lists the coefficients of \((x + 1)^{n-1}\). Here are the first few rows of Pascal’s Triangle: \[\newcommand\cn[3]{\llap{#1}#2\rlap{#3}} \begin{array}{c} &&&&&&\cn{}{1}{}\\ &&&&&\cn{}{1}{}&&\cn{}{1}{}\\ &&&&\cn{}{1}{}&&\cn{}{2}{}&&\cn{}{1}{}\\ &&&\cn{}{1}{}&&\cn{}{3}{}&&\cn{}{3}{}&&\cn{}{1}{}\\ &&\cn{}{1}{}&&\cn{}{4}{}&&\cn{}{6}{}&&\cn{}{4}{}&&\cn{}{1}{}\\ &\cn{}{1}{}&&\cn{}{5}{}&&\cn{1}{}{0}&&\cn{1}{}{0}&&\cn{}{5}{}&&\cn{}{1}{}\\ \cn{}{1}{}&&\cn{}{6}{}&&\cn{1}{}{5}&&\cn{2}{}{0}&&\cn{1}{}{5}&&\cn{}{6}{}&&\cn{}{1}{} \end{array}\] For instance, row 4 is…
Solving Simultaneous Equations: Multiple Methods
Introduction and Terms Recently, a post on the G+ Mathematics community involved how to determine \(x\) and \(y\) when: \[3x + 5y = 12 \\ x + y = 2\] This is generally referred to as simultaneous equations or a system of equations. As a general rule, for such a problem to be solvable, you…
Finding an Incenter via Formulas
Terms For every polygon, there is a largest circle that fits entirely within that polygon. If that circle touches all sides of the polygon, then it is said to be inscribed; it is called the incircle, and its center is called the incenter (which is then also called the polygon’s incenter). Every triangle has an incenter…
Equatorial temperatures
This one strikes me, and apparently others, as highly counter-intuitive, but it’s true because of mathematics! Take any two places in the world; call these points A and B. Take any two paths between A and B that are the same distance; call these paths C and D. Let C(x) be as far down path…
Pseudocode for the Russian peasant method of binary
Just for fun… Here’s the pseudocode for the method of building a binary number from a decimal number, based on the Russian peasant method of multiplication: function mybin(mydec) { mybin = “”; do while mydec > 0 { if mydec is odd: { mydec = mydec – 1; mybin = “1” + mybin; } else: mybin…
Russian peasants, number sense, and bases
Russian peasants do too much work There is a method of multiplication called the Russian peasant method. I’ve seen it mentioned here and there, but I was not explicitly educated in the process; it struck me as being more trouble than it was worth, and I didn’t previously bother to dig farther into it. I…