In tenths Triangles.

I made these for a friend whose church wanted a progress chart for raising funds to restore a triangular stained glass window. (Which I wouldn’t have believed as a textbook task.) It’s a pretty interesting problem, though.

(They went with the top one.)

5 Greatest Discoveries in Mathematics #5

Euclid’s Elements

Written around 300BC, Euclid’s work built the foundation for modern mathematics by introducing a set of axioms and proceeding to demonstrate by mathematical rigor a collection of theorems that naturally followed.

Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for over 2,000 years following its creation.

Elements influenced the thinking of great minds ranging from Dostoevsky to Einstein, and Abraham Lincoln’s inclusion of the phrase “dedicated to the proposition” in his Gettysburg address is often attributed to his readings of Euclid.

From: polymic.com

Comic: bkahn.us

Not sure if reblogging for the history or the Frank & Ernest #mathcomic…

Amazing Mathematics Teaching Activity
from: jdh.hamkins.org

A mathematical investigation of graph coloring, chromatic numbers, map coloring and Eulerian paths and circuits. By the end each child will compile a mathematical “coloring book” containing the results of their explorations.

We begin with vertex coloring, where one colors the vertices of a graph in such a way that adjacent vertices get different colors. It starts with some easy examples, and then moves on to more complicated graphs.

The aim is to use the fewest number of colors, and the chromatic number of a graph is the smallest number of colors that suffice for a coloring.

Map coloring, where one colors the countries on a map in such a way that adjacent countries get different colors, is of course closely related to graph coloring.

Next, we consider Eulerian paths and circuits, where one traces through all the edges of a graph without lifting one’s pencil and without retracing any edge more than once. We started off with some easy examples, but then considered more difficult cases.

An Eulerian circuit starts and ends at the same vertex, but an Eulerian path can start and end at different vertices.

You can discuss the fact that some graphs have no Eulerian path or circuit. If there is a circuit, then every time you enter a vertex, you leave it on a fresh edge; and so there must be an even number of edges at each vertex. With an Eulerian path, the starting and ending vertices (if distinct) will have odd degree, while all the other vertices will have even degree.

It is a remarkable fact that amongst connected finite graphs, those necessary conditions are also sufficient. One can prove this by building up an Eulerian path or circuit (starting and ending at the two odd-degree nodes, if there are such); every time one enters a new vertex, there will be an edge to leave on, and so one will not get stuck. If some edges are missed, simply insert suitable detours to pick them up, and again it will all match up into a single path or circuit as desired.

This is an excellent opportunity to talk about The Seven Bridges of Königsberg. Is it possible to tour the city, while crossing each bridge exactly once?

Get the resource: Andrej Bauer has assembled the images into a single pdf file: https://drive.google.com/file/d/0B7eG5PHUDcmZX1FnOHRhSU9ubUU/edit?usp=sharing, and filtered the color to black/white to improve printing.

Support it with an in class app: from the apple App Store OR One Touch Draw http://www.windowsphone.com/en-us/store/app/one-touch-draw/2c1b6ab4-d8c0-45f8-a2ac-b94fa721b39e

(Most amazingly…..this was a second grade class!!!)

Simon’s Sum of Cubes.

Speaking of the sum of cubes, don’t miss Simon Gregg's great blogpost with his students’ cuisinaire rod explorations. Gives me goosebumps.

• Constructing it you see new things about ∆ numbers eg the 2nd & 3rd are multiples of 3 etc
• Also, with enveloping squares it’s curious that each new layer is the minimum square it could be with that size rod

He’s making his digital images with NRICH’s cuisinaire environment.

Danger lies before you, while safety lies behind,
One among us seven will let you move ahead,
another will transport the drinker back instead.
Two among our number hold only nettle wine,
three of us are killers, waiting hidden in line.
Choose, unless you wish to stay here forevermore,

First, however slyly the poison tries to hide
you will always find some on nettle wine’s left side.
Second, different are those who stand at either end,
but if you would move onward, neither is your friend.
Third, as you see clearly, all are different size,
neither dwarf nor giant holds death in their insides.
Fourth, the second left and second on the right
are twins once you taste them, though different at first sight.

This logic puzzle appeared in Harry Potter and the Sorcerer’s Stone, without the image. Roger Howe gave a complete analysis in the February 2002 issue of the Mathematics Teacher. Can you figure out the contents of each of the seven bottles?

Here’s the NCTM’s Harry Potter and Math page, with Howe’s fun article included. Well citations, anyway. You have to muck about with the NCTM’s generally awful website to find them.

+The more you know about it, the better it gets. It also connects infinite series and trigonometry, for example!

Today’s This Day in Math lead quote:

My work always tried to unite the true with the beautiful, but when I had to choose… I usually chose the beautiful. ~ Hermann Weyl

Sum of cubes.

Saw the top image on reddit (r/math) from Matthew Broussard, and had to pretty it up. Do you like with the gaps or the filled in?

Edit: heard from Matthew who said he likes both! Good deal.

Inverse Construction.

Saw this construction at Hyrodium and wanted to make it quickly in GeoGebra. So elegant! On GGBTube.

Nice fractions with the SurdText command. Gif via RecordIt app.

E’ possibile costruire una ellisse servendosi di 2 circonferenze concentriche di raggi a e b.
Tracciata la semiretta r per O siano S e T le sue due intersezioni con le circonferenze.
L’ellisse viene tracciata dal punto P(x(T), y(S)) quando r compie un giro completo intorno ad O.

Cool construction. So many ways to construct an ellipse - makes it amazing that there’s no closed form for perimeter.

How to draw an inverse of a.

1. Draw a unit circle.

2. Draw a straight line from the point (a, 0) to the North Pole (0, 1). This is line A.

3. Mark the point of intersection between the circle and line A.

4. Draw a straight line from the point of intersection to the South Pole (0, -1). This is line B.

5. Mark the point of intersection between line B and x-axis. This point is (1/a, 0).

The reason is homothetic triangles.