My Favorite Spiral Design

Geometric designs fascinate me. I’ve been making them for years, and never get tired of them. Last month I attended the NCTM Annual meeting in Philadelphia. The first workshop I participated in was called: Transforming Algebra and Geometry into Art and Design. The workshop was led by Marilyn Dibble, a talented math coach from Topeka, Kansas. She introduced many wonderful activities that connect middle and high school mathematics with art and design.

The spiral design depicted below is my favorite geometric pattern. Ms. Dibble did not showcase this pattern in her workshop. In fact, I couldn’t find the directions or details for this pattern anywhere on the web, so I will share it here.

To create the design, begin with a regular hexagon, preferably one that students construct themselves using Euclidean tools. We will call this the Stage 0 Hexagon.

To make the Stage 1 Hexagon,  measure clockwise from each vertex n÷6 units, where “n” is the side length in centimeters of the Stage 0 Hexagon. Connect the six points and you have constructed Stage 1, a smaller hexagon constructed inside the original figure.

Continue iterating the figure this way, each time dividing the new, smaller hexagon’s side length by 6. Repeat the process until the figure is pleasing to the eye. The more carefully the figure is measured and iterated, the less distortion will result as each new stage is completed.

The math skills that are involved in this design include:

• Euclidean construction
• direct linear measurement
• decimal division
• rounding decimals
• iterations

A Mathematical Scavenger Hunt in Boston!

On Friday June 8, I look forward to taking my Geometry class on a Field Trip to Boston. We will enjoy the sights and participate in a mathematical scavenger hunt. The idea for this field trip came from an article I read in my favorite math publication: NCTM’s MTMS.

Our Field Trip will begin at the Prudential Center in the Back Bay. We’ll make our way on foot down Boylston Street. We’ll stop for a Swan Boat ride at the Boston Public Gardens, check out the Granary Burying Ground and have lunch at our destination: Quincy Market. Along the way we’ll be pausing to solve math questions pertaining to geometry, measurement and algebra.

Click on the picture below to take a virtual tour. Can you predict what the math questions might be that pertain to each picture?

How to use origami to make a stellated octahedron

This is one of my favorite origami models, introduced to me by a gifted former student. You will need 12 square sheets of paper.

 

Understanding the Points of Concurrency in a Triangle

What are the points of concurrency in a triangle and what is mathematically special about each of them?

The centroid is the point of concurrency (intersection) of the three medians of a triangle. A median is a segment joining a vertex of a triangle with the midpoint of the opposite side.

The centroid of triangle ABC is point D.

An orthocenter is the point of concurrency of the three altitudes of a triangle. An altitude is the perpendicular line drawn from a vertex to the opposite side.

The orthocenter of triangle ABC is point O.

An incenter is the point of concurrency of the three angle bisectors of a triangle. An angle bisector is the ray which bisects an angle, or divides the angle into two congruent angles.

The incenter of triangle ABC is point I.

A circumcenter is the point of concurrency of the three perpendicular bisectors of a triangle. A perpendicular bisector is a segment which bisects a segment and forms right angles.

The circumcenter of triangle ABC is point M.

To see how these points of concurrency are constructed, please watch my short Screencast video.

Now that you have seen how to construct a triangle’s points of concurrency using geometry, how could you use coordinate geometry and algebra to identify the points of concurrency?

Given points A(10,7)  B(10,1) and C(2,1), find the coordinates of the following:

• centroid
• orthocenter
• incenter
• circumcenter

Check your answers using this handy Point of Concurrency web calculator.

How to Construct the Wheel of Theodorus

To construct your own Wheel of Theodorus using Euclidean tools, follow the steps described and illustrated below.

1.  Construct right triangle ABC, with base = 1 unit and height = 1 unit.

2.  Using hypotenuse AB as the base of the next triangle, construct a perpendicular to line AB through point A.

3.  Mark off 1 unit as the height of right triangle DBA.

4.  Construct triangle DBA.

5.  Continue using the hypotenuse of the newly constructed right triangle as the base of the next triangle iteration. The height should continue to be 1 unit.

6.  Continue constructing your “Wheel of Theodorus” until the final picture is pleasing to the eye.

7.  What is the length of the hypotenuse of the final right triangle? Explain or show how you know using the Pythagorean theorem.

Russian Square Puzzle

Have you seen the Russian square puzzle? The puzzle pieces include a small square and 4 non-congruent, right trapezoids. Fit the five puzzle pieces into the square area without any overlap. Sound easy?  Clicking on the picture will link to an interactive site so you can try to solve it. Good luck!

 

Helpful Hints:

1. The pieces will not fit perfectly into the square outline.
2. There will be some space left uncovered, but none of the pieces should overlap each other.
3. Put the square piece in the middle, at a 45 degree rotated angle.

 

Directions to Retake a Test

Have you ever received a test grade that you were dissatisfied with? In Ms. Miles’ math class, you are welcome to retake any test or graded assignment. The directions are simple and can be downloaded here as a pdf file. Just follow the directions, retake the test, turn it in, and earn points back to improve your score! It will be worth the effort. Promise.