A Stained-Glass Fractal

I have been invited to participate in Harpeth Hall’s faculty art show this fall, so I’m planning on spending some time this summer creating three or four pieces that use mathematics in their design.

I just completed the first one, which uses the fractal property of self-similarity: each outer circle is split into two smaller circles. Of course, if it were a true fractal, the process would repeat ad infinitum, but due to the limitations of working with glass I had to stop after four iterations.

Artglass fractal

An Activity for Pi Day

Pi Day (3/14) is fast approaching. Here’s a simple Geogebra activity you can do with your math students to help them get a sense of what pi is.

In Geogebra, construct a segment, then construct the midpoint. Construct a circle using the midpoint as center, and an endpoint of the segment as a point on the circle. In the Input pane, enter Circumference(<your circle>). When you divide the circumference measurement by the diameter length, you get pi! Geogebra is dynamic, so you can change the size of the circle, but the ratio will remain constant at pi.

Here’s an interactive version for you to play with at Geogebra’s site.

Here’s a screenshot:

All of my students have TI-84 graphing calculators, so I am going to have them use their Geogebra constructions to generate 10-12 different diameters and circumferences. Then we will enter the diameters in L1, the circumferences in L2, and plot the data. We will calculate the regression line, the slope of which should approximate pi.

Happy Pi Day!

Update: This went really well in my geometry class. Here’s the data we collected –

lists

Here’s the regression line equation (check out the value of the slope):

LinReg

And here is the plot of the data and the regression line:

Graph

 

Calculus for Geometry Students

In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.

I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.

Riemann

Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!

What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.

 

Algebra + Area = A Nice Proof

In my Geometry classes, it is time to work with right triangles. Rather than simply present my students with the Pythagorean Theorem, I decided to have them prove it without them knowing that’s what they were doing.

I based the following activity on a proof I found at Alexander Bogomolny’s very useful site, Cut-The-Knot.

A Proof

  1. What is the total area of the four triangles below? (They are all congruent)

 4 rt. triangles

 

 

 2. If the triangles are rearranged into the square pattern below,

Pythagorean Proof Square

(a) In terms of a and b, what is the area of the “hole” in the middle?

 

 

(b) What is the overall area of the square pattern?

 

 

3. How can you use the results of parts 1 and 2 to prove a famous theorem?

 

I like this proof, because it’s very visual as well as algebra-based. The most challenging part for my students was figuring out the area of the square “hole”. Many assumed the side of the square was 1/2 of a, which is not correct. The satisfaction and excitement they had as they saw the pieces of the proof come together were well worth the effort, though! They were all familiar with the Pythagorean Theorem, but this was the first time any of them had actually proved it.

Click here for a pdf of the worksheet and here for the solution.