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.


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”.



The Golden Rectangle and Fibonacci

One of the best things about teaching math is sharing all of the connections that exist between seemingly unrelated topics. Take, for example, the Golden Rectangle. Often described as the most visually “pleasing” quadrilateral, it is a rectangle whose length, l, and whose width, w, are in the the proportion \displaystyle \frac{l}{w}=\frac{l+w}{l} .

Here’s an example:

Golden Rectangle4.69 / 2.9 = 1.62, and (4.69 + 2.9) / 4.69 = 1.62. The number they equal, 1.62, is called the Golden Ratio, and it has its own Greek letter, phi. It’s actually irrational, like pi and e, and it equals \displaystyle \frac{1+\sqrt{5}}{2} . You can derive phi algebraically by solving \displaystyle \frac{l}{w}=\frac{l+w}{l} for l in terms of w. From the quadratic formula, \displaystyle l=\left( \frac{1+\sqrt{5}}{2} \right)w, so \displaystyle \frac{l}{w}=\frac{1+\sqrt{5}}{2}.

A Golden Rectangle can be constructed with a straightedge and compass (or GeoGebra!):

1. Construct a square ABCD: Square

2. Construct the midpoint, E, of side AB.

3. Construct a circle with center at E, passing through C: Circle

4. Extend rays through AB and DC. Construct the intersection point, F, of the circle and ray AB.

5. Construct a line perpendicular to ray AB, through point F. Construct the intersection point, G, of this perpendicular line and ray DC: Constructed Rectangle

6. Quadrilateral AFGD is a Golden Rectangle.

Another way to construct a Golden Rectangle is to begin with two adjacent squares:


Add another adjacent square with dimension equal to the combined sides of the first two:


Repeat the process, adding larger and larger adjacent squares:


This not an exact Golden Rectangle, but the more squares you add, the closer it approximates a true one.

Here is where Fibonacci comes in. He was a 13th century mathematician who published one of the first math textbooks, Liber Abaci. One of the topics he covered involved a famous sequence that can represent the idealized population growth of rabbits: 1, 1, 2, 3, 5, 8, 13, 21, 34, …. Each term is obtained by adding the previous two terms together. Can you see how the rectangle above is a visual representation of the Fibonacci Sequence? What is amazing is the fact that the quotient of any two consecutive terms in the sequence approaches phi the further up the sequence you go!

Connecting opposite corners of each square with a circular arc creates a Golden Spiral, which increases a distance of phi from the origin for each square it crosses:


There are lots of exaggerated claims made about the use of the Golden Rectangle and phi in ancient architecture, but it definitely pops up in the world of nature. Flowers generally have petals in numbers of 3, 5, 8, 13, 21, etc. For an excellent explanation of the mathematical reasons behind this pattern, check out Ian Stewart’s book, The Mathematics of Life.

A rectangle, a medieval Italian mathematician, a famous sequence, and the number of petals on flowers – all brought together through the beauty of mathematics.

Big News For GeoGebra Users!

GeoGebra, the dynamic geometry software, has just released a major update: version What makes this version really special is its 3D plotting capabilities. When you first start it up, you immediately notice the Perspective menu is different:


Clicking on the 3D Graphics option brings up a new set of tools and menus across the top, as well as a 3D axis system:


I’ve just begun playing around with it, but after a few minutes, I was able to create a plane intersecting a cone:


I am really impressed with the power and features of this version, and I hope someone is putting together a manual with tutorials. This is like an entirely new piece of software!

You can download the latest version of GeoGebra here. Did I mention it’s free? 🙂


Geometry and GeoGebra, Chapter 2

Vertical angles

Continuing my incorporation of GeoGebra into my Geometry curriculum (read about my introduction of GeoGebra here), we will start slow and simple. We are learning the basics of proof, and GeoGebra is a great tool for sparking discussion of what we might want to prove.

Example: one of the first exercises every Geometry student does is to prove that vertical angles are congruent. Instead of having them look at static pictures of vertical angles, each of my students will construct two intersecting lines, measure the angles formed, and look for a relationship. They should quickly see that the vertical angles are congruent no matter how much they move the lines around. Hopefully, they will then wonder why is that always the case. And that’s where proof comes in: if they can write a proof using variables, then they have proven it for all cases, not just the one they’re looking at.

Because it is dynamic, GeoGebra is a great tool for generating lots of conjectures. Geometry is the means we use for formally proving them.