Proof. A word that strikes fear into the hearts of many geometry students. In an effort to try to ease the anxiety created by this process, I tried a new approach (inspired by my colleague here at Harpeth Hall, Maddie Waud): I typed up and printed out three simple examples of geometric proofs and cut up the statements and reasons. I used Geogebra to create the illustrations. I then put them in envelopes, gave them to pairs of students, and asked them to arrange them in proper order.
There were a few students who didn’t get the first proof correct initially, but by the third one everyone put them together in the right order in no time. Hopefully, this exercise was a confidence-booster for my girls. Their biggest concern is that they won’t know where to begin when confronted with a proof that doesn’t have any statements and/or reasons provided, but they felt better after doing this activity.
You can download the proofs by clicking Proving a Theorem , and you can print them out for your own use.
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.
After a four-year hiatus, I’m teaching Geometry once again. I have always loved this course, because it’s such a unique branch of high-school mathematics. It doesn’t deal much with arithmetic, variables, and functions; rather, it requires students to employ logic and clear thinking to prove theorems. There’s even room for some “elegance”!
We are currently learning the basic building blocks of Euclidean geometry: points, lines, and planes, and the postulates concerning them. In previous years, I have incorporated Geometer’s Sketchpad into my teaching, but this year I’m switching to GeoGebra. I love Sketchpad, and I’m very comfortable working with it (take a tour of my YouTube videos featuring it), but I can’t justify asking my students to pay a hefty license fee to use it, when GeoGebra is open source. GeoGebra also has a built-in CAS, and its developers provide major updates regularly. Throw in all the lessons and files available on GeoGebraTube, and it’s very hard to resist making the switch.
After making sure all of my Geometry students had the latest version of GeoGebra installed, we worked through the basic features of it. Within a few minutes they were creating some impressive artwork with it. Today, I issued the first of what I plan to be a regular series of GeoGebra challenges: create a rectangle (not a square) that remains a rectangle regardless of how they drag corners and sides. It’s harder than it sounds, and it requires them to think about what makes a rectangle a rectangle.
Today in Calculus, we looked at three different types of regression equations that can be used to model data. Linear regression was first. The students measured each others’ forearm lengths and their heights. You can have a good conversation about which should be independent and which dependent. We decided the forearm lengths were independent, and height was dependent. We put the data into our TI-84 Lists, determined appropriate window parameters, and plotted the data. Then we used the LinReg feature to graph a line of best fit:
The next regression equation we investigated was the quadratic one. For this one, I used the old favorite, the water fountain stream:
We pasted it into GeoGebra and put some points on the path of the water. Using those points and the QuadReg feature of the TI-84, we came up with an excellent model:
Finally, we used the sunrise and sunset times of the fifteenth day of each month in Nashville, TN to see how a sinusoidal function can model data. (I grabbed the data from the US Naval Observatory.) Students converted the times that were in hours:minutes to decimal hours (16 + 56/60) – (6+57/60). We entered the months (January = 1, February = 2, etc.) and length of daylight into the calculator’s lists and used SinReg to determine the sinusoidal function that best fits the data. If time permits, it’s a nice exercise for the students to figure out the amplitude, period, phase shift, and translation themselves, and compare their function to the calculator’s.
This set of exercises took about 50 minutes, and the girls got pretty excited when they saw how closely their functions matched the data, and how they can use them to determine other points.