Keeping Parents In The Loop

I’ve been a math teacher for 30 years now, and I’ve also been a parent of a math student. I know from experience that high school students don’t rush home every afternoon to share with their parents what they’ve learned that day! So, in order to foster parent/teacher communication and parent/child communication this year, I’ve been emailing newsletters to the parents of my students.

They aren’t long or complicated, but they always have three components:

  1. A quick summary of what we’ve covered since the last newsletter;
  2. A topic or concept parents can discuss with their child;
  3. An interesting website, article, or blog that is math-related.

The response from parents has been very positive, and it’s well worth the time it takes to put together. I use GeoGebra to create the illustrations, and if I want to link to an interactive applet, has thousands to choose from. One word of advice: make sure you put your recipients’ email addresses in the bcc address box to protect their privacy.

Here’s the latest one I just sent to the parents of my geometry students:

Greetings, Parents of my Geometry Students,

It’s hard to believe we have finished our first quarter. One down, three to go!
We are wrapping up Chapter 3, which is all about parallel lines and angles. When you have two parallel lines intersected by a transversal, all kinds of congruent and supplementary angles are formed. Ask your daughter to point out to you which angles are congruent in the figure below, and which ones are supplementary (adding up to 180 degrees):


We also learned why all the interior angles of a triangle always add up to 180 degrees. The figure below is a visual proof of this (if you don’t get it, ask your daughter for help!):

Interior angles

One of the most amazing facts in Euclidean geometry is that the sum of a polygon’s exterior angles is 360 degrees, regardless of how many sides the polygon has. It seems counter-intuitive, but to see why, click here, and drag the slider in the applet (created by Aeolus Ophion).
In our next chapter, we will be exploring triangles and all the ways we can prove that they are congruent.
For the cool math site of this edition of the HH Geometer, click here.
Have a restful Fall Break!

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? :)


Easing Into Proof

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.

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.