I was working on preparing problems for precalculus class on inverse functions. I started to type the following run-of-the-mill problem type:

If f(x) = blah and g(x) = bleh, show f(x) and g(x) are inverse functions.

And the work usually goes something like this:

f(g(x)) = f(bleh) = algebra kung fu happens here = x
g(f(x)) = g(blah) = some more algebra kung fu = x
Then conclude f(x) and g(x) are inverses.

Without thinking about it, I typed f(x) = -x - 6. Then I wrote the statement on my pad of paper, started working... exchanged the y and x, solved for y.... and got the exact same function.

y = -x - 6. Hmmm.

A part of me wondered if I had made a careless error. Double checked. Nope. No error. I wondered if the graph of the function was its own reflection across the y = x line.


Sure enough... Since y = -x - 6 is perpendicular to the line y=x, it will be its own reflection across the y=x line and consequently its own inverse. In fact, this made me think of an interesting question to pose to my students...

"Can you define a class of linear functions that are all their own inverses?"

In hindsight, perhaps I should be more mindful when constructing tasks for my students. But then again, this would be a great discussion to have with my students. Why does the function end up being its own inverse? Can we think of non-linear functions that are their own inverse? Can we define these classes of functions carefully?

We held our first bowl team session in MTPS today: an friendly, informal competition dividing the class into two groups of roughly equal 'ability.' The bowl competition we attend at the University of Nebraska-Lincoln in November consists of games of 15 questions with 30 seconds to answer each question.

Here is a question we spent some time on in class today.



Here is a screenshot from my Promethean board of the work we did as a group with students:


And here is a screenshot from the Geogebra sheet students constructed in front of the class at my computer terminal to demonstrate what is going on in this problem. If you would prefer to download the Geogebra sheet for your own tinkering, you can find it here.


No one got this question right in the allotted 30 seconds, so we spent some time developing the various parts of the expression. We color coded the edges on the cube (segments AB, AD, and AG) orange. We color coded the diagonals on faces of the cube (segments AC, AH, and AJ) green. We color coded the segment inside the cube (segment AI) black.

The kids had a lot of fun with building the Geogebra sheet and then trying to reconcile between sqrt(2) and the crudely rounded segment length 1.41. Or the sqrt(3) and 1.732.

The pink stuff at the top of Promethean board screen shot is our efforts to generalize this question. I tell the kids we are trying to "hack all possible problems." We use the term invariant and that we are looking to write a question that covers all possible question types. For example, what would happen to the question if s = 2? Or if s=3? Or if s=n? Is it possible to answer the question before the moderator finishes reading it? If so, at what point can we be confident we can buzz in and answer correctly?

I was trying to write an item for an assessment where I would give a student a graph of a piecewise function and ask them questions about the domain, range, and to evaluate the output value for a specific input value - for example, find f(-3). The purpose of this post isn't really to pit Geogebra and Desmos against one another; rather, I want to make note of some of the things I was thinking about as I tried to make a piecewise function graph in each program. (Disclaimer: I am not an expert at this. I have much more experience with Geogebra than with Desmos. I want to see what the differences are between the programs to figure out when to use each of these powerful tools to enhance my instruction in mathematics. I am sharing my thinking about this task.)

Here is the graph I made using Desmos: Desmos_Open_Circle_Piecewise

As I typed the function syntax into Desmos, I thought the editor was a little more user friendly than Geogebra. I typed <= and the editor automatically generated the less than or equal to signs for the restrictions on x. When I wasn't sure what to type, I browsed the examples of projects submitted by Desmos users found on the Desmos homepage. Ideally, I want to capture this graph and place it on an assessment. A photocopier may not pick up on the sections of the function on the graph given the lack of density (being able to make the segments and curves thicker). The way to make these curves denser was not immediately obvious to me. OpenCircleProblemThe circle centered at (1, 1/3) with radius 1/10 is my effort to place an open circle on the graph. To clean up the image, one thing I could try is modifying the restriction on x [for example, writing 1.2<=x<=4 instead of 1<=x<=4] so the user does not see the part of the function jutting into the circle shown at the left.

Here is the same graph I made using Geogebra:

GeogebraPiecewise After making the graph in Desmos, I assumed I could use similar syntax to make the graph in Geogebra. Using similar syntax, I had a problem with the restrictions in Geogebra. Each function has a default y-value of 0 for values of x outside the restriction. Pictures are, after all, worth a thousand words... here are the three functions shown individually. Take a look at the x-axis. Red_Parabola Blue_Line_with_Positive_Slope   Orange_Line

Here is the exact syntax I typed into the Geogebra input bar for each of the above pictures.
(1 / 4 (x - 1)² + 1) (-3 ≤ x ≤ -1)
(x - 2) (-1  <  x  <  1)
1 / 3 x (1 ≤ x ≤ 4)
I incorrectly assumed the syntax would be similar to that of Desmos. I knew from experience I could clean this issue up by using Condition to Show Object in the Object Properties menu if I had to, but I couldn't remember exactly how. I went to Youtube and found a video on graphing piecewise functions in Geogebra:

Below is an image of the corrected Geogebra graph using the appropriate If[ ] commands to define the rules f(x), g(x), and h(x). GeogebraPiecewiseCorrected
Here is the corrected syntax I typed into the Input Bar to define f(x), g(x), and h(x):
If[-3 ≤ x ≤ -1,1 / 4 (x - 1)² + 1]
If[-1  <  x  <  1,x - 2]
If[1 ≤ x ≤ 4,1 / 3 x]
This approach eliminated the x-axis issues from the improper syntax I used at first. These graphs show some of the thinking I do day-to-day as a mathematics teacher trying to construct examples to display in class and problems to use in assessments. If somebody reading this has any advice that could help me become more effective with using Desmos or Geogebra for this purpose, please email me at or find me on Twitter (@ShelbyAaberg). **Update! See below for additional support on Desmos use. Thanks to Eric Berger (@teachwithcode) and (@Desmos). Endpoints_In_Desmos  


Here is the additional resource from @Desmos. Desmos_Tweet

Here is a problem I have been working on this afternoon. I worked on building a model in Geogebra to help students understand the situation.

2013 Probe I Problem #3
A cylindrical can with radius of 6 inches is filled with 2 inches of water. When a smaller cylindrical can 4 inches tall is placed inside of the first with its bottom lying on the bottom of the first, the water in the first can rises a further inch to 3 inches. What is the radius of the smaller can?

A. 3   B. sqrt(12)   C. sqrt(15)   D. 4   E. sqrt(18)

Here is a screenshot of the Geogebra model I constructed.



Here's a PDF of my typed-up solution.

Nested Cylinders Problem

Here's the Geogebra Sheet (built in the Geogebra 5.0 beta with 3-D graphing) if you'd like to take a look.

Here is a prime example why Twitter is a great collaboration resource for math teachers. Last night I was killing time while waiting for a haircut. Reading through some tweets, I noticed a chat going on with the hashtag #ggbchat. With some luck, I caught the very end of the session and posted a question about something that's been bugging me about Geogebra.








I use Geogebra to analyze student summative assessment data in my classes. I like to sort the data to guide me when deciding which students I should group together for class activities. Sorting the data inside a Geogebra spreadsheet would eliminate an extra step for me (specifically, entering the data into Excel and sorting prior to copying & pasting data into Geogebra). I would think the software should allow a user to select a list, right-click, and then be given the option to sort the column of data. Here's a solution to the problem I faced, compliments of Geogebra guru John Golden (@mathhombre).



John also followed up with an idea to create a "Sort" button using a script.





Here is a screencapture of John's suggestion for making a Sort button.



I don't know how many elementary school teachers use Geogebra in their classrooms, but I suspect it's probably not many. Yesterday I led a follow-up workshop session for a group of mostly K-5 math teachers. I showed how to use Geogebra to make a number line with tick marks noting common fractions.

A thought experiment: suppose you are a K-5 math teacher and want to show students how to compare the fractions 4/9 an 3/7 on a number line. The decimal expansions show these two values are pretty close to one another. 3/7 is approximately 0.428571 and 4/9 is approximately 0.444444. Drawing these two fractions on a number bounded between 0 and 1 might be tough. They may even appear to be the same if the scale isn't discerning enough. What would a person do to find number lines to make teaching materials? Like many people, he or she might go to Google and do something like this:



We may able to find number lines with sevenths and ninths indicated, but they will probably be separate. It may take some time to dig through these images. Should a teacher be at the mercy of materials made by somebody else? What if the teacher could design his or her own materials instead? Enter Geogebra.

Here's a PDF I made with some basic instructions on how to start building number lines: Creating Number Lines with Fractions in Geogebra

In addition to a PDF showing an introduction on how to make these things, I also posted a Geogebra sheet on the Geogebra community site for teachers to download for free. An image of the sheet I built appears below.