Using Desmos to find the quadratic model of a real-world image

Desmos is at it again...

Since I started using Desmos more regularly this summer, I have found that it is gradually replacing a number of my other math software options.  The latest instance happened when my algebra 2 students were finding examples of things that take the form of a parabola.

In the past, I have had students find an example of a parabola in their everyday lives and snap a picture of it.  I've seen images of everything from the McDonald's logo to a water fountain to a mug shot of someone's jaw line.  Then I would have students upload their images into Geometer's Sketchpad and construct a number of points along the parabola.  Here is an example...

After that, I would have the students enter the ordered pairs into a TI-84+ or TI-Nspire graphing calculator, find the regression equation, and go back to Sketchpad and graph the function on top of the image.  Again, an example:

Some years I would have students explore around with vertex form a bit to try to fit a quadratic model before we found the regression equation.  But it was a painfully long process to bounce back and forth between Sketchpad and graphing calculator.

This year, I decided to direct students to Desmos to compute the regression equation.  After all, we had used Desmos to find linear regression equations earlier in the year and all students had access to Desmos w/ our 1:1 laptops.  (Not all students would have a TI-84 or Nspire to use in years past.)

It turns out the students had a great idea. Why use Sketchpad to plot the points and graph the regression function?  Why not just use Desmos?  I loved the idea and it really simplified the process.

Students simply inserted their images into Desmos, plotted points on the parabola, created a table with their ordered pairs, and found the quadratic regression equation (& graph) all in one place.

Follow this link for an example (image shown below).

Better yet, using the sliders in Desmos is a breeze.  Simply insert an image of anything parabolic and have students fit the model using vertex form.

Desmos Activity Builder

The team at Desmos has really hit the ball out of the park when it comes to user-friendly math technology.  If you haven't checked out the Desmos Activity Builder at teacher.desmos.com, you're missing out.

I've recently created some inquiry-based activities for my algebra 2 students.  The best part is that there are math educators much smarter than me working on creating activities as well.  Big names such as Michael Fenton and Andrew Stadel are sharing their ideas, and I'm able to use them!

My most recent activities for algebra 2:

Transformations of Functions

Absolute Value Graphs & Transformations

Quadratic Functions in Vertex Form

My most recent activities for geometry:

Equations of Lines - Part 1

Equations of Lines - Part 2

Again, it's great when giants like Andrew Stadel roll out activities (such as this one on rotations) right as we're working on transformations in geometry.

Keep up the great work, Desmos!

Un-structuring Word Problems

This year, I have the opportunity to teach two sections of advanced algebra 2.  My colleague, Todd, teaches the other five sections of advanced algebra 2.  Todd is an excellent educator and someone I have already learned a lot from.

We are both fans of Dan Meyer's TED talk and love the idea of un-structuring our math problems as much as possible.  In an attempt to force our students to think about what information is necessary to solve a problem, we've begun to modify a question or two each assignment in this way.


Here's a question found in a typical systems of equations lesson:

(c) Pearson Education 2015 Algebra 2

We give our students this problem instead:

*1.       You are considering renting a car from two different rental companies.  Write a function that shows the cost of renting from each company.  Which one should you choose?  Explain.

Our students are aware they do not have enough information to complete the task.  Not yet, anyway.
If we assigned this problem on a Monday, students would be expected to think about what information is important and necessary in order to correctly answer the problem.  On Tuesday, to start class, we give students 1 minute to ask anything they want about the problem.   The problem is then part of their homework for Tuesday night and is expected to finished by Wednesday.

It has been interesting to hear some of the questions students ask on problems like this one.  Often times, students ask for the necessary information very quickly and don't ask for anything unnecessary.  There have been times, however, that we get some interesting questions about things that have no effect on the problem.