Desmos (https://www.desmos.com/) is a free online graphing calculator. Students can use the free app (Android and iOS) on their phone or on any device via a browser to create an interactive graph to check their computational work and deepen their understanding through visual insight. Simply enter an equation of a function and Desmos will graph it. Change any value and immediately see the impact of the change. Enter a second function and see where the functions intersect (if they do) – a visual alternative to solving simultaneous equations. Add a slider for an undefined parameter in your equation – this creates an animation, allowing the parameter’s value to be adjusted manually, or played automatically, for a range of values. You can also add inequalities, with Desmos shading the appropriate area. And graph a derivate or integral, and much more.
For a quick overview of the app, watch this 3 min YouTube video:
Once the app is installed, students do not need wi-fi to use it which adds to its flexibility. The interface is designed to work well on phone screens as well as tablets and laptops or desktops, and graphs can be saved and shared online.
One popular alternative to Desmos is GeoGebra (https://www.geogebra.org/), which is available as an app for phones/tablets or via a browser.
Who are we?
Damien Raftery (Dept. of Business/Teaching and Learning Centre) and Emer McGann (Dept. of Built Environment), Institute of Technology Carlow.
Damien is using Desmos with first-year undergraduate business students, Emer with second-year undergraduate engineering students.
How Desmos can be used to enhance learning
I started using Desmos last academic year with business students to quickly show in class the graphs of simple functions. This included explaining slopes and tangent lines, and finding maximum and minimum values as a complement to a more formal approach using differentiation. Although students were encouraged to download the app and use Desmos to graph examples in class and for homework questions, the initial benefit came from the ease with which students could visualise the various functions when demonstrated on the digital projector (for example, using this graph of a quadratic function).
Desmos can also be used to develop conceptual understanding in statistics, for example, when introducing correlation, I give this graph (adapted from one shared on Desmos) as a homework activity. Students follow the link from the VLE and then interact with a scatter diagram to move points around to explore the impact on a correlation coefficient.
This academic year, for mathematical modelling and differentiation, I have integrated the creation of graphs using Desmos into all examples and homework questions. Students are explicitly instructed to build graphs to visualise functions and to check the results of their computations. Screencast solutions are complemented by short video explanations using Desmos graphs. The intention is that the creation and analysis of the graph is an important, rather than optional, step when solving a problem. I intend to continue this integration with other topics, in particular linear correlation and regression, as my colleague Emer is also doing.
Figure 1: Turn any Desmos graph into an animated GIF http://www.gifsmos.com/
I started last year by showing engineering students Desmos graphs in class using the digital projector. We were working on calculus at the time. The purpose of using Desmos was so students could get a tangible insight into the functions we were differentiating or integrating. Students could then visualise the points of interest for the particular function under scrutiny, for example, they could visually locate the maxima and minima. They could then satisfy themselves that it was possible to use calculus techniques to locate these points. The graph below illustrates an example of the graphical information presented to students.
Similarly, for functions that we integrated, it was possible to show students graphically the functions we were integrating and the corresponding integral function by displaying these on a Desmos graph on the digital projector. Students could identify the area under the graph and I could visually demonstrate the limits of the function to further explain this concept to them. This meant that students could have an expectation of the result a definite integral question would give.
This academic year I have introduced Desmos within the topic of Linear Regression Analysis with the students. For the first time students were required to download the Desmos mobile app to their phones and used it to draw up quick scatter diagrams and to check their answers for the linear regression line parameters and the r2 value. I plan to encourage students to use the app when we move onto the calculus topic later in the year. Last year I was the one who worked off the website application in class, but this year I want students to become more familiar with the application and work in parallel to me using the mobile app.
Student feedback was generally positive as students indicated that they found it helpful to have a visual perspective of the functions they were differentiating and integrating.
Figure 2: Illustrating a Polynomial Function with the Corresponding Derivative Function
The developers of Desmos have created a range of interactive Classroom Activities lessons and you can create your own lessons with their Activity Builder (or adapt ones shared by others). As a teacher you can monitor the progress of your learners through the lessons. We have yet to use this with our students yet, but is looks like a powerful way to engage learners with meaningful tasks. It facilities activities requiring students to make predictions and write about their mathematics. A useful feature is that the interface enables students to draw on the screen.
You can preview and use activities that have been shared by others, for example, with Marbleslides, you have to edit the function(s) so that the marbles slide down passing through the stars. Try the Preview of Marbleslides: Lines.
Figure 3: Screenshot from one of the Marbleslides activity
Summary and useful links
We find that Desmos is very useful for students to use to produce interactive visual representations of mathematical functions, helping them to better understand some complicated ideas as well as checking the results of their calculations. It can also help us to explain concepts using engaging, interactive visuals.
Damien discovered Desmos via Dan Meyer, who gave a link to a post on billsmathblog. In his post, Bill gives a calculus problem, asking students to build the model themselves in Desmos – a link to his answer is in the blog post, but if you have time to solve the problem, you can learn how to use Desmos whilst creating the model. Desmos has a User Guide, whilst Math Vault provides a useful overview Desmos: A Definitive Guide in Graphing and Computing. For some really interesting statistics examples, visit Bob Lochel’s blog including his Desmos Lessons for AP Statistics. Also check out @Desmos and their blog including The Desmos Guide to Building Great (Digital) Math Activities.
What do you think? Post your thoughts in the comments below or tweet #12appsDIT @damienraftery.
Get started: Tweet a picture of your first graph
Use Desmos to graph the functions y=x2+2x+3 and y=2x+4. Find the points that satisfy both functions. [After a productive struggle, you can watch this video cheat]
Extra challenge: Add a slider to the value of the intercept of the straight line (tip: see 1-min video on adding sliders). Use the slider to find when there is a unique point that satisfies both functions.
Tweet #12appsDIT @damienraftery with a screenshot of your graph (optional: add URL for the graph)