If asked to think back to when you first learned to graph functions, most of you probably recall hours spent with a pencil, ruler, and graph paper. Your teacher might have explained the equation of a line in slope-intercept form and sent you home with a worksheet of linear equations, expecting that my graphing them, you would come to understand the different components of the equation. But what if there was a simpler, more precise, and more effective way in which to represent graphing functions?
Desmos is a powerful graphing tool freely available online or through an app. With Desmos, users can easily graph and manipulate functions through an intuitive interface. Beyond basic graphing capabilities, Desmos includes a wide array of tools that make analyzing and manipulating graphs as easy as the click of a mouse or swipe of a finger. For example, given functions can be assigned a specific color, allowing the user to easily differentiate among multiple graphed equations, and intersection points between graphs appear with a simple click. Also, Desmos has a slider tool, by which variables in an equation can be easily changed, resulting in obvious and immediate changes to the corresponding graph.
For comparison, the TI-84 graphing calculator can also perform some of these basic tasks (i.e. graphing multiple functions or locating intersection points), but it does so in a 6″x4″ graphing window. Furthermore, unless you have one of the newest TI-84 models, the screen is only black and white, making it difficult to distinguish between pieces of different graphs. Another way in which Desmos ranks above the TI-84 is that it shows the equations of the graphs in a split screen beside the graphs themselves (and even if some TI-84 calculators have this capability, the small screen becomes even smaller). With all of the advantages that Desmos brings to the table, the charge to math teachers is finding ways to incorporate this technology in the classroom.
Teaching a lesson on graphing equations in slope-intercept form is one example of an activity that can incorporate graphing technology. Using Desmos, students can input the generic linear equation y=mx+b, enabling the sliding tool for the values m and b. As they manipulate the values of m and b using the slider, students can make connections between the values of these variables and their effect on the corresponding graph. For example, a student who increases the value of m will see that the graph of the line gets steeper, which ultimately helps give understanding of the concept of slope. In Focus in High School Mathematics: Technology to Support Reasoning and Sense Making, Dick and Hollebrands (2011) offer another thought-stimulating graphing activity for students. This activity involves graphing the successive equivalent equations that result from algebraically solving a problem for x. Consider the example 2x-3=-5x+1:
Using Desmos, a student can graph the equivalent equations in various steps of solving the problem (equivalent equations are given the same color).
From this activity, a student can see that at each step of the problem, the intersection point of equivalent equations maintained the same x-value (0.571 or 4/7). The solution (purple line) extends through each of these intersections points, because the value of x for which the equations are equivalent is 4/7. It can be noted that this same activity could be completed with a TI-84 calculator, but the tangle of monochromatic lines would make it difficult to draw conclusions.
As highlighted by Dick and Hollebrands (2011), this example activity builds students’ ability to reason, strategically problem solve, and make connections among concepts, validating the activity as a meaningful approach to incorporate technology in the classroom.
These examples have only scratched the surface of all that Desmos has to offer. Try it for yourself and see how your students will benefit!