Gifted Students + Desmos

Throughout my teacher assisting semester, I have taken a closer look into not only different resources and pedogogical theories and methods for education in general and in mathematics, but I have also examined those for two step equations in middle grades mathematics. My target audience has been the eighth grade students I have had the blessed opportunity to be their teacher assistant at Grandville Middle School. It truly has been an honor to serve the 120 or so students as they learn mathematics and grow as mathematicians.

In my mission to be the best teacher for my current and future students, I want to provide opportunities for all students to develop and extend their knowledge of the learning targets and content standards. Gifted students are one set of students that I will need to attend to. In order to provide differentiated instruction for this student population, albeit small, I can use Desmos, an online math education resource known for its online graphing calculator and activity modes for both the teacher and the student. The resource can be used either through a browser or an app.

Specifically related to two step equations, the standards are seventh grade standards and within the context of an equations and expressions unit would not necessarily mean that graphing the equations are included, as was the case with the mini-unit I designed and implemented under the guidance of my cooperating teacher. Therefore, a gifted student who would benefit from an extension could receive one in the form of exploring the graphical nature of two step equations while other students continue to practice solving them algebraically. Connections between the graphical and algebraic representations of the equations will come in subsequent units and courses.

For example, algebraic problems in the two step equation mini-unit might look something like this.To extend the gifted student's learning, I would ask the student to forgo the algebraic solving that the other students are doing and to instead graph the left side of the equation and the right side of the equation separately. Next, the student would be prompted to communicate what they noticed in completing this process. Once the student arrives at the importance of the intersection, I would prompt the student to seek out the connection between the graphical representation and the algebraic representation (i.e. the intersection point contains the input value that is the same as the value solved for algebraically).

Take the first problem as a sample problem for this extension: -15 = -4m + 5

If you were to solve it algebraically, one way to solve it looks like this:

Instead, if you were to examine it on Desmos graphically, you would arrive at something like this.

The student would be moved to see that intersection between the two lines occurs at the value m = 5, which is the same value solved for algebraically. Once this connection is noticed, I would have the student verify this observation experimentally using the practice problems sheet as a guide.

Additionally, through this exploration, students are able to better understand the concept of equality and--hopefully--further dismiss the idea that an equal sign must indicate a solution of the form m equals a number, since the items entered in Desmos are seen as lines.

This is just one way a gifted student who could benefit from an extension of their learning would experience that extension. Students could also explore multi-step equations (likely to be the next presented concept), how the graphical connections hold there, and even delve into the special cases of no solution and infinitely many solutions both graphically and algebraically, among others. Lots to offer the gifted students in the classroom, and Desmos seems to cover those bases.