# Engaging students: Finding x- and y-intercepts

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Lissette Molina. Her topic, from Algebra: finding $x-$ and $y-$intercepts.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Using this topic, students can now use word problems that involve two variables in our everyday lives. One problem that many scientists often use is population growth. In population growth, we can usually see a trend of a line and determine the slope. We initially begin with a certain population in a certain year, this is considered the y-intercept, since we start at the initial year that we consider to be at x=0. Using the slope of the line when we are speaking in terms of population decay, we may then set our y=0 to find when a population would be equal to zero. We can also consider other examples such as the depreciation of a car, or when a business’s grows out of debt and begins to profit. Word problems include, but are not limited to, problems that involve a trend and wanting to find where that trend will lead to at a certain point, x, when we are given an initially amount or reverse this operation.

How can this topic be used in your students’ future courses in mathematics or science?

This topic crosses multiple courses in mathematics. In general, knowing the x and y-intercepts of equations help students start outlining what the graph of the function might look like. This gives part of the visual representation needed to complete part of the graph. These intercepts usually also give a prediction of what the shape of the graph may look like. A fun assignment would be giving a student two points on the graph and along with the intercepts of that equation that the points belong to. Along with this, these intercepts give us the solutions of the equations. When there are not x or y-intercepts, we would now know that the solutions do not exist or at least are imaginary. Overall, x and y-intercepts help us get a better understanding of what the graphs of almost all equations must look like. This is essentially especially when we are graphing by hand.

How can technology be used to effectively engage students with this topic?

Graphing calculators is one fun essential way of finding intercepts as well as learning functions on a calculator. When a student graphs a function on a graphing calculator, for example, the sine function, we can ask the student where they believe the graph would intercept with the x-axis. We would then ask them to find the intercepts using the calculator by pressing [2nd][trace][4] function and proceed to find the approximated x-intercepts. The student would then find that the intercepts occur at every npi/2. Essentially, using this function is an interesting way of estimating the intercepts along the graph in an interactive way. Other online graphing calculators may do this as well and give students a better understanding of where the intercepts occur.