## Graphing Lines Using the Most Efficient Technique

Today I am updating a previous blog on graphing lines.

Last week I spent Wednesday covering graphing lines by finding x- and y-intercepts. I then covered graphing lines using the slope and y-intercept on this Monday. Students love to graph lines using slope because it is so accessible conceptually. If you understand what the slope of a line tells you (up or down so many units as you move this many units to the right), all that’s required to graph the line is the ability to plot a y-intercept and the ability to count (to the next point).

When students are presented two techniques, they will always drift to the technique they feel is easier and often ignore the other technique. However, there are times (equations in standard form: Ax + By = C) when finding the intercepts will be a more efficient way to graph the line. At the end of Monday’s class I put up several equations in standard form and slope-intercept form, as well as equations of vertical and horizontal lines, and asked the students to graph each line with their groups.

I was not surprised that some students tried to rewrite each equation in slope-intercept form. Fortunately, for my purposes, some students made algebra mistakes with some of the equations and I was able to point out that this would not have happened if they had chosen to find the intercepts instead. I showed them that it takes 5-10 seconds to determine the intercepts, and approximately the same amount of time to plot those two points and graph the line. We spent the rest of the class discussing how to efficiently graph the lines, depending on the equation we were given.

If an equation is presented in standard form, the algebra required to find the two intercepts can be done mentally. When finding the x-intercept by setting y = 0, the term containing y will disappear. At that point finding the x-intercept turns into a division problem. The same holds true when finding the y-intercept. Students are less likely to make an error in this fashion that solving the equation for y in order to graph using the slope. This is especially true when the coefficient of the y term is negative.

Don’t get me wrong. Students need to be able to find the slope of a line by solving the equation for y, especially later in the course when trying to determine whether two lines are parallel, perpendicular, or neither. What I am saying is that students should possess the critical thinking skills to determine which technique is the most efficient one for graphing a particular line. This comes up again in solving systems of two equations in two unknowns. There are times when substitution is more efficient than elimination, and vice versa. Students should not turn their back on substitution because elimination is “easier” for them. Although the quadratic formula can be used to solve any quadratic equation, students should realize when another technique (factoring, extracting square roots, completing the square) is more efficient.

After Monday’s class, I think my students have a complete strategy for graphing lines, no matter what form they are presented in. I think that they are now able to look at a situation and determine an optimal strategy for graphing a line, and solving a problem in general. I think that their graphing tool box contains all the necessary tools, and that they know when to use each of them. That was a good day.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post a general teaching article on my blog. If you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.