# Engaging students: Using the point-slope equation of a line

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 again comes from my former student Tiffany Jones. Her topic, from Algebra: using the point-slope equation of a line.

B1. How can this topic be used in your students’ future courses in mathematics or science?

The topic of using the point-slope equation of a line comes up in some of the early topics of Calculus 1 such as, how to find the equation of the tangent line of a curve at a given point. The slope, ­, of the tangent line of a curve at a given point, , is equal to the instantaneous rate of change or slope of the curve at that given point. The slope is calculated by evaluating the following limit:

$\displaystyle m = \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

If the difference quotient has a limit as h approaches zero, then that limit is called the derivative of the function at . Then, values of  and  are substituted into the point-slope equation of a line to determine the equation of the tangent line of a curve at a given point.

$y-y_0 = m(x-x_0)$

C1. How has this topic appeared in pop culture?

On December 31st 1965, Chuck Jones’ released an animated short titled “The Dot and The Line: A Romance in Low Mathematics”. This ten minute, Oscar-winning film explores the complex relationship between lines, dots, and disorganization. The Line as desperately in love with the Dot. Yet, the Dot is currently involved with a chaotic Squiggle. The Dot ignores the Line, disregarding him as boring and predictable. He lacks complexity. Through a montage following this rejection, the line teaches himself to create angles, form curves, and produce close-ended shapes as well. With this new confidence, he then reveals his newfound self to the Dot. The Dot sees that there is no method to the Squiggles madness.

While the topic of using the point slope equation of a line is not an explicit topic of the short, I feel that this video as an engage activity can be great conversation starter about the relationship between a point and a line. From there the lesson can go on to talk about the point-slope equation. Furthermore, this video can open discussions about the slope-intercept and the point-point forms of a line.

E1. How can technology be used to effectively engage students with this topic?

Explore Learning offers a Gizmo and worksheet on the point-slope form of a line. The Gizmo is an interactive simulator that allows the student to physically move the point around the Cartesian plane or use the sliders to adjust the point values and the slope value. The Gizmo shows the resulting line. I think that the use of such a tool can reinforce the relationship of a particular slope and a particular point to give an equation of a line.

The Gizmo offers to the slope-intercept form of the equation. So this simulator can also be used for a lesson on the slope-intercept form. Also, the Gizmo can place a right triangle along the line with leg lengths to show how the rise and run values change with the overall slope value.

Additionally, I think that this simulator can be used to allow the students to explore the equation. For instance, the students can see why when the graph is shifted to the left 2 units, the resulting equation has (x+2).

References:

http://www.imdb.com/title/tt0059122/?ref_=ttawd_awd_tt