# 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 Brittany Tripp. Her topic, from Algebra: using the point-slope equation of a line. How can this topic be used in your students’ future courses in mathematics and science?

The point-slope equation of a line can be used in a variety of different ways in mathematics classes that some students may encounter later on. It is used in Calculus when dealing with polynomials. For instance, “key concepts of calculus: limits, continuity, derivatives, and integrals are all relatively trivial for polynomial functions.” It is also seen when dealing with Linear Approximations. “A differentiable function is one for which there is a tangent line at each point on the graph. In an intuitive sense, the tangent to a curve at a point is the line that looks most like the curve at the point of tangency. Assuming that f is differentiable at a, the tangent line to the graph y = f(x) at the point (a,f(a)) is given by the equation.

y – f(a) = f ‘(a)(x – a)

This equation arises from the point-slope formula for the line passing through (a,f(a)) with slope f ‘(a).” In Pre-Calculus with discussing horizontal and vertical shifts you can easily relate back to point-slope equation of a line. You can relate point-slope equation of a line to the definition of derivative where the equation is rewritten with limits to describe the slope as the derivative. This is just a few ways that point-slope pops up in later mathematics courses. It is important to be able to form the point-slope equations of a line, as well as slope-intercept form, and being about to understand it well enough to build off of it when leading into harder concepts. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Point-slope equation of a line is used in movies in a huge way that most people probably never even realize. Point-slope equation of a line is used in pinhole cameras. A pinhole camera “is a simple optical imaging device in the shape of a closed box or chamber. In one of its sides is a small hole which, via the rectilinear propagation of light, creates an image of the outside space on the opposite side of the box.” In other words, let’s say we had an object, there is light constantly bouncing off the object. In the case of a pinhole camera, there is a small hole in the nearest wall/barrier which only allows light to pass through the hole. The light that makes it through the hole then hits the far wall, or image plane, creating a projection of the original image. The way point-slope equation of a line is used is first by adding a coordinate plane that has the origin centered at the pinhole. We can imagine that our scene is off to the right of the origin and the image plane is off to the left of the origin. We can choose some point in our scene to be a coordinate point in our coordinate plane. Some of the light bouncing off of that point in our scene will pass through the pinhole and land somewhere on our image plane. One of the ways we can find where it lands in our image plane is by using slope-intercept equation of a line. There is a really cool video on the khan academy website that talks all about the mathematics behind pinhole cameras. There is actually an entire curriculum called Pixar in a Box that goes through a variety of different topics and subject matter that is involved in the making of Pixar movies. How can technology be used to effectively engage students with this topic?

There are a ton of games online that involve point-slope equation of a line. One website that I found that has a variety of games on it is called Websites for Math. I went and tried out some of these games myself and found them to be fun and entertaining, but somewhat challenging at the same time. The website has links to different games that pertain to slope and equation of a line. You can choose games specifically by what form of an equation of a line you want to practice, among other things. The first game I tried was Algebra Vs. Cockroaches. It pops up with a coordinate plane with a cockroach on it and you have to type in the equation of the line in order to kill the cockroach, but if you take too long the cockroaches start to multiple. I liked this game because it started with just having you identify the y-intercept before leading into harder equations. However, this game focused more on slope-intercept equation of a line than point-slope equation. There were games specifically designed for point-slope equation of a line. One of those games being point-slope jeopardy. If you choose a questions for 300 points you are given a coordinate point and a slope and asked to write the point-slope equation that fits for the given data. If you choose a question for 600 points you are given two coordinate points and asked to write the point-slope equation of the line that fits the given data. Therefore, you must first use the coordinate points to calculate the slope and then plug that into your equation. What I also like about this game is that you can either play by yourself or with a friend. The things I enjoy most about this website is that it has games that don’t only pertain to slope-intercept equation of a line. There are games that focus on slope specifically, graphing equations, slope-intercept form, etc. That way if you are having issues with any of the topics that may have been discussed previously, to point-slope equations of a line, you can find a game that might help refresh your memory.

References:

http://matheducators.stackexchange.com/questions/9907/should-i-be-teaching-point-slope-formula-to-high-school-algebra-students

http://calculuswithjulia.github.io/precalc/polynomial.html

http://www.pinhole.cz/en/pinholecameras/whatis.html

https://www2.gcs.k12.in.us/jpeters/slope.htm

http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

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