The mathematics of linear growth/decay, exponential growth/decay, inverse variation, and joint variation relationships are some of the most important core concepts that high school students can learn. With these concepts mastered, students will have the necessary foundation to better comprehend more advanced work in mathematics and science. Newton’s famous inverse square law of physics, F = km1m2/d2, is an example of a joint and inverse variation relationship. I once told a class of computer programming students that knowing how to find the equation that expresses the relationship between two loop control variables is one of the more important skills they can learn. When writing computer programs, I create functions that map math coordinates to computer screen coordinates and vice versa. All of these functions are just applications of linear growth/decay.
This post discusses the mathematics of linear growth/decay from an analytic geometry point of view and how I teach these concepts to my students. The statements that I make about the slope of a line, tangent of an angle, slopes of parallel lines, slopes of perpendicular lines, and equations of lines are familiar to many. If there shall be any novelty in this post, it will be in the manner in which the content is presented. Many text books treat some of these relationships as a simple mathematical fact of life. At the teachable moment, it’s nice to be able to give a curious student a straight forward explanation of why a not so simple fact is true.
All graphs in this post were created with my program Basic Trig Functions. Equations can be entered in any format. It’s not necessary to express equations as explicit functions of x. Go to mathteachersresource.com to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.
I will start by discussing the concept of slope of a line. I tell my students that the slope of a line describes the steepness of a line. It would be easy to fall off a house roof that has a slope > 1. The slopes or grades of train tracks are gentle, and are usually well under 10% (1/10). A train on a track with a 1% (1/100) slope can only pull half or less of the load that it can pull on a zero slope track. After driving down a road with a 6% (3/50) grade, you fully appreciate the message on the warning sign. A vertical line is infinitely steep, and therefore we say that its slope is undefined or it has no slope. No slope lines are NOT the same as zero slope lines. As I discussed in a previous post, the brilliant math educator, Zalman Usiskin, had a profound effect on my understanding of slope and how I teach these concepts to my students. Diagram (1) below illustrates the concept of slope and the text box gives the definition of a slope of a line. Most students immediately understand that positive slope lines describe linear growth and negative slope lines describe linear decay.
Most students can guess that parallel lines have equal slopes. Based on theorems from Euclidean geometry, diagram (2) below illustrates why parallel lines have equal slopes. Line AF is a transversal for parallel lines AB and DE, and angles BAC and EDF are corresponding angles. Of course, each of two parallel horizontal lines have slopes equal to zero, and each of two parallel vertical lines have undefined slopes.
Almost all students can’t guess that the product of the slopes of two oblique perpendicular lines equals -1 (Oblique lines are not horizontal or vertical). Many text books treat this relationship as a simple mathematical fact of life. The reason that this relationship is true can be explained in terms of theorems in Euclidean geometry, or in terms of a trigonometric identity. A theorem in Euclidean geometry states the following: “The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.” Diagram (3) below shows the setup for showing why the product of slopes rule works for perpendicular lines. The text box below gives the formal explanation why this relationship is true. It is sufficient to demonstrate that this relationship is true for perpendicular lines that intersect at (0, 0) because sliding the intersection point to any point in the x-y coordinate plane preserves this relationship. Although the second demonstration is like using a sledge hammer to drive a tack, it’s an interesting way to demonstrate the relationship.
Diagram (4) below shows the setup for deriving the slope-intercept form and the point-slope form of the equation of a line, and the text box below gives the derivation of the equations.
The equation of a line comes in six flavors which are described in the text box below. When you know slope m and y-intercept (0, b) of a line, the slope-intercept equation is the flavor of choice. If you know the slope m and one point (x1, y1) on the line, find the point-slope equation of the line, and then you can convert it to the slope-intercept form of the equation. The standard equation Ax + By = C where the constants A, B, and C are relatively prime integers, no common factors, is useful when setting up to solve a system of linear equations. Some definitions of the standard form of an equation require the constant A to be a nonnegative integer.
Now for a short quiz to give to your students as a fun exercise with slope and equations of lines. Diagram (5) below shows the graphs of six linear equations. Have your students find the equation of each graph. If the line is oblique (not horizontal or vertical), write the equation in slope-intercept form. The answers are given at the end of this post. If you decide to take the quiz now, promise me that you will not take a peek.
Comments and Suggestions:
- Similar to the graphs above, give students the graphs of linear equations and require them to find the equations of the graphs in different formats. This exercise is a good way to improve algebraic manipulation skills. Let them test their equations by entering the equations into a computer graphing program that allows them to enter equations in any format. If their equation is incorrect, they immediately see why and fix it. They now have a reason to do some algebra, and the computer becomes an experimental tool.
- Learning to find the equation of a line in different formats is not so easy for many students. Remind them that most students initially make many errors and mistakes, and they are no different. There is no shame in that. They should think of the exercises as good mental gymnastics. It just takes time to sort things out.
- I use the handout Introduction to Slope and Equation of a Line (student version) with my beginning algebra students and as a review for more advanced students. You can also download the teacher version of the handout. To download other free math handouts and exercises in a variety of topics for your classroom, go to mathteachersresource.com/instructional-content.html.
My next post will focus on setting up a table of data values, recognizing a linear relationship pattern in the table, finding an equation that expresses the relationship between two variables, and using the equation to make predictions. A major goal, of course, should be to get students to the point where they really understand that the slope of a line is a unit rate and be able to give a simple interpretation of what the slope of a line means in a given problem situation.
Graph A: y = 6
Graph B: y = -3/4x + 3
Graph C: x = -7
Graph D: y = 3x + 18
Graph E: y = -2x + 16
Graph F: y = 4/3x – 8