Graphing Linear Equations
In this unit, students learn about graphing of linear equations in two variables. Much of this material was presented initially in seventh grade; i.e., representing linear functions by graphing, constant rate of change, slopes, and slope-intercept form of equations. I usually spend time with a brief review of basics, but with particular emphasis on the equation of a straight line (y = mx + b) and the formula for finding the slope of a line:
I point out that the order of the y and x coordinates can be reversed as long as the pattern is maintained in both numerator and denominator. With respect to the y=mx+b formula, students learn to graph by finding the y-intercept and plotting additional points based on the slope of the line. They also learn to find the equation of a line by identifying the y-intercept and determining the slope by inspection.
This unit will therefore focus on two new topics; specifically the point-slope form of a linear equation and graphing inequalities.
Point Slope Form of Linear Equations
Students have learned the y= mx+b form of linear equations, but now learn the more general point-slope form. The point-slope form allows for identification of the y=mx + b form of an equation given the slope of an equation and a point the line passes through, or given two points through which the line passes.
The formula for the point-slope form tends to be confusing at first because of the notation. It is necessary, therefore, that students get sufficient practice with the form. An alternative method is available but it also proves to be confusing. Given that confusion with either approach is inescapable, in my opinion the point-slope formula is a better value for the time spent learning it. It is more formal and provides a good introduction to notation that students will be using in subsequent math courses.
For information purposes, I present the alternative method for those who may prefer it. Given a slope and a point through which the line passes, students are to find the equation of the line. For example, the slope is given as 2 and the line passes through the point (3, 4).
Since (3, 4) represents an equation in which x = 3 and y = 4, these values and the slope (m = 2) can be plugged in to the y = mx + b equation to obtain 4 = 2(3) + b, or 4 = 6 + b. Solving for b, we obtain b = – 2, so the final equation is y = 2x – 2. The confusion comes from seeing where to place the 3 and 4 in this example; students will often get them mixed up and will write 3 = 2(4) + b. Others will repeatedly forget that the (3, 4) represents x and y respectively.
Problem 1 can be solved by seeing that the term in the numerator gets pushed down to the denominator, and the exponent changed to positive 4. Similarly the term in the denominator gets pushed up to the numerator, changing the sign of the exponent. Problem 2 is a time = time type equation which was covered in the previous lesson. Problem 4 requires application of the slope formula which students have learned in a lesson previous to this one. Problem 5 offers a review of solving a quadratic equation by factoring. It is important to keep such method front and center, so that when the unit on quadratic equations is reached, it serves as a springboard for learning how to solve equations that are not factorable.
Initial Activity. In preparation for this lesson, I make up a set of index cards on each of which is an ordered pair of numbers. Each pair is derived from the same equation. So if the equation was y = 4x -2, ordered pairs would include (0,-2), (1, 2), (25, 98), (-2, -10) and so on. I pass out the cards so each student has one.
“You all have a card with an ordered pair on it. I want you to find a partner, and between the two of you find the slope of your two ordered pairs. If you don’t have a partner, I’ll work with you.”
Students begin calculating the slope using the recently learned slope formula. It doesn’t take long before I hear students call out “I got 4”, “So did I”, and so forth. Pretty soon everyone notices that they obtained the same slope.
“Wow, how do you think that happened? Shall we try it again? Give your card to someone else, and let’s try it once more.”
The same thing happens, of course. One time, a girl totally mystified asked me “How did you do that?” as if it were a magic trick.
“What do you think is happening here?” At least two students, but usually more, have ventured that they are all ordered pairs from the same line.
“Yes, that is indeed how it is done. A straight line has the same slope, as we learned in seventh grade. So no matter what two points you pick on that line, you will always get the same slope. This is the basis for what is called the “point-slope form of a linear equation.”
Point-Slope Form Using Numbers. “You figured out that you all got the same slope because the ordered pairs on the cards were taken from the same equation. And in fact, that equation was y = 4x – 2. The fact that the slope is the same for any straight line no matter what two points allows us to solve some problems that you would think are impossible.”
I write on the board:
“I’ve written the slope of a line, and a point that the line passes through. Now I used the same equation as before, y = 4x – 2. But let’s pretend we don’t know this. What’s the slope?
I’m told it is 4.
“And since it’s a straight line, no matter what two points I pick, the slope formula will give me 4 as an answer. We already have a point, (2, 6). Now I’m going to represent any other point on the line by the ordered pair (x, y). Since I have two points, I can use the slope formula to find the slope of the line, like this.”
“How do I eliminate the x-2 from the denominator?” I might have to write the 4 on the right hand side as 4/1 to get them to see the process via cross-multiplication.
“Now solve this equation for y.” They do this and obtain:
“Let’s try another one. This line passes through (3, 1) and has a slope of 2.” I have them follow in their notebooks.
“Like we did before, we’ll pick any other point on this line and call it (x, y). Somebody tell me what we do next—it’s what we did before.”
Someone will tell me and after initial set up, I have them solve the equation:
I may have them do one more just to get into the rhythm. Then we’re ready for the actual derivation of the formula.
Derivation of Point-Slope Formula. “Now we’re ready to make this into a formula that you can use. To review: No matter what two points we pick on a straight line, what do we know about the slope?”
Answer should be “it’s always the same” and that’s usually what I hear.
“Instead of using numbers this time, let’s say our slope is m. And we are given a point: (x₁, y₁). And let’s solve for y in the same way we just did when we were using numbers. We get the following.”
“We cross-multiply like we did before and we get this.”
“This is the point-slope formula. Write it down, learn it by heart, or if you don’t like to memorize, have it tattooed on your arm so you’ll always have it handy. That way you won’t have to Google it. Now let’s put it to use.”
Writing an Equation With Two Points and No Slope. “Now let’s say you are asked to find the equation of the line that passes through (-2, 1) and (3,4). Can you do it?”
Someone might ask “What’s the slope?”
“Ah ha!” I say. “No slope is given. How can we find the slope?”
There are usually more than a few students who know: Use the slope formula.
Homework. Problems are a combination of one point and slope given, or two points given.
Subsequent lesson is not included in this book, but introduces concept of parallel lines having same slopes, and perpendicular lines having slopes such that the slope of one line is the negative reciprocal of the slope of the other. Typical problems would be:
1, Line A is parallel to the line y = 3x – 2 and passes through the point (5, 4). Find the equation of the line.
2. Line A is perpendicular to the line y = -2x + 6 and passes through (0, 3). Find the equation of the line.