LE 4 Traditional Math: Graphing Equations in Standard Form
Accelerated seventh grade math and Math 8
In the previous lesson students learned that linear functions are those which when graphed are non-vertical straight lines. They also learned how to graph them by finding three or more ordered pairs and plotting them.
In plotting points, equations were in the form of y = mx + b. When equations were in standard form (Ax + By = C) students solved for y to find ordered pairs. In this lesson, students learn how to graph equations in standard form by plotting the x and y intercepts.
The concept of x and y intercepts is an elusive one for seventh graders. Students do not automatically connect the visual definition with its mathematical interpretation. That is, they may understand that the x intercept of an equation is where the line intersects the x-axis, but have a harder time recalling that its coordinates are (x,0).
Warm-Ups.
1. For the function f(x) = 2x – 5, find f(10). Answer: f(10)=2∙10 – 5 = 20-5 = 15.
2. What does x equal if y equals zero for the equation y = -4x+8 ? Answer: 0 = -4x + 8; 4x = 8; x = 2
3. Solve for y. -3x + y = 5 Answer: y = 3x + 5
4. Find an ordered pair of numbers for y = 7x - 8 Answers may vary. Examples: (0,-8), (2,6)
5. Solve 3 – y = 8 Answer:-y=5, divide both sides by -1; y = -5
Problem 2 sets the stage for this lesson’s discussion of x and y intercepts. Problem 4 can have any number of answers. Students may ask how to do this; prompts might be “Is y dependent on what x is?”, “In an ordered pair, which number represents the x value and which one represents the y?” or the most obvious “How did you find ordered pairs for the problems yesterday?” Problem 3 will also relate to today’s lesson. Problem 5 is to remind them how to work with a negative variable like –y. Remind them that it means –1y , so that to solve Problem 5 we divide both sides by –1.
The Wind-Up: Standard Form and x and y Intercepts. Sometimes I like to start a lesson with a provocative question, like: “Anybody remember what we did yesterday?” This will usually result in the class quieting down even if that wasn’t your goal. Someone will undoubtedly remember, and just before the class decides to be silly, I’ll ask if there were any homework problems students found difficult that they want to see. I then go over the Warm-Ups.
“Problem 4 of the Warm-Ups is similar to what you did yesterday; you had to find y values for various x values, usually -1, 0, 1 but other numbers if you had a fraction in the equation you were to graph. Problem 3 of the Warm-Ups is what you did when equations were in what is called “Standard Form”. You solved for y and then found ordered pairs.”
I write the following on the board and have them write it in their notebooks:
Standard form of linear equations: Ax + By = C
“Let’s look at an equation in standard form: 3x + 2y = 6. We can actually graph this equation in this form, without solving for y and finding the ordered pairs like we did yesterday. One of your warm-up problems asked what x equals if y equals zero. In your notebooks I want you do to the same, find x if y equals zero.”
I will go around to make sure students are doing it, and seeing a proliferation of x =2 and making sure people are on board, I’ll ask them to now find y if x equals zero. They should get y = 3.
“Now I want you to give me the ordered pairs. So for the first one, when y equals zero, what is the ordered pair?” I might have to remind them that ordered pairs are written as (x, y).
Hearing (2,0), I move on to the ordered pair when x equals zero. They should get (0,3).
I now show them what the graph of 3x + 2y = 6 looks like:
I ask them to identify the y and x axes just to make sure everyone remembers that. I then ask what the coordinate of the point is where the graph crosses the y axis. Because of blank stares that have occurred when I’ve asked this question, I have developed the habit of running a pointer along the graph and then pausing where it crosses the y axis. “Can anyone name the coordinates—the ordered pair?”
Hearing (0,3), I then do the same for the coordinate where the graph crosses the x axis and this time students tend to be quicker at naming the coordinates: (2,0).
“These points where the graph crosses the y and x axes are called ‘intercepts’.” I write the following on the board and have them write it in their notebooks:
The y-intercept is the point where the graph crosses the y axis and has the coordinate (0,y)
The x-intercept is the point where the graph crosses the x axis and has the coordinate (x,0)
I explain that the x and the y in the ordered pairs indicate that it could be any number on the respective axis.
Examples. A line crosses the y axis where y equals 3. What is the coordinate of that point? (0,3).
A line crosses the x axis at x = -3. What is the coordinate of that point? (-3,0)
The Pitch: Graphing an Equation in Standard Form Using Intercepts. Before I move on I’ll ask if anyone can see where we’re going with this and how we can graph equations in standard form by using intercepts. Some students will see it, which is excellent if they do. Some won’t see it, so now I move on.
“Let’s see how to do this. Say you are told to graph the equation 4x + 2y = 4. We have to find the x and y intercepts. Look at the definition of x-intercept that is in your notebook, and think about how you would find the x intercept.”
Someone may come up with the answer, but if not, I will refer to the Warm-Up question that addressed this, which was Problem 2 above.
“When y equals zero we can solve for x like we did before. What do we get?” They should get x = 1.
“The definition says the y coordinate is zero. Did we do that?”
We did, in fact do that when we made y zero and solved for x, which some students may say. But if they don’t, I will say it. “So what’s the coordinate of the x-intercept?” Hearing (1,0), I do the same for the y-intercept and go around checking their work in notebooks offering guidance and making sure they’re getting it. They should get y = 2 and the y intercept at (0,2).
“So now we plot the points and connect the line.” The graph should look like this:
I then write the steps on the board for them to copy in their notebooks:
STEP 1: Find x intercept (make y= 0 and solve for x);
STEP 2: Find y intercept (make x = 0 and solve for y).
STEP 3: Plot points (x,0) and (0,y)
STEP 4: Graph equation by connecting the points.
Examples: We have already worked on one together, so these examples they work on by themselves while I go around to provide guidance as needed. Generally students find these problems fun. Problem 4 is in different form, so I work on this with the students as described.
1. 3x – 4y = 12 Answer: x-intercept: 3x = 12; y=4 : (4,0) y-intercept: y=-4: (0,-3)
2. 2x - y = 8 Answer: x-intercept: x=4: (4,0); y-intercept: -y=8, y=-8; (0,-8)
3. 14x+18y=126 Answer: x-intercept: 14x=126,x = 9; (9,0); y-intercept: y=7 (0,7)
4. y=4x-8 Answer: This is not in standard form. But we can still find the x and y intercepts. I work through this one at the board for the y intercept. If x = 0, then y = -1, so the coordinates are (0,-1) for the y-intercept. Students should then find the x intercept:4x-8 = 0, 4x=8, x = 2;x intercept: (2,0)
Homework. Homework is more of the same. When numbers are large, students again will have to decide how to label the axes in order to have a reasonable size. The more difficult ones I go over with them the next day.