LE 7 Traditional Math: Slope-Intercept Form of Linear Equations
Accelerated seventh grade and Math 8
In the interest of full disclosure, I confess that I’ve always been surprised at how students seem to have difficulty remembering what’s what in the slope-intercept form of linear equations: y = mx + b. I say this as someone who advocates that students need practice and repetition to get concepts and procedures into long-term memory. I have shown the derivation of the formula which students had difficulty following, and which had no observable effect on their ability to work with the formula.
What does work is practicing identifying what is the y-intercept (b) and slope (m) of the line described by the equation—as well as graphing the equation using this information.
Typically, textbooks have students transform equations in standard form (Ax + By = C) into the slope-intercept form when graphing the equation. This has its usefulness, but is somewhat limited at the seventh grade level since students have difficulty solving for y for an equation like 3x + 2y = 7. Solving it requires dividing all terms by 2, which seventh graders find confusing. Textbooks therefore limit such exercises to equations where y has no coefficient. I offer the more involved equation as a challenge for those students who can do it, and in case anyone questions whether I differentiate instruction in my classes.
Warm-Ups.
1. Express 1/40 as a percentage. Answer: 1÷40 = 0.025 = 2.5%
2. Find the slope of the graph: Answer: (-3-(-2))/(2-4) = -1/-2 = ½. Slope can also be obtained by inspection; vertical distance or rise = 1, and run = 2.
3. What is the y-intercept in the graph above? Answer: y = 4
4. A purse was discounted 15% and sold for $51. Find the original price. Answer: 0.85/1 = 51/x; 0.85x = 51; x = $60
5. Find the x and y intercepts for the equation 3x – 2y = 9. Answer: x-intercept: 3x=9, x =3; y-intercept -2y = 9, y = -9/2.
Problem 2 requires students to find the slope of the line; two of the points are indicated but unlabeled. Students will need to identify the coordinates of the points in order to apply the scope formula, or they can count the vertical and horizontal units by inspection. Problem 3 calls for identification of the y-intercept, which will be a part of the day’s lesson to follow.
The Wind-Up: The Slope and y-intercept
I go over any problems from the previous day’s homework assignment, which was a worksheet covering the various ways to graph an equation, as well as the concepts of negative and positive slopes, and horizontal and vertical lines. “As you’ve seen, there are a number of ways you’ve learned to graph an equation. Today we’ll learn yet another one.”
I wait for groans and ignoring any, proceed. “It is probably the one you will use most often.” I wait for groans again. “It’s the one you will use a lot because it’s the easiest. But let’s start out graphing an equation from a table of values.”
I write the following table of values on the board and have the graph it in their notebooks:
The graph should look like this:
“What is the slope of the line?” I usually wait for 3 or 4 hands to go up before calling on someone. Alternatively, I inspect what students have done and pick someone who calculated it correctly but rarely volunteers. The slope is 3.
“Now someone tell me what the y-intercept is.” The answer is 4.
“I happen to know what the equation of this line is. And I will tell you whether you want to know it or not. It’s y = 3x + 4.
(In the past I’ve had students try to guess the equation but generally they don’t get it, and it takes more time than I want to allow. It can be done, however, and then related to what I say next. I find it more efficient to cut to the chase.)
“We said the slope of the line is 3. What is the coefficient of x in the equation?” They see that it is 3.
“Now what do you think the 4 in the equation represents?”
At this point some are catching on, and identify the 4 as the y-intercept.
“Let’s see if the equation works. Plug in the x values from the table, and see if you get the y values.” They do this and see that it works, which is my cue to then move on.
The Pitch: The Slope-Intercept Equation. I point out what we just showed about the equation and the graph. “In the equation y = 3x + 4, we saw that the coefficient of x in the equation is the slope of the line, and the constant in the equation, 4, is the y-intercept of the line. This is not a coincidence. The equation y = 3x + 4 is in what is called Slope-Intercept form. All straight lines can be represented by equations in the following form”:
I write this on the board and they copy it in their notebooks.
“The m in the equation is the slope of the line, and b is the y-intercept. If an equation is in that form, you don’t even have to plot it to know what the slope and y-intercept are.”
I gave an example: y = -2x + 9 and ask what the slope is. There is usually a pause and someone will venture the answer: -2.
“It is -2 because the coefficient of x is -2. That leaves ‘9’ to be what?” It’s obviously the y-intercept.
“What about y = 3x -9? What are the slope and y-intercept?” The slope they get right away and pause for the y-intercept.
“Suppose I were to write it this way:” y = 3x + (-9)
“Remember, subtraction is addition of the opposite.” I usually see more students raising their hands to volunteer the answer; -9.
Examples:
I want them to tell me the slope and y-intercept of the equations.
1. y = 5x + 3 Answer: m=5, b = 3
2. y = -6x – 8 Answer: m=-6, b = -8
3. 2x + y = 5. This will give them pause. Prompt: “We need to put this in y = mx + b form? How can we do that?” Someone usually volunteers that you can solve for y. Which we then do and I advise them to put the x term first on the right hand side so it is more obvious what is m and what is b. They will get y = -2x + 5; m = -2, b = 5.
4. 3x = y + 9 Answer: y = 3x-9; m =3, b = -9
Common mistakes: Students will get the m and b mixed up. Some students may still be weak at solving for y.
Graphing Equations in Slope-Intercept Form. Having now worked through examples of the slope-intercept equation to identify slope and y-intercept, I move on to graphing equations in this form.
“Graphing these equations is fairly easy, because we have all the information we need to do so.” I start with an equation and have them work in their notebooks as I work on the board. y = 2x – 1
I then write the steps as we work through the first one.
Step 1: Identify slope (m) and y-intercept (b): m=2,b = -1
They should be able to do this fairly easily.
Step 2: Plot y intercept. (0,-1)
This is also fairly easy.
Step 3: From the y-intercept, draw the vertical and horizontal lines that represent the slope and plot that point.
This requires demonstration. I have them write the slope as 2/1, so they know that we are going to draw a vertical line 2 units up from the intercept and then 1 unit to the right. “When we do this, positive numbers mean ‘up’ for verticals, and negative numbers mean ‘down’. When we draw horizontals, positive numbers mean go to the right, and negative numbers mean go to the left.”
Step 4: From the point obtained in Step 3, use the slope in the same way to find another point. (Fig 5)
Step 5: Connect the three points with a straight line.
Examples. The first few examples I work with the students and then allow more independent work.
1. Graph y = -3x + 2 “Where do we put the y intercept?” It is plotted at (0,2)
The slope (m) is -3, which I instruct students to write as -3/1.
“We said positive numbers are up, and negative numbers are down. So from (0,2), we go 3 units in what direction?” Down.
“Now we go 1 unit to the right or left?” Right.
“Repeat the process.” They do so and connect the points.
2. Graph y = -2/3(x) + 2 We follow the same procedure by plotting the y-intercept and then plotting the points determining the slope of -2/3, and then repeating the process.
3. Graph y = 4x Is there a y-intercept? The answer will likely be “No” to which I respond “Actually there is. It’s zero. We just don’t write y = 4x + 0. So where does the intercept go?” At the origin. Then they plot the points for the slope.
4. Graph 3x + y = 5. “What must we do first?” Solve for y. Now the equation is y=-3x + 5 which they will do.
Homework. The homework is more of the same, and as usual, I leave time to provide guidance and help as needed. In the process, I might pick a problem to do together (so they think they are getting their homework done “for free”) that has a slope of, say -3/4. “We can write this as -3/4, but we can also write it as 3/-4.” We proceed with that so that they see points can be plotted in the same manner.