Finding slopes using the formula presented in the previous lesson is generally one that students learn readily, and often recite “y-two minus y-one over x-two minus x-one”. It is helpful also to show how slope can be obtained visually by finding the vertical and horizontal distance. This helps reinforce the concept of rise over run, which is the same as “change in y/change in x”. Doing this also helps suggest why any two points on the same line result in the same slope. It also sets the stage for showing why this is true via similar triangles. Students do not have enough geometric tools by which to fully understand why the triangles are similar, but they will address it again in high school geometry.
I begin this lesson with an activity that students enjoy and takes less than ten minutes. It leads into the fact that any two points on a straight line result in the same slope.
Warm-Ups.
1. If a car gets 35 mi/gallon, how many gallons per mile does it get? Answer: miles/gallon = 35/1 =1/x; 35x = 1; x = 1/35 gal/mi
2. Find the slope of the line. Answer: (8-(-1)/(-2-1) = 9/-3 = -3
3. Ian has $56. Sam has 20% more money than Ian. How much money does Sam have? Answer: 1.2×56=$67.20 OR (100+20)/100 = x/56; 1.2/1 = x/56; x = 1.2×56 = $62.50
4. Find the y-intercept of the graph of y = 4x -1. Answer: y=4(0) -1 = -1;
5. Determine whether the relation shown in the table below is a function. Explain why or why not. Answer: There are two y values for the same x value (17). A function must have only one output for each input.
Students will likely have questions on Problem 1. Prompts might be “If the ratio is miles/gallon, how do we write a proportion?” “How do we write the ratio of 35 miles to 1 gallon?” “What are we trying to find, gallons or miles?” For Problem 3, students may need to be reminded of what the proportions are; i.e., 120/100 = mark-up/original.
The Wind-Up: Finding Slopes on the Same Line. I start this lesson with an activity. Each student receives an index card with an ordered pair of numbers on it. Students pair up, and each pair figures out the slope represented by their two ordered pairs. Each ordered pair on the card represents a point on the same line, such as y = 2x + 2, so ordered pairs might be (0,2), (-2, -2), (100, 202) and so on.
It doesn’t take long for students to start shouting out the slope they obtained, and to see that everyone is getting the same slope (two in this case). They are usually surprised by this and I act surprised as well. “Wow, what a coincidence. Everyone pick new partners so no two people are using the same ordered pairs as before, and let’s try this again.” They do so, and of course end up getting the same slope. One time when I did this a girl was totally amazed as if I had done a magic trick and asked “How did you do that?”
Usually one or two students will figure out that the points are all on the same line. In any case, whether someone offers an answer or not, I tell them they are points from the same line, and show them the equation I used. “No matter what two points you pick on a straight line, the slope will always be the same. To show why this is the case, let’s take a look at the line that I used, which was based on y = 2x + 2.”
The Pitch: Slope of a Straight Line is Constant. I have them draw it in their notebooks by plotting the three points and connecting them with a straight edge.
“I’ve labeled three points on it. I pick the points (1,4) and (0,2), but instead of using the formula, I’m going to count the unit squares. I’m going to draw a line from (1,4) down to the line that (0,2) is on.” I have them do the same. “Now draw a line from (0,2) to join that line.”
I have them count the squares in the vertical and horizontal lines. “So what do we get if we have vertical distance/horizontal distance?”
Having done that, and all agreeing on “two” as the answer, I say this is “rise over run” and it’s also the change in y/change in x. The y value changes from 2 to 4, and the x value changes from 0 to 2. So when you use the formula you used yesterday, that’s what it’s doing. It’s counting the squares for you.” I sometimes but not always hear a few “Ohhh’s” at this.
Next I have them do the same for the points (2,6) and (0,2):
I have them count squares again, and they will get 4 for vertical and 2 for horizontal resulting in a slope of 2.
“No matter what two points we pick, when we draw the lines as we did, we get triangles whose vertical and horizontal sides are proportional. These kind of triangles are called ‘similar’.” (Although this book does not include geometry, students may have had some exposure to it in the unit on proportions, and/or will have it in later chapters covering geometric figures.)
I write on the board:
The slope of a straight line is constant.
Examples of Types of Slopes. I have them plot points, draw the graph and compute the slopes.
2. (3,0), (2, 1) The slope is -1 and the graph is:
3. (3,1), (2,1) The slope is 0 and the graph is:
“Notice for the first graph, the slope is positive, and it starts from lower left to upper right. What do we notice about the second one? Is the slope positive or negative? How would you describe the line?”
I want to hear that the slope is negative and the line starts from upper left to lower right. I also describe it as “From left to right, the line goes downward.”
The third graph they’ve seen from the previous lesson. “What is the slope? What can you say about the line”.
I want to hear that the slope is zero and the line is horizontal.
“One more graph; plot the points (3,2) and (3, 4).”
The previous lesson had a graph like this, so if they remember what was done the day before, they will know that the slope is undefined. The graph is:
“Vertical lines have an undefined slope.”
Homework. At this juncture, I prepare a worksheet that covers the past lessons from the beginning: functions, relations, graphing equations by plotting points, by using x and y intercepts, finding slope, and identifying the type of slope (e.g., negative, positive, zero or undefined) by inspection of the graph.