## Slope

Materials: graphs A-D prepared for Introducing the Concept, grease pencil or marker

Preparation: post or display on the overhead projector graphs A-D.

• Say: We've talked about these graphs before. Can you remind me of what you think the equations tell us about how the graphs will look?
Summarize the discussion from the end of Introducing the Concept.

• Say: Slope is rate of change. If we want to use a number to name the slope of a line, we need to find how fast y changes in comparison to changes in x.

• Point to the line y = x in graph A.

• Say: This line passes through the point (0, 0). How would you compare the steepness of the line between (0, 0) and (1, 1) and between (2,2) and (2, 2)?
Students should realize that on any given line, the slope (or steepness) is the same throughout.

• Say: You can find the vertical change between points on the line by identifying two points and subtracting their y values. How can you find the horizontal change between the same two points?
Students should readily suggest that you could subtract their x values. Be sure to emphasize that the order in which you subtract is important. If you subtract the first y value from the second, then you MUST subtract the first x value from the second. In fact, (y2y1)/(x2x1) is the standard form of the slope formula. It doesn't really matter which point you choose to be the first point as long as once you choose it, you stick with it.

• Ask: What is the slope of the line y = x?
Work this out with the slope formula, using several different pairs of points so students are convinced that as long as their points are on the line, the slope they compute will be the same as the slope anyone else computes.

• Ask: What is the slope of each line in all of these graphs?
You might assign small groups to find the slopes. Record each slope and the equation of its line on the chalkboard.

• Ask: Were you right? Does the number that multiplies x have anything to do with the slope?
When an equation is in the slope-intercept form, the coefficient of x is the slope. Discuss how this knowledge might help students to find the slope of any line, no matter the form of the equation or whether or not they have drawn the graph.

• Ask: If I told you that I had two lines, one with slope 4 and one with slope 2, what could you tell me about those lines?
Students should be able to tell you that both slopes are positive, so the lines will tend upward from left to right. They should also understand that the line with slope 4 is steeper than the line with slope 2, so the slope 4 line will stay closer to the y axis.

• Ask: If I told you that I had two lines, one with slope 4 and one with slope 2, what could you tell me about those lines?
Students should be able to tell you that both slopes are negative, so the lines will tend downward from left to right. They should also understand that the line with slope 4 is steeper, so it will stay closer to the y axis.

• Say: If a car can go 30 miles on one gallon of gas, the ratio of miles to gallons is 30:1. An equation that describes the relationship of miles traveled to gallons of gas consumed is m = 30g. Draw a coordinate grid and plot some points that represent different combinations of miles traveled and gallons of gas consumed.
This is a simple equation to graph. The scales on the axes will need to be adjusted in order to best show the coordinate pairs.

• Ask: Is this a linear equation? How do you know?
It is a linear equation because its points all lie on a line.

Wrap-Up and Assessment Hints
Offer students a variety of experiences with slope. Have them look at graphs and predict, then compute, their slopes. Then, have them look at equations and compute the slopes of the lines they represent.

Slope        Two-step Linear Equations
with Rational Numbers.