June 10, 2009

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This lesson introduces the notion of slope through intuitive, geometric, and Algebraic means. This lesson may take 2 (or even 3) class periods to allow for full student comprehension, as slope is traditionally one of the most difficult-to-acquire concepts in the Algebra curriculum.

- Mathematics > Problem Solving

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This lesson is written as a unitary lesson. However, it may well be that the students will take two, or even three, class periods to fully acquire the various concepts of slope. It is important for students to develop a robust understanding of the concept of slope, both in order for them to appropriately symbolize the situations used in this project, but also to better prepare them for Trigonometry and Calculus, where the concept of slope is critical.

**Group Size:** Any

**Learning Objectives:**

1. Students will acquire a robust conception of slope, being able to use situational, geometric, and Algebraic means to describe it.

a. Students will identify slope as the rate at which a variable changes.

b. Students will identify slope as the "steepness" of a line.

c. Students will identify slope as "the change in y over the change in x" or as "rise over run"

2. Students will determine the slope of a graphed line.

3. Students will determine the slope of a line connecting two ordered pairs.

**Materials:**

Graph paper/whiteboards

Colored pencils/dry erase markers

Rulers

**Procedures:**

Begin by reminding students of what they learned in the previous lesson. Have them verbalize the definitions of intercepts, and then have them break into groups to graph the following equation:

*y = -3x + 9*

Ask students to identify the intercepts on the graph. Next, ask them to describe the graph (i.e. is it steep? what direction does the line point in? where does it cross the axes? etc.)

Introduce the geometric notion of slope as "the steepness of a line". Reference grades on highways, where a numerical value of the steepness of a hill is used to help motorists understand how steep a particular stretch of highway is going to be. Have each group identify two separate points on the line. Have them find how far they must travel "up" from the lower point to get to the same level as the upper point. Have them record this as the "rise". Next, have them record how far "over" they must travel to actually arrive at the second point (emphasize the **negative direction** of travel). Have them record this as the "run". Have them then create the ratio of the rise over the run and simplify the ratio. Groups should report in on their results, and their results should be the same. Highlight the fact that different groups chose different points on the line, but they all returned the same result. Emphasize the notion that the slope is the same all along the line, regardless of the points chosen, *because the line has the same steepness throughout*.

Put the following equation on the board, and have them graph it and find the slope:

*-2x + 8y = 24*

Have the groups work on this and circulate through the room providing help and answering questions as necessary. Once all groups have finished, have them report back in. In the discussion, highlight the fact that the value of the slope should always be given in simplest form, emphasize the graphical difference between a positive and negative slope and between slopes with absolute value greater than 1 and those with absolute value less than 1.

**Game idea**: Play "Simon Says" with slope. Have all students face the same way (towards a central whiteboard for instance), and show a slope of 2, 1, 1/2, 0, -1/2, -1, and -2 with their arms. Then, play several rounds where you call out a numerical value of the slope, and the students must show it with their arms.

Lead a discussion on lines with slope of 0 and undefined slope. Highlight the difference between the floor (which has *no* slope) and a wall (for whom talking about its "steepness" makes no sense). The same applies for slopes where either the rise or the run is 0.

Next, challenge the student to graph the following two equations:

*y = 4*

*x = -2*

Allow some time for student groups to find the answer before providing help. If the entire class is stuck, lead a discussion about the fact that since the equation only identifies one variable, the other one does not matter. Allow the students to come to the conclusion that these equations necessarily lead to horizontal and vertical lines (respectively). Have them practice with several other examples, identifying the slope as either 0 or undefined in each case.

Next, introduce the notion that slope does not need to be determined graphically. Ask students how they have been finding the rise and run between the two points that they have been choosing. They will answer that they are finding the difference between them. Next, ask them to graph the equation *y = -1/3x + 6 * and to find two points that they want to use to identify the slope. Once they have, ask them what *mathematical* means (rather than simply counting) they could use to find the "distance" between the two x-coordinates and the two y-coordinates? Use this as an introduction into the method of finding the slope numerically by identifying the difference between the two y-coordinates as the rise and the difference between the two x-coordinates as the run.

Use the following examples on the board as practice:

*a. Find the slope of the line containing (3, 5) and (-2, 0)*

*b. Find the slope of the line containing (2, 8) and (4, 4)*

*c. Find the slope of the line containing (-2, 5) and (-2, 7)*

Highlight example (c) as being "tricky" because the run = 0.

Next, have the student groups work collaboratively to graph the cost curve from the following example and determine its slope:

*To run a cookie-baking business, the Algebra class determines that it will cost $0.25 per cookie produced to pay for the ingredients and utilities. If the rent the school charges them is $5, graph the cost curve and determine its slope.*

Ask them to also determine the equation of the cost curve, with *y* as the $ and *x* as the # of cookies produced.

Lead a discussion once they are finished where you elicit the fact that the variable cost of producing the cookies *is* in fact the slope of the line. (Try to elicit that the fixed cost is the y-intercept.) Give them a second example, and ask them to try to write an equation, graph it, and find the slope:

*The Road Runner was eating peaceably on the side of the road when he spotted the Coyote coming up over the side of a cliff. If the Road Runner started 100 meters from the side of the cliff and runs at the rate of 40 m/sec, write an equation describing his distance from the cliff ( y) as a function of the number of seconds (x) he has been running. Graph this equation and find its slope.*

For this example, make sure that students have accurately created an appropriate scale for their axes before beginning to graph. Spend the extra time necessary to ensure that all groups are able to verbalize and accurately choose an appropriate scale. After they have identified the slope (which should be 2/5 or 0.4), lead a discussion in which you help them to verbalize that the slope in this situation is the speed at which the Road Runner runs.

Use the two preceding examples as the basis for a discussion on slope as a rate of change. Focus on how the fundamental similarity between variable costs and speeds is that both represent a quantity that changes as a function of another quantity (variable cost is price varying with number, speed is position varying with respect to time).

Have the students work through the following example. However, *before* they begin working, have them try to identify the quantity that will be the slope, and *why* it represents the slope:

*The Algebra class is now doing research into their business plan. They want to model their revenue curve for charging $0.75/cookie. Write the equation for the revenue curve where y stands for the $ made and x stands for the # of cookies sold. Graph the equation and determine the slope.*

Students should be able to identify that the price charged is the slope for the revenue curve (3/4).

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