Discovering Slopes

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Objective

SWBAT compute the slope of a line, graphically and algebraically.

Big Idea

The rate at which one quantity changes with respect to another can be determined either by computation or by looking at a graph.

Accessing Prior Knowledge

15 minutes

Accessing Prior Knowledge

Certain topics that had been part of the 9th grade CCSS, are no longer included because students are expected to have achieved fluency with them by the end of 8th grade. This particular lesson targets 8.EE.B.5 & 8.EE.B.6. Yet, it is worth studying because it provides good groundwork review before taking on the 9th grade standards, particular those under Creating Equations and Reasoning with Equations domains.

To begin the lesson, I project the following problem on the whiteboard (APK_Chuck) and ask each student to read it to themselves and write their answer in their notebooks. One reason to start with this problem is that it is a rate of change problem and intuitively easier than using lines and slope.

The subtraction of the pounds and months are changes and weight vs time is a popular idea. This makes the slope formula easier to understand and it helps make sense of positive, negative, and zero rates of change.

 “Mr. and Mrs. Happychuck had a healthy baby boy named Chuck who weighed 7.5 pounds at birth. At the end of 4 months, the baby weighed 13.5 pounds. What is Chuck’s weight age ratio at 4 months?

(Call on a student to answer; the ratio is 13.5 pounds to 4 months or (13.5)/4)

It is important to make sure that the students always mention the units involved, in this case, pounds and months. Then ask the students to calculate the rate of change of Chuck’s weight, that is, how much he gained per month. Ask a volunteer to go to the board and do the work:

Students usually will say that Chuck gained 6 pounds in 4 months. Ask them to say the amount gained in one month. Most will know they must reduce (or divide)

Students should say Chuck gained weight at a rate of 1.5 pounds per month.

Tell the students that we will be computing the slope of a line, and that slope is calculated in very much the same way.

 

New Info/Application

35 minutes

 

 New Info/Application: 

 I begin the lesson by projecting the resource (plane_slope)  and I ask all the students to analyze the picture of the airplane taking off. I tell them that slope is a synonym for steepness and it is calculated much like the Chuck problem in the APK. 

I ask students to indicate how high the plane has elevated from the first snapshot to the third, and write it down in their notebooks. Each unit can be taken to equal 10 meters.  

Then ask students to determine how much has the plane has flown horizontally form the 1st to the 3rd snapshot.  The nose of the plane could be used as a reference point to find both values.

 

Common error: Some students follow an oblique path when asked to find the vertical change of the plane. Make it clear that to find vertical change is like finding the height of something and a line going up or down is used. In this case a vertical line from the lowest point chosen to the highest.

I as that students write these two values in ratio form and explain to students that the definition of slope is the ratio of vertical change to horizontal change in the steepness of a line, commonly referred to as rise over run. (students should see that the slope is 6/18 or 1/3). Affirm that the slope is positive.

Use the thumb rule to gauge students understanding and point out how to measure rise and run by counting the number of x-units and y-units between the two points on the graph.

This is a good time to demonstrate to students that if were going down from the 3rd nose and left to the 1st nose of the snapshots, we would obtain -6/-18 which is equal to 1/3.

Then I call on a voluteer student to come up and draw a straight line from the nose of plane 1 to plane 3. Ask to mark the point where the nose of “plane 1” is and the same goes for “plane 3”  Tell students that these are preferably points because they are at intersections of the vertical and horizontal white grid lines. Label them with their proper coordinate pair.  Instruct students that the formula for finding slope of a line is m=  y2 -y1 / x1-x2  where the coordinates of two points are (x1, y1) and (x2, y2).

Students should find the slope of the airplane path using the slope formula.

Have students do the same for different points on the line, for example, going from nose 1 to nose 2.  They will see that the slope is always 1/3.

Scaffold: Resource: slopeFormula_scaffold

 

Partner Activity 

  1. Divide the class into pairs of students. Each pair needs to draw x and y axes on a sheet of graph paper.
  2. Draw the following two T-charts on the board and have partners plot the points, draw the graph and find the slope of each using the slope formula.

 

x

y

1

1

0

0

-1

1

 

 

 

 

 

x

y

0

6

-3

-3

1

9

 

 

 

  

 

  1. Have students discuss their results. (the first chart does not represent a line so they will not be able to find the slope; the second chart represents a line with slope 3)
  2. Discuss solutions with the class.

 

 


Generalization

10 minutes

Generalization:

Have students remain in pairs and hand each group a ramp_slope“wrap up” card. (see resource)

The task takes us back to the objective of the lesson and includes a negative slope.

I indicate that one member should graphically find the slope of the ramp by using the image and the other partner should find two points and use the slope formula. At the end, each student should compare answers and discuss the matter if any of them made a mistake. Students should discuss, how each obtained their answer. (-1/7)

Homework:  Resource: HW_Discovering Slopes

 Extension: Ask students to find the slope of a vertical and a horizontal line and explain their answer mathematically.