Explorinq Slope

Submitted by: Barbara Roach     School: Keenan High 

Grade Level: 8-12      Subject: Algebra 1

Competency Goal /Objective(s): Find slopes of lines. (AL15-D)

Overview:  This is an exploratory lesson that assists students in defining and describing  slope visually (What does it look like?) and physically ( How is it expressed numerically?). 

First, students will listen to a descriptive passage and write an intuitive definition of slope. Second, students will study visual representations of slope. Third, students will study numeric representations of slope. The lesson includes exploratory activities (individual and collaborative pairs), teacher-led discussion and whole class discussion. 

Focus/Essential Questions: 
1. What is the slope of a line? 
2. How do you find the slope of a line? 
3. What makes the slope of a line positive? 
4. What makes the slope of a line negative? 
5. What makes the slope of a line zero? 
6. What makes the slope of a line undefined? 

Time Frame: One 90-minute class period.

Resources: 

Health Algebra One, D. C. Health Publishing Company, 1997. 

Transparencies of lines: A, B, M and N.

Graph paper or coordinate grids 

Rulers 

Graph paper transparency (Coordinate Plane) 

Written activity - What's My Slope 

Ticket to leave rubric 

Assessment:

Informal - Observation, questioning, cooperative work and individual responses.

Formal - Ticket to leave activity (summarizer) 

Instructional Activities: 

Activator: Teacher will launch lesson by reading the following passage: 

In everyday language, we use words like slope, 
Slant, steep, and tilt to describe objects that are not flat or perfectly upright. 
The slope is for advanced skiers. 
The earthquake tilted the highway. 
The roof has a steep pitch that allows snow to slide off. 
The grade of the road is 6%. 
The gutter does not slant enough to drain well. 
A line can also have slope. 

In pairs, the students will write a definition of slope. 
Each pair will read their definition to the class. The teacher will write all responses on the overhead or board and point out commonalities. Using commonalities, the teacher will write an appropriate definition for the slope of the slope of a line to be used during this and subsequent lessons. 

Example: The slope of a line is the measure of the tilt or steepness of line. 

Demonstration/Discussion/Notes: 

The teacher will display Transparency A.  The teacher will ask the following questions: 

1. Which is steeper, line A, or line B? 

2. Which is steeper, line M or line N?

3. Can you think of a way to measure the steepness using numbers? 

4. Does this method apply to all of the lines? Explain. 

Once the students have completed the intuitive notion of slope, the teacher will introduce numerical representations of slope. 

The teacher will continue to use transparency A to demonstrate/discuss the following: 

The lines in graph 1 rise upward from left to right. Lines like A and B have positive slope. 

The line of graph 2 falls downward from left to right. Lines M and N have negative slope. 

We will now explore the following: It is possible to describe the slope of a line using a single positive or negative number? Is there more than one way to determine slope? 

Using the same graphs, find the slope by using rise/run. 

Choose two points on line A. Move horizontally from the first point to directly below the second point and, then vertically to the second point. (A right triangle is formed.) Label run and rise. 

Repeat several times using different points on A, B, M, and N. 

Choose two points on line A. Write the points on the overhead and label coordinates using subscripts. (x1, y1) and (X2,Y2)

As you move from the first point to the second point, the ratio of the change in y to the change in x is the slope of the line. 

Repeat several times using different points and different lines. 

NOTE: Emphasize that subscripts are used to distinguish between different x- and y- coordinates. In other words, the subscripts provide a different way to name coordinates of the same ordered pair. A common error is the incorrect order in which the slope formula is applied. The order of subtraction is important. (Caution students to carefully label the points to be used.) 

Do other examples using all lines. Use both formulas. 
NOTE: Label coordinates of the points. 

The lines, A and B, have positive slope. The graphs rise upward to the right. 

Choose two points on line A and B and calculate the slopes respectively, to verify that the slopes are positive. 

The lines M and N have negative slope. The graphs fall downward to the right. 

Choose two points on line M and N and calculate the slopes respectively, to verify that the slopes are negative. 

Emphasize: 

• Two points on a line are all that is needed to determine slope. 

• The numerator is read as "y sub 2 minus y sub I" and is called the rise. 

• The denominator is read as "x sub 2 minus x sub I" and is called the run. 

• The slope of a straight line can be determined by writing the ratio of vertical change as you move from one point on the line to another point on the line.

The slope of a line can also be determined by determining the ratio of rise to run. It is used when a graph is given. 

The teacher will display a blank transparency of the coordinate plane. 

Draw a horizontal line and select two points on the line. Calculate the slope. Repeat several times using different lines. 

Horizontal lines have slope of zero. The lines have no rise. 

Draw a vertical line and select two points on the line. Calculate the slope. Repeat several times using different lines. 

Vertical lines have undefined slope. Division by zero is  undefined. The lines have no run.  NOTE: 0/3 is 0, while 3/0 is not defined. 

Independent/Guided Practice: 
The students will complete worksheet, What's My Slope? The teacher will circulate and give assistance. The students may use notes but must complete the worksheet individually. Once all students have completed the assignment, ask volunteers to give and explain answers to class. Note: A transparency of the worksheet could be displayed and students may come to the overhead to give explanations. If needed, make clarifications and reteach. 

Summarizer - Ticket to Leave 
Each cooperative pair will write responses to the following questions: 

1. Explain the difference between positive and negative slope. 

1. Do you think it is possible for a line to have a negative slope if it passes through the points (5,6) and (5, -3)? Why? 

NOTE: The Ticket to Leave can be used as a whole group discussion, a cooperative activity or as a homework assignment. 

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