Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 56777
What's the Going Rate?
Students discover that the unit rate and the slope of a line are the same, and these are used to compare two different proportional relationships.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Interactive Whiteboard, Basic
Calculators, LCD Projector, Overhead Projector
Instructional Time: 2 Hour(s)
Freely Available: Yes
Keywords: unit rate, rate of change, slope, proportional relationships, direct variation equation
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Intro Unit Rate and Slope of Line.pdf
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
The students will be able to:
graph a proportional relationship.
understand that a unit rate is the same as the slope of the line and be able to find the slope.
compare two different proportional relationships represented in different ways.
write an equation for a proportional relationship, using the slope.
Prior Knowledge: What prior knowledge should students have for this lesson?
MAFS.7.RP.1.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For
example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
MAFS.7.RP.1.2 Recognize and represent proportional relationships between quantities.
1. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
2. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
3. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as t = pn.
4. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is
the unit rate.
Guiding Questions: What are the guiding questions for this lesson?
1. What does a unit rate represent?
page 1 of 4 2. How do you find the slope of a line? Is there more than one method, even though there is only one slope for a line? Which method is most efficient in this problem?
3. When finding the slope of a line, does the change in y (or x), represent the vertical rise or the horizontal run?
4. What does the slope represent in the context of this problem?
5. How does the unit rate compare to the slope of the line?
6. What is the first step to graph a proportional relationship?
Teaching Phase: How will the teacher present the concept or skill to students?
Small Group Discussion: The students will be in small groups of 3 or 4. The teacher will give each group the following directions:
You have the opportunity to choose between three jobs. Job A offers you 2 hours after school, $8.40 per day, five days a week. Job B offers you 3 hours after school,
for $12.75 per day, and you can work any three afternoons a week. Job C offers you 20 hours a week, for $82 a week, and you can work the 20 hours of your
choice. Discuss with your group what should be considered when trying to choose the job they want and which job is the better choice. On the given dry erase
board, write a summary of your groups discussion, which job your group would take, and why.
The group should find the hourly rate of each job. (Job A $4.20/hour, Job B $4.25/hour, and Job C $4.10/hour) They might also discuss how their schedule after
school affects their decision, and might be more important than the hourly rate.
The teacher then provides a sheet of graph paper to each group and assigns one of the jobs to each group. They are asked to make a table and a graph that would
display the pay for each hour they work at the job they are given. The teacher suggests they make the x-axis represent the hours and the y-axis the money. The
teacher can demonstrate on the board to get the group started. Job A Graph.pdf Remind students to include a title and labels with units and equal increments for the
axes. (They can be given the job they chose, or assigned a job, so all jobs are graphed.)
This information is displayed (either the students' work or pre-prepared by the teacher) and the teacher uses these tables and graphs to move into the topics of this
lesson:
Note: If available, graphing calculators may be used for students to make tables and graphs. Or, the teacher can pre-prepare the tables and graphs on graphing
calculator software to be used to project them for students to see.
1. Unit rate: The teacher discusses unit rate as the money that each job pays per hour and which job has the highest pay per hour or higher unit rate.
2. Slope: (Display the slopes as they are used in the lesson.)
The teacher will say: (Answers to questions are in italics.)
The unit rate represents the slope of the line. The slope is often represented as a ratio, which could be expressed as a unit rate found at the point on the graph with
the ordered pair (1, 4.2) or in the table, x = 1 and y = 4.2. The ratio for the slope is frequently represented with m. For example $8.40/2 hours = the unit rate of
$4.20/1 hour. However for convenience the slope may be expressed as a ratio without 1 in the denominator. The slope of the graph for Job A, the example, could be
expressed as m = 8.4/2 or 4.2/1; both are correct.
Look at the graph for Job A. Look at what happens between the origin (0,0) no hours worked and no money earned and the point (1, 4.2) one hour worked and $4.20
earned. To find the slope (unit rate) you could count the vertical units from 0 to 4.2 for the numerator and then the horizontal unit from 0 to 1 for the denominator. m
= 4.2/1 Both numerator and denominator are positive numbers. Demonstrate that if you started at (1, 4.2) and counted to (0,0): -4.2/ -1 = 4.2/1, the same slope.
Do this from the origin to the point (2,8.4). m = 8.4/2 = 4.1/1, the same slope.
Since a line has only one slope, what generalization could you make about finding its slope, using any two points on the line? You could use any two points on the line
to find its slope.
Look at the graph and find the vertical and then horizontal distance in units from (1, 4.2) to (2, 8.4) The vertical distance some call the rise is 4.2, and the horizontal
distance which some call the run is 1. This is another method for finding the slope, counting the vertical units (the change in the y value) for the numerator, then
counting the horizontal units (the change in the x value) for the denominator. m = rise/run
Another method to find the slope without needing to look at a graph is to use the values in the ordered pairs. We determined the slope is the ratio: change of y
values/change of x values for any 2 points on a line.
In the example using the points (1,4.2) and (2,8.4) above what is the change in y values? 8.4 – 4.2 = 4.2
What is the change in the x values? 2 -1 = 1
What is the slope? m = 4.2/1
Do this with the points (0,0) and (2,8.4)? Do you have an equivalent ratio? Yes, 8.4/2
What are 3 methods to find the slope, m? Find the unit rate; use the rise/run from the graph; use the ordered pairs from any two points on the line, change in y
values/change in x values.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher, along with the students, will complete the introductory handout Unit Rate and Slope of a Line. Using the speed of a horse, the students would complete a
table and graph and then surmise that the unit rate and slope of a line are the same. This concept might have been determined in the teaching phase discussion of the
three jobs, and therefore the worksheet is a way to revisit the concept.
The handout discussion would also lead into reviewing proportional relationships:
the ratios of y/x are equivalent, since they have the same unit rate.
the graph begins at the origin.
the equation of the line is in the form y = mx.
The teacher will ask the students to look for proportional relationships in the rest of this lesson.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
The next handout, Practice: Unit Rate and Slope of a Line, allows the students to practice finding slope and unit rate given a graph or table, and writing the equation y
page 2 of 4 = mx for the relationship. This worksheet should help reinforce that when given a graph of a line, the student could find the slope using rise/run, and when using the
table, could find the unit rate by determining the y value, when x = 1, or determine ordered pairs and use the slope formula.
The handout Comparing Proportional Relationships presents graphs, tables, and equations and asks the student to choose the higher unit rate. This provides further
practice in finding the slope from graphed lines and tables, and provides real world problems where comparisons are made.
Answer Keys
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The students will be asked to complete the following questions in their journal or math notebook, etc:
1. How do you represent a proportional relationship with an equation? What does each letter represent? y = mx; m = unit rate or slope, x is the independent variable,
y is the dependent variable
2. What is the relationship between the unit rate and the slope of the line? Same value
3. When given a graph of a line, what method would you use to find the slope? Why? Find the ordered pair, when x = 1 and use the y value of the ordered pair, since
this is the unit rate; or find the rise/run; use the ratio of the change in y/change in x for two points on the line. Use the method that appears to be most efficient,
based on the graph.
4. When given a table, what method would you use to find the slope? Why? Either find the unit rate, when x = 1; or write two ordered pairs and use the ratio change
of y value/change in x value, whichever is most efficient based on the data.
5. How could you use the table to graph a proportional relationship? Use the data to write ordered pairs; graph the points represented by the ordered pairs; draw a
line through the points to the origin.
The teacher will assign partners and ask the students to compare their answers with a partner and revise their answers, if needed. The teacher will ask students to
share their responses.
Then, the students will be assigned the Closure Handout to complete independently. This handout provides the teacher the opportunity to determine which students
can apply the lesson.
Summative Assessment
The student will be able to successfully complete a handout where they must compare the unit rates in two different situations, and the student will answer questions
about unit rate, slope of a line, and the comparison between two given proportional relationships in a journal or notebook.
A Summative Assessment for this lesson may be found in the Closure section.
Formative Assessment
The first activity (job earnings) in the Teaching Phase could be used as an assessment of the students' prior knowledge. As the teacher circulates while the students
are working, the teacher should correct misconceptions and make notes of students who have difficulty completing the work. Small groups may be formed to work
with these students and their understanding should be continuously checked as this lesson progresses.
Through small group work and student discussions, the teacher will be able to recognize if students understand the concept of unit rate.
The teacher will be able to identify the students who can find the slope and unit rate correctly given a graph or table.
Through the independent work, the teacher will be able to recognize the students who are able to compare proportional relationships that are represented in different
ways.
Feedback to Students
Through the opening discussion, the teacher will be able to reinforce/correct/restate the concept of unit rate and the slope.
The class handouts will allow the teacher to give feedback to students as they apply their new knowledge, and then the students will have independent work that
allows them to apply the concepts.
Throughout the lesson, the teacher will use questions to guide the students, and provide students with opportunities to revise their work, as needed.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
1. In the first discussion of the three jobs, have the students make the tables only. Then give each group 3 graphs that represent the three jobs and ask them to
match the graphs that would match the job/table. This might be best to do to save time and allow for more discussion.
2. Complete the second worksheet in small group settings with students who need the additional support or, if needed, complete it as a whole class.
3. Provide support for students who encounter unfamiliar vocabulary with definitions, examples, and/or translations, as needed.
Extensions:
Define the equation y = mx as a direct variation equation and solve direct variation word problems.
Have the students write a real world problem that includes two proportional relationships and represent each in different ways.
Suggested Technology: Document Camera, Computer for Presenter, Interactive Whiteboard, Basic Calculators, LCD Projector, Overhead Projector
Special Materials Needed:
class sets of handouts
graph paper
optional: graphing calculators
page 3 of 4 Additional Information/Instructions
By Author/Submitter
Students who participate in this lesson will engage in the Mathematical Practice Standards:
MAFS.K12.MP.7.1 - Look for and make use of structure.
MAFS.K12.MP.5.1 - Use appropriate tools strategically.
SOURCE AND ACCESS INFORMATION
Contributed by: Tami Parish
Name of Author/Source: Tami Parish
District/Organization of Contributor(s): Washington
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional
relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to
determine which of two moving objects has greater speed.
MAFS.8.EE.2.5:
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
When students work toward meeting this standard, they build on grades 6–7 work with proportions and position
themselves for grade 8 work with functions and the equation of a line.
page 4 of 4