Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different
proportional relationships represented in different ways.
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and provide
a foreshadowing of tomorrow? List the questions.
Joe Flacco completed 12 out of 24 of his passes, Aaron Wrap Up Questions: Brainstorm some other examples of
Rodgers completed 15 out of 28 passes and Eli
direct variation. When might someone use these
Manning completed 10 out of 18 passes on their first
relationships in a career? The NFL has decided to add
game of the season. Which quarterback had a better
another football team of 52 athletes. How many more
showing on their first game? Justify your answer. If
college students would you anticipate would need to play
each player had attempted 500 passes, how many
collegiate football to fill the team?
completions would they have? (resource sheet
included)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson?
1. Begin the lesson by having the students think about comparisons and have them record then discuss quantities
they are interested in comparing. Record key discussion points on the bottom of the warm-up.
2. Present the students with the scenario comparing the number of students who play football in high school,
college and NFL.
3. Assign students to groups of 3 or 4 and have them answer the questions # 1 - 5, showing all their work and
writing a complete sentence for an answer. Have the students rotate their responsibilities of recorder,
mathematician, timekeeper and manager. Each student should have an opportunity to be a different role for each
question. Bring the class back together and discuss answers and strategies to achieve answers. Display different
ways of finding the answers on the board.
4. Bring class together for a whole group discussion for question #6. Discuss with students how to write C in terms
of D.
5. In groups have students answer question #7 and share responses.
6. Have students complete the graph and table in groups for questions #8 and #9 and #10
7. Help students fill in notes on direct variation.
8. Have students give a “thumbs up” or “thumbs down” if they think the relationship in questions #6 and #10 were
direct variation.
9. Have students complete the Problem Solving using multiple representational resource sheets independently.
This resource sheet can be modified based on the learners need by labeling the x and y axis, breaking the
resource sheet sections into separate pages to make it bigger, and having certain students complete certain
portions. Collect resource sheet to assess students understanding.
10. Ask students the closure questions.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student mastery? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened (and conceptual) understanding.
For students to be able to develop an understanding of direct variation given different scenarios. This will be
demonstrated with the thumbs up and thumbs down and by collecting the multiple representation resource sheet.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation Course: Common Core 8
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Vocabulary: Direct Variation, constant, part-to-whole ratio, part-to-part ratio.
Students will need to understand connections between proportions and direct variation.
Misconceptions: Labeling graphs correctly and comparing part to part versus part to whole ratios.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Student Resource sheets attached below and two
problems Solving Multiple Representation resource
sheets
Students will complete a Problem Solving Multiple
Representation resource sheet comparing the number of
High School Seniors who play football to the number that
make it into the NFL.
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you use to
reflect on the effectiveness of this lesson?
Have students mastered the concept of direct variation and able to explain in multiple ways?
How will I modify tomorrow’s lesson based on students’ current understanding?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
Warm-Up
Joe Flacco completed 12 out of 24 of his passes, Aaron Rodgers completed 15 out of 28 passes and Eli Manning
completed 10 out of 18 passes on their first game of the season. Which quarterback had a better showing on their
first game? Justify your answer. If each player had attempted 500 passes, how many completions would they
have?
What are some other quantities you are interested in comparing? Brainstorm some
ideas below.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
Scenario: Many young people every year set their goal of becoming an NFL football player. According
to the “NFL_Hopefuls” website for every 100 students who play their senior year of football only 9 of
them play collegiate football. Every year there are approximately 9000 students playing collegiate
football and only 215 of them make it onto an NFL roster.
In groups answer the following questions, show all your work and use complete sentences in your
answer.
1) The ratio of the number of collegiate players to the number of High School Senior players is
9 collegiate players
. This is a part-to-whole relationship. We can also represent this as a
100 High School Senior Players
part-to-part relationship. What is the ratio of the number of Collegiate Players to Non collegiate players
that played their senior year in High School?
2) How many collegiate players would there be if there were 2000 High School Senior Players?
3) How many collegiate players would there be if there were 10,000 High School Senior Players?
4) How many High School Senior players would there be if there were 100 collegiate players?
5) How many High School Senior players would there be if there were 9000 collegiate players?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
6) Let C = number of collegiate players who played High School Senior year football and
D = number of students who played their senior year of High School football but did not play collegiate
football. Follow the instructions and record how to write an equation for C in terms of D.
7) Use complete sentences to explain how you can use the information above to determine the number of
students who played their senior year of High School Football but did not play in college.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_________________________________________________________
(8 and 9) Table and graph comparing students who played collegiate football and students who did
not.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
10) Let C = number of students who played collegiate football and H = total number of students who
played football their senior year of High School. Write an equation for C in terms of H.
Notes: A _____________________ ______________ is a ratio that has a constant value.
________________________ __________________________ is when 2 quantities which we call x and
y have a constant ratio. One quantity varies directly with another. In direct variation
the constant ratio is commonly labeled ______ and the ratio is written as
y
 k or y = kx.
x
11) The ratio of collegiate football players that make it into the NFL each year is approximately

1 NFL player
. Let N = the number of NFL players and C = the number of collegiate players.
50 Collegiate players
Complete the Problem Solving Using Multiple Representations resource sheet.
Wrap Up
1) Brainstorm some other examples of direct variation. _________________________________
_____________________________________________________________________________________
_______________________________________________________________________
2) When might someone use these relationships in a career? _____________________________
_____________________________________________________________________________________
_______________________________________________________________________
3) The NFL has decided to add another football team of 52 athletes. How many more college students
would you anticipate would need to play collegiate football to fill the team?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
Answer Key
Warm UP
Joe Flacco completed 12 out of 24 of his passes, Aaron Rodgers completed 15 out of 28 passes and Eli Manning
completed 10 out of 18 passes on their first game of the season. Which quarterback had a better showing on their
first game? Justify your answer. If each player had attempted 500 passes, how many completions would they
have?
Joe Flacco completed ½ his passes for 50% or 0.5
Aaron Rodgers completed
Eli Manning completed
15
for 53.6% or 0.536
28
5
for 55.6% or 0.556
9
Eli Manning had
a better showing because he completed more passes.

What are some other quantities you are interested in comparing? Brainstorm some
ideas below.
Answers may vary
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
Scenario: Many young people every year set their goal of becoming an NFL football player. According
to the “NFL_Hopefuls” website for every 100 students who play their senior year of football only 9 of
them play collegiate football. Every year there are approximately 9000 students playing collegiate
football and only 215 of them make it onto an NFL roster.
In groups answer the following questions, show all your work and use complete sentences in your
answer.
1) The ratio of the number of collegiate players to the number of High School Senior players is
9 collegiate players
. This is a part-to-whole relationship. We can also represent this as a
100 High School Senior Players
part-to-part relationship. What is the ratio of the number of Collegiate Players to Non collegiate players
that played their senior year in High School?
9 collegiate players
91 noncollegiate players
2) How many collegiate players would there be if there were 2000 High School Senior Players?
180 collegiate players
3) How many collegiate players would there be if there were 10,000 High School Senior Players?
900 collegiate players
4) How many High School Senior players would there be if there were 100 collegiate players?
About 1,111 High School Seniors
5) How many High School Senior players would there be if there were 9000 collegiate players?
100,000 High School Seniors
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
6) Let C = number of collegiate players who played High School Senior year football and
D = number of students who played their senior year of High School football but did not play collegiate
football. Follow the instructions and record how to write an equation for C in terms of D.
C 9

D 97
97C  9D


97C 9D

97
97
C


9
D
97
7) Use complete sentences to explain how you can use the information above to determine the number of
students who played their senior year of High School Football but did not play in college.
If you know the number of noncollegiate players who played their Senior year in High School, you can
use the information that there are 9 collegiate players for every 97 noncollegiate players to find the
number of collegiate players.
(8 and 9) Table and graph comparing students who played collegiate football and students who did
not.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: NFL Football Unit Rate as Slope Direct Variation
Date: _____________ Teacher(s): ____________________
Course: Common Core 8
Start/end times: _________________________
10) Let C = number of students who played collegiate football and H = total number of students who
played football their senior year of High School. Write an equation for C in terms of H.
C
9

H 100
100C = 9H

100C 9H

100 100
C


9
H
100
Notes: A Constant Ratio is a ratio that has a constant value.
Direct Variation is when 2 quantities which we call x and y have a constant ratio. One quantity varies
directly with another. In direct variation
the constant ratio is commonly labeled K and the ratio is written as
y
 k or y = kx.
x
11) The ratio of collegiate football players that make it into the NFL each year is approximately

1 NFL player
. Let N = the number of NFL players and C = the number of collegiate players.
50 Collegiate players
Complete the Problem Solving Using Multiple Representations resource sheet.
Wrap Up
1) Brainstorm some other examples of direct variation. _________________________________
_____________________________________________________________________________________
_______________________________________________________________________
2) When might someone use these relationships in a career? Scout, coach, player, sports medicine
3) The NFL has decided to add another football team of 52 athletes. How many more college students
would you anticipate would need to play collegiate football to fill the team?
2600 college students who play football
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.