Mathematics Senior Level Capstone Course
Unit Overview
Title of Unit: To Speed or Not to Speed? That Is the Question. Unit Designers:
Amy Bianco - Fauquier County Dennis Myers – Warren County Lisa Rogers – Warren County
Editor: Diane Leighty, UVA-SCPS Office of Mathematics Outreach Context: Living in our current economic times we want to drive our vehicles
efficiently. How does the speed you travel effect your fuel efficiency and what other factors should be considered to be driving
economically? Number of Class
Hours:
3 Hours Unit
Design: X Task Based ___Project Based Other Subject
Areas/Disciplines Addressed:
Economics; Environmental Science
Driving Question: How does the speed you travel affect the fuel efficiency of your vehicle?
Mathematics Content Addressed:
Solving practical problems; Solving and graphing Absolute Value Equations and Inequalities; Transformational approach to graphing; Finding vertex and intercepts of functions; Linear Programming; Writing absolute value
equations
MPE Addressed:
Problem Solving; Understanding and Applying functions; Procedure and Calculation Assumption of Prior
Knowledge:
College and Career Readiness/21st Century Skills to be taught (T) during this unit or expectation (E) for student use during this unit and assessed (A):
Collaboration – Students work in pairs (or a group of 3 if needed) (E)
Research -
Communication (Oral and/or Written) – Discussion about the problem and written display of results (E)
Technology – Use of a graphing calculator to graph the functions (E)
Critical Thinking/Decision Making - Other: (Describe)
Major Products and/or Performances:
Group – Graph of the Linear Programming problem on Chart paper
Presentation Audience:
X Class
School Individual – Teacher walks around and ask questions of groups.
Journal Entries
Expert Community Other:
Launch: Event or experience used to engage the students interest and inquiry:
Have Gas clip
http://www.youtube.com/watch?v=s0ydZ4OsOY0
Evaluation: Formative Assessments (During the Unit)
Interview X Practice Presentations X
Mathematicians Journal Notes
Preliminary
Plans/Outlines/Prototypes
Checklists
Rough Drafts Concept maps
Field Tests Other:
Summative Assessment (End of Project)
Written Products, with a rubric Peer Evaluation, with a rubric
Oral Presentation with a rubric Self Evaluation, with a rubric
Other Product(s) or
Performance(s), with a rubric
Resources Needed: On-site people, facilities: Teacher
Equipment/Technology: Graphing Calculator; Chart Paper; Markers
Materials: Handouts; Graph paper
Community Resources:
Reflection Methods: Individual, Group, and/or Whole Class
Mathematicians Journal Small/Focus Groups X
Whole Class Discussions X Fishbowl Discussions
Survey Other:
Material Adapted From:
Virginia’s Senior Level Capstone Course Instructional Plan
Unit Title: To Speed or Not to Speed? That Is the Question.
Driving Question: How does the speed you travel affect the fuel efficiency of your vehicle?
Task: Students solve practical problems involving speed and fuel efficiency. They interpret the results and decide on the appropriate speeds for fuel efficiency.
ENGAGE
How will student’s interested be piqued so they want to engage in the inquiry in this unit?
Time: 15 min
Show the short clip of the movie “Cars”. http://www.youtube.com/watch?v=s0ydZ4OsOY0
Pose the question of what type of cars are driven by members of the class. Consider the idea that some students may not have a car or a car in their family. Ask about ideal/dream car. Ask the students if they know the fuel efficiency of their vehicle or any dream vehicle. What affects the fuel efficiency of a vehicle?
Mathematician Journal
Prompts Tell me about your dream car. Does gas mileage influence your decision? EXPLORE Teacher provides guidance for the explorations to prepare students with the
knowledge and skills to engage in the task. Students will self-assess on the prior knowledge and skills assumed for the unit
Time: 75 min
Exploration #1 Kit’s Volvo (HO#1)
The goal of this activity is discover the meanings of (h, k) and a in the general absolute value equation,
and in the real world.
Students work individually to create a solution to the
problem; the teacher directs instruction as needed to solve the problem posed. Some groups may require additional
guidance during this process.
Students review how to identify the vertex, symmetry, and the slope from a graph. Students identify and interpret the values for a, h, and k for an absolute value function.
Exploration #2 Sasha’s Truck (HO#2)
Using the skills developed in Exploration #1, students work in pairs to develop responses to the task. This should be
Mathematician Journal
Prompts
followed by a class discussion of the response.
EXPLAIN
Teacher introduces the main task of the unit and prepares students to in small group independent work...
Time: 10-15 min
Introduce Smart Car Buying (HO#3)
Prepare students to work in groups of 2 or 3 to respond to the questions. (Pairs are preferred for this activity but it may be necessary for there to be a group of three.) Review the expectations for how to share the work. Remind the students that while the pair will think and problem-solve together they will each develop a response to the questions and this will be turned into the teacher.
The teacher regroups student so there are teams of four with only one member from a single pair on the team. They first compare their responses and then develop a single group response which is recorded on poster paper to post in the room for a classroom discussion.
Mathematician Journal
Prompts
How do you plan to get started? Do you have any questions before you begin formulating your solution? ELABORATE The student groups are working independently with teacher consultations.
Time: 50 min
Teacher works with all groups, observing, listening, and asking probing questions as students work on Smart Car Buying (HO#3).
Student pairs work on the task Smart Car Buying (HO#3) for 30 minutes.
The teacher then regroups class into groups of four with only one member from a single pair on each team.
Students take their responses from the pairs work to the new group. The group records on chart paper a sketch of the feasible region of the graph. The students represent the feasible region as a system of inequalities that include absolute value inequality and a linear inequality. These are hung at the front of the room.
Mathematician Journal
Prompts Can you pull any other additional information from the graph? EVALUATE Working groups
Each group of students presents their solution using the large graphs.
Mathematician Journal
submit products or make
presentations
Time: 25 min
Individually each person responds to the prompt in their Journal: Compare your solution to the others in the class. Do you see anything that another group did that you think would improve your group’s solution?
Collect the individual’s work on the task.
Map the Unit
What do students need to know and be able to do to complete the task/project/problem successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.
Task:Living in our current economic times we want to drive our vehicles efficiently. How does the speed you travel effect your fuel efficiency and what are the other factors should be considered to be driving economically.
KNOWLEDGE AND SKILLS NEEDED
Assumed already learned
Students will self-assess
Will be taught during the unit
1. Solving Practical Problems X X
2. Solving Absolute Value Equations X X
3. Solving Absolute Value Inequalities X X
4. Transformational Approach to
graphing X X
5. Finding the vertex of a function X X
6. Finding the intercepts of a function X X
7. Solving a Linear Programming
Problem X X
What project tools will student’s use? Know/need to know lists Daily goal sheet
X Mathematician’s Journals Briefs/Memos
Task lists
Planning Calendar
□ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________ □ ________________________________
HO #1
Kit’s Volvo
The fuel efficiency of a car depends on the speed at which it is driven. For example, consider
Kit’s Volvo. When it is driven at a rate of r mph, it gets miles
per gallon.
This should be completed without the use of a graphing calculator
Graph m versus r, 0 < r ≤ 80.
Notice that this graph has a vertex. What are its coordinates? In words, what is the
meaning of this point? Is this a maximum or a minimum why? Real-world meaning?
Explain any symmetry that is found in the graph and/or the table.
Find the slope of each section of the graph. In words, what is the meaning of the slope
for each section of the graph?
If the slope of one of the sections is a and the vertex is (h,k), write the general equation
for any absolute value function.
o Does it matter whether a or –a is used in the equation? Why? Why not?
o In real world context, which points or part of the graph are not valid?
HO #2
Sasha’s Truck
The fuel efficiency m (in miles per gallon) of a truck depends on the speed r (in miles per hour) at which it is driven. The relationship between m and r usually takes the form
For Sasha’s truck, the optimal fuel efficiency is 24 miles per gallon,
attained when the truck is driven at 50 miles per hour. When Sasha drives at 60 miles per hour, however, the fuel efficiency drops to only 20 miles per gallon.
Find another driving speed r for which the fuel efficiency of Sasha’s truck is exactly 20
mpg.
Fill in the rest of the missing entries in the table.
r m
60 20
50 24
40 30 20 10
Draw a graph of m versus r, for 0 < r ≤ 80. This may be graphed either on paper or on a
calculator.
Why is zero not included?
Find the values of a, h, and k. Write the equation of the function.
Shade in the region of the graph that shows all speeds where the fuel efficiency is above
Find the x-intercepts. In real-world context, what do these points mean?
Does it make sense for an absolute value function to be used to relate fuel efficiency
HO #3
Smart Car Buying
Bob Wrench, the manager of the Wrench’s Auto Shop, is ordering hybrid cars and non-hybrid cars from a wholesaler. Hybrids cost $30,000 and non-hybrids cost $20,000. Storage space at the shop requires that the order be no more than 120 cars total. Bob knows from previous experience that at least 30 of each type are needed, and that the number of hybrids should not exceed the number of non-hybrids.
Sketch the region of a graph that satisfies all of the conditions stated above (called the
feasible region). Write each condition as an inequality.
Bob sells hybrids for $50,000 and non-hybrids for $35,000. Calculate the profit at each
vertex of the feasible region.
How many hybrid cars and non-hybrids car should Bob buy to maximize his profit?
Represent the feasible region as a system of inequalities that includes an absolute value
Solutions to HO #1
Kit’s Volvo Solution Key
The fuel efficiency of a car depends on the speed at which it is driven. For example, consider
Kit’s Volvo. When it is driven at r miles per hour, it gets miles
per gallon.
Graph m versus r, 0 < r ≤ 80.
Notice that this graph has a vertex. What are its coordinates? In words, what is the
meaning of this point?
Vertex: (55, 32); The best fuel efficiency of 32 mpg happens at 55 mph.
Explain any symmetry that is found in the graph and/or the table.
There is symmetry on either side of the vertex. The numbers in the table mirror each other on either side of (55, 32). The graph can be folded along the line r=55.
Find the slope of each section of the graph. In words, what is the meaning of the slope
for each section of the graph?
From r=0 to r=55, the slope is 0.2 = 1/5; that means that for every increase in speed of 5 mph, the fuel efficiency increases by 1 mpg. From r=55 to r=80, the slope is -0.2; that means that for every increase in speed of 5 mph, the fuel efficiency decreases by 1 mpg.
If the slope of one of the sections is a and the vertex is (h,k), write the general equation
for any absolute value function.
o Does it matter whether a or –a is used in the equation? Why? Why not?
Sasha’s Truck Solution Key
The fuel efficiency m (in miles per gallon) of a truck depends on the speed r (in miles per hour) at which it is driven. The relationship between m and r usually takes the form
For Sasha’s truck, the optimal fuel efficiency is 24 miles per gallon,
attained when the truck is driven at 50 miles per hour. When Sasha drives at 60 miles per hour, however, the fuel efficiency drops to only 20 miles per gallon.
Find another driving speed r for which the fuel efficiency of Sasha’s truck is exactly 20
mpg.
The fuel efficiency will be 20 mpg when the speed is 40 mph (due to symmetry around vertex (50, 24).
Fill in the rest of the missing entries in the table.
r m
60 20
50 24
40 20
30 16
20 12
10 8
Draw graph of m versus r, for 0 < r ≤ 80.
Find the values of a, h, and k. Write the equation of the function.
Shade in the region of the graph that shows all speeds where the fuel efficiency is above
Sasha needs to drive between 35 mph and 65 mph to keep her fuel efficiency about 18 mpg.
Find the x-intercepts. In words, what do these points mean?
The x-intercepts are (110, 0) and (-10, 0). (110, 0) means that if Sasha drives 110 mph, she will get 0 mpg. If she drives -10 mph, she will also get 0 mpg (Negative speeds do not make sense in the real world.)
Does it make sense for an absolute value function to be used to relate fuel efficiency to
speed?
It seems that this relationship might be more curved than linear (perhaps quadratic, trigonometric). The positive x-intercept does not make sense; though the fuel efficiency would be very low at 110 mph, it would probably not be 0 mpg.
Smart Car Buying Solution Key
Bob Wrench, the manager of the Wrench’s Auto Shop, is ordering hybrid cars and non-hybrid cars from a wholesaler. Hybrids cost $30,000 and non-hybrids cost $20,000. Storage space at the shop requires that the order be no more than 120 cars total. Bob knows from previous experience that at least 30 of each type are needed, and that the number of hybrids should not exceed the number of non-hybrids.
Sketch the region of a graph that satisfies all of the conditions stated above (called the
feasible region). x = non-hybrids y = hybrids
Bob sells hybrids for $50,000 and non-hybrids for $35,000. Calculate the profit at each
vertex of the feasible region. P = 15000x + 20000y
(30, 30); P = 450,000 + 600,000 = 1,050,000 (90, 30); P = 1,350,000 + 600,000 = 1,950,000 (60, 60); P = 900,000 + 1,200,000 = 2,100,000
How many hybrid cars and non-hybrid cars should Bob buy to maximize his profit?
He should buy 60 of each type of car to yield a profit of $2,100,000.
Represent the feasible region as a system of inequalities that includes an absolute value
