ALEX Lesson Plan

     

Comparing Fuel Economy (adapted from CMP "Comparing and Scaling" Investigation 4.1)

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  This lesson provided by:  
Author:Mark Baynes
System: Tallassee City
School: Southside Middle School
The event this resource created for:GEMS
  General Lesson Information  
Lesson Plan ID: 23976

Title:

Comparing Fuel Economy (adapted from CMP "Comparing and Scaling" Investigation 4.1)

Overview/Annotation:

In this lesson students will explore rates. They will use the concept of rates to compute and compare fuel economy.
This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 4
3. Determine and justify solutions for multi-step word problems, including problems where remainders must be interpreted.

a. Write equations to show solutions for multi-step word problems with a letter standing for the unknown quantity.

b. Determine reasonableness of answers for multi-step word problems, using mental computation and estimation strategies including rounding.

Unpacked Content
Evidence Of Student Attainment:
Students:
When given multi step word problems,
  • Solve a variety of multistep word problems involving all four operations on whole numbers including problems where remainders must be interpreted.
  • Explain and justify solutions using connections between the problem and related equations involving a single (letter) unknown.
  • Evaluate the reasonableness of solutions using estimation strategies.
Note: Multi step problems must have at least 3 steps.
Teacher Vocabulary:
  • Operation
  • Multi Step problem
  • Remainder
  • Unknown quantity
  • Equation
  • Rounding
  • Mental strategy
  • Partition
  • Estimation
  • Reasonableness
Knowledge:
Students know:
  • Context situations represented by the four operations.
  • How to calculate sums, differences, products, and quotients.
  • Estimation strategies to justify solutions as reasonable.
Skills:
Students are able to:
  • Solve multi-step word situations using the four operations.
  • Represent quantities and operations physically, pictorially, or symbolically.
  • Write equations to represent the word problem and use symbols to represent unknown quantities.
  • Use context and reasoning to interpret remainders.
  • Use estimation strategies to assess reasonableness of answers by comparing actual answers to estimates.
Understanding:
Students understand that:
  • Using problem solving strategies will help them determine which operation to use to solve a problem.
  • Remainders must be interpreted based on the context, and remainders are sometimes ignored, rounded up, or partitioned.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.4.3.1: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
M.4.3.2: Solve single-step word problems.
M.4.3.3: Recognize key terms to solve word problems.
Examples: in all, how much, how many, in each.
M.4.3.4: Solve division problems without remainders.
M.4.3.5: Recall basic addition, subtraction, and multiplication facts.

Prior Knowledge Skills:
  • Demonstrate computational understanding of multiplication and division by solving authentic problems with multiple representations using drawings, words, and/or numbers.
  • Identify key vocabulary words to solve multiplication and division word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
  • Recall basic multiplication facts.
  • Add and subtract within 20.
  • Represent repeated addition, subtraction, and equal groups using manipulatives.
  • Distinguish between rows and columns.
  • Use repeated addition to solve problems with multiple addends.
  • Count forward in multiples from a given number.
    Examples: 3, 6, 9, 12; 4, 8, 12, 16.
  • Recall doubles addition facts.
  • Model written method for composing equations.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.4.1 Solve one-step word problems involving real-life situations using the four operations within 100 without regrouping and select the appropriate method of computation when problem solving.


Mathematics
MA2019 (2019)
Grade: 7
1. Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Compute a unit rate for ratios that compare quantities with different units.
  • Determine the unit rate for a given ratio, including unit rates expressed as a complex fraction.

  • Example: if a runner runs 1/2 mile every 3/4 hour, a student should be able to write the ratio as a complex fraction.)
Teacher Vocabulary:
  • Unit rate
  • Ratio
  • Unit
  • Complex fractions
Knowledge:
Students know:
  • What a unit rate is and how to calculate it given a relationship between quantities.
  • Quantities compared in ratios are not always whole numbers but can be represented by fractions or decimals.
  • A fraction can be used to represent division.
Skills:
Students are able to:
Compute unit rates associated with ratios of fractional
  • lengths.
  • Areas.
  • quantities measured in like or different units.
Understanding:
Students understand that:
  • Two measurements that create a unit rate are always different (miles per gallon, dollars per hour)
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.1.1: Define unit rate, proportions, area, length, and ratio.
M.7.1.2: Recall how to find unit rates using ratios.
M.7.1.3: Recall the steps used to solve division of fraction problems.

Prior Knowledge Skills:
  • Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
  • Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  • Generate equivalent fractions.
  • Define quantity, fraction, and ratio.
  • Reinterpret a fraction as a ratio.
    Example: Read 2/3 as 2 out of 3.
  • Write a ratio as a fraction.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.
  • Calculate unit rate or rate by using ratios or proportions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.1 Calculate a unit rate (limited to whole numbers under 100).


Mathematics
MA2019 (2019)
Grade: 7
2. Represent a relationship between two quantities and determine whether the two quantities are related proportionally.

a. Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional.

b. Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions.

c. Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Decide whether a relationship between two quantities is proportional.
  • Recognize that not all relationships are proportional.
  • Use equivalent ratios in a table or a coordinate graph to verify a proportional relationship.
  • Identify the constant of proportionality when a proportional relationship exists between two quantities.
  • Use a variety of models (tables, graphs, equations, diagrams and verbal descriptions) to demonstrate the constant of proportionality.
  • Explain the meaning of a point (x, y) in the context of a real-world problem.
  • Example, if a boy charges $6 per hour to mow lawns, this relationship can be graphed on the coordinate plane. The point (1,6) means that after 1 hour of working the boy makes $6, which shows the unit rate of $6 per hour.
Teacher Vocabulary:
  • Equivalent ratios
  • proportional
  • Coordinate plane
  • Ratio table
  • Unit rate
  • Constant of proportionality
  • Equation
  • ordered pair
Knowledge:
Students know:
  • (2a) how to explain whether a relationship is proportional.
  • (2b) that the constant of proportionality is the same as a unit rate. Students know:
    • where the constant of proportionality can be found in a table, graph, equation or diagram.
    • (2c) that the constant of proportionality or unit rate can be found on a graph of a proportional relationship where the input value or x-coordinate is 1.
Skills:
Students are able to:
  • (2a) determine if a proportional relationship exists when given a table of equivalent ratios or a graph of the relationship in the coordinate plane.
  • (2b) identify the constant of proportionality and express the proportional relationship using a variety of representations including tables, graphs, equations, diagrams, and verbal descriptions.
  • (2c) model a proportional relationship using coordinate graphing.
  • Explain the meaning of the point (1, r), where r is the unit rate or constant of proportionality.
Understanding:
Students understand that:
  • (2a) A proportional relationship requires equivalent ratios between quantities. Students understand how to decide whether two quantities are proportional.
  • (2b) The constant of proportionality is the unit rate. Students are able to identify the constant of proportionality for a proportional relationship and explain its meaning in a real-world context. (2c) The context of a problem can help them interpret a point on a graph of a proportional relationship.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.2.1: Define proportions and proportional relationships.
M.7.2.2: Demonstrate how to write ratios as a fraction.
M.7.2.3: Define equivalent ratios and origin.
M.7.2.4: Locate the origin on a coordinate plane.
M.7.2.5 Show how to graph on Cartesian plane.
M.7.2.6: Determine if the graph is a straight line through the origin.
M.7.2.7: Use a table or graph to determine whether two quantities are proportional.
M.7.2.8: Define a constant and equations.
M.7.2.9: Create a table from a verbal description, diagram, or a graph.
M.7.2.10: Identify numeric patterns and finding the rule for that pattern.
M.7.2.11: Recall how to find unit rate.
M.7.2.12: Recall how to write equations to represent a proportional relationship.
M.7.2.13: Discuss the use of variables.
M.7.2.14: Define ordered pairs.
M.7.2.15: Show how to plot points on a Cartesian plane.
M.7.2.16: Locate the origin on the coordinate plane.

Prior Knowledge Skills:
  • Recall basic addition, subtraction, multiplication, and division facts.
  • Define ordered pair of numbers.
  • Define x-axis, y-axis, and zero on a coordinate.
  • Specify locations on the coordinate system.
  • Define ordered pair of numbers, quadrant one, coordinate plane, and plot points.
  • Label the horizontal axis (x).
  • Label the vertical axis (y).
  • Identify the x- and y- values in ordered pairs.
  • Model writing ordered pairs.
  • Define quantity, fraction, and ratio.
  • Reinterpret a fraction as a ratio.
    Example: Read 2/3 as 2 out of 3.
  • Write a ratio as a fraction.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.2 Use a ratio to model or describe a real-world relationship.


Local/National Standards:

NCTM Number and Operations Standard Grades 6–8 Understand numbers, ways of representing numbers, relationships among numbers, and number systems: - Understand and use ratios and proportions to represent quantitative relationships.

Primary Learning Objective(s):

Students will use rates to compute fuel economy and gas mileage.

Additional Learning Objective(s):

 
 Preparation Information 

Total Duration:

31 to 60 Minutes

Materials and Resources:

You will need transparencies (attached). These can also be used as slides in a PowerPoint.

Technology Resources Needed:

You will need a overhead projector (ELMO or LCD projector, if available) and a computer with Internet access.

Background/Preparation:

The students should be able to complete long division problems.

  Procedures/Activities: 
1.)Put the rates transparency on the overhead (see attachment). Read each statement to the class and ask students to interpret each of the statements. Ask students what is being compared? Are the quantities the same kinds of measures (or counts) or are they different? Tell the students that each statement on the transparency compares two different things.....Let the students identify the two things in each statement. Tell the students that these are called rates because they tell us the rate at which something happened.

2.)Now discuss with the students how a rate can be scaled up or down to find an equivalent rate. Read this problem: My brand new car gets 30 miles to the gallon in the city. How much gas will my car use if I drive it 240 miles in the city? Place methods transparency on overhead and discuss the two possible ways listed on the over head. Ask the question: Which way did you think was best? Why? Can you think of another way of finding this rate?

3.)Talk to students about their experiences with cars and travel. Ask them if they know any drivers who worry about gas mileage.

4.)Put the story of Johnny and Fred on the overhead. Read the story aloud and direct the students to the table above. Tell them the table shows where Johnny and Fred live. It also shows the route they will take to visit their mom and dad. Point out the distances between cities and how this could be very important later in who wins the competition. Pose the question: "Which car is more fuel efficient on the highway?" Have students work in pairs on the problem. If some are struggling redirect them to the question/problem in step #2. It's the same kind of problem. Remind students that you are interested in explanations and why they think their answers are correct.

5.)Have students report their answers and share their thought process while solving the problem. In any solution, students should take into account two quantities: gallons used and miles traveled. The ratio of total miles to total gallons is typically used to report fuel efficiency and commonly represented as miles per gallon. Some students may compute gallons per mile and make perfectly good sense out of these rates. Miles per gallon is more common, but gallons per mile (gpm) is fine as long as students can explain what rate they have computed and what it means.

6.)During this activity each student will be given a certain vehicle and a destination. They must research (in the computer lab) this vehicle to show what kind of fuel economy it gets. (for example: 2006 F-150 SuperCrew gets 18 miles to the gallon on the highway.) Use the Mapquest hyperlink to help find the car's mpg. Using the cars estimated miles per gallon, the student must find how many gallons of gas it will take to drive round trip to their destination. They will need to calculate how much they will spend in fuel cost with the price of gas being $3.95 per gallon. Example: A student is given a slip of paper with the statement: You have a 2008 Chevy Tahoe and you are going to visit your aunt in Seattle, Washington. How much fuel will be needed to travel there and back? If gas is $3.95 per gallon, what will be the cost of fuel for your trip?
(Fuel Economy)

7.)After students complete their assignment, they will make class presentations to share their results. Allow small group discussions using the following questions: How does the rising fuel cost affect our economy? What do you think about fuel shortages? Identify any changes in your lifestyle that have occurred due to the rise in fuel cost.


Attachments:
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  Assessment  

Assessment Strategies

Teacher observation and test (see attached.)

Acceleration:

If a student has mastered the primary learning objective of this lesson they can research the concept of unit rates and how they are useful in real life situations. This research could be published in their math journals or on a sheet of paper to be turned in.

Intervention:

Students who need remediation in this area can access the following website. What is Your Rate?


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.