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Starting With Definitions
Before the problem can even be written, students need to have a working vocabulary. Provide them with a dictionary or the glossary of the math book to write down the definitions with an example of the following words for this lesson:
1. Ratio - Comparisons of two quantities which can be expressed as fractions, percents, decimals are in the form x:y. An example of a ratio could be the ratio of males to females in the math classroom: 2 females/5 males or a car travels at 60mph (miles per hour) or 60 miles/1 hour.
2. Rate - A comparison of quantities with different units. The car traveling at 60 mph has the rate of 60 miles per hour should be written as 60 miles: 1 hours. Another example would be found in a grocery store: Macintosh Apples are $1.39/lb. and a Volkswagen gets 35 miles per gallon in the city.
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Ratios to Mileage Comparatives: A Warm-Up & Actual Problem
Warm-Up Problem: Student A's car gets 42 miles to the gallon on the freeway. How much gas will Student A's car use if it is driven 126 miles on the I-90 freeway?
Answer: Ratio would be to set up a scale to get the answer so have students fill in the blanks for "X."
42 miles per gallon = 42 miles : 1 gallon
X : 2 gallons
126 miles : X gallons
X = 84 miles : 2 gallons and 126 miles : 3 gallons = blanks for 'X."
Answer: Students can divide 126 miles/42 to get the correct answer which would be 3 gallons.
Next, let's use this problem to problem solve a comparing fuel economy problem for real life students: Marcy and Will.
Problem: Marcy and Will got summer jobs as counselors. They both drove to the same summer camp using alternate freeway routes. In comparing the fuel economy of their cars, each argued that their car got the best gas mileage. How would you solve this problem using the information below:
Marcy took I-90 and got to Camp Woodmark and posted the following data: 323 miles : 20 gallons of gas.
Will took I-405 and got to Camp Woodmark and posted the following data: 278 miles : 16 gallons of gas.
Answer: Marcy's car went 323 miles/20 gallons = 16.2 miles/gallon
Will's car went 278 miles/16 gallons = 17.4 miles/gallon
Additional Activities: Students can use the above data to make a prediction table and an equation to solve for the problem: Whose car, Marcy or Will's, will get the best fuel economy if the trip took 450 miles or 515 miles? This math activity can provide students an opportunity to create a rate table of gallons of gas and miles for Marcy and Will's cars which could start at 323 miles: 20 gallons and 278 miles : 16 gallons respectively and include the new data of 450 miles and 515 miles for problem solving. Students can then graph the rate table table to show which car is truly more fuel efficient and create conclusions and further predictions.
Closure: Given today's gas prices, this lesson can provide a reality check for students beginning to drive for the first time. Teachers can adapt this problem and add another problem solving category of "cost of final trip or cost of each gallon of gas or ask would regular, plus or premium gas at $2.39/galllon; $2.99/gallon or $3:09/gallon get the better mileage for Will and Marcy's trip?"