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Making Sense of Percent Concentrations: Don't Freeze the Engine

  • Lesson
9-12
1
Algebra
Martha Haehl
Location: unknown

In this lesson, students will learn to read an antifreeze chart to determine the necessary mix of antifreeze and water for protection against various temperatures. Students will then numerically analyze a particular cooling system to determine how much fluid to drain in order to get the desired percent concentration of antifreeze. Finally, students will set up a system of equations to solve the same system.

First, help students familiarize themselves with an antifreeze chart (as found on a container of antifreeze). In particular, explain the purpose of antifreeze in the cooling system of a vehicle: It prevents the water from freezing. The freezing point of water is 32°F. When antifreeze is added to the water in a cooling system, the temperature at which the mixture will freeze drops below 32°F. This protects the engine when the outside temperature drops.

When students generally understand the purpose of antifreeze, distribute the first two pages of the Don't Freeze the Engine activity sheet. That is, hand out Parts 1 and 2; Part 3 will be used at the end of the lesson, and it can be handed out later.

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Don't Freeze the Engine Activity Sheet

 As students read the chart on the activity sheet, ask the following questions: 

  • How many quarts of antifreeze will protect an 18-quart cooling system in -10°F weather?
  • If a 10-quart cooling system contains 2 quarts of antifreeze, to what temperature is it protected?
  • Why is the highest temperature on the chart 26°F and the lowest temperature ‑66°F? That is, why aren't temperatures outside of that range included in the chart? [The water in the engine is not likely to freeze when temperatures are above 26°F, and motor vehicles are not likely to be used (or function correctly) in temperatures below ‑66°F.]

Questions 2‑7 in Part 2 closely model how a mechanic would approach a problem with a cooling system. A mechanic would use a refractometer to gauge the current level of protection. He would calculate, based on the reading, how much fluid to drain and replace with antifreeze for the desired level of protection. Students model this process using a 20‑quart system with a current level of protection of 10°F. Have students use the antifreeze chart to fill in the information for the given system.

Students will be figuring percent concentrations and calculating how much water is drained when a given amount of mixture is drained. Then, students use this information to estimate the amount of fluid that should be drained to reach the desired level of protection. As students complete the activity sheet, monitor their work. Are they correctly interpreting the chart and figuring percent concentrations? Offer help, as necessary.

When all students have completed the activity sheet, conduct a class discussion. Review the answers to the questions, but also ask follow‑up questions to ensure that students understand what they did. A good question is to ask each student (or pair of students) to describe a situation in which they need to obtain a correct percent concentration. They can choose a situation similar to the one described for cars, or they can choose one that is not similar and explain how the two situations are different. A possible response to such a question could be: "I added 4 tablespoons of cocoa to 6 ounces of milk, and the result was too sweet. I therefore added 2 ounces of pure milk to reduce the percent concentration (and thereby make it less sweet). This is a little bit different from the antifreeze situation, because I wouldn't have to drain any hot cocoa before adding milk, unless the cup was too full."

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Solutions to all of the questions on the activity sheet can be found on the 

Don't Freeze the Engine Answer Key

Finally, using the last two pages of the Don't Freeze the Engine Activity Sheet, students will solve a related word problem and answer some application questions.

Read the problem at the beginning of Part 3, and ask students, "How is this problem related to the activities in Part 1 and Part 2?" This can lead to a good class discussion, and students should realize that the main difference is that they do not have to determine how much coolant needs to be drained from the engine.

For a formal assessment, you can have students solve this problem on their own. Questions 9‑13 lead students through the solution. As they work, you may want to circulate through the room and notice what they are doing. Are they correctly setting up equations using the information in the problem? Instead of using equations, are they attempting to find the solution by organizing the information in a chart? Is there yet another solution strategy that they are trying?

In closing, discuss with students the summary questions on percent concentration (14‑17).

Assessment Options 

  1. Observe student work. Are students correctly reading the antifreeze chart? Are they correctly calculating percent concentrations?
  2. Listen to student discussion. Are they relating the numerical analysis to the word problem?
  3. Are students correctly using variables and formulas to set up equations?

Extensions 

  1. You can throw a "Mixtures Party" for your class. At a local grocery store or a wholesale club, buy several quarts of juice concentrate. Make up two batches of juice: one that is very weak (maybe a 5-10% solution), and another that is very strong (perhaps an 80-100% solution). Then, based on the package directions, tell students what percent concentration is needed; for instance, if the package says that one part concentrate should be mixed with three parts water, then a 1/(1 + 3) = 25% concentration is needed. Allow students to solve the problem individually and then discuss the solution as a class. When the class agrees on a solution, you can let the students mix appropriate amounts together and drink the resulting beverage. If they did not solve the problem correctly, they'll know when they try the result!

Questions for Students 

1. How do you determine percent concentration of antifreeze in a cooling system?

[Divide the amount of antifreeze by the total amount of fluid in the system, then multiply by 100.]

2. How do you change the percent concentration of a solution?

[Add some amount of a solution that has a different percent concentration.]

Reflection 

  • Are students of different learning styles engaged in the lesson?
 
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Algebra

Making Sense of Percent Concentrations: Mix It Up

9-12
In this lesson, students will use two colors of beads to form two different percent mixes, and then form a final mix using scoops from the two mixes. From this tactile start, students will develop and use a formula to determine the final percent mix from two source mixtures.

Learning Objectives

Students will be able to:

  • Read an antifreeze chart to determine mix of antifreeze and water for engine protection at various temperatures
  • Achieve a desired percent concentration of fluid by calculating the amount of fluid to drain and replace with antifreeze
  • Solve related word problems

NCTM Standards and Expectations

  • Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts.
  • Draw reasonable conclusions about a situation being modeled.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP3
    Construct viable arguments and critique the reasoning of others.
  • CCSS.Math.Practice.MP4
    Model with mathematics.