Before There were more positive responses for Kit 2. Why doesn t the drummer just pick Kit 2?

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1 The Percent Equation LAUNCH (7 MIN) Before There were more positive responses for Kit 2. Why doesn t the drummer just pick Kit 2? During How else can you compare two ratios besides using fractions and a common denominator? After Compare the ratios for the kits. Is one preferred much more than the other? How does the number of total reviews affect your decision? KEY CONCEPT (5 MIN) In each example, ask students to show how each word in the sentence translates to form the percent equation. Note that the whole in every problem is 100%. There are three basic types of percent problems that students can use the equation to solve. Each involves knowing two of the three parts and finding the third. PART 1 (6 MIN) Before solving the problem How many parts are there in any percent equation? Javier Says (Screen 1) Use the Javier Says button to emphasize the idea that when solving real-world problems involving percents, students will need to readily recognize both the word form and the mathematical form of the equation, and transition between the two. Some statements are sentences, and others are equations. Which form do you find easier to identify the parts of the percent equation? While solving the problem Which statement is not in the same order as the percent equation in the Key Concept (y m x)? PART 2 (7 MIN) Before solving the problem How are these three statements different? Which percent problem is easiest to estimate? While solving the problem For the first problem, how can you solve the equation? For the second problem, how can you show 4% in the percent equation? For the third problem, how can you use the simplest form of 40/100 to make solving easier? PART 3 (7 MIN) Javier Says (Screen 1) Use the Javier Says button to emphasize the importance of understanding the language of percents to make sure that you protect yourself. Use the language of the percent equation. What do you know, and what are trying to find by solving this problem? After solving the problem The raise amount is short 5 cents, which seems like very little money. How would that affect the amount of money he makes over an entire year (about 2,000 hours)? CLOSE AND CHECK (8 MIN) How might the hairdresser use information about his 8% raise to plan, predict, or make a decision?

2 The Percent Equation LESSON OBJECTIVES 1. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships. 2. Represent proportional relationships by equations. FOCUS QUESTION How do percents and the percent equation help describe things in the real world? MATH BACKGROUND Previously, students used proportional reasoning to analyze relationships and to solve real-world problems. They studied ratios and equivalent ratios and worked with the concept of the constant of proportionality. They used unit rates to solve problems involving total cost and number of items purchased. Students are familiar with solving one-variable equations and writing ratios in the form of percents. In this lesson students use percents to represent parts of wholes in order to describe relationships, solve problems, and make comparisons. The term percent is derived from Latin, meaning per 100. Students will use their knowledge of percents as the ratio of a given number to 100. Every percent can be expressed as a fraction or as a decimal, and vice versa. This lesson also introduces students to the percent equation: part % whole, which has the form y mx. In this equation, m, the percent, is the constant of proportionality. The Key Concept uses a percent model to provide an especially helpful visual demonstration of how to find each piece of the percent equation given the other two. Students will learn to solve percent equations, and in the process they will recognize three basic percent problems: find a part of a number, find what percent one number is of another, and find a whole given the part and percent. They will use the constant of proportionality in the equation, and they will identify and write the percent equation when it appears in word problems. Many of the problems in this lesson can make use of the Fraction and Percents tool in Proportionality & Percents mode. Make sure you are familiar with the tool to visually illustrate, solve, and check the percent problems in the lesson. In the lessons that follow, students will apply their understanding of percents and the percent equation to common everyday kinds of problems, particularly those involving taxes, tips, and commission. LAUNCH (7 MIN) Objective: Use ratios to compare quantities. Author Intent Students compare two real-world situations involving a part and a whole. They describe the situations using ratios and make a decision based on their comparison. Questions for Understanding Before Why would the drummer make a decision based on buyers reviews? [Sample answer: The drummer wants to buy a kit that the users of the website appear to prefer. You can make a decision based on how frequently buyers are happy with their purchase of each kit.]

3 The Percent Equation continued There were more positive responses for Kit 2. Why doesn t the drummer just pick Kit 2? [Sample answer: There were also more reviewers overall for Kit 2.] During How can you compare results from two different sets of reviewers? [Sample answer: You can rewrite each ratio as a fraction and use equivalent fractions to write both ratios using the same denominator. The ratio with the greater numerator has a larger portion of positive responses.] How else can you compare two ratios besides using fractions and finding a common denominator? [Sample answers: You can write each ratio as a decimal or percent, and then compare those ratios.] After Compare the ratios for the kits. Is one preferred much more than the other? How does the number of total reviews affect your decision? [Sample answer: No; the ratios are too close to decide which kit is preferred. The drummer is also using too small of a sample size to make a good decision.] What other factors might the drummer consider when deciding on a drum kit? [Sample answers: price, colors, brand, friends opinions, number of drums in each kit] Solution Notes Websites often provide user reviews that tell how many people approve of an item or service. The Companion solution to this problem purposely doesn t include percents. Instead, it uses the concept of equivalent ratios to compare the reviews of the two drum sets. Some students may appreciate that there are less cumbersome ways to make the comparison. You can use students solution methods as an opportunity to discuss the usefulness of percents in this situation. Invite students who converted the ratios to decimals or percents to show and explain their computations. Have students talk about which method using equivalent ratios or using percents is more practical for helping the drummer make his choice. Invite any students who used mental math to solve the problem to demonstrate their methods. Show that using mental math is challenging in this problem because the ratios are very close in value. Connect Your Learning Move to the Connect Your Learning screen. Students should remember that percents are another form for ratios. Don t let students become overwhelmed by the term percent equation. For now, just look for them to connect percents with purchasing decisions. As needed, suggest the key part that percents play in the decisions coaches, salespersons, politicians, and people in marketing or finance routinely and carefully make. Use the Focus Question to help students think of other situations in which ratios are used, such as a commercial in which 4 out of 5 dentists prefer a toothpaste. Ask students to try to use percents to describe their situations. KEY CONCEPT (5 MIN) Teaching Tips for the Key Concept Because this is the first time percents are mentioned in Grade 7, you may want to review what percents are before going into this topic. One way is to click on the hyperlink to the glossary and read the definition. You can call on students to click on each radio button and connect the percent equation to the equation that describes a proportional relationship. In each example,

4 The Percent Equation continued ask students to show how each word in the sentence translates to form the percent equation. Note that the whole in every problem is 100%. You can use the Fraction and Percents tool in Proportionality & Percents mode to demonstrate the three examples shown in this Key Concept. Stress that each situation relates to ratios they learned in Grade 6 as well as proportional relationships they worked with in the previous topic. There are three basic types of percent problems that students can use the equation to solve. Each involves knowing two of the three parts and finding the third. Remind students that they worked with this equation in the topic Proportional Relationships and that x is the independent variable, y is the dependent variable, and m is the constant of proportionality. PART 1 (6 MIN) Objective: Identify parts of a proportional relationship represented by the percent equation. Author Intent Students identify whether the unknown value in each statement is the part, the percent, or the whole. Stress that when solving percent problems involving the percent equation, they will always be finding one of those three parts. This activity prepares students for solving the percent equation and writing their own to solve real-world problems. Questions for Understanding Before solving the problem How many parts are there in any percent equation? [three] Javier Says (Screen 1) Use the Javier Says button to emphasize the idea that when solving real-world problems involving percents, students will need to readily recognize both the word form and the mathematical form of the equation, and transition between the two. Some statements are sentences, and others are equations. Which form do you find easier to identify the parts of the percent equation? [Sample answer: The mathematical statements are easier because you can figure out which piece of the equation matches up with each of the variables y, m, and x. It is easier to understand the word form if you translate it to an equation.] While solving the problem Which statement is not in the same order as the percent equation in the Key Concept (y m x)? [part (f)] Solution Notes You can call on students to label the portions of each statement. Use three different colors to distinguish the part, percent, and whole. Some students may find it easier to identify the answer by first identifying the two known portions. For example, in part (c), they can identify 126 as the whole and 79% as the percent. Therefore, What must be the part. Students may have an easier time identifying the answer to part (f) if they rearrange the statement as 12 is 4% of what number? As an extension, consider using the Fractions and Percents tool to find the solution to one or more of the equations. Students will formally solve the percent equation in Part 2.

5 Differentiated Instruction The Percent Equation continued For struggling students: Ask students to describe the part, the percent, and the whole in their own words. Remind students that percents can be written as decimals or as fractions. Encourage students to convert any fractions or decimals to percents (parts (b) and (e) only). If students are having trouble, encourage them to draw a model of the problem, following the model in the Key Concept. For advanced students: Get students to find 10 of their own models around the classroom that show a part of a whole and ask them to estimate the percent. For instance, a student might estimate that 18% of the area of the whiteboard is filled with markings. Got It Notes If you show answer choices, consider the following possible student errors: Students who choose A are mistaking the part for the whole. They may be thinking that a large number, such as 190, must represent the whole. You can show them that the solution to this equation is a much greater whole, If students select B, point out that one of the numbers provided has a percent symbol. PART 2 (7 MIN) Objective: Solve the percent equation. Author Intent Now that students are familiar with both the word form and mathematical form of the percent equation, they translate the word form into the mathematical form and then solve the percent equation. Students solve one equation for each piece of the percent equation: part, percent, and whole. Questions for Understanding Before solving the problem How are these three statements different? [Sample answer: Each statement can be modeled with the percent equation; however, one asks for the part, one asks for the percent, and one asks for the whole.] Which percent problem is easiest to estimate? [Sample answer: It is easiest to estimate a percent of the whole, like 40% of 60. Since it s less than half, the answer should be slightly less than 30.] While solving the problem For the first problem, how can you solve the equation? [You can isolate the variable by dividing each side by 50.] For the second problem, how can you show 4% in the percent equation? [You can rewrite 4% as a fraction ( 4 ) or a decimal (0.04).] 100 For the third problem, how can you use the simplest form of to make solving the equation easier? [Sample answer: If you rewrite in simplest form as 2 5, you can solve the equation using smaller numbers, which generally reduces the potential for computational errors.] Solution Notes There are two opportunities to rewrite a percent in another form, so consider showing both a fraction and a decimal. Students may want to reduce the fraction immediately or wait until the end to simplify the answer.

6 Error Prevention The Percent Equation continued Encourage students to check their answers for reasonableness. For instance, in the second problem, they should know that 4% is a small part of the whole. They can check that their solution is a much greater number than 25. Encourage students to use number sense whenever possible it may avoid having to solve a percent equation. For example, in the first problem, some students may reason as follows: 11 is 11% of 100, and 50 is half of 100, so 11 must be twice the percentage of 50 as it was for 100, or 22%. Be careful that students do not incorrectly reason that halving the whole would also halve the percent. Another example for the third problem is that 10% of 60 is 6, so 40% of 60 is 4 6, or 24. Got It Notes Unlike the Example, this problem asks students to translate each arithmetic sentence to the percent equation and then determine whether the statement is true. Statement IV is the only one that is not already in the form originally presented to students in the Key Concept. Watch for students who have trouble translating sentences into equations. PART 3 (7 MIN) Objective: Use the percent equation to solve a problem. ELL Support On the Student Companion page for the Part 3 Got It, there are two tasks for students to complete and discuss: What are gift certificates and how do they work? What are tips and how do they work? Beginning and Intermediate Have students share their descriptions of gift certificates with other student pairs and listen for the accuracy of their descriptions. Review as a large group, and make a list of key words and phrases that students used in their descriptions. Then have them complete the second task. Compare their descriptions of tips and review them as a group similar to what was done for gift certificates. Students may not be as familiar with tips, so allow time for students to share their descriptions with other students before sharing as a large group. Advanced Have students include in their descriptions the impact that gift certificates and tips have on the price you pay for something. Author Intent Students solve a real-world problem involving percents and decide whether information in the problem is correct. It is an introduction to the kinds of problems they, as consumers and workers, will face in everyday life. Questions for Understanding Javier Says (Screen 1) Use the Javier Says button to emphasize the importance of understanding the language of percents to make sure that you protect yourself from errors others might make. You may wish to discuss with students how frequently percent applications arise in everyday life, particularly in money matters such as interest rates for loans or mortgages, raises, and shopping. Use the language of the percent equation. What do you know, and what are trying to find by solving this problem? [You know the whole, $7.50, and the percent, 8. You need to find the part and see whether it should be $.55.]

7 The Percent Equation continued After solving the problem The raise amount is short 5 cents, which seems like very little money. How would that affect the amount of money he makes over an entire year (about 2,000 hours)? [The error would cost the hairdresser $100 over a year.] Solution Notes This problem has an animated solution that uses colors to distinguish the part, percent, and whole. Consider returning to the Key Concept to ask students which of the three situations this problem represents. Consider having a volunteer demonstrate how to solve this problem by first rewriting the percent as a decimal rather than as a fraction. Let students decide which method they prefer. Some students may want to find whether $.55 is indeed 8% of $7.50. Although they will correctly determine that it is less than 8%, this method does not say what the correct amount should be. Differentiated Instruction For struggling students: Help students better understand what information is given and what they need to find using a Know-Need-Plan organizer. Clarify that the current wage, $7.50, and the percent of increase, 8% raise, are correct information. The letter gives the raise amount of $.55, which students need to verify. If students are having trouble, encourage them to draw a model of the problem, following the model in the Key Concept. For advanced students: Ask students to calculate the percent of time they spent using technology on the previous day. First, they must write down the times of day they used technology, then add up the number of hours, and finally calculate what percent of the day they used technology. Have students share their percent with a partner and tell the partner to find the amount of time (the part). Got It Notes Encourage students who are able to solve this problem using number sense. Since 20% 1/5, then 3 is 1 of the haircut. Therefore, the price of the haircut is 5 times as 5 much as $3, or $15. Compare this method to the provided solution to show that in both cases you divide the part by the percent to find the whole. (Note that multiplying by 5 is equivalent to dividing by 1, the percent written as a fraction.) 5 If you show answer choices, consider the following possible student errors: Students who choose B are finding the total of the haircut and tip. If students are using the given value 20% as the whole, they may need help writing the percent equation from a real-world problem. If students select D, they are probably multiplying the two quantities and might need to review solving an equation or setting up the percent equation. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer Percents are a common language that helps to communicate and compare information. The percent equation helps to describe proportional relationships in the real world. You can use the percent equation to find or verify numbers in the real world.

8 Focus Question Notes The Percent Equation continued Listen for students to connect percent and proportional relationships: a percent is a number divided by 100; it is a ratio. Two equivalent ratios have a proportional relationship. You can always use the percent equation to find a part, a whole, or a percent, as long as you know the other two pieces of information. Essential Question Connection Percents are convenient when discussing money. In this lesson, students saw percents in problems involving a hairdresser's raise and leaving a tip. Use the following question to connect to the Essential Question about percents and making plans, predictions, and decisions. How might the hairdresser use information about his 8% raise to plan, predict, or make a decision? [Sample answer: He may decide whether he can afford something in the coming year, or predict what his salary might be in two years.]