Problem Solving: Percents

Transcription

1 Problem Solving: Percents LAUNCH (7 MIN) Before Why do the friends need to know if they have enough money? During What should you use as the whole when you find the tip? After How can you find the total bill, including tax and tip, in fewer steps? Do you think the friends have enough money for the bill? PART 1 (8 MIN) Why can t you subtract the ticket cost from the total to find the service fee? Does it matter whether you divide the total amount by 2 and work with one ticket, or double the ticket cost and work with both tickets? After solving the problem Why do you think you are charged a service fee for the tickets? How might you avoid paying service fees? PART 2 (8 MIN) Without referring to specific numbers, describe what you need to find. While solving the equation Why is finding the result of a 20% increase the same as finding 120% of the original number? After viewing the animated solution How do the percent diagrams help you understand the solution? PART 3 (7 MIN) Javier Says (Screen 1) discuss the importance of looking at sales closely when comparing prices for different things or for similar things at different stores. How can you begin solving this problem? What else do you need to know to find the total cost of Board 3? How is Board 1 related to multiplying fractions? For Board 2, what are two ways to find 40% off of a selling price? CLOSE AND CHECK (8 MIN) Describe the plans, predictions, or decisions that someone might make based on the problems involving percent in this lesson. In the real world, you may need to estimate percents. When does it make sense to underestimate percents? When does it make sense to overestimate percents?

2 Problem Solving: Percents LESSON OBJECTIVE 1. Use proportional relationships to solve multi-step ratio and percent problems involving percent increase and decrease. 2. Use proportional relationships to solve multi-step ratio and percent problems involving taxes and gratuities (tips). 3. Use proportional relationships to solve multi-step ratio and percent problems involving markups and markdowns. FOCUS QUESTION How do percents help you compare, predict, and make decisions? MATH BACKGROUND This lesson concludes the topic by presenting students with real-world situations, all of which can be solved using percents. Since each problem in this lesson involves multiple steps, encourage students to plan before solving by figuring out which strategy, method, or formula applies to each situation. Guide them to draw upon personal experiences, make diagrams, and use simpler numbers or compatible numbers to understand the problem. Students might use estimation or guess and check in addition to solving a proportion or equation. The central theme of the problems in this lesson is decision making. For example, students solve the Launch by deciding whether two friends have enough money to pay for a meal, including tax and tip. This problem invites a discussion about the percent food is taxed, what an acceptable percent for tip is, and how the price of a meal can be significantly greater than the cost of food. Part 3 is also a complex problem that involves making a decision as a consumer. Examining all three different forms of store sales involves knowing how to convert fractions, decimals, and percents comfortably. A firm understanding of the percent equation is also helpful to solving this problem, as is recognizing and distinguishing the meanings of the many terms associated with markups and markdowns. In this lesson, however, the problem that is most likely to cause students to think critically about the applications of percent is Part 2, which involves both percent increase and decrease. This problem demonstrates that increasing an original quantity by a certain percent and then decreasing it by the same percent does not yield the original quantity. It instead results in a quantity less than the original quantity. Here is an opportunity for students to investigate why this misconception is incorrect and also why the order in which you apply the two changes does not matter. In future topics, students will work with percents greater than 100, continue to relate percents to decimals, and apply percents to probability, including simple and compound events. LAUNCH (7 MIN) Objective: Use percents to solve a real-world problem involving tax and tips. Students find a percent of a whole and calculate a total. They compare the total to a quantity of money to decide whether two friends have enough money. This problem emphasizes what quantity you use to find tip in a real-world situation that students will often face.

3 Questions for Understanding Problem Solving: Percents continued Before Why do the friends need to know if they have enough money? [Sample answer: If they do not have enough, they will not leave a reasonable tip for the wait staff.] During What should you use as the whole when you find the tip? [You can use either the subtotal or the total of the check to calculate the tip.] After How can you find the total bill, including tax and tip, in fewer steps? [Sample answer: The tax on the meal is 10%, so the tax and tip combined is 30%. Instead of multiplying the subtotal by 30% and adding the two numbers, you can multiply the subtotal by 1.3 and find the total bill directly.] Do you think the friends have enough money for the bill? [Sample answer: Yes; they should be calculating tip based on the subtotal.] Students may find the tip using either the subtotal or the total as the whole. Discuss which is the proper way to calculate the tip and how you can find the tax and tip at the same time (by adding the percents). Students may recognize that increasing the bill by 20% involves multiplying the cost of food by 0.2 and adding that quantity to the cost of food. Using the Distributive Property, you can show a faster way to find the new total. total bill cost of food tax tip total bill cost of food 0.1(cost of food) 0.2(cost of food) total bill 1.3(cost of food) Connect Your Learning Move to the Connect Your Learning screen. Use the Launch to discuss how a good grasp of percents can help you make smart purchasing decisions, and avoid situations like not having enough cash for a tip. Encourage students to think about the Focus Question as they talk about the role estimating with percents plays in everyday shopping decisions. PART 1 (9 MIN) Objective: Use proportional relationships to analyze percent problems involving percent increase and decrease. This problem introduces students to another real-world application of percents: the processing (service) fees charged by ticket sellers. Students find the additional charges, both as a quantity and as a percent, and see how it impacts the price that consumers pay for products or services. Questions for Understanding Why can t you subtract the ticket cost from the total to find the service fee? [Sample answer: The ticket cost is for each ticket, but the total is for two tickets.]

4 Problem Solving: Percents continued Does it matter whether you divide the total amount by 2 and work with one ticket, or double the ticket cost and work with both tickets? [Sample answer: No; if you divide the total amount by 2, you will get the service fee per ticket. If you multiply the ticket cost by 2, you will need to divide the service fee by 2 at the end to find the service fee per ticket.] After solving the problem Why do you think you are charged a service fee for the tickets? [Sample answer: The company needs to pay to maintain the website and ticketing software.] How might you avoid paying service fees? [Sample answers: You can purchase tickets before a certain date. You can buy tickets at the ticket booth in person instead of online. You can purchase a group of tickets at once.] You can use the color in the provided solution to help students substitute values into the percent equation. Error Prevention Students may divide the fee by the cost of two tickets or divide the total fee by the cost of one ticket. Highlight key terms in the problem (such as per ticket ) to help students correct their error. Got It Notes If you show answer choices, consider the following possible student errors: If students are dividing the total cost, including the fees by the price of one ticket, they may choose A. Students who select C are likely dividing the cost of three tickets by the total cost (0.93) and writing the number incorrectly as a percent. Students who choose D found the fee but did not express it as a percent rate. PART 2 (8 MIN) Objective: Use proportional relationships to solve multi-step percent problems involving sales. Students solve a problem involving both percent increase and decrease. This problem fights a misconception that percent increase and decrease are inverse operations. Working through this problem can help students understand how the original quantity in the percent of change formula can change within a problem. Instructional Design You can have students predict the answer to this problem. Take a vote on whether the answer will be greater than, less than, or equal to the original quantity. Have a student record the results on the whiteboard. You could also use the Fractions and Percents tool and create your own corresponding percent diagrams, similar to the ones provided in the animated solution. Questions for Understanding Without referring to specific numbers, describe what you need to find. [Sample answer: I need to find out whether increasing a number by a certain percent and then decreasing it by the same percent will result in the original number.]

5 Problem Solving: Percents continued Why is finding the result of a 20% increase the same as finding 120% of the original number? [Sample answer: You are adding 20% of the number to 100% of the number, which is 120% of the number.] After viewing the animated solution How do the percent diagrams help you understand the solution? [Sample answer: The diagrams show that the second part of the problem has a greater whole than the first part. Although you are finding 20% in both parts, you are finding 20% of different numbers.] Use the animated solution to demonstrate the relative size of the quantities involved in the two steps of the solution. This solution may help students visualize how the whole has changed even though the percent is the same. Some students may know that 20% 1 5 and quickly see that 1 of 100 is 20. They can 5 use mental math to find the new quantity, 120. This time, however, they should realize that 1 5 of 120 will not be 20. Differentiated Instruction For struggling students: Students may benefit from seeing what happens when you increase a number by 100% and then decrease the new number by that same percent. They may recall that increasing a number by 100% is the same as doubling the number, but decreasing a number by 100% is reducing the quantity to zero. For advanced students: Ask students which process will result in a greater value: a 10% increase followed by a 10% decrease (like the Example), or a 10% decrease followed by a 10% increase. Suggest they experiment with an original quantity of 100. They should realize that the order does not matter. Got It Notes Students should recognize that 100% of any number is the number itself and that increasing it by 100% is the same as doubling it. Ask students to explain why decreasing a number by 50% means halving the number. Help students understand that doubling a number and halving a number are inverse operations. Therefore, increasing a number by 100% and then decreasing the result by 50% gives the original number. Invite students to repeat this process with another starting number. Got It 2 Notes This problem presents another way that businesses may define percent markup to make sure that the original quantity is the same when calculating both percent markup and markdown. Point out that neither of these ratios is expressed as a percent because this formula for percent markup is not true percent of change. Stores are more interested in comparing markup and markdown to track pricing and profits than using the correct formula for percent of change.

6 Problem Solving: Percents continued PART 3 (7 MIN) Objective: Use proportional relationships to solve multi-step percent problems involving fees. This problem presents three different ways that stores commonly advertise sales and special deals. Students organize a great deal of information and apply what they have learned about markdowns to solve a series of multi-step problems. They also use the percent equation in order to calculate and add tax to each item. Instructional Design Although students learned about sales tax while using the percent equation, you can show that sales tax is an example of a percent increase. Questions for Understanding Javier Says (Screen 1) Use the Javier Says button to discuss the importance of looking at sales closely when comparing prices for different things or for similar things at different stores. How can you begin solving this problem? [Sample answer: You can work on each board separately. Start by finding the total cost for Board 1. First find the sale price and then add tax to find the total cost.] What else do you need to know to find the total cost of Board 3? [Sample answer: You need to consider two situations: both if you trade your board and if you do not.] How is Board 1 related to multiplying fractions? [Sample answer: You need to find a part of a whole. You multiply the selling price by 1 4.] For Board 2, what are two ways to find 40% off of a selling price? [Sample answer: You can find 40% of 450 and subtract that number from 450. You can find 60% of 450, which is the remaining part of the selling price left.] Because there are three parts to this problem, and there are two methods to solve each part, you should consider showing the provided solution to organize the work for this problem and save time. Method 2 in the provided solution is new to this topic and challenges students to consider how percent of change relates to percent of a number. For example, 1 4 off a price is equal to 3 of that price. Ask them which method they prefer. Some may 4 explain that they prefer Method 1 because it uses the percent of change formula that they have used many times in this topic. Others may prefer Method 2 because it has fewer steps. Error Prevention Remind students that the sales tax is calculated using the sale price, not the selling price. You must find the markdown before you add sales tax.

7 Got It Notes Problem Solving: Percents continued Encourage students to use estimation to solve this problem. For Leash A, they should round the selling price to $15 and can use mental math to find the sale price of $12. Since the sales tax will be less than 1 dollar, this eliminates choices B and D. If students select C, they may have forgotten to add tax. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer Percents are a common language for comparing proportional relationships. They are a standard way to communicate rates like taxes, tips, commissions, fees, interest, and sale amounts so that you can compare your options and plan your finances in the real world. Focus Question Notes Good student answers should reference ways percents are used to compare quantities in this lesson: choosing a surfboard, deciding if you can afford a board leash, and determining a ticket fee that is expressed as a percent rate. Essential Question Connection This lesson directly relates to the Essential Question because percents are used to compare numbers and make decisions. Use the questions below to help students understand the role percents play in real-world situations. Describe the plans, predictions, or decisions that someone might make based on the problems involving percent in this lesson. [Sample answers: A surfer may decide which board to buy, whether to sell his old board, and whether to buy a board leash. The surfer may then plan how to market his old board in order to get the best price possible. A consumer may decide whether to buy tickets through a certain agency based on the percent rate of the fee they charge. The ticket buyer may plan to save money for the tickets.] In the real world, you may need to estimate percents. When does it make sense to underestimate percents? When does it make sense to overestimate percents? [Sample answer: It makes more sense to overestimate the tip or tax on a dinner so you can be certain you have enough money to pay the total bill. It makes sense to underestimate how much interest you'll have earned in your bank account if you want to make sure you'll have enough money for a special item by a certain date.]