Unit 5: Percents. Learning Objectives 5.2

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1 Unit 5 Table of Contents Unit 5: Percents Learning Objectives 5.2 Instructor Notes The Mathematics of Percents Teaching Tips: Challenges and Approaches Additional Resources Instructor Overview Tutor Simulation: What s the Best Price? Instructor Overview Puzzle: Transit Trouble Instructor Overview Project: Is a Percent a Percent? Common Core Standards 5.21 Some rights reserved. See our complete Terms of Use. Monterey Institute for Technology and Education (MITE) 2012 To see these and all other available Instructor Resources, visit the NROC Network. 5.1

2 Unit 5 Learning Objectives Unit 5: Percents Lesson 1: Introduction to Percents Topic 1: Convert Percents, Decimals, and Fractions Learning Objectives Describe the meaning of percent. Represent a number as a decimal, percent, and fraction. Lesson 2: Solving Percent Problems Topic 1: Solving Percent Problems Learning Objectives Identify the amount, the base, and the percent in a percent problem. Find the unknown in a percent problem. 5.2

3 Unit 5 Instructor Notes Unit 5: Percents The Mathematics of Percents Instructor Notes The word "percent" will be familiar to students they'll all have seen sales offering 20% off, or heard that only 5% of the class received an A on the final exam. But it's very possible that they will arrive at Unit 5 without knowing exactly what these numbers mean or how to calculate them. This unit will teach students just what percents are, and how they're related to fractions and decimals. They'll see how to convert between those three different, but equivalent, forms of a number and learn techniques for solving percent problems. Real life application problems, such as calculating sales tax and tips, will show students the importance of understanding percents. Teaching Tips: Challenges and Approaches Although students are familiar with the idea of percents, they may be uncertain of the mechanics. Many have already been stumped by % off tags in a store, or shocked by how much sales tax added to the cost of a purchase. We suggest starting with a firm grounding in the meaning of percent, using visual models and making a clear connection to fractions and decimals, numbers they've already learned to be comfortable with. Once students know what percents are and how to work with them, you can introduce the "percent/base/amount" technique for writing and solving percent equations. Percents, Fractions, and Decimals When introducing the concept of percents, explain that per cent means per 100 or out of 100. This will help them understand that a percent is actually a ratio of a number to 100 and is just another way of expressing part of a whole. The meaning of percents and their comparison to fractions and decimals will be much clearer with the use of illustrations that divide a physical something into 100 parts. For example, the sketch below of a plate of spaghetti shows how a single serving can be broken into portions and described by a percentage, a decimal, and a fraction of the whole: 5.3

4 [From Lesson 1 Topic 1, Presentation] Visuals like this will help students grasp the meaning of percent. Once students understand what a percentage is, they'll have an easier time expressing one in fractional and decimal form, and vice versa. There are just a few common errors to warn them about. Conversion between percents and decimals is simply a matter of moving the decimal point two places. But students should be encouraged to check their answer to make certain that their answer makes sense and that they've moved the decimal point in the proper direction. Remind them that sometimes zeros need to be added in order to move the decimal point correctly. Let students know that converting a fraction to a percent is easier if they convert the fraction to a decimal first. Then the decimal point just has to be moved two places to the right to get the equivalent percent. Explain that after converting a percent to a fraction, they need to treat it like every other fractional answer and simplify it if possible. The process of converting between decimals, fractions, and percents is more likely to stick if students are given lots of practice with basic mechanical problems like this one: 5.4

5 [From Lesson 1 Topic 1, Worked Example 3] Once students understand the conversion process, the concept of percents greater than 100% should be explained and illustrated. Solving Percent Problems Solving percentage problems can be difficult for students. To help them succeed, start by making sure that they can identify the three parts of a percent problem the percent, the base, and the amount. Provide students with a lot of practice identifying these parts as well as putting them into the basic equation "Percent Base = Amount," as seen in the example below: 5.5

6 [From Lesson 2 Topic 1, Presentation] Once they get that down, students will need to learn how to solve the equations they've written, like 20% n = 30. They aren't used to seeing % signs in equations, and it may not occur to them to convert percents into fractions or decimals. Try comparing this problem to a similar but more familiar equation, like 2 n = 30 (or two times what number is 30 ) to help them figure out that they can divide the amount by the percent to find the base. Some students may be more comfortable solving percent problems another way. The example above relied on the formula Percent Base = Amount. This unit also discusses a variation on that formula: Percent = amount base can then be written as cross multiplying.. Show students how to rewrite an equation like 20% n = 30 as 30 20% n. This That makes it a familiar proportion problem that can be solved by 100 n It is very easy in any of these percent problems to have the decimal point in the wrong spot. Once again encourage students to see if their answers make sense, perhaps by comparing their answer to a more obvious percent like 10% or 50%. For example, in the case of 20% n = 30, they might calculate that n is 150. Is that reasonable? Well, figure out what 10% of 150 is: it's 15, which means that twice that, 20% of 150, is indeed 30. But if a student had found n to be 15, they'd quickly see that 10% of 15 is 1.5, so 20% of 15 can t possibly be 30. Keep in Mind As always, problems that are relevant to students' own experiences make learning about math easier and more appealing. A good application problem to discuss with students is figuring out a tip. It is customary to give a 15-20% tip for good service at a restaurant. Checking the bill for the sales tax rate, usually given as a percent, can be a shortcut for figuring a tip. Most state sales taxes are in the neighborhood of 7-10%. If the sales tax is 8% and amounts to $3.65, then a 16% tip would be double the $3.65, or $7.30. Even though students typically have a good understanding of percents, there are still several common mistakes that are made. For example, when representing 6% as a decimal, it should correctly be written as.06 not.6. Another common mistake that is made is when there are two discounts on a given item. For example, if a store takes 10% off the price of an item, then takes another 20% off this new price, there is not a 30% total discount. Another common misconception is that students will think that it is better to take the larger discount first when in fact it doesn t matter. Students may remember learning in elementary school the is over of method for finding unknowns in a percent problem. You should discourage the use of this method and promote 5.6

7 equations or proportions to solve for the unknown. While the is over of method also relies on proportions, students tend to forget how to set this problem up correctly. Additional Resources A good website to review percents is After a brief review, an applet tests knowledge on many different percent topics, including converting between fractions and percents, as well as decimals and percents. Also included are practical uses of percents, such as commissions, sales taxes, and estimating tips. Knowing the percent, decimal and fractional equivalents for a given number is a very useful skill. This can be practiced at and ame.shtml. Additional practice for finding a percent of a number is available at Summary After completing this unit, students will understand what percents are and how to calculate them. They'll also be able to write any number as an equivalent fraction, decimal, or percent. 5.7

8 Unit 5: Percents Instructor Overview Tutor Simulation: What s the Best Price? Unit 5 Tutor Simulation Purpose This simulation allows students to demonstrate their understanding of percents. Students will be asked to apply what they have learned to solve a problem involving: Percentages Percentages in Decimal Form Percentages in Fraction Form Applying Percentages to Real World Problems Problem Students are presented with the following problem: Rohan, a drummer in the school's jazz band, wants to buy a new drum kit. He's found the kit he wants at four different stores, but each store sells the drums at a different price with a different discount. He needs your help to figure out which store has the best deal. Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation, Make sure they have completed all other unit material. Explain the mechanics of tutor simulations. o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. Emphasize that this is an exploration, not an exam. 5.8

9 Unit 5: Percents Objectives Instructor Overview Puzzle: Transit Trouble Unit 5 Puzzle Transit Trouble is a game that tests a player's ability to calculate and apply fractions and percents. To be successful, students will have to quickly transition between numbers in written form and their visual equivalents. Figure 1. Players use their knowledge of percentages and fractions to put the right number of passengers on each bus. Description 5.9

10 Each puzzle in the game begins with a crowd of people waiting to board two buses. Both buses are labeled with the portion of the crowd they can hold. Players are given a limited amount of time to drag the correct number of passengers to each bus. If they answer correctly within the allotted time, they earn points. There are three levels of play. In Level 1, the desired distribution of passengers is given as a fraction of the crowd size. In Level 2, one bus's capacity is given in fractional terms and the other as a percentage. In Level 3, both buses are labeled with the percentage of passengers they can hold. The problems in Transit Trouble are simple and better suited to individual than group play. They could also be explored in a classroom setting where teacher-directed questions and answers can expand on the work. 5.10

11 Unit 5: Percents Student Instructions Instructor Overview Project: Is a Percent a Percent? Unit 5 Project Introduction In today s world, you are often confronted with advertisements for percent-off sales. In this project, you will explore various coupons offered by a retailer and determine how to save the most money on your purchases. Be prepared to become an expert coupon shopper! Task Working together with your group, you will explore the best way to purchase items from a sporting goods store. Based on what you have learned, you will go on a shopping spree to purchase the most goods with a limited amount of money. Finally, you will analyze your shopping to help the storeowner figure out the best way to price his or her products. Instructions Solve each problem in order and save your work as you progress as you will create a professional presentation at the conclusion of the project. 1. First problem: You will be making several purchases at a sporting goods store. You can use an advertisement flyer from a store, make a trip to a local sporting store, or shop online. You will need to have prices for at least six products with at least one or two higher priced items in the range of between $200 - $500. Record the relevant information in the chart below. Item Price per Item 5.11

12 You were mailed a coupon for 20% off your purchases. Before you leave home, you want to make sure that you know the discounted price. Select two items from your price list and calculate the discounted price. [Hint: 20% off is the same as paying for 80% of cost of the merchandise.] You go to the local store to purchase your equipment and notice a sign at the front of the store stating that you can receive an additional 10% off all store items. When you checkout, you present your 20% off coupon and the store clerk states that both discounts can be used together. Select the same two items as before and calculate the new discounted price. For the two items, use the new discounted price to calculate the actual percentage off that you received. [Hint: First calculate the difference between the original price and the discounted price and then determine the percentage off. The formula would be: 2. Second problem: When you get home, you begin to wonder if it makes a difference which coupon is used first. You decide to calculate the cost if the order was reversed and the 10% coupon was used first and then the 20% off coupon. Decide if the order makes a difference and calculate the actual percentage off that you received. Be able to defend your idea. You may want to try calculations with several items. 5.12

13 You decide to purchase a big-ticket item (one that costs between $200 and $500), so you search the Internet and find multiple discount coupons (see table below). Some of these coupons can be used in combination with others. Consider the different coupons and their requirements listed in the table below. List the possible combinations of coupons that may be used. Coupon Requirements $25 off Can only be used on items costing more than $200. Can be used with ONE other coupon. 10% off Can be used with other coupons. 20% off Can be used with other coupons. 25% off Cannot be used with any other coupons. Determine the lowest price you can obtain on your big-ticket item. [Hint: When using the $25 off coupon, the order may make a difference.] 3. Third Problem: With all of your purchases at the sporting goods store, you have won a contest that allows you to go on a spending spree. You will have $1000 to spend at the store and will be able to use each coupon only once. Using your price list from before and the available coupons determine how to spend the $1000. Be sure that you stay below $1000 with the discounts. You can purchase more than one of each item. Try to get the most for your money. Determine the total value of the items you purchased and calculate the actual percent off that you received. 4. Fourth problem: Finally, your group will use your coupon knowledge to make a presentation to the owner of the store to determine how best to manage coupon sales. Choose one bigticket item on your price list to help you to calculate, for any coupon that applies, how prices could be marked up to offset the reduction in prices. Fill in the Marked-Up Price column in the table below. Coupon Marked-Up Price Percent of Original Price $25 off 10% off 20% off 25% off 5.13

14 [Hint: For example, if you consider the 20% off coupon for an item that costs $ You would then calculate (.8)x = $ Then divide both sides by.8, which results in $ The storeowner should price the item at $374.99, so the discount takes the price back down to $ ] Next, determine the actual percent mark-up for each of the coupons. Fill in the Percent of Original Price column in the table above. [Hint: For example, when the $ item was marked up to $374.99, which results in ( ) = 1.25 or 125% of the price or a 25% mark-up. The formula would be: Finally, using your list of possible coupon combinations from Question # 2, determine marked-up price and percent of original price for your big-ticket item. (You have already made the calculations for the single coupons above.) Make a table for the storeowner, so he or she can understand the different combinations and how prices should be marked up. Collaboration Get together with another group to compare your answers to each of the four problems. Discuss how your group decided to spend the $1000 in the spending spree. Exchange your list of items with the other group and see if you can come up with a purchase plan that saves them more money. Discuss your plans to help the storeowner. Some groups may have chosen to eliminate using certain coupons or not allowing coupons to be combined on purchases. If you owned a store, how could you promote discounts and still make money? What is a reasonable discount? Does the overall price matter? What is the difference between dollar-amount coupons and percent-off coupons? Is sales tax computed on the original or discounted price? Does it change how the discounts should be applied? Do some Internet research to determine how much your state charges in sales tax. Include the sales tax in your presentation to the storeowner. Conclusions Your presentation to the owner of the store should demonstrate how coupons influence pricing. It should be on a poster board highlighting the differences in coupons. It should include all of the 5.14

15 mathematics used to solve the four problems above. You may either neatly write out the tables or use software such as Microsoft Word to create a professional computer generated product. You may want to include pictures of the actual items to make it colorful. Instructor Notes Assignment Procedures Problem 1 An example of a price list is included below. PLEASE NOTE THAT ALL ANSWERS WE CALCULATE ARE BASED ON THIS DATA. If students use different data, their answers will of course vary from ours, but the overall results should be comparable. Item Description Price per Item In-line Skates $59.97 Tennis Racket $ Golf Bag $ Air Hockey Table $ Electronic Dartboard $69.99 Pogo Stick $

16 Badminton Kit $69.99 Croquet Set $79.99 Trampoline $ In the case of a 20% off coupon, if the pogo stick and croquet set were selected, the total price would be $ The discounted price would be (0.80) = $ In the case of an additional 10% off, if the pogo stick and croquet set were selected, the new discounted price would be (.90)91.98 = $ The actual percent off would be found by ($ $82.78)/ =.28, so the actual percent off is 28%. Problem 2 The order of the discount does not matter since it is a multiplication problem and multiplication is commutative. If the price is x, the discounted price would be found by calculating, x (.90)(.80) or x(.80)(.90), which are mathematically equivalent. Both result in a 28% discount. The possible combinations are listed in the table below. Students at this point may not realize that the order matters when using the $25 off coupon with the others. Single Coupons Possible Combinations $25 off $25 off and 10% off 10% off $25 off and 20% off 20% off 10% off and $25 off 25% off 20% off and $25 off 10% off and 20% off 5.16

17 Students should try different combinations in order to figure out the lowest price. For example, the trampoline costs $299. The lowest discount can be found by using the 20% off coupon and then applying the $25 off coupon, resulting in a lowest price of $ Problem 3 There are numerous possibilities, which is why we have not asked them to find THE best option. Our spending spree looked like this: Purchase #1: Air hockey table and the pogo stick ($514.98). We used the 20% off coupon ($411.98) followed by the 10% off coupon, bringing the cost to $ Purchase #2: Golf bag, tennis racket, croquet set, and electronic dartboard ($459.96). We used the 25% off coupon to bring the cost to $ Purchase #3: Trampoline ($299.99). We used the $25 off coupon bringing the cost to $ In total, we spent $990.74, leaving $9.26. We have no additional items we can purchase with this amount. The total value of our items is $ We paid $ for them, for a savings of $ Our discount was 22.3%. Problem 4 For example, using the trampoline as our item, we have the following mark-ups: Coupon Marked-Up Price Percent of Original Price $25 off $ $25 = $ % off $ = $ % off $ = $ % off $ = $ For example, using the trampoline as our item, we have the following percents: Coupon Marked-Up Price Percent of Original Price $25 off $ $25 = $ $ $ = 1.08=108% 10% off $ = $ $ $ = 1.11 =111% 20% off $ = $ $ $ =

18 =125% 25% off $ = $ $ /$ = 1.33 =133% For example, using the trampoline as our item, we have the following percents: Coupon Marked-Up Price Percent of Original Price $25 off and 10% off $ =$ % $25 off and 20% off $ =$ % 10% off and $25 off ($ ) 0.9 = $ % 20% off and $25 off ($ ) 0.8 = $ % 10% off and 20% off $ (0.8 x 0.9)= $ % Perhaps to avoid high mark-ups, we would suggest that the percent off coupons not be used in combination. Recommendations Have students work in teams to encourage brainstorming and cooperative learning. Assign a specific timeline for completion of the project that includes milestone dates. Provide students feedback as they complete each milestone. Ensure that each member of student groups has a specific job. Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students instructions list several websites that provide information on numbering systems, game design, and graphics. The following are other examples of free Internet resources that can be used to support this project: An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. 5.18

19 or Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work. Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose. Rubric Score Content Presentation/Communication 4 The solution shows a deep understanding of the problem including the ability to identify the appropriate mathematical concepts and the information necessary for its solution. The solution completely addresses all mathematical components presented in the task. The solution puts to use the underlying mathematical concepts upon which the task is designed and applies procedures accurately to correctly solve the problem and verify the results. Mathematically relevant observations and/or connections are made. There is a clear, effective explanation detailing how the problem is solved. All of the steps are included so that the reader does not need to infer how and why decisions were made. Mathematical representation is actively used as a means of communicating ideas related to the solution of the problem. There is precise and appropriate use of mathematical terminology and notation. Your project is professional looking with graphics and effective use of color. 5.19

20 3 2 1 Developmental Math An Open Program The solution shows that the student has a broad understanding of the problem and the major concepts necessary for its solution. The solution addresses all of the mathematical components presented in the task. The student uses a strategy that includes mathematical procedures and some mathematical reasoning that leads to a solution of the problem. Most parts of the project are correct with only minor mathematical errors. The solution is not complete indicating that parts of the problem are not understood. The solution addresses some, but not all of the mathematical components presented in the task. The student uses a strategy that is partially useful, and demonstrates some evidence of mathematical reasoning. Some parts of the project may be correct, but major errors are noted and the student could not completely carry out mathematical procedures. There is no solution, or the solution has no relationship to the task. No evidence of a strategy, procedure, or mathematical reasoning and/or uses a strategy that does not help solve the problem. The solution addresses none of the mathematical components presented in the task. There were so many errors in mathematical procedures that the problem could not be solved. There is a clear explanation. There is appropriate use of accurate mathematical representation. There is effective use of mathematical terminology and notation. Your project is neat with graphics and effective use of color. Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics. There is some use of appropriate mathematical representation. There is some use of mathematical terminology and notation appropriate to the problem. Your project contains low quality graphics and colors that do not add interest to the project. There is no explanation of the solution, the explanation cannot be understood or it is unrelated to the problem. There is no use or inappropriate use of mathematical representations (e.g. figures, diagrams, graphs, tables, etc.). There is no use, or mostly inappropriate use, of mathematical terminology and notation. Your project is missing graphics and uses little to no color. 5.20

21 Unit 5 Correlation to Common Core Standards Unit 5: Percents Common Core Standards Unit 5, Lesson 1, Topic 1: Convert Percents, Decimals, and Fractions Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and critique the reasoning of others. Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and critique the reasoning of others. Unit 5, Lesson 2, Topic 1: Solving Percent Problems Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP.8. Mathematical Practices CATEGORY / CLUSTER MP.8.3. Construct viable arguments and critique the reasoning of others. Grade: Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and critique the reasoning of others. 5.21