Percent and Proportional Relationships

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Unit 3 – Lesson Plan – Percent and Proportional Relationships

Week Four        Math (7th Grade)

**Written by Cheryl Faye Jackson, Georgia State University

Grade: 7             Subject: Mathematics                   Lesson Titles:  Proportional Reasoning.

Stage 1 – Desired Results
Established Goal(s): GPSMCC7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Enduring Understanding(s): Students will … apply proportional reasoning to solve multistep ratio and percent problems, e.g., simple interest, tax, markups, markdowns, gratuities, commissions, fees, percent increase and decrease, and percent error. recognize situations in which percentage proportional relationships apply. convert between fraction, decimal and percent.	Essential Questions: How are proportions used to solve problems? What are some reasons that prices increase over time? What are some reasons that prices decrease over time? What does it mean when a price changes by more than 100%?
Students will know… Key Terms · equivalent ratios: Ratios that represent the same fractional number, value, or measure. · Percent: a ratio that is expressed per 100.  The symbol % means per 100 or divided by 100.  100% represents 1 whole in context. · unit rate: A comparison of two measurements in which the second term has a value of 1.	Students will … Solve problems involving percent using computation for percents. Calculate percent change. Develop meaning of percents greater than 100 by putting them in real life context. Estimate and compute using percent.
Stage 2 – Assessment Evidence
Performance Tasks: Match Them Up (MCC7.RP.3)Materials: One set of cards per group Order Matters (MCC7.RP.3) Materials: Order matters cards Order matters tray Pencil Paper Now and Then  (MCC7.RP.3) Materials: Now and Then Activity Sheet Whiteboards Erasable markers Erasers Calculators Toy Cars (optional) Blank Now and Then Activity Sheet Internet access Metric ruler	Other Evidence: · Assessment – Quiz (Friday) · Math folder, notes, examples, classwork, and homework. · Class discussions. · End of Unit Project: Mathdonalds, Chez Mathématiques, and Was 7×3, Is 10 + 11, Always Will Be 30 – 9.
Stage 3 – Learning Plan – Monday
Warm-up: Solve the proportions to find the missing value.  Round to the nearest hundredth if necessary.1. 5/100 = x/12                2. 15/100 = x/9Guided Practice: Percent Increase or DecreaseSolutions to precent problems can be found by using the same method as for solving proportions. Look for equivalent ratios and remember your understandings of decimals. You may encounter percents above 100.To solve problems involving percent increase or percent decrease requires that you understand that the original amount is always 100% or the whole amount before the discount or increase has been applied. Consider the difference in these two percent decrease and percent increase problems. Misconceptions: Students struggle to estimate solutions to problems involving percentages greater than 100% or less than 1%. The mistake students might make is changing the percents to decimals. For example, students might think that 1055% is the same as 1.055. If I notice some students committing this error, I will stop them and review how to convert percents to decimals. When writing percent of increase or decrease proportions, students misplace the variable because they do not understand the variable’s relationship to either the whole (100%) or the percent of increase or decrease.   When finding the percent increase or decrease in a problem, first you identify the original number and then the change. The original number is identified by the value that takes place first in the problem. Since the value may or may not be mentioned first in the problem, you will need to read the problem carefully to be able to identify the values. The change is the difference between the earlier value and the current value in the problem. The change can be solved by subtracting the two numbers. To find the percent increase or decrease, divide the change by the original value. Change Original Example 1: The population of Collier Heights was 3,200 one year and increased to 3,520 the next year. What is the percent increase of the population of Collier Heights? Explanation: Find the difference in the two numbers. This equals the change in population 3,520 – 3,200 = 320 Divide the difference by the original population. 320 ÷ 3,200 = 0.10 The decimal 0.10 can be changed to a percent. 0.10 = 10% The population increased by 10%.  Example 2: Mr. St. Cyr had 25 students in his 4th period math class. The next day, he only had 20 students. What is the percent decrease in the number of students in Mr. St. Cyr’s math class? Explanation: Find the difference in the two numbers. This equals the change in the number of students. 25 – 20 = 5 Divide the difference by the original number of students. 5 ÷ 25 = 0.20 The decimal 0.20 can be changed to a percent. 0.20 = 20% The number of students in Mr. St. Cyr’s math class decreased by 20%. Independent Practice: Harper-Archer Middle School’s baseball team, the Jaguars, won 20 games last year. This year they won 22 games. What is the percent increase of games from this year to last year? Tony bought a printer on sale for $45.00. The original price was $60.00. What is the percent decrease in price on the printer? If a loaf of bread was priced $1.50 on one day and $1.20 on the next, what is the percent decrease of the price of the loaf of bread? The Toys of Life store had 240 customers on October 24th . They had 264 customers on October 25th . What was the percent increase in customers from October 24th to October 25th ? Dr. Baler’s veterinary practice is booming. Last year he had 1,500 appointments. This year he had 2,250 appointments and had to hire more help. What is the percent increase in appointments from this year to last year? Lisa’s little brother weighed 8 pounds when he was born. He now weighs 36 pounds. What is the percent increase in weight of the little boy? John used to weigh 270 pounds and now weighs 216 pounds. What is the percent decrease in weight in John? Rita Rose had 120 buttons her button collection. She now has 150 buttons. What is the percent increase in the number of buttons in Rita Rose’s collection? Mathew had $50.00 in his special hiding place. He decided to spend $15.00 on his mom’s birthday present. What percent decrease in money will this be for Mathew?
End of Unit Project – To Work on Monday through Friday. Completed by FridayYou are going to participate in this task where you will use percent. The three parts are: Mathdonalds, Chez Mathématiques, and ‘Was 7×3, Is 10 + 11, Always Will Be 30 – 9’.” (Note to teacher: Chez Mathématiques means “House of Math” and ‘Was 7×3, Is 10 + 11, Always Will Be 30 – 9’ is a math-y way of saying the popular store Forever 21. MathDonalds is a fast-food restaurant where you will be ordering food from a menu and figuring tax on the items you ordered. Chez Mathématiques is a fancy restaurant where you will be ordering food, figuring tax and a 15% tip. At the clothing store, you will be estimating your total purchase by estimating tax and sale prices.” When you are at the restaurants, you will be allowed to use a calculator to figure tax and tips and attend to precision. When you are at the store, you will be using your estimation skills because that is what shoppers usually do. They estimate the prices before they arrive at the checkout. MATHDonalds  You may use a calculator.                                  You and your group do not need to choose the same items, but may work together to help each other. Choose 2 items from the menu: _____________________ and ______________________ Find the total cost of the two items.  _______________________ Find the tax for the total cost of the two items. Show work. _________________________  Find the total for the whole bill (items + tax).  __________________________ Chez Mathématiques You may use a calculator.                                          You and your group do not need to choose the same items, but may work together to help each other. Choose 2 items from the menu: ______________________ and _____________________ Find the total cost of the two items: _______________________  Find the total for the whole bill. (1. Add the tax and tip percentage together.  2. Find that percent of total.  3. Add to item total.)  Show all work: __________________________ Was 7 x 3, Is 10 + 11, Always Will Be 30-9 Calculator may not be used. a. For Item A, estimate 10% sales tax: __________________ b. Find the estimated total for Item A with tax (item + 8a): ________________ a. For Item B, estimate 9% sales tax: _______________ b. Find the estimated total for Item B with tax (item + 9a): _______________ a. For Item C, estimate the price with the 30% off sale (rounded item – 30%): ____________ b. Estimate a 10% sales tax on the sale price (10% of 10a): _______________ c. Estimate the total (10a + 10b) ______________ a. For Item D, estimate the price with the 25% off sale (rounded item – 25%): ____________ b. Estimate a 10% sales tax on the sale price (10% of 11a): _______________ c. Estimate the total (11a + 11b): ______________(Kacie Travis, Middle Level Resources, 2013).
Closing: Farmer Jake harvested 2,500 pounds of carrots this fall. He sold them out of the back of his truck and brought home only 250 pounds after a full day of selling. What is the percent decrease in the number of carrots Farmer Jake has left? Each student will be given this as an exit ticket they must complete before leaving the classroom.
Homework: MondayWrite a math notebook entry about what it means to have a percent less than 1 or a percent greater than 100. When does it not make sense to have a percent less than 1 or greater than 100? Write down at least three things you think are the most important when solving problems involving percents.Bring in newspaper articles that contain percentages. Check your work to make sure that the article does not contain any mathematical errors.Continue to work on End of Unit Project
Stage 3 – Learning Plan – Tuesday
Warm-up: Estimate.1. Find 5% of $14.50.2. Find 20% of %10.19.Guided Practice:800 People came to Harper-Archer Middle School’s football game. Sixty-five percent of the people came to cheer for the Jaguars home team. How many people came to cheer for the Jaguar’s home team?Explanation of Warm-up: Change 65% to a decimal.  65% = 0.65 Multiply the percent by the number of people who attended the game. 800 × 0.65 = 520 520 people came to cheer the Jaguar’s home team. Task: Now and Then – Rules The game task Now and Then is based on The Price is Right game show. The contestants are given a year, an object, and a price. The contestant must then guess whether the object’s price represents the price now or the price from the given year. For this version of the game the students will use calculations involving percents. This task reviews computational procedures for percents and percent change while putting percents less than 1 and greater than 100 in a real‑life context. The Now and Then activity sheet contains researched values from a specified date. On the activity sheet the percent increase is included for the practice problem but not for any of the others. Students will copy the information from the PowerPoint. Students will fill in the blanks as I proceed to prevent them from moving ahead of the rest of the class. Students will work in pairs or they may work alone. Student pairs will check each other’s work and reason through their final answer by sharing ideas even if working individually. One representative for each group will collect the necessary materials: Model: I will read the practice problem to the students and have them complete the practice problem on their activity sheets. Practice Problem: The average price of chewing gum in 2004 was $0.25. What is the price of chewing gum now? Students will work in pairs to find the price of gum when it is increased by 20% and then write it in the blank box provided. They will also calculate the price of gum when it is increased by 35% and then write it in the box. Students will show their work and thought processes on their activity sheet. When students have both prices, they will debate and decide which value truly represents the current price of gum. When they agree upon a price, they will write their choice on the whiteboard and keep it turned over until every group has reached a decision. Before giving the answer, go over the solutions so the empty boxes. Model: I will lead the students through the solution and ask questions for understanding. The students will think about what they know, whether the price would be bigger or smaller, and what calculations they should use. I will Show students the solution. $0.25 increased by 20% → $0.25 × .2 = $0.05, so the increased price would be $0.25 + $0.05 = $0.30 I will repeat the same process for the other percent increase. $0.25 increased by 35% → $0.25 × .35 = $0.0875 = $0.09 then $0.25 + $0.09 = $0.34 I will ask, What happens when you have a value of money that has more than two decimal places? Students will decide and write on their whiteboards if the price of gum is now $0.30 or $0.34. The groups will hold up the answer they put on their whiteboard. I will reveal the correct answer. I will tell the students that during the game, I will record points for each correct answer. If some students ask what the winner gets I will tell them what the prize will be a new car (a toy car from the materials). I will check for the students’ understanding by asking them if there are any questions. I will address all misconceptions and make it clear that I am officially starting the game. I will reveal the first question and each question one at a time using the Now and Then PowerPoint. Students will need to copy information from the PowerPoint onto their activity sheets. Independent Practice: Students will work in groups as contestants on a fictitious game show, Now and Then. They should use their knowledge of percent computations and percent change to answer each game show question. Question 1 follows the same process as the practice problem. Students should correctly change the percentage to a decimal before multiplying. Since this percentage is greater than 100, they will be multiplying the price by a number that includes a whole number part of 1. Students will be chosen randomly to show their solutions in order to create greater individual accountability. Each group will hold up its choice for which is the real answer. I will reveal the correct answer from the PowerPoint, then tally up the team scores. Question 2 will be shown. Students will fill in the blanks on their activity sheet, and then identify how it is different than Question 1 I will give them a brief review of percent change then I will repeat the process from Question 1 until I have worked through all 6 problems. Question 3 is similar to Question 1. Question 4 can be solved by using a proportion or an algebraic equation. I will explain to students that they already know what the price is at the end, but they do not know what the price was to begin with. I will check for their understanding by asking them how can we represent something that we don’t know? The price now is 14% of the original price, so 0.14x represents the current cost of the laptop. Then solve 0.14x = 249 and 0.14x = 430. Differentiation: For students who have not had experience with solving one-step equations, they can use the proportions 249/x = 14/100 and 430/x = 14/100. Once the class has completed the six questions, I will tally the results and present the winning team with their new car.
Closing:Explain in your own words how to correctly change a percentage to a decimal before multiplying when the percent is greater than 100. Each student will be given this as an exit ticket they must complete before leaving the classroom.
Homework: The students will create their own Now and Then game show and host it in front of the rest of the class. Regroup students into groups of three. In one class period, the groups will create two of their own questions. The Leader will retrieve Blank Now and Then overheads for the group. The Researcher will find information on prices in particular years using the Internet or other reference materials. Students will find prices from the year they were born. The Question Creator selects the questions and writes them on the overhead. Each student has her or his own specific title, but each group member helps the group with each part of the preparation. On Wednesday students will host a student-run Now and Then show. The Researcher will present her or his group’s question to the class, and the Leader will present the solution. Continue working on MathDonald’s Project
Stage 3 – Learning Plan – Wednesday
Warm-up: Explanation of Warm-up:Guided Practice:Some sales people are paid on commission. This means that they are paid on a percent of the total sales they make. Waiters and Waitresses are usually tipped a percentage of the total price of the meal and drinks their customers purchased.Example 1: Mrs. Hanson made 2% commission on a house she sold. The house sold for $150,000. How much was Mrs. Hanson’s commission?Explanation: Change the percent of the commission to a decimal. Multiply the percent commission by the total sale. $150.000 × 0.02 = $3,000 Mrs. Hanson earned $3,000 commission on the sale of the house. Independent Practice: Miss Sally waited on a large party at the Chesterfield Restaurant. The party’s bill totaled $110.00. As the party was pleased with the service they received, they tipped Miss Sally 20%. What amount did Miss Sally receive as her tip? Ed makes 6% commission on every case of widgets he sells. This week, Ed sold 16 cases for $800.00 each. How much did Ed make on commissions this week? Maria makes $6.00 an hour plus 5% commission for telephone sales. Yesterday, Maria worked 8 hours and had sales of $600.00. How much did Maria make yesterday? Alonzo earns 8% commission on vacuum cleaner sales. This past month, Alonzo sold 70 units at $150.00 each. How much did Alonzo make in commissions for the past month? Rick was given a tip of 15% waiting on a table of customers who ordered $60.00 in food and drinks. How much did Rick earn in tips waiting on tis table? Acme Auto Parts gives all of their sales people an hourly wage and 3% commission on all the sales they ring up. Daniel rang up $3,200.00 last week. How much did Daniel earn in commissions for the week? Tanner and Brad split the 6% commission they made selling a house for $120,000. How much did each of them receive? Bridget’s mom offered to pay her $5.00 to clean out the refrigerator and freezer, and if she got it done before dinner time, 6:00 p.m., she would receive a 20% bonus. Not one to pass up an opportunity, Bridget cleaned the refrigerator and freezer by 5 p.m., an hour to spare! What is the total amount Bridget received from her mom? A salesperson set a goal to earn $2,000 in May. He receives a base salary of $500 as well as a 10% commission for all sales. How much merchandise will he have to sell to meet his goal?
Closing: After eating at a restaurant, your bill before tax is $52.50.  The sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip you leave for the waiter? How much will the total bill be, including tax and tip? Express your solution as a multiple of the bill.Solution: The amount paid = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x $52.50 Each student will be given this as an exit ticket they must complete before leaving the classroom.
Homework: Wednesday Giving 110% is a common expression so, have students explain why it’s impossible. Then find other common, but mathematically impossible, expressions. Explain how to determine which stores are the most popular or which stores sell the most merchandise. Continue working on MathDonald’s Project
Stage 3 – Learning Plan – Thursday and Friday
Warm-up Thursday: Wayne has spent 40% of his allowance on paper for his computer printer. If Wayne spent exactly $3.60 including tax on the paper, how much is his allowance, and how much does he have left to spend?Explanation of Warm-up: Turn the percentage from the problem into a decimal. 40% = 0.4 Divide the dollar amount by the percent. $3.60 ÷ 0.4 = $9 Wayne’s allowance is $9.00 and he has $9.00 – $3.60 = $5.40 left over. Warm-up Friday: A business is growing and needs to increase their building size. Their current space is 13,500 square feet. When the increase is done they want to have 18,000 square feet. Find the percent of increase in the amount of floor space. Explanation of Warm-up:                               New – Original        %      Original        = 100  18,000 – 13,500      x        10,500        = 100                 4,500             x       13,5000       = 100             33.3%  = x   For the next two days you are going to apply your recent understanding of proportional reasoning to real-world situations with percents.  Percents are used every day in the form of tax on goods, tipping at restaurants, salesman commission, goods on sale, and more. Example:  Skateboard problem 1. After a 20% discount, the price of a SuperSick skateboard is $140. What was the price before the discount? Explanation: In this problem, the beginning whole price is not known.  We know that the beginning price was the 100% price.  Because there is a 20% discount, we only paid 100% – 20% or 80% of the original price. We have our equations  . Example: Skateboard problem 2. A SuperSick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the increase? In this case, we know the original price or 100% is $140.  This price will increase, which means we will be paying more.  In fact, we will pay 100% + 20% more or 120% of the original price for the skateboard.  These equations are . Explanation: The distributive property is indirectly involved in working with percent decrease and increase. In the first problem, if x is the original price of the skateboard (in dollars), then after the 20% discount, the new price is 80%x or 0.80x.   Using the distributive property: x (100% – 20%) = 100%x – 20%x = 80%x or 0.80x.   In the second problem, the new price is 120%x.  Using the distributive property: x(100% + 20%) = 100%x +20%x = 120%x  or 1.2x.  Guided Practice: Gas prices are projected to increase 124% by April 2015. A gallon of gas currently costs $4.17. What is the projected cost of a gallon of gas for April 2015? (7RP3) Solution: The answer is that the original cost of a gallon of gas is $4.17. An increase of 100% means that the cost will double. I will also need to add another 24% to figure out the final projected cost of a gallon of gas. Since 25% of $4.17 is about $1.04, the projected cost of a gallon of gas should be around $9.40. $4.17 + 4.17 + (0.24 · 4.17) = 2.24 x 4.17 100%    100%    24%                                                                                                        $4.17    $4.17      ?       2. A sweater is marked down 33%. Its original price was $37.50. What is the price of the sweater before sales tax? (7RP3) Solution: The discount is 33% times 37.50. The sale price of the sweater is the original price minus the discount or 67% of the original price of the sweater, or Sale Price = 0.67 x Original Price.                     $37.50 Original Price of Sweater                                          33% of $37.50 Discount     67% of $37.50  Sale Price of Sweater 3. A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was the amount of the discount? (7RP3) Solution:  Discount 40% of original price       Sale Price $12 60% of original price Original Price (p) Independent Practice: An item is discounted 30% and then reduced another 20%. Use an example to demonstrate if the resulting discount is equivalent to a discount of 50%? Does taking a 6% discount on an item, and then adding 6% sales tax result in the original price of an item? Support your answer with an example. At a discount furniture store, Chris offered a salesperson $600 for a couch and a chair. The offer includes the 8% sales tax. If the salesperson accepts the offer, what would be the price of the furniture, to the nearest dollar, before tax? (7RP3)           A $552           B $556           C $592 4. Appliances at Discount City Store are on sale for 70% of the original price. Eli has a coupon for an 18% discount on the sale price. If the original price of a        microwave oven is $500, how much will Eli pay for the oven before tax? (7RP3)           A $440           B $287           C $260           D $240 5. While on vacation, a group can rent bicycles and scooters by the week. They get a reduced rental rate if they rent 5 bicycles for every 2 scooters rented. The reduced rate per bicycle is $15.50 per week and the reduced rate per scooter is $160 per week. The sales tax on each rental is 12%.The group has $1600 available to spend on bicycle and scooter rentals. What is the greatest number of bicycles and the greatest number of scooters the group can rent if the ratio of bicycles to scooters is 5:2?
Closing: ThursdayExplain in words how you estimate the tax on a sale of $1,500 when the tax rate is 5.8%. Each student will be given this as an exit ticket they must complete before leaving the classroom.
Homework: Thursday                                    Write an essay about Tax, Tip, and Sale percentages. Include: An introduction A summary An explanation of why this is important. A conclusion how you will use this learning in the future.
Independent Practice: FridayFind each total.  Show all work.  Round to the nearest hundredths place.1. $256 with a 5.5% sales tax                           2. $21.99 with a 8% sales tax$270.08                                                           $23.75Find the total payment, given the cost, tax, and tip rate. $42.75, 6% tax, 15% tip $51.73 4.   You buy a game for $39.99 in a store that has a city tax of 0.5% and a state tax of 7.25%.  How much will you pay for the book?      $43.09 Estimate a 15% tip for each amount. $9.89 for lunch                                                                       6. $22 for a haircut ≈$1.50                                                                         ≈$3.30 Find the cost of each service, given the percent and the amount of the tip. 7.  20% tip, $19.70                                                                    8. 15% tip, $20.04             $98.50                                                                         $133.60 You go to a restaurant with three other people.  The total for the food you all ordered is $58.94.  There is a 7% food tax.  You want to give a 15% tip.  You decide to share the bill equally.  Find how much you will each pay. (Ask yourself, “How many of us went to dinner together?”)      $17.98
Closing: Friday Write several percent problems in which the solution is 35%.Turn in the Mathdonalds, Chez Mathématiques, and Was 7×3, Is 10 + 11, Always Will Be 30 – 9 Project.