Ohio Standards Connection Number, Number Sense and Operations Benchmark I Use a variety of strategies including proportional reasoning to estimate, compute, solve and explain solutions to problems involving integers, fractions, decimals and percents. Indicator 14 Use proportional reasoning, ratios and percents to represent problem situations and determine the reasonableness of solutions. Indicator 15 Determine the percent of a number and solve related problems; e.g., find the percent of markdown if the original price was $140, and the sale price is $100. Benchmark D Use models and pictures to relate concepts of ratio, proportion and percent. Mathematical Processes Benchmark A Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods. Lesson Summary: In this multiple day lesson, students encounter the use of percentages in the real world. The lesson begins with finding sales tax and discounts on purchases. Using 10 by 10 grids, students represent percent to develop conceptual understanding of the real world applications and relate fractional parts to the percentages. The model helps students determine and understand the use of an algorithm for finding percent of a given number. Students use reasoning skills to estimate and show understanding. Estimated Duration: Six to seven hours Commentary: Students should use a variety of methods to study percents and their relationship to fractions and decimals. Using models to determine percent of a given number help students understand percent as a ratio and develop proportional reasoning. By finding the percent of a number for real world situations, students see the relevance of the mathematics studied. A reviewer stated that using shaded models would give students a visual concept so that they see what a percent really is. Pre-Assessment: This pre-assessment reveals students prior knowledge of percent. Students describe percent, explain the reasonableness of given statements and identify the decimal form of given percents. Distribute Percent Pre-Assessment, Attachment A to each student. Have students complete the worksheet individually or with a partner Select students to share their responses with the class. Scoring Guidelines: Percent Pre-Assessment Answers, Attachment B is provided as a reference. Students ready for instruction provide explanations for percent such as out of 100, shaded models, numbers or words. They may relate percents to equivalent fractional form or provide specific examples such as; if there are 10 pencils, five of them would be red because 50 percent is the same as a half. They also provide accurate decimal equivalencies. Provide intervention to students based on results of their work. 1
Post-Assessment: Informally assess students understanding of percents throughout the lesson by observing the strategies used, listening to responses given for problem situations and reading journal entries. Summative assessment options may include; Percent of a Given Number Post-Assessment, Attachment C Project about percents which includes: 1. Identifies a percent such as 25 percent. 2. Explains what 25 percent means or represents. 3. Shows how to find 25 percent of a given number using models such as 10 by 10 grids, words or numbers. 4. Creates and solves real world problem situations using 25%. 5. Presents orally and/or on a poster to the class. Scoring Guidelines: Use the rubric to determine levels of understanding. Meets Expectations Approaching Expectations Progressing Toward Expectation Intervention Needed Provides appropriate strategies to find percent of a given number in a variety of contexts including sales tax, discounts and a part of a whole. Strategies include grid models, fractional equivalencies and algorithms. Estimates using reasonable strategies and proportional thinking. Clearly communicates understanding of percent of a number in response to problem situations. Provides appropriate strategies to find percent of a given number in a variety of contexts including sales tax, discounts and/or a part of a whole. Strategies include grid models and algorithms. Estimates using reasonable strategies, however may not be as precise. For example, finds sales tax for $2.50 and provides answer such as between 14 cents and 21 cents or 15 cents because it is more that two dollars. Communicates understanding with some minor flaws in reasoning. Provides a strategy to find percent of a given number in limited contexts including sales tax, discounts and/or a part of a whole. Strategies include grid models and algorithms and may contain minor flaws. Estimation is limited. May rely on calculator to find exact answer, then estimates. Communicates understanding with minor flaws in reasoning. Provides inappropriate strategies to find percent of a given number. Strategies include incorrect grid models and algorithms. Estimation strategies show little or no reasoning. Communicates flawed understanding using mathematical vocabulary inappropriately. Instructional Procedures: Part One 1. Ask students if they have ever gone to a store with 10 dollars, tried to buy something which costs $9.99 and found that they did not have enough money. Ask why? (Sales tax) 2
2. Discuss sales tax. Ask questions such as; What is sales tax? (A tax which provides funds for government.) When is sales tax added? (For goods and services, when eating in a restaurant) When is sales tax not added to the purchase? (Food and some beverages at a store, when food is taken or carried out of the restaurant.) What is the sales tax in Ohio? (Six percent, 2005) Is there a local sales tax? (Depends on the local government.) Instructional Tip: Use the local sales tax to complete problem situations. Six percent and seven percent are used interchangeably throughout the lesson. If local and state sales tax falls between whole number percents, such as seven and one-quarter percent, explore estimation strategies once students show understanding of percents of a number. 3. Distribute 10 by 10 grids to students. Ask questions to clarify use of the visual representation. What can the grid represent when used for money? (one dollar) If the grid represents one dollar, what does each small square represent? (one cent) One cent is what percent of a dollar? (one percent) 4. Have students shade in six percent of one dollar and pose a problem situation such as; you purchase a small package of pencils for one dollar, how much will it cost with sales tax? ($1.06) Allow students to discuss the situation, then select students to share their responses. Make sure students understand that six percent of the dollar is added to the cost. 5. Pose the problem situation. Dalia had $2.15 in her purse. She wanted to buy a cool new notebook for $2.00. Does she have enough money to purchase the notebook after sales tax is included? a. Have students shade two 10 by 10 grids, each representing six percent of a dollar. b. Explain that for every one dollar, six cents is added. c. Ask, What is the sales tax for two dollars? d. Provide time for partners to determine the sales tax, the total cost and a response to the problem situation. Observe the methods students use, clarify misunderstandings. e. Select students to share their responses. Record strategies students use. Expect strategies such as multiplying two and six, adding $1.06 and $1.06, and adding $2.00 and $0.12. 6. Pose the problem situation. Harold found a CD on sale for $5.50. He had $6.00 in his wallet. Does he have enough to purchase the CD after sales tax of seven percent is included? a. Have partners discuss and determine a response for the problem situation. b. Observe the strategies students use. Strategies may include shading seven percent on five or six grids. Observe how students determine half of seven and whether to round to three or four cents. Also, students figure the tax on $6.00 to be $0.42 and determine he has enough or multiply five and seven and add half of seven. c. Select students to share their responses and record new strategies on chart paper or the board. 7. Present additional scenarios using store advertisements and products interesting to sixth grade students. Encourage students to use estimation strategies with prices that have 3
unfriendly cents such as $8.36. Include scenarios purchasing food or a situation not requiring tax. Have students create scenarios of items they would like to purchase and determine the sales tax. 8. Have students write a journal entry about finding sales tax. Have them use pictures, words and numbers to show their understanding. Collect and read through the entries, noting misconceptions and misuse of vocabulary in communication. Part Two 9. Ask students what it means to get a discount or when a store advertises a sale such as Everything is 25 percent off! a. Have them discuss with a partner and share examples where they have purchased something on sale or received a discount. b. Select students to share responses and experiences. 10. Pose the problem situation. Brady needed a new folder for school. He went to the dollar store and saw a sign that said Everything is 25 percent off! He finds a folder with his favorite football team marked $1.00 and decides to purchase it. What is the cost of the folder before tax is added? a. Distribute a sheet of 10 by 10 grids to each student. b. Determine how the grid should be used. (Each grid represents one dollar.) c. Have students shade 25 percent of the grid. Ask; What does the shaded area represent? (the discount) What does the non-shaded area represent? (the cost after the discount) How much is the discount? ($0.25) the folder? ($0.75) d. About how much sales tax (six percent) will be added to the cost? ($0.05) 11. Pose the problem situation. Deidra wanted to purchase a hair brush at same dollar store. The price of the hair brush is $2.00. How much will Deidra save on the hair brush? a. Have partners discuss and solve the problem situation. Observe strategies used such as; use two grids each shaded 25 percent add $0.25 and $0.25 or multiply $0.25 and two add $0.75 and $0.75, then subtract from $2.00 divide two dollars by four divide two dollars by four, then multiply by three, then subtract from $2.00 b. Select students using different strategies to share their responses and record the strategies on chart paper or the board. 12. Pose the problem situation. Kenny went to the Movie-Mania store with a $20 gift card. They are having a sale on DVDs and videotapes. DVDs are on sale for 30 percent off and the videotapes are 50 percent off. He wants to purchase one of each. The DVD he wants costs $12 and the videotape costs $16. How much will each item cost after the discount? What is the sales tax(seven percent) on the amount? Can he purchase both? (DVD costs $8.40. Videotape costs $8.00. The sales tax is about $1.15. The total is about $17.55. Yes, he has enough.) a. Have partners discuss and solve the problem situation. Observe the strategies used and assist as necessary. b. Select students to share their responses with the class. 4
13. Present additional problem situations or store advertisements to determine discount and new price of items. Have students create original problem situations and share with partners. 14. Assign a journal entry about finding discounts and sale prices. Have students use pictures, words and numbers to explain their understanding. Collect and read the entries to determine level of understanding and necessary intervention or re-teaching. Part Three 15. Pose problem situations for students to think about percents of a given number. Have students gather in groups of four to discuss the situation and write ideas generated. For example: Tyler made 75 percent of his free throws in a basketball game. There is a 50 percent chance of rain today. Sheena bought a pair of shoes which were on sale at 30 percent off. Jill bought a sweatshirt on sale at 25 percent off. Anna bought a sweatshirt on sale at 40 percent off. Jill paid less than Anna. 16. Ask students what each of the statements mean and ask questions such as; Did Tyler make 75 free throws? (Do not know, but probably not in one game) How do you know? (Too many for one game) Could have he made 75 free throws? How can 75 percent made be the same as making 75 free throws? (He attempted 100 free throws and made 75.) What does 50 percent chance of rain mean? (It may or may not rain; there is an equal chance of both happening.) Did Sheena pay 30 dollars for her shoes? How much did she pay? (Seventy percent of the cost. Can not give an exact cost, do not know the original price.) How is it possible that Jill paid less, when Anna received a larger discount? (The original cost of Jill s sweatshirt was less than the original cost of Anna s sweatshirt.) What does percent of a number mean? (It is a part of the number, something out of 100, the part of the number that is in the same proportion to the number as the percent is to 100.) 17. Pose the situation: The Principal of Oakdale Middle School surveyed the 20 members of the Mathematics Club and found the following results: 50% of the members were in fifth grade. 25% of the members received an A in mathematics on their last grade card. 10% of the students loved to do algebra problems. 75% of the students played a musical instrument. How many students does each of these percentages represent? 18. Allow students to discuss and solve the problem. Observe the methods students use and assist as needed. Ask questions to guide groups, such as: What percent represents all of the students in the club? (100%) Is 50% smaller or larger than 100%? (smaller) Do you think the number of members who are in fifth grade is smaller or larger than 20? (smaller) What is another way to represent 50%? (as a fraction, one-half or as a decimal, 0.5) 5
What is one-half of 20? (10) Is this a reasonable answer? Instructional Tip: Other strategies to observe may include converting the percent to a fraction equivalency, then figuring the fractional part. Dividing 20 by the denominator of the fraction equivalency may be used such as 20 divided by 10, the denominator for 10 1. Fifty percent may be used as a benchmark to find 25 and 75 percent. 19. After students complete the problems, select students to share the solution and strategies with the class. Clarify the responses and strategies students use. Record the different strategies on chart paper or the board. 20. Continue the scenario. The Principal at Pinedale Middle School found the same results for his 40 member Mathematics Club. What number of students represents each statement? Allow students to practice individually, asking for peer or teacher assistance when necessary. Select students to share their responses. 21. Pose the situation. Write the statements on the board or overhead projector. The Principal at Ferndale Middle School found some interesting facts about students he surveyed. About how many students does each of the survey statements represent? About 32% of the 19 students in the drama club take acting classes. About 18% of the 40 students in the Cooking Club went out to eat last night. About 54% of the 35 students in the Ski Club have broken legs. About 80% of the 49 students on the football team could not catch a ball. Guide students through the first survey statement. Use questions such as: a. The question says About how many students ; what does that infer? (estimating) It also says About 32% ; what does that infer? (rounding) b. How many students do you think take acting classes? Allow students to discuss this with a partner or in small groups, then share their response. If students have difficulty continue to investigate using questions such as; c. About what fraction of the students in the drama club are taking acting classes? (32% is about one-third) d. What multiple of three is closest to 19? (18) e. What is one-third of 18 or 18 divided by three? (6) f. Is this reasonable? Six is less than one-third of nineteen and 32% is a little less than onethird. 22. Explain to students that they can use multiplication and the decimal form of the percent to find the actual number. Model the situation. a. Find the decimal equivalency of 32 percent (0.32). b. Multiply 0.32 and 19. Choose either to complete the computation using a standard algorithm or calculator. The answer is 6.08. Discuss the meaning of this answer: does this mean there are 6.08 students taking acting classes? (No, it means that 6.08 is exactly 32% of 19. So 6 students is very close to 32% of 19.) c. Note that this agrees with our previous estimate. 6
23. Have students complete the remaining three situations, then provide more examples for practice. 24. Ask questions to reinforce that the percent means different things based on the whole. Which turned out to be greater: 32% of 19 or 18% of 40? If Gabriel gets 90% of the problems correct on a quiz and Dahren gets 70% of the questions correct on an exam, does this mean that Gabriel answered more questions correctly than Dahren did? (No, because the exam may have had more questions than the quiz.) Which is larger: 80% of my salary or 20% of Bill Gates salary? Part Four 25. Distribute two sheets of 10 by 10 grids to each small group of students. Write the following scenario on the overhead projector or board. Two hundred students at Elmdale Middle School were asked interest questions for a survey. The principal reported the following results: One percent of the students chose liver and onions as their favorite school lunch. Three percent of the students chose the Merry-Go-Round as their favorite ride at the amusement park. Twenty-five percent of the students enjoy camping in the jungle. Eighty-eight percent of the students own a polka music CD. 26. Direct students to use the grids to determine the number of students who chose liver and onions, chose the merry-go-round, enjoy camping in the woods and have a polka music CD. Discuss what the 10 by 10 grids represent. Have them shade the number of squares which represent the number of students. Observe and note the processes students discuss and use to solve the problem. Pose questions to students in need of assistance such as; How many grids represent the number of students in the survey? (Use two grids.) Why? How many squares would you shade to represent one percent? (A possible response may be one on each grid.) Why? (One percent means one out of a hundred, so one square out of 100 is shaded.) How many students chose liver and onions? (2) Why does one out of 100 or one percent accurately describe two out of 200? (They are ratios that show the same relationship or are proportional, or they are equivalent fractions.) 27. Select students to share their solutions and show the models of the grids. Ask questions to clarify the meaning of percent and relate it to ratios and proportions. What is being compared when percent is used? (Percent identifies a part to a whole.) What are these comparisons called? (ratios) Describe percent as a ratio. (Percent shows a part to a whole, and the ratio of the part to the whole is the same as the ratio of the percent to 100.) Instructional Tip: Students may suggest that they multiplied the number of squares in one grid by two. This is appropriate when using the models. If students try to use multiplication for finding the percent of the number, multiplying the percent and the number, question the reasonableness of the result. 7
For example, if students multiply 3 and 200 to solve for 3 percent of 200, ask if 600 makes sense when the whole is 200. 28. Pose the situation. It was reported by the principal at Mapledale Middle School that he found the same results as the Elmdale Middle School Survey, but Mapledale Middle School has 300 students. Ask students to determine how many students chose liver and onions, chose the merry-goround, enjoy camping in the jungle and have a polka music CD. Observe the methods students use and provide assistance as needed. Use questions from step 26. 29. Select students to share their solutions and reasoning. Ask students questions to develop understanding. Why are 10 by 10 grids more appropriate to use than five by five or other grids? (Each 10 x 10 grid has one hundred squares. It is like percent, a ratio comparing parts to 100.) 30. Pose a situation involving a third school with 250 students reporting the same survey results. a. Have groups determine the number of students representing one percent, three percent, 25 percent and 88 percent. Observe students and look for evidence that students use proportional reasoning when determining the percentage of 50. Evidence includes using half of the number out of one hundred, because 50 is half of 100. b. Ask questions related to proportional skills such as; Between what numbers will your answers fall? (Between the results for 200 and the results for 300. More precisely, where does 250 lie in relation to 200 and 300? (It is half way between 200 and 300.) Can you find the numbers half way between the 200 results and the 300 results? Use a similar example with friendly numbers such as 10 percent of 250. Make a visual model using two 10 by 10 grids and shade to represent 10 percent and ask for 1 0 percent of 200. What is the relationship of 50 to 100? (Fifty is half of 100 or two 50 s is 100.) What is half of 10? (five) What is the next step? (Add 10 + 10 + 5 to get 25. 10 percent of 250 is 25) c. To deal with the problem of non-whole numbers of people (e.g. one percent of 250 is 2.5 people), ask questions such as: Three percent of 200 was 6 and three percent of 300 was 9. So what do we think three percent of 250 might turn out to be? (6, 7, 8 or 9 probably 7 or 8). What number is exactly 3% of 250? (7.5) Does this mean that there are exactly 7.5 students whose favorite ride is the merry-go-round? (No, can t have half a person) So how do we interpret this? (Most likely number is 7 or 8) What is 7 out of 250 in decimal form? In percent form? (.028 = 2.8%) What is 8 out of 250 in decimal form? In percent form? (.032 = 3.2%) What are these percentages rounded to the nearest whole number? (3%) So either 7 or 8 is a possible answer to this problem. 31. Select students to share their solutions and strategies. Clarify the strategies using proportional reasoning by modeling each percent out of 100 and the percent out of 50. 32. Provide problem situations for practice and homework. Examples of situations include: 8
One hundred twenty-five people attended the chorus concert. Twenty percent of the people were grandparents. How many grandparents attended? (Twenty percent of 100 is 20 and one-fifth of 25 is 5. Twenty-five grandparents attended.) James collected 480 coins last month. One-third of the coins were nickels, 25 percent were dimes and 40 percent were quarters. What number of nickels, dimes and quarters did he collect? The class set a goal of reading 600 books during the school year. By the end of the second quarter they had read 288 books. About what percent of the goal had they reached? Was the class on track to reach the goal for the year? Jim s meal was 20 dollars. He knew he would leave a 10 percent tip for poor service, 15 percent tip for good service and 20 percent for excellent service. What are the amounts of the different tips? Three classes were collecting coins for a charity. Each class had a goal. The table shown shows how much money has been collected and the goal. Class Collected Goal Grade Five $210 $400 Grade Six $180 $300 Grade Seven $165 $300 a. Which class is closest to reaching the goal? b. How do you know? c. What percent of the goal has each class reached? d. What percent of the goal remains to be collected? e. Which classes are easier to compare? Why? 33. Assign a problem situation for students to determine the percent of a given number such as those in the lesson. Have students record their response in their mathematics journals. The response includes explaining their strategy and answering the problem. Part Five 34. Distribute two index cards and a copy of Reasonableness Cards, Attachment E, to each student. Explain the directions and review the example on the attachment. Each student is to make two Reasonableness Cards. Instructional Tip: If students still have trouble reaching a reasonable estimate, a different approach is to place percents on a number line: 0% 50%? 100%? = 75% 35. As students complete the cards check them for accuracy. 36. Pair students. a. Let one person go first and read his/her problem. The person reads his/her questions, one at a time, giving the second person time to answer each question. 9
b. Allow the second person to use paper and pencil to answer the questions. c. Have the two people settle any disagreements and decide if the estimate is reasonable. d. Have the pair of students follow the same directions for the other Reasonableness Cards. Have students repeat this procedure with a different partner, if necessary. e. Monitor the process and observe if students show understanding or need more practice. Clarify the concept and answer questions students may have. 37. Distribute and assign Percent Problems, Attachment F for students to practice the concept. Have students complete problems one through four. Check for understanding of the algorithm for finding a percent of a given number. Have them work the rest of the problems together as a class or in pairs, but work at least one of the problems all together as a class. Check problems for correctness. Clarify any misunderstanding or confusion. Instructional Tip: Add another class period if more practice is necessary. Work with students who do not understand the concepts by providing direct instruction to work through misunderstandings. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). Use 10 by 10 grids to model problems. Provide calculators to find the percent of a given number. Use common percents and friendly numbers to develop conceptual understanding. Provide opportunities for discussion of problem situations to clarify meaning of the situation and determine what is being asked. Post strategies for students to refer to throughout the lesson. Real-world applications, such as sales tax and discounts, provide a purpose for learning and are accessible to students Extensions: Use mail order catalogues to create an order including tax and shipping. Conduct surveys of students or family members to find percents. Have students graph the percents in a circle graph, analyze the data and draw conclusions. Home Connections: Have students create a discount problem that they ask a parent or adult to solve. The adult should explain his/her solution. The student determines if the adult solved the problem correctly. Have students write in their journals about this experience. Have students, during a shopping trip, determine the best sale price for a purchase from two different stores. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not 10
endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: overhead centimeter grids, markers, overhead calculator, overhead projector or a way to display a centimeter grid For the student: calculators, 10 10 grids, 3 x 5 index cards (two per student), newspapers, and restaurant menus Vocabulary algorithm discount number sense percent reasonableness sales tax Technology Connections: Calculators allow the student to concentrate on the process of solving the problem and develop their problem-solving skills. Check educational or mathematical Web sites for problem-solving situations involving percent and for data to create percent problems. Research Connections: Cawletti, Gordon. Handbook of Research on Improving Student Achievement. Arlington, Va.: Educational Research Service, 1999. Edelson, Daniel C., Douglas N. Gordin and Roy D. Pea. Addressing the Challenges of Inquiry- Based Learning, Technology and Curriculum Design. Journal of the Learning Sciences, 8(3-4), 391-450, 1999. Schoenfeld, Alan. Cognitive Science and Mathematics Education. Hillsdale, N.J.: Erlbaum Associates, 1987. General Tips: Prepare to have students with varying levels of understanding about the concept of percent. Adjust the amount of time needed for each part of the lesson as needed. Becoming proficient at the reasonableness of an answer is important to understanding the concept of percent. 11
Attachments: Attachment A, Percent Pre-Assessment Attachment B, Percent Pre-Assessment Answers Attachment C, Percent Post-Assessment Attachment D, Percent Post-Assessment Answers Attachment E, Reasonableness Card Example Attachment F, Percent Problems Attachment G, Percent Problem Answers 12
Directions: Answer the following questions. Using Percents in the Real-World Grade Six Attachment A Percent Pre-Assessment 1. Describe what percent means to you. Provide real-world examples of percent and how it is used. 2. What does 50% of the pencils are red mean? 3. What does 25% of the horses are black mean? 4. 75% of 60 = 80. Discuss the reasonableness of this statement. 5. 10% of 75 = 40. Discuss the reasonableness of this statement. Provide the decimal equivalency for the given percents. 10% 45% 83% 5% 100% 13
Attachment B Percent Pre-Assessment Answers Answers: 1. Percent: out of 100 2. Half of the pencils are red. 3. One-fourth of the horses are black. 4. This statement is not reasonable because 80 would be more than 100% of the number. The answer needs to be less than the number 60. 5. This statement is not reasonable because 40 is more than half (50%) of 75. Provide the decimal equivalency for the given percents. 10% 0.1 45% 0.45 83% 0.83 5% 0.05 100% 1 14
Attachment C Percent Post-Assessment Directions: Solve the following problems. Show your work. Include explanations as proof that each answer is reasonable by using pictures, numbers, diagrams, grids or number lines. 1. Seventy percent of the 200 concert tickets were sold on Thursday. a. How many tickets were sold on Thursday? Explain the reasonableness of the answer using pictures, words or numbers. b. On Friday, 40 more tickets were sold. What percent of the tickets had not yet been sold? Show your work using pictures, words or numbers. 2. Julie has been hoping to purchase a shirt that costs $30, but her mom says $30 is too much to pay for the shirt. Today the shirt is on sale for 40% off the regular price. a. How much will the shirt cost today? Explain the reasonableness of the answer using pictures, words or numbers. b. What is the sales tax for this purchase? 15
Attachment C (continued) Percent Post-Assessment 3. John and Chad are looking at CDs. John wants to purchase a CD for $15.99. He has $18, but when he figures the 6% sales tax, he says he does not have enough money. Chad disagrees and says the sales tax is less than $1.20. Who is correct, Chad or John? Use pictures, words or numbers to show your work. What reasoning did Chad use to solve the problem? 4. Brittany s bill at the restaurant is $7.49. There is a 7% sales tax. Brittany plans on leaving a 15% tip for the waitress. About how much is the total cost of the meal, including tax and tip? 16
Attachment D Percent Post-Assessment Answers 1. a. 140 tickets. The answer is reasonable they sold 70 out of each 100. 70 + 70 = 140. b. Ten percent, because 20 tickets remain after Friday. Uses an acceptable strategy(ies). 2. a. The shirt will cost $18.00. Accept appropriate strategies. For example: 30 x 0.4 = 12 $30 - $12 = $18 b. The sales tax is $1.26. For example: Sales tax for $20 is $1.40. The sales tax for $2 is $0.14. Subtract $0.14 from $2.00. 3. For example: $15.99 0.06 = 0.9594 or $0.96. Chad is correct. He determined the sales tax on $20 ($1.20) which is more than $15.99. He knew that $15.99 + $1.20 is less than $18. 4. For example: $7.49 0.07 =.5243 or 0.52 (tax) $7.49 + 0.52 = $8.01 $8.01 0.15 = 1.2015 or $1.20 (tip) $8.01 + 1.20 = $9.21 OR $7.49 0.15 = 1.1235 or $1.12 (tip) $7.49 0.07 = 0.52 (tax) $7.49 +1.12 +0.52 = $9.13 OR round $7.49 to $7.50 since the question says About how much is the total cost of the meal? 17
Attachment E Reasonableness Card Example Directions: On index cards, create two Reasonableness Cards. Do not use 10%, 25%, 50% or 75%. Three or four statements must be made before a reasonable range for an estimate is established. Questions for narrowing the reasonable range are written on the front of the card, while answers to the questions are on the back of the index card. Use complete sentences for questions and answers. An example follows. Front view of card: Problem: 80% of 60 1. Will the answer be greater or less than 30? 2. Will the answer be closer to 30 or closer to 60? 3. In between what two numbers should your answer fall? 4. What is your estimate for 80% of 60? Back view of card: Answer in complete sentences. 1. The answer will be greater than 30 because 50% of 60 would be 30. 2. The answer will be closer to 60 because 60 is 100% and 30 is 50%, and 80% is closer to 100% than to 50%. 3. The answer will be between 45 and 60. 4. My estimate is around 50. 18
Attachment F Percent Problems Directions: Solve the following problems. Show your work. Explain the reasonableness of the answer. Use pictures, words or numbers. 1. Find 45% of 80. Prove the reasonableness of your answer. 2. Find 13% of 65. Prove the reasonableness of your answer. 3. Find 73% of 115. Prove the reasonableness of your answer. 4. Find 60% of 250. Prove the reasonableness of your answer. 5. A skateboard costs $75 at store A. Store B sells the same type of skateboard for $70. Store A is having a sale with 25% off the price of the skateboard. Store B is also having a sale. They are giving a 15% discount on the skateboard. Which store, A or B, has the better bargain? Give proof for your choice. 6. Cree is shopping for a pair of jeans. The pair she likes is regularly $28.50. The jeans are on sale for 40% off the regular price. Sales tax is 6%. How much will Cree pay for the jeans? Explain your answer. 7. The Jones family went to a restaurant for supper. Their bill came to $32.90. There is a 7% sales tax. They decide to leave a 20% tip. What was the total amount of the bill, including tax and tip? Show your work. 19
Attachment G Percent Problem Answers Answers may vary. These are possible solutions and explanations. 1. 36; It is close to 40, which is 50% of 80. 2. 8.45; It is a little more than 6.5, which is 10% of 65. 3. 83.95; 75% of 100 is 75; 75% of 15 is close to 12; so 73% of 115 is close to 75 + 12, but a little less. 4. 150; 50 % of 250 is 125; 10% of 250 is 25; so 60% of 250 =125 + 25. 5. Store A: $75 0.25 = $18.75. $75 18.75 = $56. 25. Store B: $70 0.15 = $10.50. $70 10.50 = $60.50. Store B has the better bargain. 6. $28.50 0.40 = $11.40. $28.50 11.40 = $17.10 $17.10 0.06 = 1.026 = $1.03. $17.10 + 1.03 = $18.13. If a student combines the 40% discount with the 6% sales tax to get a 34% discount, have students think about the context and from what amount percentages are taken. For example 40% off is taken from $28.00 where the 6% sales tax is $17.10. The amounts are different, therefore the percentages cannot be combined. 7. $32.90 x.07 = 2.3030 = $2.30 tax. $32.90 + 2.30 = $35.20 $35.20 x.15 = $5.28 tip. $35.20 + 5.28 = $40.48. 20