Unit 4 Table of Contents Unit 4: Ratios, Rates and Proportions Learning Objectives 4.2 Instructor Notes The Mathematics of Ratios, Rates and Proportions Teaching Tips: Challenges and Approaches Additional Resources Instructor Overview Tutor Simulation: Tracking Soccer Team Standings Instructor Overview Puzzle: Out of Proportion Instructor Overview Project: Painting Your Way to Profit 4.3 4.8 4.9 4.11 Common Core Standards 4.24 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. 4.1
Unit 4 Learning Objectives Unit 4: Ratios, Rates and Proportions Lesson 1: Ratio and Rates Topic 1: Simplifying Ratios and Rates Learning Objectives Write ratios and rates as fractions in simplest form. Find unit rates. Find unit prices. Lesson 2: Proportions Topic 1: Understanding Proportions Learning Objectives Determine whether a proportion is true or false. Find an unknown in a proportion. Solve application problems using proportions. 4.2
Unit 4 Instructor Notes Unit 4: Ratios, Rates and Proportions Instructor Notes The Mathematics of Ratios, Rates and Proportions This unit introduces students to ratios. They'll learn to recognize and apply these numbers in familiar situations and to describe them both verbally and symbolically. By the time they complete the unit, they'll be able to define, write, simplify, and evaluate ratios, rates, unit rates and prices, and proportions. Teaching Tips: Challenges and Approaches Students all deal happily with ratios every single day, they just don't realize it. Once they learn a few definitions and techniques, put in some practice, and see how useful ratios are, this unit should go smoothly for most of them. Preparation Much of the work is an extension of the ideas and skills learned in Unit 2, Fractions and Mixed Numbers. Before tackling ratios, make sure that students have mastered that material. The emphasis on this unit should be the practical application of fractions. Recognizing Ratios Students experience ratios all the time they read speed limit signs on the way to school, they hear commercials claiming that nine out of ten dentists prefer a particular type of toothpaste, and they figure out how much money they'll make working over the weekend. They just don't call these values ratios. Make that connection use lots of examples and problems to show how ordinary and everyday ratios are, and students will be more interested and comfortable in working with them. Laying out all the different ways to write ratios will help students realize they've seen them before: 4.3
[From Lesson 1, Topic 1, Topic Text] It will also help to identify some of the key words and symbols that signal a ratio is at hand, such as: per for every to / : Ratios vs. Rates Students should pick up the concept of ratios easily comparing two numbers is something they've already done plenty of. But they may struggle a bit with rates. Many think that rate means speed, and at least at first, they'll be confused that prices or wages can also be expressed as rates. Giving a clear definition of rate will help, as will more examples, like the following: 4.4
[From Lesson 1, Topic 1, Worked Example 4] This shows students that rates compare two numbers measured in different units to be a rate, all that matters is that the units are different. Be sure to point out that all rates are ratios but not all ratios are rates. Cross-multiplying This unit teaches students several new ideas and terms, but really only one fresh technique cross-multiplication. Students are shown how to use it to decide if a proportion is true, and also to calculate an unknown, as seen below: 4.5
[From Lesson 2, Topic 1, Presentation] It is important to emphasize that cross-multiplying should be done only after checking that the units are consistent. In fact, it s important to stress that students must watch units carefully throughout these lessons. It s easy for them to focus so much on getting the numbers correct that they forget to pay attention to the units. Make a point of checking units when working through problems in the classroom. One word of warning: For whatever reason, students really like cross-multiplying. They remember learning this in elementary school and it sticks with them. Once they relearn this skill, they may try to apply it where it shouldn t be used, for example when multiplying or dividing two fractions. Keep In Mind As always, application problems should be used frequently. Show students explicitly how ratios, rates, and proportions are useful in everyday life. We've all stood indecisively in the grocery store trying to figure out which package is the better buy: 4.6
Additional Resources [From Lesson 1, Topic 1, Topic Text] In all mathematics, the best way to really learn new skills and ideas is repetition. Problem solving is woven into every aspect of this course each topic includes warm-up, practice, and review problems for students to solve on their own. The presentations, worked examples, and topic texts demonstrate how to tackle even more problems. But practice makes perfect, and some students will benefit from additional work. A good website to review all the skills learned in this unit can be found at http://www.321know.com/rat.htm#topic7. Summary Unit 4 teaches students the practical applications of ratios and proportions. They'll learn how to write and simplify ratios, find unit rates and prices, and evaluate proportions. Because ratios are really fractions, this material reinforces what was learned Unit 2: Fractions. 4.7
Unit 4: Ratios, Rates and proportions Purpose Instructor Overview Tutor Simulation: Tracking Soccer Team Standings Unit 4 Tutor Simulation This simulation allows students to demonstrate their understanding of ratios. Students will be asked to apply what they have learned to solve a problem involving: Ratios Proportions Applying Ratios and Proportions to Real-World Situations Problem Students are presented with the following problem: It's soccer season, and somebody is tracking team standings. Guess what... that somebody is you. Enjoy the games. 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. 4.8
Unit XX: Ratios, Rates and proportions Instructor Overview Puzzle: Out of Proportion Unit 4 Puzzle Objectives Out of Proportion lets students see how well they understand ratios and proportions. In order to succeed, they'll have to write a proportion based on a verbal description, and then calculate the missing value that will make the equation true. Description This puzzle is made up of 20 word problems. In each one, players are given enough information to set up a proportion with one unknown value. Once they arrange the relationship correctly, 4.9
they're asked to choose the value that will satisfy the proportion. Students can figure the answers in their heads or on paper using the method they prefer, such as finding the equivalent fraction or cross-multiplying. Out of Proportion is suitable for both individual and group play. It could also be used in the classroom to illustrate the horizontal and vertical symmetry of proportions. 4.10
Unit 4: Ratios, Rates and Proportions Student Instructions Instructor Overview Project: Painting Your Way to Profit Unit 4 Project Introduction There are numerous cases in which proportional reasoning helps to clarify thinking and illuminate the actual costs associated with doing business. Task In this project, your group will draft a plan for financing the start-up of your own painting company. The owner of a local apartment complex is interested in hiring you to paint all of the apartments in his complex. By using proportional thinking, your group will minimize costs, determine the terms of a contract with the owner of the apartment complex, and make a presentation to the bank from which you will seek a loan. 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: Make a trip to the local paint store or, alternatively you can shop online at www.homedepot.com, which provides prices and specifications on paint products. You will be painting inside, so select a single brand of interior paint. Be sure that the paint is sold in quarts, gallons, and 5-gallon sizes. Record the relevant information in the chart below. Use a proportion to determine the number of gallons in 1 quart, expressing your answer as a decimal. Note that there are 4 quarts in 1 gallon. In the table below, record all the volumes in gallons. [Hint: To convert quarts to gallons, consider a proportion of the form 1 g a lo n x g.] a lo n s 4 q1 u aqr ut as r t 4.11
Developmental Math An Open Program Now compute the unit price of the paint. Be sure to justify the way in which you chose to compute the unit price and explain why it is the most relevant and informative. [Hint: When computing unit prices, there are two ways to compute them depending on whether you use the information on volume or coverage.] Information on Paint Purchase Options Option #1 quart container Option #2 gallon container Option #3 5-gallon container Price Volume (in gallons) Coverage (in sq. ft.) Unit Price 2. Second problem: You will now consider a simple scenario before attempting to calculate the cost of the paint needed for the entire apartment complex. First, we will calculate the best purchase option for a wall that is 8 feet high and 12 feet long. Make a sketch and determine the area of this wall (length x height). From the information in your chart, determine whether to purchase paint in quarts (1, 2, or 3) or whether to purchase a gallon. 4.12
Developmental Math An Open Program Based on your proportional thinking, you suspect that the coverage listed on the cans is not correct. You reason that since the amount of coverage depends on volume of paint in the can, the coverage amounts should be proportional. Determine whether, for each option, the ratio of coverage to volume is the same. If two of the ratios match and a third does not, determine the amount of coverage that would make the coverage rates proportional. Does this new information change your decision to purchase paints in quarts (1, 2, or 3) or a gallon? With your new coverage amount(s), re-compute your unit prices and record the new results in the table below. Information on Paint Purchase Options Option #1 quart container Option #2 gallon container Option #3 5-gallon container Price Volume (in gallons) Coverage (in sq. ft.) Unit Price You decide to calculate the cost for various coverage areas using the unit prices. Fill in the tables below using the unit prices you found in the preceding (revised) table. Now determine the lowest cost for each coverage area. Cost Using Quarts Coverage Area 4.13
Cost using unit price Number of quarts needed Actual Cost of Paint Purchased Developmental Math An Open Program 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost Using Gallons Coverage Area 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost using unit price Number of gallons needed Actual Cost of Paint Purchased Cost Using 5-Gallon Cans Coverage Area 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost using unit price Number of 5-gallon cans Actual Cost of Paint Purchased 4.14
Developmental Math An Open Program You discover that unit cost is not the only consideration when calculating the cost of a job. With this new information, your group needs to make a plan for purchasing paint. In the table below, there are various coverage areas. For each, determine two different purchase options using combinations of quarts, gallons, and/or 5-gallon cans. You should determine which option results in the least cost for each coverage area. Remember to use the numbers from the Revised Information on Paint Purchase Options for your calculations. Coverage Area (sq. ft.) Paint Purchase Plan Purchase Option #1 (dollars) Purchase Option #2 (dollars) Least Cost (dollars) 150 sq. ft. 250 sq. ft. 500 sq. ft. 800 sq. ft. 1500 sq. ft. 3. Third Problem: Next, your group will determine the cost of painting walls in one apartment in the complex. They are studio apartments (one large room) that are 33 feet by 15 feet and have 8-foot high walls. So, there are two walls that are 33 feet by 8 feet and two walls that are 15 feet by 8 feet. Make a sketch of each wall and calculate its area. Based on your group s previous plan, determine the best paint purchase option for this apartment. Calculate the actual cost of painting one apartment. Your group should make sure that your plan produces the lowest priced option for purchasing paint. Remember to use the numbers from the Revised Information on Paint Purchase Options for your calculations. 4.15
4. Fourth problem: Your group will estimate the cost of painting 3 apartments in the complex. Adjusting your plan from Problem 2 above, determine the least cost for paint needed. Next, determine the amount of labor needed for the job. Your group will assume four people can paint three apartments per eight-hour day and that you are paying each of them $10 per hour. Therefore, based on your paint price and cost for labor, compute the total amount your group would charge the owner for the three-apartment job. Finally, your group will estimate the cost of painting the entire apartment complex, which contains 174 studio apartments. You will use this information to determine the terms of your agreement with the owner and make a presentation to a bank for a start-up loan for your painting company. [Hint: You may want to use a proportion for this problem, such as $? $ x.] 3 a1 p7 a4 r t a mp a er nt t ms e n t s Your group will need to purchase paint up front to begin the job before you receive any payment from the owner. You plan to ask the bank for a $1,000 start-up loan. Determine the number of apartments you could paint with that start-up money, before you would need to ask the owner for a partial payment, which you would then use to purchase more paint. [Hint: You may want to use a proportion for this problem, such as $? $ ] 1 0 0 0 3 a x p a ar pt am r t e mn t es n t s Collaboration Get together with another group to compare your answers to each of the four problems. Discuss how your group decided to purchase paint and explain your plan. Some groups may have chosen to go with extra paint, while some may have chosen to purchase exactly the paint they needed. What if you were able to obtain a profit from painting an apartment? What would be a reasonable amount to charge per apartment? Would that change how many apartments you are able to complete with your start-up money? What elements are missing from the plan? 4.16
Developmental Math An Open Program Do some Internet research to determine how much extra you may need for miscellaneous items. Include your extras in the final presentation. Conclusions Your final presentation will be a professional analysis and report of the job to present to the bank in order to make your case in applying for the start-up loan. It should be in a binder or folder that will be presented to a bank. It should include all of the mathematics used to solve the four problems above. You may either neatly write out the tables and draw the studio apartment or use software such as Microsoft Word to create a professional computer-generated product. You may want to use headings to separate your plan into two parts: Labor and Materials. Instructor Notes Assignment Procedures Problem 1 In the table below are data obtained on Glidden paint from Home Depot. The relevant unit price is in square feet per area of coverage. PLEASE NOTE THAT ALL SUBSEQUENT ANSWERS WE CALCULATE ARE BASED ON THIS DATA. If students use different data, their answers will of course vary from that given, but the overall results should be comparable.] Information on Paint Purchase Options Option #1 quart container Option #2 gallon container Option #3 5-gallon container Price $9.97 $21.97 $99 Volume(in gallons).25 gallons 1 gallon 5 gallons Coverage(in sq. ft.) 150 sq. ft. 350 sq. ft. 1750 sq. ft. Unit Price 0.066 dollars/sq. ft. 0.063 dollars/sq. ft. 0.057 dollars/sq. ft. Problem 2 4.17
The wall would be 8 ft. x 12 ft. = 96 sq. ft. Since this is less than the coverage for a quart, we would purchase a quart of paint. The coverage amounts are not proportional, and in general the coverage amounts listed on quarts of paint are not proportional to those listed on gallons or 5-gallon containers. (One reason for this is that the expectation is that those ordering quarts will be painting trim and not walls, and walls generally absorb more paint than does trim.) Coverage estimates for quarts usually range from 100-150 sq. ft., and the numbers chosen in Problem #2 above would indicate a purchase of only one quart if the coverage were at or above 100 sq. ft. but two quarts if the coverage is less. However, two quarts cost almost as much as a gallon in most cases. Information on Paint Purchase Options Option #1 quart container Option #2 gallon container Option #3 5-gallon container Price $9.97 $21.97 $99 Volume (in gallons).25 gallons 1 gallon 5 gallons Coverage (in sq. ft.) 87.5 sq. ft. 350 sq. ft. 1750 sq. ft. Unit Price 0.114dollars/sq. ft. 0.063 dollars/sq. ft. 0.057 dollars/sq. ft. 4.18
Here are the calculations, when done in unit price per unit of coverage. The lowest cost for the 150 sq. ft. area is $19.94 (quarts), for the 500 sq. ft. area is $43.94 (gallons) and for the 1500 sq. ft. area is $99.00 (5-gallon can). Cost Using Quarts Coverage Area 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost using unit price 150 sq. ft. x 0.114 dollars/sq. ft. = $17.10 500 sq. ft. x 0.114 dollars/sq. ft. = $57.00 1500 sq. ft. x 0.114 dollars/sq. ft. = $171.00 Number of quarts needed 2 qts. 6 qts. 18 qts. Actual Cost of Paint Purchased $9.97 x 2=$19.94 $9.97 x 6=$59.82 $9.97 x 18=$179.46 Cost Using Gallons Coverage Area 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost using unit price 150 sq. ft. x 0.063 dollars/sq. ft. = $9.45 500 sq. ft. x 0.063 dollars/sq. ft. = $31.50 1500 sq. ft. x 0.063 dollars/sq. ft. = $94.50 Number of gallons 1 gal. 2 gal. 5 gal. Actual Cost of Paint Purchased $21.97 x 1=$21.97 $21.97 x 2=$43.94 $21.97 x 5=$109.85 4.19
Cost Using 5-Gallon Cans Coverage Area 150 sq. ft. 500 sq. ft. 1500 sq. ft. Cost using unit price 150 sq. ft. x 0.057 dollars/sq. ft. = $8.55 500 sq. ft. x 0.057 dollars/sq. ft. = $28.50 1500 sq. ft. x 0.057dollars/sq. ft. = $85.50 Number of 5-gallon cans Actual Cost of Paint Purchased 1 can 1 can 1 can $99.00 x 1=$99.00 $99.00 x 1=$99.00 $99.00 x 1=$99.00 Below are some purchase options, including the least purchase option for each coverage area using the Glidden data from Home Depot. Paint Purchase Plan Area Needed to Cover (sq. ft.) Purchase Option #1 (dollars) Purchase Option #2 (dollars) Least Cost (dollars) 150 sq. ft. 2 quarts @ $9.97=$19.94 1 gal. @ $21.97=$21.97 2 quarts for $19.94 250 sq. ft. 3 quarts @ $9.97=$29.91 1 gal. @ $21.97=$21.97 1 gal. for $21.97 500 sq. ft. 1 gal. @ $21.97 and 2 quarts @ $9.97=$41.91 2 gal. @ $21.97=$43.94 1 gal. and 2 quarts for $41.91 800 sq. ft. 2 gal. @ $21.97 and 2 quarts @ $9.97=$63.88 3 gal. @ $21.97=$65.91 2 gal. and 2 quarts for $63.88 1500 sq. ft. 4 gal. @ $21.97 and 2 quarts @ $9.97=$107.82 5 gal. @ $99.00 = $99.00 5 gal. for $99.00 4.20
Problem 3 The total area is (33 feet x 8 feet x 2 walls) + (15 feet x 8 feet x 2 walls)=768 sq. ft. The least cost will be obtained from 2 gallons and 1 quart for a total cost of $53.91. Problem 4 The total coverage area is 768 sq. ft. per apartment x 3 apartments = 2,304 sq. ft. The least cost option for this job will be $142.94 (one 5-gallon can and 2 gallons). Student answers may be different if they did not use the Glidden data from Home Depot. 4 people x 8 hours x $10 per hour = $320.00 The total cost for the three-apartment job would be labor + paint = $320.00 + $142.94 = $462.94. The total cost for the three-apartment job would be labor + paint = $320.00 + $142.94 = $462.94. The ratio problem of $? $ x can be solved to obtain a cost of 3 apartments 174 apartments $26,850.52. Student s answers may vary depending on whether they used the data from Home Depot and on what choice they made for the lowest cost option for the three-room apartment job. Solving the ratio problem with? = $462.94 gives x = 6.48 apartments. So, we could paint 6 apartments before asking the owner for the first partial payment. 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: http://www.moodle.org 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 4.21
among educators around the world as a tool for creating online dynamic websites for their students. http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview 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 with parents. http://www.docs.google.com 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. http://why.openoffice.org/ 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. 4.22
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. 4.23
Unit 4 Correlation to Common Core Standards Unit 4: Ratios, Rates and Proportions Common Core Standards Unit 4, Lesson 1, Topic 1: Simplifying Ratios and Rates 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: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and critique the reasoning of others. Unit 4, Lesson 2, Topic 1: Understanding Proportions 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: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP-3. Construct viable arguments and critique the reasoning of others. 4.24