The Mathematics of Ratios, Rates and Proportions Teaching ...

Developmental Math—An Open Program

Instructor Guide

4.1

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

4.3

Instructor Overview Tutor Simulation: Tracking Soccer Team Standings

4.8

Instructor Overview Puzzle: Out of Proportion

4.9

Instructor Overview Project: Painting Your Way to Profit

4.11

Common Core Standards 4.24

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Unit 4 – Table of Contents

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Instructor Guide

4.2


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.

Unit 4 – Learning Objectives


Instructor Guide

4.3


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:

Unit 4 – Instructor Notes


Instructor Guide

4.4

[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:


Instructor Guide

4.5

[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:


Instructor Guide

4.6

[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:


Instructor Guide

4.7

[From Lesson 1, Topic 1, Topic Text]

Additional Resources

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.

http://www.321know.com/rat.htm#topic7


Instructor Guide

4.8

Unit 4: Ratios, Rates and proportions

Instructor Overview

Tutor Simulation: Tracking Soccer Team Standings

Purpose

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.

Unit 4 – Tutor Simulation


Instructor Guide

4.9

Unit XX: Ratios, Rates and proportions

Instructor Overview

Puzzle: Out of Proportion

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,

Unit 4 – Puzzle


Instructor Guide

4.10

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.


Instructor Guide

4.11


Instructor Overview

Project: Painting Your Way to Profit

Student Instructions

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 llo n

4 q u a r t sx g a llo n s

1 q u a r t.]

Unit 4 – Project


Instructor Guide

4.12

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.


Instructor Guide

4.13

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.


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


Instructor Guide

4.14

150 sq. ft. 500 sq. ft. 1500 sq. ft.

Cost using unit

price

Number of quarts

needed

Actual Cost of

Paint Purchased

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


Instructor Guide

4.15

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.

Paint Purchase Plan

Coverage Area

(sq. ft.)

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.


Instructor Guide

4.16

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

$ ?

3 a p a r t m e n t s

$ x

1 7 4 a p a r t m 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

$ ?

3 a p a r t m e n t s

$ 1 0 0 0

x a p a r t m e 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?


Instructor Guide

4.17

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.]


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


Instructor Guide

4.18

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.


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.


Instructor Guide

4.19

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


Instructor Guide

4.20

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 1 can 1 can 1 can

Actual Cost of

Paint Purchased $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


Instructor Guide

4.21

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

3 apartments 174 apartments can be solved to obtain a cost of

$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

http://www.moodle.org/


Instructor Guide

4.22

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.

http://www.wikispaces.com/site/for/teachers

http://pbworks.com/content/edu+overview

http://www.docs.google.com/

http://why.openoffice.org/


Instructor Guide

4.23

3

• 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.

• 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.

2

• 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.

• 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.

1

• 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 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.


Instructor Guide

4.24


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.

Unit 4 – Correlation to Common Core Standards

Learning Objectives

The Mathematics of Ratios, Rates and Proportions Teaching ...

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