Unit 2: Ratio and Proportion

Have you ever been on a long trip and asked “are we there yet”? Have you ever tried to decide which store or brand has a better deal on something you want to purchase? In this unit, you’ll get the opportunity to learn how to answer questions that involve unit rates, including constant speed and pricing. You will learn about how quantities are compared, how speed and time can be used to figure out your estimated arrival time, and how using a percent to find the part of a whole can help when you have to pay taxes or leave a tip.

Completing this unit should take approximately 6 hours and 45 minutes.

☐ Subunit 2.1: 2 hours and 30 minutes

☐ Subunit 2.2: 2 hours and 15 minutes

☐ Subunit 2.3: 2 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to:

• Describe a comparison of two quantities as a ratio using the appropriate ratio language.

• Define a ratio as a unit rate.

• Convert within-measurement units using ratio reasoning.

• Make a table to represent equivalent ratios.

• Identify ratio and rate reasoning by using tables, graphs, diagrams, or equations.

• Explain how ratios and rates fit in the context of a real-world situation.

• Solve for a percent of a quantity.

• Solve problems that involve finding the whole, given the part and a percent.

2.1 Ratio   2.1.1 What Is a Ratio?

When you compare two quantities you are using a ratio. We use ratios all the time to describe situations and events. Perhaps you are having some friends over and you plan to order pizza. You know your six friends will eat two pizzas. So the ratio of friends to pizza is 6 to 2. As you will learn in this subunit, there are three ways to write a ratio, and specific rules govern the order in which ratios are written.

• Explanation: CK-12: “Equivalent Ratios”

Instructions: Read about Casey looking to buy milk at the store. As you read, write down the definition of the word ratio when it is given to you. Notice there are three different ways to write a ratio. You should become familiar with each of these ways and understand that they are used interchangeably. Also notice that a ratio can compare a part to a part as well as a part to a whole. It is very important that you recognize that the order in which the quantities are mentioned in the question is the order in which you write your answer. Continue reading through the examples and complete the guided practice. Watch the first video, Khan Academy’s “Introduction to Ratios.” While watching, solve the problems with the instructor (notice his simplifying mistake with the dogs:horses ratio that had to be fixed with the quotation bubble). The instructor also introduces to you the idea that three quantities can make up a ratio. The second video is more of the same type of example. Feel free to skip this video if you are feeling confident with the idea of ratios.

Reading about ratios and watching the video should take you approximately 30 minutes.

2.1.2 Ratio Language to Describe a Relationship

When you are talking about a ratio it is important to use the correct language. As you noticed in the previous subunit, the word to is used to compare the quantities. In this subunit you will have the opportunity to practice word problems involving ratios. As you read each question, think about if the ratio is a part-to-part ratio or a part-to-whole ratio.

• Did I Get This? Activity: Khan Academy’s “Ratio Word Problems”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Notice the use of the word to as each item is compared. Practice the ratio word problems until you feel confident that you understand how to use ratios in various forms (recommended a minimum of 10 minutes of practice).

Completing this activity should take you approximately 15 minutes.

2.1.3 Making Tables of Equivalent Ratios

A ratio, like a fraction, can be simplified or expanded, but still be equivalent. We often use ratios to figure out bigger quantities. Remember the pizza party you had in the last subunit? What if you were planning an even bigger event hosting 48 people? In this next subunit you will learn about how a ratio table will be used to help organize your equivalent ratios.

• Explanation: YouTube: Will Kimbley’s “Using Ratio Tables”

Instructions: As you watch the video, make the tables and fill them out as the instructor does the same.

Watching this video and taking notes should take you approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Mixing Concrete”

Link: Illustrative Mathematics: “Mixing Concrete” (PDF)

Instructions: Read the question about mixing sand and cement to make concrete. It is important to note that the sand to cement ratio of 5:3 will combine to make 8 parts concrete. You do not need to account for any cement settling between particles of sand. Solve the problem before you scroll down to the answer. Read the commentary piece about how various contexts expect us to assume that the ratio parts combine to make a whole. Notice the two strategies shown in the solution.

Completing this activity should take you approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Ratio of Boys to Girls”

Link: Illustrative Mathematics: “Ratio of Boys to Girls” (PDF)

Instructions: Read the question about the ratio of boys to girls in a class. Solve parts a and b before scrolling down to see the answers. The commentary piece gives you some insight into ratios and how you can solve. There are multiple solution strategies. Read through all of them and compare the various ideas to how you solved the problem.

Completing this activity should take you approximately 15 minutes.

2.1.4 Ratio Reasoning with Measurement Conversions

Ratios can help convert between various measurement types. Whether you are familiar with the American Unit Conversion or the Metric Unit Conversion, the idea is the same. A smaller measurement (distance or weight) can convert to a larger measurement using the ratio reasoning. This subunit will show you some examples of how to convert between various measurement units. The expectation here is that you see that we can convert between units, so don’t feel like you need to memorize all the various conversions.

• Explanation: YouTube: Mathispower4u: James Sousa’s “American Unit Conversion”

Instructions: As you watch the video notice how the instructor is always going back to the conversion table to help with the ratio reasoning. Use this as tool. Similar conversion tables can be found easily on the Internet. Your goal is to be able to set up a unit fraction as shown in the examples.

Watching this video and taking notes should take you approximately 15 minutes.

• Explanation: Khan Academy’s “Metric Unit Conversion”

Instructions: This video shows how you convert between metric units. Take notes of the prefixes and their values given in the first two minutes of the video. Having these written down will be helpful. The video goes through a few different strategies to convert within metric units. Pick a strategy that makes the most sense to you.

Watching the video and taking notes should take you approximately 15 minutes.

• Web Media: Paula Swanson’s MrsSwansonsClass Wiki: “Metric System”

Instructions: Scroll down to the image of a chalkboard with each of the prefixes listed as stair steps. This is a common way to view the metric system and to help remember the conversions. Notice the mnemonic device listed under each word “King Henry Does Usually Drink Chocolate Milk.” The first letter in each of the words stands for the prefix in the metric system. There are also some hints about how multiplying or dividing by 10 makes conversions pretty simple. Copy this image into your notebook for future reference.

Taking notes should take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Units”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Focus on the prefix of the units given to you, as they will guide you to know if the unit is getting smaller or larger (recommended a minimum of 10 minutes of practice).

Completing this activity should take you approximately 15 minutes.

2.2 Unit Rate

Have you ever seen a speed limit sign? It might read 70 MPH. This is a form of a unit rate. Specifically, it means you are allowed to drive 70 miles in one hour. The units used here are miles and hours. Unit rates are very helpful when you are trying to compare items. You have the choice of buying 12 packs of gum for \$10.50 or 4 packs of gum for \$3.10. Which is a better deal? If you found the unit rate (price per one pack) for each situation, you would know which choice is a better deal. This subunit will also teach you the difference between a rate and a unit rate.

2.2.1 Finding a Unit Rate   - Explanation: YouTube: Mathispower4u: James Sousa’s “Rates and Unit Rates”

``````Link: YouTube: Mathispower4u: James Sousa’s [“Rates and Unit

Instructions: When looking at the second slide in the video, notice
the similarities and differences of rates and unit rates. Write down
the definition of each, as well as an example of each. Throughout
the video a calculator is used to do the division in each problem.
Sixth-grade students are expected to know how to do this division
(often involving decimals) without a calculator. While a calculator
is a valuable tool and speeds up the process, you should do the
mathematical calculations using your long division skills. As you
watch the video and become familiar with how to convert a rate into
a unit rate, pause the video before the solutions are given, solve
on your own, and then check your solution with the video instructor.

Watching this video and working through the examples should take you
approximately 30 minutes.

-   [CCSS.Math.Content.6.RP.A.3b](http://www.corestandards.org/Math/Content/6/RP/A/3/b)

attributed to James Sousa.
``````
• Explanation: CK-12: “Comparison of Unit Rates”

Link: CK-12: “Comparison of Unit Rates” (HTML)

Instructions: Read through the explanation on unit rates. Remember a unit rate is when you simplify a rate down to one. Go through the “Guidance,” “Examples,” “Vocabulary” and “Guided Practice” sections. Skip the first two videos; however, watch the third video, “James Sousa, Example of Determining Unit Rate.” Try to solve the unit rate question before he gives you the answer. Do the practice questions at the end of the lesson.

Reading this explanation, watching the video, and working the example problems should take you approximately 30 minutes.

2.2.2 Using Unit Rate for Unit Pricing and Constant Speed   - Did I Get This? Activity: Illustrative Mathematics: “Price Per Pound and Pounds Per Dollar”

``````Link: Illustrative Mathematics: [“Price Per Pound and Pounds Per
Dollar”](http://s3.amazonaws.com/illustrativemathematics/illustration_pdfs/000/000/549/original/illustrative_mathematics_549.pdf?1343857011) (PDF)

grocery store. Answer questions a - d before scrolling down the
work.

Completing this activity should take you approximately 15 minutes.

-   [CCSS.Math.Content.6.RP.A.2](http://www.corestandards.org/Math/Content/6/RP/A/2)

``````
• Did I Get This? Activity: Illustrative Mathematics: “Running at a Constant Speed”

Link: Illustrative Mathematics: “Running at a Constant Speed” (PDF)

Instructions: Answer the questions a - d. Then, scroll down to find the solutions. This activity has a lot of “teacher talk” between the question and the solution. Feel free to read as little or as much of this as you would like.

Completing this activity should take you approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “The Escalator”

Link: Illustrative Mathematics: “The Escalator” (PDF)

Instructions: Read through the question. Notice that this question is a multiple-choice question where having more than one correct answer is possible. When you have answered the question, scroll down to find the solution. This activity has a lot of “teacher talk” between the question and the solution. Feel free to read as little or as much of this as you would like. However, do read the first two paragraphs of “Mathematical Content.” The paragraphs explain why the correct answers are correct and why the distractors are included, yet incorrect. After you read these two paragraphs, describe some key mathematical words that lead you to believe that those answers were correct or incorrect. Include a few sentences that could be used as words of advice when dealing with multiple-choice problems like this one.

Completing this activity should take you approximately 30 minutes.

2.3 Percent of a Quantity

At a recent sports event, the broadcaster announced the stadium attendance of 59,800. You overhear a fellow fan state. “That means this stadium is 92% full.” Can you figure out the number of people it would take to fill the stadium to capacity? This is a situation where you would use what is called the percent equation. During this subunit you will learn various strategies to solve problems involving finding the whole.

• Explanation: YouTube: Mathispower4u: James Sousa’s “Solving Percent Problems Using the Percent Equations”

Instructions: This video goes through the various types of problems you might see while using the percent equation. The instructor talks about the importance of the words of, is, and what. These are key words not only in the percent equation but in all math problems. You might remember the use of the word of when you completed the multiplying fractions subunit in unit one.

Watching this video should take you approximately 15 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Example 1: Solve a Percent Problem Using a Percent Equation”

Instructions: This video has an example for you to practice and check your solution. Watch the first 30 seconds of the video as the instructor introduces the problem. Pause the video at 0:30 and solve the problem. Watch the remainder of the video to check your solution. Notice how the instructor’s last step is to verify that the solution makes sense. Be sure to do this, as well, once you solve a problem.

Watching this video and solving the problem should take you approximately 15 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Example 2: Solve a Percent Problem Using a Percent Equation”

Instructions: This video has an example for you to practice and check your solution. Watch the first 30 seconds of the video as the instructor introduces the problem. Pause the video at 0:30 and solve the problem. Watch the remainder of the video to check your solution. Notice how the instructor’s last step is to verify that the solution makes sense. Be sure to do this once you solve a problem.

Watching this video and solving the problem should take you approximately 15 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Example 3: Solve a Percent Problem Using a Percent Equation”

Instructions: This video has an example for you to practice and check your solution. Watch the first 30 seconds of the video as the instructor introduces the problem. Pause the video at 0:30 and solve the problem. Watch the remainder of the video to check your solution. Notice how the instructor’s last step is to verify that the solution makes sense. Be sure to do this once you solve a problem.

Watching this video and solving the problem should take you approximately 15 minutes.

• Checkpoint: College of the Sequoias: Ross Rueger’s Math 360 - PreAlgebra: “Chapter 5, Chapter Summary and Review”

Link: College of the Sequoias: Ross Rueger’s Math 360 - PreAlgebra: “Chapter 5, Chapter Summary and Review” (PDF)

Instructions: Do review exercises 1 - 10 ALL, 63 - 73 ODD. You can find the answers here, near the end of the document on pages 420 - 421.

Completing these problems and checking your answers should take you approximately 1 hour.

`````` Instructions: Test your knowledge by completing this checkpoint.