Unit 2: Ratio and Proportional Reasoning   Imagine you are building a new tree house. The blueprint has the design, but on a much smaller scale. You will be able to use proportional reasoning to help solve problems like this one and change the small design to life-size dimensions.

In this unit, proportional reasoning is the focus. You will begin by taking a look at ratios and rates and how they help us to build proportions. You will analyze proportional relationships including scale drawing and similar figures. Once you have built the foundational skills, you will model life situations using proportions and graphs. This will help you build a definition of slope and begin to distinguish proportional relationships from other relationships.

Completing this unit should take approximately 12 hours and 30 minutes.

☐    Subunit 2.1: 3 hours and 15 minutes

☐    Subunit 2.2: 5 hours and 30 minutes

☐    Subunit 2.3: 45 minutes

☐    Subunit 2.4: 3 hours

Unit2 Learning Outcomes
Upon successful completion of this course, you will be able to: - Solve problems involving ratios and rates.
- Calculate the missing side of a proportion and determine whether two ratios create a proportion.
- Use proportional reasoning to model and solve real-world problems.
- Analyze shapes to determine if they are similar and calculate the missing side.
- Extend the use of proportions to analyze scale drawings.

Standards Addressed (Common Core): - CCSS.Math.Content.7.RP.A.1 - CCSS.Math.Content.7.RP.A.2 - CCSS.Math.Content.7.G.A.1

2.1 Ratios and Rates   We make comparisons every day. You might hear someone mention the number of boys to girls, apples to oranges, or even missed basketball shots to total shots taken. Sometimes we make more complex comparisons with two different units like miles per hour or pages per minute. Theses comparisons are ratios and rates and will be the focus of the first section of Unit 2.

2.1.1 Ratios   2.1.1.1 What Is a Ratio?   - Explanation: CK-12: “Ratios” Link: CK-12: “Ratios” (PDF)

Instructions: Watch the first video for an introduction to the idea of ratios. Then complete the reading and practice questions. You do not need to watch the second video.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Ratios/lesson/Ratios/).
``````

2.1.1.2 Three Ways to Write a Ratio   - Explanation: CK-12: “Equivalent Ratios” Link: CK-12: “Equivalent Ratios” (PDF)

Instructions: Read through the passage on Casey’s errand. Take note of the three different ways ratios can be written. Then try the practice problems at the end. Be sure to write your answers three times using each of the different forms. You do not need to watch the videos.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Equivalent-Ratios/lesson/Equivalent-Ratios/).
``````

2.1.1.3 Reducing a Ratio   - Explanation: CK-12: “Ratios in Simplest Form” Link: CK-12: “Ratios in Simplest Form” (PDF)

Instructions: Read through the passage to see why simplifying ratios is important. Then watch the review video to see the steps for reviewing more difficult examples. Finally, solve the 17 practice questions at the bottom of the page.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Ratios-in-Simplest-Form/lesson/Ratios-in-Simplest-Form/).
``````

2.1.2 Rates   2.1.2.1 What Is a Rate?   - Explanation: CK-12: “Identification and Writing of Equivalent Rates” Link: CK-12: “Identification and Writing of Equivalent Rates” (PDF)

Instructions: Read through the passage to see how rates are different than ratios. Then watch the review videos to see solved examples. After you watch the videos, try the 15 practice questions that follow.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Identification-and-Writing-of-Equivalent-Rates/lesson/Identification-and-Writing-of-Equivalent-Rates/).
``````

Instructions: Watch the video for examples of unit rate questions. Take note of the key words that let you know the question is asking for a unit rate. Then try rate problems 0.5, 1, and 2, which can be found in the left-hand menu.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

``````

2.1.2.3 Better Buy   - Explanation: CK-12: “Comparison of Unit Rates” Link: CK-12: “Comparison of Unit Rates” (PDF)

`````` Instructions: Read through the passage on comparing unit rates to
see how they can tell us which product is a better purchase. Then
solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 30 minutes.

-   [CCSS.Math.Content.7.RP.A.1](http://www.corestandards.org/Math/Content/7/RP/A/1)

attributed to CK-12 Foundation. The original version can be found
``````

2.2 Proportions   When two ratios or rates are equal to each other we have a proportion. Proportions become very useful when trying to make predictions or conversions. You might be designing a model for a new product or a blueprint for the next building in the city. Both of these require creating a smaller image of a real-life product. Then a proportion can be used to make conversions from the models to the real-life object.

2.2.1 How to Write a Proportion   - Explanation: CK-12: “Proportions” Link: CK-12: “Proportions” (PDF)

Instructions: Read through the passage, which discusses how proportions are formed and how to tell if two ratios form a proportion. Then complete the 15 practice questions. You do not need to watch the video. Note that this resource covers the material you need to know for Subunit 2.2.1 and Subunit 2.2.2.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

attributed to CK-12 Foundation. The original version can be found
``````

2.2.2 Is it a Proportion?   Note: This subunit is covered by the resource assigned beneath Subunit 2.2.1. Focus specifically on how to determine if two ratios will form a proportion. Practice questions 1 through 8 in Subunit 2.2.1 will give you a chance to work on this skill.

Instructions: Watch the video. Take note of the introduction to why cross multiplication works and how to use it to solve for missing variables in proportional relationships. When you reach the four-minute mark, there are problems that include variables. Please stop to video and solve each proportion before watching the solution.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

is attributed to YouTube and Shaunteaches. The original version can
``````
• Did I Get This? Activity: Vimeo: Will Kimbley’s “Using Cross Multiplication” Link: Vimeo: Will Kimbley’s “Using Cross Multiplication” (YouTube)

Instructions: You will be able to see five practice questions on the screen before hitting the play button. Solve questions 2, 4, and 7 in your notebook using cross multiplication. Then press the play button to view the solutions. (Note that you may solve questions 3 and 6 as well, but the solutions are not shown in the video.)

Completing this activity should take approximately 30 minutes.

2.2.2.2 Scale Factors   - Explanation: CK-12: “Proportions to find Scale Ratios” Link: CK-12: “Proportions to Find Scale Ratios” (PDF)

Instructions: Begin by scrolling down and watching the “Video Review” to be introduced to the idea of a scale factor and how to find the scale factor. Be sure to know how scale factors between 0 and 1 will change the size of a similar shape. Then solve the practice questions below the video.

Completing this activity should take approximately 30 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Proportions-to-Find-Scale-Ratios/lesson/Proportions-to-Find-Scale-Ratios/).
``````

2.2.3 Solving Proportions for Missing Value   - Did I Get This? Activity: XP Math: “Solving Proportions” Link: XP Math: “Solving Proportions” (Flash)

Instructions: This timed game will help you learn to solve proportions. You will be given 1 minute to solve as many proportions as possible. Play the game five times, each time try to increase the number that you are able to answer correctly on the first try. If you are getting the questions correct on the first try, you can change to a higher level for the next round of the game.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

``````

2.2.3.1 Solving Using Cross-Multiplication   - Explanation: CK-12: “Proportions Using Cross Products” Link: CK-12: “Proportions Using Cross Products” (PDF)

Instructions: Read the “Guidance” and “Guided Practice” sections that explain how you can multiply the number diagonally to solve for missing values in a proportion. Next watch the “Video Review” to see the steps in action. Finally, solve the 16 practice questions that are below the video.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

attributed to CK-12 Foundation. The original version can be found
``````

2.2.3.2 Solving Using Scale Factors   - Explanation: YouTube: Wikispaces: EOC Pre-Algebra: “Using Scale Factor to Solve Ratio Problems” Link: YouTube: Wikispaces: EOC Pre-Algebra: “Using Scale Factor to Solve Ration Problems” (YouTube)

Instructions: Watch the video to see what a scale factor is and how it is used with proportions.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

is attributed to Wikispaces: EOC Pre-Algebra. The original version
can be found
[here](http://eocprealgebra.wikispaces.com/Using+Scale+Factor+to+Solve+Ratio+Problems+%282%29).
``````

Instructions: Watch the first two examples in the video. While watching, take note of how the labels can be used to help write a proportion. When the time reaches 4:15, pause the video and write two different proportions to represent example number 3. Then complete the video to see if your answers are correct.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

It is attributed to the Khan Academy.
``````
• Activity: Beginning and Intermediate Algebra: Tyler Wallace’s “Rational Expressions - Proportions” Link: Beginning and Intermediate Algebra: Tyler Wallace’s “Rational Expressions - Proportions” (PDF)

Instructions: Solve word problems #31 - 40 on page 5 and page 6 using proportions. The answers can be found on page 7.

Completing this activity should take approximately 30 minutes.

2.2.5 Representing Proportional Relationships   - Explanation: CK-12: “Problem Solving Plan, Proportions” Link: CK-12: “Problem Solving Plan, Proportions” (PDF)

Instructions: Read the entire passage to see how you can find clues in the word problem to set up your proportion for solving. Then view the “Video Review.” Finally, try the 15 practice problems that follow the video, using proportions.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.RP.A.2](http://www.corestandards.org/Math/Content/7/RP/A/2)

attributed to CK-12 Foundation. The original version can be found
``````
• Activity: Gwinnett County Public Schools: “Ratios and Proportions” Link: Gwinnett County Public Schools: “Ratios and Proportions” (HTML)

Instructions: Read the “Engage” section to learn about graphing proportional relationships. Then, before scrolling down to the “Explain” section, make a table that would represent the time in hours and the amount of money earned if you were told someone makes \$9 an hour. Next, return to the “Explain” section and compare your table with the table provided. Finally, create a graph of the values to form a straight line.

Completing this activity should take approximately 45 minutes.

2.3 Similar Figures   - Explanation: CK-12: “Scale Factor of Similar Polygons” Link: CK-12: “Scale Factor of Similar Polygons” (PDF)

Instructions: Read through the “Guidance” and “Guided Practice” sections to see how ratios and proportions can be used with geometric shapes. As you are reading, take note of how the angles in the similar figures are related and how the sides of similar figures are related. Finally, solve practice problems 1 - 4 only at the bottom of the page.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.G.A.1](http://www.corestandards.org/Math/Content/7/G/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/geometry/Scale-Factor-of-Similar-Polygons/lesson/Scale-Factor-of-Similar-Polygons/).
``````
• Activity: GeoGebra: “Similar Polygons” Link: GeoGebra: “Similar Polygons” (HTML)

Instructions: Read through the interactive worksheet. Then use your mouse with the points of the shapes to see how the similar figures change. Pay special attention to how the angles change and how the lengths change.

Completing this activity should take approximately 15 minutes.

• Activity: orgLib: “Preview Questions for Test: 7.G.1 - Worksheet” Link: orgLib: “Preview Questions for Test: 7.G.1 - Worksheet” (HTML)

Instructions: Scroll down to the six questions and click on “Consume” beneath them. You should be brought to a screen where you can type in your answers. After each question you will need to click on “Submit Answer.” When you have all six questions done, click on “Submit Test.” At this point, you will be brought to a screen that will show you whether your answers are correct.

Completing this activity should take approximately 15 minutes.

2.4 Scale Drawings   - Explanation: CK-12: “Proportions to Find Dimensions” Link: CK-12: “Proportions to Find Dimensions” (PDF)

Instructions: Read the information in the “Guidance” and “Guided Practice” sections, and take note of how proportions can relate scale drawings to actual objects. You do not need to watch the video. Then complete the 15 practice problems at the bottom of the webpage.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.G.A.1](http://www.corestandards.org/Math/Content/7/G/A/1)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/arithmetic/Proportions-to-Find-Dimensions/lesson/Proportions-to-Find-Dimensions/).
``````
• Did I Get This? Activity: Gwinnett County Public Schools “Geometry: Introduction to Scale Drawings” Link: Gwinnett County Public Schools “Geometry: Introduction to Scale Drawings” (HTML)

Instructions: Scroll down under the images of the flags to where it says “Scale Drawings and Geometric Figures.” Located here are four practice problems. Solve each of the practice problems provided. If the “Check My Answer” feature is not working and you are unable to input any numbers, please write your answers on paper.

Completing this activity should take approximately 15 minutes.

2.4.1 Maps   - Checkpoint: Wikispaces: romanpercentage: “Land Expansion with Proportions” Link: Wikispaces: romanpercentage: “Land Expansion with Proportions” (PDF)

Instructions: Use proportions to solve the four problems posed on Romans using a map to help them travel. You can ignore the directions above the questions.

Completing this activity should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.G.A.1](http://www.corestandards.org/Math/Content/7/G/A/1)

attributed to Wikispaces: romanpercentage. The original version can
be found
[here](http://romanpercentage.wikispaces.com/2+Land+Expansion+with+Proportion).
``````

2.4.2 Blueprints   - Activity: Commonwealth of Australia: “Building Scales” Link: Commonwealth of Australia: “Building Scales” (HTML)

Instructions: Read from the “Introduction” section to the “Working with a Scale” section to see how ratios are used to relate blueprints to the actual building. Then solve the three “Determining Length” questions. Hit the submit button for each problem to see if your answers are correct.

`````` Note to Educators: The lesson entitled [“Developing a Sense of
to the Common Core State Standards for mathematics. In addition,
examples of student work are included, for example [“Proportion and
Non-Proportion
Situations”](http://www.svmimac.org/images/SVMIPD.091312.ProportionNonPropotion.pdf).

Note to Students and
Educators: [orgLib](http://www.orglib.com/7.RP-Ratios-Proportional-Relationships-displayFolderContents_0d1520c2bb_A1CD0C5C74C44858A6FF25CD48CD9945.html) offers
some wonderful practice questions related to
CCSS.Math.Content.7.RP.1 and 7.RP.2.
Completing this activity should take approximately 15 minutes.

-   [CCSS.Math.Content.7.G.A.1](http://www.corestandards.org/Math/Content/7/G/A/1)

``````
• Did I Get This? Activity: orgLib: “Preview Questions for Test: 7.RP.1 - Worksheet” Link: orgLib: “Preview Questions for Test: 7.RP.1 - Worksheet” (HTML)

Instructions: Scroll down to the six questions and click on “Consume” beneath them. You should be brought to a screen where you can type in your answers. After each question you will need to click on “Submit Answer.” When you have all six questions done, you will click on “Submit Test.” At this point you will be brought to a screen that will show you whether your answers are correct.

Completing this activity should take approximately 15 minutes

• Did I Get This? Activity: orgLib: “Preview Questions for Test: 7.RP.2 - Worksheet” Link: orgLib: “Preview Questions for Test: 7.RP.2 - Worksheet” (HTML)

Instructions: Scroll down to the six questions and click on “Consume” beneath them. You should be brought to a screen where you can type in your answers. After each question you will need to click on “Submit Answer.” When you have all six questions done, you will click on “Submit Test.” At this point you will be brought to a screen that will show you whether your answers are correct.

Completing this activity should take approximately 15 minutes.