Unit 5: Statistics   Is there a relationship between the hours students spend studying for a test and their test scores? We could use a scatter plot to graph this situation: hours spent (x-axis) and test score (y-axis). The coordinates on the graph would show if there is a relationship. In this unit, you will learn about scatter plots as well as how to model situations with bivariate data. Bivariate data is the term used in math to explain that two variables (hours and test scores) are being used.

This unit should take you approximately 10 hours and 45 minutes to complete.

☐    Subunit 5.1: 5 hours and 45 minutes

☐    Subunit 5.2: 5 hours

Unit5 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Construct and interpret scatter plots. - Make predictions and analyze trends in scatter plots. - Analyze data to demonstrate relationships between bivariate data sets. - Construct and interpret a two-way table summarizing data on categorical variables. - Solve problems in the context of bivariate measurement data.

Standards Addressed (Common Core): - CCSS.Math.Content.8.SP.A.1 - CCSS.Math.Content.8.SP.A.2 - CCSS.Math.Content.8.SP.A.3 - CCSS.Math.Content.8.SP.A.4 - CCSS.ELA-Literacy.RST.6-8.4 - CCSS.ELA-Literacy.WHST.6-8.1

5.1 Graphing Bivariate Data   Have you ever wondered if having larger hands would improve your video gaming skills? Or maybe you’ve speculated if your height has any correlation with how high you can jump? A correlation is a relationship between two variables. Mathematicians look for both a positive and a negative correlation between variables. This subunit will teach you about how to collect, organize, and graph data from two variables (bivariate).

• Explanation: Pascack Hills Honors Math Analysis: “Scatterplots” Link: Pascack Hills Honors Math Analysis: “Scatterplots” (HTML)

Instructions: In the first paragraph, read about scatterplots and what having a correlation means. You will see a list of seven general types of scatter plots ranging from “Perfect positive correlation” to “No evident correlation.” As an eighth-grade student, you will want to know about the types of correlation and how to graph correlations; however, you do not need to know about a correlation coefficient or the “R value” of a correlation. Skip the R Value (regression) paragraph. Read the remaining paragraphs on the page. Take notes on the examples of each type of correlation. Sketch/label each of the graphs. You will not need to know the information presented in the SAT Questions.

Reading this lesson and taking notes should take approximately 15 minutes.

• Explanation: CK-12: “Displaying Bivariate Data” Link: CK-12: “Displaying Bivariate Data” (HTML)

Instructions: The purpose of this resource is to give you a general idea about what you will see and learn in this subunit. During this subunit, focus on learning what a scatter plot looks like, how it is created, and how a mathematician can read a scatter plot. You won’t spend a lot of time on making graphs or practicing skills in this resource.

Read the short paragraph under the “Guidance” section and take notes to help you understand what bivariate means. Look at the data in the table. Do you see how it is organized and easy to follow? The data is graphed in a scatter plot. Notice that the data points are not connected in a scatter plot. However, mathematicians do often draw a single line to represent the general trend of the data. You will work more with this line (called the line of best fit) in a future subunit. For now, read the paragraph under the scatter plot that has the line of best fit. Take notes on how mathematicians discuss bivariate data. Continue reading and taking notes until you reach the section titled “On the Web.” You will start practicing these skills in the next resource.

Reading this lesson and taking notes should take approximately 30 minutes.

• Explanation: CK-12: “Make a Scatterplot to Represent Data” Link: CK-12: “Make a Scatterplot to Represent Data” (HTML)

Instructions: Read about Mr. Watson’s idea that there is a correlation between height and speed. What do you think? Would you expect taller athletes to be faster? Throughout this resource you are going to learn how to make a scatter plot. However, the creation of the scatterplot is not the only learning objective. You will want to make sure you understand how to organize your data in a table. Also, you want to start to make sense of the scatter plot to determine if there is a correlation between the variables.

Read the “Guidance” section and take notes on the bolded definitions in the text and in the “Vocabulary” section. Pay attention to how to recognize which variable is the input data that goes on the x-axis and which is the output data on the y-axis. Also, notice that when you make a scatter plot you can choose the range of the numbers on the axes based on the collected data. Make the scatter plot in your notebook that represents the data for studying math and the grade earned. Do you notice how all the data points trend upward to the right? That is called a positive correlation. Conversely, if the data points trended down to the right, mathematicians call this a negative correlation. You will work more with positive and negative correlations in future resources.

Read about the graphing calculator opportunity. Try it out on a graphing calculator if you have one. Complete examples A, B, and C and be sure you understand the solutions. Finally, you will complete the scatter plot that displays height and speed – do you see a correlation?

Reading this lesson, taking notes, and working through the examples should take approximately 30 minutes.

Instructions: While watching the video, sketch the scatter plot with the instructor. Listen closely from the 0:45 mark to the 1:05 mark as the instructor explains why the axes are labeled with period on the x-axis and average score on the y-axis. It will be important for you to know which variable goes on which axis when graphing.

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

Instructions: At the beginning of the video, press pause and read ahead so you can start thinking about the situation. Complete the sentence about which axis exercise should go on. Continue watching the video as the instructor talks through the answer to both the axis question and the range of axis question. These are important considerations before starting to construct scatter plots.

Watching the video should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Constructing Scatter Plots” Link: Khan Academy’s “Constructing Scatter Plots” (HTML)

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. Practice constructing scatter plots until you feel confident that you understand how to set up and create a scatter plot (practice for a minimum of 10 minutes).

Completing these practice problems should take approximately 15 minutes.

• Explanation: Shodor’s Interactivate: “Bivariate Data Relations” Link: Shodor’s Interactivate: “Bivariate Data Relations” (HTML)

Instructions: Read the dialogue between the student and mentor. It might help to do a quick sketch of what the scatter plots would look like for the situations concerning temperature versus people at the beach, and time jogging versus time running a mile. The purpose of this resource is for you to be able to read and understand the conversation about the relationship between bivariate data.

Reading the discussion and sketching the graphs should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Hand Span and Height” Link: Illustrative Mathematics: “Hand Span and Height” (PDF)

Instructions: The first page asks you collect data from each student in your class. If you have 12-15 people to collect data from, then go ahead and do this with the people around you. However, if you do not have people to use for data collection, scroll down to page 2 and use the sample data. Answer questions a-c on page 1. Check your solutions on page 3.

Completing this activity and checking your solutions should take approximately 30 minutes.

5.1.1 Interpreting Scatter Plots   - Explanation: Khan Academy’s “Studying, Shoe Size, and Test Scores Scatter Plots” Link: Khan Academy’s “Studying, Shoe Size, and Test Scores Scatter Plots” (YouTube)

Instructions: Begin watching the video as the instructor explains the two displayed scatter plots. Pause the video at the 0:30 mark and answer the question using the choices at the right. Continue watching the video as the instructor explains the solution.

As a model of good test-taking skills, the instructor considers each example even though he sees the relationship quickly. It is a good strategy to read all the multiple-choice questions and contradict or agree with each of them before selecting the best choice.

Watching the video should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.SP.A.1](http://www.corestandards.org/Math/Content/8/SP/A/1)

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• Did I Get This? Activity: Illustrative Mathematics: “Texting and Grades I” Link: Illustrative Mathematics: “Texting and Grades I” (PDF)

Instructions: Read about the situation and interpret the scatter plot on page 1. Use your mathematical vocabulary to answer the question on the graph. Read the solution on page 2. Do a self-assessment on the quality of information you gave in your answer compared to the paragraph on page 2. Add to your answer if necessary.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Birds’ Eggs” Link: Illustrative Mathematics: “Birds’ Eggs” (PDF)

Instructions: Use the scatter plot to answer questions a-e on page. Printing the page for yourself would be beneficial. However, these questions can be solved without printing the document. Check your solutions on page 2-3.

Completing this activity and checking your solutions should take approximately 30 minutes.

5.1.2 Line of Best Fit   In the previous subunits, you made and interpreted scatter plots. Often, mathematicians will use the trend of the scatter plot to draw a line that represents the trend. As an eighth grader, you want to be able to draw a line of best fit, use your prior knowledge of writing linear equations to find an equation for the line of best fit that you drew, and finally, use your line and equation to estimate additional data points on the scatter plot.

• Activity: Shodor’s Interactivate: “Line of Best Fit” Link: Shodor’s Interactivate: “Line of Best Fit” (HTML)

Instructions: Begin reading the dialogue between the student and mentor. The third time the mentor speaks, click on “Regression” to access the hyperlink. In the box beneath the graph, enter: (1, 2) (2, 3) (3, 4). Then click on “update plot.” Select the button “Display line of best fit.” Notice how the line touches all the points in a straight line. Now, add (9, 3) to the list of coordinates. Notice how the line of best fit changes drastically; it also isn’t touching each coordinate anymore.

Finish reading the dialogue between the student and the mentor. Lastly, use the regression program to enter various coordinate points, create your own line of best fit, and then check it with the “Display line of best fit”button.

Reading the discussion and completing this activity should take approximately 15 minutes.

• Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Investigation of Scatter Plots” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Investigation of Scatter Plots” (PDF)

Instructions: Use Janae’s swimming data found here, on the second page, to complete the scatter plot. Answer questions 1-7 about the swimming situation. You can learn how to use a graphing calculator to check your solution here.

Completing this activity and checking your solutions should take approximately 45 minutes.

• Did I Get This? Activity: Dr. Ted Coe’s Mathematics with Geogebra: “Line of Best Fit” Link: Dr. Ted Coe’s Mathematics with Geogebra: “Line of Best Fit” (HTML)

Instructions: Follow the numbered instructions. To adjust the line of best fit, slide the black dot along the intercept and slope line. Check your line of best fit by selecting the “My Fit” box and comparing it with the line of best fit created when you check the “Geogebra Line” box.

Uncheck each box, move the lettered points to various spots on the grid, and complete steps 1-4. Your line doesn’t have to be identical to the Geogebra Line. However, it should be fairly close. Using different points, create 8-10 lines to check your understanding.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Laptop Battery Charge” Link: Illustrative Mathematics: “Laptop Battery Charge” (PDF)

Instructions: Use the screenshots of the laptop battery charge to answer questions a-d on page 1. Check your solutions on pages 2-3.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Animal Brains” Link: Illustrative Mathematics: “Animal Brains” (PDF)

Instructions: Using the data on page 1, answer question a-f. Check your solutions on pages 2-3.

Completing this activity and checking your solutions should take approximately 30 minutes.

5.2 Interpreting Bivariate Categorical Data   Data collection, graphing, and looking for patterns do not always happen just when you have a number as a variable, as with the time spent studying versus test score situation. Sometimes people want to collect and compare data that involve categorical variables such as eye color, pets, sports, or school subjects. This subunit will focus on collecting categorical data in two-way frequency tables and understanding relative frequency in the data.

• Explanation: Shodor’s Interactivate: “Numerical and Categorical Data” Link: Shodor’s Interactivate: “Numerical and Categorical Data” (HTML)

Instructions: Read the discussion between the students and mentor. This discussion should help you understand the difference between categorical and numerical data. The remainder of this unit will focus on categorical data. Make sure you are able to understand the difference between the two data types.

Reading the discussion should take approximately 15 minutes.

• Explanation: SOPHIA: Ryan Backman’s “Two-Way Tables/Contingency Tables” Link: SOPHIA: Ryan Backman’s “Two-Way Tables/Contingency Tables” (Flash)

Instructions: This video gives you an introduction to two-way tables. Pause the video at the 1:10 mark and copy the definition. Take notes as you watch the example about burritos and drink choices. At the 1:50 mark the instructor shows you how to find marginal totals. Take notes on how to do this and be sure you understand where the numbers come from.

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

• Explanation: SOPHIA: Jeff Franz’s “Association in Two-Way Frequency Tables” Link: SOPHIA: Jeff Franz’s “Association in Two-Way Frequency Tables” (Flash)

Instructions: This video shows you how to use two-way frequency tables to make associations and find trends in data. As you watch, listen as the instructor explains how the numbers in the table represent the total number of people surveyed. Also, pay attention to how the instructor is able to make some statements about the association of the data. Take notes on how to find a percent of specific data in the table. Converting numbers to a decimal and a percent help mathematicians make general statements about the overall data.

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

• Explanation: SOPHIA: Katherine Williams’s “Two Way Tables/Contingency Tables” Link: SOPHIA: Katherine Williams’s “Two Way Tables/Contingency Tables” (Flash)

Instructions: Watch as the instructor introduces the data in the table regarding shakes and fries. Pause the video at 0:40 mark. Copy the two-way table in your notebook and fill in the marginal data. Continue watching the video and check your numbers with the instructor. Make sure you also find the total number of people surveyed.

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

• Did I Get This? Activity: The Mathematics Vision Project’s Secondary One Mathematics: An Integrated Approach: “Module 7, Modeling Data” Link: The Mathematics Vision Project’s Secondary One Mathematics: An Integrated Approach: “Module 7, Modeling Data” (PDF)

Instructions: Read Part I of the “After School Activity” section. Study the table, called a two-way frequency table. Notice that the numbers in both the rows and columns add up to the totals. Can you figure out the total number of boys and the total number of girls Rashid asked about after-school preferences? Use the bottom totals to help you.

Read Part II. Complete the problems in Data Set 1 and Data Set 2.

Continue practicing by completing the “Ready,” “Set,” and “Go” problems on pages 15-17 of the text (pages 19-21 of the document).

Check the accuracy of your two-way tables here.

Reading this selection and answering the practice problems should take approximately 45 minutes.

• Explanation: Centre for Innovation in Mathematics Teaching: “Two Way Tables” Link: Centre for Innovation in Mathematics Teaching: “Two Way Tables” (HTML)

Instructions: Read about Emma’s classmates and their pets beneath the “Example Question” heading. Be sure you understand each of the variables and how the table is set up. Fill in the numbers of each of the dogs and cats as found in the table next to the “Practice Questions” section. Answer each of the practice questions and check your solution with the “Show Me” button. Under the “Exercises” section, answer each of the questions and check your solution.

Completing this activity and checking your solutions should take approximately 30 minutes.

• Explanation: Maxfield Foundation: Susan Dean and Barbara Illowsky’s “Sampling and Data: Frequency, Relative Frequency, and Cumulative Frequency” Link: Maxfield Foundation: Susan Dean and Barbara Illowsky’s “Sampling and Data: Frequency, Relative Frequency, and Cumulative Frequency” (HTML)

Instructions: The two-way tables you learned about in the past few resources often use relative frequencies. In this resource, you will learn how to find a relative frequency from data in a frequency table.

Read through the data about how many hours students worked each day. Take notes on the bold-faced vocabulary words. Read and understand how to find a relative frequency. Notice that each of the relative frequencies uses the value 20 as the denominator and divisor because 20 students were surveyed.

You can stop at the “cumulative relative frequency” definition, as this does not pertain to the eighth-grade curriculum. You will have more opportunities to calculate relative frequency in the next resource.

Reading this lesson, taking notes, and studying the examples should take approximately 30 minutes.

• Explanation: The Mathematics Vision Project’s Secondary One Mathematics: An Integrated Approach: “Module 7, Relative Frequency” Link: The Mathematics Vision Project’s Secondary One Mathematics: An Integrated Approach: “Module 7, Relative Frequency” (PDF)

Instructions: The two-way tables you have learned about in the past few resources often use relative frequencies. Read Rachel’s data that compares the number of text messages for adults and teenagers.

Relative frequency can be calculated for different parts of the data. For example, you could calculate the relative frequency of the row data or the relative frequency of the column data. Each of these separate strategies would change the total amount.

Calculate the relative frequency of the row data for text messaging. Read the paragraph that starts with “Part II,” which should contain information similar to the previous resource about relative frequency. Take any notes as you read. Scroll down to the top of the second page to check your work for the relative frequency of the row data.

Look at the table of relative frequency of column data on the bottom of the second page. Write two statements about this data.

Calculate the relative frequency of the whole table, using the idea that 48 people were surveyed. Check your work on the top of page

1.
Completing this activity and checking your solutions should take approximately 30 minutes.

2. CCSS.Math.Content.8.SP.A.4

• Did I Get This? Activity: Illustrative Mathematics: “What’s Your Favorite Subject” Link: Illustrative Mathematics: “What’s Your Favorite Subject?” (PDF)

Instructions: Using the table on page 1, answer the question below the table. Try to make 2-3 associations for favorite subject and grade level for this school. Check your solutions on page 2.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Explanation: University of Leicester’s Learning Development: “Bar Charts” Link: University of Leicester’s Learning Development: “Bar Charts” (HTML)

Instructions: Scroll to the bottom of the page for a quick look at a segmented bar graph (also called stacked bar chart). You probably don’t have a lot of experience with this type of graph. The next resource includes a question asking you to create a segmented bar graph, so study this example and do your best.

Studying this example should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Music and Sports” Link: Illustrative Mathematics: “Music and Sports” (PDF)

Instructions: Using the data found on the top of page 2, answer questions a-e on page 1. If you need help with a visual representation for part e, read the commentary on the top of page 2. See if you can create a segmented bar graph. Do your best on this and learn from the solution on page 3. Check your solutions on pages 2-3.

Completing this activity and checking your solutions should take approximately 30 minutes.

• Checkpoint: Mathematics Assessment Project: “Scatter Diagram” Link: Mathematics Assessment Project: “Scatter Diagram” (PDF)

Instructions: Complete questions 1-3. Make sure you read each of the statements on page 2 carefully before deciding if it is true or false. Check your answers with the student work here. This student received full points on the assignment.

Completing this activity and checking your solutions should take approximately 15 minutes.