Unit 5: Scatter Plots & Trend Lines (4...

Unit 5: Scatter Plots & Trend Lines (4 weeks)

UNIT OVERVIEW

Students will begin the unit by exploring measures of central tendency and spread and displays of one-variable data including, dot plots, histograms, and box-and-whisker plots. They will use the five number summary to create box-and-whisker plots and identify outliers with the 1.5 X IQR rule. They will be introduced to using the STAT menu on the graphing calculator.

In investigation two, students will be introduced to scatter plots and trend lines. They will fit a trend line to a scatter plot by hand and find its equation.. They will use the equation of the trend line to make predictions by interpolating or extrapolating. The students will develop a deeper understanding about the meaning of the slope and intercepts in context. These ideas are revisited in subsequent inestigations. In Investigation three, students will continue to explore trend lines and predictions. They will use technology (either a graphing calculator or a spreadsheet) to calculate the linear regression equation and to find the correlation coefficient. The students will be able to interpret the meaning of the correlation coefficient and explain the difference between correlation and causation. During investigation four, students will perform experiments in which they collect and analyze data using linear models. In this investigation, students will apply their knowledge from the previous two investigations. In this investigation the teacher will get the class prepared and organized, and then will walk around to observe and ask questions. Investigation five, students will work with data sets that contain outliers to identify the influence that outliers have on the calculation and interpretation of the slope, y-intercept, linear regression equation, and correlation coefficient. In the last investigation students will explore situations in which the data represents more than one trend, will fit a line to each section of the data set, and will use the lines to make predictions. In this way they will be introduced to piecewise linear functions.

Essential Questions

• How do we make predictions and informed decisions based on current numerical information?

• What are the advantages and disadvantages of analyzing data by hand versus by using technology?

• What is the potential impact of making a decision from data that contains one or more outliers?

Enduring Understandings • Although scatter plots and trend lines may reveal a pattern, the relationship of the

variables may indicate a correlation, but not causation. Unit Content Investigation 1: One Variable Data (three days) Investigation 2: Introduction to Scatterplots and Trend Lines (two day) Investigation 3: Technology and Linear Regression (two days) Investigation 4: Explorations of Data Sets (four days) Investigation 5: Exploring the Influence of Outliers on Trend Lines (two days) Investigation 6: Piecewise Functions (two days) Performance Task: Linearity is in the Air — Can You Find It? (three days) (Note: The performance task should begin early in the unit to give students time to collect data.) Suggested Time line: Day 1- Following investigation 3 - brainstorming and research Day 2- Following investigation 5 - collecting and analyzing data Day 3- Following investigation 6 – work day Review (one day) End-of-Unit Test (one day) Common Core Standards Mathematical Practices #1 and #3describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning. Practices in bold are to be emphasized in the unit.

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Standards Overview

• Analyze functions using different representations • Summarize, represent, and interpret data on a single count or measurement variable • Summarize, represent, and interpret data on two categorical and quantitative variables • Interpret linear models

Standards with Priority Standards in Bold 8-SP 1. Construct and interpret scatter plots for bivariate measurement data to investigate

patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8-SP 2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

8-SP 3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

S-ID 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

S-ID 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). S-ID 6. Represent data on two quantitative variables on a scatter plot, and describe how the

variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. c. Fit a linear function for a scatter plot that suggests a linear association.

S-ID 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S-ID 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID 9. Distinguish between correlation and causation. Vocabulary Boxplot causation correlation correlation coefficient data data set dependent variable distribution domain extrapolation graphical representation histogram

independent variable interpolation inter quartile range (IQR) line of best fit linear regression linear relationship/model mean (average) median measures of central tendency mode nonlinear relationship/model ordered pair

outlier piecewise function prediction regression equation scale scatter plot skewed distribution slope trend line variable x-intercept y-intercept

Assessment Strategies Performance Task: Linearity is in the Air — Can You Find It?

During the unit, have students develop a hypothesis about a real-world ‘nearly’ linear situation interesting to them, find relevant data, model the data, analyze the mathematical features of the model, and make and justify a conclusion. By the end of the unit, all students will present their findings to the class. NOTE: The performance task should be spread out over the entire unit.

Other Evidence (Formative and Summative Assessments)

• Exit Slips • Class work • Homework assignments • Math journal • Unit 5 Test

Unit 5 Materials List CT Algebra I Model Curriculum Version 3.0

Unit 5 Materials List

For all investigations students will need graphing calculators. In addition, here are some materials specific to each investigation. Investigation 2

• Raw spaghetti • Measuring tapes or yard sticks • Projector for Power Point • Rulers

Investigation 3

• Projector for Power Point • Computers (for Excel or other statistical software) • Rulers

Investigation 4

• Projector for Power Point • Computer lab • Rulers • Several yard/meter sticks or several tape measures • Rubber bands (400-500) • Several stopwatches or the ability to project online stopwatch (or use a cell phone) • Masking tape • Several pieces of 2-foot long rope. Different diameters • 9-inch balloons for every student in the class. • 12-inch balloon for the teacher. • Small aerobic exercise equipment.

Investigation 6

• Clear cylindrical container and rice Performance Task

• Computer lab for student research

Unit 5 Websites CT Algebra I Model Curriculum Version 3.0

CT Algebra I Model Curriculum Web Sites for Unit 5

Where used Web site address Last checked

Activity 5.1.1 Hurricane Isaac www.youtube.com/watch?v=uYZzrgRm4nw. 9/30/12

Activity 5.1.2 Home Run Hitters www.espn.go.com/mlb/statisitcs 9/30/12

Activity 5.1.3

Mac Donald’s Nutrition Information http://www.mcdonalds.com/us/en/food/food_quality/nutrition_choices.html

9/30/12

Activity 5.1.3

Gasoline Prices http://fuelgaugereport.aaa.com/?redirectto=http://fuelgaugereport.opisnet.com/index.asp

9/30/12

Activity 5.1.5 NBA Basketball Statistics www.espn.go.com/mlb/statisitcs 9/30/12

Activity 5.1.6 Weather statistics (Farmington, CT) http://www.weather.com/weather/wxclimatology/daily/USCT0075?climoMonth=9&x=12&y=4

9/30/12

Activity 5.1.6 Baseball Salaries http://espn.go.com/mlb/team/salaries/_/name/nyy/new-york-yankees

9/30/12

Exit Slip 5.1.1 Data on Marine Animals http://www.elasmo-research.org/ 9/30/12

Activity 5.2.2 NBA Players http://www.nba.com/history/players/ 9/30/12

Activity 5.3.2 Evolution of the telephone http://www.youtube.com/watch?v=JcnXOhrmDB8 9/30/12

Activity 5.3.4

Shark Attacks http://www.flmnh.ufl.edu/fish/sharks/statistics/FLmonthattacks.ht

m

9/30/12

Activity 5.4.1

Forensic Anthropoloy: Dr. Trotter’s Equations http://www.ehow.com/how_5611616_determine-height-through-skeleton.html.

9/30/12

Activity 5.6.1 Women’s 100m Freestyle Swimming Records http://en.wikipedia.org/wiki/World_record_progression_100_metr 9/30/12

Unit 5 Websites CT Algebra I Model Curriculum Version 3.0

es_freestyle

Activity 5.6.2 Triathlons http://www.trifind.com/ct.html. 9/30/12

Performance Task

Age of Crawling http://lib.stat.cmu.edu/DASL/Stories/WhendoBabiesStarttoCrawl.html.

9/30/12

Performance Task

Economic Data http://www.cyberschoolbus.un.org/infonation3/basic.asp.

9/30/12

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Unit 5 – Investigation 1 Overview CT Algebra I Model Curriculum Version 3.0

Unit 5: Investigation 1 (3 Days)

One Variable Data CCSS: S-ID 1; S-ID 2; S-ID 3 Overview Students will explore and define measures of center and measures of spread and display data in dot plots, histograms, and box-and-whisker plots. Students will analyze real world data by hand and using a graphing calculator. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able to find and understand measures of center as well as measures of spread. They will be able to create and interpret a dot plot, histogram and box-and-whisker plot. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.1.1 asks students to construct a frequency table and histogram. Exit Slip 5.1.2 asks students to find the five-number summary, range, and interquartile range and construct a box-and-whisker plot. Journal Entry asks students to describe the difference between the mean and the median and how changing the data set affects each measure of center.

Launch Notes Begin the investigation by discussing Hurricane Isaac, which hit the city of New Orleans in August 2012 on the seventh anniversary of Hurricane Katrina. You may want to show a video of the destructive power of the hurricane such as the one at www.youtube.com/watch?v=uYZzrgRm4nw. Closure Notes Activity 5.1.7 ties together the all of the concepts developed in this investigation and is designed to be done in groups followed by a whole class discussion. As you wrap up the discussion, ask the class what have they have learned about measures of center and measures of spread and how histograms and box-and-whisker plots convey information about a set of data. Teaching Strategies

I. 2005 was a particularly bad year for hurricanes in the Atlantic Ocean. Introduce the data table in Activity 5.1.1 Hurricanes. Review the definitions of mean, median, and mode, which students should be familiar with from previous courses. As students compute these statistics for the hurricane data they will find that three

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numbers (75, 80, 85) are tied for the mode. That is why mean and median are more frequently used as measures of center. Hurricanes are classified by category according to their maximum wind speed. Students will classify each of the fifteen hurricanes of 2005 by category and calculate the measures of center on the category variable. In this data set there is an unambiguous mode for category 1 hurricanes, which appears as the tallest bar when a dot plot is constructed.

At this point you may introduce students to the graphing calculator. The hurricane data are entered into L1 under the STAT menu. Students will then use the calculator to compute “one variable statistics.” From the maximum and minimum values they will compute the range. Tell the students to save the data in their calculator as the data will be used later to create a histogram. Students first create a histogram by hand in questions 14 and 15 in the Activity 5.1.1. Then in questions 16 and 17 they create a histogram on the calculator. One advantage of using the calculator is that the width of the bins may be easily changed. Students may observe the effect of changing bin width on the number of and height of the bars.

As an extension, some students may perform a similar analysis for the 2012 hurricane season (pages 7 and 8 of Activity 5.1.1).

Differentiated Instruction (Enrichment) Assign pages 7 and 8 of Activity 5.1.1. The instructions are much less structured than the previous ones. Question 22 asks them how the statistics and data displays can be used to draw conclusions about the difference between the two hurricane seasons.

Activity 5.1.2 Home Run Hitters provides additional practice with measures of

center, the range, and histograms. You may want to assign the first two pages as homework and save the third page, which involves looking at the shape of histograms for class discussion the next day. When students compare the homeruns in the American League and National League they see that the American League data is somewhat skewed with a tail to the right and the National League data is closer to being mound shaped.

Activity 5.1.3 More Histograms may be used at any time during this investigation

to give students more practice constructing and analyzing histograms. The one new skill introduced here is to use the Trace feature to read the frequency distribution from the display of the histogram in the Graph window.

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Differentiated Instruction (For Learners Needing More Help) To lessen the load, you can provide data lists that are already sorted from least to greatest. The lists can also be minimized so there are fewer numbers in the tables for the students to work with. Many students may find multiple columns difficult to deal with so one long column may provide more structure. If students need help sorting data sets, they may try writing numbers on slips of paper and moving them around until they are in order.

Exit Slip 5.1.1 assesses students’ ability to construct a frequency table and histogram. It may be given at the end of the first or second day of the investigation.

Journal Entry Describe the difference between the median number and the mean number of a data set. What would happen to each measure if you added more numbers to your data set? What if you took away numbers? Make sure to include when each one would be most useful. You may use one of the examples you have completed to support you answer.

II. Discuss the difference between measures of center and measures of spread. So far students have learned three measures of center (mean, median and mode) and one measure of spread (range). Discuss the advantages and disadvantages of using the range to characterize the spread of a data set. On the one hand, knowing the range helps determine the scale when you are making a histogram since all of the data lies between the minimum and maximum. On the other hand, the range can give a distorted picture of how spread out most of the data is. You may use the example of gasoline prices from Activity 5.1.3 to illustrate this. The range is almost a dollar ($.979). Yet it is strongly influenced by the maximum value (Hawaii, $4.605), which is $0.245 above the next highest value (Alaska, $4.360). For this reason statisticians look for other ways to measure spread. The two most common measures are the standard deviation (SD) and the interquartile Range (IQR). You may show students that the calculator will find the standard deviation with the 1-Var Stats command. It actually shows both the population and sample standard deviations but for our purposes we will use the sample standard deviation designated by Sx. In the case of the gasoline price data, Sx = $0.182. Tell students they will learn more about standard deviation in another course; for now they should simply recognize it as a measure of spread.

The new measure of spread we are now going to focus on is the IQR. You may introduce IQR through a class exercise. Have the class line up in order according to some criterion. If you don’t think students will be embarrassed, an obvious criterion

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is height. Or if you prefer, assign each student a random number by drawing slips of paper out of a hat. Find the median of the data set by having students step forward in pairs beginning with the two at the ends and moving inward until only one or two students are left in the original row. If it is one student, that student’s number is the median. Otherwise it is the average of the two numbers in the middle. Now find the quartiles by finding Q1, the median of the group that lies below the median of the entire set, and Q3the median of the group that lies above the median of the entire set. Explain that the median and the quartiles divide the entire set into four (approximately) equal groups. The median may be thought of as Q2. The interquartile range is the difference Q3 – Q1. Have all students whose numbers lie between Q1 and Q3 step forward. The interquartile range shows the spread over the middle half of the data set.

The discussion should lead into Activity 5.1.4 The Five-Number Summary, where students find Q1 and Q3. In later activities they may find these statistics on the calculator using the command 1-Var Stats and scrolling down. Remind students that all data values must be ordered in order to determine the five-number summary by hand. You may want to do one or two examples from this activity in class and assign the rest for homework. Students are now ready for Activity 5.1.5 Outliers and the 1.5 × IQR Rule. In this activity the concept of outlier is developed. Although there is no formal definition for outlier, it is a member of a data set that does not seem to fit the prevailing pattern. In the case of the test scores in the first example, the score of 33 is an obvious outlier, and if this score belongs to the student who missed four days of class before the test, we have an explanation why this score differs so much from the others. One method statisticians use to identify outliers is the 1.5 × !"# rule. Outliers are defined as values that are more than 1.5 times IQR units below Q1 or above Q3. Show students how to compute the lower fence and the upper fence using the 1.5 × !"# rule and then determine whether a point in the data set is an outlier. Before doing questions 2 and 3 in Activity 5.1.5 you may want to discuss the controversy that arose in 2010 when basketball star LeBron James left the Cleveland Cavaliers to join the Miami Heat where he could team up with two other superstars, Dwayne Wade and Chris Bosh. How did James’s performance compare with those of his other teammates on the Cavaliers? In other words, was he an outlier? The 1.5 × !"# rule can help answer this question.

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III. Once students are comfortable finding and interpreting the five-number summary, they are ready to display the five numbers in a special graph called a box-and-whisker plot. Activity 5.1.6 Box-and-Whisker Plots introduces students to this type of graph. The box shows the two quartiles and the median. Its length is equal to the IQR. The whiskers connect the edges of the box to the extreme values (minimum and maximum). Students first construct the plot by hand and then with the calculator. One variation of the box-and-whisker plot allows the whiskers to extend only to data points that lie within the lower and upper fences. In this display outliers are indicate as discrete points, with special symbols such as an asterisk or small square or cross on the TI graphing calculators. There are two versions of Activity 5.1.6. The questions are the same but the order is different. In Activity 5.1.6a students examine several data sets and make box-and-whisker plots by hand before using the calculator to make the plots. Toward the end of the activity they return to data sets used earlier. In Activity 5.1.6b students work with one data set at a time and answer all questions pertaining to that data set before moving on to another one. Use whichever version you prefer. At this point you can compare and contrast histograms with box-and-whisker plots. Ask students as in question 8(e) how to identify an outlier from a histogram. (In the case of Thriller it will appear in a bar all by itself at one end of the distribution.) Exit Slip 5.1.2 assesses students’ ability to find the five-number summary and construct a box-and-whisker plot. Activity 5.1.7 Test Grades is designed as a culminating activity for this investigation. It is particularly suited for group work as described below.

Group Work Activity 5.1.7 Test Grades may be given to students to work on in groups. Students can then present their findings to the entire class. Since the different groups will be analyzing different data sets, the class as a whole may then compare and contrast the results from each group.

Resources and Materials

• Activity 5.1.1 Hurricanes • Activity 5.1.2 Homerun Hitters • Activity 5.1.3 More Histograms • Activity 5.1.4 Five-Number summary • Activity 5.1.5 Outliers and the 1.5 × IQR Rule • Activity 5.1.6 Box-and-Whisker Plots

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• Activity 5.1.7 Test Grades • Exit Slip 5.1.1 Marine Animals • Exit Slip 5.1.2 Calories in Fruit • Graphing Calculators • Bulletin board for key concepts • Student Journals

Name: Date: Page 1 of 8

Activity 5.1.1 Algebra I Model Curriculum Version 3.0

Hurricanes Each year tropical storms that form in the Atlantic Ocean are given names. The first named storm starts with “A”, the second starts with “B”, and so on. A tropical storm becomes a hurricane if its wind speed reaches 74 miles per hour. 2005 was the most active year for hurricanes on record. In July of 2005, Cindy was the first tropical storm to become a hurricane. In August of 2005, Katrina made headlines worldwide as it wreaked havoc on the city of New Orleans. This chart shows the maximum wind speed, in miles per hour, for each of the fifteen Atlantic Ocean hurricanes of 2005.

2005 Atlantic Ocean Hurricanes

Name Dates Max Wind Speed (mph)

Cindy 7/3 - 7/7 75 Dennis 7/4 - 7/13 150 Emily 7/11 - 7/21 160 Irene 8/4 -8/18 105

Katrina 8/23 - 8/30 175 Maria 9/1 - 9/10 115 Nate 9/5 - 9/10 90

Ophelia 9/6 - 9/17 85 Philippe 9/17 - 9/23 80

Rita 9/18 - 9/26 180 Stan 10/1 - 10/5 80

Vince 10/8 - 10/11 75 Wilma 10/15 - 10/25 185 Beta 10/26 - 10/31 115

Epsilon 11/29 - 12/8 85 In previous courses you learned about three statistics called measures of center. These statistics are described in the box at the right. 1. For the maximum wind speeds, find the:

a. mean

b. median

c. mode

Three Measures of Center

The mean is the average that you're used to, where you add up all the data values and then divide by the number of values. The median is the "middle" value in a list of numbers. To find the median, first list the numbers in numerical order. Then, if the number of values is odd, the median is the number in the middle. If the number of values is even, the median is the mean of the two numbers in the middle A mode is a value that occurs most often. If no number is repeated, then there is no mode for the list. Some lists of numbers may have more than one mode.



2. For these data, which measure of center is larger, the mean or the median? Why do you suppose this is?

3. What difficulty did you have answering question 1(c) above? What does that tell you about

the mode of a set of values? Hurricane Categories Hurricanes are classified based on their maximum wind speed according to the Saffir-Simpson Hurricane Scale shown in this chart.

Saffir-Simpson Hurricane Scale

Category Max Wind Speed (mph)

1 74–95 2 96–110 3 111–130 4 131–155 5 155+

4. Use the Saffir-Simpson Hurricane Scale to categorize the hurricanes in the chart below.


Name Dates Max Wind Speed (mph) Category

Cindy 7/3 - 7/7 75 1 Dennis 7/4 - 7/13 150 4 Emily 7/11 - 7/21 160 Irene 8/4 -8/18 105

Katrina 8/23 - 8/30 175 Maria 9/1 - 9/10 115 Nate 9/5 - 9/10 90

Ophelia 9/6 - 9/17 85 Philippe 9/17 - 9/23 80

Rita 9/18 - 9/26 180 Stan 10/1 - 10/5 80

Vince 10/8 - 10/11 75 Wilma 10/15 - 10/25 185 Beta 10/26 - 10/31 115

Epsilon 11/29 - 12/8 85



5. Find the mean, median, and mode of the category data. 6. Compare your results from questions 1 and 5. Describe any patterns you observe. 7. Display the category data with a dot plot. The

categories are shown on the number line. For every hurricane, place a dot above the appropriate location on the number line. (An example of a dot plot is given at the right. This dot plot shows the major league home run leaders in September 2011.)

Distribution of Hurricane Categories in 2005

8. On the dot plot locate the three measures of center for the category data. Place an asterisk (*)

on the mean, put a square around the median, and put a circle around the mode. 9. Write three sentences about the conclusions you can make from the dot plot and your

analysis of the data.

1 2 3 4 5Category



Using Technology to Calculate Statistics Enter the maximum wind speed data into L1 as shown above. (Press STAT then select EDIT. Then select STAT CALC and 1-Var Stats. Find the statistics for the list L1.) The screen on the left will appear:



10. The first number to appear is the mean. (It appears as !, which is called “x bar.”) Does this number agree with the mean you calculated in question 1(a)?

11. For now, ignore the four numbers below the mean. The last number on the screen is n. It

tells you how many data values you have. What is the value of n? Is this correct? Notice the arrow to the left of n. It suggests that you can scroll down. Scroll down until you see the statistics shown in the image at the bottom right of the previous page.

12. On this screen you will find the minimum (Min), the maximum (Max) and the median (Med).

Do these results make sense? Explain. 13. The range of a dataset is the difference between the maximum and minimum values. The

range measures the amount of spread in a dataset.

!"#$% = !"#$%&% −!"#"$%$ Find the range for the maximum wind speed data.

Histograms One way to display one-variable data is with a histogram. A histogram is like a dot plot, except that the values are grouped into intervals called bins. To create a histogram, we must have a frequency table. A frequency table contains a set of bins and the number of data values contained in each bin. The number of data values in a bin is called the frequency of the bin. We can draw a graph and represent each bin with a bar. The height of a bar shows the frequency of the bin. It is important that all the bins are the same width. All data values must be placed in one of the bins. 14. A frequency table for the maximum wind speeds is shown below. Each bin has a width of 20

miles per hour. We must identify the number of data values in each bin. The data values contained in the first two bins and the frequency of the first two bins have already been identified. Fill in the rest of the table.



Bin Maximum Wind Speeds Frequency 60 ≤ x < 80 75, 75 2

80 ≤ x < 100 90, 85, 80, 80, 85 5

100 ≤ x < 120

120 ≤ x < 140

140 ≤ x < 160

160 ≤ x < 180

180 ≤ x < 200 15. Use the data in the table to make a histogram of the maximum wind speeds. Notice the bins

are on the x-axis and the frequencies are on the y-axis. The first bar is drawn for you.

Maximum Wind Speeds in 2005

16. Now use your calculator to make the same histogram. Follow these steps:

• In the Stat Plot menu, select Plot 1, turn it on, and select the histogram icon. Xlist should be L1 and Freq = 1.

• In the Window menu set Xmin = 60, Xmax = 200, Xscl = 20, Ymin = 0, Ymax = 8, Yscl = 1.

• Press Graph. Describe what you see.

17. Now draw a histogram with a narrower bin width of 10 miles per hour. To change the bin width to 10, go to the Window menu, and set Xscl = 10. Then press Graph. What do you notice? How are the two histograms alike? How are they different?

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Maximum Wind Speed (mph)

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1 2 3 4 5Category

Hurricanes in 2012 Now let’s do the same analysis with the hurricanes of 2012. The data in the table below are incomplete. Your teacher will give you the complete data set or you can find it yourself at Wikipedia.org/wiki/2012_Hurricane_Season.


Name Dates Max Wind Speed (mph) Category

Chris 6/19-6/22 75 Ernesto 8/1-8/10 85 Gordon 8/15-8/20 110

Isaac 8/21-9/1 80 Kirk 8/28-9/2 105

Leslie Michael Nadine Rafael Sandy

18. Find the mean, median, and range for the maximum wind speeds of the 2012 hurricanes. 19. Make a dot plot of the categories of the 2012 hurricanes.

Distribution of Hurricane Categories in 2012

20. Make a histogram for the maximum wind speeds of the 2012 hurricanes.

Maximum Wind Speeds in 2012



21. Write a paragraph contrasting the 2005 and 2012 hurricane seasons. In your paragraph refer

to the tables, the statistics (mean, median, and range), and your data displays (dot plots and histograms).

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Home Run Hitters The following table lists the American League and National League top 20 home run leaders for the 2011 Season. (Source: espn.go.com)

1. Compute the statistics below for homeruns. You may use a calculator to find the mean.

American League National League

Mean= Mean=

Median = Median =

Mode = Mode =

Minimum = Minimum =

Maximum = Maximum =

Range = Range =

AL Homerun Leaders NL Homerun Leaders Rank Player Team HR Rank Player Team HR

1 Jose Bautista TOR 43 1 Matt Kemp LAD 39 2 Curtis Granderson NYY 41 2 Prince Fielder MIL 38 3 Mark Teixeira NYY 39 3 Albert Pujols STL 37 4 Mark Reynolds BAL 37 4 Dan Uggla ATL 36 5 Ian Kinsler TEX 32 5 Mike Stanton FLA 34 Jacoby Ellsbury BOS 32 6 Ryan Braun MIL 33 Adrian Beltre TEX 32 Ryan Howard PHI 33 8 Paul Konerko CHW 31 8 Jay Bruce CIN 32 Evan Longoria TB 31 9 Lance Berkman STL 31

10 Miguel Cabrera DET 30 Justin Upton ARI 31 J.J. Hardy BAL 30 Michael Morse WSH 31 Mike Napoli TEX 30 12 Troy Tulowitzki COL 30

13 David Ortiz BOS 29 13 Joey Votto CIN 29 Nelson Cruz TEX 29 14 Carlos Pena CHC 28 Mark Trumbo LAA 29 15 Corey Hart MIL 26 Josh Willingham OAK 29 Alfonso Soriano CHC 26

17 Robinson Cano NYY 28 Aramis Ramirez CHC 26 18 Adrian Gonzalez BOS 27 Carlos Gonzalez COL 26 Carlos Santana CLE 27 19 Brian McCann ATL 24

20 Adam Lind TOR 26 20 Andrew McCutchen PIT 23



2. Make a frequency table for the homeruns in each league.

American League National League Interval Frequency Interval Frequency

20 ≤ x < 24 20 ≤ x < 24 24 ≤ x < 28 24 ≤ x < 28 28 ≤ x < 32 28 ≤ x < 32 32 ≤ x < 36 32 ≤ x < 36 36 ≤ x < 40 36 ≤ x < 40 40 ≤ x < 44 40 ≤ x < 44

3. Use the frequency tables to make a histogram for each league.

American League Home Runs in 2011

National League Home Runs in 2011

4. Now have your calculator make the histograms. Use StatPlot 1 for the American League and

StatPlot 2 for the National League.

What numbers did you use for the window?

XMin, = _______ YMin = ________

XMax, = _______ YMax = _______

XScl, = _______ YScl = ________ 5. Change the value of XScl on your calculator. Describe what happens to the histograms.

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Classifying Histograms Some histograms may be classified by their shape. Three common shapes are shown below.

Mound Shaped

The bars are symmetric

about a line in the center.

Skewed Right

A tail extends to the right of the

highest bar.

Skewed Left

A tail extends to the left of the

highest bar.

6. Which of the home run histograms is mound shaped? 7. Which of the home run histograms has a skewed shape? 8. In a mound shaped distribution, the mean and the median are about the same. Is this true for

the histogram you found in question (6)? Explain. 9. In skewed distribution, the mean is usually closer to the end of the tail than is the median. Is

this true for the histogram you found in question (7)? Explain. 10. Based on the data, which league had the stronger home run leaders in 2011? Base your

answer on the statistics and the histograms.



More Histograms

The table below shows the number of calories in various sandwiches sold at McDonalds Restaurants. 1. Find the measures of center, the minimum, the maximum, and the range of the calories.

2. Choose a bin width and make a frequency table for the calories. Make sure that every data

value is contained in a bin.

McDonalds Sandwiches Calories Mean

Big Mac 540 Median

Quarter Pounder with Cheese 620 Mode

Double Quarter Pounder with Cheese 760 Minimum

Hamburger 250 Maximum

Cheeseburger 300 Range

Double Cheeseburger 440

Filet-O-Fish 380 Bin Frequency

Big N' Tasty® with Cheese 510

Crispy Chicken Classic Sandwich 510

Grilled Chicken Classic Sandwich 350

McChicken 360

Honey Mustard Snack Wrap® (Crispy) 330

Honey Mustard Snack Wrap® (Grilled) 250

Ranch Snack Wrap® (Crispy) 350

McRib ®† 500

(Source: http://www.mcdonalds.com/us/en/food/food_quality/nutrition_choices.html)



3. Create a histogram using the frequency table you created in (2). Label and scale your axes.

4. The average gas prices in April 2012 for a gallon of regular gasoline for each state and

Washington, D.C. is shown below. Enter the prices into a list on your calculator.

State Price ($) State Price ($) State Price ($) Alaska 4.360 Kentucky 3.918 New York 4.149 Alabama 3.790 Louisiana 3.801 Ohio 3.789 Arkansas 3.764 Massachusetts 3.910 Oklahoma 3.676 Arizona 3.866 Maryland 3.964 Oregon 4.077 California 4.221 Maine 3.996 Pennsylvania 3.967 Colorado 3.886 Michigan 3.931 Rohde Island 3.980 Connecticut 4.167 Minnesota 3.724 South Carolina 3.720 District of Columbia 4.175 Missouri 3.672 South Dakota 3.821

Delaware 3.894 Mississippi 3.778 Tennessee 3.771 Florida 3.938 Montana 3.774 Texas 3.813 Georgia 3.827 North Carolina 3.890 Utah 3.713 Hawaii 4.605 North Dakota 3.839 Virginia 3.894 Iowa 3.754 Nebraska 3.844 Vermont 3.988 Idaho 3.773 New Hampshire 3.869 Washington 4.127 Illinois 4.064 New Jersey 3.796 Wisconsin 3.843 Indiana 3.948 New Mexico 3.788 West Virginia 3.941 Kansas 3.714 Nevada 3.963 Wyoming 3.626 (Source: http://fuelgaugereport.aaa.com/?redirectto=http://fuelgaugereport.opisnet.com/index.asp)



5. Use the 1-Var Stats command to find the mean, the median, the minimum, the maximum, and the range of the average gas prices. a. Fill in the table below.

b. Based on the range that you calculated in (a), choose a bin width for your histogram. Explain how you arrived at your choice.

c. Use the calculator to make a histogram. Sketch the histogram below.

Mean Median

Minimum Maximum

Range



d. In the graph window, press TRACE. Scroll across the bars to find the boundaries of each bin and the frequencies. Record this information in the table.

Bin Frequency

6. Describe the shape of the histogram for the calories in McDonalds’ sandwiches. Is it mound

shaped or is it skewed? Explain. 7. Describe the shape of the histogram for average gasoline prices. Is it mound shaped or is it

skewed? Explain.

8. On your sketch of the average gas process histogram, place an X on the bar that includes

Connecticut. 9. How would you describe Connecticut gasoline prices in comparison with the rest of the

United States?



The Five-Number Summary

We often are interested in how much spread there is in a data set. The spread, or variability, of data describes how far apart the data values are. A set of statistics that help us see the amount of spread in a data set is the five-number summary. The five-number summary consists of the minimum, Q1, median, Q3, and maximum. Q1 and Q3 are the first and third quartiles. The median equals Q2. Quartiles divide a data set into four quarters. To create the five-number summary of a data set, start by ordering the data set into increasing or decreasing order. Then, find the median (middle) of your data set. The median divides the data set into two halves. To find the quartiles, find the median of the lower half and find the median of the upper half. Example: Below are the arm-spans (in cm) of 15 Algebra I students: 148 152 152 152 154 154 154 162 163 164 164 170 172 180 181 Solution: The data values are already ordered. There are an odd number of values, so the

median is the middle number in the list. The lower half and upper half are shown in boxes. Each half has 7 data values. The median of the lower half is 152, and the median of the upper half is 170.

148 152 152 152 154 154 154 162 163 164 164 170 172 180 181

Median of the lower half (Q1) Median of the upper half (Q3) Middle of the data set (Median = Q2) 1. Use the statistics above and the minimum and maximum to complete the table.

minimum Q1 median Q3 maximum

Arm-spans

Minimum Q1 Median Q3 Maximum



2. Find the five-number summary for the maximum wind speeds of the named hurricanes in 2005.

a. Write the maximum wind speeds in increasing order.

b. Fill in the five-number summary.

Rules for Finding the Median & Quartiles

When you have an even number of data values, the median equals the average of the middle two numbers. If the lower half and upper half of the data set also have an even number of values, Q1 and Q3 will be the average of the middle two numbers in the lower half and upper half, respectively.

Name Max Wind

Speed (mph)

Name

Max Wind Speed (mph)

Cindy 75 Philippe 80 Dennis 150 Rita 180 Emily 160 Stan 80 Irene 105 Vince 75 Katrina 175 Wilma 185 Maria 115 Beta 115 Nate 90 Epsilon 85 Ophelia 85


Maximum Wind

Speeds



3. Find the five-number summary of the speeds of the land animals below.

Land Animals Speed (mph) Cheetah 70

Pronghorn antelope 61 Thomson’s gazelle 50

Quarter horse 48 Elk 45

Coyote 43 Ostrich 40

Greyhound 39 Rabbit(domestic) 35

Giraffe 32 Reindeer 32

Cat(domestic) 30 Grizzly bear 30

White-tailed deer 30 Human 28

Elephant 25 Black mamba snake 20

Squirrel 12 Pig (domestic) 11

Chicken 9

a. Write the speeds in increasing order.


minimum Q1 median Q3 maximum Speeds of

Land Animals



4. The following table lists the top 25 all-time highest grossing movies as of 9/16/11. Find the five-number summary of the 25 highest box office revenues (in millions of dollars).

a. Write the movie revenues in increasing order.


# Movie Title and Year $ # Movie Title and Year $

1 Avatar (2009) 761 14 The Lord of the Rings: Return of King (2003) 377

2 Titanic (1997) 601 15 Spider-Man 2 (2004) 373 3 The Dark Knight (2008) 533 16 The Passion of the Christ (2004) 370 4 Star Wars: Episode IV (1977) 461 17 Jurassic Park (1993) 357 5 Shrek 2 (2004) 436 18 Transformers: Dark of the Moon (2011) 351 6 E.T.: The Extra-Terrestrial (1982) 435 19 The Lord of the Rings: 2 Towers (2002) 340 7 Star Wars: Episode I (1999) 431 20 Finding Nemo (2003) 340 8 Pirates of the Caribbean (2006) 423 21 Spider-Man 3 (2007) 337 9 Toy Story 3 (2010) 415 22 Alice in Wonderland (2010) 334 10 Spider-Man (2002) 404 23 Forrest Gump (1994) 330 11 Transformers (2009) 402 24 The Lion King (1994) 328 12 Star Wars: Episode III (2005) 380 25 Shrek the Third (2007) 321

13 Harry Potter - Deathly Hallows (2011) 377


Movie Revenues

(in millions)



Outliers and the !.!×!"# Rule

1. Ms. Sanchez gave a test to one of her algebra classes. Here are the scores:

83, 86, 91, 78, 80, 33, 75, 91, 72, 82, 88, 84, 93, 99, 74, 79, 90

a. Look at the list of scores. Is there one score in the list that looks very different from the others? Which one? Explain.

b. One student in class was absent for the four days before the test was given. Which score most likely belonged to this student? Explain.

c. Sort the data and record the five-number summary in the chart below (You can use the 1-Var Stats command on your calculator to find these statistics).

d. Find the interquartile range (IQR). The interquartile range equals Q3 – Q1.

e. Statisticians have observed that in most data sets, almost all data values lie between a “lower fence” and an “upper fence.” The fences are at the first quartile minus 1.5 times the IQR, and the third quartile plus 1.5 times the IQR. Calculate the fences according to the formulas given below:

!"#$% !"#$" = Q1 – 1.5 × IQR = __________________ !""#$ !"#$" = Q3 + 1.5 × IQR =__________________


Algebra Test Scores



f. Any data point that does not lie within the two fences is called an outlier. Which test score is an outlier?

Switching Teams

2. When LeBron James played for the Cleveland Cavaliers during the 2009-2010 season, was

he an outlier? Here are the statistics for all players on the team that year.

Rank Player Points per Game

1 LeBron James 29.7 2 Antawn Jamison 15.8 3 Mo Williams 15.8 4 Shaquille ONeal 12.0 5 Sebastian Telfair 9.8 6 Delonte West 8.8 7 Anderson Varejao 8.6 8 J.J. Hickson 8.5 9 ZydrunasIlgauskas 7.4 10 Anthony Parker 7.3 11 Daniel Gibson 6.3 12 Jamario Moon 4.9 13 Jawad Williams 4.1 14 Leon Powe 4.0 15 Danny Green 2.0 16 Darnell Jackson 0.8 17 Cedric Jackson 0.2 18 Coby Kari 0.0

a. Record the five-number summary of the points per game in the chart below:

b. Find the IQR.


Points Per Game



c. Find the fences: !"#$% !"#$" = Q1 – 1.5 × IQR = __________________ !""#$ !"#$" = Q3 + 1.5 × IQR = __________________

d. Any data value outside the fences is an outlier. Are there any outliers on this team? If so, who?

3. Now apply the same analysis to the Miami Heat for the 2010-1011 season. Here are the

statistics.

Rank Player Points per Game

1 LeBron James 26.7 2 Dwyane Wade 25.5 3 Chris Bosh 18.7 4 UdonisHaslem 8.0 5 Mike Bibby 7.3 6 Eddie House 6.5 7 Mario Chalmers 6.4 8 James Jones 5.9 9 Carlos Arroyo 5.6 10 Mike Miller 5.6 11 ZydrunasIlgauskas 5.0 12 Erick Dampier 2.5 13 Juwan Howard 2.4 14 Joel Anthony 2.0 15 Jamaal Magloire 1.9 16 Jerry Stackhouse 1.7 17 Dexter Pittman 1.0

a. Record the five-number summary of the points per game in the chart below:

b. Find the IQR.

minimum Q1 median Q3 maximum Points Per

Game



c. Find the fences: !"#$% !"#$" = Q1 – 1.5 × IQR = __________________ !""#$ !"#$" = Q3 + 1.5 × IQR =__________________

d. Any data value outside the fences is an outlier. Are there any outliers on this team? If so, who?


Activity 5.1.6a Algebra I Model Curriculum Version 3.0

Box-and-Whisker Plots

A box-and-whisker plot is a convenient way to display the five-number summary. To draw a box-and-whisker plot:

a. Mark the minimum, maximum, median, Q1, and Q3 above the numbers on your number line.

b. Draw a box that represents the middle 50% of the data by drawing a box from Q1 to Q3. The length of the box represents the interquartile range (IQR).

c. Draw a vertical line segment inside the box to show the median. d. Draw “whiskers” to represent the lowest 25% (by connecting Q1 to the minimum value)

and highest 25% (by connecting Q3 to the maximum value) of the data.

1. The data set on the right lists the high, average, and low temperatures in Farmington, CT from September 8 to September 23, 2011. Complete the table, using your calculator. (Suggestion: enter the data in L1, L2, and L3 and save for question 4.)

Min Q1 Med Q2 Max IQR SD High Avg. Low

The statistics for the high temperatures are displayed in a box-and-whisker plot below.

September High Average Low

8 78 67 57 9 77 68 55 10 71 62 52 11 63 59 55 12 69 64 60 13 82 72 64 14 78 68 57 15 81 68 57 16 70 63 55 17 63 56 50 18 75 60 48 19 71 60 51 20 73 58 43 21 77 60 46 22 75 64 53 23 81 74 66

2. Draw box-and-whisker plots below for the average and low temperatures.



3. Match the box-and-whisker plots on the right to the descriptions they match.

a. Which plot has the greatest range?

b. Which plot has the greatest IQR?

c. Which plot has the greatest Q1?

d. Which plot has the greatest median?

e. Which plot has the greatest Q3?

4. The following set of data lists the top 18 New York Yankees Salaries for 2011. Find the five-

number summary, the range, and the interquartile range. Then make a box-and-whisker plot.

RK Player Salary (millions)

1 Alex Rodriguez 32.0 2 CC Sabathia 24.3 3 Mark Teixeira 23.1 4 A.J. Burnett 16.5 5 Mariano Rivera 14.9 6 Derek Jeter 14.7 7 Jorge Posada 13.1 8 Robinson Cano 10.0 9 Nick Swisher 9.1


10 Rafael Soriano 9.0 11 Curtis Granderson 8.3 12 Russell Martin 4.0 13 Phil Hughes 2.7 14 Eric Chavez 1.5 Andruw Jones 1.5 Freddy Garcia 1.5 17 Boone Logan 1.2 18 Bartolo Colon 0.9

(Source: http://espn.go.com/mlb/team/salaries/_/name/nyy/new-york-yankees)

0 5 10 15 20 25 30 35 40

Salary (millions)



5. Michael Jackson’s Thriller is the top-selling album of all time, with 110 million albums sold world-wide. The next 24 top-selling albums are below.

Album Albums

sold (millions)

Thriller, Michael Jackson 110 Back in Black, AC/DC 49 The Dark Side of the Moon, Pink Floyd 45

The Bodyguard, Whitney Houston 44

Bat Out of Hell, Meatloaf 43 Eagles: Their Greatest Hits, 1971–1975 42

Dirty Dancing, Various Artists 42 Millennium, Back Street Boys 40 Saturday Night Fever , Bee Gees 40

Rumours, Fleetwood Mac 40 Come On Over, Shania Twain 40 Led Zeppelin IV 37 Jagged Little Pill, Alanis Morisette 33

Album Albums

sold (millions)

Sgt. Pepper, The Beatles 32 Falling Into You, Celine Dion 32 Music Box, Mariah Carey 32 Dangerous, Michael Jackson 32 1, The Beatles 31 Let’s Talk About Love, Celine Dion 31

Goodbye Yellow Brick Road, Elton John 31

Spirits Having Flown, Bee Gees 30

Born in the U.S.A., Bruce Springsteen 30

Brothers in Arms, Dire Straits 30 Immaculate Conception, Madonna 30

Bad, Michael Jackson 30

a. Find the five-number summary, the range, and the interquartile range of the albums sold data.

b. Apply the 1.5 times IQR rule to find the fences for this set of data.

Upper fence = ____________________ Lower fence = ______________________

c. According to the 1.5 times IQR rule, can Thriller be considered an outlier? Explain.

min Q1 median Q3 max range IQR

Albums sold (millions)



d. Find the five-number summary, the range, and the interquartile range of the albums sold data without Thriller.

e. Compare the charts made in parts (a) and (d). Which statistics were most affected when the Thriller’s albums sold was removed from the list?

f. Which statistics were least affected when Thriller’s albums sold was removed from the list?

g. Statisticians often prefer a statistic that is not affected by outliers. Which measure of spread do you think they prefer? The range or the IQR? Explain.

h. Enter the data for all 25 albums in your calculator. (Suggestion: use L4 so you can keep the temperature data in L1, L2, and L3). Now use the 1-Var Stats command to find the mean and standard deviation.

Mean = ________________ Standard Deviation = ________________

i. Remove the data value “110” for Thriller from your list and recalculate the mean and

standard deviation for the remaining albums. Mean = ________________ Standard Deviation = ________________

j. Describe the effect of an outlier on the mean and standard deviation of a data set.

min Q1 median Q3 max range IQR




Box-and-Whisker Plots on the Calculator 6. Use the calculator to make the three box &

whisker plots for the temperature data from problem 1. You may display them side by side if you use L1 in StatPlot 1, L2 in StatPlot 2, and L3 in StatPlot 3 and turn all plots on. Select either of the two icons for Box & Whiskers shown to the right.

Indicate what values you used in the Window menu: Xmin = _____________ Xmax = _____________ Xscl = ________________ Ymin = _____________ Ymax = _____________ Yscl = ________________ 7. Now change the values of Ymin and Ymax. Does this affect the way the box-and-whisker plots

are displayed? Explain. 8. Now make a box & whiskers plot for the album data from problem 5. Turn off all plots except

for one and use the data in L4. Adjust the values in the Window menu so that the entire plot shows.

a. First select the icon “Box and Whiskers Showing Outliers.” Make a sketch of what you see

in the space below.

b. Then select the icon “Box and Whiskers without Outliers.” Make a sketch of what you see in the space below.



c. When there is an outlier, what is the effect of changing the box-and-whisker display on the length of the whiskers?

d. Now have the calculator make a histogram for the albums sold data. Make a sketch of what you see in the space below.

e. How can you tell from the histogram that there is an outlier in this set of data? 9. Make up a data set with ten values, listed in order, which could have the box-and-whisker

plot shown below.

10. Write a story (context) to go along with your data in (9).


Activity 5.1.6b Algebra I Model Curriculum Version 3.0

Box-and-Whisker Plots

A box-and-whisker plot is a convenient way to display the five-number summary. To draw a box-and-whisker plot:

a. Mark the minimum, maximum, median, Q1, and Q3 above the numbers on your number line.

b. Draw a box that represents the middle 50% of the data by drawing a box from Q1 to Q3. The length of the box represents the interquartile range (IQR).

c. Draw a vertical line segment inside the box to show the median. d. Draw “whiskers” to represent the lowest 25% (by connecting Q1 to the minimum value)

and highest 25% (by connecting Q3 to the maximum value) of the data.

1. The data set on the right lists the high,

average, and low temperatures in Farmington, CT from September 8 to September 23, 2011. Complete the table, using your calculator. (Suggestion: enter the data in L1, L2, and L3)

Min Q1 Med Q3 Max IQR Sx

High

Avg.

Low

The statistics for the high temperatures are displayed in a box-and-whisker plot below.

2. Draw box-and-whisker plots below for the average and low temperatures.

September High Average Low 8 78 67 57 9 77 68 55 10 71 62 52 11 63 59 55 12 69 64 60 13 82 72 64 14 78 68 57 15 81 68 57 16 70 63 55 17 63 56 50 18 75 60 48 19 71 60 51 20 73 58 43 21 77 60 46 22 75 64 53 23 81 74 66



Box-and-Whisker Plots on the Calculator 3. Use the calculator to make the three box &

whisker plots for the temperature data from problem 1. You may display them side by side if you use L1 in StatPlot 1, L2 in StatPlot 2, and L3 in StatPlot 3 and turn all plots on. Select either of the two icons for Box & Whiskers shown to the right.

Indicate what values you used in the Window menu: Xmin = _____________ Xmax = _____________ Xscl = ________________ Ymin = _____________ Ymax = _____________ Yscl = ________________ 4. Now change the values of Ymin and Ymax. Does this affect the way the box-and-whisker plots

are displayed? Explain.

5. Match the box-and-whisker plots on the right to the descriptions they match.

a. W

Which plot has the greatest range?

b. WWhich plot has the greatest IQR?

c. WWhich plot has the greatest Q1?

d. WWhich plot has the greatest median?

e. WWhich plot has the greatest Q3?



6. The following set of data lists the top 18 New York Yankees Salaries for 2011. Find the five-number summary, the range, and the interquartile range. Then make a box-and-whisker plot.


1 Alex Rodriguez 32.0 2 CC Sabathia 24.3 3 Mark Teixeira 23.1 4 A.J. Burnett 16.5 5 Mariano Rivera 14.9 6 Derek Jeter 14.7 7 Jorge Posada 13.1 8 Robinson Cano 10.0 9 Nick Swisher 9.1


10 Rafael Soriano 9.0 11 Curtis Granderson 8.3 12 Russell Martin 4.0 13 Phil Hughes 2.7 14 Eric Chavez 1.5 Andruw Jones 1.5 Freddy Garcia 1.5 17 Boone Logan 1.2 18 Bartolo Colon 0.9

(Source: http://espn.go.com/mlb/team/salaries/_/name/nyy/new-york-yankees) Five Number Summary:



7. Michael Jackson’s Thriller is the top-selling album of all time, with 110 million albums sold world-wide. The next 24 top-selling albums are below.

a. Enter all of the data into L1. Using the 1-Var Stats command, find the five-number summary, the range, and the interquartile range, mean and standard deviation of the albums sold.

With Thriller

b. Apply the 1.5 times IQR rule to find the fences for this set of data.

Upper fence = ____________________ Lower fence = ______________________

Album Albums

sold

(millions)

Thriller, Michael Jackson 110

Back in Black, AC/DC 49

The Dark Side of the Moon, Pink Floyd 45

The Bodyguard, Whitney Houston 44

Bat Out of Hell, Meatloaf 43

Eagles: Their Greatest Hits, 1971–1975 42

Dirty Dancing, Various Artists 42

Millennium, Back Street Boys 40

Saturday Night Fever , Bee Gees 40

Rumours, Fleetwood Mac 40

Come On Over, Shania Twain 40

Led Zeppelin IV 37

Jagged Little Pill, Alanis Morisette 33

Album Albums

sold (millions)

Sgt. Pepper, The Beatles 32

Falling Into You, Celine Dion 32

Music Box, Mariah Carey 32

Dangerous, Michael Jackson 32

1, The Beatles 31

Let’s Talk About Love, Celine Dion 31

Goodbye Yellow Brick Road, Elton John 31

Spirits Having Flown, Bee Gees 30

Born in the U.S.A., Bruce Springsteen 30

Brothers in Arms, Dire Straits 30

Immaculate Conception, Madonna 30

Bad, Michael Jackson 30

min Q1 median Q3 max range IQR mean Sx Albums sold

(millions)



c. According to the 1.5 times IQR rule, can Thriller be considered an outlier? Explain.

d. Enter the data without Thriller in L2. Using the 1-Var Stats command, find the five-number summary, the range, and the interquartile range, mean and standard deviation of the albums sold for the data without Thriller.

Without Thriller

e. Compare the charts made in parts (a) and (d). Which statistics were most affected when the Thriller’s albums sold was removed from the list?

f. Which statistics were least affected when Thriller’s albums sold was removed from the list?

g. Statisticians often prefer a statistic that is not affected by outliers. Which measure of spread do you think they prefer? The range or the IQR? Explain.

h. Compare the values for the mean and standard deviation (Sx) for the data with and without the Thriller album. Describe how the outlier affected the mean and standard deviation of the data set.

min Q1 median Q3 max range IQR mean Sx




8. Now make a box & whiskers plot on your calculator using the album data from problem 7. Turn

off all plots except for one and use the data in L1 (all albums including Thriller). Adjust the values in the Window menu so that the entire plot shows.

a. First select the icon “Box and Whiskers Showing Outliers.” Make a sketch of what you see

in the space below. Remember to scale your graph.

b. Then select the icon “Box and Whiskers without Outliers.” Make a sketch of what you see in the space below. Remember to scale your graph.

c. When there is an outlier, what is the effect of changing the box-and-whisker display on the length of the whiskers?

d. Now have the calculator make a histogram for the albums sold data. Make a sketch of what you see in the space below.

e. How can you tell from the histogram that there is an outlier in this set of data?



9. Make up a data set with ten values, listed in order, which could have the box-and-whisker plot shown below.

10.

Write a story (context) to go along with your data in (9).



Test Grades

Ms. Smith gave an Algebra test to her three classes. The distribution of scores is as follows:

First Class Second Class Third Class 70 60 60 75 65 65 76 70 66 77 75 66 78 76 66 78 77 67 79 78 69 79 79 72 80 79 75 80 80 77 80 80 83 80 81 85 81 81 88 81 82 91 82 83 93 82 84 94 83 85 94 84 90 94 85 95 95 90 100 100

In today’s activity you will compare the statistics for the three classes. Directions: Each group will first look at data from one of Ms. Smith’s classes. Create a poster to display your results. Include all of the statistics on the poster paper. 1. Calculate the mean, median, mode, range, and standard deviation for your data.

2. Determine the first quartile and the third quartile.

3. Calculate the IQR.

4. Use the 1.5×!"# rule to find the upper and lower fences. Use the fences to determine if there are any outliers.



5. Create a box and whisker plot on a piece of poster paper. Mark any outliers that you found. 6. Construct a histogram on the poster paper. Be sure to label the axes and set an appropriate

scale. 7. Draw conclusions from your data explaining what you have learned about the class’s test

scores. Data Center

Collect data from the other groups and fill in the table.

First Class Second Class Third Class Mean

Median Mode Range

Standard Deviation

Upper Quartile Lower Quartile

IQR Upper Fence Lower Fence

Outliers

8. Which statistics are the same for all three classes? Which are different?

9. What set of test scores has the most spread? Which statistics show this?

10. What do the box-and-whisker plots and histograms tell you about the differences among the three classes?

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Unit 5 Investigation 2 Overview Algebra I Model Curriculum Version 3.0


Introduction to Scatterplots and Trend Lines CCSS: 8-SP 1; 8-SP 2; 8-SP 3; S-ID 6 a, c; S-ID 7 Overview Students are introduced to scatter plots and trend lines, use the equation of trend lines to make predictions, and learn the meaning of interpolation and extrapolation. Students also develop a deeper understanding of slope by interpreting the slopes of trend lines. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able to fit a trend line to data, write an equation for the trend line, and use the equation to interpolate or extrapolate. They should be able to understand the contextual meaning of the parameters of the trend line equation. If the students have not mastered these topics in two days, the upcoming investigations will reinforce these concepts. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.2.1 asks students to determine the equation of a trend line and use it to make a prediction. Exit Slip 5.2.2 asks students to draw a trend line through points, determine the equation of their trend line, interpret the slope of the trend line in the context of the problem, and use the equation of the trend line to make a prediction. Journal Entry prompts students to describe the difference between interpolation and extrapolation.

Launch Notes You may begin this investigation by presenting the Activity 5.2.1 Sea Level Rise (PowerPoint). The presentation contains photographs of planet Earth and Glaciers in Alaska. Lead a class discussion on sea level rise and melting icecaps to increase their awareness of these environmental concerns and stimulate interest in the Sea Level Rise activity. Closure Notes Summarize the concepts that students learned in this activity by asking students to define the following terms: scatter plot, trend line, prediction, interpolation, and extrapolation. Point out that the predictions which arise from individually selected trend lines will vary because each trend line is different due to our individual judgments on what line is the best fit. In the next

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investigation we will learn about the unique line of “best fit” and how to find it using the graphing calculator. Teaching Strategies

I. You my introduce Activity 5.2.1 Sea Level Rise (PowerPoint) prior to using the Activity 5.2.1 Sea Level Rise activity. The PowerPoint presentation presents students with the problem of predicting the change in sea level between 1888 and 2010, and between 1888 and 2020. Data are presented and the presentation models the process of fitting a trend line to data, finding the equation of the trend line, and using the trend line to make predictions. When displaying slide 2 where the variables are defined, check that students understand the variable assignments and can correctly interpret the points in the table. Some students may have trouble understanding that the sea level is being measured since 1888, but the variable is being measured since 1900. You also can ask students whether they see any trends in the data. Slides 5 & 6 prompt students to think about how a scatter plot or trend line can be used to make predictions. Explain to the class that drawing a trend line to fit data is analogous to taking a piece of raw spaghetti and laying it on the graph and adjusting it until it seems to fit the data. You may distribute spaghetti to the students so they can try it for themselves. Slide 8 shows students how to find the equation of a trend line. Before showing students this slide, ask students how we can find an equation of a trend line. Mention that students can use slope-intercept form or point-slope form to find an equation. Slides 9 & 10 present predictions from the trend line. Make sure that students understand that a prediction from a trend line is an estimate and is subject to a person’s selection of a trend line. You may also want to point out that the passage of time did not cause the sea level to rise. The distinction between correlation and causation will be fully addressed later in this unit. Following the PowerPoint presentation, or during the presentation, you can assign students problems in the accompanying worksheet Activity 5.2.1 Sea Level Rise.

Differentiated Instruction (For Learners Needing More Help) Provide students who have difficulty remembering formulas needed to calculate the slope and/or equation of a line a formula reference sheet as a memory aid. Or, students can maintain a “Formula Reference” section in their notebook that includes formulas used throughout the course along with examples of how to use the formulas.

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During the course of Activity 5.2.1 students will have learned that more than one trend line may fit a data set. The only criterion they have been given to find a trend line is in question 4 which states, “Find a line that comes as close as possible to the plotted points and has points on both sides of the line. “ In Activity 5.2.2 Scatter Plots and Trend Lines, students examine more precisely what this means. The first data set relates the heights and weights of 8 NBA superstars. Four different trend lines are drawn, and students must choose which one is best. Discuss the examples given and help dispel common misconceptions, for example that the line must pass through two of the data points or that there must be exactly the same number of data points on either side of the line. After discussion the class should agree that Student 4’s trend line is the best.

Activity 5.2.2 gives students additional practice in finding the equation of the trend line from two points on the graph and using the equation to make predictions. Part of this activity may be assigned as homework.

Differentiated Instruction (For Learners Needing More Help) Have students create a mnemonic device or “rap” for the steps in plotting data and calculating the equation of a trend line.

II. Activity 5.2.3 Television, Homework and Test Scores focuses student attention on

making predictions from trend lines and understanding the distinction between interpolation and extrapolation. Introduce Activity 5.2.3 by having the students predict the correlations of hours of TV watched and percent of homework completed, hours of TV watched and test scores, and percent of homework completed and test scores. After having students share their informal understanding of the relationship between these variables, instruct students to analyze some hypothetical student data. Group Activity Form groups of three or four. Have the students compare their trend lines by comparing the points they chose and their resulting equations. Then have students brainstorm about the definition of interpolation and extrapolation based on context clues. Have one person from each group write their definition on the board.

Differentiated Instruction (For Learners Needing More Help) Have students make a procedure card that lists the steps calculating the equation of a trend line. Allow students to use the card. For Activity 5.2.4 Height and Shoe Size, students collect data on their height and shoe size. The class data is then used to create a scatterplot and students find trend lines to fit the data. Students may provide an estimate of their height. If measuring devises are available, students can instead measure each other’s height. Students may

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want to keep their shoes on. If this happens you should mention that the height measurements will be less accurate. At this point you may use Exit Slip 5.2.1 or 5.2.2 to assess student understanding.

Differentiated Instruction (Enrichment) Question 11 in Activity 5.2.4 asks students to research the difference between women and men’s shoe sizes. Based on their research, they should come up with a recommendation on how to improve the analysis of the data based on these differences.

Differentiated Instruction (Enrichment) You may choose to have students use the Internet to research the definition of the word interpolation and extrapolation in mathematics. They should obtain a definition from at least three websites and bring the definitions and each source to class the next day. Students may share their definitions in small groups and then write a definition in their own words to share with the class.

Differentiated Instruction (Enrichment) Have students come up with data about a relationship they are interested in. For example, the number of calories and grams of fat in breakfast cereals. Suggest that they use the Internet to find data. They should create a scatterplot, draw a trend line, find the equation of the trend line, and use the trend line to interpolate and extrapolate.

Journal Entry Describe the difference between interpolation and extrapolation. Which do you believe is more accurate and why? Which do you believe is more useful and why? Your response should be at least 5 sentences.


• Activity 5.2.1 – Sea Level Rise • Power Point for Activity 5.2.1 – Sea Level Rise • Activity 5.2.2 – Scatter Plots and Trend Lines • Activity 5.2.3 – TV Watching, Homework and Test Scores • Activity 5.2.4 – Height and Shoe Size • Exit Slip 5.2.1—Barometric Pressure • Exit Slip 5.2.2– College Students • Bulletin board for key concepts

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• Student Journals • Raw spaghetti • Measuring tapes or yard sticks • Computer & Projector • Rulers



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The table at the right shows the annually averaged change in the sea level since 1888. Let x be the number of years since 1900 and y be the increase in sea level between 1888 and year x. For example, in 1920 the sea level had risen 3 centimeters over what it was in 1888. Scientists warn that when the sea level reaches 50 centimeters above the 1888 level, catastrophic effects may occur to our seashores and low-lying cities. If present trends continue, when will this occur? 1. Why might we want to predict the sea level

rise? Who might be affected?

Annually Averaged Sea Level Change Since 1888

Year (Since 1900)

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Sea Level Change

since 1888 (in cm)

y

Year, Sea Level Change

(x, y)

0 1.5 (0, 1.5) 10 1 20 3 30 4.5 40 7 50 9 60 11.5 70 12.5 80 13.5 90 14 100 17.5

2. Graph the data on the coordinate axis at the right.

3. Does the data set appear linear or not? Explain.

4. Sketch a trend line on your scatter plot by hand. Find a line that comes as close as possible to the plotted points and has points on both sides of the line. Use a straight edge to draw your trend line.

5. From the graph of your trend line, predict how much the sea level had risen by 2010.

6. From the graph of your trend line, predict how much the sea level will have risen by 2013.



7. Find two points on the trend line and label them A and B on the graph. Write their coordinates below.

A ( , ) B ( , )

8. Use the coordinates of A and B to find the slope of the trend line and an equation for the trend line.

9. Is your trend line equation similar to that of your classmates? Why or why not?

10. Use your equation to check your answers to questions 5 and 6.

11. Use your equation to make this prediction. By which year will the sea level have risen 23

centimeters above the sea level height in 1888?



Scatter Plots and Trend Lines The table and graph give the height (in inches) and weight (in pounds) of some of the NBA’s greatest players.

Sample of Top 50 All-Time NBA Players Player Height Weight Kareem Abdul-Jabbar 86 266 Larry Bird 80 220 Wilt Chamberlain 85 275 Patrick Ewing 84 255 Magic Johnson 83 255 Michael Jordan 78 215 Scottie Pippen 79 228 Isiah Thomas 73 182 (Source: www.nba.com/history/players/50greatest) Several students drew trend lines to fit the data. 1. Student 1 connected the points for Isiah Thomas and Kareem Abdul-Jabbar. Student 2

connected the points for Isiah Thomas and Wilt Chamberlin. Explain why neither of these lines fit the data well.

Trend Line from Student 1 Trend Line from Student 2

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2. Student 3 and Student 4 drew lines that do not pass through any of the data points. Which line is a better fit for the data? Explain.

Trend Line from Student 3 Trend Line from Student 4

3. Find the coordinates of two points on the trend line you chose in question (2) and use them to

find the slope of that line.

a. What does the slope represent in this situation?

b. Find the equation of the line you chose in question (2).

c. Use your equation from part (b) to predict the weight of an NBA player whose height is 71 inches.

d. Use your equation from part (b) to predict the weight of an NBA player whose height is 82 inches.

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City and Highway Fuel Efficiency 4. The table below gives the miles per gallon (MPG) used by some of 2010’s most fuel-efficient

cars in the city and on the highway.

a. Graph the data, using city MPG for x and highway MPG for y.

b. Draw a trend line through the points. Use two points on the trend line to find the

equation of the line.

c. What does the slope represent in this graph?

d. Use your equation from part (b) to predict the highway MPG for a car that gets 38 MPG in the city.

e. Use your equation from part (b) to predict the city MPG for a car that gets 38 MPG on the highway.

Car City MPG

Highway MPG

Toyota Prius 51 48 Honda Civic Hybrid 40 45 Honda Insight 40 43 SmartCar 33 41 Volkswagon Jetta 30 41 Toyota Yaris 29 36 MINI Cooper 28 37 Kia Rio 28 34 Hyundai Elantra 26 35 Honda Accord 22 31



Temperature and Ice Cream Sales The scatter plots below show the relationship between temperature and ice cream sales. The same scatter plot is shown below with four different trend lines.

A B

C D

5. Which of the four lines (A, B, C, or D) do you think is the best fit for these data? Justify

your choice.

6. What does the slope of the trend line tell us about the relationship between the two variables? 7. Does this relationship make sense in the real world context? 8. What was the approximate Fahrenheit temperature on the day ice cream sales were greatest?

(A formula relating Fahrenheit and Celsius temperatures was introduced in Unit 2. Or, if you have forgotten the formula, find it in a reference book or on line.)

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Is the amount of television students watch related to how much homework they complete or how well they score on tests? Scatter plots and trend lines can help answer these questions. 1. The following data were collected on nine students in an algebra class.

a. Create a scatter plot that compares

the hours of TV watched per week (x) to the percent of homework completed (y).

b. Draw a trend line that represents

the data.

c. Write an equation for the trend line.

d. What is the slope of the trend line? What does the slope mean in context of this problem?

e. Use your equation to predict the percent of homework completed by a student who watches 24 hours of television in a week.

f. The prediction in part (e) was an interpolation. What does interpolation mean?

Hours of TV watched per week 32 13 28 19 11 21 15 11 15 Percent of homework completed 58 82 65 87 98 78 75 92 75 Test score 66 85 75 85 100 88 85 90 90



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2. Use the set of data collected on nine students in an algebra class.

a. Create a scatter plot that compares the hours of TV watched per week (x) to the test score (y).

b. Draw a trend line that represents

the data.


d. What is the slope of the trend line? What does it mean in the context of this problem?

e. Use your equation to predict the test score of a student who watches 40 hours of TV in a week.

f. The prediction in part (e) was an extrapolation. What does extrapolation mean?

Hours of TV watched per week 32 13 28 19 11 21 15 11 15 Percent of homework completed 58 82 65 87 98 78 75 92 75 Test score 66 85 75 85 100 88 85 90 90



3. Use the set of data collected on nine students in an algebra class.

a. Create a scatter plot that compares the percent of homework completed (x) to the test score (y).

b. Draw a trend line that represents the data.


d. What is the slope of the trend line? What does it mean in the context of this problem?

e. A student scored an 80 on the test. Use your equation to predict the percent of homework this student completed.

f. Is the prediction in part (e) an interpolation or an extrapolation? Explain. 4. Use the results from questions 1–3 to complete this statement:

“The more a student watches television,

Hours of TV per week watched 32 13 28 19 11 21 15 11 15 Percent of homework completed 58 82 65 87 98 78 75 92 75 Test score 66 85 75 85 100 88 85 90 90

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Height vs. Shoe Size This activity is based on collecting data on height and shoe size from members of your class. Individual Data 1. What is your height in inches? 2. What is your shoe size? Group Data 3. Enter in a table on the board with other members of your class. Then copy the class data in

the table below.

Height (inches) Show size

Height (inches) Shoe size

Height (inches) Shoe size

4. Plot the data points on the graph.

Don’t forget to label your axes.

5. Draw a trend line through the data points.

6. Find an equation for your trend line. Round the slope to the nearest hundredth.

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7. What does the slope mean in the context of this problem?

8. Substitute your shoe size into the equation, to find your predicted height. How close is your prediction to your actual height?

9. Substitute your height into the equation to find your predicted shoe size. How close is it your prediction to your actual shoe size?

10. Use the equation to answer the following questions.

a. Calculate the shoe size of someone who is 61 inches tall. Is this interpolating or extrapolating?

b. Calculate the shoe size of someone who is 41 inches tall. Is this interpolating or

extrapolating? c. Calculate the shoe size of someone who is 81 inches tall. Is this interpolating or

extrapolating? d. Calculate the height of someone who has a shoe size of 8. Is this interpolating or

extrapolating? e. Calculate the height of someone who has a shoe size of 15. Is this interpolating or

extrapolating? f. Calculate the height of someone who has a shoe size of 5. Is this interpolating or

extrapolating?

11. Research the relationship between women and men’s shoes sizes. How do you think the differences in the male and female shoe sizes may affect the conclusion you have reached? What could you do to improve your predictions?


Technology and Linear Regression CCSS: 8-SP 1; 8-SP 2; 8-SP 3; S-ID 6 a, c; S-ID 7; S-ID 8, S-ID 9 Overview Students will develop a deeper understanding of trend lines and predictions. They will use technology (either a graphing calculator or a spreadsheet) to calculate the linear regression equation and to find the correlation coefficient. The students will be able to interpret the meaning of the correlation coefficient. The students will also be able to explain the difference between correlation and causation. Be creative about collecting and finding data – local newspaper, internet searches, etc. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able to find the equation for the line of best fit using technology. They should be able to identify the strength and direction of a trend line using the correlation coefficient. Student should be able to explain the difference between one variable being correlated to the other and one variable causing the other to occur. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.3 asks students to match a graph with a possible value of r. Journal Entry probes students’ understanding of correlation and causation.

Launch Notes Try to start off this investigation with a video to get the students attention and interest. If you don’t have the ability to show videos in your classroom bring in objects related to the upcoming lesson that may get them interested (several old cell phones, fake money, college pamphlets, or pictures of basketball players) Closure Notes Ask students to reflect on how technology is helpful in finding the line of best fit and the correlation coefficient. Then ask them how much information they can obtain from a graph made by hand. They should recognize that without knowing a precise value of r, they can obtain a qualitative understanding of the strength and direction of the correlation just by viewing the graph. Teacher Notes – Background Information on the Correlation Coefficient What is the correlation coefficient? This information is for the teacher’s background knowledge. The “r” value is Pearson’s Sample Correlation Coefficient or just the correlation coefficient. The word correlation refers to the “co-relation” between the two variables being analyzed. The correlation coefficient is one indication of how well the linear regression fits the data. The value

of r is always a number between -1 and 1. It provides two pieces of information: the direction and strength of the linear relationship between the two variables. If r is positive, then the relationship is increasing: as x values increase, so do y. If r is negative, then the data is negatively correlated: as x values increase, y values decrease. The closer r is to 1 or -1, the stronger the linear relationship between the two variables. If all the data points are collinear, non-horizontal, then r = 1 or -1. If there is no linear correlation between the two variables, then r = 0. If r = 0, that does not mean that there is no relationship between the variables, only that the relationship is not linear. For example, data that is in the shape of a circle, a v, or a parabola will have r = 0. Data that is horizontal will also have r = 0, even collinear horizontal data. The magnitude of the correlation coefficient indicates to what extent for any value of x the linear regression is a better estimator of the corresponding value y than is the simple arithmetic mean of all the y data. So, for example, if the slope of the linear regression is zero, then the linear regression can do no better predicting the y variable than the average of the y data, because, for horizontal data, the linear regression is “y = average of the y data.” Some students may ask about the formula for r. If “n” is the number of ordered pairs, ! is the mean of the x values, ! is the mean of the y data, sx is the standard deviation of the x values, sy is the standard deviation of the y values in the ordered pairs that are your data, then ! =∑(!! ¯ˉ!)(!! ¯ˉ!)(!!! )!!∙ !!

. Do not ask students to calculate r by hand except in a statistics course.

It is very important that the diagnostics is turned on for all the graphing calculators. If this is not turned on, then you will not get an r-value. When they run the linear regression Prepare The Calculator

1. Turn on the Diagnostics so that the correlation coefficient r will appear. a. Clear the home screen. b. Press 2nd 0 CATALOG and scroll down to DiagnosticOn then press

ENTER. c. Home screen now shows DiagnosticOn. Press ENTER. d. Home screen shows Done.

There are different sets of graphing calculator directions. The file titled Unit 5 Complete Graphing Calculator Directions is generic, in-depth, and useful for all situations. There are other graphing calculator directions that are specific for certain lessons. You have the choice on how you want to distribute the directions. If your students do not have graphing calculators at home, you can always choose an activity from the previous investigation that you did not do. Teaching Strategies I. Activity 5.3.1 Fitting Lines with Technology is intended to piggyback off the previous investigation. It begins with the same set of data for the height and weight of basketball players as in Activity 5.2.2. This time students find the line of “best fit” with the linear regression program. Students learn that in this program the slope is represented by a (rather than m) and the

y-intercept, as before with b. They may compare the regression equation with the equations they derived by hand in the previous activity.

Technology note: Screens [1] through [8] on pages 1 and 2 of Activity 5.3.1 show students the steps needed to find and plot the regression equation. In screen [2] note that you may choose from Plots 1, 2, and 3, and that the data does not necessarily have to be in lists L1 and L2. You also have a choice of which symbol to use for the data points on the graph. In screen [3] we have instructed students to examine the data to determine the window parameters. The short cut for doing this is Zoom 9 ZoomStat. You may want to avoid introducing this shortcut until you are confident that students can figure out the window parameters on their own. When you select the LinReg (ax +b) feature as shown on screen [5] you should specify which lists the data are in. On some calculators the default is L1 and L2, but students should be encouraged to name the lists instead of relying on the default option. In Screen [7] students have entered the regression equation from the information from screen [6] into the Y= menu, with the parameters rounded off. A short cut is to tell the calculator where to store the regression equation. On some calculators this is done by adding the location to the home screen with the command LinReg(ax + b) L1, L2, Y1. On the Ti-84 Plus there is a prompt on the screen for the regression equation. In both cases you access Y1 by pressing VARSY-VARS- 1 Function 1: Y1.

Students are then introduced to the meaning of r. The scatter plots shown on page 2 are designed to develop an intuitive understanding of how the value of r indicates the strength and direction of the correlation. Students should then classify the correlation between basketball players’ heights and weights as strong and positive. If time permits, students may then look at two other data sets and again find the regression equation and the correlation coefficient. Note that in both of these examples the correlation is again strong although it is positive in problem 7 and negative in problem 8. Point out that the direction of the correlation (positive or negative) is related to the slope of the regression line. You may then introduce Activity 5.3.2 Evolution of the Telephone by showing a video showing the history of the cell phone: http://www.youtube.com/watch?v=JcnXOhrmDB8. If you can project videos, but you are blocked from YouTube, then search for “evolution of the cell phone” on another website that your school does allow. If you cannot show videos, then print out some pictures of old cell phone and put them around the room. Have volunteer students read about the history of the telephone. Have students circle or highlight the row for corded phone the sales in the first table before starting the table at the bottom of page 1 and then transferring the data to lists in the STAT menu. Go through the accompanying slideshow with the students. If you do not have access to

PowerPoint, all the information can be done as board notes. The students may get hung up on question 11 in which they write the equation of the trend line by hand. If the students are struggling, then give them the steps to writing the equation as guidance as was done in the previous investigation.

Differentiated Instruction (For Learners Needing More Help) Have students make a procedure card that lists the keystrokes for plotting data and calculating the regression. Allow students to use the card.

As in the previous activity students are asked to interpret the value of the correlation coefficient r. And as in the previous investigations students continue to make predictions based on the trend line. The last two pages of Activity 5.3.2 ask students to pick another product and perform and analysis similar to the one done for corded phones. Questions 14–20 may be assigned as homework even if students do not have access to calculators at home. They may then use calculators to answer questions 21–24 in class the next day. II. As students continue to find regression lines and interpret correlation coefficients they can now begin to think about the distinction between correlation and causation. Activity 5.3.3 Correlation and Causation provides and opportunity to develop this concept. In the first example one can argue that higher temperatures cause an increase in sales of ice cream, since in warm weather more people are outdoors and looking for cool refreshments. In the second example sales of sunglasses and ice cream may both be related to the weather, but one does not cause the other. In the third example Target sales increase in the later months of the year (probably due to holiday shopping) but in no way to the high sales cause the decrease in sunlight (nor does the decrease in sunlight cause sales to rise). In the final example, however, the decrease in sunlight with the changing seasons does cause a drop in average temperature.

Group Activity Do the first example in Activity 5.3.3 with the entire class and then assign the remaining examples to small groups. Each group can then report their findings to the class. They should report the correlation coefficient and their conclusion as to whether causation is present. The courtroom scene for Shark Attacks (Activity 5.3.4) described below is also suitable for students working in groups.

An alternative way to introduce the distinction between correlation and causation found in Activity 5.3.4 Shark Attacks. There is a strong positive correlation (r ≈ 0.93) between ice cream sales and the number of shark attacks, but confusing correlation with causation could lead to the absurd conclusion that ice cream sales should be banned! There are two variations of this activity. If you want to set up a courtroom scene, use Activity 5.3.4a. Otherwise you can use the worksheet format Activity 5.3.4b, possibly as a homework assignment.

Differentiated Instruction (For Learners Needing More Help) The wording for Activity 5.3.4 Shark Attacks can be simplified. Vocabulary and key words can be defined in the students’ notes. Instead of grouping by math ability, group the students by reading ability.

Differentiated Instruction (Enrichment) To make questions 7-10 more challenging, ask students to come up with their own examples and have other groups give their opinions of them.

Activity 5.3.5 Regression Equation Practice may be used as needed for additional class work or homework. The examples include the relationship between the area of a home and its sale price, the battle against malaria, and how the cost of attending college has increased over time.

Differentiated Instruction (Enrichment) Have the students research the cost of the “top choice” college they want to attend. Is the college they like private or public? Have the students compare their prediction to price of this school.

Journal Entry How do you determine the difference between correlation and causation? Give an example of two variables that will be highly correlated and one variable causes the other to happen. You may come up with your own variables or you may use examples from previous classes (but not today’s class). Explain why these variables demonstrate correlation and causation. Your response should be no less than 5 sentences.


• Activity 5.3.1 Fitting Lines with Technology • Activity 5.3.2 Evolution of the Telephone • Power point for Activity 5.3.2 • Activity 5.3.3 Correlation and Causation • Activity 5.3.4a Shark Attacks Courtroom scene • Activity 5.3.4b Shark Attacks Worksheet • Activity 5.3.5 Regression Equation Practice • Exit Slip 5.3 - Scatter Plots and Correlations • Bulletin board for key concepts • Graphing Calculators • Student Journals • Projector • Computers • Rulers



Finding the Line of Best Fit with Technology We have learned how to fit a trend line to a scatter plot and find an equation of the trend line. In this activity we will use the calculator to perform both tasks. The trend line you will find with the calculator is called the regression line or line of best fit. The table below, from Activity 5.2.2, gives the height (in inches) and weight (in pounds) of some of the NBA’s greatest players. a. Enter the height data in L1 and the weight data in

L2. Do not sort the data because the height and weight of each player need to stay together. Your lists should look like figure 1 below.

b. You can graph these points by turning on a STAT

PLOT, choosing a scatter plot, and using L1 and L2 for the lists. See figure 2 below.

c. To see your scatter plot, you first need to set your

window. We put the height data in L1, which the calculator is using as the x-coordinates, so x needs to go from 73 to 86. Set the x-scale to 2.

Since y contains the weights, it needs to go from 182 to 266. Set the y-scale to 10. It is always a good idea to set your window a bit bigger to see all the points. See figure 3 below.

d. Hit the GRAPH button to see the scatter plot. See figure 4 below. Figure 1 Figure 2 Figure 3 Figure 4

e. Your calculator can find the equation of the line of best fit through a process called linear

regression. This is a statistical calculation, so the command is found in the STAT ► CALC menu ▼ 4:LinReg(ax+b). Note that the calculator uses a for the slope and b for the y-intercept. Hit ENTER to bring the command to the home screen and ENTER again to run the LinReg(ax+b) command. This command is shown in figure 5 on the next page.

Player Height Weight Kareem Abdul-Jabbar 86 266

Larry Bird 80 220 Wilt Chamberlain 85 275 Patrick Ewing 84 255 Magic Johnson 83 255 Michael Jordan 78 215 Scottie Pippen 79 228 Isaiah Thomas 73 182



Figure 5 Figure 6 Figure 7 Figure 8 The output of the LinReg(ax+b) command is shown in Figure 6. a ≈ 7.03 is the slope of the regression line and b ≈ –332.45 is the y-intercept of the regression line. Using these values, the equation of the regression line is ! = 7.03! − 332.45. To graph the regression line, type the equation into the Y= menu as shown in figure 7. Hit Graph. The regression line will appear in the same window as the scatter plot as shown in figure 8. The statistics r2 and r in the LinReg(ax+b) measure the correlation between the two variables. r is called the correlation coefficient. It measures the strength and direction of the linear relationship between two variables. The sign (positive or negative) of r indicates the direction of the linear relationship. If r is positive, the slope of the regression line is positive, and if r is negative, the slope of the regression line is negative. The value of r measures the strength of the linear relationship. If r is close to 1 or -1, the data has a strong linear relationship. Scatterplots are shown below with their corresponding values of r.

Perfect Positive

Correlation

High Positive

Correlation

Low Positive

Correlation No

Correlation Low

Negative Correlation

High


Perfect


! = 1 ! = 0.8 ! = 0.3 ! = 0 ! = −0.3 ! = −0.8 ! = −1 1. Circle the word(s) that describe the correlation between basketball players’ height and

weight. Explain your choice.

strong weak none

positive negative



For each data set below, state an appropriate window, make a scatter plot, and find and graph the line of best fit using your calculator. Note the correlation of the two variables in each example. Show your teacher each completed screen. 2. The table below from Activity 5.2.2 gives the city and highway miles per gallon (MPG)

used by some of 2010’s most fuel-efficient cars. Enter city MPG in L1 and highway MPG in L2.

Car City MPG Highway MPG

Toyota Prius 51 48 Honda Civic

Hybrid 40 45

Honda Insight 40 43 SmartCar 33 41

Volkswagon Jetta 30 41 Toyota Yaris 29 36 MINI Cooper 28 37

Kia Rio 28 34 Hyundai Elantra 26 35 Honda Accord 22 31

a. Window: Xmin _____ Xmax _____ Xscl _____

Ymin _____ Ymax _____ Yscl _____

b. Equation of the Line of Best Fit:

y = ___________ x + ____________

c. What is the value of r?

d. Circle the word(s) that describe the correlation between city MPG and highway MPG. Explain your choice.

strong weak none

positive negative

e. What is the predicted highway MPG for a car that gets 38 MPG in the city?



3. The table below shows the number of viewers (in millions) and the rank for a sample of shows from the 2007-2008 season. Enter millions of viewers in L1 and rank in L2.

Name of Show Viewers (millions) Rank

How I met your Mother 8.21 70

New Amsterdam 8.85 60 Deal or no Deal 9.72 50 Sunday Night NFL Pre-Kick Off 10.64 40

Law and Order: SVU 11.33 30

Without a Trace 13.13 20 Grey’s Anatomy 15.92 10

a. Window: Xmin _____ Xmax _____ Xscl _____

Ymin _____ Ymax _____ Yscl _____


y = ___________ x + ____________

c. What is the value of r?

d. Circle the word(s) that describe the correlation between viewers (in millions) and rank. Explain your choice.

strong weak none

positive negative

e. What is the predicted rank of a show with 5 million viewers?



Evolution of the Telephone Most people know that Alexander Graham Bell is commonly credited as the inventor of the telephone. In 1876, Bell was the first to obtain a patent for an "apparatus for transmitting vocal or other sounds telegraphically". Most people don’t know that the first cordless phone was patented in 1959 by Dr. Raymond P. Phillips Sr., an African American inventor from Terrell, Texas. The first cellular network in the world was built in 1977 in Chicago, IL. In 1979 the first cellular network (the 1G generation) was launched in Japan. With the evolution of cell phones, the sales of corded phones slowly phased out. Consumer Electronics and Electronic Components – Factory Sales by Product 1990 to 2005 [In millions of dollars (11,021 represents $11,021,000,000).]

CATEGORY 1990 1995 2000 2001 2002 2003 2004 2005 Home office products, total 11,021 24,140 36,854 34,924 33,505 38,282 41,770 45,032

Cordless telephones 842 1,141 1,562 1,960 1,261 1,268 1,134 1,017

Corded telephones 638 557 386 294 266 256 259 261 Telephone answering devices 827 1,077 984 1,062 1,060 1,210 1,247 1,426

Caller ID devices (NA) (NA) 54 35 20 12 13 9

Personal computers 4,187 12,600 16,400 12,960 12,609 15,584 18,233 18,215

Computer printers (NA) 2,430 5,116 5,245 4,829 4,734 4,053 3,858 Aftermarket computer monitors (NA) 879 1,908 2,173 1,670 1,856 2,214 2,315

Modems/fax modems (NA) 770 1,564 1,564 1,445 1,419 1,465 1,525

Computer peripherals 1,980 816 1,950 2,150 2,256 2,707 3,032 3,575

Computer software 971 2,500 4,480 5,062 4,961 5,060 5,162 5,250

Personal word processors 656 451 240 97 36 13 6 5

Fax machines 920 919 386 349 297 242 186 128

Digital cameras (NA) (NA) 1,825 1,972 2,794 3,921 4,739 7,468 Source: 2007 Statistical Abstract of the United States, U.S. Census Bureau 1. Use the corded telephones sales (in millions of dollars) to complete the table below.

Year Years since 1990 Sales of Corded Phones (in millions of dollars)



2. Use technology to find the linear regression line. Identify the slope and y-intercept below. Round each number to the nearest hundredth. (Slope) a = (y-intercept) b =

3. Identify the equation of the linear regression line. The equation is of the form ! = !" + !.

4. Use the equation to estimate the sales in 1997. 5. Is the prediction in question (4) an interpolation or an extrapolation? Explain.

6. What is the value of the correlation coefficient r?

7. What does the value of r indicate about the strength and direction of the data?

8. What does r tell us about the use of corded phones since 1990? Explain.

A graph of the corded phone sales over time is shown below.

9. Draw a trend line on the graph above that you feel fits the data points.

150 2 4 6 8 10 12

700

0

100

200

300

400

500

600

Years since 1990

Cor

ded

Pho

ne S

ales

(in

mill

ions

)



10. Identify two points on your line.

11. Find the equation of your line.

12. Compare the line you just created to the linear regression line which you calculated in question (3) by answering the following questions: a. Compare the y-intercepts. How close is your line’s y-intercept to the regression line’s

y-intercept?

b. Compare the slopes. How close is your line’s slope to the regression line’s slope?

c. Is your line to steep or too flat? Explain how you came to your conclusion.

d. Based on these answers, do you believe you drew a good line of fit? Explain.

13. Use the equation you created to determine the year when the sales of corded phones will drop

to 100 million dollars? Is this interpolation or extrapolation?



Go back to the table on page 1 and select a different product. Complete the table below using the sales information for this product. Write the name of the product above the table.

Product:

Year Years since 1990 Sales (in millions of dollars)

14. Draw a scatterplot of the sales of the product over time in the coordinate plane below. Label

and scale the axes appropriately.

15. Draw a trend line on the graph above that you feel fits the data points.

16. Circle the word(s) that describe the correlation between the sales of the product and the years

since 1990. Explain your choice. strong weak none

positive negative



17. Identify two points on your line.

18. Find the equation of your line.

19. Predict the sales of this product in the year 2015.

20. If you are offered a job to be the head of a marking department that sells this product, would

you accept it based on research you just did? Do you think your job at this company would be secure for a long time?

21. Use technology to find the equation of the linear regression line.

22. Calculate the correlation coefficient r. Does this value of r confirm your prediction in question (14)?

23. How does your line compare to the linear regression line which you found in question (16)?

24. Did you make a good trend line? Explain.



Correlation and Causation As you have seen in the previous activities, the linear correlation between two variables can be measured by the correlation coefficient, r. The correlation coefficient may indicate that the linear relationship is strong or weak, and it may indicate that the trend has a positive or negative slope. The graphs below, introduced in Activity 5.3.1, show values of r for different scatterplots.

Perfect Positive

Correlation

High Positive

Correlation

Low Positive

Correlation No

Correlation Low


High


Perfect


! = 1 ! = 0.8 ! = 0.3 ! = 0 ! = −0.3 ! = −0.8 ! = 1 In addition to knowing whether two variables are correlated, statisticians often want to know if there is causation in the relationship. Causation occurs if changes in one variable cause a change in the other variable? It is difficult to prove causation. To prove causation we must perform a carefully designed experiment that measures the effect of changing a variable on another variable. All other variables must be fixed. 1. An ice cream shop keeps track of how much

ice cream they sell versus the temperature for 12 days.

a. Imagine that a trend line is drawn. Is the

slope positive or negative?

b. Look at the graphs at the top of the page and describe the strength of the correlation (strong, weak, or none).

c. Do you think the hotter temperatures caused an increase in ice cream sales? Explain.

2610 12 14 16 18 20 22 24

700

0100200300400500600

Temperature (degrees Celsius)

Ice C

ream

Sales

($)



2. The Ice Cream shop tracks how many sunglasses were sold by a big store on several days and they compare the number of sunglasses sold each day to

their ice cream sales on that day.

a. Describe the direction of the correlation.

b. Describe the strength of the correlation.

c. Do you think that the sales of sunglasses could have caused an increase in ice cream sales? Explain.

3. The table shows the 2010 monthly sales (in billions) at Target Stores and the length of

sunlight in hours and minutes on the last day of each month in Hartford, CT.

Target Sales vs. Daylight Hours Month Sales

(in billions) Daylight

(hours: min) Daylight

Hours July 4.6 14:26 August 5.0 13:10 September 5.6 11:48 October 4.6 10:25 November 6.0 9:24 December 9.9 9:12

a. The amount of daylight is given in hours and minutes. Fill in the last column with the

daylight hours, rounded to the nearest tenth.

b. Enter the data from the “Sales” column into L1 and the data from the “Daylight Hours” column into L2. Find the regression line and the value of r.

c. Describe the strength and direction of the correlation.

250100 125 150 175 200 225

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100

200

300

400

500

600

Sunglasses Sold

Ice

Cre

am S

ales

($

)



d. Why might these variables be correlated?

e. Do you think that the increase in sales at Target could have caused shorter days? Explain.

f. Do you think that shorter days could have caused an increase in sales at Target? Explain. 4. The table shows the length of sunlight in hours and minutes on the last day of each month in

Hartford, CT and the average daily temperatures (2010).

Daylight Hours vs. Average Temperature Month Daylight

(hours: min) Daylight

Hours Average

Temperature July 14:26 79 August 13:10 75 September 11:48 69 October 10:25 55 November 9:24 44 December 9:12 30

a. Fill in the column for daylight hours from the table in problem 3.

b. The data from the “Daylight Hours” column should be in L2. Enter the data from the “Average Temperature” column in L3. Using lists L2 and L3, find the regression equation and the value of r.

c. Describe the strength and direction of the correlation.

d. Do you think that the decrease in daylight could have caused colder days? Explain.



Shark Attacks Courtroom Scene

Judge (Leader): _______________________ Plaintiff (Materials): ______________________

Defendant (Liaison): ___________________ Dr. Data (Recorder): ______________________

This court case involves a lawyer (the plaintiff) in a fight to ban ice cream shops at the beach. Please assign the roles to everyone in your group, if you haven’t done so yet.

Judge [Cough] I see in the reports that the plaintiff wants to ban ice cream from all beaches. What is your opening statement?

Plaintiff I strongly believe that ice cream causes shark attacks. The more often people buy

ice cream, the more it makes sharks want to attack us. This is why it is necessary for us to ban ice cream from all beaches. I have data to support this. I would like to call my first witness to the stand, Dr. Data.

Dr. Data I have with me a table that gives the last year’s sales for an ice cream shop in

Florida and the number of shark attack in Florida from 1926-2011.

Month Ice Cream Sales ($)

Number of Shark Attacks

January 430 5 February 570 13 March 810 36 April 940 61 May 1100 47 June 1240 58 July 1340 73 August 1390 84 September 1240 95 October 1030 68 November 810 43 December 430 10

(Source: http://www.flmnh.ufl.edu/fish/sharks/statistics/FLmonthattacks.htm)

Dr. Data I have created a scatter plot from the table. (The scatter plot is on the next page.) (Task 1) Everyone in the courthouse needs to draw a trend line on the scatter plot

below. This will summarize the relationship between ice cream sales and shark attacks.



1,6000 400 800 1200

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Ice Cream Sales ($)Sh

ark

Atta

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Plaintiff As the data clearly shows, there is a strong positive correlation between ice cream and shark attacks.

Defendant Objection, your honor! Scatter plots drawn by hand are just estimations. Therefore,

this does not prove that ice cream causes shark attacks. Judge Sustained. We need to use graphing calculators to get an exact correlation. Dr. Data When everyone in the group has a graphing calculator we could use the data to

find the coefficient of correlation, r. This number would show that ice cream needs to be banned. [At this point, everyone in your group has a graphing calculator. Enter the data to find the regression line and the correlation coefficient.]

Plaintiff (Task 2) We all got a correlation coefficient value, r = ________.

(Task 3) Thus, the more people buy ice cream, the

________________________________________________________________.

So, ice cream causes shark attacks. The plaintiff rests, your honor.

Judge Does the defendant have anything to add?

Defendant Yes. I am here to tell you that correlation is not the same as causation. Just

because two variables are related does not mean that one caused the other to occur. There may be a third factor, which may or may not be known.



Dr. Data Now I see. Ice cream sales don’t cause shark attacks to occur! Ice cream is usually sold during the summer, and it is during the summer that people are more likely to go swimming. The increased shark attacks are simply caused by more time in the water, not ice cream.

Judge I have come to a verdict. I rule in favor of the defendant. Just because two variables are correlated does not mean that one caused the other to happen. Let’s all try a practice problem to see if we understand.

Dr. Data I have lots of data to look at! The table contains records on the number of car

accidents compared to the amount of snowfall for the town of Rocky Hills.

Inches of Snow 0 1.5 4 5 0.5 7 12 4 1.5 2.5 5.5 5.5 9 3 4.5

Accidents 2 5 6 5 1 10 15 3 5 2 6 7 11 4 8 Judge I would really like to see what the value of the coefficient of correlation is.

[Everyone enters this data into their graphing calculator and finds r] Dr. Data (Task 4) There is a strong correlation. r = ________. But now I’m confused.

Does this mean that that the snow causes more accidents or not?

Defendant It appears as if causation seems to require more than just correlation. Plaintiff What are you talking about? Judge Suppose that someone gets into an accident. He or she thinks that the cause was

the snow. To prove this, thinks about the same person driving the same road with similar distractions, but it is not snowing.

Dr. Data If we could rewind history and change one small thing (making the snow stop),

then causation could be observed. Plaintiff But we can’t rewind history. It is impossible to directly observe causation. I could

argue that aliens cause the ice to form and the cars to crash.

Defendant But now you are arguing something that is unlikely. The icy roads probably

caused the increase in car crashes. In fact, it’s this ability to make assumptions that make people able to “conduct experiments”.



Dr. Data Wow I learned a lot today! I think we should play a game of true or false! Everyone Hooray! Judge (Task 5) The shorter the time someone holds a driver’s license, the more likely

she is to have an accident. Therefore, new drivers cause more accidents. True/False? Why?

Plaintiff (Task 6) The younger a driver is, the higher his or her car insurance premiums.

Therefore, young age causes high premiums. True/False? Why? Defendant (Task 7) Increase of flu is correlated with lower outside temperature. Therefore,

cold temperatures cause flu. True/False? Why? Dr. Data (Task 8) Higher incidence of lung and mouth cancer is correlated with increased

use of tobacco products. Therefore, tobacco causes cancer. True/False? Why? Judge Let’s go ask a teacher if our answers are correct! Everyone Hooray again!



Shark Attack Worksheet

Shark Attacks Did you know that selling ice cream causes shark attacks?

Month Ice Cream Sales ($)

Number of Shark attacks

January 430 5 February 570 13 March 810 36 April 940 61 May 1100 47 June 1240 58 July 1340 73 August 1390 84 September 1240 95 October 1030 68 November 810 43 December 430 10

(Source: http://www.flmnh.ufl.edu/fish/sharks/statistics/FLmonthattacks.htm) 1. Draw the trend line in the graph provided. 2. How would you describe the correlation between ice cream sales and shark attacks?

3. In case there are any nay-sayers still in the room, use the graphing calculator to calculate the

correlation coefficient, r.

4. Complete the statement below:

“This means that the more people buy ice cream, the

5. Does this mean that buying more ice cream causes shark attacks?

1,6000 400 800 1200

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Ice Cream Sales ($)Sh

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Car Accidents Let’s try another example that will help separate correlation from causation. The table contains records on the number of car accidents compared to the amount of snow fall for the town of Rocky Hill.

Inches of Snow 0 1.5 4 5 0.5 7 12 4 1.5 2.5 5.5 5.5 9 3 4.5 Accidents 2 5 6 5 1 10 15 3 5 2 6 7 11 4 8 6. Use the graphing calculator to calculate the correlation coefficient, r. 7. Does this mean that more snow on the ground causes accidents? Explain your reasoning.

Here are four more examples to practice identifying the difference between correlation and causation. 8. The shorter the time someone holds a driver’s license, the more likely she is to have an

accident. Therefore, new drivers cause more accidents. True/False? Why?

9. The younger a driver is, the higher his or her car insurance premiums. Therefore, young age causes high premiums. True/False? Why?

10. Increase of flu is correlated with lower outside temperature. Therefore, cold temperatures cause flu. True/False? Why?

11. Higher incidence of lung and mouth cancer is correlated with increased use of tobacco products. Therefore, tobacco causes cancer. True/False? Why?



Regression Equation Practice

1. The table shows the square footage and purchase price of several recently sold homes in Connecticut.

Date of Sale Square Feet Price ($ in thousands) December 16, 2010 3654 705 December 17, 2010 4590 650 December 21, 2010 2011 245 December 21, 2010 1514 275 December 29, 2010 1822 318 January 28, 2011 2480 385 January 31, 2011 2278 278 February 14, 2011 1917 183 February 15, 2011 1361 212 February 18, 2011 2617 385

a. Use your calculator to make a scatter plot and find the regression line for these data.

b. Interpret the slope of the regression line.

c. Describe the strength and direction of the regression line.

d. Use the regression equation to predict the sales price of a house with 800 ft2. Is this interpolation or extrapolation?

e. Use the regression equation to predict the sales price of a house with 3000 ft2. Is this interpolation or extrapolation?

f. Do you think the increase in square footage might cause the increase in price? Explain.

g. What other variables might cause an increase in the price of a home? Explain.



2. The table below gives the average costs of higher education (cost of college) for several years in between 1986 to 2007. Complete the table below by filling in the blank column. Select a good graphing window, draw a scatter plot of the data, calculate equations for the lines of best fit, and answer the questions about the situation.

Year Years since 1986 Cost of Public Colleges Cost of Private Colleges 1986 3859 9228 1996 7014 17612 2001 8653 21856 2002 9196 22896 2003 9787 23787 2004 10674 25083 2005 11426 26257 2006 12108 27317 2007 12805 28896

3. Average Cost of Public Colleges vs. Years Since 1986 a. Window: Xmin ________ Xmax ________ Xscl ________

Ymin ________ Ymax ________ Yscl ________


c. Describe the direction and strength of the correlation.

4. Average Cost of Private Colleges vs. Years Since 1986 a. Window: Xmin ________ Xmax ________ Xscl ________

Ymin ________ Ymax ________ Yscl ________


c. Describe the direction and strength of the correlation.



5. Use the equations from questions (3) and (4) to answer the following questions: a. Find the average cost of a public college for the year after you graduate high school. Is

this an interpolation or an extrapolation?

b. Find the average cost of a private college for the year after you graduate high school. Is this an interpolation or an extrapolation?

c. Find the average cost of a public college in 2045. Is this an interpolation or an extrapolation?

d. Find the average cost of a private college in 2045. Is this an interpolation or an extrapolation?

e. Find the year when the average cost of a public college was $5000. Is this an interpolation or an extrapolation?

f. Find the year when the average cost of a private college was $19,000. Is this an interpolation or an extrapolation?

g. Find the average cost of a private college in 1990. Is this an interpolation or an extrapolation?


Explorations of Data Sets CCSS: 8-SP 1; S-ID6; S-ID 8 Overview In this investigation, students will be collecting and analyzing in interdisciplinary activities. In this investigation, students will apply all of their knowledge from the previous two investigations. The role of the teacher is to get the class prepared and organized, and then will walk around to observe and ask questions. There are many activities to choose from, but it is important that you do at most 4 days of activities. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able to answer a question about the world that can be analyzed with bivariate data. For student generated data, the students will be able to use technology to calculate the regression equation and correlation coefficient. The students will be able to solve an equation for y given x and solve for x given y. They will be able to explain the meaning of slope and intercepts in context and distinguish between data that is correlated compared to casual. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.4 asks students to make a prediction based on the regression equation they have found for the ulna-height data collected in class. Journal Entry asks students to describe the process of making a prediction based on a set of data that have been collected.

Launch Notes All of these lessons include manipulative that should be displayed in the front of the class so as the students enter that ask “Are we doing something with those ______ today?” Closure Notes Once the investigation is complete be sure they reflect on the work they have done. Is it valid? How useful was technology? Did our correlation guarantee causation? As interesting as the applications may be, continue to emphasize and review the math content and skills that were applied. This may be done with Exit Slip 5.4 – 321 blastoff and Journal Entry 5.4.

Teaching Strategies I. Begin this investigation with Activity 5.4.1 Forensic Anthropology. This will take two days to complete. Start by showing the Power point activity to give student background information. The PowerPoint concludes by posing a problem to the students: How do you find the height of a person whose skeletal remains include an ulna and no other long bones? Instead of or in addition to the PowerPoint, you could bring in a variety of washed and boiled bones from the butcher. Another prop might be dolls and action figures. Once the slide show is completed, distribute the activity sheet. Before breaking into small groups, read the introductory paragraph, complete step 1, and discuss step 2. Then break up into groups of 4 To gather data, you might want to give each group a tape measure (for height) and a ruler (for ulna length). Have each student write the average of his or her height and ulna data at a central collection place such as the blackboard, interactive whiteboard, on a computer spreadsheet or on an overhead transparency. It may help your class if you have other data included (i.e. principle, other teachers, or your own data). Tell the students to record all their classmates’ data on their activity sheets. Let the small groups struggle for short while with the scale. This will lead to a discussion on how the tight clustering of the data points will lead to inaccuracy when getting the equation by hand. If you have traditional classes, all students should finish the first two pages before coming into class the second day. When the students begin using the calculators, they are allowed to use their sheets from previous lessons. If the students do not have calculator directions, then you may want to give them some. Compare the results from the calculations by hand with those from the technology. Discuss the advantages and disadvantages of using technology. Note that you would want to use technology if you had “messy” data, if you were making a presentation, and to find as good a linear model as possible. Technology will find the line of best fit. If your school has access to the internet, or your students have access at home, then the extension questions on page 4 will be a great way to verify their work. The linear regression equations developed by Professor Trotter are found at http://www.ehow.com/how_5611616_determine-height-through-skeleton.html. One example of the Trotter equations for determining stature is “Stature of white female = 4.27 · Ulna + 57.76 (+/- 4.30)” where m = 4.27 centimeter change in height for every centimeter change in stature and b = 57.76. There are many other examples online as well. If your students do not have this access readily available, then you should skip the extension and have the students work on the letter to the commissioner of the police department.

Differentiated Instruction (Enrichment) Which is better correlated with height? Length of foot, shoe size or length of the ulna? Students may measure foot length and record shoe size, then plot height as a function of each. Should female data be grouped with male data for shoe size? Research the history of detective work and how footprints or shoe prints are used to help identify criminals and solve crimes.

Exit Slip 5.4 asks students to make predictions based on the data they have collected in Activity 5.4.1. II. Following the activity on forensic anthropology, you should have students work on at least one other activity that requires them to collect data and apply a linear model. Here are several options. Activity 5.4.2 Rubber Bands requires a lot of room for the students, so this may have to be done in a hallway. For the students’ safety be sure that they stand clear of the flying rubber bands. This activity will take a long time to collect the data. Discuss with the students why it is more accurate to take the average of three trials as opposed to just shooting the rubber band once. This is a good time introduce the concept of “outlier” in the context of two-variable data; this will be more fully developed in the next investigation. Students have the option of using the calculator or finding a line of best fit by hand. As the students are working on questions 4 and 5, have them identify examples that involve interpolation and those that involve extrapolation. You may conclude by discussing sources of error that might make their predictions inaccurate.

Differentiated Instruction (For Learners Needing More Help) Lessen the time it takes to input data by transferring data to student calculator lists by linking or to student computer with a flash drive. This will enable students to focus on the essential tasks.

Activity 5.4.3 Stadium Wave. There are two versions to this activity. Activity 5.4.3a is more open ended and less scaffolded than Activity 5.4.3b. This activity becomes very funny while collecting data. It is much harder to do the wave than you would anticipate. Once all the data are collected they may work in groups to come up with an equation. They have the choice of creating an equation by hand or using technology. To address Problem 2 in Activity 5.4.3a, they will need to use the internet to find the number of seats that are around the top row of Fenway Park. Require students to submit individual reports. As in a lab in a science class, all the data collection and computation may be done in teams, but the lab reports must all be done individually. Activity 5.4.5 Balloons requires a 9-inch (diameter) balloon for every student, one 12-inch balloon, string for each group, and meter stick for each group. Read the active question and procedure with the students. Demonstrate one example so the students know the approximate location of the circumference. Let the groups work on the rest on this activity with little guidance. The students can find the answer to number 5 either by creating an equation by hand or with the use of technology. When you think every group has made their predicted time, then gather the class’s attention and take out the 12-inch balloon. Even if every group is not up to number 8, you should still perform the actual test and then have students go back to questions 7–9 if they need to. For Activity 5.4.5 Walking Away, put the students into groups of 3. They can take steps any size they want, but remind them to try to stay consistent with every trial. To expand this, you

may want to make the students find the equations both by hand and with technology. Although all the questions are written on the paper, the teacher should ask each group different questions. Is this an example of interpolation or extrapolation? How would you describe the relationship between the two variables? What is the strength and direction of the correlation? Why did you choose to get the equation the way you did?

Group Activity Every example in this investigation is a group activity. Try to rotate the way you make your groups. Do one activity with high students grouped with low students. Do another activity with students grouped by ability in which higher levels have less scaffolding then lower level groups.

III. Two additional activities are provided for this Investigation. Unlike the previous ones, Activity 5.4.6 Population and Representation does not involve experimental data and can be assigned for homework if students have computer access. As an alternative you can provide them with this data set showing the population of each state and the number of representatives it elects to the U.S. House of Representatives.

State Population Rep. State Population Rep. Alabama 4,779,735 7 Montana 989,415 1 Alaska 710,231 1 Nebraska 1,826,341 3 Arizona 6,329,013 9 Nevada 2,700,551 4 Arkansas 2,915,921 4 New Hampshire 1,316,472 2 California 37,253,956 53 New Jersey 8,791,894 12 Colorado 5,029,196 7 New Mexico 2,059,180 3 Connecticut 3,574,097 5 New York 19,378,104 27 Delaware 897,934 1 North Carolina 9,535,475 13 Florida 18,801,311 27 North Dakota 672,591 1 Georgia 9,687,653 14 Ohio 11,536,502 16 Hawaii 1,360,301 2 Oklahoma 3,751,354 5 Idaho 1,567,582 2 Oregon 3,831,074 5 Illinois 12,830,632 18 Pennsylvania 12,702,379 18 Indiana 6,483,800 9 Rhode Island 1,052,567 2 Iowa 3,046,350 4 South Carolina 4,625,364 7 Kansas 2,853,118 4 South Dakota 814,180 1 Kentucky 4,339,362 6 Tennessee 6,346,110 9 Louisiana 4,533,372 6 Texas 25,145,561 36 Maine 1,328,361 2 Utah 2,763,885 4 Maryland 5,773,552 8 Vermont 625,741 1 Massachusetts 6,547,629 9 Virginia 8,001,024 11 Michigan 9,883,635 14 Washington 6,724,540 10 Minnesota 5,303,925 8 West Virginia 1,852,996 3 Mississippi 2,967,297 4 Wisconsin 5,686,986 8 Missouri 5,988,927 8 Wyoming 563,626 1

Since the U.S. Constitution provides for representation based on population, students should find a strong positive correlation between the two variables. In class discussion you may draw upon what students have learned from their study of American government in their social studies classes. (You may want to point out that there are 435 Representatives. Based upon its proportion of the total U.S. population Connecticut had six representatives up through the year 2000, but based on the 2000 and 2010 censuses it now has only five.) Finally Activity 5.4.7 Conducting an Experiment provides students with the opportunity to design and conduct their own experiment. Each group will need a stop watch and possibly other materials such as a tape measure, jump rope or step stool. If aerobic items are not available they can always do jumping jacks, push-ups, run in place, or any other exercise. Explain to the class that they are going to be put into groups and that each group need to think of a way to create an experiment that shows the relationship between one specific exercise and teenagers’ heart rate. Split the class into 4 groups A, B, C, and D and give students no more than 10 minutes to reach an agreement of what type of exercise they will do. In this experiment, the group may choose either to have one person do every trial or to have different students conduct the trials. Either approach is acceptable; just make sure that the groups include in their report all variables that may affect the results. There are three important pieces to this activity that the students must pay attention to. 1) The groups must vary the amount of exercise, either by varying the time or the number of number of reps for each trial, 2) each student should have a resting heart rate before doing the next trial, and 3) they have to count the heart rate for the same number of seconds after every trial. (always 10 seconds, 30 seconds, 1 minute, etc.)

Journal Entry Select an activity in class in which you collected data and describe how you were able to use the data to make a prediction.


• Activity 5.4.1 Forensic Anthropology • Power point for Activity 5.4.1 • Activity 5.4.2 Rubber Bands • Activity 5.4.3a Stadium Wave • Activity 5.4.3b Stadium Wave (scaffolded) • Activity 5.4.4 Balloons • Activity 5.4.5 Walking Away • Activity 5.4.6 Population and Representation • Activity 5.4.7 Conducting an Experiment • Exit Slip 5.4

• Bulletin board for key concepts • Student Journals • Graphing Calculators • Projector • Computer lab • Rulers • Several yard/meter sticks or several tape measures • Rubber bands (400-500) • Several stopwatches or the ability to project online stopwatch (or use a cell phone) • Masking tape • Several pieces of 2-foot long rope. Different diameters • 9-inch balloons for every student in the class. • 12-inch balloon for the teacher. • Small aerobic exercise equipment.



Forensic Anthropology

While excavating for the new school building, construction workers found partial skeletal human remains. Who is this person? Is the person a male or a female? When did the person die? Forensic anthropologists were called in on the case to analyze the bones and help answer these questions. Police want to know the person’s height in order to match the person with missing persons on file. Can you estimate the person’s height from his or her skeleton bones?

The long bones such as the femur (thigh), tibia (shin) and ulna (forearm) predict height better than the shorter bones. The only intact long bone is the ulna, which is 28.5 centimeters in length. Your job as the forensic anthropologist’s assistant is to estimate the height of the mystery person whose bones were found.

Step 1: STATE in a full sentence what you are going to find.

Step 2: COLLECT DATA on the height and ulna length from everyone in your class.

• Measure and record the ulna (forearm) lengths and heights of each of your classmates.

• Place your elbow on the table with the thumb pointed toward your body.

• Have the classmate measure from the round bone in your wrist, just below your pinky finger, to the bottom of your elbow, which is resting on the desk.

• People may measure differently, so it is best to have two different people measure your ulna. Use the average of the two measurements.

Measurement 1 = Measurement 2 =

Average of 2 Measurements =

Name Ulna Length (cm) Height (cm) Name Ulna Length (cm) Height (cm)



Step 3: PLOT the data and find a trend line by hand from the graph.

a. Which is the independent variable?

b. Which is the dependent variable?

c. Graph the data on the coordinate axis below. Create a nice window for your graph with an appropriate scale. Label the axes and give the graph a title.

d. Does the data appear linear or not? Explain.

e. Sketch a trend line on your scatterplot by hand or using a straight edge. Find the equation of the trend line by hand. Show your work here.

f. Is your trend line equation similar to that of your classmates? Why or why not?



Step 4: USE TECHNOLOGYTO CALCULATE the regression line and correlation coefficient.

a. Write the regression line you found using technology.

b. Write the correlation coefficient r you found using technology.

c. Based on your correlation coefficient, comment on the direction and the strength of the linear relationship between ulna lengths and height.

d. Compare the hand-calculated trend line from step 3e with the regression line you found using technology in step 4a. How close are the two slopes and y-intercepts? What are the advantages or disadvantages of doing the work by hand or using technology?

Step 5: Make a PREDICTION.

a. Use the linear regression line that you found using technology to estimate the height of the missing person. Remember that the found ulna bone is 28.5 centimeters long. Show your work.

b. How accurate or reliable is your prediction? Explain your answer by referring to the

graph and the correlation coefficient r.



EXTENSIONS:

1. Your boss asked you to do some research online to see if there is already a formula or maybe a table that can be used to determine someone’s height from his or her ulna bone length. What is this formula?

2. Use the anthropologist’s formula to predict the height of the mystery person. How close was your prediction to the anthropologist’s prediction?

3. What might explain the differences in the height that you found using your regression line and the height you found using the anthropologists formula?

Present Your Results Write a letter to the commissioner of the police department informing him of the estimated height of the mystery person.

Describe the steps you took to find the height of the person. Analyze how well the model scatterplot and your equation represent the data you

collected. Is there a correlation or causation between ulna length and height? Explain Discuss whether or not you could use your model to estimate or predict the height of

the mystery person. Compare your estimation of the height of the person to the estimation using the

anthropologist’s equation. Include in your report equations, graphs, estimates, and predictions.



Rubber Bands Materials: Rubber bands, ruler, pen, masking tape (optional) Jobs: Ruler/rubber band holder, rubber band measurer, distance measurer/recorder Directions: Hold the ruler parallel to the floor with the 0 end pointing away from you. You

will need to select 7 different lengths to stretch the rubber band back and then let fly. Don’t pull the rubber band farther than 12 inches. At each length you select, you will need to do 3 trials and average the distances for that length. Rotate Jobs!

Length Pulled Back (in), x

Distance, Trial # 1

Distance, Trial # 2

Distance, Trial # 3

Mean Distance (ft), y

1. Create a scatter plot (from the shaded boxes) showing the association of the length of the

rubber band (independent variable) with the average distance traveled (dependent variable). 2. Summarize your scatter plot.

Discuss the direction and strength of the correlation.

3. Fit a line to the data and find its

equation. (You may do this by hand or with a calculator)

120 2 4 6 8 10

36

0

6

12

18

24

30

x

y



4. Use your equation to predict the distance a rubber band would travel if stretched the following length.

a. 4.65 inches b. 7.21 inches

c. 14 inches d. 1 inch 5. Use your equation to predict the length a rubber band would have to be pulled in order to

travel the following distance.

a. 6 feet b. 20 feet c. 25 feet d. 35 feet

6. Would it be reasonable to estimate how far a standard size rubber band would travel if we

stretched it 25 inches? Why or why not? 7. You have learned that correlation and causation are not the same. In this case does the length

the rubber band is pulled cause the distance it travels to vary? Explain.



Stadium Wave

Suppose you have an internship one summer with a company that makes TV commercials. You are told to find the solutions to the following two problems:

Problem 1: A client who sells automobiles wants a 15 second wave made by people from all walks of life standing single file along a roadway as the car drives along in front of them. How many people do you need to hire to create a 15 second wave?

Problem 2: A soft drink client wants a commercial to have a video clip of a stadium wave at Fenway Park. Will the 30 second commercial be able to include a video that shows the wave sweeping all the way around the people in top row of seats at Fenway Park? If not, what fraction of the park will a 30 second wave include? (You will have to do a little research on Fenway Park.)

You can find information on stadium waves at the following Wikipedia page. http://en.wikipedia.org/wiki/Wave_(audience)

PROCEDURE:

Create a simulation in the classroom and develop a mathematical model for the wave:

1. Have 5 classmates stand up in the room and do a wave. A practice run would be useful as you decide on how to do your wave. Time how many seconds it takes to complete a wave with 5 people. Record the data in the table. Then try the wave with different numbers of students.

Number of people doing the wave 5 6 8 10 13 16

Time to do the wave (Nearest tenth of a second)

2. Use technology or the graph on the next page to develop a mathematical model (linear equation) relating the number of people doing the wave and the time to do the wave.

3. Write a report for you employer to answer Problem 1 and Problem 2. Be sure to include the mathematical justification for your answers to both questions. Explain if the two variables are correlated or if the number of people causes the time of the wave to change.



Stadium Wave

Suppose you have an internship one summer with a company that makes TV commercials. A client who sells automobiles wants a 15 second wave made by people from all walks of life standing single file along a roadway as the car drives along in front of them. How many people do you need to hire to create a 15 second wave? You can find information on stadium waves at the following Wikipedia page. http://en.wikipedia.org/wiki/Wave_(audience) PROCEDURE:

1. Have 5 classmates stand up in the room and do a wave. A practice run would be useful as you decide on how to do your wave. Time how many seconds it takes to complete a wave with 5 people. Record the data in the table. Then try the wave with different numbers of students.

Number of people doing the wave 5 6 8 10 13 16

Time to do the wave (Nearest tenth of a second)

2. Graph the data from your table as a scatter plot on the coordinate grid. Label and scale the axes appropriately.



3. Fit a line to the scatter plot. Find the equation of the line by hand or using technology.

4. How does the slope in your equation relate to the context of the problem?

5. How does the y-intercept in your equation relate to the context of the problem?

6. Use your equation to predict the amount of time it would take for all students in your class to complete a wave.

7. Using all of the students in your class, find the actual time it takes for the wave.

8. Why do you think your predicted time was not exactly the same as the actual time?

9. Write a report for you employer to answer the question, “how many people it will take to create a 15 second wave?” Explain whether or not the two variables are correlated and if the number of people performing the wave causes the time of the wave to change.



Balloons Have you ever blown up a balloon, let it go, and watched it fly around? Have you ever done this in a math classroom? Today we will! Here is the question we need to answer: How long will it take for a balloon with a 40 inch circumference to deflate? It would be easy to answer this question if we have the right sized balloon, but I only have 9-inch diameter balloons. Procedure You are going to work in a group of size three or four. Each person will get a balloon, but you will each take turns blowing your balloon up and letting it go! 1. Person 1 will blow up his or her balloon. Someone else will measure the circumference on

the balloon to the nearest quarter of an inch and record it in the table below. 2. Have the stopwatch ready to go and have whoever is holding the balloon release it. Time

how long it takes in seconds for the balloon to deflate and hit the ground. Record the time it took for the balloon to deflate and hit the ground in the table below. Also, do not “throw” the balloon – just release it and let the balloon deflate and fall to the ground.

3. Repeat steps 1 and 2 with your group. Each person in your group must inflate the balloon to a different circumference.

4. Write your group’s data in the table below and on the table on the board. Record each coordinate pair on the class table. Since this is a scatter plot, it doesn’t matter if people in different groups have the same circumference.

Name Circumference (in inches)

Time (in seconds)

Name Circumference

(in inches) Time

(in seconds)



5. Use the data to create a scatterplot. You may do this by making a graph by hand or by using technology. Label and scale the axes appropriately.

6. Based on these data, find a linear equation to predict how long it will take for a balloon with a circumference of 40 inches to deflate. Show all of your work.

7. Use the equation to predict how long it will take for a balloon with a circumference of 40

inches to deflate.

8. Interpret the slope of your linear equation. What does it mean in context of the situation?

9. Is your prediction from question (7) an interpolation or an extrapolation? Explain



10. Is there causation between the circumference of the balloon and the time it takes to deflate? Explain.

Now it is time to do the experiment. We will need volunteers to measure this one balloon that is capable of inflating to 40-inches in circumference! We also need a timekeeper. The actual time for a balloon with a 40-inch circumference to deflate is __________________.

11. How close was your predicted time to the actual time?

12. What other variables might affect the results of this experiment?



Walking Away Choose your roles: recorder, measurer, and walker.

Distance is to be measured to the nearest inch. 1. Mark a starting point. The walker will take the number

of steps indicated in the table below. For each number of steps record the distance in inches from the starting point.

2. Use the grid below to make your scatter plot. Label and

scale the axes appropriately.

3. Fit a line to the data.

Number Of Steps

Distance (inches)

0

3

5

7

8

11

4. Find an equation for the trend line either by hand or with technology. 5. Use your equation to determine how far the walker traveled after taking 9 steps.



6. How strong is the correlation between the number of steps and the distance? Explain. 7. Is there causal relationship between the number of steps and the distance? Explain.



Population and Representation Each person (or pair) is going to research the relationship between the population of a state and the number of representatives that state elects to the United States House of Representatives.

Doing the Research You will go to a search engine (i.e. Google, Yahoo, etc.) and research two topics:

• The current population for each state • The current number of elected representatives for each

state (To save time you only have to find the population and number of elected representatives for 15 states. If you find the data for more states, your predictions will be more accurate.)

Writing about the Results 1. Explain the two variables you are comparing. What did you type into the search bar to get

your results? Cite the websites you used.

2. Make a table of the data points that you collected.

3. Create a scatter plot on a piece of graph paper or using technology. Draw the regression line

on the scatterplot.

4. Describe the correlation. 5. Write the equation of the regression line.



Predict y, given x

6. Choose a value for your x-variable. 7. Use your equation to calculate the approximate number of elected representatives (y) for the

value of x you chose above.

8. Describe your prediction in a sentence.

Predict x, given y 9. Choose a value for your y-variable. 10. Use your equation to calculate the approximate population of a state (x) for the value of y you

chose above.

11. Describe your prediction in a sentence. 12. Write a paragraph explaining what you learned about the relationship between your variables

and how this knowledge could be useful to someone.



Conducting an Experiment Each group is going to examine the relationship between a certain type of aerobic exercise and a person’s heart rate.

Create an Experiment Your group needs to think of an aerobic exercise that you can do in the classroom or in the hallway. Your group’s job is to come up with an experiment that shows how a teenager’s heart rate changes based on a type of aerobic exercise. Describe the Experiment 1. Explain the two variables you are comparing. Then, in detail, list the steps that will allow you

to perform an experiment to study the relationship between these two variables.

Write about the Results 2. Make a table of the data points that you collected.

3. Create a scatter plot on a piece of graph paper or using technology. Draw the regression line

on the scatterplot.

4. Describe the correlation. 5. Write the equation of the regression line.



Predict y, given x

6. Choose a value for your x-variable. 7. Use your equation to calculate the heart rate of a person (y) for the value of x you chose

above.

8. Describe your prediction in a sentence.

Predict x, given y 9. Choose a value for your y-variable. (Do not choose a heart rate value that is less than your

resting heart rate.) 10. Use your equation to calculate the value of x for the value of y you chose above.

11. Describe your prediction in a sentence. 12. Write a paragraph explaining what you learned about the relationship between your variables

and how this knowledge could be useful to someone.

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Outliers CCSS: S- ID 6; S-ID 8 Overview Students will work with data sets that contain outliers to identify the influence that outliers have on the calculation and interpretation of the slope, y-intercept, linear regression equation, and correlation coefficient. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able define an outlier, identify whether a potential outlier is present on a scatter plot and name the coordinates of the outlier, draw regression lines and provide a general description of the influence that outliers have on the slope as well as the direction and strength of the relationship between two variables; and describe the impact that outliers have on linear regression equations, their related components (i.e., slope, y-intercept, correlation coefficient), and the conclusions drawn from an analysis of a data set in which they are included. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.5 requires students to identify if an outlier exists and then make a prediction using the data set. Journal Entry asks students to explain how to determine whether or not a data point is an outlier.

Launch Notes Begin with a whole class discussion about outliers. Have students think about what they learned in Investigation 1. Ask students how they would identify an outlier from a dot plot, a histogram, or a box-and-whiskers plot. Then ask them to think about what an outlier would look like if they had a scatter plot. They may suggest that an outlier would appear at a greater distance from a regression line than the other points on the graph. This leads into Activity 5.5.1, which introduces outliers in two variable data. Closure Notes In a wrap up discussion you may ask the class, “Why are outliers important?” Students should realize that outliers can affect the accuracy of a prediction made with a regression line. Exit Slip 5.5 may be used to assess their ability to take the effect of an outlier into account when making a prediction.

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Teaching Strategies

I. After reviewing the concept of outlier from Investigation 1, use Activity 5.5.1 Outliers to introduce outliers in the context of two-variable data. Students may have difficulty identifying an outlier from the table; some may select “Crazy, Stupid, Love” since it has been out the longest time from its release. Others may focus on “The Help” since it leads in gross sales. Suggest that making a scatter plot will help resolve the issue. You may choose to have them make a scatter plot by hand or use the calculator. (Note that these data present an opportunity to discuss the degree of precision that is appropriate to this situation. In making a graph and using it to make predictions, it makes sense to round the sales figures to the nearest million dollars.)

Once students have seen a scatter plot of the data they are likely to agree that “The Help” is indeed an outlier. “Crazy, Stupid Love” has the largest value of x but it appears to be close to a line passing through the other data points. An explanation for the outlier is that “The Help” was an extremely popular movie. When students address question 5, there is no correct answer. Including the outlier takes into account that a few of the future movies may also turn out to be big hits. Excluding the outlier will give a more accurate prediction for the sales of a typical movie.

Students may then follow up with the questions on page 2, which ask them to make predictions based on the regression line they choose in question 6.

In Activity 5.5.2 Barry Bonds’ Home Runs, students are asked to analyze a table and a graph to identify and explain two possible outliers. They are asked to create a regression line and find the equation and correlation coefficient for the data. Students should use a graphing calculator; however, if that is not an option, instead of having them find the correlation coefficient, have them describe the type of correlation between the data. Students will then remove the outlier and repeat the process. Students should then look at the two different sets of information they have and determine the differences between them.

Activity 5.5.2 can lead to a rich discussion about how much weight statistical evidence can be given in determining a legal case. When looking at his previous performance, Bonds’ 73 home runs in 2001 do appear to be an outlier, which requires some explanation. However, the fact that it is an outlier does not prove that the cause was his use of performance enhancing drugs.

Activity 5.5.3 Home Prices involves a less complex set of data than Barry Bonds’ homeruns and is suitable for individual work either in class or for homework. In this activity, students are asked to identify an outlier and then find a trend line. They can find a trend line by hand or find the line of best fit using a graphing calculator. Students will then be asked to make predictions and determine whether the prediction involved interpolation or extrapolation.

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II. The second day of this lesson gives you a chance to consolidate students’ understanding of outliers. Activity 5.5.4 Chicago Bulls may be assigned for work in groups followed by a whole class discussion. Students will discover that one super star (Michael Jordan) does not fit the regression line relating number of minutes played in a season to total points scored in a season. This activity also reinforces the importance of the correlation coefficient and how to interpret the slope of the line in a real world context (in this case, points per minute).

Group Activity Students may play the “Outlier Game.” Students work in pairs, each with a calculator. Each student enters 10 pairs of data in L1 and L2. They should try to find a set with strong positive correlation, but must not place all the points on a single line. Each student then computes the value of r. Students then switch calculators. Each student now enters an eleventh ordered pair. The object is to make this pair an outlier. They now re-compute the value of r. The winner is the student whose outlier reduces the opponent’s value of r the most. You may want to restrict the values they can choose to lie within a specified window.

Differentiation (Enrichment) Activity 5.5.5 Crickets Chirping may be assigned. This is activity has a minimal amount of structure, asks students to use their judgment concerning outliers, and then has them research other formulas relating temperature and cricket chirps to the one they have derived.

Differentiation (For Learners Needing More Help) Activity 5.5.4 Sea Glass may be assigned. This is a highly scaffolded activity that will help students master the steps needed to analyze as set of data with an outlier.

Exit Slip 5.5 may be used to assess students’ understanding of outliers and their effect on regression lines.

Journal Entry In your own words, describe how you would determine if an ordered pair is an outlier. You may use examples from class to support your explanation.

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• Activities 5.5.1 Outliers • Activities 5.5.2 Barry Bonds’ Home Runs • Activities 5.5.3 Home Prices • Activities 5.5.4 Chicago Bulls • Activities 5.5.5 Crickets Chirping • Activities 5.5.6 Sea Glass • Exit Slip 5.5 Cooling Off • Instructions for the Outlier Game • Bulletin board for key concepts • Calculators



Outliers As we’ve seen with one-variable data, sometimes a value doesn’t fit the trend (like the number of Michael Jackson’s Thriller albums sold). The same can be true when we look for correlation between two variables. The values that don’t seem to fit the pattern are called outliers. Statisticians use a calculation to identify outliers, but we will just use our common sense to identify them. 1. If the data has an outlier, what do you think happens to your ability to make predictions from

the data?

The following table lists the top movies in theaters on September 14, 2011. The data represents the number of weeks since the movie’s release and the total box office gross income. 2. Examine the table and state which movie is possibly an outlier? 3. Make a scatter plot (by hand or with a calculator). Does the scatter plot help you determine

which movie is possibly an outlier. 4. What might explain this outlier?

Movie Weeks Total Gross Contagion 6 $27,923,641 The Help 36 $139,978,511 Warrior 6 $6,706,702 The Debt 15 $23,188,820 Colombiana 20 $30,730,482 Our Idiot Brother 20 $22,115,877 Apollo 18 13 $15,583,986 Shark Night 3D 13 $15,315,710 Crazy, Stupid, Love 48 $78,967,645

Source: the-numbers.com



5. Do you think this outlier should be included in the data when trying to predict future movie sales?

6. The dashed line is the line of best fit with the outlier included, and the solid line is the line of

best fit without the outlier.

500 5 10 15 20 25 30 35 40 45

140

0

20

40

60

80

100

120

Number of Weeks since Release

Tot

al S

ales

(in

mill

ions

of

$)

Which line would be a better predictor of total gross box office sales for movies in general? Explain.

7. Based on your choice in question 6, predict the total gross sales for a movie 30 weeks after its release.

8. Based on your choice in question 6, predict how many weeks it will take for a film to have total gross sales of $50 million.



Barry Bonds’ Home Runs

Many consider San Francisco Giants’ slugger Barry Bonds to be one of the greatest baseball players of all time. In 2001, he hit 73 home runs and broke the record for the most home runs hit in a single season. However, in 2011, Bonds was found guilty of obstruction of justice due to his testimony on steroid use which was found to be evasive and misleading. His possible steroid use has led many to question whether his home run record should be allowed to stand. Barry Bonds played in the major leagues for 22 years. His home run and games played statistics for each season are displayed in the table below.

Season Home Runs Games Played 1986 16 113 1987 25 150 1988 24 144 1989 19 159 1990 33 151 1991 25 153 1992 34 140 1993 46 159 1994 37 112 1995 33 144 1996 42 158 1997 40 159 1998 37 156 1999 39 102 2000 49 143 2001 73 153 2002 46 143 2003 45 130 2004 45 149 2005 5 14 2006 26 130 2007 28 126



Below is a scatterplot showing the number of home runs Barry Bonds hit each season.

230 2 4 6 8 10 12 14 16 18 20

80

0

10

20

30

40

50

60

70

Years since 1985

Num

ber

of H

ome

Runs

1. From the table and the graph, identify two points that appear to be outliers. 2. Give a possible cause for each outlier. 3. In 2005 Barry Bonds missed most of the season due to a knee injury. Use the table or the

scatter plot to explain whether this might have affected his performance in 2006 and 2007. 4. Calculate the regression line for 1986 through 2004, which are the years before his knee

injury. (Suggestion: Let the independent variable be years since 1986.)

5. Determine the correlation coefficient r for the data from 1986 to 2004.



6. Use your regression line to predict the number of home runs Barry Bonds would hit in 2001? How does this compare with the actual value?

7. Now eliminate the data for the year 2001. Calculate the regression line for 1986 through

2004, excluding the year 2001. (Suggestion: Let the independent variable be years since 1986.)

8. Determine the correlation coefficient r for the data from 1986 to 2004, excluding the year 2001.

9. Compare the two regression lines:

a. Which has the greater slope?

b. Which shows a stronger correlation?

c. Explain the differences. 10. If we include all 22 years of Bonds’ career in finding a regression line, how would this affect

the slope and the strength of the correlation? Make a prediction and then test it using a calculator or appropriate software.



Home Prices The table shows the square footage, purchase price, and sale date of several homes that sold in Farmington, Connecticut. 1. Do you think that there is a linear relationship

between square footage and purchase price? Explain.

2. Are there any outliers? What might explain these

outliers?

3. Using technology, calculate a line of best fit.

4. What does the slope of the line of best fit represent

in the context of the problem? 5. What is the value of the correlation coefficient? Explain the meaning of this value.

6. Imagine the house that is an outlier never sold. Perform linear regression without using the outlier. Give the regression equation, and state the value of the correlation coefficient.

7. Think about the two linear functions you created to model this data. Which model do you believe is a stronger predictor of the price of a home and its square footage? Explain.

Date Sold Sq. Ft. Price (in thous.)

12/16/2010 3654 705 12/17/2010 4590 650 12/21/2010 2011 245 12/21/2010 1514 275 12/28/2010 1878 310 12/29/2010 1822 318 1/28/2011 2480 385 1/31/2011 2278 278 1/31/2011 1618 140 2/3/2011 961 95 2/9/2011 3298 1 2/14/2011 1917 183 2/14/2011 2676 575 2/15/2011 1361 212 2/18/2011 2617 385



8. Using the model you chose in question 7, predict the sale price of a house with 800 ft2. Is this interpolation or extrapolation? Explain.

9. Using the model you chose in question 7, predict the sale price of a house with 4000 ft2. Is

this interpolation or extrapolation? Explain. 10. Using the model you chose in question 7, predict the square footage of a house that sold for

$290,000. Is this interpolation or extrapolation? Explain.



3,5000 500 1000 1500 2000 2500 3000

3,000

0

500

1000

1500

2000

2500

Minutes Played

Tot

al P

oint

s

Chicago Bulls The data below is from the Chicago Bulls basketball team. The table shows the total points scored and the total minutes played by a sample of players in the 1987 – 1988 National Basketball Association season.

Player Minutes Played

Total Points

Brown 591 197 Corzine 2328 804 Grant 1827 622 Jordan 3311 2868 Oakley 2816 1014 Paxton 1888 640 Pippen 1650 625 Sellers 2212 777 Sparrow 1044 260 Vincent 1501 573

1. Create a scatterplot showing the total minutes played and total points scored for the sample

of players in the coordinate plane above. Does there appear to be an outlier? If so, what player represents the outlier?

2. Find the line of best fit and the correlation coefficient, r. Plot the regression line on your scatterplot. Equation:________________________________ Correlation coefficient, r = _________

3. Interpret the strength and direction of the correlation coefficient. Using the r-value, how well does the equation represent the data?

4. Suppose you remove the point (3311, 2868) from the data set and recalculate the regression

line. What do you expect will happen to the regression line?



5. Remove (3311, 2868) and create a new scatter plot. Recalculate the correlation coefficient and the line of best fit. Plot the regression line on the graph.

3,5000 500 1000 1500 2000 2500 3000

3,000

0

500

1000

1500

2000

2500

Minutes Played

Tot

al P

oint

s

Equation:______________________________ Correlation coefficient, r = _______

6. Interpret the strength and direction of the correlation coefficient. Using the r-value, how well

does the equation represent the data? 7. What effect did the outlier have on the regression equation? What effect did the outlier have

on the correlation coefficient? 8. If you were going to use an equation to predict a player’s total points based on the number of

minutes played, which regression equation would be a more accurate representation for most players on this team? Why?



9. Using the equation from question 8, interpret the real-world meaning of the slope and the real-world meaning of the y-intercept.

10. Using the equation from question 8, calculate and interpret the x-intercept. Is this an

example of interpolation or extrapolation? 11. Using the equation from question 8, predict the total points for a player who has 3500

minutes of playing time. Is this an example of interpolation or extrapolation?

12. Using the equation from question 8, calculate the playing time for someone who has scored a

total of 2000 points. Is this an example of interpolation or extrapolation?



Crickets Chirping

It is a well-known fact that crickets tend to chirp faster in warmer weather than they do in cooler weather. The following data shows the relationship between chirps per minute of some crickets and the corresponding temperature.

Chips per Minute

Temperature (in F°)

20 89 16 72 20 93 18 84 1 81 16 75 15 70 1 82 15 69 17 83 16 81 17 84 14 77

1. Analyze the data. Determine which points are outliers. Suggest a possible explanation

for the outliers.

2. Exclude the outliers and find a regression line to fit the remaining data points. Determine the strength and direction of the correlation. Use your regression line to predict the temperature if a cricket chirps at a rate of 21 chirps per minute.

3. Search the internet for information on the relationship between cricket chirps and temperature. Find at least two different formulas. Compare the formulas you find on the internet with the equation of your trend line. Summarize your conclusions in a written report.



120 2 4 6 8 10

8

0

1

2

3

4

5

6

7

Hours on the Beach

Piec

es o

f Se

a G

lass

Sea Glass

Sea glass, pieces of glass tumbled smooth by the ocean, are considered to be a treasure by many people. The last time Alicia visited the ocean, she looked for sea glass on the beach every day. The table below shows how much sea glass she found based on the number of hours that she spent on the beach.

1. Graph the data by hand on the coordinate plane below.

2. Enter the data into L1 and L2 on your calculator. 3. Use your calculator to find the regression line. Record the equation below.

Equation:

4. Find the correction coefficient.

Correlation coefficient:

5. Graph the regression equation on the axes above. Circle the outliner on the graph. 6. Remove the outlier from the lists in your calculator. Now find the new regression line and

record it below.

Equation: 7. Find the new correlation coefficient and record it below.

Correlation coefficient:

Hours on the Beach each Day

Pieces of Sea Glass Found

1 2

2 8

4 3

6 5

8 7

10 8



8. Use both regression equations to predict how many pieces of sea glass Alicia would find after 7 hours on the beach.

a) Prediction with outlier:

b) Prediction without outlier:

9. Which regression equation do you think is a better predictor of the amount of sea glass Alicia will find? You may want to plot the points you found in question (8) on the scatterplot to see which prediction is “better”.

10. Use the regression equation which you think provides a better prediction to predict how

many hours Alicia would need to be on the beach to find 3 pieces of sea glass.


Unit 5 Investigation 5 The Outlier Game Algebra I Model Curriculum Version 3.0

The Outlier Game

This is a game for two players. Each player should have a graphing calculator. Names of Players: Player 1: ___________________________________________ Player 2: ___________________________________________ Before playing the two players should agree to use the same window. Write your choices in the space below. Xmin = ___________ Xmax = ____________ Xscl = _______________ Ymin = ___________ Ymax = ____________ Yscl = _______________ Now each player enters ten ordered pairs in L1 and L2. Each of the points MUST lie within the window. The ordered pairs may not all lie on the same line but they should have a strong positive correlation. Calculate the equations of the regression lines and the value of r Player 1: Equation: ____________________ r = __________ Player 2: Equation: ____________________ r = __________ Now switch calculators. Enter an ordered pair in the 11th row of L1 and L2. You must pick a point that lies within the window. Your object is to reduce the value of r by as much as possible. Again calculate the equations of the regression lines and the value of r. Player 1’s calculator: Equation: __________________ r = ____________ Player 2’s calculator: Equation: __________________ r = ____________ Which player succeeded in reducing the value of r on the other player’s calculator the most? This player is the winner!

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Piecewise Functions CCSS: 8-SP 1; 8-SP 2; 8-SP 3; S-ID 6 a, c; S-ID 7, F-IF 7b Overview Students will explore situations in which the data are best represented by a piecewise linear function, will fit a line to each section of the data set, and will use the lines to make predictions. Assessment Activities

Evidence of Success: What Will Students Be Able to Do? Students will be able to identify two points on each line segment and use them to calculate the equation of the line that contains that segment. They will be able to identify the domain for which the line segment fits the data, and write the piecewise function given the graph. Students will be able to create a story that describes a piecewise graph. Assessment Strategies: How Will They Show What They Know? Exit Slip 5.6 asks students to create a piecewise function to represent the height of an elevator during an elevator ride. Journal Entry 1 probes students’ reaction to encountering piecewise functions and their comfort level with this new concept. Journal Entry 2 asks students to explain how to write a rule for a piecewise function if they are given a graph.

Launch Notes Display Activity 5.6.1 Swimming Records. This activity will provide a smooth transition from scatterplots to piecewise functions. Focus the students’ conversations on how the data is similar to and different from the data sets previously studied in Unit 5. Closure Notes Ask students to describe real world situations for which piecewise functions are good models. Draw out the fact that in each situation they have encountered the slope of the line (that is the rate of change) is different for the separate pieces of the graph. You may want to go back to specific examples such as the triathlon (speeds differ for different events) or the dog food activity (rates differ depending upon the size of the dog).

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Teaching Strategies

I. Begin with Activity 5.6.1 Swimming Records. Make sure that students understand that the data refer to individuals who have set a new swimming record, not the individual with the best time during that particular year. The gap between 1936 and 1952 results from the fact during that time period no one broke the record set by Willy den Ouden in 1936. Have students fit a trend line by hand to the graph on page 1. Statistically it gives a fairly good fit; the regression line shows a strong negative correlation with r = –0.96. You may want to show the regression line with your calculator on the overhead projector. Point out that points are not evenly distributed around the regression line: those at either end of the data set lie above the line and those in middle lie below. This suggests that two lines might produce a better fit than just one line. Ask students to use the graph on page 2 to fit two trend lines by hand. Go over the questions on this page in class to introduce students to the concept of a piecewise function.

Technology Tip: If you decide to use the calculator to show regression lines for each piece of the data, you may enter the data into lists L1 and L2. Then copy L1 into L3 and L5 and copy L2 into L4 and L6. In L3 and L4, delete ordered pairs that have x-values (years since 1910) greater than 26. In L5 and L6, delete ordered pairs that have x-values (years since 1910) less than or equal to 26. Then find the regression lines for L4 vs. L3 and for L6 vs. L5. Use Activity 5.6.2 Paychecks & Triathlons to strengthen student understanding of piecewise functions. Here the data are perfectly linear, so each piece of the function will be a perfect fit. When discussing the domain, you can either have the students write it in terms of x (i.e. 0 ≤ x ≤ 40 or x > 40) or you may have students use words (i.e. working between 0 and 40 hours or working more than 40 hours). You may do the first part of this activity (Paychecks) in class and assign the second part (Triathlons) for homework.

Differentiated Instruction (For Learners Needing More Help) Have students write down the steps for writing an equation and how to determine the domain for each part of a piecewise function on a notecard. Allow them to use this card throughout the investigation.

Differentiated Instruction (Enrichment) Students may research the history of the triathlon and the different kinds of races and their distances. Ask them to consider how realistic it is to model each segment of the race with a linear function, when in fact the racer’s speed may very. Students who are interested may be encouraged to participate in a local triathlon event http://www.trifind.com/ct.html. If they do, they may construct a piecewise function to model their performance.

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II. Activity 5.6.3 Dog Food presents another real world situation that can be represented by a piecewise function. This activity can be assigned for students to complete in groups. You could assign one type of dog (small, medium, or large) to each group. (If you have more than three groups, two or more groups will have the same type of dog.) After the groups have derived an equation for their dog type, they can put their answers to questions (2), (3), and (5) on the board. In a whole class discussion, have students figure out how to represent the function in the format shown in question (8). Then have students return to their groups to answer the remaining questions.

Group activity Activity 5.6.3 is particularly suited for grouping high level students with low level students so that someone in the group will remember how to find an equation of a line given two points. As you circulate, make sure that all students in each group understand the derivation of their equation.

An optional follow up to Activity 5.6.3 is Activity 5.6.4 Feeding the Birds. This may be started in class or part of it may be assigned as homework. Several features make this activity a bit more challenging. First, the dependent variable is not a traditional unit of measure; in this case our unit is the amount of food in a birdfeeder. Second, students need to reason about how the height of the bird feed determines the number of perches available, which in turn determine the rate at which the height decreases. Third, when calculating the slope, fractions will appear in the numerator. Since the units for y are fractions, the equations should be in fractions, not rounded decimals. The first five questions on the 4-perch feeder are more scaffolded than the remaining ones on the 6-perch feeder. The extension questions may be assigned for enrichment.

Differentiated Instruction (Enrichment) Students can build a physical model with rice as suggested in the Extension questions. They should compare how well the mathematical model corresponds with the physical one.

If you find that your students are doing very well with piecewise functions, then give Exit Slip 5.6 after the first three activities. If your class has a good grasp of piecewise functions, then you may be able to skip the third day. At this point you may use the following Journal Entry.

Journal Entry 1 Reflect on your participation in class today and complete two or more of the following statements: I learned that I…

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I was surprised that I… I noticed that I …. I discovered that I… I was pleased that I…

III. Two additional activities are appropriate for a third day. Students create and interpret

graphs of piecewise functions, write rules for piecewise function, evaluate the functions, and finally write stories that correspond to a graph. Activity 5.6.5 Bike Tours presents two scenarios involving piecewise functions and distance-time graphs. You may choose to use one or both scenarios. In class the activity may be done as a teacher-led lesson or as a group work activity. If Scenario 1 is done in class, Scenario 2 may be assigned for homework. In question (23), stress to the students that “45 minutes” needs to be converted to hours because the piecewise function rule is in terms of hours. Set up a ratio 45/60 and have them simplify to either ¾ or 0.75 hours.

Differentiated Instruction (For Learners Needing More Help) Remind students of the graphs that they created in Unit 4 using a motion detector. Many of them could have been modeled by piecewise functions. You might set up the motion detector and ask someone to “walk a graph” that would look like Jackie’s bike trip.

Activity 5.6.6 Creating Stories provides additional practice with distance-time graphs and has students create their own story and write a function to match it.

Journal Entry 2 If you are given a graph of a piecewise function, explain how you would find the function rule and the domain for each piece of the function.


• Activity 5.6.1 Swimming Records • Activity 5.6.2 Paychecks & Triathlons • Activity 5.6.3 Dog Food • Activity 5.6.4 Feeding the Birds • Activity 5.6.5 Bike Tours • Activity 5.6.6 Creating Stories • Exit Slip 5.6 - Elevator Ride • Bulletin board for key concepts • Individual white boards, markers and erasers • Clear cylindrical container and rice



Swimming Records

The following table lists the Women’s Long Course 100m Freestyle World Records from 1912 to 2009. (Source: wikipedia.org)

The data from the table are shown in the graph.

1. Draw a trend line on the graph. Do you think this line fits the data well? Explain.

1000 10 20 30 40 50 60 70 80 90

90

50

55

60

65

70

75

80

85

Years since 1910

Wor

ld R

ecor

d T

imes

(in

sec

onds)

Year since 1910

Time (seconds) Athlete (Country)

54 58.9 Shane Gould (Australia) 62 58.5 Shane Gould (Australia) 63 57.54 Kornelia Ender (E. Ger) 64 56.96 Kornelia Ender (E. Ger) 65 56.22 Kornelia Ender (E. Ger) 66 55.65 Kornelia Ender (E. Ger) 68 55.41 Barbara Krause (E. Ger) 70 54.79 Barbara Krause (E. Ger) 76 54.73 Kristin Otto (E. Ger) 82 54.48 Jenny Thompson (US) 84 54.01 Jingyi Le (China) 90 53.77 Inge de Bruijn (Neth) 94 53.52 Jodie Henry (Australia) 96 53.3 Britta Steffen (Germany) 98 52.88 LisbethTrickett (Australia) 99 52.07 Britta Steffen (Germany)

Year since 1910

Time (seconds) Athlete (Country)

2 78.8 Fanny Durack (Australia) 5 76.2 Fanny Durack (Australia)

10 73.6 Ethelda Bleibtrey (US) 13 72.8 Gertrude Ederle (US) 14 72.2 MariechenWehselau (US) 16 70 Ethel Lackie (US) 19 69.4 Albina Osipowich (US) 20 68 Helene Madison (US) 21 66.6 Helene Madison (US) 23 66 Willy den Ouden (Neth) 24 64.8 Willy den Ouden (Neth) 26 64.6 Willy den Ouden (Neth) 46 62 Dawn Fraser (Australia) 48 61.2 Dawn Fraser (Australia) 50 60.2 Dawn Fraser (Australia) 52 59.5 Dawn Fraser (Australia)



2. There is a gap in the scatter plot between the years 1936 and 1956 (26 and 46 years from 1910). What might explain this gap?

3. Sometimes it takes more than one function to model a set of data. Split the graph into two

pieces. Draw a trend line for each piece. This new graph is a piecewise function.

4. Find an equation for each of your trend lines.

a. Equation for first trend line:

b. Equation for second trend line:

1000 10 20 30 40 50 60 70 80 90

90

50

55

60

65

70

75

80

85

Years since 1910

Wor

ld R

ecor

d T

imes

(in

sec

onds)



5. Find the domain for each trend line, that is, for which values of x does each line fit the data.

a. First trend line:

b. Second trend line:

6. Which trend line would you use to predict the Women’s Long Course Swimming Record for

the year 2030? Why?

7. Use the trend line to predict the record for 2030. Is this interpolation or extrapolation?

8. What might cause your prediction in 2030 to be inaccurate?



Paychecks & Triathlons

Steve works for a theatrical lighting design company. He works many hours during the summer

months when the company is very busy. The following table lists his gross pay (before taxes and

other deductions) based on the number of hours he worked in a week.

Hours

worked

Gross

Pay

30 855

34 969

35 997.50

38 1083

40 1140

41 1182.75

46 1396.50

48 1482

54 1738.50

1. Does one line fit the data well, or will it take more than one line to obtain a good fit?

2. When does the pattern change? What do you know about pay rates that might explain this

change?

3. Use technology to find two regression lines. Line 1 contains the points with x-values from 30

to 40, and Line 2 contains the points with x-values from 41 to 54.

Line 1: Line 2:

What does the slope mean? What does the slope mean?

4. A piecewise function contains two or more rules with each rule acting on a different part of

the function’s domain. Label Steve’s weekly pay function and write each rule on its

own line, along with the part of the domain where that rule is applied.



5. Find and . Make sure you use the correct rule.

Helen Jenkins, of Great Britain, finished first in the Dextro Energy Triathlon in London on

August 6, 2011. The race consisted of a 1.5 km swim, a 42.9 km bike ride, and a 10 km run.

Below is a table and graph of her time and distance as she completed each leg of the race.

Time Distance

0 0

24 1.5

24.6 1.5

78.2 44.4

89.3 44.4

129.3 54.4

6. Find an equation for the total distance Helen traveled from the time she started during each

event (swimming, biking, running). Find each equation by hand. (Assume that she traveled

at a constant rate during each event.)

a. Swimming b. Biking c. Running

7. How much time did Helen have to rest:

a. Between swimming and biking?

b. Between biking and running ?

8. Complete the piecewise function that models the distance Helen traveled during the race

9. Find .

10. Find .



11. Find .



Dog Food Kathy is dog sitting for her neighbor who is on vacation. Her first assignment is to feed the dogs. The neighbor gave Kathy a checklist of the dog names and how much food they eacheat, but one of the dogs thought it was a snack and bit a corner off!

1. Based on what you noticed, how many ounces do you think Kathy should give:

a. Spot: _____

b. Thor: _____ c. Zeus: _____

d. How did you come up with these predictions? Spot finished his food, but Thor and Zeus had most of their food left over. Kathy called a vet who said: “Thor and Zeus have different feeding formulas than Buster, Spencer, and Spot. Adult dogs that weigh less than 20 pounds are classified as small dogs. Dogs that weight 20 to 100 pounds are classified as mid-size dogs. Dogs that weight more than 100 pounds are large dogs.”

Dog Food Chart Small Mid or

Large

Weight of the Dog

(pounds)

Daily Amount of

Food (ounces)

S 5 5 S 10 10

S / M 20 20 M 50 35

M / L 100 60 L 125 62.5 L 150 65

.

2. Use the table provided to find an equation for feeding small dogs:

3. Use the table provided to find an equation for feeding mid-size dogs.

4. Thor is a mid-size dog. Use your equation from question 3 to determine how many ounces of dog food Thor should be fed. How many ounce did Kathy overfeed Thor?



5. Use the table from the previous page to find the equation for feeding large dogs.

6. What does the slope mean in the equation for large dogs?

7. Zeus is a large dog. Use your equation from question 5 to determine how many ounces of dog food Zeus should be fed. How many ounces did Kathy overfeed Zeus?

Since there are 3-pieces to formula for feeding the dog, the data can be written as a piecewise function. The different feeding rules are considered to be the pieces of a single function.

8. Help Kathy complete the piecewise function for any weight of a dog.

9. Complete the table using the piecewise equation you wrote in question 8.

Dog’s Weight (Pounds)

Amount of Dog Food (ounces)

71

120 16 75



10. It is necessary to graph each piece of the function only for its appropriate domain. Graph the feeding function on the axes below. When you’re finished, the graph should consist of 3 line segments. (Small, midsize, and large).

11. Kathy has so much fun with the dogs that she decided to buy a 130-pound Great Dane. Kathy determines that she should feed the dog 63 ounces of food daily. Do you agree? Show your work to justify your answer.

12. If Kathy is giving 40 ounces of food to a dog, how much does that dog weigh? Show your work and explain your answer in a sentence.



13. By using the graph (not the equations), what is the approximate value of f(30)? Explain the meaning in a sentence.

14. Using the graph (not the equations), what is the approximate value of w if f(x) = 30. Explain the meaning in a sentence.

15. What is the range for small dogs?

16. What is the range for midsize dogs?

17. What is the range of large dogs?



Feeding the Birds Background Your neighbor is an ornithologist. She has to leave for a weekend. She has asked you to make sure her birdfeeder always has food in it. Refilling too seldom will cause the birds to look elsewhere for food. Refilling too much will scare off the birds. Leading Question How often should you feed the birds so they keep coming back? The birdfeeder has 4 holes - one pair near the bottom and one pair about halfway up, as shown in the diagram above. 1. When you go to the bird feeder early in the morning, the birdfeeder is nearly full. You check

back in 40 minutes and it is about half full. You come back 40 minutes later and it’s still not empty. The birds are still coming by consistently to eat, so they are still hungry. Why isn’t it empty?

2. Answer the next two questions to help determine the amount of time it will take for the birds to eat the final half of the food in the birdfeeder.

For 40 minutes, birds ate nonstop off 4 perches. How many total minutes were birds on a perch?

How long would it take the birds to eat the same amount of food off of 2 perches? 3. If the birdfeeder is initially filled to the top, how many total minutes will it take to get

empty? 4. Complete the table below.

Total time (minutes), x

Fraction of the birdfeeder that is full, y

1 ½ 0



5. Find a piecewise function for the amount of food in the birdfeeder after any number of minutes. a. Write an equation for a line through the first two points in the table.

b. Write an equation for the line through the 2nd and 3rd points in the table. c. Now write the piecewise function.

! = ! ! = _______________________!ℎ!"____________________________________________!ℎ!"_____________________

You did so well taking care of your neighbor’s birdfeeder that she recommended you for a weekend job watching her coworker’s birdfeeder. This birdfeeder has 6 feeding holes – 2 at the bottom, 2 about one-third of the way up, and two holes about two-thirds of the way up. 6. The first morning you get there you notice that the feeder is about 2/3 full.

You wait a while and notice that it takes about 30 minutes before the feeder is about 1/3 full. How many more minutes do you think it will take before you need to refill the feeder?

7. Answer the next three questions to help determine the amount of time it will take for the birds to

eat the first third of food in the birdfeeder and the final third of food in the birdfeeder.

For 30 minutes, birds ate nonstop off 4 perches. How many total minutes were birds on a perch?

How long would it take the birds to eat off 6 perches? (Remember the time should be faster!) How long would it take the birds to eat off 2 perches? (Remember the time should be slower!)



8. If the birdfeeder starts off full, how many total minutes will it take for the feeder to become empty? How did you get your answer? Show your work. State your answer in a sentence.

9. This table that shows the birdfeeder started full. Complete the table by showing the total

amount of time it will take to get each fraction of the birdfeeder to be full.

Total time (minutes), x

Fraction of the birdfeeder that is full, y

0 1 2/3 1/3 0

10. Write a piecewise function that can be used to find the amount of feed in the birdfeeder after

any number of minutes. Show all steps in the space below.



11. Graph your piecewise function on the coordinate plane below.

12. If it has been 30 minutes since you refilled the bird feeder, approximately how much of the

birdfeeder still has food? 13. If there is ¼ of the bird food remaining in the birdfeeder, approximately how many minutes

has it been since you last refilled? EXTENSION 14. Build a mock birdfeeder like the one pictured on page 2. Use a clear cylindrical container as

the birdfeeder. Use rice as the bird food. Make holes in the container to allow the rice to spill out. How well did you mathematical model agree with your physical model?

15. Write a mathematical description of how to determine how quickly the birdfeeder will empty. 16. Can you generalize the description above for any number of feedholes?

1600 20 40 60 80 100 120 140

1

0

1/6

2/6

3/6

4/6

5/6

Time (minutes)

Frac

tion

of

Bird

Foo

d R

emai

ning



60 1 2 3 4 5

25

0

5

10

15

20

Number of hours since 9 a.m.D

ista

nce

from

sch

ool (

mi)

Bike Tours Jackie has started a bike club. To increase membership Jackie plans a leisurely bike tour (including lunch) for those interested in joining. The tour begins and ends at their school. The graph at the right represents Jackie’s travel around the route. Interpreting the Graph 1. What was Jackie’s average speed

(miles per hour) for the first hour? 2. During what time interval is Jackie

going the fastest? When is she going the slowest?

3. When was lunch? How long was lunch? 4. What is the furthest Jackie gets from school? 5. What time does the group end the tour? 6. What is the first time that Jackie is ten miles from school? 7. How many total miles did Jackie travel? 8. During what time interval(s) was Jackie’s distance more than 15 miles from school?



Writing a Piecewise Function Let !(!) represent Jackie’s distance from the school (measured in miles) at time t (measured in hours since 9 a.m.). 9. Write a rule for !(!).

! ! =

_________________________________!ℎ!"______________________________________________________!ℎ!"______________________________________________________!ℎ!"_______________________________________________________!ℎ!"________________________________________________________!ℎ!"______________________

10. On what intervals is the slope of the line segment positive? What does this mean in the

context of the bike trip? 11. On what intervals is the slope of the line segment negative? What does this mean in the

context of the bike trip? 12. Find !(1.25). In other words, find the distance that Jackie is from school at 10:15 a.m.

(10:15 a.m. is 1.25 hours after 9 a.m.)



Using a Piecewise Function

The week after their first trip, the bike club decided on another outing. They started at 9 a.m. and did not take a break from lunch. They started at the school and returned to the school at 1 p.m.

13. What was their average speed (mph) for the first hour? 14. What was the average speed for the next hour? 15. What was the farthest the bikers got from school? 16. What time did the group end the tour? 17. What is the first time that they were ten miles from school? 18. How many total miles did the group travel? 19. During what time interval were they going the fastest? When were they going the slowest? 20. Did the group ever return to the school before the end of the trip?

50 1 2 3 4

25

0

5

10

15

20

Number of hours since 9 a.m.

Dis

tanc

e fr

om s

choo

l (m

i)



Writing a Piecewise Function Let !(!) represent the distance that the bike club is from school (measured in miles) at time t (measured in hours since 9 a.m.). 21. Write a rule for !(!).

! ! =

____________________________!ℎ!"_________________________________________!ℎ!"_________________________________________!ℎ!"_________________________________________!ℎ!"______________

Applying a Piecewise Function 22. Use the piecewise function to determine exactly how far from school the bike club was after

3.25 hours? (12:15 p.m. is 3.25 hours after 9 a.m.) 23. Use the piecewise function to determine exactly how far from school the bike club was at

9:45 a.m.? (9:45 is 0.75 hours after 9 a.m.)



Creating Stories Lisa kept a diary of her family’s vacation trip. Her family drove along an interstate that had mile markers posted along the way.

Vacation Trip Clock Time Time since 8 a.m. Total Miles

8 am 0 10 am 100

10:30 am 100 12 190

3 pm 310

1. Complete the table above and draw the graph of this function. Label and scale the axes appropriately.

2. How far did the family travel after noon? 3. What was their speed (in miles per hour) in the afternoon? 4. What was their speed (in miles per hour) during the first two hours?

Time since 8 a.m.

Mile

Mar

ker (

in m

iles)



5. Using the graph, how long were they driving when Lisa’s family passed the 260-mile marker?

6. Using the graph, around how much time was Lisa’s family less than 50 miles away from

home? 7. Did they stop to eat (or stop for another reason)? How long did they stop for? 8. Create a story describing Lisa’s family trip. 9. Write a piecewise function to represent the distance Lisa’s family travelled during their

family trip. Show your work.

! ! =

____________________________!ℎ!"_________________________________________!ℎ!"_________________________________________!ℎ!"_________________________________________!ℎ!"______________



10. Use the piecewise function to determine the mile marker Lisa’s family passed at 1:30 p.m. (1:30 p.m. is 5.5 hours after 8 a.m.) Show your work.

11. Use the piecewise function to determine the mile marker Lisa’ family passed at 11 a.m. Show

your work.

12. Write a brief new story that can be modeled with a piecewise function. Then write the function. Your function should have at least three parts.

Unit 5: Scatter Plots & Trend Lines (4...

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