Teacher Resource for Unit 1 Lesson 1: Linear Models Refresher

Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1

Math Practices 2, 3, 4

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Teacher Resource for Unit 1 Lesson 1: Linear Models Refresher

Time Frame: 3-4 Days

Math Content Standards: S-ID.6, 7, S-IC.1

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. Use

given function or choose a function suggested by the context. Emphasize linear, quadratic, and

exponential models.

b) Informally assess the fit of a function by plotting and analyzing residuals.

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.IC.1 Understand statistics as a process for making inferences about population parameters based on a random

sample from that population.

Math Practices: 2, 4, 5

2. Reason abstractly and quantitatively. This is observed when students label independent and dependent

variables on the scatter plot and when they graph the line of best fit.

4. Model with mathematics. This is observed when students complete the table of data, make the scatter

plot, and determine the line of best fit.

5. Use appropriate tools strategically. This is observed when students make measurements of distances

dropped and determine the line of best fit.

Prior Knowledge - Determine line of best fit

- Make a scatter plot

- Interpret meaning of slope

- Interpret meaning of (r)

- Able to use “LINREG” on graphing calculator

Student Friendly Targets - I can create a scatter plot.

- I can determine a line of best fit using technology and graph it.

- I can interpret slope and intercepts in a context.

- I can make predictions using a line of best fit.

- I can interpret an (r) value.

- I can determine residuals and use them to assess fit.

Essential Question: From a scatter plot, how are two quantitative variables related?

Academic Language: Residual, slope, model, inference, line of best fit, scatter plot.

Differentiation: Possible method would be to let students choose their own set of data to explore.



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Procedures:

Sample Activity: (Possible pre-assessment – ask students to complete 1-4 in groups of 2-4 before debriefing as

a whole class)

The table below represents data collected from putting together a shopping cart train.

1) Create a scatter plot; then determine a line of best fit with your

calculator. Graph the line on the scatter plot. Record the equation for

the line.

2) Interpret the y-intercept and slope of your line of best fit within the

context.

3) Interpret the meaning of (r) (the correlation coefficient) and use it to

explain the validity of your line.

4) Your brave friends decide to make a shopping cart train of 12 carts.

Data from www.inspiration.com/freetrial/inspiredata How long do you predict the train to be?

Another data set for students to analyze and discuss:

5) Create a scatter plot; then determine a line of best fit with your

calculator. Graph the line on the scatter plot. Record the equation for

the line.

6) Interpret the y-intercept and slope of your line of best fit within the

context.

7) Interpret the meaning of (r) (the correlation coefficient) and use it to

explain the validity of your line.

8) A Snickers bar contains 57 grams per serving. How does your

prediction for calories compare to the 271 actually in the bar?

Data from www.inspiration.com/freetrial/inspiredata

Through whole-group, direct instruction, discuss residual values with the class with the following information.

Using the scatter plot and line of best fit, we can find the residual values.

The residual for a point is the difference between the observed value of the dependent variable and the value

predicted by the line of best fit.

Residual = observed y – predicted y

Shopping Cart Train

Number of Carts Length (cm)

2 138

3 169

4 202

5 234

7 295

10 390

14 517

20 709

Candy Bar Serving

Size (g)

Calories

Almond Joy 45 220

Baby Ruth 60 275

Butterfinger 60 270

Caramello 45 208

Heath 39 210

Hershey 43 210

Kit Kat 42 218

Krackel 41 210

Milky Way 58 262

Mounds 53 258

Nutrageous 51 260

Payday 52 240

http://www.inspiration.com/freetrial/inspiredata




3

Residuals can be used to determine whether a linear model is a good model for describing the data. The sum of

residuals for a set of data is 0 for the ideal mathematical model.

Using the candy bar data, groups of students will find the following:

The discussion questions will ask groups of students to explain what a residual value means (for instance, -3.14

shows that the observed value/point is approximately 3 below the predicted value/line of best fit). While still in

groups, students will need to conclude that the sum of residuals for a line of best fit is 0.

Ask students to work in groups of 2-4 on the correlation coefficient handouts. Each group will receive one set of

data, and each student receives the task explanation handout; the groups will then share their findings with the

class and discuss the general outcomes. Allow each group to justify whether they think their data is most linear.

Ask the class to organize the groups’ correlation coefficients in order of increasingly best fit. Emphasize that the

correlation coefficient indicates whether or not a linear model is the best fit for set of data.

Be sure to address the following questions during the discussion.

- Which correlation coefficient values indicate a “good fit?”The values that are closest to 1 or -1 are

better than ones that are not.

- What does a negative correlation coefficient indicate? A negative slope for the regression line. A

positive correlation coefficient? A positive slope for the regression line. A correlation coefficient that is

zero? A linear model would not be appropriate for the data.

- What consistencies do you observe about the residual plots? They should display a random scatter.

Close the discussion by asking groups to complete the error analysis in which students used only the residual or

the correlation coefficient to foster a discussion on the necessity of using the values together to determine

whether a data set should be modeled with a linear equation.

After students are comfortable with the above procedures, the class moves on to the Barbie Bungee Jump

Activity that follows. Student handouts for the pre-activities as well as for the Barbie Bungee Jump follow. A

video for introducing Barbie Bungee Jump can be found at

http://www.youtube.com/watch?v=CwS3MvJE7l4&NR=1&feature=endscreen

Candy Bar Serving

Size (g)

Calories Observed y – Predicted y Residual

Almond Joy 45 220 220 – 223.14 -3.14

Baby Ruth 60 275 275 – 273.15 1.85

Butterfinger 60 270 270 – 273.15 -3.15

Caramello 45 208 208 – 223.14 -15.14

Heath 39 210 210 – 203.13 6.87

Hershey 43 210 210 – 216.47 -6.47

Kit Kat 42 218 218 – 213.13 4.87

Krackel 41 210 210 – 209.8 .2

Milky Way 58 262 262 – 266.48 -4.48

Mounds 53 258 258 – 249.81 8.19

Nutrageous 51 260 260 – 243.14 16.86

Payday 52 240 240 – 246.47 -6.47

http://www.youtube.com/watch?v=CwS3MvJE7l4&NR=1&feature=endscreen



4

Bungee Jump Barbie Lesson Scoring Guide

Standard Completely meets standard Partially meets

standard

Rarely meets

standard

MP5 Correctly measures distance dropped and

records in table

Meets 6-8 of

the completely

meets criteria.

Meets fewer than

6 of the

completely meets

criteria. S-ID.6/MP2 Scatter plot created with axes labeled

S-ID.6a/MP4/MP5 Line of best fit determined and recorded

S-ID.7 Slope interpreted correctly

S-ID.7 Y-intercept interpreted correctly

S-ID.6 Line of best fit is graphed on the scatter plot

S-ID.6b Residuals calculated and recorded correctly

S-ID.6b Residuals explained correctly

S-ID.8 Meaning of r explained correctly

S-ID.6a Correct prediction made

Final Bungee Jump prediction successful

Student Handouts for the multi-day lesson follow.



5

Student Self-Assessment for Lesson 1

Ratings:

1: I’ve never seen this topic and wouldn’t even know how to begin.

2: I’ve heard or seen this before, but don’t know how to start or complete the problem.

3: I know the topic and can work through the problem but am unsure whether I am correct.

4: I feel confident that I could present my work and solutions to the class.

5: I feel that I could correctly teach this topic to another student if asked.

Target Self-Assess on

Shopping Cart

Train

Self-Assess

on Candy Bar

Self-Assess on

Barbie Bungee

Jump

I can create a scatter plot.

I can determine a line of

best fit using technology

and graph it.

I can interpret slope and

intercepts in a context.

I can determine residuals

and use them to assess fit.

I can interpret an (r) value.

I can make predictions

using a line of best fit.

Directions: Respond to the following in complete sentences with correct academic vocabulary.

1. Explain the contextual meaning of the y-intercept of a model fit to data.

2. Explain the contextual meaning of the slope of a model fit to data.

3. Explain how the analysis of residuals and the correlation coefficient is used to verify the validity of a

line of best fit.



6

Student Resources

Learning Targets:

- I can create a scatter plot.




The table below represents data collected from putting together a shopping cart train.


1) Create a scatter plot; then determine a line of best fit with your calculator. Graph the line on the scatter

plot. Record the equation for the line.

2) Interpret the y-intercept and slope of your line of best fit within the context.

3) Interpret the meaning of (r) (the correlation coefficient) and use it to explain the validity of your line.

4) Your brave friends decide to make a shopping cart train of 12 carts.

How long do you predict the train to be?

Shopping Cart Train

Number of Carts Length (cm)

2 138

3 169

4 202

5 234

7 295

10 390

14 517

20 709




7


5) Create a scatter plot; then determine a line of best fit with your calculator. Graph the line on the scatter

plot. Record the equation for the line.

6) Interpret the y-intercept and slope of your line of best fit within the context.

7) Interpret the meaning of (r) (the correlation coefficient) and use it to explain the validity of your line.

8) A Snickers bar contains 57 grams per serving. How does your prediction for calories compare to the 271

actually in the bar?

Candy Bar Serving

Size (g)

Calories

Almond Joy 45 220

Baby Ruth 60 275

Butterfinger 60 270

Caramello 45 208

Heath 39 210

Hershey 43 210

Kit Kat 42 218

Krackel 41 210

Milky Way 58 262

Mounds 53 258

Nutrageous 51 260

Payday 52 240




8

Learning Target:


- I can calculate correlation coefficients and use them to assess fit.

Using the scatter plot and line of best fit, we can find the residual values.

The residual for a point is the difference between the observed value of the dependent variable and the value

predicted by the line of best fit.

Residual = Observed y – Predicted y

Discuss the following with your partner. Record your answers below.

How are residuals related to the data and the line of best fit?

Explain what residual values of -3, 0, and 15 mean.

Predict what the residuals of an ideal model would be.

What is the sum of the residuals? Why do you think this occurs?

Give a summary of what a residual value shows.

Candy Bar Serving

Size (g)

Calories Observed y – Predicted y Residual

Almond Joy 45 220

Baby Ruth 60 275

Butterfinger 60 270

Caramello 45 208

Heath 39 210

Hershey 43 210

Kit Kat 42 218

Krackel 41 210

Milky Way 58 262

Mounds 53 258

Nutrageous 51 260

Payday 52 240



9

The following are the data sets to be used in the exercise.

Height and weights of students:

Height (in) Weight (lb)

76 200

70 185

68 170

69 175

70 200

65 160

66 160

67 175

71 205

74 215

Number of people in a party and total bill for

dinner:

People Bill ($)

1 8.50

3 29.30

5 63.75

10 92.55

6 60.35

4 48.75

2 42.35

2 32.55

3 50.65

5 85.25

Hours of sleep the night before the ACT and

score on the ACT:

Sleep

(hours)

Score

8 25

9 28

7 21

10 26

8.5 18

6.5 16

5.5 25

11 28

9 29

7 19

6 26

7.5 23

8.5 24

6.5 27

8.5 21

10 19

9 20

Years since 1995 and student enrollment:

Years Students

1 537

2 577

3 625

4 665

5 720

6 770

7 832

8 895

9 960

10 1030



10

Elevation of Arizona cities and average daily

temperature.

Elevation (feet) Temperature ( )

Phoenix: 1106 84.5

Page: 4288 69

Flagstaff: 6923 59.9

Winslow: 4856 70.4

Tucson: 2402 83.6

Casa Grande: 1404 86.5

Grand Canyon: 6923 62.8

Yucca: 1831 82.7

Age of an individual and number of contacts

contained in their cell phone.

Age # of phone contacts

11 15

20 230

32 126

66 102

12 503

16 465

18 856

23 25

46 1288

44 465

33 105

10 86

55 42

50 560

25 432

Horsepower of an automobile and gas mileage.

Horsepower Gas Mileage (mpg)

145 18.2

510 7.2

445 11.6

320 12.7

286 8.8



11

Task: You will receive a set of data where you are to:

1. Use the table and formula given to calculate the correlation.

2. Describe the relationship using this numerical value.

3. Prepare to discuss as a group why (or why not) your set of data is more linear than the other groups.

4. Using technology, calculate the line of best fit.

5. Compute the residuals for your data.

6. Create a residual graph that plots the independent value versus the residual calculated for that particular

data point.

7. Be prepared to show your residual graph to the discuss the choice made in task #3. Would you stick

with the class first choice? Why?

Describing relationships numerically.

When describing relationships between two variables, you should address each of the following:

Direction (positive or negative)

Strength of the relationship

Deviations from the overall pattern

Correlation measures the direction and the strength of a linear relationship. Suppose we have data on variables x

and y for n individuals ),(,),,(),,( 12211 nn yxyxyx and we have the means x , y and standard deviations xs ,

ys for each variable. The correlation ® is defined as:

y

i

x

i

s

yy

s

xx

nr

1

1

Wrapping It Up:

Students were given data for the average salary of major league baseball players from 1970 to 2010. Jimmy’s

group calculated the correlation coefficient to be 0.94 and concluded that a strong positive linear relationship

was present. John’s group made a residual graph of the same data which is shown below. John’s group claimed

that the data was not linear. Which group do you agree with and why?

Data Source: http://www.baseball-almanac.com/charts/salary/major_league_salaries.shtml

http://www.baseball-almanac.com/charts/salary/major_league_salaries.shtml



12

Use the formula to calculate the correlation between height and weight of students.

x = y =

xs = ys =

Height (in) Weight (lb)

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

76 200

70 185

68 170

69 175

70 200

65 160

66 160

67 175

71 205

74 215

y

i

x

i

s

yy

s

xx

nr

1

1=



13

Use the formula to calculate the correlation between hours of sleep and the score on the ACT.

x = y =

xs = ys =

Sleep (hrs) Score

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

8 25

9 28

7 21

10 26

8.5 18

6.5 16

5.5 25

11 28

9 29

7 19

6 26

7.5 23

8.5 24

6.5 27

8.5 21

10 19

9 20

y

i

x

i

s

yy

s

xx

nr

1

1=



14

Use the formula to calculate the correlation between number of people in a party and total bill for dinner.

x = y =

xs = ys =

People Bill ($)

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

1

8.50

3

29.30

5

63.75

10

92.55

6

60.35

4

48.75

2

42.35

2

32.55

3

50.65

5

85.25

y

i

x

i

s

yy

s

xx

nr

1

1=



15

Use the formula to calculate the correlation between years since 1995 and student enrollment.

x = y =

xs = ys =

Years Students

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

1 537

2 577

3 625

4 665

5 720

6 770

7 832

8 895

9 960

10 1030

y

i

x

i

s

yy

s

xx

nr

1

1=



16

Use the formula to calculate the correlation between elevation of Arizona cities and average daily temperature.

.

x = y =

xs = ys =

Elevation (feet) Temperature ( )

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

Phoenix: 1106

84.5

Page: 4288

69

Flagstaff: 6923

59.9

Winslow: 4856

70.4

Tucson: 2402

83.6

Casa Grande: 1404

86.5

Grand Canyon: 6923

62.8

Yucca: 1831

82.7

y

i

x

i

s

yy

s

xx

nr

1

1=



17

Use the formula to calculate the correlation between age of an individual and number of contacts contained in

their cell phone.

x = y =

xs = ys =

Age # of phone

contacts

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

11

15

20

230

32

126

66

102

12

503

16

465

18

856

23

25

46

1288

44

465

33

105

10

86

55

42

50

560

25

432

y

i

x

i

s

yy

s

xx

nr

1

1=



18

Use the formula to calculate the correlation between horsepower of an automobile and gas mileage.

x = y =

xs = ys =

Horsepower Gas Mileage

(mpg)

x

i

s

xx

yx

i

s

yy

y

i

x

i

s

yy

s

xx

145

18.2

510

7.2

445

11.6

320

12.7

286

8.8

y

i

x

i

s

yy

s

xx

nr

1

1=



19

BARBIE BUNGEE JUMP

- I can create a scatter plot.




- I can interpret an (r) value.


Overview: You will start by attaching 1 rubber band to a Barbie doll's ankles and then measuring how far she

will fall before rebounding. Then add additional rubber bands and measure the drop. Continue to add rubber

bands and measure the drops then create a mathematical model or equation that can predict the distance a

Barbie doll will fall for a given number of rubber bands. Last of all, a final bungee jump spot will be

announced. Using the prediction equation, determine the number of rubber bands to give Barbie the greatest

thrill. Let Barbie take the plunge, and see how well your prediction model worked.

(Disclaimer: no Barbies were injured during the creation of this activity. Other action figures such as Storm

Troopers, G.I. Joe may be substituted.)

# of rubber bands Distance Dropped

(to the top of head)

1

2

3

4

5

1. Complete the table.

2. Plot the points on the grid. Label and scale the graph.

3. Find the equation of the line of best fit. Record it here.

(distance) = a + b(rubber bands)

r = 2r

4. Give an interpretation of the slope for the equation.



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5. Give an interpretation of the y-intercept for the equation.

6. Graph the line of best fit on the scatter plot of your data. Calculate and record the residuals for 1 through

5 rubber bands. Explain how these residuals assess the fit of the function.

7. Interpret r for your line of best fit.

8. Use the line of best fit to predict how far she would fall with 10 rubber bands.

9. Your instructor will specify a location for the final bungee jump. Use your equation to determine the

number of rubber bands to give Barbie the greatest thrill in this bungee jump. This means she should

come as close as possible to the ground without hitting her head. When you have determined the

number of rubber bands, make this bungee cord and let Barbie jump.

Write a report that reflects your understanding of the Barbie Bungee Jump outcomes. Answer the following

questions within your paragraphs.

Directions: Respond to the following in complete sentences with correct academic vocabulary.

1. Explain the contextual meaning of the y-intercept of a model fit to data.

2. Explain the contextual meaning of the slope of a model fit to data.

3. Explain how the analysis of residuals and the correlation coefficient is used to verify the validity of a line of

best fit.

Teacher Resource for Unit 1 Lesson 1: Linear Models Refresher

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