teacher resource for unit 1 lesson 1: linear models refresher
TRANSCRIPT
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
1
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.
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
2
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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.
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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.
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
6
Student Resources
Learning 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.
The table below represents data collected from putting together a shopping cart train.
Data from www.inspiration.com/freetrial/inspiredata
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
7
Data from www.inspiration.com/freetrial/inspiredata
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
8
Learning Target:
- I can determine residuals and use them to assess fit.
- 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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
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=
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
19
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 make predictions using a line of best fit.
- I can interpret an (r) value.
- I can determine residuals and use them to assess fit.
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.
Math 3 Unit 1 Lesson 1 S-ID.6, 7, S-IC.1
Math Practices 2, 3, 4
20
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.