Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Targeted Content Standard(s):
Student Friendly Learning Targets
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.IC.1 Understand statistics as a process for making inferences about
population parameters based on a random sample from that population.
I can…
 Create a scatter plot.
 Determine a line of best fit using
technology and graph it.
 Interpret slope and intercepts in a
context.
 Make predictions using a line of best fit.
 Interpret an r value.
 Determine residuals and use them to
assess fit.
Targeted Mathematical Practice(s):
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for an express regularity in repeated reasoning
Supporting Content Standard(s): (optional)
S.ID.7 Interpret the slope (rate of change) and the intercept (constant
term) of a linear model in the context of the data.
Purpose of Lesson:
Explanation of Rigor: (Fill in those that are appropriate.)
Conceptual:
Students will describe how two
variables are related (S-ID.6).
Procedural:
Students will create scatter plots
representing two variables (S-ID.6).
Students will understand that
statistics is a process that attempts to
draw conclusions about a larger
population (S-IC.1).
Students will analyze residual values
to determine whether a function is a
good fit for the data (S-ID.6b).
Application:
Students will gather data from a realworld context and then use
technology to fit a linear function to
the data (S-ID.6a).
Students will use technology to
calculate a linear function that fits
data in a scatter plot (S-ID.6c).
Vocabulary:
Residual
Slope
Inference
Mode
Line of best fit
Scatter plot
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Evidence of Learning (Assessment):
Pre-Assessment: Shopping Cart Activity (Segment 2)
Formative Assessment(s): Candy Bar Activity, including components covering correlation coefficient (Segment 3)
Summative Assessment: Barbie Bungee Jump (Segment 4)
Self-Assessment: Use Self-Assessment document (Segment 1) directly following the Shopping Cart Activity, Candy Bar
Activity, and Barbie Bungee Activity (Segments 2 through 4)
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Lesson Procedure:
Segment 1
Approximate Time Frame:
2-4 minutes during each of the
following segments
Focus:
This segment focuses on student
reflection and meta-cognition as well
as fostering their ability to write about
the conceptual components of the
lesson.
Lesson Format:
Resources:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
Differentiation for Remediation: n/a
MP3 - Through determining their level of mastery of each
learning target, students will display Math Practice 3 by
constructing an argument regarding their understanding.
MP4 - Answering the conceptual questions throughout the
lesson will display Math Practice 4 through their ability to
explain the contextual meaning of the mathematics.
Differentiation for English Language Learners:
Potential Pitfall(s): n/a
Independent Practice (Homework): n/a
Differentiation for Enrichment: Students can write a onesentence summary reflection on their mastery of each
learning target. They can explain what learning strategies
helped them learn and how improvements can be made to
the learning strategies (either the strategies themselves or
the students’ approaches to completing them) for more
learning to occur.
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Steps:
Students will self-assess their mastery of the learning targets after each
segment is complete. (Note: students can also use the document and
ratings to pre-assess their understanding before each segment.)
Students will answer the conceptual questions as the lesson progresses;
these questions will be a part of the summative assessment.
Student Self-Assessment for Barbie Bungee Jump Lesson
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 ad solutions to the
class.
5: I feel that I could correctly teach this topic to another student if
asked.
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.
Segment 2
Approximate Time Frame:
45-50 minutes
Focus:
Students will use technology to
calculate the line of best fit for a given
set of data and interpret the meaning
of the mathematics in the given
context. Some questions, especially
regarding (r) (the correlation
coefficient), serve as pre-assessments
to determine students’ prior
knowledge.
Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Resources:
Handout
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Math Practice Look For(s):
While working on this segment, students will display Math
Practice 2 when they are able to accurately explain what a
given value means in the context and explain what a
contextual piece of information means symbolically.
Differentiation for Remediation: If students cannot
complete the segment, reference Math 1 Units 1 and 8 for
more materials on linear equations and modeling with
technology.
Differentiation for English Language Learners:
Differentiation for Enrichment:
Potential Pitfall(s):
Ensure students remember the steps for finding a linear
model with technology and the components of a linear
equation.
Independent Practice (Homework):
The Candy Bar data set and its corresponding questions 58 could be used as in-class or out-of-class independent
practice.
Steps:
The Shopping Cart Train data set and corresponding questions can be
used as a pre-assessment for the lesson. Ask students to complete
questions 1-4 in groups of 2-4 before debriefing as a whole class. After
completing the Shopping Cart Train, ask students to self-assess their
mastery of the learning targets on the Self-Assess sheet. The Candy Bar
data set and questions 5-8 can be used as further practice in class or as
homework. Another method for fostering more student ownership
throughout the lesson segment by asking students to generate their
own data, either about candy bars or another topic.
Teacher Notes/Reflection:
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Segment 3
Approximate Time Frame:
45-50 minutes
Focus:
Through direct instruction, students
will learn the meaning of residual
values and correlation coefficients (r)
as well as how to calculate them.
Lastly, students will learn how to use
the two values to determine how well
a linear function fits a set of data.
Lesson Format:
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Resources:
Handouts with residual values,
correlation coefficients (r), and data
sets (p. 7-17 in the Resource
Document)
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
Students will display Math Practice 3 when they discuss as
a class which correlation coefficients indicate the best fit.
Creating models for the data provided will display Math
Practice 4; using technology to do so will display Math
Practice 5. Using appropriate vocabulary, symbols, and
appropriate rounding will be vital and will display Math
Practice 6.
Differentiation for Remediation:
Potential Pitfall(s):
Ensure students realize the difference between residual
and the correlation coefficient, emphasizing that (r) does
not stand for residual.
Independent Practice (Homework):
Differentiation for English Language Learners:
Differentiation for Enrichment:
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Steps:
Through whole-group, direct instruction, discuss residual values with
the class using the following information.
Teacher Notes/Reflection:
By 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
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.
Using the candy bar data, groups of students will find the following:
Candy Bar
Almond Joy
Baby Ruth
Butterfinger
Caramello
Heath
Hershey
Kit Kat
Krackel
Milky Way
Mounds
Nutrageous
Payday
Serving
Size (g)
45
60
60
45
39
43
42
41
58
53
51
52
Calories
220
275
270
208
210
210
218
210
262
258
260
240
observed y –
predicted y
220 – 223.14
275 – 273.15
270 – 273.15
208 – 223.14
210 – 203.13
210 – 216.47
218 – 213.13
210 – 209.8
262 – 266.48
258 – 249.81
260 – 243.14
240 – 246.47
residual
-3.14
1.85
-3.15
-15.14
6.87
-6.47
4.87
.2
-4.48
8.19
16.86
-6.47
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. As before,
another method for fostering more student ownership throughout the
lesson segment by asking students to generate their own data.
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
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.
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Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
9
Math 3 - Lesson Title: Linear Models Refresher
Unit 1: Statistics (Lesson 1 of 3)
Time Frame: 4-5 Days
Essential Question: From a scatter plot, how are two quantitative variables related?
Segment 4
Approximate Time Frame:
120 minutes
Focus:
This activity serves as the summative
assessment for this lesson, asking
students to apply their understanding
of lines of best fit, residuals,
correlation coefficients, and
interpreting the mathematics in terms
of the context of the experiment.
Lesson Format:
Resources:
Video
Barbie Bungee Activity handouts (p.
19-20 of Resource Document)
Barbie Dolls
Rubber bands
Whole Group
Small Group
Independent
Modeled
Guided
Collaborative
Assessment
Modalities Represented:
Concrete/Manipulative
Picture/Graph
Table/Chart
Symbolic
Oral/Written Language
Real-Life Situation
Math Practice Look For(s):
While working on this segment, students will display Math
Practice 2 when they are able to accurately explain what a
given value means in the context and explain what a
contextual piece of information means symbolically.
Differentiation for Remediation:
Potential Pitfall(s):
Independent Practice (Homework):
Differentiation for English Language Learners:
Differentiation for Enrichment:
Steps:
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
Teacher Notes/Reflection:
http://www.youtube.com/watch?v=CwS3MvJE7l4&NR=1&feature=end
screen
Students’ answers to the Barbie Bungee Jump activity and their
paragraph answering the conceptual questions from the selfassessment serve as the summative assessment for this lesson.
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