5E Lesson Plan Math
Grade Level: 8
Lesson Title: Statistics with Bivariate Data
Subject Area:
Unit Number: 6
Lesson Length: 10
days
Lesson Overview:
This unit bundles student expectations that address representing bivariate sets of data with
scatterplots and representations of linear situations. According to the Texas Education
Agency, mathematical process standards including application, a problem-solving model, tools
and techniques, communication, representations, relationships, and justifications should be
integrated (when applicable) with content knowledge and skills so that students are prepared
to use mathematics in everyday life, society, and the workplace.
During this unit, students continue to examine characteristics of linear relationships through
the lens of trend lines that approximate the relationship between bivariate sets of data.
Students contrast graphical representations of bivariate sets of data that suggest linear
relationships with bivariate sets of data that do not suggest a linear relationship. Scatterplots
are constructed from bivariate sets of data and used to describe the observed data.
Observations include questions of association such as linear (positive or negative trend), nonlinear, or no association. Students extend previous work with linear proportional and linear
non-proportional situations to trend lines as they continue to represent situations with tables,
graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0, respectively. Within a
scatterplot that represents a linear relationship, students use the trend line to make
predictions and interpret the slope of the line that models the relationship as the unit rate of
the scenario.
Unit Objectives:
Students will…
examine characteristics of linear relationships through the lens of trend lines that approximate
the relationship between bivariate sets of data
contrast graphical representations of bivariate sets of data that suggest linear relationships
with bivariate sets of data that do not suggest a linear relationship
use scatterplots to describe observed data including questions of association such as linear
(positive or negative trend), non-linear, or no association
extend previous work with linear proportional & non-proportional situations to trend lines as
they continue to represent situations with tables, graphs, and equations in the
form y = kx or y = mx + b, where b ≠ 0
use trend lines to make predictions
interpret slope that models relationships as unit rate.
Standards addressed:
TEKS:
8.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
8.1B Use a problem-solving model that incorporates analyzing given information, formulating a
plan or strategy, determining a solution, justifying the solution, and evaluating the problem-
Unit 6: Statistics with Bivariate Data
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solving process and the reasonableness of the solution.
8th Grade
8.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as
appropriate, and techniques, including mental math, estimation, and number sense as
appropriate, to solve problems.
8.1D Communicate mathematical ideas, reasoning, and their implications using multiple
representations, including symbols, diagrams, graphs, and language as appropriate.
8.1E Create and use representations to organize, record, and communicate mathematical
ideas.
8.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
8.1G Display, explain, and justify mathematical ideas and arguments using precise
mathematical language in written or oral communication.
8.4B Readiness Standard Graph proportional relationships, interpreting the unit rate as the slope
of the line that models the relationship.
8.5A Supporting Standard Represent linear proportional situations with tables, graphs, and
equations in the form of y= kx.
8.5B Supporting Standard Represent linear non-proportional situations with tables, graphs, and
equations in the form of y = mx + b, where b ≠ 0.
8.5C Supporting Standard Contrast bivariate sets of data that suggest a linear relationship with
bivariate sets of data that do not suggest a linear relationship from a graphical representation.
8.5D Readiness Standard Use a trend line that approximates the linear relationship between
bivariate sets of data to make predictions.
8.5I Readiness Standard Write an equation in the form y = mx + b to model a linear relationship
between two quantities using verbal, numerical, tabular, and graphical representations.
8.11A Supporting Standard Construct a scatterplot and describe the observed data to address
questions of association such as linear, non-linear, and no association between bivariate
data.
ELPS:
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her
own learning processes in all content areas.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic
media to gain an increasing level of comprehension of newly acquired language in all content
areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of
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English language acquisition in listening.
8th Grade
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of
different language registers (formal/informal) using vocabulary with increasing fluency and
accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate,
advanced, or advanced high stage of English language acquisition in speaking.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of
comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or
advanced high stage of English language acquisition in reading.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a
specific purpose and audience in all content areas. ELLs may be at the beginning,
intermediate, advanced, or advanced high stage of English language acquisition in writing.
Misconceptions:
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Students may think that the trend line has to begin at the origin rather than
understanding that a trend line is not always proportional.
Students may think that if both numbers in the data set are decreasing, then it
represents a negative trend.
Students may confuse a positive trend with a negative trend.
Some students may attempt to connect the dots of a scatterplot rather than realizing the
data is discrete and not continuous.
Underdeveloped Concepts:
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Some students may think that the slope in a linear relationship is m =
,
since the x-coordinate (horizontal) always comes before the y-coordinate (vertical) in an
ordered pair, instead of the correct representation that slope in a linear relationship
is m =
.
Students may use (y,x) as the ordered pair instead of (x,y)
Some students may not associate the unit rate of a problem situation to the slope of the
line that represents the problem situation.
Some students may not relate the constant rate of change or unit rate to m in the
equation y = mx + b.
Some students may not relate the constant of proportionality or unit rate as k in the
equation y = kx or m in the equation y = mx + b, when b = 0.
Some students may think that a constant rate of change always means the situation is
proportional.
Vocabulary:

Bivariate data – data relating two quantitative variables that can be represented by a
scatterplot
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Unit 6: Statistics with Bivariate Data
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 Data – information that is collected about people, events, or objects
 Discrete data – data with finite and distinct values, no inclusive of in-between values
 Linear relationship – a relationship with a constant rate of change represented by a
graph that forms a straight line
 Rate – a multiplicative comparison of two different quantities where the measuring unit
is different for each quantity
 Scatterplot – a graphical representation used to display the relationship between
discrete data pairs
 Similar shapes – shapes whose angles are congruent and side lengths are
proportional (equal scale factor)
 Slope – rate of change in y (vertical) compared to the rate of change


in x (horizontal),
or
or
, denoted as m in y = mx + b
Trend line – the line that best fits the data points of a scatterplot
y-intercept – y-coordinate of a point at which the relationship crosses the y-axis
meaning the x-coordinate is equal to zero, denoted as b in y = mx + b
Related Vocabulary:
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Association
Constant
Constant rate of change
Correlation
Dependent
Independent
Linear association
Negative trend
No association
No trend
Non-linear association
Non-linear relationship
Non-proportional relationship
Ordered pair
Origin
Positive trend
Prediction
Proportional relationship
Rate of change
Scale factor
Unit rate
x-axis
y-axis
List of Materials:
Day 1
Relatable Data Personal Whiteboards or Large Construction Paper
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Unit 6: Statistics with Bivariate Data
Day 2-3
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8th Grade
http://www.shmoop.com/probability-statistics/bivariate-data.html
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8thdata/cc-8th-scatter-plots/e/constructing-scatter-plots
Scatterplots[1].ppt
Day 4
www.mathworksheetsland.com
Day 5-7
http://worksheets.tutorvista.com/scatter-plot-worksheet.html
www.ixl.com
www.mathworksheetsland.com
Scatter-Plot-Works Scatter-Plot-Works Scatter-Plot-Works
heet 2.pdf
heet.pdf
heet 3.pdf
Day 8-9
INSTRUCTIONAL SEQUENCE
Phase One: Engage the Students
Day 1
Activity: Scatter Plots with Grades, Sample Activity
Relate the concept to the students using material that is relevant to them. In this activity, you
will need access to test grades (sports scores, recent temperatures in your area, shoe sizes or
any other data sets that will allow students to relate to personally could be used instead). The
example shown will use students’ previous and current test scores to predict possible future
test scores.
Have the students construct a scatterplot using the data you’ve acquired. This can be done in
a variety of ways depending on availability of material and time allowed. Students plot
previous test averages for their class in relation to the week assessed. They then describe
any observations they can make from their graph (Note: This can be done in front of the entire
class, in small groups, or individually by providing each student with the necessary data).
Ask students to observe the data and to make connections. Are the test scores plotted linear
or non-linear? Are they proportional or non-proportional? Do the scores represent a function?
Have the students draw a line as close as possible to represent the scatter plot. Are the
grades improving? Decreasing? Or are there no real correlations to make with their test
scores.
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Unit 6: Statistics with Bivariate Data
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Have students manipulate and create other scatterplots using test scores from other classes
or pre-generated scores to show other possible outcomes. Allow them to come up with their
own conclusions on future scores based on the scatterplot and data provided.
Examples used should include a positive trend, a negative trend, and no trend. Students
should be able to determine a linear or non-linear pattern from the examples used.
Vocabulary: Introduce and explain vocabulary as needed, relating to the material shown
above. The students used bivariate data (weeks and their test scores) in order to create a
scatterplot that may or may not have shown a trend.
What’s the teacher doing?
What are the students doing?
Guiding students to create their own
scatterplots using data obtained relating to
the students.
Reinforcing prior knowledge of linear and nonlinear, graphing coordinate points, and relating
material to real world scenarios.
Allowing students to make correlations
between the data and to make connections
to predict possible future outcomes.
Plotting data in the form of scatter plots and
using the graph to develop an understanding of
trend patterns.
Phase Two: Explore the Concept
Day 2-3
Activity: Guided Practice & Lecture
Lead the class with a power point presentation or lecture relating material to the previous day’s
scatterplot. Add and reinforce vocabulary terminology and allow students additional practice
creating their own scatterplots by guiding them.
Have them label appropriate titles, subtitles, x-axis, y-axis, and scales as needed. Make sure
to stay with simple whole numbers until the students grasp the material. Incorporate a variety
a rational numbers for more advanced students.
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Unit 6: Statistics with Bivariate Data
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In the example shown above, students should be able to draw the shown trend line (make
sure to address students that are connecting dots to form their trend line) to indicate it’s a
positive trend, and to predict grades based on the homework they’ve done. They should be
able to correlate the hours of homework having a direct effect on grades for the class. They
should also be aware that trend lines are NOT required to start at the origin, nor are they
always proportional.
This is also a perfect time to explain independent and dependent variables along with direct
and inverse variation. Does one variable rely on the other? Are the two variables
independent of one another?
Direct variation means as __item 1__ increases, then __item 2__ will increase as well.
An example would be the distance a vehicle travels related to how long it’s been traveling.
Have students give other such examples showing direct variation in the real world.
Indirect variation shows that as ___item 1__ increases, then ___item 2___ will decrease. If
___item 1___ decreases, then ___item 2___ will increase. An example would be how long it
takes for a vehicle to travel X miles. As the vehicle’s speed increases, the time to arrive
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Unit 6: Statistics with Bivariate Data
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8th Grade
decreases, and as the vehicle’s speed decreases, the time before arriving increases.
Have students give other examples of inverse variation.
Independent variables are often placed on the x-axis while dependent variables are place on
the y-axis. When students create their own graphs, they need to be aware of this.
Other possible topics to use: time of day and temperature, years of experience and income,
age and height.
Free premade power point presentations are available online along with videos and sample
problems. Links and one power point are provided below.
http://www.shmoop.com/probability-statistics/bivariate-data.html
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-data/cc-8th-scatterplots/e/constructing-scatter-plots
Lesson 11.4: Scatter Plots
Standards: SDP 1.0 and 1.2
Objective: Determine the correlation of
a scatter plot
Scatterplots[1].ppt
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Unit 6: Statistics with Bivariate Data
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Day 4
Activity: Alternate variations of slope and k.
Repeated exercises using various representations of slope and ensuring the students
recognize that the different vocabulary terms/variables/equations should be used on a rotating
basis until students are comfortable interchanging them.
y = kx and k = y/x represent the constant of proportionality which also denotes slope in an
alternate form and creates linear, proportional equations. Worksheets can be created to
reteach this material (7th grade) if needed at www.mathworksheetsland.com (constant of
proportionality search).
What’s the teacher doing?
What are the student’s doing?
Lecturing, discussing power point
presentation, introducing vocabulary terms,
and relating scatterplots to data sets.
Following along with teachers lecture and
guided examples. Learning appropriate
mathematical terminology.
Providing guidance in labeling tables and
charts appropriately.
Phase Three: Explain the Concept
Day 5-7
Activity: Worksheets, Online Practice
Students that have access to individual computers or a computer lab can benefit from practice
problems assessing trend behavior, plotting and creating scatterplots, reading scatterplots,
charts, graphs, and other visual forms of data, making correlations between two sets of data
and predicting possible future outcomes.
http://worksheets.tutorvista.com/scatter-plot-worksheet.html is free and offers multiple
worksheets to complete online.
www.ixl.com under 8th grade N14 offers practice with scatterplots but requires a membership.
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Unit 6: Statistics with Bivariate Data
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8th Grade
Several worksheets are provided below but different representations of the data provided from
the websites above would allow the students a broader view and a variety of ways to solve for
solutions involving bivariate data. The websites also give the students immediate feedback as
opposed to late feedback offered from paper assessments & practice.
Day 5
Day 6
Day 7
Scatter-Plot-Works Scatter-Plot-Works Scatter-Plot-Works
heet 2.pdf
heet.pdf
heet 3.pdf
www.mathworksheetsland.com
What’s the teacher doing?
What are the students doing?
Monitoring students working in pairs, small
groups or individually on practice with
scatterplots and other forms of bivariate
data.
Developing technical skills while reinforcing
knowledge of graphs on a variety of computer
websites.
Completing additional worksheets on
interpreting bivariate data.
Phase Four: Elaborate on the Concept
Day 8-9
Activity: Group work, creating real world problems using bivariate data.
Questions to ask each student, informally assessing them on each one.
Have the students tell you what bivariate data is in their own words.
Can students make predictions of both independent and dependent variables when giving the
other based on their trend lines?
Can students write an equation in the form y = mx + b to model a linear relationship given
verbal, numerical, tabular, or graphical representations? This should be review, can review
Unit 4 material as needed.
Can students successfully create their own scatterplot from observed data? And determine
whether the association is linear, non-linear, or represents no association?
Partner up students in pairs and allow them to create their own word problems. Have students
depict two problems: one showing a linear trend and one showing a non-linear trend. Offer
suggested topics for those having difficulties choosing their own, or allow them to collaborate
together for a few minutes with classmates before starting the project.
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Unit 6: Statistics with Bivariate Data
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Samples of non-linear bivariate data shown below.
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8th Grade
Unit 6: Statistics with Bivariate Data
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8th Grade
What’s the teacher doing?
What are the students doing?
Providing students with in-depth real world
problems and guiding them through solving
both linear, and non-linear examples.
Independently solving real world problems by
evaluating bivariate data provided in a variety
of forms such as graphs, charts, tabular, and
numerical.
Assessing students’ capabilities and
adjusting them for difficulty with regards to
number sets used and steps required.
Creating their own scatterplots from data with
appropriate labeling.
Phase Five: Evaluate Students’
Understanding of the Concepts
Day 10
Activity: Assess students based on IFD (PI, performance indicator now PA)
1) A minimum wage is the lowest hourly, daily, or monthly amount that employers must legally
pay to workers. The United States Department of Labor introduced the Fair Labor Standards
Act in 1938. Today the minimum wage is at the same hourly rate as it was in 2009. The table
below shows the minimum wage rate and its respective year.
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Unit 6: Statistics with Bivariate Data
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a) Construct a scatterplot that accurately represents the minimum wage data and the year.
b) Describe the association of the observed data as linear, non-linear, or no association.
c) Draw a trend line that approximates the relationship between the year and minimum wage.
d) Describe the trend of the graph as positive, negative, or no trend.
e) Use the trend line to predict the minimum wage in the year 2020.
2) As minimum wage increased, the price of certain items increased as well.
a) Compare and contrast the relationships with the sets of data displayed in the graphs below
and describe the associations in the graphs as linear, non-linear, or no association.
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Unit 6: Statistics with Bivariate Data
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2) Audie constructed a scatterplot to display the amount she spent on gasoline for the past 6
months. Sometimes she would put a few gallons of gas in her car, and sometimes she would
fill her tank completely. Audie drew a trend line to approximate the relationship between the
total price spent on gasoline and the number of gallons of gas she put in her car.
a) Describe the trend of the graph as positive, negative, or no trend.
b) Use the graph and trend line to write an equation in the form y = mx + b to represent the
linear relationship between total price spent on gasoline and the number of gallons of gas
Audie put in her car.
c) Describe the meaning of slope of the trend line in terms of a unit rate that represents the
relationship between the total price spent on gasoline and the number of gallons of gas Audie
put in her car.
d) Use the trend line to predict the amount Audie would pay for 3 gallons of gas, 12 gallons of
gas, and 16 gallons of gas.
2) Audie used her credit card to pay for her gasoline and other purchases. She usually makes
bimonthly payments large enough to pay off any recent charges and still decrease her overall
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Unit 6: Statistics with Bivariate Data
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credit card balance. Her credit card company provided her with a scatterplot displaying her
monthly balance for January (month 1) through June. Audie added a trend line to the graph to
approximate the relationship between her monthly credit card balance and the year.
a) Describe the trend of the graph as positive, negative, or no trend.
b) Use the graph and trend line to write an equation in the form y = mx + b to represent the
linear relationship between Audie’s monthly credit card balance and the year.
c) Use the trend line to predict Audie’s credit card balance in July, September, and the
following January assuming she continues her spending and payment habits.
What’s the teacher doing?
What are the students doing?
Assess the students’ understanding of
statistics with bivariate data through PI and a
formal assessment.
Demonstrating mastery and understanding of
statistics with bivariate data through
assessments involving real-world critical
thinking questions.
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