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TRANSCRIPT
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Christine Bolen
E00889469
CURR 305 UNIT: MATH
APPLIED LINEAR EQUATIONS
December 9, 2009
9:30 a.m
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Unit Project Christine Bolen
CURR 305 12/09/2009
Lowenstein
Table of Contents
Conceptual Framework
Overview - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Page 3
Rational - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Pages 3 - 6
Concept Map - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Page 7
Unit Goals/Mi Content Standards - - - - - - - - - - - - - - - - Pages 8 - 9
Scope and Sequence Chart/Multiple Intelligences Chart Page 10
Lesson Plans
Graphing Linear Equations (Direct Lesson) - - - - - - - - - Pages 11 - 18
Rise and Run (Cooperative Lesson) - - - - - - - - - - - - - - Pages 19 - 28
Linear Regression (Inquiry Lesson) - - - - - - - - - - - - - - Pages 29 - 37
Challenge Scenario/Authentic Assessment - - - - - - - - - - Pages 38 - 44
Bibliography and Resources - - - - - - - - - - - - - - - - - - - - Page 45
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Unit Project Christine Bolen
CURR 305 12/09/2009
Lowenstein
Overview
This unit is designed for students at a 9th
/10th
grade math level in an Algebra 1
classroom. The unit topic is linear equations, with a focus on slope formulas. The unit is
entitled “Applied Linear Equations.” The essential question is, “In what ways can we use
data to address ecological and cultural issues?” The purpose of this unit is to gain an
understanding and appreciation for how data can be represented and used to support
arguments and predict future events. The concepts of linear equations can be applied to
any subject area that utilizes data, such as science, physics, biology, economics, finance,
and political science.
Rationale
This unit plan, titled, “ Applied Linear Equations,” is designed for a 9th
/10th
grade
algebra 1 course. The unit topic is linear equations, with the essential question, “In what
ways can we use linear equations to address ecological and cultural issues?” The big idea
of the unit is data, how it can be represented, and how it can be used to express and solve
problems.
This unit explores algebraic concepts, such as linear functions, linear equations, slope and
constant rate of change, linear regression, and predicting outcomes with linear models.
The students will apply these concepts to real world ecological problems by expressing a
problem mathematically using linear equations in various forms. Students will develop
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an appreciation for the simplicity and power of mathematics as a universal tool that can
be used to express and solve real life problems.
Classroom work will involve both group and individual activities that explore the
concepts of linear equations using hands on approach. The students will be working with
real life data as they explore the concepts of linear equations, such as slope and rate of
change. The final project, titled, "Applying Math to Real World Problems," is designed
to act as the culminating experience for this unit. This assessment is designed to enable
students to apply math to real world problems. For this assessment, students will design a
mathematical model using data associated with the ecological and cultural crisis and
present their presentations to the community. To prepare for this event, students will be
introduced to linear regression, where they will inductively find the algorithm used in
linear regression.
Prior to this unit students have had experience evaluating expressions, simplifying
equations, writing equations in function from, using formulas, making a table of values
for a function. All of these skills prepared the students for this unit. This unit requires the
students to build on these skills as they explore the elements of linear functions, such as
slope, rate of change, intercepts, various forms of equations, and fitting a line to data.
This unit prepares the students for the next unit where they will solve linear systems of
equations and inequalities. Moreover, this lesson lays the foundation for students who
will study calculus, which is the study of change.
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This unit helps prepare students for the current ecological crisis by providing them the
tools to interpret data associated with the ecological data. As the students analyze the
trends related to oil supply and demand, peak oil, arctic sea ice extent, rising sea levels,
etc, they will discover patterns, draw conclusions and make predictions based on the data.
Such knowledge is a powerful force towards changing the trajectory we find ourselves
on.
This unit is differentiated to accommodate the diverse needs of the students. Specifically,
to accommodate students who are English second language learners, Dyslexic, have a
hearing impairment, or read at a 4th
grade level, visual aids will be used to explain both
abstract and concrete concepts, and information will be read to them. To accommodate
students with ADHD, the lesson will facilitate an appropriate amount of activity, change
and movement. Gifted students will be given appropriate challenges to cultivate a deeper
understanding of the concepts. For my South American students, ecological data from
their respective countries will be made available for analysis and comparison. Lessons
will include both group work and individual work. Lesson activities will be tiered and
assessments will include at least three options to accommodate different intelligence
types.
My educational philosophy is that, for the purpose of a better society, the primary
objective of public education is to teach students to think critically and develop their
ability to make a logical and cohesive argument. I believe that students learn best in an
environment where they feel accepted as part of a community. Finally, lessons should be
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designed in a way that is relevant to the students. This unit plan embodies my philosophy
by its design. It is designed around topics that are relevant to all of the students, and
culminates with the ecological and cultural crisis. The essential questions and lessons in
this unit develop students’ ability to think critically. The unit plan is designed to develop
students’ ability to understand and critique data, identify patterns, and make predictions,
developing their ability to make a logical and cohesive argument. Engaging the students
in a discussion about an ecological issue that affects each of them, and designing lessons
where they work together in groups, helps to create a community within the classroom.
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!Unit Project Christine Bolen
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CURR 305 10/12/2009
Lowenstein
Unit Goals/Objectives
Students will be able to…
Cognitive-
• Plot data points on an x-y axis. (A2.1.3)
• Identify the slope of a line and use the slope to describe the rate of change in the
data. (A2.1.7)
• Find the equation of a line when given the coordinates of two points. (A3.1.1)
• Express a linear equation in slope-intercept, point-slope, and standard forms.
(A3.1.1)
• Describe the meaning of constant rate of change. (A2.3.2)
• Apply linear equations to real life problems such as oil usage, sea level change, etc.
(L1.2.4)
• Summarize data in a graph or chart. (A2.1.3)
• Predict an outcome using a linear equation. (L1.2.4)
Affective-
• Develop an appreciation for the simplicity and power of mathematics as a
universal tool that can be used to express and solve real life problems.
High School Content Standards and Expectations
This unit will meet the following high school content standards and expectations for
algebra:
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A2.1.3 - Represent functions in symbols, graphs, tables, diagrams, or words and translate
among representations.
A2.1.7 - Identify and interpret the key features of a function from its graph or its
formula(s).
A2.3.2 - Describe the tabular pattern associated with functions having a constant rate of
change (linear); or variable rates of change.
A3.1.1 - Write the symbolic forms of linear functions (standard, point-slope, and slope-
intercept) given appropriate information and convert between forms.
L1.2.4 - Organize and summarize a data set in a table, plot, chart, or spreadsheet; find
patterns in a display of data; understand and critique data.
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Scope and Sequence Chart/Multiple Intelligences Chart
Day 1 Day 2 Day 3 Day 4 Day 5
Direct Lesson:
“Graphing
Linear
Equations”
MI:
Kinesthetic
Visual
Interpersonal
Extended
Activity:
Homework
sheet
Cooperative
Lesson: “Rise
and Run”
MI:
Kinesthetic
Spatial
Logical
Linguistic
Extended
Activity: Group
Evaluation and
Student activity
sheets
*Direct Lesson:
“Linear
Equations in
Slope-Intercept
Form”
*Direct Lesson:
“Linear
Equations in
Point-Slope
Form”
* Direct
Lesson: “Linear
Equations in
Standard Form”
Day 6 Day 7 Day 8 Day 9 Day 10
Inquiry Lesson:
“Linear
Regression”
MI: Logical
Intrapersonal
Spatial
Linguistic
Interpersonal
Review for
Unit Test
Unit Test Authentic
Assessment
Work Day:
Watch short
ecological
video.
Assignment
Instructions
Groups formed
Topics selected
Authentic
Assessment
Work Day:
Computer Lab
Gather data
Day 11 Day 12 Day 13 Day 14 Day 15
Authentic
Assessment
Work Day in
Computer Lab:
Graph data
Build the math
models
Authentic
Assessment
Work Day in
Computer Lab:
Create
presentations
Authentic
Assessment
Work Day in
Computer Lab:
Finish
Presentations
Finalize Math
Models.
Presentations
for those unable
to attend
community
event.
Authentic
Assessment
Presentations
MI: Linguistic
Logical
Spatial
Kinesthetic
Interpersonal
Intrapersonal
Naturalistic
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I. Framing the Lesson
The purpose of this lesson is for students to learn the relationship between linear
equations in written form and linear equations graphed on a coordinate plane. Students
should be familiar with number lines, a single dimensional system. They will now work
with a two dimensional system, namely a coordinate plane, and ordered pairs, where they
will express linear functions on a graph. Graphing linear equations lays the foundation for
slope and rate of change and comparing families of graphs.
Michigan Standards
Algebra I –
A2 Functions
A2.1 Operations and Transformations with Functions
A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and
translate among representations.
A3 Families of Functions
A3.1 Lines and Linear Functions
A3.1.2 Graph lines (including those of the form x = h and y = k) given
appropriate information.
Lesson Objectives
Students will be able to:
! Identify the x and y axis on a coordinate plane
! Identify the four quadrants of a coordinate plane
! Identify points in a coordinate plane
! Plot points on a coordinate plane
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! Create a table to represent x and y values from an equation
! Identify the intercept points of a line on a graph
! Represent a linear equation on a graph
Materials
Overhead projector or chalkboard
Multi-colored markers or different colored chalk
Pencil
Paper
Graph paper
II. Engage (8 minutes)
The teacher will say to the class, “Can someone explain the purpose of the coordinate
system used in the game Battleship and how it is used?”
The teacher will show the students a Battleship game brought into the classroom and pass
the game around the room so all the students can see it. As we will discuss the
coordinate system used to play the game Battleship, the teacher will introduce an x-y
coordinate plane, by drawing it on the board and labeling the four quadrants. The teacher
will explain how this coordinate system is slightly different than the one used in the game
Battleship, explaining that instead of letters and numbers, they will be using only
numbers, where the first number of an ordered pair represents the x coordinate and the
second number of an ordered pair represents the y coordinate.
By the end of this lesson, you will be able graph a linear equation.
The teacher can ask the following questions to guide through a discussion.
• What applications are there for a coordinate system in the real world?
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• What are the benefits of using numbers instead of letters in a coordinate system?
• What would the equation, x = y look like when graphed on a coordinate plane?
The purpose of this lesson is for students to learn the relationship between linear
equations in written form and linear equations graphed on a coordinate plane. Students
should be familiar with number lines, a single dimensional system. They will now work
with a two dimensional system, namely a coordinate plane, and ordered pairs, which lays
the foundation for visually expressing functions.
The objectives are to:
* Identify the x and y axis on a coordinate plane
* Identify the four quadrants of a coordinate plane
* Identify points in a coordinate plane
* Plot points on a coordinate plane
* Identify the intercept points of a line on a graph
* Represent a linear equation on a graph
The teacher will then take a few minutes to review concepts familiar to the students, such
as the use of a number line; and the domain and range of a function.
III. Explore/Enable/Explain (25 minutes)
Using either an overhead projector or the board, the teacher will introduce a two-
dimensional coordinate plane by drawing an x-y axis, labeling the four quadrants. The
teacher will ask the students to draw the x-y axis on their sheet of graph paper and label
the quadrants. The teacher will then demonstrate how to plot ordered pairs on the plane
while involving participation from the students and monitoring for understanding. The
teacher will then create a table that represents the values of x and y from an equation,
such as y = x, with the domain -3, -2, -1, 0, 1, 2, 3. The teacher will guide the students to
fill in the values of x and y in the table on their sheets of graph paper. The teacher will
guide the students as they plot the points on their sheet of graph paper. The teacher will
have the students draw a line through the plotted points and announce to the students that
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they just graphed an equation. The teacher will ask the students to identify the range of
the function.
The teacher will introduce the equation y = 2x - 1, with a domain of -3, -2, -1, 0, 1, 2, 3.
The teacher will ask the students to describe what they think the graph of this equation
would look like. The teacher will have the students work together in groups to fill in the
values on the table.
The teacher will ask for a volunteer to use the table they created and plot the points either
on the board or on the transparencies. Different colors of marker or chalk will be used to
plot the points. The teacher will use a different color of marker or chalk to draw a line
through the points. The teacher will ask the students to compare the graphs of the two
equations and explain how they are different.
The teacher will discuss some key concepts of linear equations by asking the following
questions:
1) At Friday night’s football game, what did it mean when the opposing team
intercepted the football?
2) Where does the x intercept occur on the graph?
3) Where does the y intercept occur on the graph?
4) Looking at graph #1, if the x-axis represents time in minutes, and the y-axis
represents distance in miles, what was the distance traveled at 1 hour?
5) Looking at graph #2, what is how long did it take to travel 3 miles?
The teacher will either put practice problems on the board or on the overhead. While the
students are working on this exercise, either independently or in small groups, the teacher
will circulate around the classroom to assist anyone who might be having problems and
answer any questions. During this time the teacher will assess the students as they solve
the problems. When all students are finished, the teacher will go over the answers with
the students, checking for understanding.
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IV. Enact/Evaluate (15 minutes)
After the teacher explained, modeled, and guided the class in graphing a linear equation,
the students will be given a worksheet with three choices of problems to solve
independently. This worksheet will be turned in at the end of the class period. The
teacher will assess the students by reviewing the process the students used to graph an
equation.
The in-class worksheet is attached.
V. Extend (3 minutes)
Homework problems will be assigned to reinforce and give practice in the lesson
concepts and skills. The assigned homework is attached.
VI. Differentiation
This lesson was differentiated to accommodate the diverse needs of the students. The
lesson was abstract in that it asked students to imagine what a function would look like
when graphed on a coordinate plane. Creating tables and plotting points on graph paper
helped the kinesthetic learner. The use of different colors appealed to the visual learner.
Working in groups was good for the interpersonal learning style. Moreover, giving
students the option of which problems to solve accommodated various learning styles.
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In-class Activity
Solve from one set of the following problems.
1) Graph the following equations.
a) y + x = 2
b) x – 2y = 3
c) y = 0
d) x = 2
2) The Weather Service releases weather balloons twice daily at over 90 locations in the
United States in order to collect data for meteorologists. The height h (in feet) of a
balloon is a function of the time t (in seconds) after the balloon is released.
a. Make a table showing the height of a balloon after t seconds for t = 0 through
t = 10.
b. A balloon bursts after a flight of about 7,200 seconds. Graph the function and
identify the domain and range.
3) Suppose the point (a,b) lies in Quadrant IV. Describe the location of the following
points: (b, a), (2a, -2b), and (-b, -a). Explain your reasoning.
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Rise and Run
I. Framing the Lesson
Objectives and Lesson Context
The purpose of this lesson is for students to learn to calculate the slope of a line and
interpret slope as rate of change. Students should be familiar with graphing linear
equations. This lesson serves to introduce to students the concept of rate of change and its
application to real-world problems. This lesson prepares students for the next lessons
where they will write symbolic forms of linear equations; create mathematical models
and using linear models to solve problems. Moreover, this lesson lays the foundation for
students who will study calculus, which is the study of change.
Michigan Standards
Algebra I –
A2 Functions
A2.1 Operations and Transformations with Functions
A2.1.7 Identify and interpret the key features of a function from its graph or its
formula(s).
A3 Families of Functions
A3.1 Lines and Linear Functions
A3.1.2 Graph lines (including those of the form x = h and y = k) given
appropriate information.
Lesson Objectives
Students will be able to:
! Identify the slope of a line
! Find the y-intercept of a linear equation
! Interpret the slope of a line as a rate of change
! Work cooperatively in groups to accomplish a shared goal
Materials
Group Exercise Instructions and Guidelines
Role Description sheets (one for each student)
Overhead projector or chalkboard
21 Textbooks (3 for each group)
14 rulers (2 for each group)
Pencil
Paper
Graph paper
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II. Engage
Anticipatory Set/Motivation/Connection to Student Life Experience
The teacher will have the following question written on the board as students enter the
classroom,
“In what ways can you measure changes to your individual grade point averages
from one grading period to the next?”
The teacher will announce that today’s topic is measuring change. The purpose of this
lesson is to calculate the slope of a line and interpret slope as rate of change. The teacher
will briefly review concepts familiar to the students, such as the graphed linear equations.
• The teacher will ask the students to ponder the question on the board, individually
for five minutes.
• The students will be asked to list their ideas either as a written list, a graph, a
chart, or a diagram.
• As the students are thinking about this question, the teacher would pose the
following questions to guide them:
o How would you model change?
o How could you use math to describe change?
• After five minutes the teacher will ask the students to spend the next few minutes
comparing their ideas with their neighbor’s.
• After a few minutes are completed, the teacher will ask for volunteers to share
their ideas with the rest of the class.
• As the ideas are shared, the teacher will encourage students to ask questions of the
presenters.
III Explore/Enable
Group Composition
The teacher will introduce the group activity, titled “Rise and Run,” by explaining the
team roles from the Role Descriptions handout. The teacher will have each student
choose one of the five roles to fill. Six groups of five will be formed to include each role
per group. Dividing the groups based on students’ role choices make the groups
somewhat homogeneous, by attracting different intelligent types to each group.
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Role Descriptions
Each group will have
• A Team Presenter – The team presenter will be responsible to direct the activities
of the team. The presenter will also be responsible to record the team’s
observations at the end of the exercise and report the team’s findings to the rest of
the class.
• A Builder – The builder will be responsible to construct the ramps, using a stack
of books and a ruler that will serve as the ramp.
• A Measurer – The measurer will be responsible for measuring the rise and run of
the constructed ramps and communicating the values to the recorder.
• A Recorder – The recorder will be responsible to make a table to record the rise,
run, and slope for seven ramps. The recorder will be responsible for recording the
measurements in the table. The recorder will be responsible for calculating and
recording the slope of the ramps in the table.
• A Graph Maker – The graph maker will be responsible to graph each of the slopes
on a sheet of graph paper, labeling each graph correctly.
Procedures
The teacher will have the classroom set up with seven stations, with each station set up
for five students. The instructions for the exercise will be posted at each station. After the
students are seated in their groups, the teacher will review the instructions for the
exercise, by reading and demonstrating how to construct a ramp, how to measure the
vertical rise and horizontal run, how to calculate the slope, and how to graph the slope.
The students will then begin the activity, with the team presenters assuming the role of
assisting and overseeing their group’s work.
Step 1.
a. The builder will use a stack of books and the ruler to construct a ramp.
b. The measurer will measure the vertical rise and horizontal run of the ramp and
communicate the values to the recorder
c. The recorder will record the measurements in the table.
d. The recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope value in the table.
f. The graph maker will graph the slope on a sheet of graph paper, indicating the
slope value.
Step 2.
a. Without changing the vertical rise, the builder will construct three ramps with
different runs by moving the lower end of the ruler.
b. The measurer will measure the vertical rise and horizontal run of each of the three
ramps and communicate the values to the recorder.
c. The recorder will make record the measurements of each of three ramps in the
table.
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d. For each of the three ramps, the recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope values for each of three ramps in the table.
f. The graph maker will graph each of the three slopes on a sheet of graph paper,
indicating the slope value.
Step 3.
a. Without changing the horizontal run, the builder will construct three ramps with
different vertical rises by adding or removing books
b. The measurer will measure the vertical rise and horizontal run of each of the three
ramps, communicating the values to the recorder.
c. The recorder will make a table and record the measurements of each of three
ramps in the table.
d. For each of the three ramps, the recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope values for each of three ramps in the table.
f. The graph maker will graph each of the three slopes on a sheet of graph paper,
indicating the slope value.
The team presenter will direct and oversee the above steps
Step 4.
a) The team presenter will collect and record the groups’ observations as they use
the data collected from the activity to complete the following exercises as a group.
b) Describe how the slope of the ramp changes given the following conditions.
• The run of the ramp increases, and the rise stays the same.
• The rise of the ramp increases, and the run stays the same.
c) Describe the relationship between the rise and the run of a ramp:
• if its slope = 1
• if its slope > 1
• if its slope < 1
Step 5.
a) The team presenters will present their group’s findings to the rest of the class.
Social Skills Training
Students are developing their social skills throughout this entire lesson by learning to
work together toward a common goal. This lesson requires participation from each group
member for the group to be successful. The teacher will stress the importance of
teamwork and cooperation. The teacher will briefly review the group rules, many of
which are the classroom rules, developed by the students at the beginning of the year.
The teacher will call on students to model a few of the rules.
Applicable rules include:
1. Treat each other with respect. (If you can’t say something nice, say nothing.)
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2. Listen to others attentively as they express themselves (We can learn from
others, even if we do not agree.)
3. Encourage participation from all team members
4. Give reasons for your opinions
5. Address each other by their first name
IV. Evaluate/Enact
As the students are working in their groups, the teacher will walk around the room
monitoring their progress, answering questions, and guiding them if necessary. After the
groups have finished presenting each of their findings, the teacher will pose the following
questions on the overhead or on the board:
• How do you calculate the slope of a line?
• How would you calculate changes to your grade point average from one
grading period to the next?
• How do you interpret slope as a rate of change?
• What can you conclude about a line if the slope of the line is zero?
• How can you tell if the slope of a line is positive, negative, zero, or
undefined?
The teacher will ask a question from each group to open the discussion.
In closing, the students will reflect on what they learned from the activity by journaling in
a learning log.
Each student will be accountable for the role they played in their group. Each member of
the group will assess the performance of the team and the team members by completing a
group evaluation sheet, which will be submitted at the end of class. A group evaluation
sheet is attached.
Student learning will be extended through the use of slope-intercept form in the next
section.
V. Differentiation
This lesson is differentiated to accommodate the diverse needs of the students. Students
are given the opportunity to choose from several team roles that would appeal to their
particular learning style. The group work appeals to the interpersonal learner.
Constructing and measuring the ramps appeals to the bodily-kinesthetic learner. Graphing
the results appeals to the spatial learner. Performing the calculations appeals to the
logical-mathematical learner. Organizing and presenting the results appeals to the verbal-
linguistic learner. Reflecting on the results appeals to the intrapersonal learner.
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Name____________________
Group Evaluation
Please complete the following questions and hand in before leaving class.
Each team member contributed to the group activity Yes No
Each team member followed the posted group rules Yes No
Our group worked well together Yes No
I enjoyed this group activity Yes No
One thing that our team did well was _________________________________________
One thing that your team could have done better was _____________________________
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Role Descriptions
Each group will have
• A Team Presenter – The team presenter will be responsible to direct the activities
of the team. The presenter will also be responsible to record the team’s
observations at the end of the exercise and report the team’s findings to the rest of
the class.
• A Builder – The builder will be responsible to construct the ramps, using a stack
of books and a ruler that will serve as the ramp.
• A Measurer – The measurer will be responsible for measuring the rise and run of
the constructed ramps and communicating the values to the recorder.
• A Recorder – The recorder will be responsible to make a table to record the rise,
run, and slope for seven ramps. The recorder will be responsible for recording the
measurements in the table. The recorder will be responsible for calculating and
recording the slope of the ramps in the table.
• A Graph Maker – The graph maker will be responsible to graph each of the slopes
on a sheet of graph paper, labeling each graph correctly.
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Group Instructions
The team presenter will direct and oversee the following steps
Step 1.
a. The builder will use a stack of books and the ruler to construct a ramp.
b. The measurer will measure the vertical rise and horizontal run of the ramp and
communicate the values to the recorder
c. The recorder will record the measurements in the table.
d. The recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope value in the table.
f. The graph maker will graph the slope on a sheet of graph paper, indicating the
slope value.
Step 2.
a. Without changing the vertical rise, the builder will construct three ramps with
different runs by moving the lower end of the ruler.
b. The measurer will measure the vertical rise and horizontal run of each of the three
ramps and communicate the values to the recorder.
c. The recorder will make record the measurements of each of three ramps in the
table.
d. For each of the three ramps, the recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope values for each of three ramps in the table.
f. The graph maker will graph each of the three slopes on a sheet of graph paper,
indicating the slope value.
Step 3.
a. Without changing the horizontal run, the builder will construct three ramps with
different vertical rises by adding or removing books
b. The measurer will measure the vertical rise and horizontal run of each of the
three ramps, communicating the values to the recorder.
c. The recorder will make a table and record the measurements of each of three
ramps in the table.
d. For each of the three ramps, the recorder will calculate the slope: Slope = rise/run
e. The recorder will record the slope values for each of three ramps in the table.
f. The graph maker will graph each of the three slopes on a sheet of graph paper,
indicating the slope value.
Step 4.
The team presenter will collect and record the groups’ observations as the group
uses the data collected from the activity to complete the questions on the graphic
organizer handout.
Step 5. The team presenters will present their group’s findings to the rest of the class.
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Graphic Organizer
Team Presenter: _____________________ Builder: ________________________
Measurer: __________________________ Recorder: _______________________
Graph Maker: _________________________ Grade: ______/______
Graph #1 Graph #2 Graph #3
Vertical Rise: _______ Vertical Rise: _______ Vertical Rise: _______
Horizontal Run: ______ Horizontal Run: ______ Horizontal Run: ______
Slope (rise/run): _____ Slope (rise/run): _____ Slope (rise/run): _____
Graph #4 Graph #5 Graph #6
Vertical Rise: _______ Vertical Rise: _______ Vertical Rise: _______
Horizontal Run: ______ Horizontal Run: ______ Horizontal Run: ______
Slope (rise/run): _____ Slope (rise/run): _____ Slope (rise/run): _____
Graph #7
Vertical Rise: _______
Horizontal Run: ______
Slope (rise/run): _____
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Directions: Complete the following questions as a group.
1. Describe how the slope of the ramp changes given the following conditions:
a) The run of the ramp increases, and the rise stays the same. _________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
b) The rise of the ramp increases, and the run stays the same. __________________
_________________________________________________________________
__________________________________________________________________
__________________________________________________________________
2. Describe the relationship between the rise and the run of a ramp:
a) If its slope = 1 ____________________________________________________
_________________________________________________________________
b) If its slope > 1 ____________________________________________________
_________________________________________________________________
c) If its slope < 1 ____________________________________________________
_________________________________________________________________
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Linear Regression
I. Lesson Framing
Michigan Standards/Benchmarks Addressed
This assignment includes the following state standards:
S2.1.1 – Construct a scatter plot for a bivariate data set with appropriate labels.
S2.2.1 – For bivariate data that appear to form a linear pattern, find the least squares
regression line by estimating visually and by calculating the equation of the
regression line. Interpret the slope of the equation for a regression line.
S2.2.2 – Use the equation of the least squares regression line to make appropriate
predictions.
L1.2.4 – Organize and summarize a data set in a table, plot, chart, or spreadsheet; find
patterns in a display of data; understand and critique data displays in the media.
Lesson Objectives
Students will:
• Use real world data to inductively explore linear regression.
• Differentiate between positive correlation and negative correlation.
• Create a scatter plot and perform linear regression on a data set.
• Make predictions by visually estimating best-fitting lines.
• Write an equation to model data.
• Create a mathematical model using real world data.
• Calculate the equation and slope of the regression line.
• Test predictions against a mathematical model.
This lesson is given at the end of the unit and prepares the students for the challenge-
based scenario where they will be creating and presenting mathematical models to the
community. This lesson will be completed in one class period. Prior to this lesson,
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students will have had practice graphing and plotting points, graphing linear equations
and functions, writing linear equations in various forms, and calculating slopes and rates
of change. This lesson demonstrates how to express real world problems in mathematical
terms, through the use of linear functions, using a best fitting line. During this lesson,
students will discover inductively, with limited guidance, how linear regression is derived
and how it is useful in mathematical models. As students use the equation of the least
squares regression line to make appropriate predictions, their predictions will be more
accurate, thus improving the quality of their mathematical models. During this lesson,
students will develop their skills in organizing, summarizing, and critiquing real world
data sets, preparing them for their project at the end of the unit. This the last lesson in the
unit prior to the student’s culminating project. The lesson to follow in the next unit
explores non-linear mathematical models.
Materials
• Graph paper
• Graphing Calculators
• Computers with Microsoft Excel software
• Overhead Projector
• Student Instruction Sheet
II. Anticipatory Set
The lesson will begin by introducing a study completed by Algebra II students that
explores the relationship between television time and test scores, found at the following
web address:
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http://www.utdanacenter.org/k12mathbenchmarks/tasks/29_tvtestgrades.php.
This study captures the students’ attention, because their peers created it, and it involves a
topic that is relevant to the students. After discussing the study, the students will be
introduced to the inquiry question for this unit, which is “how would you best fit a line to
plotted data?” The students will be asked to take a few minutes to study the two scatter
plots below and each student write a short paragraph describing how would they would
best fit a line to the plotted data?
After dividing the students into groups of four or five, the students will be asked to
discuss within their groups answers to the following questions:
Graph #1 Graph #2
1. What is the correlation between hours of studying and test scores?
2. What is the correlation between hours of television and test scores?
3. Predict a reasonable test score for 4.5 hours of studying.
4. Predict a reasonable test score for 4.5 hours of television watched.
5. What method was used to make the predictions?
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III. Explore/Enable/Explain
Students will use the 1st clue set to answer the following questions.
1. Can you think of a method that would provide a more accurate prediction of the
test score for 4.5 hours of studying?
2. How can the your method of fitting a line to the data result in a better fit?
After the groups share their answers, each group will complete the following questions
using clue set #2.
1. Use graph paper to draw a scatter plot and use a linear equation to fit the line to
the data. (See instruction sheet)
2. How could you find the sum of the distances between each point on the scatter
plot and the line?
3. How would we mathematically determine the line that best fits the data?
4. Define residue.
5. Define the sum of squared residues.
6. Compare your prediction of a reasonable test score for 4.5 hours of studying
against a prediction using linear regression. (See Student Instruction Sheet)
a. What is the slope of the line of the equation given by linear regression?
b. Using the equation given from linear regression, find y when x = 4.5
c. How does the answer compare to your prediction?
IV. Enact/Evaluate
Conclusion and Justification:
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• Using your prediction comparisons before and after you made the revisions to
your answers of how to best fit a line to the data, what is your conclusion for a
best fit a line to the data in a scatter plot?
• How close was your original prediction to the value derived from linear
regression?
• Explain how you came to your final conclusion and how the data supports your
conclusion about lines that best fit the data.
Class Discussion
Explain how you reached your conclusions about lines that best fit the data. The class
will discuss how the equation derived from linear regression. The teacher will ask the
students how this equation can be used to describe the data.
Assessment of Student Learning
Students will be assessed on the use of linear regression during their culminating project.
This authentic project requires them to use math to model a real world problem, and more
specifically, requires them to use linear regression to determine the line equation that best
fits the data. Moreover, the project provides options for their presentations. The reflection
assignment is an assessment tool to determine the level of students’ understanding.
V. Differentiation
Spatial – The use of graphs and charts will appeal to these students.
Linguistic – The writing assignments will appeal to these students.
Interpersonal – The group orientation of this assignment will appeal to these students
Intrapersonal – The reflection portions of this assignment will appeal to these students.
Logical – Using induction to solve a problem will appeal to these students.
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Student Instruction Sheet
Inquiry Question
Individually, take a few minutes to study the two scatter plots below and write a short
paragraph describing how would you would best fit a line to the plotted data?
Graph #1 Graph #2
Divide into groups of 4 or 5 and complete the following questions.
Correlation
1. What is the correlation between hours of studying and test scores?
2. What is the correlation between hours of television and test scores?
Make a Prediction
1. Predict a reasonable test score for 4.5 hours of studying.
2. Predict a reasonable test score for 4.5 hours of television watched.
3. What method was used to make the predictions?
Working in your groups, use the 1st clue set to complete the following questions.
1. Can you think of a method that would provide a more accurate prediction of the
test score for 4.5 hours of studying?
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2. How can the your method of fitting a line to the data result in a better fit?
Using clue set #2, each group will complete the following questions.
Develop an Algorithm
1. On a sheet of graph paper, construct an x, y-axis and draw a line representing the
equation, y = x. Plot the following six points on the graph:
(1, 1), (2, 4), (3, 1), (4, 3), (4, 5), (6, 6)
2. How could you find the sum of the distances between each point on the scatter plot
and the line?
3. How would we mathematically determine the line that best fits the data?
4. Define residue.
5. Define the sum of squared residues.
Test Your Prediction
1. Compare your prediction of a reasonable test score for 4.5 hours of studying
against a prediction using linear regression as follows:
Using your graphing calculator, create a scatter plot of the following data:
Hrs of
studying
1 2 3 3 3 4 5 5 6 7 7 8 8
Test
Score
54 59 58 65 67 69 70 77 80 82 88 85 82
a. Press stat and select Edit. Enter hrs studying (1, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8)
into list 1(L1). These will be the x-values. Enter score into list 2 (L2). These
will be the y values.
b. Press 2nd
and Y= and select Plot1. Turn Plot1 on. Select scatter plot as the
type of display. Enter L1 for the Xlist and L2 for the Ylist.
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c. Press zoom 9 to display to scatter plot so the points are visible.
d. Perform regression by pressing stat. From the CALC menu, choose
LinReg(ax+b). Write down the equation given. This is the equation that
models the line using linear regression.
e. Draw the best fitting line by pressing Y= and enter the equation given from
previous step for y1. Press graph.
f. Using the equation given from linear regression, find y when x = 4.5
g. What is the slope of the line given by the equation given by linear regression?
h. How does the answer compare to your prediction?
As a group answer the following questions:
1. Using your prediction comparisons before and after you made the revisions to your
answers of how to best fit a line to the data, what is your conclusion for a best fit a line to
the data in a scatter plot?
2. How close was your original prediction to the value derived from linear
regression?
Individual Reflection:
Explain how you came to your final conclusion and how the data supports your
conclusion about lines that best fit the data.
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Clue set – Part 1
• How would you mathematically measure the accuracy of the line fit?
• How would you determine if there is a line that would be a better fit?
• Could you develop an algorithm to measure the line’s fit to each of the plotted
points? An algorithm is a sequence of instructions used to solve a problem.
• From what you have learned, how does your original answer to the question of
finding a line that best fits the data compare to your new ideas of finding the
line that best fits the data?
• Revise your answer to the question, how would you best fit a line to data?
Clue set – Part 2
• Hint: draw a vertical line from the point to the line, and calculate the observed
value of y minus the predicted value of y. The observed minus the predicted is
called a residue. We can calculate the sum of the residues for each point.
• How could you change the algorithm to account for the distorting affects of
summing negative residues?
• How would squared residues for each point in the scatter plot and the line
eliminate the effect of negative values?
• How would the squared distance between each point and the line affect the
accuracy of the line fit?
• Compare your original answer of how to find the line that best fits the data to
the method of linear regression or sum of squared residuals.
• Use the comparison to revise your answer to the question of how you would
best fit a line to plotted data.
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Applying Math to Real World Problems
The authentic assessment, "Applying Math to Real World Problems," is designed to act
as the culminating experience for this unit titled, Applied Linear Equations. This
assessment is designed to enable students to apply math to real world problems. For this
assessment, students will have the option of working in groups or individually to design a
mathematical model using data associated with the ecological and cultural crisis.
Students will present their models to their parents and others at a community-sponsored
event scheduled for the end of the semester. The students will be given approximately
one week of in-class time to complete their projects.
Before students begin this project, they will view a short video related to the ecological
and cultural crisis. Individually each student will write a journal entry reflecting on the
video. The students will then be presented with recent ecological data. The data will be
supplied in the form of tables. One class period will be spent demonstrating how to use
data to develop a mathematical model. As students become familiar with the data
available, they will have an opportunity to form groups and decide on which data they
will use to develop their model.
As students develop their models, they will apply lessons from the unit that explores
linear functions in the real world. To prepare for this project, the students learned to
interpret, graph, and analyze linear equations and functions, apply linear regression, and
make predictions. They will use these lessons to create mathematical models, by
researching at least three approved websites, and represent the data in the form of linear
equations. As part of their model, they will make predictions based on historical data.
The students will be asked to explain how the data supports their prediction. Students will
then create a power point presentation of their math model, which they will present to the
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community at the community event. Those who are unable to attend the event will
present their math models during class time the day prior to the event.
After the presentations are complete, the guests in attendance will be asked to complete a
short exit survey for the purpose of measuring changes in their beliefs regarding the
ecological and cultural crisis, as a result of the presentations. During the next class
period, the students will compile the data and analyze the results of the surveys.
The essential question or focus of this unit asks students to consider, "How can we use
the data associated with ecological and cultural crisis?" This authentic assessment
challenges students to apply math to ecological and cultural crisis by using the ecological
data to model linear equations. This assessment promotes critical thinking skills because
it asks students to make predictions and to use math to analyze their predictions. The
students must explain the rationale behind their predictions, developing their ability to
make a logical coherent argument.
This project promotes social and civic skills by having students work together in groups
and an issue of importance to themselves and their community. The project is meaningful
and relevant to their everyday lives because it teaches students how to apply skills they
have learned to solve everyday problems. It is important that students are encouraged to
use their creativity and skills acquired in the classroom to engage in critical analysis and
problem solving outside of the classroom.
This culminating experience is differentiated to accommodate the needs of diverse
students. Differentiation is built into this assessment by giving students options on what
they model to the community. Linguistic learners will benefit with the written aspects of
this project, and logical learners will benefit by the analysis required for this project.
Spatial learners benefit when constructing the visual representation of their models and
kinesthetic learners will benefit by presenting during the civic event. Interpersonal
learners will benefit by working on this project in a group, and intrapersonal learners will
benefit by completing this project individually. Naturalistic learners might benefit by
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focusing on an environmental element of this project. Moreover, The range of complexity
provided by this assessment accommodates students with differing learning abilities.
Student Instruction Sheet/Checklist
Assignment: Create a mathematical model that expresses an ecological
problem in mathematical terms. Prepare a power point presentation to
showcase your mathematical model to the community.
Self-Evaluation Criteria Checklist:
Individual Assignment
Complete
1
Incomplete
0
Research/Preparation • Select an ecological topic for your model from
the list provided.
• Research at least three different websites related to your chosen topic.
• Gather the data to be used for your model.
• Graph the data using Microsoft Excel, including at least one scatter plot.
• Using linear regression, write an equation that
best fits the data.
Power Point Slides
• Title of your math model and names. • A description of the ecological topic chosen. • Graphs that clearly depict the data with
appropriate labels.
• Your prediction or hypothesis, based on a visual
estimate of a best fitting line to the data.
• A graph showing the scatter plot with a best-fit line derived from linear regression.
• A slide displaying the equation of the line that
best fits the data.
• A second prediction using the equation derived
from linear regression.
• A slide comparing your original prediction to the prediction made from the equation.
• A summary of your findings
• Sources sited
Organization: • Power Point slides are readable from the back of
the classroom.
• Power Point is visually interesting. • Power Point flows logically and makes sense.
Mechanics: • Correct spelling errors. • Correct punctuation errors.
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What did you learn from this project? _________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
I have assessed my project and it meets all described criteria.
(Student’s Signature)
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Letter to Parents October 30, 2009
Dear Parents/Guardians, Your students in my Algebra class have been invited to participate in a community- sponsored event scheduled for Friday, January 16, at 6:30 p.m., at the Civic Center.
At this event, the students will demonstrate their application of math to real world problems during group presentations. For this event, the students will apply lessons from a unit that explores linear functions in the real world. In this unit the students will learn to interpret, graph, and analyze real life data. They will learn to create mathematical models, using the data they collect, and represent the data in the form of linear equations. They will also learn to calculate predictions based on historical data. To prepare for this event, your students will be working with a variety of ecological
data that will be collected from various sources. The students will work in groups and each group will have the opportunity to choose from several ecological topics for their project. Each group of students will represent their analysis in the form of a mathematical model and each group will create a presentation that explains and supports their findings. The students will be provided everything they need to do their projects and will be using class time to create their mathematical models. You are encouraged to attend the December 16th event, where each group of
students will present their model to the community. Students unable to attend this event will be given the opportunity to present their project on Thursday, December 15, during class. You are welcome to attend. If you have any questions, please feel free to contact me at 734-555-1234. Sincerely,
Christine Bolen Algebra I Teacher
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Rubric for Applying Math to Real World Problems
1
Below the Standard
2
Approaching the Standard
3
Meets the Standard
4
Exceeds the Standard
Preparation Topic selected for the
model was both
inappropriate and not
approved.
Topic selected for the model was
appropriate, not on the list, and
not approved.
Topic selected for the
model from the list
provided.
Discovered an appropriate
topic that was not on the
list and received approval
to use for your model.
Research
The websites used were
not relevant to the topic
selected.
Used less than three relevant
websites to gather data.
Used three different
relevant websites to
gather data.
Used more than three
relevant websites to gather
data.
Graph Data Did not graph the data
in the form of a scatter
plot.
Did not successfully use
Microsoft Excel to plot the
collected data in the form of a
scatter plot. Data not represented
appropriately.
Successfully used
Microsoft Excel to plot
the collected data in
the form of a scatter
plot. Graph
appropriately
represented the data.
Successfully used
Microsoft Excel to plot the
collected data into more
than one scatter plot to
represent alternative
interpretations of the data.
Power Point
Slides
Presentation includes
less than 5 of the slides
that meet the specified
requirements.
Presentation includes at least 5
of the slides that meet the
specified requirements.
Presentation includes
10 slides that meet the
specified requirements.
Presentation includes more
than 10 slides that meet the
specified requirements.
Organization Power point slides
satisfy one of the
following three
standards:
1. Slides flow
logically.
2. Slides are readable
from the back of
the classroom
3. At least one power
point
enhancement was
used to add visual
interest.
Power point slides satisfy at
least two of the following three
standards:
1. Slides flow logically.
2. Slides are readable from
the back of the classroom
3. At least one power point
enhancement was used to
add visual interest.
Power point slides
satisfy each of the
following three
standards:
1. Slides flow
logically.
2. Slides are
readable from the
back of the
classroom
3. At least one
power point
enhancement was
used to add
visual interest.
Slides satisfy all of the
required standards and
provide an in depth
analysis of the results.
Mechanics Several spelling and
punctuation errors.
Few spelling or punctuation
errors.
No spelling or
punctuation errors.
Use of proper
mathematical notation
included in slides.
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Guest Exit Survey
Circle the selection which best describes the impact of the presentations on your beliefs:
1. The presentations convinced me that there exists an ecological and cultural crisis.
Not convinced somewhat convinced convinced
3. The presentations convinced me that global warming is real.
Not convinced somewhat convinced convinced
3. The presentations convinced me that a significant amount of climate change is man
made.
Not convinced somewhat convinced convinced
4. The presentations convinced me that oil usage at the current rate is unsustainable.
Not convinced somewhat convinced convinced
5. The presentations convinced me to be more conscious of the environment.
Not convinced somewhat convinced convinced
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Bibliography and Resources
Larson, Boswell, Kanold, and Stiff (2008). Teacher’s Edition McDougal Lettell,
Algebra I
Television and Test Grades, Mathematics Benchmarks, Grades K-12
http://www.utdanacenter.org/k12mathbenchmarks/tasks/29_tvtestgrades.php.
http://mi.gov/documents/mde/AlgebraI_216634_7.pdf
Hopkins, Rob (2008). The transition handbook: From oil dependency to
local resilience. White River Junction, VT: Chelsea Green Publishing.
Weir, Giordano, and Fox (2008). A First Course in Mathematical Modeling,
Brooks Cole Publishing