CURR 305 UNIT: MATH€¦ · Day 1 Day 2 Day 3 Day 4 Day 5 Direct Lesson: “Graphing Linear...

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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”


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


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


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.





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.


ramps and communicate the values to the recorder.

c. The recorder will make record the measurements of each of three ramps in the

table.





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.





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


Presentation includes more

than 10 slides that meet the


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.


3. The presentations convinced me that a significant amount of climate change is man

made.


4. The presentations convinced me that oil usage at the current rate is unsustainable.


5. The presentations convinced me to be more conscious of the environment.


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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

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