Unit 6: Systems of Linear Equations
3 weeks
Unit Overview
Essential Questions:
What does the number of solutions (none, one or infinite) of a system of linear equations represent?
What are the advantages and disadvantages of solving a system of linear equations graphically versus
algebraically?
Enduring Understandings:
A system of linear equations is an algebraic way to compare two equations that model a situation and find the
breakeven point or choose the most efficient or economical plan.
UNIT CONTENTS
Note:
The bolded investigations are model investigations for this unit.
Investigation 1:
Solving Systems of Linear Equations (3 days)
Investigation 2:
Solving Systems of Linear Equations Using Substitution (2 days)
Investigation 3:
Solving Systems of Linear Equations Using Elimination (3 days)
Performance Task: Community Park (4 days)
End-of-Unit Test (1 day)
Appendices:
Unit 6 Performance Task Sample Student Handout, Sample Checklist, and Sample
Evaluation Rubric, Unit 6 Calculator Directions, and End-of-Unit Test
Course Level Expectations
What students are expected to know and be able to do as a result of the unit
1.1.9 Develop, compare and apply functions using a variety of technologies (i.e. graphing calculators,
spreadsheets, and on-line resources).
1.2.1 Develop and apply linear equations and inequalities that model real-world situations.
1.2.2 Represent functions (including linear and nonlinear functions such as square, square root, and
piecewise defined) with tables, graphs, words and symbolic rules; translate one representation of a
function into another representation.
1.2.3 Create graphs of functions representing real-world situations and label with appropriate axes and
scales.
1.2.5 Recognize and explain the meaning and practical significance of the slope and the x- and y-intercepts
as they relate to a context, graph, table or equation.
1.2.7 Explain how changes in the parameters a and b affect the graph of an exponential function and validate
the practical significance of the parameters in a real-world problem.
1.3.1 Simplify and solve equations and inequalities.
1.3.2 Use functional notation to evaluate a function for a specified value of its domain.
1.3.5 Solve systems of linear equations that model real world situations using both graphical and algebraic
methods.
2.2.1 Use algebraic properties, including associative, commutative and distributive, inverse and order of
operations to simplify computations with real numbers and simplify expressions.
2.2.3 Choose from among a variety of strategies to estimate and find values of formulas, functions and roots.
2.2.4 Judge the reasonableness of estimations and computations.
4.1.1 Collect real data and create meaningful graphical representations of the data with and without
technology.
4.1.2 Estimate strong and weak and positive and negative correlations from tables and scatter plots.
4.2.1 Analyze models of functions using trend lines and the graphing.
Vocabulary
Addition Property of Equality
Profit
Breakeven Point
Revenue
Cost
Solution to a System of linear Equations
Elimination
Substitution
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Unit Plan 6, 8 13 09
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Fixed Cost
Multiplication Property of Equality
System of Linear Equations
Transitive Property
Variable Cost
Assessment Strategies
Performance Task(s)
Other Evidence
Authentic application in new context
Formative and Summative
assessments
Community Park
 Warm-ups, class activities, exit
NOTE: Prior to the end of the unit, you may wish to describe the
slips, and homework have been
Performance Task and have students investigate the historical and cultural
incorporated throughout the
aspects of the design and use of public parks and gardens in various urban,
investigations.
suburban and rural settings. Students might share their own experiences at
 End-of-Unit Test
various parks and gardens.. After the three investigations are complete,
students will use systems of equations to design a community park.
INVESTIGATION 1 – Solving Systems of Linear Equations (3 days)
Students create the appropriate visual or graphical representation of data from the U.S. Census Bureau about
annual wage gaps. They analyze real world problems with statistical techniques and model real data
numerically, algebraically, and graphically using appropriate tools, technology and strategies.
See Model Investigation 1.
INVESTIGATION 2 – Solving Systems of Linear Equations Using Substitution (4 days)
Data from the Heifer International, a non-profit organization that provides seeds and breeding animals to
impoverished peoples is used to begin this investigation. Students will model the real world problem as a
system of equations and solve the system of equations using the substitution method. Students then solve
problems in two other contexts – car racing and a popcorn fundraiser.
See Model Investigation 2.
INVESTIGATION 3 – Solving Systems of Linear Equations Using Elimination (3 days)
Here students solve systems of linear equations using the elimination method, identify the characteristics of a
system of linear equations that lend themselves to the elimination method, and interpret the solution to a system
of equations within the context of the problem.
Suggested Activities
3.1 As a launch to this investigation you might show the 5 minute Computer Problems vignette. Go to
http://www.thefutureschannel.com/hands-on_math/computer_problems.php from The Futures Channel
which introduces students to the Dell Computers call center in Round Rock, Texas. The video shows the
different facets involved in ordering and assembling a computer from component picking to shipping. Based
on information in the video, you and the class may develop a number of simultaneous equation scenarios
based around cost and time constraints to use as part of this investigation. The mathematical focus of this
investigation is to introduce students to how the elimination method utilizes the addition and multiplication
properties of equality to solve a system of equations. To help students develop an understanding of the
sequence of steps involved in the elimination method, you may elicit from them what they already know
about how to solve equations that contain one or two variables. Emphasize during the discussion the idea
that when solving an equation what is done to one side of an equation must be done to the other side for the
equation to remain in balance and the solution to the equation to remain unchanged. Use probing questions
to guide students in the development of the elimination method. Students should recognize that the
elimination method can be justified in terms of the substitution property of equality in conjunction with the
addition property of equality. Emphasize, as you did with the substitution method, that the values of both
variables satisfy both equations, and thus provide the solution to the system. You may use an exit ticket
which asks students to solve a system of equations using the elimination method and explain their reasoning
or algebraic properties that support the various steps.
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Unit Plan 6, 8 13 09
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Differentiation: Exploration and discussion of the elimination method should focus on both the process used
as well as the algebraic principles that support the process. If students understand the algebra that supports
the elimination method but have difficulty remembering and/or following the sequence of steps involved,
you might have them work in pairs or small groups to develop a notes card that describes the sequence of
steps in their own language and includes one or more examples of how to implement the elimination
method. Use simpler equation pairs, such as x + y = 12 and x – y = 16, as examples.
3.2 You may continue with more explorations that use the elimination method to solve systems of equations
that have none, one, or many solutions (for example: 2x + 8y = 6 and x + 4y = 3). To emphasize the
connection between the number of solutions to a system of equations and the number of intersections
between the lines, write the two equations in the slope-intercept form and then graph the equations. By
putting both equations in the slope-intercept form, the students can readily identify the slope and y-intercept
of the lines, graph the lines as needed and make connections between the graphical and algebraic nature of
the lines. As part of this activity you may wish to incorporate a collection of problems that involve fractions
and decimals to reinforce and practice computation skills as needed. As an exit slip you might ask students
to explain graphical characteristics of systems of equations that have none, one, or an infinite number of
solutions.
In the reference section of their journal, in their own language, students might add information about how
graphical features correspond to algebraic solutions of simultaneous equations. Students may also include
one or more examples of how to implement the elimination method when equations contain fractions or
decimals.
3.3 Once students understand the algebraic principles that support the elimination method and can clearly
describe the relationship between the number of solutions and the number of intersections of the graphs, you
may lead students in an exploration of the characteristics of systems of equations that lend themselves to the
substitution versus the elimination method. You may use systems of equations students have already solved
in Investigations 2 and 3. Ask students to work independently, in pairs, or small groups to identify
characteristics of systems of equations that lend themselves to each method. When identifying common
characteristics it is important to remind students that they may use either the elimination or substitution
method. However, by looking at the characteristics of the equations within a system one method might be
easier to work with then the other. Students may design a graphic organizer to list the characteristics.
Students can then apply the criteria they develop to solve a collection of word problems similar to the
following which utilize Connecticut as a backdrop.
a. At the upcoming school fair, your class is planning to raise money for a class trip to Washington, DC.
You plan to sell your own version of Connecticut Trail Mix. After doing research on the cost of various
ingredients, you find you can purchase a mixture of dried fruit for $3.25 per pound and a nut mixture for
$5.50 per pound. The class plans to combine the dried fruit and nuts to make their unique Connecticut
Trail Mix that sells for $4.00 per pound. After researching the number of people who attended last
year’s fair, you anticipate you will need 110 pounds of trail mix. How many pounds of dried fruit and
how many pounds of mixed nuts do you need to make the trail mix and earn a reasonable profit?
b. Your family is planning to take the Amtrak train from Hartford to New York City for a day trip. As a
result of some research you learn that your friend Jackie took the train with a group of 3 adults and 5
children and it cost them $269.50. A cousin also took the train to the city with a group of 2 adults and 3
children and it cost them $171.50. Find the price of an adult’s ticket and the price of a child’s ticket.
c. During the 2008-2009 basketball season the UConn women’s team had an incredible undefeated season
(39-0) and won the NCAA championship. Maya Moore and Renee Montgomery were the top scorers
during the year and together they scored 1,398 points. If Maya scored 110 more points than Renee, how
many points did each player score during the season? (Retrieved May 21, 2009 from
http://www.uconnhuskies.com/sports/w-baskbl/stats/2008-2009/teamcume.html)
d. During the 2008-2009 men’s basketball season, UConn’s Hasheem Thabeet and Jeff Adrien had a total
of 746 rebounds and Jeff had 30 fewer rebounds than Hasheem. How many rebounds did Hasheem and
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Unit Plan 6, 8 13 09
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Jeff each have during the season? (Retrieved May 21, 2009 from
http://www.uconnhuskies.com/sports/m-baskbl/stats/2008-2009/teamcume.html)
As an exit ticket you might ask students to identify the method they used to solve a specific system of
equations and explain why they chose that method. Or have students identify the main characteristics of
systems of equations that lend themselves to the substitution and the elimination methods respectively. For
homework students may solve a series of word and non-word problems that involve systems of equations
and explain why they chose either the elimination or substitution method.
Differentiation: Students may write or select more problems that draw on a theme or area of interest.
Students may present them to the class to solve as a review of the entire unit.
Assessment
By the end of this investigation students should be able to:
 Use the elimination method to solve a system of equations;
 Explain the algebraic properties upon which the elimination method is based;
 Explain the relationship between the number of solutions to a system of equations and the relationship
between the slopes and y-intercepts of the equations within a system; and
 Identify the characteristics of systems of equations that lend themselves to the substitution and
elimination methods.
Unit 6 Performance Task - Community Park (4 days)
This Performance Task should be used after the three investigations are complete. Community Park provides an
opportunity for students to apply what they learned in Unit 6.
Suggested Activities
You may choose to use one of the following vignettes on The Futures Channel website that deal with landscape
architecture to raise student interest and launch the task:
1. http://www.thefutureschannel.com/dockets/hands-on_math/landscape_architecture/ focuses on designing a
picnic area in the Cibola National Forest in New Mexico and is about 4 minutes in length; and
2. http://www.thefutureschannel.com/dockets/hands-on_math/landscape_architects/ focuses on geometrical
features architects incorporated into their design of Millennium Park, a botanical garden in Chicago, IL, and
is about 6 minutes in length.
Another way to begin the project would be to have students do an Internet search about Frederick Law
Olmstead. Then, the class could begin with a discussion of park design and the works of Olmstead, including a
notable urban park and a cemetery park, both in Connecticut. Alternatively, this could be used as an extension
to the project.
In whole class discussion, let students know that they will work in small groups to design a community park
(see the Unit 6 Performance Task - Sample Student Handout, Sample Checklist, and Sample Evaluation
Rubric). Tell them that the dimensions of the park are outlined by four boundary lines. They will need to find
the four corners of the park algebraically and sketch the outline of their park on a large piece of graph paper.
Then, they will need to follow the provided instructions to complete the map of the park as the developers have
requested.
Depending on time and student interest, this project may be opened up and extended. You might facilitate a
whole class discussion on what features the students would like to have in their community park. Once students
have brainstormed a list of ideas, they may research one of the features. The students could read local
community rules regarding zoning, building, handicap accessibility, and other park regulations. In addition,
they could find the cost of each feature, such as the cost of the gravel pathways after determining the
appropriate depth of gravel needed and the volume of gravel required for the project; or cost of the asphalt
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Unit Plan 6, 8 13 09
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needed for the basketball courts; or choose the kind of trees to plant, how many, and the cost per tree. Each
group could use the scoring rubric to assess their work. As a way to share and compare results, the class might
create one large park on graph paper. Each group could use the scoring rubric to assess their work.
End-of-Unit Test (1 day)
Technology/Materials/Resources/Bibliography
Technology:
 Classroom set of graphing calculators and whole-class display for the graphing calculator
 Graphing software
 Computer
 Overhead projector with view screen or computer emulator software that can be projected to whole class, and
interactive whiteboard
On-line Resources:
 http://www.youtube.com/watch?v=l5Hxw_Jf2B4
 http://www.thefutureschannel.com/dockets/hands-on_math/landscape_architecture/
 http://www.thefutureschannel.com/dockets/hands-on_math/landscape_architects/
Materials:
 Chalk, colored pencils, white board markers
 Index Cards
 Vocabulary Cards
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Unit Plan 6, 8 13 09
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Unit 6 Performance Task
Sample Student Handout, p. 1 of 1
Community Park
Name:
Date:
Your town has decided to build a new community park. Your task is to create a graph that the construction
workers and landscapers will use as a guide. Your graph should be neat and detailed. In addition, you must
show how to solve the systems of linear equations algebraically, explain why you chose the method you did to
solve each system, and include any additional work to complete the requirements listed below.
The community park will sit on a piece of land which measures 150 meters by 180 meters. The x-axis is the
southern border of the property (180 meters long) and the y-axis is the western border of the property (150
meters long). Due to parking and other restrictions, the community park only takes up a portion of the plot of
land. The boundaries for the park are described by the four equations below.
5 x  12 y  300
12 x  5 y  125
5 x  12 y  1990
19 x  22 y  1140
The developers are not too far along in their planning, but they have decided the placement of the fenced in
basketball court. The northeast corner of the rectangular basketball court is at the intersection of 2 x  5 y  255
and 2 x  3 y  585 . The court extends 20 meters directly south from the given corner (this is the shorter side of
the basketball court). The sides of the rectangular basketball court are in a 3:2 ratio. Sketch the perimeter of the
basketball court on your park grid. Then, find the percentage of the total area of the park the basketball court
covers. Be sure to include all of the computations required to find the percentage.
As another requirement of the park design, you must include two pathways that go along the diagonals of the
park (connecting the vertices of the quadrilateral). Include the equations of the diagonal lines that run down the
center of the two pathways and the point of intersection of the center of the two pathways. Choose a feature (i.e.
fountain, rose garden, bench, tree, fish pond, gazebo, etc.) to place at the intersection of the two pathways.
The developers will get back to you with more details for the community park after the next meeting. Your task
at this point in the planning stage is to create the graph containing the four sides that outline the community
park and include a sketch of the location of the basketball court and the two pathways. They will use this outline
to continue planning!
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Unit Plan 6, 8 13 09
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Unit 6 Performance Task
Sample Checklist, p. 1 of 1
_____ 1) The x- and y-axes of the coordinate plane are labeled and appropriate scales are indicated.
_____ 2) The four lines that create the boundaries of the park are correctly graphed and labeled.
_____ 3) The basketball court is correctly placed, labeled, and is in proportion to the given scale of the park
design.
_____ 4) You have found the percentage of the area of the basketball court compared to the total area of the
park. You have included all the work needed to calculate the percentage.
_____ 5) The pathways are correctly placed and labeled and are the correct length given the scale of the
graph. The equations of the two pathways are written and the intersection is labeled with the
coordinate pair.
_____ 6) You have attached work that show how you solved each system of equations.
_____ 7) You have explained why you chose the method you used to solve each system of equations.
_____ 8) You have used correct spelling, grammar and sentence structure in the written explanations of
your work.
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Unit Plan 6, 8 13 09
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Unit 6: Project – Community Park
Sample Evaluation Rubric
The Unit Performance Task is scored based on the following rubric on a scale of 0 to 3 for a total of 24 possible points.
Component
0 = Missing
1. The x- and yaxes of the
coordinate plane
are labeled and
an appropriate
scale is
indicated.
2. The 4 lines that
create the
boundaries of
the park are
correctly
graphed and
labeled.
3. The basketball
court is
correctly placed,
labeled, and
sized given the
scale of the
graph.
4. You have
calculated the
percentage of
the total area of
the basketball
CT Algebra One for All
Unit Plan 6, 8 4 09
2 = Proficient
3 =Advanced
The x- and y-axes
are not labeled, and
an appropriate scale
is not indicated.
1 = Needs
Improvement
Only one of the
axes is labeled,
and/or the scale is
not indicated.
Both the x- and yaxes are labeled but
a scale is not
indicated.
Both axes are
labeled and an
appropriate scale is
indicated.
None of the 4 lines
that create the
boundaries of the
park are correctly
graphed and
labeled.
One or 2 of the 4
lines that create the
boundaries of the
park are correctly
graphed and
labeled.
Three of the 4 lines
that create the
boundaries of the
park are correctly
graphed and
labeled.
All 4 lines that
create the
boundaries of the
park are correctly
graphed and
labeled.
The basketball court
is not included in
the graph.
The basketball court
is included in the
graph but it is not in
the correct place.
The basketball court
is correctly placed
but is not correctly
sized given the scale
of the graph.
The basketball court
is correctly placed,
labeled, and sized
given the scale of
the graph.
The area and the
percentage of the
area of the
basketball court
compared to the
The area of the
basketball court is
included, but the
percentage of the
area of the
The area of the
basketball court and
the percentage of
the area of the
basketball court to
The area of the
basketball court and
the percentage of
the area of the
basketball court to
Page 8 of 22
Student SelfReflection
Points
Earned
Component
court compared
to the total area
of the park.
0 = Missing
total area of the
park is not included.
1 = Needs
2 = Proficient
Improvement
basketball court to
the total area of the
the total area of the park are both
park is not included. included, but work
to support the
percentage is not
included.
The pathways are
The pathways are
included in the
correctly placed in
graph, but they are
the graph but are
not in the correct
not labeled.
place.
3 =Advanced
the total area of the
park are both
included, and work
supporting the
answer is included.
5. The pathways
are correctly
placed and
labeled and are
the correct size
given the scale
of the graph.
The pathways are
not included in the
graph.
6. You have
attached your
work to show
how you solved
each system of
linear equations.
7. You have
explained why
you chose the
method you
used to solve
each system of
linear equations.
8. You have
correct spelling,
grammar and
No work is attached
to support how the
systems of
equations were
solved.
Work is attached to
support 1 or 2 of the
4 systems of linear
equations.
Work is attached to
support 3 of the 4
systems of linear
equations.
The pathways are
correctly placed,
labeled in the graph,
the equations of the
pathways are
correctly found, and
the point of
intersection is
correct.
Work is attached to
support all 4
systems of linear
equations.
No explanation of
the method used to
solve any of the
systems of
equations or why
each method was
used is provided.
The written
explanation is
difficult to follow
An explanation to
support the method
used to solve 1 or 2
of the 4 systems of
equations and why
each method was
used is provided.
The written
explanation is
difficult to follow
An explanation to
support the method
used to solve 3 of
the 4 systems of
equations and why
each method was
used is provided.
The written
explanations have a
logical sequence a
An explanation to
support the method
used to solve all 4
systems of
equations and why
each method was
used is provided.
The written
explanations are
logical and have an
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Unit Plan 6, 8 4 09
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Student SelfReflection
Points
Earned
Component
sentence
structure in the
written
explanations of
your work. You
have followed
the rules of
Standard
English.
0 = Missing
due to unacceptable
use of segues and
transitions.
The written
explanation contains
awkward and
confusing sentence
structure throughout
and is devoid of
transitions.
There are 6 or more
spelling and/or
grammatical errors,
and the written
explanations do not
follow rules of
Standard English.
1 = Needs
Improvement
due to sentence or
paragraph structure
issues.
The written
explanation
demonstrates
acceptable use of
segues and
transitions.
The written
explanation has 3 to
5 grammatical and
/or spelling errors
and follows rules of
Standard English.
2 = Proficient
reader can follow.
The written
explanations
demonstrate
acceptable use of
segues and
transitions.
The written
explanations use
appropriate word
choices, but is
occasionally
awkward in its
phrasing.
Student SelfReflection
interesting sequence
which the reader
can easily follow.
The written
explanations
demonstrate
excellent use of
segues and
transitions.
The written
explanations use
sophisticated and
appropriate word
choices.
The written
The written
explanations have 1 explanations have
to 2 grammatical
no grammatical or
and/or spelling
spelling errors and
errors and follows
follows rules of
rules of Standard
Standard English.
English.
Total Points Earned = _____ out of 24 possible points = _______%
Comments:
CT Algebra One for All
Unit Plan 6, 8 4 09
3 =Advanced
Page 10 of 22
Points
Earned
Unit 6 Calculator Directions
This document contains calculator directions for:
 Turn on the Diagnostics
 Set up List Editor and Clear Lists
 Enter Data in the Lists
 Graph a Scatter Plot
 Calculate and Graph the Linear Regression Equation
 Using Multiple Lists
 Finding the Intersection of Two Lines
PREPARE THE CALCULATOR
1. Turn on the Diagnostics so that the correlation coefficient r
will appear.
a. Clear the homescreen
b. Press “2nd 0 CATALOG”, scroll down to “DIAGNOSTICS
ON”. “ENTER”
c. Homescreen now shows “Diagnostics on.” “ENTER”
d. Homescreen shows “Done.”
2. Set up List Editor and Clear Lists.
a. Press “STAT”
b. Press “5 SET UP EDITOR”
(You will only need to set up editor if you are missing a list or if your
lists are out of order – you needn’t set up the editor every time.)
c. Press “STAT” again and press “1.EDIT”
If there is any data in the lists, you can enter new data by typing over
the existing data OR you can highlight” L1” at the VERY TOP of the
list to clear the entire list.
d. When you place your cursor over the list name at the top so
the list name L1 is highlighted, you can operate on the
entire list. Press “CLEAR – ENTER” to clear an entire list.
Note: you must highlight the list name at the top, not the
first entry.
PLOT THE DATA
3. Enter Data in the Lists.
a. Press “STAT –EDIT” to get to the list editor. Move the
cursor to the place you wish to enter data. Type in a value.
Press “ENTER”.
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Unit Plan 6, 8 4 09
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b. Repeat entering data and pressing enter until all data in List
1 is entered. Then enter all data in List 2.
c. You can amend each single entry with the “delete” key.
d. Be sure the length (also called the “Dimension”) of one list
is the same as the dimension of the other list. Otherwise,
when you try to plot the data you will see the error message
“Dimension Mismatch”
4. Graph a scatter plot of the data.
a. Press “2nd Y= STAT PLOT”
b. Highlight Plot 1 by moving the cursor and press
ENTER. You can also just press “1”.
c. Set up the STAT PLOTS by using your cursor to
highlight ON, the first TYPE which is a scatter plot, the
XList which is L1, the YLIST which is L2, and the
MARK you wish to use.
d. Choose a scale or “window” by observing the
minimum and maximum values of the data in the
XList. Similarly with the data in the YList. Press
“WINDOW” and enter the maximum and minimum
values for x, and y.
e. Then choose the scale for each axis. (Do you wish to
put tick marks on the axes by 1’s, 5’s or ?)
The Xres tells the calculator to use every pixel on the screen if
Xres=1. The calculator will use 1/5 of the pixels if Xres=5,
and so on. Leave the Xres at 1 unless you are graphing a very
complicated, time-consuming function.
f. Press GRAPH.
g. An automatic window setting can be used instead of
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Unit Plan 6, 8 4 09
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setting the window manually. Press “ZOOM” then
press “9 ZOOM STAT”.
FIT A MODEL TO THE DATA
5. Calculate and Graph the Linear Regression Equation
a. Start with a clear homescreen.
b. Press “STAT”, cursor to the right one click to highlight
“CALC”
c. Scroll down to 4:LinReg(ax+b), press ENTER or press key
“4”
d. On the homescreen will be the command “LinReg(ax+b)”.
e. Press “2nd 1 L1” the name of the list that contains the
independent variable ( i.e., the X list which is L1)
f. Press the Comma key “,” which is to the right of the x2 key.
g. Enter the list that contains the dependent variable, by
pressing “2nd 2 L2 which is the Y list.
h. Comma “,”
i. The calculator will paste the regression equation into the
Y1= screen so you can graph it easily. To call up Y1, press
“VARS”, move the cursor to the right to “Y-VARS”.
Press “1. FUNCTION”. Then press “1. Y1”.
j. The homescreen should now show the command to
calculate the regression using the values of the independent
variable from list 1, the values of the dependent variable
from list 2, and the Y1 where the equation will be pasted.
k. Press “ENTER” to execute the command. You can see the
linear regression, the correlation coefficient r, and the
coefficient of determination r2. You can verify that 0.9769
is approximately 0.98842. With Linear Regressions, we are
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Unit Plan 6, 8 4 09
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concerned with r, the correlation coefficient.
l. Press the “Y=” key to verify that the regression equation is
indeed pasted into Y1. Note that “Plot 1” at the top is
highlighted. This gives you an easy way to turn on or off
the plot. Just move the cursor to the plot number, highlight
it, click enter to toggle between choosing or not choosing
to graph the scatterplot. There is no need to go back into
the 2nd STAT PLOT menu.
m. Press Graph – BEAUTIFUL!
ADDITIONAL INFORMATION:
1. Any list may be used for the independent variable or the dependent
variable. In fact, you can create and name a list if you wish, just insert
at the top of the lists. Press “2nd INS” and type the name. To indicate
which List contains the independent (XLIST) data, press the keys “2nd
3 L3” to indicate List 3, for example.
2. To save one List into another List, move the cursor to the top of the
new List where you will store the data, and enter the old list name,
then press “ENTER”. For example if the data is in L3 and you wish to
store it in List 6, move the cursor to the very top of List 6, highlighting
the list name, press “2nd 3 L3”, “ENTER”.
3. A common error on the calculator screen is “DIM MISMATCH”.
You can correct this error by turning off any unused STAT PLOT, or
by making sure that the lengths of the two lists being graphed as a
scatter plot are equal in length.
4. Use the link cable that connects two calculators or a calculator to a
computer to send and receive data. This saves the trouble of each
person typing in their own data. With connected calculators, both
calculators press 2nd LINK (the second function of the X variable key).
The receiving calculator presses receive first. The sending calculator
can now choose what to send and then press “SEND”
PLOTTING AND ANALYZING TWO SETS OF DATA
6. Using multiple lists
a. At times, it is desirable to view two scatter plots on the
same axes, or to keep one set of data and work with
another set of data. You may use many lists, and insert
more lists than the 6 lists readily available on the STAT
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 14 of 22
EDIT. The calculator will simultaneously graph up to 3
scatter plots. To reset Lists 1-6 without erasing them,
press STAT 5: SET UP EDITOR ENTER.
b. The following data will be used to illustrate plotting 2
scatter plots. Press “STAT 1: EDIT” to access the lists
and enter the following data in Lists 1 through 4.
(Source:
http://en.wikipedia.org/wiki/World_record_progression
100_metres_freestyle)
List 1
Year
(since
1910)
2
5
10
13
14
16
19
20
21
23
24
26
List 2
Time
(seconds)Women’s
Long Course
Swimming World
Record
78.8
76.2
73.6
72.8
72.2
70
69.4
68
66.6
66
64.8
64.6
List 3
Year
(since
1910)
0
2
2
8
1
10
12
14
24
25
List 4
Time
(seconds)Men’s
Long Course
Swimming
World Record
62.8
62.4
61.6
61.4
60.4
58.6
57.4
56.8
56.6
c. Create a scatter plot and set the window for the two sets
of data by pressing “2nd Y= STAT PLOT” turning on
Plot 1 and then Plot 2. Use one mark for Plot 1 and
different mark for Plot 2.
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 15 of 22
d. Set a window for the Scatter Plot manually by Pressing
“WINDOW” or automatically by pressing
“ZOOM9:STAT”.
e. Press “GRAPH”
f. Calculate the regression for Lists 1 and 2. Press “2nd
QUIT” to go to the homescreen. Clear the homesreen
and press “STAT CALC 4:Lin Reg ENTER 2nd 1 L1,
2nd 2 L2, VARS YVARS 1:FUNCTION 1: Y1
ENTER”
g. Calculate the regression for Lists three and four. Press
2nd QUIT to go to the homescreen. Clear the
homescreen and press “STAT CALC 4:Lin Reg
ENTER 2nd 3 L3, 2nd 4 L4, VARS YVARS
1:FUNCTION 1: Y2 ENTER” This regression equation
will be stored in Y2.
h. Graph the data and the regression equations by pressing
“GRAPH”
7. Finding the Intersection of Two Lines
a. Finding the intersection of two lines is easier if the
STAT PLOTs are OFF. Though you can turn off the
STAT PLOT in the “2nd-STAT PLOT” key and
toggling to OFF, the quickest way to turn on and off the
STAT PLOTS is by pressing the “Y=” key. Arrow up
to the Plot 1 which is highlighted. Press “ENTER” to
toggle it off (un-highlight it). Then arrow over to Plot
2, “ENTER,” and both Plot1 and Plot 2 are not
highlighted.
b. In the “Y=” screen, enter the two equations whose
intersection you will find. The screen shot to the right
has the two linear regressions that represent the 100
meter free style swimming world record in seconds as a
function of year since 1910. Women’s data are in Y1
and men’s are in Y2.
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 16 of 22
c. Find a window so that the point of intersection of the
two lines is near the center of the screen. As you can
see in the graph in 6h above, the lines do not intersect
on the screen. They appear to intersect to the lower
right of the screen, so choose to extend the x maximum
and the y minimum so you can see more to the right
and lower on the graph. A possible window and its
graph are shown to the right.
d. Now that the intersection of the lines is visible on the
graph, press “2nd CALC” on the row of graphing key
just below the calculator screen. Select “5.
INTERSECT”.
e. You will be prompted to select the first curve or
equation you wish to find the intersection of. Arrow up
or down to select the first curve (line), notice that the
equation is at the top of the screen. When the correct
equation is listed, press “ENTER.”
f. You will be prompted to select the second curve or
equation you wish to find the intersection of. Arrow up
or down to select the second curve (line), notice that the
equation is at the top of the screen. When the correct
equation is listed, press “ENTER.”
g. The calculator instructs you to enter a guess, and,
again, the cursor is usually close enough to the point of
intersection that you want, so just press “ENTER” yet a
third time. If not, use the left and right arrow keys to
move the point closer to the point of intersection.
h. Finally, the coordinates of the point of intersection are
displayed at the bottom of the screen. In this case, the x
represents the years since 1910, and the y represents the
seconds it takes to swim 100 meters. According to this
model, halfway through the year 1960, women and men
will both hold the same world record of 49.6 seconds.
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 17 of 22
UNIT 6 TEST - SYSTEMS OF LINEAR EQUATIONS
NAME: ________________________________ DATE: ___________________________
1. Two pizza parlors in Darien are offering special rates to attract birthday party business.
Gino’s Pizza Parlor charges $16.00 to cover both the room rental fee and party favors and an
additional $8.00 for each individual pizza. The Venetian Shores Pizza Parlor charges $24.00
to cover the rental fee and party favors and an additional $6.00 for each individual pizza. The
birthday person gets a free pizza at each restaurant.
a. If you and two friends celebrate your birthday, which pizza parlor would you choose if
you wanted to save money? Justify your answer.
b. Define your variables. Then, write a system of linear equations to model the total cost of
a pizza party at each pizza parlor.
_________________________
c. Solve the system of equations by graphing. Choose the appropriate scale and label both
axes.
_________________________
d. What is the solution to the system of linear equations and what does it mean in the
context of this problem?
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 18 of 22
e. Under what conditions would it be best to have another party at Gino’s Pizza Parlor and
under what conditions would it be best to have your party at The Venetian Shores Pizza
Parlor?
2. The Blast Off Music Company makes CDs and DVDs. The company can make a total of
500 CDs and DVDs each day. It costs $8 to make each CD and $10 to make each DVD. The
company can spend $4800 for production costs each day. How many individual CDs and
DVDs can The Blast Off Music Company make in one day?
a. Define your variables and write a system of linear equations to model the situation.
b. Solve the system of equations and explain your solution in terms of the context of the
problem.
3. Fast Photo and Pretty Pictures both print photos taken with digital cameras. Fast Photo
charges a setup fee of $10.00 and $0.69 per print. Pretty Pictures charges a $6.00 setup fee
and $0.99 per print. Tell whether each statement is true or false and explain how you found
your answer.
a. There is no number of prints for which the cost at both stores is the same.
b. The equation y = 0.99 + 6x can be used to find the cost of having the prints made at
Pretty Pictures.
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 19 of 22
c. It is always more expensive to have your prints made at Fast Photo.
d. If Jorge paid $20.85 to have 15 photos printed, he must have used Pretty Pictures.
4. Choose a method and solve each system of equations. Show your work and check your
solutions.
a. y = -2x + 3
2x + y = 3
c. 4x – 3y = 1
y = 1.5x + 0.5
b. x – 3y = - 18
-x + 2y = 10
d. 2x + 6y = -1
4x – 3y = 3
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 20 of 22
5. Students at Southern Connecticut State University (SCSU) were given two monthly cellular
phone options to choose between (Note: In the rate quotes, m represents minutes).
Campus CT Plan: C = $0.30m
University Phones Plan: C = $0.10m + $25
a. How are the two plans different?
b. Explain the meaning of the slope of the two plans.
c. Explain the meaning of the y-intercept of the two plans.
d. Which plan do you think is a better deal? Explain your thinking.
e. When do the two monthly plans cost the same?
f. The two companies decide to merge and form a third company, the University CT Phone
Plan. The new company will charge $0.35 per minute after the first 40 minutes. Thus, the
first 40 minutes are free. Complete the table below.
University CT Phone Plan
# of Minutes Used
Cost
0
10
20
30
40
50
60
70
80
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 21 of 22
g. If x is the number of minutes used, write an equation that models the cost of the new
phone plan.
h. What is the domain of x for your model in part g?
i. Is this plan better than either of the two original plans? Show your work or explain your
reasoning..
CT Algebra One for All
Unit Plan 6, 8 4 09
Page 22 of 22