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
Breakeven Point
Cost
Elimination
Fixed Cost
Multiplication Property of Equality / Profit
Revenue
Solution to a System of linear Equations
Substitution
System of Linear Equations
Transitive Property
Variable Cost
Assessment Strategies
Performance Task(s)
Authentic application in new context / Other Evidence
Formative and Summative assessments
Community Park
NOTE: Prior to the end of the unit, you may wish to describe the Performance Task and have students investigate the historical and cultural aspects of the design and use of public parks and gardens in various urban, suburban and rural settings. Students might share their own experiences at various parks and gardens.. After the three investigations are complete, students will use systems of equations to design a community park. / · Warm-ups, class activities, exit slips, and homework have been incorporated throughout the investigations.
· End-of-Unit Test
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.
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 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 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
Unit 6 Performance Task