MATHEMATICS 10C:

Systems of Linear Equations

Eastglen, J. Percy Page, M.E. LaZerte,

Strathcona, W.P. Wagner

Teacher Team:

EastglenHigh School: Chris Hammond, Mark Dobko

J. PercyPageHigh School: Rebecca Lyons, Deanna Matthews

M.E.LaZerteHigh School: Lori Lepatsky, Phuong Nguyen, Victoria Wisheu

StrathconaCompositeHigh School: Darlene Scammell, Maureen Selk

W.P.WagnerHigh School: Kim Burnham

Facilitator: John Scammell (Consulting Services)

Editor: Rita Feutl

Textbook Resources:

McAskill, Bruce, Wayne Watt, Eric Balzarini, Scott Carlson, Blaise Johnson, Ron Kennedy, Terry Melnyk, Harold Wardrop. Mathematics 10. Toronto: McGraw Hill Ryerson, 2010. Print.

Van Bergeyk, Chris, Dave Van Bergeyk, Garry Davis, Jack Hope, Delcy Rolheiser, David Sufrin, David Zimmer, David Ellis. Foundations and Pre-Calculus Mathematics 10.Pearson Canada Inc., 2010. Print.

The use of Understanding by Design: MATHEMTICS 10C: Systems of Linear Equations content is expressly and solely forEdmonton Public School Board teacherson anon-profit, non-commercial, internal basis.

2010

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS
Big Idea / 4
Enduring Understanding / 4
Essential Questions / 4
Knowledge / 5
Skills / 5
Stage 2 ASSESSMENT EVIDENCE
General Teacher Notes / 6
Transfer Task #1 My Cellphone is my Life / 6
Transfer Task #2 Money Matters / 11
Rubric for Transfer Task / 15
Stage 3 LEARNING PLANS
Lesson #1 Modelling Real Life Situations / 16
Lesson #2 Solving Systems Graphically with and without Technology / 20
Lesson #3 Determining the Number of Solutions / 24
Lesson #4 Solving Systems Algebraically / 27
APPENDIX
1. Balance Problems / 32
2. Foxtrot Cartoon / 33
3. Modelling Systems / 34

Unit: Systems of Linear Equations

STAGE 1 Desired Results

Big Idea

Real world applications can be modeled and analyzed using more than one linear equation to find solution(s).

Enduring Understandings

Students will understand:

  • that there are connections between the real world and systems of equations.
  • that there are multiple ways to solve problems involving systems of linear equations.
  • the relationship between the graph of a system of equations and the meaning of the solution.
  • that solutions can be communicated in a variety of ways.

Essential Questions

  • Which method of solving systems of equations is best in a given situation?
  • When is the solution to the system not the answer to the problem?
  • In what ways will technology help or hinder our understanding of systems?
  • When is it possible for “no solution” to be the solution to the problem?
  • What real life situations modelled by systems of equations have one solution, no solution or an infinite number of solutions?

Knowledge

Students will know:

  • what a system of equations is and what comprises its solution.
  • that there are 0, 1 or infinitely many solutions to a linear system and what conditions lead to each.
  • that solutions can be verified.
  • that the solution to a linear system can be found graphically or algebraically.
  • that systems of equations can be used to model real world situations.

Skills

Students will be able to:

  • solve a system of equations algebraically and graphically with and without technology.
  • explain their strategy for solving a system of equations.
  • verify solutions.
  • interpret and communicate a solution within a context.
  • determine the number of solutions to a system from a graph or equations.
  • model a situation requiring a system of equations algebraically and/or graphically.


STAGE 2 Assessment Evidence

1Desired Result

General Teacher Notes:

You may choose from two transfer tasks. Only one transfer task is needed to evaluate student understanding of the concepts relating to systems of linear equations. The rubric will evaluate either completed task.

Transfer Task #1: MY CELLPHONE IS MY LIFE

Criteria:

Each student will:

  • recognize what the rate and initial value in a real-life situation represent in an equation.
  • graph two or more equations representing the situation.
  • potentially describe the slope of horizontal lines.
  • write equations for application problems.
  • use the intersection of the graphs and solution to the equation to solve the problem.
  • use a graphing calculator or graphing software to graph linear equations.
  • change the view window to see the part of the graph that is useful for solving a problem.

*When work is judged to be limited or insufficient, the teacher makesdecisions about appropriate intervention to help the student improve.

Teacher Notes forTransfer Task #1: My Cellphone is my Life

This is a list of some cell phone providers that students may research: Telus, Bell, Fido, Koodo, Shaw, Rogers, Virgin, Solo, Speakout, etc.

Students could be given tasks over the course of the unit. For instance, count the number of texts they receive and send in a day or week. Research cell phone plans available in their community.

Going Beyond

  • speak to the meaning of the lines below and above the point of intersection
  • research the average texts per month for teenagers in North America and adjust your plan if necessary

MY CELLPHONE IS MY LIFE

Student Assessment Task

Your parents are currently paying for your cellphone. But this month, your parents received a HUGE cellphone bill and freaked out about your charges. They’ve taken away your cell phone and cancelled your plan. You need a phone! You decide to find your own plan.

You have a part-time job and your parents will only let you work six hours a week at most. Your job pays minimum wage. Assume you get paid every two weeks.

Your parents found this plan for you at $20 per month and $0.20 per text after the first thousand texts (incoming and outgoing).

Goal:

Find a better plan than your parents’ plan for your needs and budget.

Role:

You are to research two different plans to compare with your parents’ plan.

Product/Performance:

Decide what your texting needs are. Then present the following information on a poster:

the maximum you want to pay for your plan within your budget

the two plans and your parents’ plan

the fixed cost of the plans and any costper text over the fixed cost

the equations of the plans, one of which is your parents’ plan

comparison and solution of three plans, represented graphically and algebraically, as a system of equations

the slope and y-intercepts and what they mean

the point of intersection and what it means

a defence of the plan that most suit your needs that will not go over budget

Teacher Key

(This sample is based on research done on February 2010)

Maximum desired cost is $50 per month

Parent Plan$20 plan with 1000 free texts

Telus Plan$30 plan with 1000 free texts

Telus Fixed Planflat rate of $40 including unlimited texts

Slope of parents’ plan is the price per text after 1000 texts.

Slope of the Telus plan is 15¢ per text.

Slope of the fixed plan is 0 as texts are included in the plan.

The y intercept for Telus is $30 per month.

The y intercept of the parents’ choice is $20 per month.

The y intercept of the flat rate is $40 per month.

The intersection of these two lines represents where the cost and number of texts is the same for both plans. This intersection occurs at (100, 40).

Monthly Cost of Cell Phone Plans

Algebraic Solutions

 texts (over 1000)

-

texts

This means for $40 per month I would get an additional 100 texts for a total of 1100 texts. Therefore, I would choose the fix plan for $40 because I send and receive more than 1100 text messages combined.

Alternate Project:

Transfer Task #2: MONEY MATTERS

Criteria:

Each student will:

  • recognize what the rate and initial value in a real-life situation represent in an equation.
  • graph two or more equations representing the situation.
  • potentially describe the slope of horizontal lines.
  • write equations for application problems.
  • use the intersection of the graphs and solution to the equation to solve the problem.
  • use a graphing calculator or graphing software to graph linear equations.
  • change the view window to see the part of the graph that is useful for solving a problem.

Teacher Notes forTransfer Task #2: Money Matters

Prior to transfer task have students define the following vocabulary:

Mathematics 10C Systems of Linear Equations 1

  • being on call
  • contract work
  • domain
  • hourly wage
  • hourly wage plus commission
  • graduated commission
  • piece work
  • range
  • salary
  • straight commission

Mathematics 10C Systems of Linear Equations 1

*When work is judged to be limited or insufficient, the teacher makesdecisions about appropriate intervention to help the student improve.

Mathematics 10C Systems of Linear Equations 1

Going Beyond

Working at Tec Wizard, your sales are $11490. What would be your rate of commission at In Fashion in order to earn the same income? Also, express the equation that represents the new rate of commission with the same base pay.

What would be the equation, in slope y-intercept form, that would represent the base salary for In Fashion in order to earn the same income of $600 at Tec Wizard?

If you cannot change the base salary, what would be the equation, in general form, that would represent the same income earned for In Fashion as Tec Wizard?


MONEY MATTERS

Student Assessment Task

You are looking for a job and you have applied to many different places.

  • Your uncle is offering $400 a month to be on call working for the family business.
  • Tec Wizard is offering 5% straight commission on total sales.
  • In Fashion is offering $200 a month salary plus 3% commission on total sales.

Goal:

Compare the three job offers.

Role:

Evaluate the pros and cons of each job for you as an individual and choose the job that best fits your needs.

Product/Performance:

For your poster:

Identify the variables and determine the three equations that represent the situations.

Give graphical representations of the equations.

Determine the three intersection points on the graph and interpret their meaning.

Interpret the y-intercepts and slopes for all three equations

Discuss which job you would choose and why. What are the advantages and disadvantages of each?

Algebraically solve for the three points of intersection.

Define the domain and range in each equation.

Identify the dependent and independent variables.

Teacher Key

Analysis of Different Jobs (1)

If the base stays the same the new equation is y = 0.033x +200

If the commission stays the same the new equation is y = 0.03x + 240 in general form is y = 3x -100y+24000

Going Beyond:

Analysis of Different Jobs (2)

Systems of Linear Equations Unit Rubric

Assessment


Level
Criteria / Excellent
4 / Proficient
3 / Adequate
2 / Limited*
1 / Insufficient / Blank*
Performs Calculations / Performs precise and explicit
calculations. / Performs focused and accurate
calculations. / Performs appropriate and generally accurate
calculations. / Performs superficial and irrelevant calculations. / No score is awarded because there is no evidence of student performance.
Presents Data / Presentation of data is insightful and astute. / Presentation of data is logical and credible. / Presentation of data is simplistic and plausible. / Presentation of data is vague and inaccurate. / No data is presented.
Explains Choice / Shows a solution for the problem; provides an insightful explanation. / Shows a solution for the problem; provides a logical explanation. / Shows a solution for the problem; provides explanations that are complete but vague. / Shows a solution for the problem; provides explanations that are incomplete or confusing. / No explanation is provided.
Communicates findings / Develops a compelling and precise presentation that fully considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism. / Develops a convincing and logical presentation that mostly considers purpose and audience; uses appropriate mathematical vocabulary, notation and symbolism. / Develops a predictable presentation that partially considers purpose and audience; uses someappropriate mathematical vocabulary, notation and symbolism. / Develops an unclear presentation with littleconsideration of purpose and audience; uses inappropriate mathematical vocabulary, notation and symbolism. / No findings are communicated.
STAGE 3 Learning Plans

Lesson 1

Modelling Real Life Situations

STAGE 1
BIG IDEA
Real world applications can be modeled and analyzed using more than one linear equation to find solution(s).
ENDURING UNDERSTANDINGS
Students will understand:
  • that there are connections between the real world and systems of equations.
  • that there are multiple ways to solve problems involving systems of linear equations.
/ ESSENTIAL QUESTIONS
  • When is it possible for “no solution” to be the solution to the problem?
  • What real life situations modelled by systems of equations have one solution, no solution or an infinite number of solutions?

KNOWLEDGE
Students will know that systems of equations can be used to model real world situations. / SKILLS
Students will be able to model a situation requiring a system of equations algebraically and/or graphically.

Lesson Summary

This lesson will teach students to go from concrete to pictorial to symbolic when modelling real life situations.

Lesson Plan

Hook: Finding solutions to pictorial problems that model systems of equations with two and/or three variables.

Have students look at Balance Problems document (see Appendix). Note: The weights for each question are different.

Have students find solutions to the problems and discuss the way they solved the problems. Strategy is more important than the solution. Students can work in small groups, then put their answers on the board or make posters/flip chart paper that outlines how they came up with their answers. Ask them: How do you know you’re right?

Now move the strategies towards a more mathematical discussion. Turn the pictures into equations (modelling) with variables.

Have a teacher-guided discussion on what just occurred. Talk about what a system is, and that the x and the y have a meaning, etc. At this point, define solution.

Have the students create their own modelling situation on an index card and have their elbow partner turn the question into symbols. Limit them to two objects. Note: they must have a system that works. Have them put their answers on the back.

Show students Foxtrot cartoon (see Appendix) and discuss. Or go to:

Put up a word problem on the board. Have students draw a pictorial representation, and then change the pictures into equations with two variables.

Example: Three shirts and two pairs of jeans cost $230 and two shirts and four pairs of jeans cost $340. How much is a shirt and how much is a pair of jeans?

Now use a larger sample so that students will not want to draw a picture and will go right from words to symbols.

Example: Omar and his brother Mohammed go to the local candy shop to buy some treats. Omar has $5 and buys 40 sours and 18 licorice. Mohammed has $8 and buys 50 sours and 40 licorice. How much does each type of candy cost?

Going Beyond

  • Have students write systems with more than two equations and/or more than two variables. They can try problems on these sites:

Smiles:

Gone Fishing:

  • Give students pictorial examples and have them write contextual word problems to go with them.
  • Give students a system of equations and have them write contextual word problems to go with them.
  • Let students work on Level-3 questions at:
  • For strong students, you may put up a question with infinite solutions

E.g.: 2 burgers + 3 gingerbread men = $5.50

4 burgers + 6 gingerbread men = $11.00

And ask the question: How many solutions can you find?

  • For extra practice, similar to worksheets:

Supporting

  • Provide manipulatives for students who are working at the concrete level.

Examples: Cube-a-Links, pattern blocks, pencils, erasers, rulers, markers,

coins, or whatever is available

  • Let students work on Level-1 or -2 questions at:

Assessment

Modelling Systems document (see Appendix)

Resources

Websites:

Smiles:

Gone Fishing:

Additional source of pictorial questions:

Balance Problems: Teaching Student-Centred Mathematics Grades 5-8 by John Van de Walle, Lou Ann H Lovin Publisher: Allyn & Bacon, Copyright: 2006, Published: 06/17/2005

Lesson 2

Solving Systems of Equations Graphically with and without Technology

STAGE 1
BIG IDEA
Real world applications can be modeled and analyzed using more than one linear equation to find solution(s).
ENDURING UNDERSTANDINGS
Students will understand:
  • that there are connections between the real world and systems of equations.
  • that there are multiple ways to solve problems involving systems of linear equations.
  • the relationship between the graph of a system of equations and the meaning of the solution.
  • that solutions can be communicated in a variety of ways.
/ ESSENTIAL QUESTIONS
  • When is the solution to the system not the answer to the problem?
  • In what ways will technology help or hinder our understanding of systems?
  • What real life situations modelled by systems of equations have one solution, no solution or an infinite number of solutions?

KNOWLEDGE
Students will know:
  • that solutions can be verified
  • that the solution to a linear system can be found graphically or algebraically
/ SKILLS
Students will be able to:
  • solve a system of equations algebraically and graphically with and without technology
  • verify solutions

Lesson Summary