English Language Arts 10 – 2
MATHEMATICS 10C:
Systems of Linear Equations
Eastglen, J. Percy Page, M.E. LaZerte,
Strathcona, W.P. Wagner
[pic]
Teacher Team:
Eastglen High School: Chris Hammond, Mark Dobko
J. Percy Page High School: Rebecca Lyons, Deanna Matthews
M.E. LaZerte High School: Lori Lepatsky, Phuong Nguyen, Victoria Wisheu
Strathcona Composite High School: Darlene Scammell, Maureen Selk
W.P. Wagner High 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 for Edmonton Public School Board teachers on a non-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 |
[pic] Big Idea
Real world applications can be modeled and analyzed using more than one linear equation to find solution(s).
[pic] 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.
[pic] 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?
[pic] 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.
[pic] 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 |
1. Desired 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.
[pic] 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.
o 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 makes decisions about appropriate intervention to help the student improve.
Teacher Notes for Transfer 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.
[pic] Going Beyond
o speak to the meaning of the lines below and above the point of intersection
o research the average texts per month for teenagers in North America and adjust your plan if necessary
[pic] 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 cost per 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 [pic] $20 plan with 1000 free texts
Telus Plan [pic] $30 plan with 1000 free texts
Telus Fixed Plan [pic] flat 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
[pic]
Algebraic Solutions
[pic] ( [pic] ([pic] ([pic]texts (over 1000)
[pic]
[pic]
-[pic]
[pic] ([pic] ([pic]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:
[pic] 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.
o change the view window to see the part of the graph that is useful for solving a problem.
Teacher Notes for Transfer Task #2: Money Matters
Prior to transfer task have students define the following vocabulary:
• being on call
• contract work
• domain
• hourly wage
• hourly wage plus commission
• graduated commission
• piece work
• range
• salary
• straight commission
*When work is judged to be limited or insufficient, the teacher makes decisions about appropriate intervention to help the student improve.
[pic] 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?
[pic] 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)
[pic]
Systems of Linear Equations Unit Rubric
[pic] Assessment
| | | | | | |
|Level |Excellent |Proficient |Adequate |Limited* |Insufficient / Blank* |
| |4 |3 |2 |1 | |
|Criteria | | | | | |
|Performs Calculations |Performs precise and |Performs focused and |Performs appropriate |Performs superficial |No score is awarded |
| |explicit |accurate |and generally accurate |and irrelevant |because there is no |
| |calculations. |calculations. |calculations. |calculations. |evidence of student |
| | | | | |performance. |
|Presents Data |Presentation of data is |Presentation of data |Presentation of data is|Presentation of data is|No data is presented. |
| |insightful and astute. |is logical and |simplistic and |vague and inaccurate. | |
| | |credible. |plausible. | | |
|Explains Choice |Shows a solution for the|Shows a solution for |Shows a solution for |Shows a solution for |No explanation is |
| |problem; provides an |the problem; provides|the problem; provides |the problem; provides |provided. |
| |insightful explanation. |a logical |explanations that are |explanations that are | |
| | |explanation. |complete but vague. |incomplete or | |
| | | | |confusing. | |
|Communicates findings |Develops a compelling |Develops a convincing|Develops a predictable |Develops an unclear |No findings are |
| |and precise presentation|and logical |presentation that |presentation with |communicated. |
| |that fully considers |presentation that |partially considers |little consideration of| |
| |purpose and audience; |mostly considers |purpose and audience; |purpose and audience; | |
| |uses appropriate |purpose and audience;|uses some appropriate |uses inappropriate | |
| |mathematical vocabulary,|uses appropriate |mathematical |mathematical | |
| |notation and symbolism. |mathematical |vocabulary, notation |vocabulary, notation | |
| | |vocabulary, notation |and symbolism. |and symbolism. | |
| | |and symbolism. | | | |
| |
|STAGE 3 Learning Plans |
Lesson 1
Modelling Real Life Situations
|[pic]STAGE 1 |
| |
|BIG IDEA |
| |
|Real world applications can be modeled and analyzed using more than one linear equation to find solution(s). |
| | |
|ENDURING UNDERSTANDINGS |ESSENTIAL QUESTIONS |
| | |
|Students will understand: |When is it possible for “no solution” to be the solution to the |
| |problem? |
|that there are connections between the real world and systems of |What real life situations modelled by systems of equations have |
|equations. |one solution, no solution or an infinite number of solutions? |
|that there are multiple ways to solve problems involving systems | |
|of linear equations. | |
| | |
|KNOWLEDGE |SKILLS |
| | |
|Students will know that systems of equations can be used to model|Students will be able to model a situation requiring a system of |
|real world situations. |equations algebraically and/or graphically. |
| | |
[pic]Lesson Summary
This lesson will teach students to go from concrete to pictorial to symbolic when modelling real life situations.
[pic]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?
[pic] 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:
[pic] 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:
[pic]Assessment
Modelling Systems document (see Appendix)
[pic] 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
|[pic]STAGE 1 |
| |
|BIG IDEA |
| |
|Real world applications can be modeled and analyzed using more than one linear equation to find solution(s). |
| | |
|ENDURING UNDERSTANDINGS |ESSENTIAL QUESTIONS |
| | |
|Students will understand: |When is the solution to the system not the answer to the problem?|
| |In what ways will technology help or hinder our understanding of |
|that there are connections between the real world and systems of |systems? |
|equations. |What real life situations modelled by systems of equations have |
|that there are multiple ways to solve problems involving systems |one solution, no solution or an infinite number of solutions? |
|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. | |
| | |
|KNOWLEDGE |SKILLS |
| | |
|Students will know: |Students will be able to: |
|that solutions can be verified |solve a system of equations algebraically and graphically with |
|that the solution to a linear system can be found graphically or |and without technology |
|algebraically |verify solutions |
| | |
[pic]Lesson Summary
Students will solve systems of equations graphically with and without technology.
[pic]Lesson Plan
Hook: Tortoise and Hare Demo
Put a line of masking tape on the floor with one-foot increments marked along it for at least 12 feet (or use a measuring tape). Have one student be the hare and the other the tortoise. Have the hare stand on the 0 mark of the tape and the tortoise stand on the four-foot mark (the hare is giving a head start to the tortoise). At each clap of the hands the hare will move two feet and the tortoise will move one foot along the line. Have the students predict when the hare will catch the tortoise. Then run the demo by clapping slowly as the students move. Have the students note where the hare catches up.
On the board draw a graph of what just happened – this is another way to draw a picture (tying it to previous lesson). Put claps on the x-axis and position on the y-axis. Draw a dot for the tortoise and the hare for each clap. Circle the place where the hare passed the tortoise. The dots will form lines; have the students come up with the equations of the lines.
Have the students notice that this is a system of equations and the “solution” is the intersection point of the lines. Graphing and finding the intersection point is another way to solve systems. Also ask some questions about the graph such as who is ahead at a certain point, how you can tell, etc. This leads nicely into more examples for solving systems of linear equations by graphing.
Practise working through a couple of graphing examples – non-contextual and contextual.
Suggestion: Use third example from Balance Problems (see Appendix) to show another way to solve this.
[pic]
In classroom discussions, teachers should be sure to cover the following topics:
• What is a solution?
• How do you know if you are correct? (verify solution)
• What does it mean to satisfy the system?
Give each pair of students a system of equations and a grid on 11x17 paper. Have them graph the solution on the graph paper and swap with another group. They should check each other’s work.
Do examples of graphing using the graphing calculator – non-contextual and contextual. Choose questions that fit in the standard window settings and ones where the window settings need to be adjusted. You should discuss what window settings make sense in the context of the question (that is, domain and range).
Check for understanding:
Essential Question: In what ways will technology help or hinder our understanding of the system?
Do a contextual example where the answer to the question is not the point of intersection. For example:
• Compare pay schemes.
• Which job should you choose if…..?
Essential Question: When is the solution to the system not the answer to the problem?
[pic] Going Beyond
• Check “Extend” questions in Mathematics 10, textbook, p.431.
• Give students graphs of nonlinear systems and have them determine the solutions.
• Give questions where the intersection point is not in the domain of the context of the question.
[pic] Supporting
• Give students graphs of linear systems and their equations and have them find the solutions and verify.
[pic]Assessment
Foundations and Pre-calculus Mathematics 10, textbook, pp. 403-415
Mathematics 10, textbook, pp. 416-431
Exit slips
[pic]Resources
Graphing calculator
Materials for Tortoise and Hare Demo: measuring tape, masking tape
Lesson 3
Determining the Number of Solutions
|[pic]STAGE 1 |
| |
|BIG IDEA |
| |
|Real world applications can be modeled and analyzed using more than one linear equation to find solution(s). |
| | |
|ENDURING UNDERSTANDINGS |ESSENTIAL QUESTIONS |
| | |
|Students will understand: |In what ways will technology help or hinder our understanding of |
| |systems? |
|that there are connections between the real world and systems of |When is it possible for “no solution” to be the solution to the |
|equations. |problem? |
|that there are multiple ways to solve problems involving systems |What real life situations modelled by systems of equations have |
|of linear equations. |one solution, no solution or an infinite number of solutions? |
|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. | |
| | |
|KNOWLEDGE |SKILLS |
| | |
|Students will know: |Students will be able to: |
|what a system of equations is and what comprises its solution. |verify solutions. |
|that there are 0, 1 or infinitely many solutions to a linear |interpret and communicate a solution within a context. |
|system and what conditions lead to each |determine the number of solutions to a system from a graph or |
|that solutions can be verified. |equations. |
|that systems of equations can be used to model real world | |
|situations. | |
| | |
[pic]Lesson Summary
Students will determine the number of solutions to a system in a variety of ways.
[pic]Lesson Plan
Choose a student’s example from the first lesson (choose one that has a simple solution) to demonstrate an equation with one solution. Have students work in pairs on a pictorial example of a system with infinitely many solutions (example below). Discuss the results, challenges, patterns, and different processes used. Write these as a system of equations, rearrange in terms of y and graph them. Then do the same with a system with no solutions (example shown below).
1. Infinitely many solutions: How much does a goldfish or a seahorse cost?
[pic][pic][pic][pic][pic][pic]=$132
[pic][pic][pic]=$66
2. No solutions: How many minutes does it take to bake a pie or a gingerbread man?
[pic][pic][pic]=90 minutes
[pic][pic][pic][pic][pic][pic]=150 min
Lead the discussion back to graphing from the last lesson. Change the questions into equations and have students graph them, discussing the slope and y-intercept of each line and what situation creates the number of solutions in each system.
Individually or in groups, students will create three systems (one solution, no solution, infinitely many solutions) and represent them three ways: graphically, algebraically and pictorially. Introduce the word coincident.
Discussion may occur about how to recognize the number of solutions without having to rearrange the equations. This would lead into the game of Last Man Standing.
Last Man Standing Game:
Have all the students stand up. Put a system of equations up on the board. Much like the paper/rock/scissors game, have students simultaneously show the number of solutions by holding up a fist (no solutions), one finger (one solution) or all five fingers (infinitely many solutions). If students get the answer wrong they sit down and become judges. Put up a new question and continue the game until one student remains. Have several systems ready to go for this game.
[pic] Going Beyond
Leading questions:
• Can we think of an example with two (or more than one) solutions?
• Can we think of a real-world situation that has no solution or infinitely many solutions?
[pic]Assessment
Informal assessment: Last Man Standing Game (above)
[pic] Resources
Foundations and Pre-calculus Mathematics 10, textbook, pp. 442-449.
Number of Solutions for Systems of Linear Equations, Mathematics 10, textbook, pp. 446-459
[pic] Vocabulary
• coincident lines
• intersecting lines
• infinite
• parallel
Lesson 4
Solving Systems Algebraically
|[pic]STAGE 1 |
| |
|BIG IDEA |
| |
|Real world applications can be modeled and analyzed using more than one linear equation to find solution(s). |
| | |
|ENDURING UNDERSTANDINGS |ESSENTIAL QUESTIONS |
| | |
|Students will understand: |Which method of solving systems of equations is best in a given |
|that there are connections between the real world and systems of |situation? |
|equations. |When is it possible for “no solution” to be the solution to the |
|that there are multiple ways to solve problems involving systems |problem? |
|of linear equations. |When is the solution to the system not the answer to the problem?|
|that solutions can be communicated in a variety of ways. | |
| | |
|KNOWLEDGE |SKILLS |
| | |
|Students will know: |Students will be able to: |
|that solutions can be verified. |solve a system of equations algebraically and graphically with |
|that the solution to a linear system can be found graphically or |and without technology. |
|algebraically. |explain their strategy for solving a system of equations. |
|that systems of equations can be used to model real world |verify solutions. |
|situations. |interpret and communicate a solution within a context. |
| | |
[pic]Lesson Summary
Students will be able to solve a system of equations algebraically.
[pic]Lesson Plan
Part 1
Introduce the first pictorial example on the board, without writing the matching equations. Have the students discuss how to solve these without graphing and write in words the steps that they used to come up with a solution. Does their answer work? Pick out the steps that are more like the elimination method and focus on them. Let the students know that the other steps (substitution method) will be studied in Part 2 of this lesson.
At this point, write the equations on the board and work through the steps that the students suggested.
+ = 20 2x + y = 20
+ = 15 x + y = 15
Take up another example and work through the same process to solve this system:
+ = 100 3x + 2y = 100
+ = 80 x + 2y = 80
Let the students know that this process is called elimination and demonstrate more traditional questions with subtraction, different coefficients for both variables, fractions and word problems.
Give students practice time with more questions.
Part 2
Introduce the first pictorial example on the board, without writing the matching equations. Pull out the student work from Lesson 1 that was more like substitution and go over them. Write the equations on the board and work through the steps that the students suggested.
= x = 2y
+ = 30 x + 3y = 30
Take up another example and work through the same process to solve this system:
= + 3 x = 2y + 3
+ = 11 2y + x = 11
Let the students know that this process is called substitution and demonstrate more traditional questions with subtraction, different coefficients for both variables, fractions and word problems.
Give students practice time with more questions.
Part 3: Wrap Up
Spend a few minutes reviewing the different ways that we have solved equations. Split the students into groups and give them three different systems of equations. Have them discuss how to solve them and answer the questions “Which method of solving systems is best for each equation?” Results can be shared in a number of different ways such as group presentation, expert-based sharing, etc.
1. Six pencils and four crayons cost $3.40. Three similar pencils and ten similar crayons cost $4.90. How much would you expect to pay for a set of eight pencils and twelve crayons?
2. You have been offered two jobs. Job A pays you $300 per month and 2% commission of sales. Job B pays a straight commission of 10% of sales. Under what circumstances would you choose A? B?
3. A ring costs three times as much as a necklace. Two rings and one necklace cost $140. How much does each item cost?
As a follow-up to the second question, discuss the essential question “When is the solution to the system not the answer to the problem?”
[pic] Going Beyond
Provide the students with questions in three variables, pictorial and symbolic. Some examples of these can be found in the worksheets from Lesson 1.
[pic] Supporting
Some students will benefit from having manipulatives to solve systems and may need more practice with simpler questions.
[pic]Assessment
The practice time and the closing presentation will give a chance for the teacher to do a formative assessment.
[pic] Resources
Foundations and Pre-calculus Mathematics 10, textbook, pp. 416 – 439.
Mathematics 10, textbook , pp. 468 – 501
[pic] Vocabulary
• substitution
• elimination
Math 10 Unit C
Systems of Linear Equations
APPENDIX
Lesson 1
Balance Problems
[pic]
[pic]
[pic]
[pic]
Source: Van de Walle, John, Lou Ann H. Lovin. Teaching Student-Centred Mathematics Grades 5-8 . Allyn & Bacon, 2006. Print.
Lesson 1
[pic]
Lesson 2
Modelling Systems
A. Solve the following picture problems.
1. Find the value of one ladybug.
[pic]
2. Find the value of one guitar.
[pic]
3. Find the value of one umbrella.
[pic]
4. Find the value of one gear.
[pic]
5. Find the value of one pear.
[pic]
6. Find the value of three gears.
[pic]
Source:
B. For each picture above, write a system of equations.
1.
2.
3.
4.
5.
6.
C. Read the following word problems, draw pictures for the systems, and then write them as equations.
1. Two hoodies and three T-shirts cost $140. Four hoodies and one t-shirt costs $180.
2. For $15, Murphy downloaded five songs and two music videos. Sania downloaded three songs and three music videos for $18.
3. John and his sister went to the pet store to buy some fish. He bought six Nemo fish and two goldfish for $94. She only paid $50 to buy two Nemo fish and ten goldfish.
4. Homer and Marge decided that they need to get out more and buy tickets to Oilers games and Eskimo games. Mr. Winter bought two Eskimos tickets and one Oilers ticket for $130. For $260 Ms. Burnham bought four Eskimos tickets and two Oilers tickets.
5. Doughboy and Mr. Tim meet every day to buy snacks for lunch and they each spend $6.50. Doughboy always buys four doughnuts and one cinnamon bun. Mr. Tim gets two doughnuts and three cinnamon buns.
D. Write each system of equations in words.
1. How much money is each item worth?
[pic] [pic] [pic] [pic] [pic]=$250
[pic] [pic] [pic][pic][pic][pic] =$220
2. How much money is each item worth?
[pic][pic][pic][pic][pic][pic]=$132
[pic][pic][pic]=$66
3. How much does each item weigh?
[pic] [pic] [pic] =32200 lbs
[pic][pic][pic][pic][pic] =64300 lbs
4. How much baking time for each item?
[pic][pic][pic][pic][pic] =88 min
[pic][pic][pic][pic] =132 min
5. How many points is each item worth?
=32
=28
6. How many calories in each item?
[pic][pic][pic][pic][pic][pic] =3080 cal
[pic][pic][pic][pic][pic][pic] =3160 cal
Modelling Systems Answer Key
A. Solve
1. ladybug = 9
2. guitar = 6
3. umbrella = 8
4. gear = 6
5. pear = 5
6. three gears = 36
B. Systems of Equations
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
C. Word Problems
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
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1.
SALES ($)
MONTHLY PAY ($)
SALES ($)
MONTHLY PAY ($)
Implementation note:
Teachers need to constantly ask: what performances & products will reveal evidence of understanding?
What other evidence will be collected to reflect the desired results?
Implementation note:
Students must be given the transfer task & rubric* at the beginning of the unit. They need to know how they will be assessed and what they are working towards.
This is where they are equal
NUMBER OF TEXTS OVER 1000
Plan B
Best
Parents Plan
COST PER MONTH ($)
Implementation note:
Teachers need to constantly ask, what performances & products will reveal evidence of understanding?
What other evidence will be collected to reflect the desired results?
Implementation note:
Students must be given the transfer task & rubric* at the beginning of the unit. They need to know how they will be assessed and what they are working towards.
Implementation note:
Teachers need to continually ask themselves: what knowledge and skill(s) will the students leave with at the end of the unit?
Implementation note: At least every second lesson, pause & ask students to consider one of the essential questions. Has their thinking changed, evolved?
Implementation note:
Post the BIG IDEA in a prominent place in your classroom and refer to it often
2.
5.
4.
3.
D. Write equations in words.
Answers are not intended, but provided for reference
for high-end kids.
1. jacket: $70, shoe: $20
2. infinite solutions
3. scooter: 100lbs, truck: 32000 lbs
4. pancake: 40 min, gingerbread man: 12 min
5. square: [pic]; smiley face: [pic]
6. hamburger: 540 cal; fries: 500 cal
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