Course Profile Principles of Mathematics,
Grade 10, Academic, Catholic
Unit 1: Modelling Linear Systems
Time: 13 hours
Activity 1 | Activity 2
| Activity 3 | Activity
4 | Activity 5
Unit Description
In this unit, linear systems will be analysed both graphically and algebraically, with and without the use of technology. Activities in this unit provide a context for finding and interpreting points of intersection and lead students to solve linear systems by the methods of substitution and elimination. In preparation for unit two, students will explore a maximization problem that introduces the concept of quadratic functions and involves expanding and simplifying polynomial expressions.
Strand(s) and Expectations
Ontario Catholic School Graduate
Expectations: CGE1d, 2b, 2c, 4a, 4f, 5a, 5b 5.g, 7b.
Strand(s): Analytic Geometry, Quadratic
Functions
Overall Expectations: AGV.01D,
QFV.04.
Specific Expectations: AG1.01D,
AG1.02D, AG1.03D, QF1.01D, QF4.02D.
Activity Titles (Time + Sequence)
The activities in this unit, which is an extension of the linear functions topics covered in the Grade 9 course, provide students with the opportunity to develop algebraic and graphical models in the context of realistic problems. The timelines provided are suggestions to guide teacher planning and can be modified to suit the needs of individual classrooms. Teachers may consider including locally relevant applications of the topics covered in these activities, where appropriate. The activities provided involve considerable group work, and thus provide the students with opportunities to demonstrate Catholic values as collaborative contributors and Christian leaders. Many of the activities also provide opportunities for students to demonstrate their understanding of responsible citizenship and societal awareness.
|
Activity 1.1 |
The Peace and Development
Fundraiser: Intersection of Linear Models [construction of linear graphs from tables of values: interpolation: intersection of lines: formula development and evaluation (word and algebraic equations)] |
75 minutes |
|
Activity 1.2 Follow-up Skills |
Bottled Water Dilemma [use of technology in graphing; intersection of lines; subtraction of equations] Finding the intersection of lines using graphing calculators or graphing software; solving contextual problems |
75 minutes 75 minutes |
|
Activity 1.3 Follow-up Skills |
The Calculator Workshop [algebraic method of elimination] Practice with the elimination method; introduce the substitution method; practice with realistic situations |
75 minutes 255 minutes |
|
Activity 1.4 Follow-up Skills |
Battle of the Bands: Introduction to
Quadratics [construction of graphs from tables of data: maximization of revenue and area; graphing a product of binomials; quadratic regression; connect product of binomials with ax2 + bx + c] Expand and simplify second degree polynomial expressions (squaring and expanding binomials) |
75 minutes 75 minutes |
|
Activity 1.5 |
Summative Activity: Paper and Pencil
Test |
75 minutes |
Prior Knowledge Required
Number Sense and Algebra
· manipulate 1st- degree polynomial expressions to solve 1st- degree equations
· add and subtract polynomials; multiply a polynomial by a monomial; expand and simplify polynomial expressions involving one variable
· solve problems, using the strategy of algebraic modelling
Relationships
· determine relationships between two variables by collecting and analysing data
· compare the graphs of linear and non-linear relations
· collect, organize, and analyse data using appropriate equipment and/or technology
· describe trends and relationships in data
· construct tables of values, graphs, and formulas from descriptions of realistic situations and from data collected experimentally
· use interpolation and extrapolation to gather information from a graph
· distinguish between linear and non-linear relations by calculating finite differences
Analytic Geometry
· identify the properties of line segments (direction, positive/negative slope, parallelism, perpendicularity)
· calculate slope using various formulae
· identify slope as a constant rate of change
· graph lines by hand and using graphing technology
· determine the equation of a line given slope and y-intercept, slope and a point on the line, and two points on a line, in the form y = mx +b and Ax + By + C = 0
Unit Planning Notes
· Practise using spreadsheets and graphing calculators in the context of the activities presented.
· Prepare to diagnose prior learning skills throughout the unit. Skill development activities should be developed to meet the needs that arise.
· Some students may require extended or enrichment activities to challenge their learning.
Teaching/Learning Strategies
· It is expected that direct, teacher-lead instruction will be integrated within the framework of the activities as required to facilitate student learning and success. Independent practice of new skills will be necessary throughout the course.
· It is recommended that students be assigned to groups with any special needs and strengths considered. Appropriate peer grouping to benefit those students requiring extra help is suggested.
· Students will be involved in a considerable amount of group work. It would be beneficial to take some time to review appropriate group work dynamics, sharing of work responsibilities, assigning group roles, etc.
· Students should be encouraged to take ownership and responsibility for their own learning.
· Appropriate opportunities for students to communicate solutions, ideas, concepts should be provided throughout the course.
Assessment and Evaluation
· It is recommended that students be involved in the development of some of the rubrics used in the assessment/evaluation process. Students should be encouraged to self-assess and to identify areas that need improvement.
· Sample generic rubrics for oral presentations, written reports, and various learning skills are provided in this profile. They may be adapted to suit the needs of your classroom.
· Rubrics should be used where appropriate to assess student work. They are particularly appropriate to assess the expectations under the thinking/inquiry/problem-solving, communication and application sections of the Achievement Chart.
· When rubrics are used in assessment, students should be provided with the specific rubric that will be used prior to completing the assigned task.
· Care should be taken in the design of traditional paper and pencil tests to ensure that a Level 4 performance can be demonstrated.
· This unit is assessed using a paper and pencil test.
· Learning skills should be assessed in conjunction with the academic skills for each activity.
Resources
Graphing calculators; graphing software
NCTM Addenda Series: Example 3 and Activity 2 from “Algebra in a Technological World”
The Mathematics Teacher, Vol. 88 #3, March 1995: “Gas Bill Mathematics”
Lappan, Glenda, et al. Moving Straight Ahead Linear Relationships. Dale Seymour Publishing, 1988.
NCTM. Activities for Active Learning
and Teaching.
Activity 1.1: The Peace and Development Fund Raiser - Intersection of Linear Models
Time: 75 minutes
Description
By developing tables of values, sketching graphs, writing linear equations, and interpreting the point of intersection, students will analyse a situation with variations. In this activity, the student council of Holy Mary High School has decided to do a walk-a-thon as one of its many fund raising activities and are faced with a decision of an appropriate, consistent donation plan for the whole student body to follow.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
- a discerning believer formed in the Catholic faith community who develops attitudes and values founded on Catholic social teaching and acts to promote social responsibility, human solidarity and the common good;
- a self directed, responsible, life long learner who demonstrates a confident and positive sense of self and the respect for the dignity and welfare of others;
- a collaborative contributor who works effectively as an interdependent team member.
Strand(s): Analytic
Geometry
Overall Expectations
AGV.01D - model and solve problems involving the intersection of two straight lines.
Specific Expectations
AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation.
Planning Notes
· The teacher will need a class set of graphing calculators and graph paper.
· The teacher will take the time to discuss the particular charity their school usually supports and the responsibility of all citizens to reach out to others in need. The chaplaincy office or the religion department in your school would be a place to obtain more information on this topic.
· There is opportunity to extend this activity later in this unit by using the other fund raising suggestions given by the students and solving systems of linear equations by algebraic methods.
Prior Knowledge Required
Relationships
· use a graphing calculator or graphing software on a computer
· construct tables of values and graph linear relations
· construct formulas derived from descriptions of realistic situations
· describe the effect on the graph and the formula of varying the conditions of the situation they represent
· determine the values of a linear relation by formula and by interpolating or extrapolating from the graph
Analytic Geometry
· determine the point of intersection of the two linear relations, by hand for simple examples, and using graphing calculators or graphing software for more complex examples; interpret the intersection point in the context of an application.
Teaching/Learning Strategies
Teacher Facilitation: The students will work in small groups of two or three. The teacher will help any group experiencing difficulties in Part A of the activity. Students should be able to complete Part B with very little help from the teacher. During Part B the teacher will be able to observe and encourage students to work efficiently within a group setting. The students should be prepared to hand in this report and/or to give a quick oral presentation using chart paper or an overhead sheet to illustrate their charts, graphs and equations.
Student Activity
The students will work in small groups of two or three to complete the following handout:
The Peace and Development Fund Raiser - The Student Handout
This year the student council of Holy Mary High School decided to encourage students to raise funds for the Peace and Development Fund. The student council decided to encourage the students to participate in a ten-kilometre walk-a-thon as one of the many fund raising activities the school will be involved in during the year. During an organizational meeting, Andrew the student council president, asked what would be a fair donation rate per kilometre to ask of the sponsors, assuming each walker must have a minimum of 5 sponsors on their pledge sheet. Andrew then stated that the students in his home room suggested 75 cents per kilometre. Beth, who is the representative of another class, stated that perhaps the students can ask for a $5.00 donation plus 25 cents per kilometre.
Part A
1. Make a table showing the amount of money which would be pledged under each plan if the students walk up to 10 kilometres.
2. Using different colours, graph each pledge plan on the same coordinate axes.
3. At what point do the lines intersect each other? Explain what this means in the context of this situation.
4. For each plan, write a formula that will help the volunteer calculate the amount of money one sponsor owes, given the distance the student completed. Write a formula in words first and then in algebraic form.
5. If the student completed 7.5 kilometres, how much would the sponsor owe under each plan? Explain how you calculated your answer using two different methods.
6. a) If the sponsor owes $6.00, how many kilometres would the student have walked under each plan? Explain how you found the distance using three different methods.
b) If the sponsor owes $7.20, how many kilometres would the student have walked under each plan? Explain how you found the distance using two different methods.
7. Beth suggested a $5.00 donation and then 25 cents per kilometre.
a) How is this $5.00 donation represented on the graph?
b) If the rate of 25 cents per kilometer was changed to 50 cents per kilometre, how would this change the graph? Using a different colour draw the line on the same axes as Question 2. Don’t forget to show a table of values for this situation and state the formula.
c) The line in part b) intersects the line representing Andrew’s plan. Explain what the point of intersection means in the context of this situation.
8. By changing the initial donation or the donation rate or both, find a new formula which will give an $18 donation if the student completes 7 kilometres. Illustrate this new plan on a graph.
9. Which pledge plan would you suggest Holy Mary High School use in this walk-a-thon? Give reasons for your choice.
Part B
1. Your home room does not like the plans suggested so far.
a) Describe two other plans which could be used to raise funds in the walk-a-thon. The only restriction student council has placed on the pledge forms is that no sponsor should pay more than $20.00 for the completed distance.
b) Make tables of values and graphs to illustrate these two plans.
c) Write a formula for each plan.
d) Do the two lines intersect each other? Hopefully so! Where? Explain what the intersection point means.
e) Pick a distance before the intersection point and describe which plan is better. Explain why.
f) Pick a distance after the intersection point and describe which plan is better. Explain why.
g) From these two plans, which one would you prefer and why?
2. Using your chosen plan, what would the minimum amount of pledges be, if 20% of 1500 students from Holy Mary High School finished the walk-a-thon.
3. What other activities could this school use to raise money for the Peace and Development Fund?
Assessment/Evaluation Techniques
A variety of assessment tools should be used to properly evaluate the student:
· Through observation, make anecdotal comments on independent work, teamwork, organization skills, work habits, communication and initiative.
· A written report rubric (Appendix D) could be used on Part B, due to the variety of rate plans students would suggest.
· If the teacher wishes to have presentations instead of a written report, verbal presentation rubric (Appendix C) could be used.
· A paper and pencil quiz would be used to assess whether the student can set up a table, graph, find the intersection point and explain what it means in the context of another realistic situation.
Accommodations
The teacher should be available to help students who are experiencing difficulties in completing Part A of this activity. In Part B the teacher can encourage students, reword the question if not clearly understood, and provide examples from Part A as a guide.
Resources
Lappan,
Glenda, et al. Moving Straight Ahead Linear
Relationships. Dale
Seymour Publications, 1998.
Activity 1.2: Bottled Water Dilemma
Time: 75 minutes
Description
This activity provides students with a context for exploring graphical representations of a linear system of equations in two variables and interpreting the point of intersection of two linear relations. Using graphs they will examine and compare two different reward structures, for supply and sale of bottled water to a school, offered by competing companies. They will be looking for information and results to help them make a recommendation regarding which company will better serve the school’s bottled water consumption and fund-raising needs. The introduction of a pricing war provides an interesting extension which allows students to interpret the subtraction of equations in a graphical way and provides a good lead into the algebraic solution of linear systems.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
- a collaborative contributor who works effectively as an interdependent team member.
Strand(s): Analytic Geometry
Overall Expectations
AGV.01D - model and solve problems involving the intersection of two straight lines.
Specific Expectations
AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation;
AG1.03D - solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.
Planning Notes
· The use of graphing tools, either graphing calculators or graphing software, is recommended since these allow students to apply and quickly see the effects of changes made to the reward schemes. They also enable students to easily and accurately find the coordinates of the point of intersection of the two lines representing the reward schedules. It is, however, also possible to carry out the activity without technology, but it would be more time-consuming.
· This activity would be more meaningful for the students if they were able to use real data. Perhaps your school is considering the installation of a vending machine. Your Student Council might be pleased to recruit the help of Grade 10 students in the decision-making process.
· This activity could provide an opportunity to discuss some of the ethical issues around the use and abuse of water in the world.
Prior Knowledge Required
Number Sense and Algebra
· add and subtract polynomials and multiply a polynomial by a monomial;
· expand and simplify polynomial expressions involving one variable.
Relationships
· graph lines by hand or using graphing calculators or graphing software.
Teaching/Learning Strategies
Student Activity
Working in pairs, students will use graphing to investigate the two fundraising schedules in the following problem.
Problem
The Student Council must decide between two companies tendering to supply bottled water in a vending machine for the cafeteria. They want to make sure they get the “best deal.” On offer are the following
· Each month, Moose Country Water will pay the school 5 cents for every bottle sold after the first 1000 bottles.
· Northern Crystal Water will pay 7 cents a bottle after 2000 have been sold each month.
Which company should the Student Council go with to raise as much money as possible?
1. Write a word equation and an algebraic equation to describe the relationship between funds raised and number of bottles sold under each company’s scheme.
2. Graph the equations and describe the graphs.
3. Use the graphs to compare the fundraising possibilities under each scheme.
4. Based on the graphs, what advice would you give to the Student Council about which company they should choose? Explain the significance of the point of intersection of the graphs for the two companies.
5. Write your recommendations to the Student Council in the form of a brief report. Include details of your graphical analysis.
Now consider
the following development:
Moose Country Water, in an effort to secure the contract, offers an additional incentive of a $50/month donation to the school fund. On hearing about this, Northern Crystal immediately responds with a matching offer of a $50/month donation. How do these additional payments affect (a) the equations and graphs of funds raised vs. number of bottles? (b) your recommendations to the Student Council?
Teacher Facilitation: If students are struggling with Questions 3 and 4, prompt them with probing questions like "Which company should they choose if, on average, 2500 bottles of water are bought from the machine each month? Which company should they choose if, on average, 5000 bottles are bought each month? It is recommended that students use estimation skills to develop an initial response.
Extension: Have students investigate then describe, graphically and algebraically, some strategies that each company could apply to improve their chances of winning the contract to supply water to the school.
Teacher Facilitation: Students may need more support with this part. They should notice that both lines shift up by the same amount, but that where they intersect, the number of bottles sold is the same as before the incentive donations were added.
|
Follow-up Skills: 75 minutes · finding the intersection of lines using graphing calculators or graphing software; solving contextual problems |
Assessment/Evaluation Techniques
· Assessment in the Learning skills areas of independence and initiative is possible as students work on the activity.
· Individual student written reports will provide evidence of learning in all four Achievement Chart categories and could be evaluated using a rubric.
Resources
This activity is an adaptation of an example in “Algebra in a Technological World”, NCTM Addenda Series
Activity 1.3: The Calculator Workshop
Time: 75 minutes
Description
The analysis of production costs of two calculator models serves as a context for discussion of linear equations in the form Ax + By = C. Students will then carry out a graphical investigation which will lead to solution of equations by the elimination method.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
- an effective communicator who reads, understands, and uses written materials effectively;
- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;
- a collaborative contributor who works effectively as an interdependent team member.
Strand(s): Analytic Geometry
Overall Expectation
AGV.01D - model and solve problems involving the intersection of two straight lines.
Specific Expectations
AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation;
AG1.02D - solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;
AG1.03D - solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.
Planning Notes
· Graphing calculators or graphing software may be used for this activity. Also, since students will be working with equations in the form Ax + By + C = 0, software such as Zap-a-Graph would be useful, especially for the exploration in the second part of the activity. Some work with graph paper and pencil should be encouraged.
· Plan to have students work in pairs so that they may discuss what they see as they progress through the exploration.
Prior Knowledge Required
Analytic Geometry
· graphing lines by hand (using a variety of techniques) and using technology
Number Sense and Algebra
· solving first-degree equations, including equations with fractional coefficients, using an algebraic method
Teaching/Learning Strategies
Student Activity:
An electronics company is producing two types of calculators, the PC1900 (the cheaper model) and the RC2000 (more expensive model). Two machines are required to make these calculators (Machine S and Machine T). The PC1900 requires 1 minute at machine S and 5 minutes at machine T. The RC2000 requires 6 minutes at machine S and 2 minutes at machine T. Machine S is available for 360 minutes per day and Machine T is available for 400 minutes per day (they are required for other purposes throughout the day).
1. Complete each chart:
|
PC1900 |
|
RC2000 |
||||
|
Number of Calculators |
Time on Machine S |
Time on Machine T |
|
Number of Calculators |
Time on Machine S |
Time on Machine T |
|
1 |
|
|
|
1 |
|
|
|
2 |
|
|
|
2 |
|
|
|
3 |
|
|
|
3 |
|
|
|
4 |
|
|
|
4 |
|
|
|
. . . |
|
|
|
. . . |
|
|
|
x |
|
|
|
y |
|
|
2. The bottom row of the above charts gives you the time for the day’s production. Use this information to complete the following chart:
|
|
Time
on Machine S |
Time
on Machine T |
|
PC1900 |
|
|
|
RC2000 |
|
|
|
Total Time Available |
|
|
3. Let x represent the number of PC1900s and y represent the number of RC2000s. State the production equations for each machine. It will be in the form Ax + By + C = 0.
Machine S: ____________________
Machine T: ____________________
4. Graph the above equations (you may use graphing software). What do the points on each line represent?
5. At what point do these two lines intersect? What does this intersection point mean?
Teacher Facilitation: When most students have completed this part of the activity, the teacher should take up the findings so far, as a whole class discussion. Emphasis should be placed on:
· the Ax + By + C = 0 form of the equation of a straight line.
· interpretation of the point of intersection as the solution that satisfies the necessary conditions for both Machine S and Machine T.
Student Activity
Note: This activity is intended to use technology! Do not graph by hand!
Investigating Intersections
In this activity you will be discovering an algebraic process to find the intersection point. Let’s first look at some simple cases:
1. 7x - y = 2 2x - y = -3
(I) Graph the above equations.
(II) The coordinates of the point of intersection are:
(III) Add together the two equations above.
(IV) Graph the new equation on the same set of axes.
(V) What do you notice about these three lines?
2. 8x + 5y = 1 3x + 2y = 1
(I) Graph the above equations.
(II) The coordinates of the point of intersection are:
(III) Add together the two equations above.
(IV) Graph the new equation on the same set of axes.
(V) What do you notice about these three lines?
3. x - 4y = 6 -x - y = 4
(I) Graph the above equations.
(II) The coordinates of the point of intersection are:
(III) Add together the two equations above.
(IV) Graph the new equation on the same set of axes.
(V) What do you notice about these three lines?
4. In your own words, describe what is always true about the graph of the resulting equation when you add two equations together.
5. 2x + 3y = 6
(I) Graph the above equation.
(II) Multiply
each term of the equation by 5.
Graph the new equation on the same axes.
(III) Multiply
each term of the equation by -4.
Graph the new equation on the same axes.
(III) Describe what you noticed about these graphs.
6. 2x + y = 3 Î
x + y = 1 Ï
(I) Graph these equations.
(II) Graph Î + Ï
(III) Graph Î + 2 ´ Ï
(IV) Graph Î + (-2) ´ Ï
(V) Describe what you notice about each of these lines.
(VI) Which lines, other than the original two, are the most important?
7. 3x
- y = 2 Ð
x
+ 2y = 10 Ñ
(I) Graph the above equations.
(II) Graph Ð + Ñ
(III) Graph 2 ´ Ð + Ñ
(IV) Graph Ð + (-3) ´ Ñ
(V) Which lines, other than the original two are most important?
8. Without graphing, find the point of intersection of the following pairs of lines:
(I) 2x
+ 3y = 6 (II) x + 3y = -1
2x + y = -4 2x
-y = 12
9. Recall the calculator problem you solved graphically:
An electronics company is producing two types of calculators, the PC1900 (the cheaper model) and the RC2000 (more expensive model). Two machines are required to make these calculators (Machine S and Machine T). The PC1900 requires 1 minute at machine S and 5 minutes at machine T. The RC2000 requires 6 minutes at machine S and 2 minutes at machine T. Machine S is available for 360 minutes per day and Machine T is available for 400 minutes per day (they are required for other purposes throughout the day). Now use the algebraic method to find the numbers of each type of calculator you can produce in a day.
10. Revisit
the Bottled water problem
The Student Council must decide between two companies tendering to supply
bottled water in a vending machine for the cafeteria. On offer are the following:
· Each month, Moose Country Water will pay the school 5 cents for every bottle sold after the first 1000 bottles.
· Northern Crystal Water will pay 7 cents a bottle after 2000 have been sold each month.
Use an algebraic method to determine the point at which the amount of money raised would be the same for both companies.
Teacher Facilitation: Initially students will need help with the concept and process of adding two equations.
Extension
· Ask students to investigate the number of solutions to a linear system. What are the conditions for a system to have no solution? one solution? an infinite number of solutions?
· An investigation of linear programming techniques would provide another extension.
|
Follow up-Skills: 255 minutes · Students will need to practise this algebraic method of solution by elimination. The teacher should provide examples of systems for students to solve, as well as more problems in which it is necessary to formulate the system first. Following practise with elimination, students should solve systems by the algebraic method of substitution. Again, realistic applications should be provided. |
Assessment/Evaluation Techniques
· A paper and pencil quiz which contains some questions of the type in Question 8 above and a communication component in which students describe the algebraic method of solution they have discovered in the exploration.
Accommodations
· Pair students with reading or writing difficulties with other students who will be able to help them.
Resources
The above exploration is an adaptation of an activity in “Activities for Active Learning and Teaching” NCTM.
Activity 1.4: Battle of the Bands – Introduction to Quadratics
Time: 75 minutes
Description
In this activity, students will multiply linear expressions to obtain a quadratic expression. Students will also use the properties of a quadratic relationship to solve optimization problems. They will use the regression menu from the graphing calculator to find the curve of best fit and find that the product of binomials is also an expression in the form of ax2 + bx + c. The follow up to this activity will be expanding and simplifying polynomial expressions. The area high schools have decided to continue a band competition which was started last year. The High School Band Promoter has decided to change the ticket prices in order to maximize his revenue needed to pay for the advertising display in a mall close to the auditorium which he has rented for this event. He sets up spreadsheets to determine the best price to charge and the maximum floor space he can use at the mall.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
- an effective communicator who reads, understands and uses written materials effectively;
- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;
- a collaborative contributor who works effectively as an interdependent team member;
- a responsible citizen who accepts accountability for one’s own actions.
Strand(s): Quadratic
Functions
Overall Expectations
QFV.04D - solve problems involving quadratic functions.
Specific Expectations
QF3.02D - fit the equation of a quadratic function to a scatter plot, using an informal process and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software;
QF4.02D - determine the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software.
Planning Notes
· This activity should be done in pairs so students can discuss the problem and share ideas; however they should present an individual report with the aid of computer programs and/or graphing calculators.
· Graph paper is needed for this activity.
Prior Knowledge Required
Relationships
· How to use a graphing calculator or graphing program on a computer.
· Construct tables of values and graph a nonlinear relation derived from descriptions of realistic situations.
Teaching/Learning Strategies
Teacher Facilitation: Introduce the situation about finding the proper price to charge or amount of space to be used in mall as not a random choice. Point out that there is a considerable amount of research and mathematics which is used to decide on proper pricing or the amount of floor space allotted to any display. A guest speaker from some retail consultant firm would be helpful in talking about this to students and allowing them time to investigate possible careers and jobs in this area. After some meaningful discussion, the activity can be presented to small groups of two or three students. Make the students aware that they will present an individual report.
Student Activity
Battle of the Bands – Student Handout
1. Doug, who is the promoter of this event has rented an auditorium to allow high school bands an opportunity to show their talent. Usually this event attracts 1000 people at $15 for each person. At this price, all the tickets were sold last year. Doug decided to set up a display at a local mall to encourage students to participate and register their bands. To pay for this display Doug was forced to increase the price of the ticket. In order to help him decide what would be a fair price to charge, he conducted a business survey and found that the number of tickets sold will decrease by 50 for every dollar increase.
a) Form a hypothesis about the best price to charge.
b) Complete the chart in your report.
|
Number of Increases |
Ticket Price |
Number of Tickets Sold |
Revenue |
|
0 |
|
|
|
|
1 |
|
|
|
|
2 |
|
|
|
|
3 |
|
|
|
|
4 |
|
|
|
|
. . . |
|
|
|
|
x |
|
|
|
c) State the revenue equation. Revenue = ( ) ( )
d) Use a graphing program and/or graphing calculator to create a graph of Revenue vs. Number of the Increases. Include the graphs in your report. Describe the shape of the graph in words.
e) What price should Doug charge for a ticket to maximize the revenue? Explain how you found the best price.
f) How many people would he expect to be in the auditorium with the new ticket price?
g) What is the maximum revenue? Explain how this can be found using the graph.
h) How much profit was made by the new price arrangement?
i) Enter the data from the chart in part b) L1 as Number of Increases and L2 as Revenue to produce a scatter plot on the calculator. Using the regression menu on your calculator find the equation of the curve of best fit through the points.
j) Compare the two equations from part c) and part i). Use your graphing calculator to input the two equations. Do they represent the same curve? Explain why.
2. The local mall has given Doug a rope of 8 metres for the perimeter of his display and will allow him to place the display in one of two places in the mall. In the north end of the mall, there is only one wall. So he would have to use the rope for the other three sides. The south end of the mall, has an area which consists of two walls and the rope would be used for two sides.
a) Form a hypothesis about the best location and the best dimensions of the display area.
b) For each location set up a table with the following headings: Length of Display, Width of Display, Length of Rope, Area of Display. Include a line for the general case in your chart by letting x be the length of the display. Include at least three diagrams of the display for each mall location.
c) State the area equation for each location. Area= ( )( )
d) For each location, construct a graph Area vs. Length of Display.
e) Which location gives the maximum area for the display. Explain how you decided which location was the best.
f) What is the maximum area? State the dimensions for the display.
g) Enter the data to produce a scatter plot on the calculator. Using the regression menu on your calculator, find the equation of the curve of best fit for both curves.
h) Input the two equations for the northern location on the calculator. Do they represent the same curve?
i) Input the two equations for the southern location on the calculator. Do they represent the same curve?
j) Why do two equations written in different form give the same curve?
|
Follow-up Skills: 75 minutes · QF1.01D - Expand and simplify second degree polynomial expressions |
Assessment/Evaluation Techniques
· Through observation, make anecdotal comments on independent work, teamwork, organizational skills, work habits, communication and initiative.
· A written report rubric (Appendix D) could be used to evaluate such areas as the clarity of communication and correctness of computation.
Accommodations
Place students having difficulties with written work or language with students who will assist them. Extra time may be given for students demonstrating difficulties in language skills.
Activity 1.5: Summative Assessment
Time: 75 minutes
Description
Students apply the skills they have developed during this unit to solve a problem about prices of two Driver’s Education courses that can be modelled by a system of linear equations. As well as the activity, students will write a paper and pencil test of skills.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
- an effective communicator who reads, understands, and uses written materials effectively;
- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;
- a collaborative contributor who works effectively as an interdependent team member.
Strand(s): Analytic Geometry
Overall Expectation
AGV.01D - model and solve problems involving the intersection of two straight lines.
Specific Expectations
AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation;
AG1.02D - solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;
AG1.03D - solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.
Planning Notes
· The summative assessment for the unit should reflect the type of activities that were conducted during the unit. Thus, the assessment of this unit should include two parts:
· a realistic scenario that involves students working in pairs or groups to solve a linear system
· a paper and pencil test.
· A variety of application questions that require solving a linear system can be found in the textbook. As well, some of the suggested extensions in this unit could also be used as assessment activities.
Prior Knowledge Required
· All the skills developed in this unit for using Linear Systems to solve problems
Teaching/Learning Strategies
Teacher Facilitation: Present the following problem to the class:
The Safe-T-First Driver’s Ed Company offers two different packages of driver’s education courses: group classes or private lessons. The number of hours of classroom instruction and the number of hours of in-car driving instruction are the same for both groups, but the two packages are priced differently.
Group Lessons Package: Student lessons cost $10 per classroom hour plus $6 per driving hour, for a total cost of $170.
Individual Lessons Package: Private lessons cost $30 per classroom hour plus $20 per driving hour, for a total cost of $550.
Do each of the courses offer enough in-car driving time to provide a safe and complete driving course?
Provide a graphical solution to the problem, and support your answer using an algebraic solution.
Student Activity
Students work in pairs to discuss and solve the problem, completing individual solutions for submission at the end of the task.
Teacher
Facilitation: Circulate around the class and prompt students who
are having difficulty. Some students may choose to gather data on a table to
use for the graphical solution. Unless the students determine equations for each
package, they will be unable to support their graphical solution algebraically.
Students may need to be guided to use a system of two equations to determine
the number of hours that are planned for classroom instruction and in-car
instruction. Encourage students to use variables (h = number of classroom hours, d = driving hours) (Solution
uses two equations
10h + 6d = 170 and 30h + 20d = 550 to give 5 classroom hours and 20 driving
hours.)
Assessment/Evaluation Techniques
Assess the student’s written report using the rubric for Assessing Written Reports (Appendix D) with the following criteria placed in the stated categories in the left column of the rubric:
· Assess Thinking/Inquiry and Problem Solving using Expectation AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation.
· Assess Written Communication using Expectation AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions.
· Assess Application using Expectation AG1.03D - solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.
Sample Questions For the Paper and Pencil Test
1. Below are three systems of equations:
A) y = - 6x + 10 (B) 4x – 3y + 24 = 0 (C) 4x + 5y + 10 = 0
y = 
x + 7 2x
+ 5y + 50 = 0 y
= 3x + 5
Do not solve these systems! State which method (elimination, substitution) would be most appropriate to solve each system. Explain why you chose each method.
2. (i) Develop a system that has:
(a) exactly one solution;
(b) no solution;
(c) an infinite number of solutions.
(ii) For each system you have developed in (i) describe a realistic situation that would lead to the system. Explain the meaning of the solution of each system in the context of the situation.
3. Two rental stores have a weekend special on posthole diggers;
U-Dig-It store charges $50 for a damage deposit on its equipment plus $10 for each hour
ABC Rental charges a $60 damage deposit and $9 for each hour.
a) Using two different methods, find the number of hours for which the charge is the same?
b) If you are using the equipment for only 6 hours, which rental store would be the best deal? Explain your choice.
c) Which algebraic method do you use and why?
4. An electrical company charges a flat fee for a small electrical job, plus an hourly rate. A 2-hour job costs $76. A 6-hour job costs $130. What is the flat rate and the hourly rate charged by this company?