Course Profile College and Apprenticeship Mathematics (MAP4C), Grade 12, College Preparation, Combined

Unit 5: Modelling

Time: 18 hours

Unit Description

Students examine and work with a variety of mathematical models in various forms: tables, graphs, and formulas. Students investigate linear, quadratic, and exponential models in a context that allows them to develop an understanding of how models are used and created. Connections are made to occupations, such as game warden and personal trainer, and sectors, such as travel and tourism, construction, municipal planning, and business.

Unit Synopsis Chart

Activity	*Time	Learning Expectations	Assessment Categories	Tasks
5.1 Math Super Model	1.25 hours	MMV.01, MMV.03, MM1.01, MM1.03, MM3.04	Knowledge/ Understanding Communication	Examine and make predictions from a variety of models: linear, quadratic and exponential graphs and tables
5.2 See the Light	2.5 hours	MMV.01, MMV.03, MM1.01, MM1.03, MM1.05, MM3.04	Application Communication	Gather and graph light intensity data and work with exponential models
5.3 We’re Movin’ On Up	1.25 hours	MMV.01, MMV.02, ASV.04, MM1.01, MM1.03, MM2.01, AS4.03	Thinking/Inquiry/ Problem Solving Communication	Examine CPI as an example of exponential growth, create and analyse graphical and formulaic models, and make predictions
5.4 Tracing Your Roots	1.25 hours	MMV.01, MMV.03, MM1.01, MM1.03, MM1.04, MM3.04	Thinking/Inquiry/ Problem Solving Application	Make predictions, analyse quadratic graphs, and find roots graphically or by examining table of values
5.5 Stop Right Now	1.25 hours	MMV.01, MMV.03, MM3.04, MM1.03, MM1.04	Application Communication	Graph data on stopping distances under various conditions, analyse model, and predict how model would change if conditions change
5.6 Getting to the Root of the Problem	2.5 hours	MMV.01, MMV.02, MMV.03, MM2.06, MM2.07, MM3.04, MM1.05	Knowledge/ Understanding Application	Create graphs from a given quadratic equation for various applications, factor, find roots, and make decisions
5.7 The Ultimate Fan	2.5 hours	MMV.03, MM3.01, MM3.02, MM3.03	Thinking/Inquiry/ Problem Solving Application	Plan a trip to visit to see various sports teams play in their hometown while meeting budget constraints
5.8 All Hands on Deck – Summative Assessment	2.5 hours	MMV.01, MMV.02, MMV.03, MM1.05, MM2.01, MM2.02, MM2.04, MM3.03, MM3.04	Application Thinking/Inquiry/ Problem Solving Communication	Choose appropriate models to create a formula for calculating costs for building a deck

* An additional 3 hours are available for skill consolidation.

Activity 5.1: Math Super Models

Time: 1.25 hours

Description

Students model three different situations. Using graphing technology, students interpolate and extrapolate to answer questions posed for these scenarios.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM1.01 - interpret a given linear, quadratic, or exponential graph to answer questions, using language and units appropriate to the context from which the graph was drawn;

MM1.03 - make and justify a decision or prediction and discuss trends based on a given graph;

MM3.04 - enter data or a formula into a graphing calculator and retrieve other forms of the model (e.g., enter data and retrieve a scatter graph or a table of values; enter a formula and retrieve a table of values or the graph of a function).

Prior Knowledge & Skills

· use of graphing technology to create scatter plots;

· use of the regression features of the calculator (knowledge from previous units in this course. teachers may need to spend more time on this if they approach the course in a different order);

· interpolation and extrapolation.

Planning Notes

· Have a class set of graphing calculators available for this lesson.

Teaching/Learning Strategies

Student Activity

· Use graphing calculators to create scatter plots and graphs;

· Interpolate and extrapolate from graphs to interpret and analyse models.

Teacher Facilitation

· Introduce the topic of the activity – The Use of Models.

· Review with students the use of the List and the Regression features of the calculator. The teacher may want to create a reference sheet that students can refer to throughout the unit.

· Discuss the importance of r² in determining the appropriateness of a model. Students look for models with r² as close to 1 as possible.

· The teacher must turn on the Diagnostic function of the calculator to get r².

· Create one list of data (in the first quadrant only) that produces a cubic graph.

· Using the graphing calculators, demonstrate the use of the regression feature in testing linear, quadratic, exponential, and cubic relationships. Students notice that the cubic regression is the only one producing an r² of 1. (Be careful to have the precision on the calculators in floating mode as rounded values cause students to make incorrect choices.)

· Have students work through the sample worksheet to investigate the three models.

· The teacher may use cooperative learning groups. For example, use a jigsaw strategy, in which a student from each group learns a particular model, then uses it as an example to teach the other students in their home group.

Sample Worksheet

Mathematical modelling is important in many professions. A mathematical model provides a way of analysing what is happening and helps in making predictions. Presenting a collection of data in chart or co-ordinate form is one way to model a situation.

Use the data given in the three scenarios to produce scatter plots, determine the type of model demonstrated in each example (remember to look for r² = 1), and then answer the questions based on the graph you created.

Scenario 1 – Rescue 911

In a rural community, fire and paramedic crews must service a large region. They often cover two or three towns several kilometres apart. These crews travel, on average, 70 km/h to their destination. The Chief of Station 42 wants to determine the time required to reach the patient’s home (response time) for his region. The station is located next to the local hospital. The Chief knows the distances to each community that he services, including Deerborn, the furthest community, at 55 km away. Using the data, create a scatter plot showing the relationship between distance to patient and response time.

Distance to Patient (km)	Response Time (minutes)
10	8.6
15	12.9
40	34.3
50	42.9
55	47.1

The Chief wants to analyse several situations quickly so he decides to use a graph. Using your scatter plot, answer the Chief’s questions:

1. Mrs. Klein lives 52 km from Station 42; determine the approximate response time.

2. If the crew took 15 minutes to arrive at a patient’s home, what distance have they travelled?

3. If someone is having a heart attack and must make it to the hospital in 30 minutes, what is the furthest distance the crew can travel and still arrive back at the hospital 30 minutes after the call is made?

4. A second dispatch location is built in Deerborn. Both stations cover the areas previously serviced by Station 42 and take their patients to the hospital located beside Station 42. Describe how this would change the original graph. (When a call comes in, emergency crews leave from a dispatch location.)

Scenario 2 – Surf’s Up

A weather reporter makes predictions based on data collected by the National Weather Centre. These predictions can be very important to different people and professions. For instance, the wind speed has a direct effect on the height of waves in the ocean where many fishermen (anglers) work everyday. It could also be of importance to a ferry boat company taking people to the 1000 Islands near Kingston, or for parents planning a fishing trip with their children.

The data below is an example of the monthly averages for the speed of peak gusts on Iles de la Madeleine in Quebec between 1984 and 1993. Create a scatter plot showing the relationship between the month and the mean peak gusts (knots). (*A gust is a quick burst of wind; the peak gust would be the maximum speed it reaches in that burst.)

Months	Mean (average) Peak Gusts (knots)
1 – January	59
2 – February	52
3 – March	48
4 – April	46
5 – May	42
6 – June	41
7 – July	41
8 – August	42
9 – September	44
10 – October	48
11 – November	56
12 – December	57

Using your graph, answer the following:

1. Which month(s) have the lowest average peak gust speeds?

2. Which month has the highest average peak gust speeds?

3. Any gusts of wind with speed greater than 50 knots can be dangerous to small crafts. Draw a line on your graph showing this speed. State all months with average peak gusts greater than 50 knots.

4. If it was discovered that November’s average peak gusts were 5 knots below the mean this year, would you, as a fisherman, go out on the lake and do some late season fishing? Justify your response.

Scenario 3 – Gas it Up!

A consumer activist group wants the government to regulate gas prices. They investigated a 10-week period, looking at the price of gas each Monday. They discovered that gas had been going up by 1.0% each week. In the first week, gas was $0.65/L. Assuming the price of gas continued to increase at this rate, create a scatter plot showing the relationship between the week and the cost of gas per litre, over
the 10-week period.

Week Number	Price per Litre of Gas
1	0.65000
2	0.65650
3	0.66307
4	0.66970
5	0.67639
6	0.68316
7	0.68999
8	0.69689
9	0.70386
10	0.71090

Using your graph, answer the following:

1. Assuming the price keeps rising at this rate, determine when gas will cost $1.00/L.

2. In week 8, John needs fuel. If his tank is empty and can hold 50L, how much will it cost him?

3. A tour group planner creates a budget for a bus trip to Stratford to see a play. They plan on
spending $100 on fuel for the bus. If the bus requires 75L for a one-way trip to Stratford, what is the most expensive price per litre they can pay for gas and still stay within their budget?

4. If they want to take the trip in the 8th week, can the tour group stay within their budget? Explain.

Assessment & Evaluation of Student Achievement

As this is the first activity of the unit, a teacher may want to discuss the solutions with the class rather than collect the work. This early stage is a good chance to provide diagnostic assessment. Student responses could be informally assessed for Knowledge/Understanding, Application, and Communication to provide the teacher with an idea of students’ background knowledge.

However, if the teacher collects student work, the worksheet could be assessed for Knowledge/Understanding, Communication, Application, and Problem Solving:

· In Scenario 1, Questions 1 and 2 focus on knowledge of the basic concepts, while Questions 3 and 4 require students to problem solve with linear equations as well as to communicate their responses.

· In Scenario 2, Questions 1 – 3 demonstrate students’ understanding of the concepts and properties of quadratic graphs, while Question 4 asks students to communicate and reason an answer based on the model given.

· In Scenario 3, Question 1 focuses on students’ skills in extrapolation (Knowledge/Understanding), Question focuses on application, while Questions 3 and 4 focus on problem solving situations requiring students to make a series of choices.

The teacher may wish to create a rubric to mark the communication and application questions in the worksheet. It may be easier to assess the knowledge using a more traditional marking scheme.

Accommodations

Students’ skills in using the graphing technology may vary. It may be appropriate to pair or group students accordingly, as the focus of the investigation should be on the discovery of the mathematical models.

Resources

Environment Canada Website – http://weatheroffice.ec.gc.ca/index_e.shtml

OAME/OMCA “CARE Package” (download) – http://www.oame.on.ca

Activity 5.2: See the Light

Time: 2.5 hours

Description

Students gather data on the effect of layers of tinting on light intensity. Graphs are created and students interpolate and extrapolate data from the graphs. Connections are made to the tinting on vehicle windows and windows on buildings. Further connections can be made to the change in light intensity with water depth.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM1.01 - interpret a given linear, quadratic, or exponential graph to answer questions, using language and units appropriate to the context from which the graph was drawn;

MM1.03 - make and justify a decision or prediction and discuss trends based on a given graph;

MM1.05 - communicate the results of an analysis orally, in a written report, and graphically;

MM3.04 - enter data or a formula into a graphing calculator and retrieve other forms of the model.

Prior Knowledge & Skills

· entering and graphing data using a graphing calculator.

Planning Notes

· Obtain tinted plexiglas (MIRA’s will work) or coloured acetates and an incandescent light source.

· Students need graphing calculator, light sensor, Computer-based laboratory (CBL) and program for gathering light, or a CBL 2. Groups could take turns if only one is available or the teacher could do a demonstration and have students transfer data onto their own calculators.

Teaching/Learning Strategies

Teacher Facilitation

Introduce the activity with a discussion about the various uses and issues around window tinting.

· Tinting is used on car windows to reduce glare.

· Many provinces/states have regulations about the degree of tinting allowed on vehicle windows.

· Studies have shown that drivers of vehicles equipped with tinted windows were more likely to engage in aggressive driving than other drivers.

· Tinting is used on office windows to keep some of the sunlight out in order to reduce air conditioning costs in the summer.

Students work in groups to gather data for Part 1, or complete it as a demonstration and then share data with their peers.

Tips for setting up the demonstration:

· Use an incandescent light bulb.

· Use the DataMate program with the CBL 2 (preferable) or use the BULB program with the CBL.

· For help with the DataMate program, refer to Resources.

· The instructions on the BULB program are written for gathering light data at changing distances; students should ignore these instructions and instead add layers of “tinting” after each reading.

· The BULB program gathers more data than necessary. The teacher or students should remove the extra data after the experiment is performed (i.e. you will likely only use five layers of “tinting” but the program will take 11 readings; the last six readings should be deleted).

Students could work in groups to analyse the model and then present their solutions to the class on an overhead. If groups come up with different models they can be compared.

Student Activity

Part 1: Gathering Data

Students:

· turn the class lights off and turn on the light source (preferably a single light bulb);

· make the appropriate connections between the light sensor, the CBL or CBL 2, and the calculator;

· start the program to read light intensity;

· place the light sensor at a fixed distance (about 1–2 m) from the light source;

· make a prediction about the change that will occur as each layer is added;

· place an increasing number of layers of tinted plexiglas directly in front of the light SENSOR and measure the light intensity;

· graph the data and make a prediction about whether an exponential, quadratic, or linear model would be best;

· determine an appropriate model for the data (i.e., linear, exponential, quadratic) using the regression feature of your calculator or by “guess and check”.

Part 2: Analysing Data

1. In one province, the law states that any tinting on the side or rear windows of a vehicle must have a light transmittance of 35% or more. Using your model, how many layers of the tinting you examined would be acceptable under this law?

2. A construction company building an office has been requested to tint the windows to lower air conditioning costs. If the tinting applied reduces light in the same way as your tinting, how many layers would be required to reduce light transmittance by 40%? by 65%?

Part 3: Making Connections

In the ocean, light intensity decreases with depth following a similar model to the tinting model. The “layers” of water act like layers of tinting. The table below shows data for the per cent of light transmitted at various depths of the ocean on a sunny day.

Depth (m)	% of Light Transmitted
0	100
75	90
150	81
225	72.9
300	65.6
375	59

1. Graph this data and compare the graph to your graph from Part 1. How are they the same? How are they different?

2. At approximately what depth would there only be 5% of light transmitted? Note: you will need to change the WINDOW of your graph to extrapolate these values.

3. If a certain species of fish requires at least 50% light transmitted, what would be the greatest depth it could live in?

4. If a certain species of fish does not respond well to more than 40% light transmittance, what depth of water should it live in?

5. Who might find this data useful? Give examples of people who could use this information either in their jobs or in their lives outside of their jobs.

Teacher Facilitation

Assist students in making the connection between tinting and the changes in light intensity at various depths of the ocean. Other applications may include:

· creating environments with appropriate lighting for various fish species in large-scale aquariums;

· scuba diving.

Assessment & Evaluation of Student Achievement

Parts 1 and 2 could be completed as a class or in groups and taken up by the teacher or students. Individual reports for Part 3 can be submitted and assessed.

· Assess Application based on the student’s ability to apply the model to answer Questions 2, 3, and 4.

· Assess Communication based on the student’s answers to Questions 1 and 5.

Resources

Websites

San Jose University – http://geosun1.sjsu.edu/~dreed/105/exped7/13.html
(Provides data on changing light intensity with water depth.)

NC State University – http://www.ncsu.edu/sciencejunction/route/usetech/index.html
(Provides instructions for using the CBL 2.)

Brueningsen, C., B. Bower, L. Antinone, and E. Brueningsen-Kerner. Real-World Math with the CBL System: Activities for the TI-83 and TI-83 Plus. Texas: Texas Instruments Incorporated, 1999.
ISBN 1-886309-28-0

Randall, Jack. Sensor Sensibility. Berkeley: Key Curriculum Press, 1997. ISBN 1559532874

Activity 5.3: “We’re Movin’ on Up”

Time: 1.25 hours

Description

Students examine CPI (Consumer Price Index), including how it is calculated and its use in measuring inflation. Inflation rates are used to compare prices and salaries from various time periods.

Strand(s) & Learning Expectations

Strand(s): Applications of Statistics, Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.02 - interpret and analyse given formulaic models;

ASV.04 - evaluate the validity of the use of statistics in the media.

Specific Expectations

MM1.01 - interpret a given linear, quadratic, or exponential graph to answer questions, using language and units appropriate to the context from which the graph was drawn;

MM1.03 - make and justify a decision or prediction and discuss trends based on a given graph;

MM2.01 - evaluate any variable in a given formula drawn from an application by substituting into the formula and using the appropriate order of operations on a scientific calculator;

AS4.03 - explain the meaning and the use in the media, of indices based on surveys.

Prior Knowledge & Skills

· use of compound interest formula to calculate A;

· enter and graph data using a graphing calculator.

Planning Notes

· Prepare a review of the compound interest formula.

Teaching/Learning Strategies

Teacher Facilitation

· Discuss how the Consumer Price Index (CPI) is a measure of the rate of price change for goods and services bought by Canadian consumers.

· Explain how it is obtained by comparing, through time, the cost of a fixed basket of goods and services.

· CPI is used to determine increases in Old Age Security pensions, CPP payments, other social and welfare payments, spousal and child support payments, and wage increases.

Student Activity

Your uncle tells you “…back in 1971, when I was a kid, it only cost me $1.50 to go see a movie.” The cost seems pretty cheap compared to today’s prices. The chart contains CPI data for the years 1971
to 1980.

Year	CPI for All Items	% Change from Previous Year (round to 1 decimal place)
1971	24.9	>
1972	26.1	>
1973	28.1	>
1974	31.1	>
1975	34.5	>
1976	37.1	>
1977	40.0	>
1978	43.6	>
1979	47.6	>
1980	52.4	>

1. Create a scatter plot of the data, CPI vs Year, on a graphing calculator.

2. Examine your graph and make a prediction about the type of growth this appears to be. Linear, quadratic, exponential? How do you know?

3. Complete the % Change column (e.g., % change = (CPI for 1972 – CPI for 1971)/(CPI for 1971)).

4. Find the average percentage increase in prices over these years.

5. Assume the prices continue to increase at the same rate and use this rate as an approximate rate of inflation for the price of goods in Canada. Use this value to calculate what the $1.50 movie should cost now. Use the compound interest formula to help you.

A = P(1 + i)ⁿ

in this case:

A – is the new price

P – was the price in 1971

i – is the average annual inflation rate

n – is the number of years

6. How does your answer from Question 5 compare to the current price of movies? Based on this data, does the current price seem reasonable?

7. In this example, we used an average inflation rate based on the years 1971 to 1980. Why might this give us an inaccurate picture of inflation?

8. Using CPI data from 1971 to 2000, an average annual inflation rate of 5.4% was calculated.

a) Use this average to calculate what the cost of a movie should be today. How does this value compare to the actual price?

b) Assuming this inflation rate remains the same, what will the price of a movie be in 10 years? 20 years?

Extension 1

Choose a profession and investigate the increase in income in this profession for past years. Make a comparison between the rate of increase of wages with the rate of increase of CPI. Write a report to summarize your findings and offer possible explanations for any differences in the rates.

Problem: Suppose that your parents saved $30/month for your college education, beginning in the month you were born. An investment counsellor has managed to earn about $5,000 interest in total with that money over the last 17 years. This year it costs about $4,000/year to attend your local college; this covers tuition and books. You are planning on starting college next year. You have heard that inflation will be 5.6 % per year. If so will you have enough money to pay for your college education? State any assumptions you are making when working with this problem.

Extension 2

Students investigate college fees over the last decade and make predictions for the future, i.e., What will tuition for their children be?

Assessment & Evaluation of Student Achievement

· The college funding problem could be used as a performance task and assessed for Application, Thinking/Inquiry/Problem Solving, and Communication.

· Use Rubric (Appendix A – Rubric for Evaluation of Report on Wage Increases) to assess Thinking/Inquiry/Problem Solving and Communication for Extension 1.

Accommodations

· Provide scaffolding for the problem and extension.

· Allow students to present their reports orally.

Resources

Websites

Bank of Canada – http://www.bank-banque-canada.ca/en/inflation_calc.htm (An inflation calculator which uses CPI)

Canadian Council on Social Development – www.ccsd.ca/fs_avgin.html (Income data for Canada)

Statistics Canada – www.statcan.ca (Income data)

Activity 5.4: Tracing Your Roots

Time: 1.25 hours

Description

Students find the roots of quadratics using the graph and its table of values. Using the data from the graph, students make decisions based on trends they discover.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM1.01 - interpret a given linear, quadratic, or exponential graph to answer questions, using language and units appropriate to the context from which the graph was drawn;

MM1.03 - make and justify a decision or prediction and discuss trends based on a given graph;

MM1.04 - describe the effect on a given graph of new information about the circumstances represented by the graph (e.g., describe the effect of a significant change in population on a graph representing the size of the population over time);

Prior Knowledge & Skills

· use of graphing technology to graph equations, trace graphs, and read tables;

· given the vertex form of the equation, identify the vertex.

Planning Notes

· Students should briefly review the vertex form of an equation.

· Have a class set of calculators available for this lesson.

Teaching/Learning Strategies

Student Activity

Students work through the activity to discover the meaning of roots in a practical problem.

Teacher Facilitation

· Review definitions of Grade 10 concepts. Students review their knowledge of terms, such as x-intercept, root, vertex, quadratic, etc. Students need these terms to complete the worksheet.

· Some students need more direction from the teacher than others, but the teacher should allow students to work independently first, as this is an important skill they will need in college or the workforce they intend to enter.

· Class work should include problems that ask students to graph (using technology) different problems. They should identify the roots to answer specific questions for each problem. Homework should include situations where the graphs are given so that students can complete these questions without technology.

Sample Worksheet

In many urban centres, the population is growing rapidly. As a result, more and more homes are being built. Builders create subdivisions in the neighbourhood. The hydro company provides the builder with a layout of the poles needed to hang the electrical wires. The builder uses the poles to provide temporary electricity to each home during the building process.

In one subdivision in Oakville, the following equation was created to model how the wire must hang between two hydro poles.

Note: y = amount of dip (distance below the horizontal)

x = location of pole
origin is at the first pole

Problem: How far apart are the two poles? Follow the directions to solve the problem.

1. To answer this question we would like to know the position of the second pole. That distance will indicate how far apart any two poles should be. The value we are trying to find is called a <<root>> (students fill in the blank).

2. Using your graphing calculator, graph the equation.

3. Using the TRACE feature, find the position of the second pole. What point(s) are you looking for on your graph? <<x-intercepts>>.

4. Another way to find the root is to use the TABLE feature and look for a y-value of <<zero>>.

5. What is the vertex of the equation? <<(p, q) from the equation in y = a (x-p)²+q. from>>

6. Identify the relationship between the distance between the poles and the vertex. Explain your findings. <<(5, 2) where 5 is half the distance between two poles and 2 is the depth of the dip>>.

7. This Oakville subdivision has a street that is 210 m long. How many posts would you place on the street? Assume the depth of the “dip” between poles is similar to the model.

8. Given the following guidelines:

· Poles cannot be more than 12 m apart and no less than 8 m apart.

· The depth of the dip cannot be more than 2.5 m and must not be taut (straight across).

Create a quadratic equation to model the positioning of poles on a street that is 115 m long.

Assessment & Evaluation of Student Achievement

The teacher could use the activity for diagnostic assessment of students’ knowledge of quadratics and their ability to apply those concepts to different scenarios. Use a similar activity to evaluate students but use this as an opportunity to provide helpful feedback, using the following guidelines:

· Questions 1–5 are knowledge based, helping students discover the real-life application of the roots of an equation. Question 7 is an application that asks students to work with the given equation and find a solution to practical problems. Questions 6 and 8 assess Thinking/Inquiry/Problem Solving because they require students to make a series of choices and/or provide reasoning in their answers.

Accommodations

· Pair students in ESL/ELD courses with students who have strong language skills for interpreting some of the more complicated questions.

Activity 5.5: Stop Right Now

Time: 1.25 hours

Description

Students use graphing calculators to create scatter plots and equations of speed vs. stopping distances on wet pavement. Students predict equations of graphs for other surfaces (dry pavement, gravel, packed snow, and ice).

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse graphical models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM3.04 - enter data or a formula into a graphing calculator and retrieve other forms of the model;

MM1.03 - make and justify a decision or prediction and discuss trends based on a given graph;

MM1.04 - describe the effect on a given graph of new information about the circumstances represented by the graph.

Prior Knowledge & Skills

· use of graphing calculators to create scatter plots;

· use of regression features on graphing calculators.

Planning Notes

· Graphing calculators are required for each student (for writing and estimation of equations, it would be best to have the decimal point set to two places).

Teaching/Learning Strategies

Student Activity

Part 1 – Wet Pavement

When the police arrive at the scene of a collision, they can make deductions about the speed of the vehicle from the length of the vehicle’s skid marks. Use the following data for stopping distances vs. speeds on wet pavement:

Speed (km/h)	48	64	80	96	112
Stopping Distance (m) on Wet Pavement	51	76	116	169	231

1. Create a scatter plot for stopping distances vs. speeds on wet pavement.

2. Using a differences chart to help you, guess an equation for a curve of best fit. Test your guess on the calculator.

3. Perform various regressions (linear, quadratic, exponential) for this data. Which regression appears to best fit the data? Use this regression to determine the equation of the curve of best fit. How does it compare to your guess in Question 2?

4. Using the trace function, obtain an estimated speed of a vehicle with a stopping distance
of 100 m, 216 m, and 300 m on wet pavement.

5. Use the equation to find the “exact” speed of a vehicle with a stopping distance of 100 m, 216 m,
and 300m on wet pavement.

6. Why is your answer to Question 5 not really exact? <<An equation is an “exact” answer only if the algebraic model is exact>>.

Part 2 – Predicting Distances on Dry Surfaces

1. Describe how the stopping distances will change if the surface is gravel? Dry pavement?

2. Predict the stopping distances for the same vehicle on dry pavement.

Speed (km/h)	48	64	80	96	112
Stopping Distance (m) on Dry Pavement

(Note: actual distances given below are not intended to be on the worksheet. Questions 1 and 2 could be discussed and then display actual distances for students to make comparisons.)

Speed (km/h)	48	64	80	96	112
Stopping Distance (m) on Dry Pavement	31	48	75	113	171

3. Create a scatter plot for stopping distances vs. speeds on dry pavement.

4. What model do you think would best fit this data?

5. Test several models by performing regressions for the data. Choose the best model and record the equation, explaining why that model is best.

Part 3 – Predicting the Equation for an Icy Surface

1. Graph the two equations from Part 1 and Part 2 simultaneously on the graphing calculator.

2. Compare the two graphs and describe the similarities and differences.

3. Compare the two equations and describe their similarities and differences.

4. Predict how the graph of stopping distances vs. speeds on icy surfaces will look compared to the previous two graphs.

5. Predict the equation for stopping distances vs. speeds on icy surfaces. Explain your reasoning.

6. Graph your predicted equation using the graphing calculator.

7. Determine the actual equation for stopping distances vs. speeds on icy surfaces using the following data:

Speed (km/h)	48	64	80	96	112
Stopping Distance (m) on Icy Surfaces	135	223	372	553	818

8. Graph the predicted and actual curves together.

9. Compare your prediction with the actual curve.

Part 4 – Class Discussion

1. In order for police to use this data, what assumptions have been made?

Discuss such items as size and mass of vehicle, condition of vehicle, number of passengers, type of tires on vehicle, driver competence, ABS brakes.

2. Why is the data unreasonable?

Discuss that it would be tough to create a 300-m-long skid mark and the fact that it would be tough to keep a skidding car straight for that long.

Assessment & Evaluation of Student Achievement

Application and Communication could be assessed by having students complete Part A, Part B, or both Parts A and B.

· Part A – Students predict their own data for the graph of stopping distances vs. speeds on packed snow. Students then create a scatter plot on graph paper and perform a quadratic regression on their data.

· Part B – Students use the data to create a scatter plot, perform a quadratic regression, and describe their confidence in the quadratic model.

Speed (km/h)	48	64	80	96	112
Stopping Distance (m) on Packed Snow	62	96	164	243	370

Accommodations

· Students could work in partners to assist each other.

Resources

Flagging Instructions (stopping distances)
– www.dot.ca.gov/hq/traffops/signtech/signdel/flagging/stopping.htm

Activity 5.6: Getting to the Root of the Problem

Time: 2.5 hours

Description

Students create graphs from a given quadratic equation. They then look for a way to find the roots algebraically. This will lead to factoring equations of the form ax² + bx + c = 0. After determining the roots, students analyse the equation and make decisions/conclusions.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.02 - interpret and analyse given formulaic models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM1.05 - communicate the results of an analysis orally, in a written report, and graphically;

MM2.06 - factor expressions of the form ax² + bx + c;

MM2.07 - solve quadratic equations by factoring;

Prior Knowledge & Skills

· factoring expressions of the form ax² + bx + c;

· solve simple linear equations (3x – 5 = 0).

Planning Notes

· Student will learn how to factor quadratics of the form ax² + bx + c to find roots. Review Grade 10 factoring techniques and solving of quadratic equations of the form 0 = ax² + bx + c.

Teaching/Learning Strategies

Student Activity

Students:

· graph a quadratic function on graphing technology to identify the roots of the corresponding equation;

· identify the roots by factoring equations in the form ax² + bx + c = 0;

· interpret the roots to answer questions based on the given equation.

Teacher Facilitation

Day 1

· Begin with a review of finding roots of a simple trinomial using the following example:

Sarah is watching her younger sister play with her friends outside. They are skipping. They have one end of a rope tied to a tree and the other is held by one of the girls. The rope appears to hang in a parabolic shape. The height of the rope, with a horizontal axis occurring where the rope is tied/held, m metres from a “spotter,” is closely modelled by the quadratic equation: h = m² – 6m + 5. Determine the position of the two ends of the rope (i.e., h = 0).

· Present students with the following problem (on an overhead):

A golfer, standing on an elevation 12 m above the hole, hits a ball into the air. The height of the ball in metres is given by h = -5t² + 17t + 12 where t is the number of seconds after it is hit. When does the ball reach the ground (h = 0m)?

Create a graph of the function. What special point(s) on the graph identifies the time when the ball reaches the ground? Describe how you came to that conclusion.

Is it always necessary to graph the function in order to find the roots? What might be some disadvantages to using a graph?

· Take the equation from the golfing scenario and use it to develop a lesson around factoring using decomposition (the teacher may also want to introduce other methods of factoring to accommodate different learning styles.)

· Once you have the factored form, you need to talk about how to SOLVE for the ROOTS. Once roots are determined you can analyse the graph (e.g., to solve (5t + 3)(t – 4) = 0, solve both 5t + 3 = 0 and
t – 4 = 0 to find the two answers for t). Students now reason to determine the time when the ball hits the ground.)

· Follow-up with examples out of context to consolidate the concept (e.g., solve 3x² -17x + 10 = 0).

· Provide students with factoring “drill” type questions from their textbooks for homework.

Day 2

· Students work through a practical example and then present or discuss their solutions.

Example: A specialty ice cream shop sells their 2-L ice cream pails for $7. They have decided to drop the price in an attempt to increase sales. The equation P = -5x² + 23x + 84, where P represents the Profit made in one week on the pails and x represents the number of $1 decreases, models this situation. If the company wants to make $108 profit on their ice cream pails during the next week, how much should they sell the pails for?

· Homework/class work on Day 2 may contain some out of context solving of questions but should FOCUS on problems within a context. (Students have practised the concept of factoring and should apply it to solve a practical problem.)

Assessment & Evaluation of Student Achievement

· A quiz on factoring to assess Knowledge would be beneficial to students after they have had an opportunity to practise this new skill.

· As this is the last day for quadratic models, it may be appropriate to have a short open-book formative assessment. It would be a good opportunity to provide students with feedback on the skills they have developed and those they need to practise before the summative assessment activity. You may want to use the sample question below.

Parks and Recreation crews are setting up rectangular skating rinks for the city of Ottawa. The length of a skating rink should be twice the width less 5 m. The equation they use to model this situation is:
A = 2w² – 5w, where A is the area of the rink and w is the width in metres.

1. Create a graphical model of this equation on your graphing calculator.

2. Using your graph, if the width of the rink is 4 m, what is the approximate area?

3. For the rink in Question 2, what is its approximate length?

4. Using your graph, if they want an area of 150 m² what is the approximate width of the rink?

5. Verify your answer to Question 4 algebraically to determine the exact width of a rink with an area
of 150 m².

Activity 5.7: The Ultimate Fan

Time: 2.5 hours

Description

Students plan a trip that visits four professional sports teams’ cities in seven days. Students collect information from various charts and plan the trip within a given budget.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM3.01 - retrieve information from various sources;

MM3.02 - identify options that meet certain criteria, using more than one chart, spreadsheet, or schedule;

MM3.03 - make informed decisions, using data provided in chart, spreadsheet, or schedule format and taking into account personal needs and preferences.

Prior Knowledge & Skills

· interpolation and extrapolation.

Planning Notes

· Use the Internet to obtain team schedules for a sport and to obtain flight schedules.

Teaching/Learning Strategies

Student Activity

Students:

· plan a trip that goes to games of sports teams (e.g., hockey, basketball, etc.) in four different cities.

· stay within a budget.

Teacher Facilitation

· Prepare for students to collect the information regarding games and flights from the Internet or have pre-printed data available for them.

· Brainstorm how much time is really spent when you take a plane – arriving at the airport early, etc.

· Have each student create a Trip Plan (see sample worksheet), which indicates which cities they will attend in the chosen order.

· Students must make sure that their flights arrive three hours prior to the start of a game so they can get from the airport to the arena and pick up their tickets. [Tickets are prepaid.]

· Students must be given a budget (make note of current costs).

· Teachers may opt to include hotel accommodations in the process. This may be done through a random draw for each city they stay in.

· Teachers may choose to make a game of it and find the student: Who travels the most km?, Who completes the trip on the smallest budget?, Who completes the trip in the least number of days?, etc.

· This activity is a fantasy trip and should be presented that way. If the teacher is concerned that the high costs involved in the “trip” are inappropriate for the class, they may choose a smaller version of the activity. This could include arranging bus/train schedules to watch OHL hockey teams play in various cities in Ontario. Their budget could be the money collected in a school fundraiser.

· Another way to “hide” costs would be to have the Local Radio station sponsor the trip as a contest prize. The budget would be a set donation from a sponsor – students could then be told to see as many games as the schedule and budget will allow during a seven-day period. (Students who can plan a trip to see the most games win a prize!)

Sample Worksheet

Your mission should you choose to accept it…

You and a friend must travel to four different cities to watch <<your favourite sport>>. Each time you will watch a different team play in their hometown. You must complete this mission on a budget of <<$$$$$>>. You must arrange flights between each city you travel to and make sure that you arrive at least three hours before the start of the game you wish to attend. You can stay overnight in any city, for any number of nights, but you must return home within seven days of departure.

Collecting Information

1. Determine the following, given the game schedule you have found/been given.

Game	City/Team	Date/Time of Game	Airport
1
2
3
4

2. Find flights that will connect you from home to your first city and then to each city after that, remembering to return home no later than seven days after your first flight.

Departure City	Departure Time	Arrival City	Arrival Time	Cost

3. Given a list of ticket prices for each team in the league, determine the cost for you and your friend to attend each of the games you have chosen.

Game	Cost for Two Tickets

4. Determine the total length of time in days for your trip.

5. Determine the total cost of your trip.

6. Determine the average cost per day for your trip.

7. Write a summary of the trip given in a day-by-day outline.

8. What factors affected your decisions in choosing cities to visit? Explain.

9. If you could have one of: more money or more time, which would make the mission easier and why?

Assessment & Evaluation of Student Achievement

The purpose of this activity is to assess students’ ability to make connections based on given criteria using more than one chart or spreadsheet. Question 7 could be assessed under Thinking/Inquiry and Problem Solving, as students must make choices of cities and flights that must be correctly sequenced to fit the criteria. Questions 8 and 9 could assess Communication and Reasoning. A partial rubric is provided (see Appendix B – Sports Fantasy Trip Rubric).

Accommodations

· This activity may be of difficulty for students who are new to the country or who do not follow professional sports. You may wish to create a sequence of cities that students must travel between in a given timeframe to simplify the problem (i.e., you could change the scenario to “following a band” to watch concerts in different cities).

· For students who have difficulty processing several items at one time, it might be better if they could print the Internet information or receive hard copies to work from.

· Explain the entire project orally, and provide chunking suggestions and intermediate deadlines. If student has difficulty at any point, provide support so they may continue on, so they will not give up.

Resources

Air Canada – www.aircanada.ca (for flight information)

Official Website of MLB – www.mlb.com (contains schedules by month or team)

Official Website of the NBA – www.nba.com (contains schedules by month or team)

Official Website of the NHL – www.nhl.com (contains schedules by month or team)

Ontario Hockey League – www.ontariohockeyleague.com (contains schedules by month or team)

Activity 5.8: All Hands on Deck

Time: 2.5 hours

Description

In this summative activity, students develop formulas to determine the amount of materials required for various deck sizes. They use quadratic and linear models, combined with prices, to create a formula to determine the cost of various decks.

Strand(s) & Learning Expectations

Strand(s): Analysis of Mathematical Models

Overall Expectations

MMV.01 - interpret and analyse given graphical models;

MMV.02 - interpret and analyse given formulaic models;

MMV.03 - interpret and analyse data given in a variety of forms.

Specific Expectations

MM1.05 - communicate the results of an analysis orally, in a written report, and graphically;

MM2.01 - evaluate any variable in a given formula drawn from an application by substituting into the formula and using the appropriate order of operations on a scientific calculator;

MM2.02 - construct formulas to solve multi-step problems in particular situations;

MM2.04 - judge the reasonableness of answers to problems;

MM3.03 - make informed decisions, using data provided in chart, spreadsheet, or schedule format and taking into account personal needs and preferences;

Planning Notes

· Graphing calculators are required for some parts of the assessment.

· Diagrams of deck parts and deck instructions – Appendix C.

· Manipulatives such as straws, cube-a-links, and popsicle sticks may be helpful to some students.

Teaching/Learning Strategies

Teacher Facilitation

· Set up the problem: Students are starting their own business that sells do-it-yourself deck systems. Clients can order a kit to build their own square deck in any size (1 m × 1 m, 5 m × 5 m,
21 m × 21 m, etc.). Students develop methods for determining how many of each piece they will need for each size deck and for calculating the total cost for the customer.

· Have students work in groups at first to explore the problem.

· Provide each group with a copy of the page showing the deck pieces and the diagrams of 1 m x 1 m and 4 m x 4 m decks.

· Students may benefit from having manipulatives available to help explore the problem.

Student Activity

In your group:

1. Examine the deck parts and the diagrams showing requirements for a 1 m × 1 m deck
and a 4 m x 4 m deck (see Appendix C – Deck Pieces and Diagrams).

2. Complete the chart to show the number of parts required for each size deck (1 m × 1 m
up to 5 m × 5 m). You may want to draw diagrams for the other sizes.

Size	Base Piece	Support Board	Surface Board	End Board
1m × 1m
2m × 2m
3m × 3m
4m × 4m
5m × 5m

n × n

On your own:

3. Try to develop a formula that you could use to calculate how many of each piece you would need for a square deck of any size. You may want to use a graphing calculator to help you. Look for patterns that indicate whether a linear, exponential, or quadratic model would work best. Write the formula for each piece in the last row of the chart in Question 2.

4. The wholesale cost of each item is given in the following chart. You will likely want to charge your clients more for each piece so that you will make a profit. Decide how much you will mark up your prices and calculate what you will charge your clients for each piece.

Percentage Mark-up = %

	Your Cost	Client’s Price
Base Piece	$5.70
Support Board	$3.20
Surface Board	$9.70
End Board	$3.75

5. Using your prices above, calculate the cost of a 2 m × 2 m deck. Show your work.

6. Thinking about how you calculated the cost in Question 5, create a formula that you could use to calculate the cost of any size deck.

7. A client wants to purchase a system to build an 8 m × 8 m deck. Calculate what you will charge him. Show your work.

8. If a client has $2000 to spend, what size deck could they get? Show your work.

9. You also recommend that your clients apply two layers of stain to their deck. A can of stain
costs $18.00 and covers 18m² with one coat. How will you factor this into your formula?

10. Create an instruction sheet to help someone working for your company. The instruction sheet should explain how to calculate the cost of any size deck.

Assessment and Evaluation of Student Achievement

· Problem solving can be assessed as students explore the problem, develop a model for the number of each piece required, and manipulate the models to create a cost formula.

· Application can be assessed in students’ ability to use their formula to answer Questions 6-8.

· Communication can be assessed in the clarity of the instructions to calculate the cost of any size deck.

Accommodations

· Allow students to present their instructions orally to the teacher.

· Provide manipulatives so students can visualize the problem.

· Provide additional scaffolding (such as structured worksheets) to help students break the problem down further.

Appendix A

Rubric for Evaluation of Report on Wage Increases

Criteria

Level 1

(50-59%)

Level 2

(60-69%)

Level 3

(70-79%)

Level 4

(80-100%)

Problem Solving

- collection of data

- forming a model

- making conclusions and comparisons

- gathers limited relevant data on income

- creates a model that has limited effectiveness

- makes conclusions and comparisons with limited effectiveness

- gathers some relevant data on income

- creates a model that is somewhat effective

- makes conclusions and comparisons with some effectiveness

- gathers generally relevant data on income

- creates a model that is generally effective

- makes conclusions and comparisons with considerable effectiveness

- gathers consistently relevant data on income

- creates a model that is thorough and effective

- makes conclusions and comparisons with high degree of effectiveness

Communication

- integration of mathematical forms with narrative

- clarity in explanations and justifications

- limited use of correct mathematical forms

- explanations and justifications have limited clarity

- some use of correct mathematical forms

- explanations and justifications have some clarity

- most often uses correct mathematical forms

- explanations and justifications have considerable clarity

- consistently uses correct mathematical forms

- explanations and justifications are consistently clear

Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this assignment or activity.

Appendix B

Sports Fantasy Trip Rubric

Criteria

Level 1

(50-59%)

Level 2

(60-69%)

Level 3

(70-79%)

Level 4

(80-100%)

Communication

- clarity in explanations and justifications of choices

- explanations and justifications have limited clarity

- explanations and justifications have some clarity

- explanations and justifications have considerable clarity

- explanations and justifications are consistently clear

Problem Solving

- collection of data

- understanding the problem to make realistic timelines

- gathers limited appropriate data on trip costs

- timelines created demonstrate limited understanding of the problem

- gathers some appropriate data on trip costs

- timelines created demonstrate some understanding of the problem

- gathers considerable appropriate data on trip costs

- timelines created demonstrate general understanding of the problem

- gathers consistently appropriate data on trip costs

- timelines created demonstrate thorough understanding of the problem

Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this assignment or activity.

Appendix C

Deck Pieces and Diagrams

Deck Pieces

Diagram and Instructions for a Basic 1 m × 1 m Deck

· Secure four base pieces into the ground.

· Attach one support board across every two bases as shown.

· Attach surface board on top of support boards.

· Attach end pieces at either end (perpendicular to support boards).

Diagram and Instructions for a Basic 4 m × 4 m Deck

· Secure nine base pieces into the ground.

· Attach one support board across every two bases as shown.

· Attach one surface board between every four base pieces.

· Attach end pieces.

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