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