Course Profile Mathematics for College Technology (MCT4C), Grade 12, College Preparation, Combined

Unit 4: Exponential and Logarithmic Functions

Time: 20 hours

Activity 4.1 | Activity 4.2 | Activity 4.3 | Activity 4.4 | Activity 4.5 | Activity 4.6 | Activity 4.7

Unit Description

Students investigate properties of exponential and logarithmic functions. The relationship between exponential and logarithmic functions is explored both graphically and algebraically. Students use the laws of logarithms to simplify and evaluate logarithmic expressions, and to solve problems. A wide variety of exponential and logarithmic applications and models are examined.

Unit Synopsis Chart

Activity	Time	Learning Expectations	Assessment Categories	Tasks
1. Exploring Exponential Functions	3 hours	ELV.01, EL1.01, EL1.02, EL1.03 CGE2a, 2d, 5a	Knowledge/Understanding Communication	Investigating properties of exponential functions
2. Modelling Population	3 hours	ELV.01, ACV.01, EL1.04, EL1.05, AC1.01, AC1.02, AC1.03, AC1.04 CGE3c, 5g	Thinking/Inquiry/Problem Solving Communication	Modelling exponential growth and decay
3. Ed’s Exponential Existence	3 hours	ELV.01, ACV.04, EL1.05, AC4.04 CGE2b, 2c, 7b	Knowledge/Understanding Thinking/Inquiry/Problem Solving Application Communication	Applications of exponential growth and decay
4. Bridging the Gap with Logs	3 hours	ELV.01, ELV.02, ACV.04, EL1.01, EL2.01, EL2.02, AC4.04 CGE2c, 2d	Knowledge/Understanding Communication	Determining connections between exponential functions and logarithmic functions
5. A Log “Jam Session”	3.5 hours	ELV.02, ACV.01, EL2.04, AC1.01, AC1.02, AC1.03, AC1.04 CGE3c, 5g	Thinking/Inquiry/Problem Solving Communication	Modelling logarithmic functions
6. Logarithms and Applications	3 hours	ELV.02, ACV.04, EL2.03, EL2.04, AC4.04 CGE2b, 2c	Knowledge/Understanding Thinking/Inquiry/Problem Solving Application Communication	Laws of logarithms and applications of logarithmic scales
7. Summative Assessment: Springfield and Shelbyville’s History	1.5 hours	ELV.01, ELV.02, EL1.04, EL1.05, EL2.02, EL2.03, EL2.04, AC4.04 CGE2b, 2c, 5g, 7b	Knowledge/Understanding Thinking/Inquiry/Problem Solving Application Communication	Summative assessment

Note: The order of exercises 4.1 and 4.2 can be switched if the teacher would like to introduce modelling before algebra.

Activity 4.1: Exploring Exponential Functions

Time: 3 hours

Description

Students use graphing technology to explore properties of exponential functions. The graphical effect of the parameters a, b, and c in the equation y = ca^x + b is examined. The rates of change of exponential functions are compared to the rates of change of non-exponential functions.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2a - an effective communicator who listens actively and critically to understand and learn in light of gospel values;

CGE2d - an effective communicator who writes and speaks fluently one or both of Canada’s official languages;

CGE5a - a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.01 - demonstrate an understanding of the nature of exponential growth and decay.

Specific Expectations

EL1.01 – identify, through investigations, using graphing calculators or graphing software, the key properties of exponential functions of the form a^x (a > 0, a ¹ 1) and their graphs (e.g., the domain is the set of the real numbers; the range is the set of the positive real numbers; the function either increases or decreases throughout its domain; the graph has the x-axis as an asymptote and has
y-intercept = 1);

EL1.02 - describe the graphical implications of changes in the parameters a, b, and c in the equation
y = ca^x + b;

EL1.03 - compare the rates of change of the graphs of exponential and non-exponential functions (e.g., those with equations y = 2x, y = x², y = x, and y = 2^x).

Prior Knowledge & Skills

· Knowledge of graphing calculators and graphing a function

· Rates of change of a graph and properties of graphs, including domain, range, increasing, decreasing, asymptotes, and intercepts

· Use of First Differences chart;

· Understanding of positive and rational exponents, and exponent laws.

Planning Notes

· Students require graph paper and graphing calculators. Provide students with window settings for all graphing calculator activities.

· An overhead and a graphing calculator projection unit are required for class demonstrations.

· Prepare worksheets and a review of the relevant concepts of transformation of functions.

Teaching/Learning Strategies

Teacher Facilitation

· Review various types of graphical relationships, i.e., linear, quadratic, exponential, etc. with the class by projecting the graphical relationships using a graphing calculator and an overhead projecting unit. Then, discuss the properties of each type.

· Students may either work in pairs or individually to complete graphing activities.

· Distribute graphing calculators and graph paper.

· Do a numerical example of exponential growth with the class to review this concept from Grade 11 e.g., If you were investing in a mutual fund, describe the rate of change of the accumulated amount of your investment over time if the rate of growth is exponential.

· Distribute Student Activity 4.1.1.

Student Activity 4.1.1

1. Using graphing technology, graph the following functions: y = 2x, y = 10x, y = x², y = x,
y = 2^x, and y = 7^x. Use the information from the graph to complete the following chart.

Equation	What type of relationship is this?	Sketch the graph	Describe the shape of the graph
y = 2x
y = 10x
…

2. Describe a situation that could be modelled by each of the following equations:

a) y = 10x

b) y = x²

c) y = 2^x

Teacher Facilitation

· Project the graphs in #1 on an overhead and take up the chart. In a teacher-directed discussion, discuss students’ answers to #2.

· Review domain and range and then distribute Student Activity 4.1.2.

Student Activity 4.1.2

1. Using graphing technology, graph the following functions: y = 2^x, y = 3^x; y = 6^x; y = 10^x;

y = ()^x, y = ()^x, y = , and y = . Use the information from each graph to complete the following chart.

Equation	Describe the shape of the graph	Domain	Range	Intervals where increasing (if any)	Intervals where decreasing (if any)	Equation of asymptotes	Intercepts
y = 2^x
y = 3^x
…

2. Use the results in your chart to summarize your findings for exponential functions y = a^x, a > 0,
a ¹ 1.

a) State the domain of y = a^x.

b) State the range of y = a^x.

c) State the equation of the asymptote of y = a^x.

d) State the intercepts of y = a^x.

e) Describe the graphical effects of different values for a for the equation y = a^x (a > 0, a ¹ 1). Be specific about the increasing or decreasing nature of the function.

f) Describe the graph y = a^x if a = 1. How does this graph differ from the graphs where a ¹ 1?

3. a) Without graphing, explain how the graphs of y = 5^x and y = differ, and describe what they have in common.

b) Verify your answer using graphing technology.

4. a) Without graphing, explain how the graphs of y = 4^x and y = 7^x differ, and describe what they have in common.

b) Verify your answer using graphing technology.

5. a) Without graphing, explain how the graphs of y = ()^x and y = differ, and describe what they have in common.

b) Verify your answer using graphing technology.

6. Summarize your conclusions about the graphical significance of changes in parameter a for the equation y = a^x, a > 0, a ¹ 1.

Teacher Facilitation

· In a teacher-directed discussion, discuss students’ conclusions as a class.

· Summarize the key properties of y = a^x for a > 0, a ¹ 1 from question #2. Demonstrate the properties visually using the overhead and graphing calculator projection unit.

· Summarize the graphical effect of parameter a for the equation y = a^x for a > 0, a ¹ 1, from question #6. Demonstrate the conclusions visually using the overhead and graphing calculator projection unit.

· Introduce and distribute Student Activity 4.1.3.

Student Activity 4.1.3

1. a) Using graphing technology, graph the following functions: y = 2^x, y = (4)(2^x), y = (2)(2^x),
y = ()(2^x), y = (-4)(2^x), y = (-3)(2^x), y = (-)(2^x) . Use the information from the graphs to complete the following chart.

Equation	Describe graphical changes to the graph of y = 2^x	Domain	Range	Intervals where increasing (if any)	Intervals where decreasing (if any)	Equation of asymptotes	Intercepts
y = 2^x
y = (4)(2^x)
…

b) Using graphing technology, graph the following functions: y = ()^x, y = (3)()^x, y = (5)()^x,
y = (-1)()^x, y = (-3)()^x, y = (-5)()^x. Use the information from the graphs to complete the following chart.

Equation	Describe graphical changes to the graph of y = ()^x	Domain	Range	Intervals where increasing (if any)	Intervals where decreasing (if any)	Equation of asymptotes	Intercepts
y = ()^x
y = (3)()^x
etc.

2. Use the results in your chart to explain your findings for exponential functions y = ca^x, a > 0, a ¹ 1. Describe how changes in the parameter c in y = ca^x affects the equation y = a^x for the following properties:

a) shape of the graph;

b) the domain;

c) the range;

d) intervals of increasing or decreasing;

e) equations of asymptotes;

f) intercepts.

3. Summarize your conclusions by describing the graphical effects of parameter c for y = ca^x,
a > 0, a ¹ 1.

4. a) Without graphing y = (5)(3^x), state the shape of the graph, domain, range, intervals of increase/decrease, equation of asymptote, and intercepts.

b) Verify your answer using graphing technology.

5. a) Without graphing y = (-2)(4^x), state the shape of the graph, domain, range, intervals of increase/decrease, equation of asymptote, and intercepts.

b) Verify your answer using graphing calculator technology.

6. a) Without graphing y = (3) , state the shape of the graph, domain, range, intervals of increase/decrease, equation of asymptote, and intercepts.

b) Verify your answer using graphing technology.

7. a) Use a graphing calculator to graph the following functions on the same set of axes:

i) y = 2^x , y = 2^x + 1, and y = 2^x - 1

ii) y = 3^x , y = 3^x + 8, and y = 3^x – 8

iii) y = ()^x, y = ()^x - 2, and y = ()^x + 2

b) Summarize your findings by describing the graphical effects of parameter b for y = a^x + b,
a > 0, a ¹ 1.

Teacher Facilitation

· In a teacher-directed discussion, discuss students’ conclusions as a class.

· Summarize the graphical implications of parameter c for the equation y = ca^x for a > 0, a ¹ 1 (question #3). Demonstrate the conclusions visually using the overhead and graphing calculator projection unit.

· Summarize the graphical implications of parameter b for the equation y = a^x + b, for a > 0, a ¹ 1 (question #7b). Demonstrate the conclusions visually.

· Distribute Student Activity 4.1.4. This worksheet consolidates skills learned from Student
Activities 4.1.1 to 4.1.3. Student Activity 4.1.4 can be handed in for assessment; teachers may wish to instruct students to work individually rather than in pairs.

Student Activity 4.1.4

1. a) Without using graphing technology, use the base graph y = 4^x to sketch the graphs of the following functions: y = (-1)(4^x), y = (3)(4^x), y =4^x + 5, y = 4^x – 2, y = ()(4^x).

b) Verify your results using graphing technology.

c) What properties do all of the graphs have in common?

d) What graphical properties are different between these graphs?

2. Explain why a ¹ 1 for y = a^x. Use words and graphs in your explanation.

3. a) Use graphing technology to graph each of the following functions: y = (2)(6^x) + 4,
y = (-1)()^x + 3, y = (-5)(3^x) - 8.

b) State the domain, range, and intercept of each of the functions.

4. a) Is the domain for all exponential functions the same? Use examples to support your explanation.

b) Is the range for all exponential functions the same? Use examples to support your explanation.

5. Consider all of the graphical properties of the equation y = (3)(4^x). Could this equation be used to model:

a) the depreciation of a speed boat? Explain why or why not.

b) the increase of a city’s population? Explain why or why not.

c) the depletion of the balance of a bank account that decreases exactly $100.00 each week? Explain why or why not.

Assessment & Evaluation of Student Achievement

· If students work in pairs, they can be assessed for Teamwork.

· Students write a summary of their findings from the activity.

· Focus on formative assessment and self-assessment, rather than on marks.

Accommodations

· Enlarge the student activity charts for any student with spatial difficulties.

· Use computer technology rather than a small screen calculator.

Activity 4.2: Modelling Population

Time: 3 hours

Description

Students expand their knowledge of exponential functions to model exponential growth and decay in various population contexts. Data representing both growth and decay are analysed graphically and algebraically. Students develop equations to model the data and then interpret their findings.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE5g - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.01 - demonstrate an understanding of the nature of exponential growth and decay;

ACV.01 - analyse models of linear, quadratic, polynomial, exponential, or trigonometric functions drawn from a variety of applications.

Specific Expectations

EL1.04 - describe the significance of exponential growth or decay within the context of applications represented by various mathematical models (e.g., tables of values, graphs, equations);

EL1.05 - pose and solve problems related to models of exponential functions drawn from a variety of applications, and communicate the solutions with clarity and justification;

AC1.01 - determine the key features of a mathematical model (e.g., an equation, a table of values, a graph) of a function drawn from an application;

AC1.02 - compare key features of a mathematical model with the features of the application it represents;

AC1.03 - predict future behaviour within an application by extrapolating from a given model of a function;

AC1.04 - pose questions related to an application and use a given function model to answer them.

Prior Knowledge & Skills

· Understanding of positive and rational exponents, and exponent laws

· Knowledge of graphs of exponential functions and solving exponential equations algebraically

· Geometric sequences and related formulas

· Use of a graphing calculator

Planning Notes

· Prepare worksheets.

· Students require graph paper and graphing calculators. Provide students with window settings for all graphing calculator activities.

· An overhead and a graphing calculator projection unit are required for class demonstrations.

· Students require access to the Internet to gather data for Student Activity 4.2.3. Research websites that contain population data to recommend for data collection.

Teaching/Learning Strategies

Teacher Facilitation

· Student Activity 4.2.1 models linear population growth.

· For Student Activities 4.2.1, 4.2.2, and 4.2.3, students can work individually or in pairs.

· Review concepts related to geometric series to assist with Student Activity 4.2.1, question #2.

· Set a context for the exploration of exponential growth and decline by discussing reasons why population research is of interest to government agencies and to private industries. Discuss ways that population growth or decline would affect students’ neighbourhoods, e.g., roads, traffic, industry, housing, school size, etc.

· Distribute graphing calculators, graph paper, and Student Activity 4.2.1.

Student Activity 4.2.1 - Population Growth

The following chart records the population of a city from 1997 to 2001.

Year	Population (thousands)	First Differences	Ratio of Growth
1997	29 987.2
1998	30 248.2		r = 30 248.2 ÷ 29 987.2 =
1999	30 499.2
2000	30 769.7
2001	31 081.0

1. a) Complete the first differences column in the chart.

b) Is the graph of Population versus Year linear or non-linear? Explain your reasoning.

c) Use the graphing calculator to graph Population versus Year to determine the accuracy of your answer to (b).

d) What type of equation, i.e., linear, quadratic, or exponential, represents this graph?

2. a) Complete the last column of the chart to calculate the ratio for the population increases each year. What is the significance of this ratio?

b) Calculate the average ratio over the 5 years. What does this ratio represent?

3. Write an equation, in the form y = ca^x, that represents the population, P, after n years.

4. Describe how your original equation would change if the initial population were 15 000 000.

5. Describe how your original equation would change if the population doubled every year.

6. a) Use the equation to predict the population in the year 2005.

b) Use your graph to verify your solution.

7. a) Use your graph to predict how long it would take the population to increase to 33 million.

b) Use your equation to verify your solution.

8. Is it reasonable to use your graph or equation to accurately predict the population in the year 2050? Why or why not? Suggest reasons for the limitations of both the graphical and algebraic models.

Teacher Facilitation

· In a teacher-directed discussion, discuss students’ conclusions as a class. Take up the questions, providing visual demonstrations using the overhead and the graphing calculator projection unit. Alternatively, students could briefly present their solutions to the class.

· Direct students to the conclusion that for exponential growth, the equation y = ca^x has a > 1.

· Lead into Student Activity 4.2.2 by discussing/brainstorming possible reasons for population growth and population decline. Have students predict the differences in both the algebraic models and graphical models between exponential growth and exponential decay. Distribute Student
Activity 4.2.2.

Student Activity 4.2.2 – Population Decline

1. Suppose that the population of a city, recorded in the chart below, began to decrease after the year 2001. Complete the chart.

Year	Population (thousands)	Ratio of Decline
2001	31 081.0
2002	30 366.1
2003	29 637.3
2004	28 985.3
2005	28 260.7

2. Determine the average ratio over the 5-year period.

3. a) Use the graphing calculator to graph Population versus Year.

b) Explain how the shape of this graph differs from the graph in Student Activity 4.2.1 that represented population growth.

4. a) Write an equation that represents population, P, after n years.

b) Use the graphing calculator to confirm the accuracy of your equation by graphing it and comparing it with the graph in #3a.

5. a) Use the equation to determine the population after 10 years.

b) Use your graph to verify your solution.

6. a) Use the graph to determine when the population will have decreased to 27 000 000.

b) Use your equation to verify your solution.

7. Describe how your original equation would change if the population in the year 2001 was
50 000 000.

8. Describe how your original equation would change if the population declined 5% each year. Will the population ever reach 0? Explain.

9. Discuss possible limitations of both the graphical model and the algebraic model for population decline.

Teacher Facilitation

· In a teacher-directed discussion, discuss students’ conclusions as a class. Take up the questions, providing visual demonstrations. Alternatively, students could present their solutions to the class.

· In a class discussion, compare the algebraic models of exponential growth (Student Activity 4.2.1) and exponential decay (Student Activity 4.2.2). Direct students to the conclusion that for exponential decay, the equation y = ca^x has 0 < a < 1; for exponential growth, the equation y = ca^x has a > 1.

· In a class discussion, compare the characteristics of the graphical models of exponential growth and exponential decay. Similarities and the differences should be explored.

· Distribute Student Activity 4.2.3.

Student Activity 4.2.3 – Population Research

1. Use the websites provided by your teacher to locate population data for any country, province, or city over a 50-year span. Record your data in a chart.

2. Graph the relationship between Population and Year.

3. Determine an equation, in the form y = ca^x, that represents population, P, of your chosen region after n years. Show your exploration of algebraic models and justify your choice of model.

4. a) Use your algebraic model to predict the population of your chosen region in the year 2010.

b) Use your graph to determine the accuracy of your prediction.

5. a) Describe any restrictions on your algebraic model.

b) Describe any restrictions on your graphical model.

6. Despite limitations on your algebraic and graphical models, they are still useful sources of information. City planners are interested in population trends. Based on your chosen location, write a letter to a city planner outlining the population trends for this location. Include in your letter several recommendations about what should be done to prepare for the upcoming population trends in that region. Provide data to support your conclusions and recommendations.

Assessment & Evaluation of Student Achievement

· Students write a summary of their findings in this activity. The summary can be assessed formatively by the teacher or shared with a peer.

· Focus on formative assessment and peer or self-assessment, rather than on marks. Assess Thinking/Inquiry/Problem Solving and Communication using a rubric (see Appendix 4.1).

Accommodations

· Students can work in pairs if they are having difficulty with the investigation. If students require further guidance, it may be beneficial for the teacher to complete Student Activity 4.2.1 as a class and have the students complete Student Activities 4.2.2 and 4.2.3 independently or in pairs. Alternatively, the teacher can provide students with specific data.

· Use computer technology rather than a small screen calculator.

Resources

Useful websites for data collection for Part C include:

· www.statcan.ca

· http://www.region.peel.on.ca/planning/stats/popproj.htm

Activity 4.3: Ed’s Exponential Existence

Time: 3 hours

Description

Students expand on concepts investigated in the previous activity to include a variety of growth and decay applications. Problem-solving skills are developed as students work through applications. Students further develop an understanding of restrictions on both algebraic models and graphical models.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written material effectively;

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE7b - a responsible citizen who accepts accountability for one’s own actions.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.01 - demonstrate an understanding of the nature of exponential growth and decay;

ACV.04 - demonstrate facility in carrying out and applying key manipulation skills.

Specific Expectations

EL1.05 - pose and solve problems related to models of exponential functions drawn from a variety of applications, and communicate the solutions with clarity and justification;

AC4.04 - demonstrate facility in using manipulation skills related to solving linear, quadratic, and polynomial equations, simplifying rational expressions, and operating with exponents.

Prior Knowledge & Skills

· Understanding of positive and rational exponents, and exponent laws

· Knowledge of graphs of exponential functions, and solving exponential equations algebraically and graphically

· Geometric sequences and related formulas

· Use of graphing calculator

Planning Notes

· Prepare worksheets.

· Students require graph paper and graphing calculators. Provide students with window settings for all graphing calculator activities.

· If students are presenting their solutions to the class, chart paper and markers are required.

· An overhead projector and projection unit for the graphing calculator is needed to complete Student Activity 4.3.1 as a class. Prepare overhead transparencies.

Teaching/Learning Strategies

Teacher Facilitation

· Students may need to be reminded how to determine a growth/decay ratio.

· Complete Student Activity 4.3.1 together as a class in a teacher-directed activity. Use an overhead projection unit for the graphing calculator to facilitate demonstration and class discussion. Alternatively, Student Activity 4.3.1 could be completed by the students in pairs, and then taken up as a class prior to beginning the next activity, or teachers can structure a Jigsaw in which students teach others in small groups.

· Distribute graphing calculators, graph paper, and Student Activity 4.3.1.

Student Activity 4.3.1

When Ed was born, his town of Edenville had a population of 35 000. The average yearly growth rate since then has been 1.5%.

1. Assuming this growth rate continues, construct a table of values relating the population, P, of Edenville and time, t.

2. a) Use your graphing calculator to graph the Population versus Year. Use window settings
.

b) Explain how the characteristics of the graph indicate exponential growth.

3. Determine an equation, in the form y = ca^x, for the population of Edenville.

4. Explain how your equation would change if the population were declining at a rate of 1.5%.

5. a) Use your equation to determine the population of Edenville on Ed’s 15th birthday.

b) Use your graphical model to verify your solution.

6. Consider the limitations of both the graphical model and the algebraic model in this context. Summarize these restrictions.

Teacher Facilitation

· For Student Activity 4.3.2, students are placed in groups of two to four.

· The problem sets can be set up as a circuit; students can work through the stations in any order. An alternative is to have each group work through one or two problem sets and present their solutions to the class.

· Distribute Student Activity 4.3.2.

Student Activity 4.3.2

1. Ed got very sick one day and decided to go to the doctor. Dr. B. Better told him that he had a bacterial infection and put Ed on penicillin. The doctor hypothesised that he presently had 10 000 bacterium in his body. The net effect of penicillin killing the bacteria and the bacteria growing results in an overall decrease of bacteria 5% every hour.

a) Construct a table of values relating the number of bacteria and time.

b) Use your graphing calculator to graph the relationship between the number of bacteria and time.

c) Determine an equation, in the form y = ca^x, to represent the number of bacteria remaining, N, after t hours. Enter this equation into a graphing calculator to determine how well it fits your data.

d) Explain how your original equation would change if the bacteria decreased by 5% every 3 hours.

e) Use your equation to determine the number of bacteria remaining after 16 hours. Verify your answer using your graphical model.

f) Can you use your equation to determine the number of bacteria present after 7 days? Explain why or why not.

g) Use your graph to determine when the number of bacteria in Ed will be reduced to 0. Explain why this is or is not realistic.

2. Ed decided to invest $1 000 for college. Presently, the banks are offering 4%/a compounded yearly.

a) Construct a table of values relating the amount of money, A, and the number of years n.

b) Use your graphing calculator to graph the relationship between the accumulated amount of Ed’s money and the number of years the money is invested.

c) Determine an equation, in the form y = ca^x, to represent the accumulated amount of money, A, over n years of investment.

d) Explain how your original equation would change if the interest were compounded quarterly.

e) Use your original equation to determine how much money Ed is predicted to have after 3 years. Verify your solution using your graphical model.

f) State any limitations of your algebraic model and on your graphical model.

3. At basketball practice, Ed noticed that when he drops the basketball it only bounces back up to 60% of its original height.

a) Construct a table of values relating the height of the ball, h, and the number of bounces, n. Use a starting height of 2 m.

b) Use your graphing calculator to graph the relationship between the height of the ball and the number of bounces.

c) Determine an equation, in the form y = ca^x, to represent the height, h, of the ball after n bounces.

d) Use your equation to determine the height of the ball after 4 bounces. Verify your solution using your graphical model.

e) Is it realistic to use your equation or graph to determine the height after 80 bounces? Explain why or why not.

4. Ed’s parents bought a car for $25 000. He was told that in any given year, this particular car depreciates to 70% of its value of the previous year.

a) Construct a table of values relating the value of the car, V, and the number of years, n.

b) Use your graphing calculator to graph the relationship between the value of the car and the number of years after its purchase.

c) Determine an equation, in the form y = ca^x, to represent the value of the car n years after it is purchased.

d) Explain how your original equation would change if the car depreciated to 70% of its value every second year, rather than every year.

e) Explain how your original equation would change if the car depreciated to 80% of its value in the previous year.

f) Use your equation to determine the value of the car after 10 years. Verify your answer using your graphical model.

g) Explain any restrictions or limitations of either your algebraic model or graphical model.

5. Using his microscope, Ed counted 30 bacterium in his petri dish at the beginning of biology class. After carefully watching the bacteria, he observed that they double every hour.

a) Construct a table of values relating the number of bacteria, n, and time, t.

b) Use your graphing calculator to graph the relationship between the number of bacteria and time.

c) Determine an equation, in the form y = ca^x, for the number of bacteria after t hours.

d) Explain how your original equation would change if the bacteria doubled every 3 hours.

e) Use your equation to determine how many bacteria there will be after 24 hours. Verify your answer using your graphical model.

f) Would your equation be able to accurately predict the number of bacterium in the petri dish after six months? Explain why or why not.

6. The amount of time it takes for a radioactive element to decay to one half of its original amount is known as half-life. In 1970, Ed’s science teacher purchased 500 g of cobalt-60 to show students. The half-life of cobalt-60 is 5 years.

a) Construct a table of values relating the amount of cobalt-60, A, and number of years, n.

b) Use your graphing calculator to graph the relationship between the amount of cobalt-60 and the number of years.

c) Determine an equation, in the form y = ca^x, for the amount of cobalt-60 after n years.

d) Explain how your equation would change if the half-life of cobalt-60 were 10 years.

e) Use your equation to determine how much cobalt was present in the year 2000. Verify your answer using your graphical model.

f) Explain any restrictions or limitations of either your algebraic model or graphical model.

Assessment & Evaluation of Student Achievement

· Students can be assessed on Teamwork throughout the activity.

· If students present their solutions in groups, these can be assessed formatively on Application of knowledge as well as Communication skills.

Activity 4.4: Bridging the Gap with Logs

Time: 3 hours

Description

Students investigate the connection between exponential functions and logarithmic functions. Students extend their knowledge of inverses and apply it to exponential functions to discover the logarithmic function.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE2d - an effective communicator who writes and speaks fluently one or both of Canada’s official languages.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.01 - demonstrate an understanding of the nature of exponential growth and decay;

ELV.02 - define and apply logarithmic functions;

ACV.04 - demonstrate facility in carrying out and applying key manipulation skills.

Specific Expectations

EL1.01 - identify, through investigations, using graphing calculators or graphing software, the key properties of exponential functions of the form a^x(a > 0, a ¹ 1) and their graphs (e.g., the domain is the set of real numbers; the range is the set of the positive real numbers; the function either increases or decreases throughout its domain; the graph has the x-axis as an asymptote and has y-intercept = 1);

EL2.01 - define the logarithmic function log_ax (a > 1) as the inverse of the exponential function a^x, and compare the properties of the two functions;

EL2.02 - express logarithmic equations in exponential form, and vice versa;

AC4.04 - demonstrate facility in using manipulation skills related to solving linear, quadratic, and polynomial equations, simplifying rational expressions, and operating with exponents.

Prior Knowledge & Skills

· Understanding of properties of exponential functions and exponential graphs

· The concept of the inverse as an interchange of x co-ordinates with y co-ordinates of the original function

· Use of the zoom and trace features of a graphing calculator to determine points on a graph

Planning Notes

· Students require graph paper, MIRAs, and graphing calculators. Provide window settings for graphing calculator activities, e.g., for Student Activity 4.4.1;
use , , , .

· Prepare worksheets.

· Do a class activity using the temperature probe to practise gathering and graphing authentic data. For example, use a temperature probe to measure the temperature of a cup of coffee as it cools. Then graph the data to determine the pattern of cooling, and compare this with Newton’s law of cooling. Refer to Resources for a variety of calculator-based laboratory (CBL) activities appropriate for this purpose.

Teaching/Learning Strategies

Teacher Facilitation

· Review with the class the characteristics of exponential functions and real-life models that are exponential in nature (population growth, compound interest, radioactive decay, etc.).

· Review the key properties of the graphs of exponential functions.

· Distribute graphing calculators, graph paper and Student Activity 4.4.1.

· Students work through the questions independently or in pairs.

Student Activity 4.4.1

1. Create a table of values for each of the following functions

a) y = 2^x b) y = 3^x c) y = 5^x

2. Graph each of the functions on a separate grid.

3. Use your graphs to answer the following questions:

a) What point is common to all graphs?

b) Do these graphs represent functions? Explain why or why not.

c) State the Domain and Range of each function.

4. Graph the line y = x on each grid.

5. Place the MIRA along the line y = x. Graph the reflection on the same grid as the original function. Describe the relationship between the graph of the original exponential function and its reflected graph.

6. Identify several co-ordinates of the reflected graph and record them in a table of values.

7. Use your reflected graphs to answer the following questions:

a) What point is common to all of the reflected graphs?

b) Do these graphs represent functions? Explain why or why not.

c) State the Domain and Range of each function.

8. For each of the original graphs and its reflected graph, compare the co-ordinates from the table of values created in #1 with the table of values created in #6. Is there a relationship? Explain your findings.

9. Determine the inverse of the function y = 7^x by interchanging the x and y co-ordinates. Graph the original equation and its inverse on the same set of axes. Use the MIRA to confirm the accuracy of your inverse.

Teacher Facilitation

· Discuss students’ findings in a teacher-directed discussion.

· The teacher facilitates consolidation of students’ findings regarding the following properties of the graph that is the inverse of the exponential function: Domain, Range, the x-intercept (1,0) as a common point of each of the graphs for the inverse function, the shape of the graph, and the behaviour of the function around the x-axis.

· Discuss the term logarithm and its meaning.

· In Student Activity 4.4.2, students are introduced to the definition of the logarithm, “log”, as the inverse of the exponential function. Graphing calculators are used to visually illustrate exponential and logarithmic functions as inverse functions of each other. Students also compare the properties of exponential and logarithmic functions.

· Distribute graphing calculators and Student Activity 4.4.2.

Student Activity 4.4.2

1. Use graphing calculators to graph the function y = 10^x. Sketch the graph in your notes.

2. Use the zoom feature from the graphing calculator or the trace function to fill in the missing values in the following table.

x	0	1	2	3	4
y

Confirm the co-ordinates in the completed table of values using the equation.

3. a) Determine the inverse of the function y = 10^x by interchanging the x and y co-ordinates in the table in question #2. Complete the table of values for the inverse function:

x	1		100		10 000
y		1		3

b) Examine the table of values and describe the shape of the inverse function.

c) Graph y = log₁₀x on the graphing calculator. Compare the co-ordinates of this graph to the table in (a). What relationship exists between the functions y = 10^x and y = log₁₀x?

d) Verify that the inverse of y = 10^x is y = log₁₀x by graphing both functions on the same set of axes with the graphing calculator.

4. Given the equation y = 8^x:

a) What is the equation of the inverse function?

b) Verify your inverse function visually by graphing both functions on the same set of axes.

5. a) Use graphing technology to graph y =4^x, y = 9^x, y = 11^x, and y = 20^x on the same set of axes.
Use your graphs to complete the following chart:

Exponential Equation	Describe the shape of the graph	Domain	Range	Intervals where increasing (if any)	Intervals where decreasing (if any)	Equation of asymptote	Intercepts
y = 3^x
y = 7^x
y = 11^x
y = 20^x

b) For each function in (a), determine the equation of the inverse function (the logarithmic equation).

c) Use the line y = x to graph the inverse of the equations in (a) on the same set of axes.
Use your graphs to complete the following chart:

Logarithmic Equation

Describe the shape of the graph

Domain

Range

Intervals where increasing (if any)

Intervals where decreasing (if any)

Equation of asymptote

Intercepts

d) Compare the properties of the graph of the exponential functions with the properties of the graphs of the inverse, logarithmic functions. How are they alike? How are they different?

Teacher Facilitation

· In a teacher-directed discussion, discuss students’ findings. Students must be clear that the logarithmic function y = log_ax, a > 1, a ¹ 1, is the inverse of the exponential function y = a^x. During the discussion, provide visual demonstrations using the overhead.

· Discuss students’ results for question #5. Students must be clear about the graphical properties of
y = log_ax and how these graphical properties compare with the exponential functions y = a^x. During the discussion, provide visual demonstrations.

· Provide context for the usefulness of determining the inverse function through a discussion of compound interest where the time of an investment must be determined instead of accumulated amount.

· Discuss the following concepts relating to the relationship between exponential equations and logarithmic equations:

1. To find the inverse of y = a^x, switch the x and y co-ordinates: x = a^y.

2. The function x = a^y is called a logarithmic function, and x = a^y y = log_ax. Note that log_ax is the exponent to which the base a must be raised to give x.

3. Examples

a) The exponential equation x = 5^y can be written in logarithmic form y = log₅x.

b) The logarithmic equation y = log₇x can be written in exponential form x = 7^y.

c) The exponential equation 2⁴ = 16 can be written in logarithmic form 4 = log₂16.

d) The logarithmic equation 2 = log₃9 can be written in exponential form 9 = 3².

· Discuss why logarithms are useful (they were created to help us solve exponential equations; they were the only way to isolate the exponent).

· In Student Activity 4.4.3, students solve exponential equations both graphically (with graphing technology) and logarithmically.

· Distribute Student Activity 4.4.3.

Student Activity 4.4.3

1. a) Use graphing technology to graph y = 6^x and its inverse on the same set of axes. Remember that the graph of the inverse function can be determined by interchanging the x and y values of the original function, or by reflecting the original function in the line y = x.

b) The graph of the inverse can be used to solve for the exponent of the original function. Look at the original graph and its inverse, and explain how to use the graph of the inverse to solve
36 = 6^x.

c) Using the equation of the inverse function, approximate a solution for the original function y = 6^x for the following values of y:

i) y = 6 ii) y = 1 iii) y = 88 iv) y = 15

d) Explain how to solve exponential equations graphically.

2. Find an approximate solution by solving graphically:

a) 2^x = 6 b) 5^x = 11 c) 8^x = 23

3. Use the relationship x = a^y y = log_ax to solve each of the following:

a) log₅25 = x b) log₄64 = x c) log₂128 = x

d) log₁₀100 = x e) log₃x = 0 f) log₆x = 2

g) log₇x = 4 h) log₈x = 3 i) log₅x = 5

4. The equation y = 15^x represents the growth of a population of dust mites, where x represents the number of days. Determine, to the nearest day, how long it will take for the population to reach:

a) 150 b) 600 c) 3000

What assumption did you make? (started with one dust mite.)

5. The equation y = 4^x represents the population increase of the number of algae, where x represents the number of weeks. Determine, to the nearest week, how long it will take for the population to reach:

a) 80 b) 400 d) 2000

What assumption did you make?

Teacher Facilitation

· Additional questions like #3 should be assigned if further practice is needed for students to work with the relationship x = a^y y = log_ax.

· Extension

· Students could explore the restrictions of logarithmic functions.

· Students could explore the graphs of logarithmic functions of different bases including
a = 1, a < 0, and 0 < a < 1.

Assessment & Evaluation of Student Achievement

· Learning Skills, such as Works Independently, or Teamwork can be assessed in Student
Activity 4.4.1 and Student Activity 4.4.2.

· Questions in Student Activity 4.4.3 that require explanations can be assessed for Communication.

· Student Activity 4.4.3 can be assessed for Knowledge/Understanding using a marking scheme.

Resources

Brueningsen, C., et al. Real-World Math with the CBL System – 25 Activities Using the CBL and TI-82. Texas Instruments, 1994.

Brueningsen, C., et al. Real-World Math with the CBL System – Activities for the TI-83 and TI-83 Plus. Texas Instruments, 1994.

Activity 4.5: A Log “Jam Session”

Time: 3.5 hours

Description

Students extend their knowledge of logarithms and logarithmic graphs, and apply logarithms in context to real-life situations involving sound. Students experiment with different sound levels from a portable radio or CD/tape player to determine the intensity of sound at various levels of volume.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE5g - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.02 - define and apply logarithmic functions;

ACV.01 - analyse models of linear, quadratic, exponential, or trigonometric functions drawn from a variety of applications.

Specific Expectations

EL2.04 - solve simple problems involving logarithmic scales;

AC1.01 - determine the key features of a mathematical model (e.g., an equation, a table of values, a graph) of a function drawn from an application;

AC1.02 - compare the key features of a mathematical model with the features of the application it represents;

AC1.03 - predict future behaviour within an application by extrapolating from a given model of a function;

AC1.04 - pose questions related to an application and use a given function model to answer them.

Prior Knowledge & Skills

· Understanding of the key properties of logarithmic functions and their graphs;

· How to identify the model (graphical or algebraic) used in an investigation;

· How to interpolate, extrapolate, and apply their results from an investigation.

Planning Notes

· Prepare worksheets.

· Discuss with students the definitions of decibel, Richter Scale, and pH.

· Make sure that students know how to use and read a sound-level meter.

· Students need access to the Internet to complete Student Activity 4.5.2. Teachers may investigate websites to recommend to students. Refer to Resources for suggested websites.

· To collect data, students require a radio or a CD/tape player (one per group) and a sound-level meter (one per group). If the Mathematics Department does not have a sound-level meter, sound-level meters may be available through the Science or the Technology Department.

· Note: An alternate method of collecting data is to use a calculator-based laboratory (CBL) and a microphone.

· Students require graph paper.

· Students should work in groups of two to four for Student Activity 4.5.3.

Teaching/Learning Strategies

Teacher Facilitation

· Review with the class applications of logarithms and real life models that are logarithmic in nature (compound interest, pH levels, Richter scale, decibel scale). Briefly explain the decibel scale and discuss its usefulness in a variety of contexts.

· Discuss the function of logarithms, i.e., they allow us to make very large or very small numbers more manageable to work with, and discuss the function of logarithmic scales, i.e., they measure quantities that can have a very large range.

· Discuss the concept of the intensity of sound. Relate different levels of sounds to music concerts, machinery, a car horn, barely audible whispers, and to sounds not detectable by the human ear.

· Prepare students for the activity by discussing hearing: The human ear is capable of hearing a wide range of sounds. The intensity of sounds, and related electronic measurements are often expressed in decibels (abbreviated as dB). The dB is not an absolute measurement; it is based upon the relative intensity between two sounds. Furthermore, it is a logarithmic concept, so that when comparing very large ratios, it can be expressed with small numbers.

· Students need practice and instruction on how to use a sound-level meter properly.

· Student Activity 4.5.1 provides exercises designed to introduce decibels and how decibels relate to different levels of loudness.

· Divide students into small groups of two to four. Provide each group with a sound-level meter to gather data and distribute Student Activity 4.5.1.

· Complete #1(a) together as a class; students complete the remaining questions in their groups.

Student Activity 4.5.1

The formula for computing the decibel relationship between two sounds of intensity A and B is given by the formula X =10 log, where A is the intensity at sound level X and B is a standard reference intensity near the lower level of human hearing.

Questions

1. Table 1 shows a comparison of intensity ratios and their sound level equivalents measured in decibels. Note that if the intensity of a sound is increased by a multiple of 10, the sound level increases by 10 dB, but if the intensity is multiplied by 100, the sound level only increases by 20 dB.

Table 1 – Ratio of Intensity of Sound Compared to dB level

	2	3	5	10	20	32	100	1000
	3	5	7	10	13	15	20	30

Using the table, how many times more intense is a sound of 20 dB than a sound of

a) 15 dB, b) 10 dB, c) 5 dB

2. a) Choose two additional noisy events in your daily life at school to measure. Add them to the first column. Use sound-level meters to measure the noise level (both low and high level readings) of each of the following:

Event measured	Low Reading (dB)	High Reading (dB)	Average Reading (dB)
Conversational Speech in the classroom
Cafeteria during a lunch period
Announcements

b) Determine an average reading for each event measured.

c) Which aspects of your daily life at school are the noisiest? Propose measures that could be used to reduce noise levels in your environment. Provide justification why these measures would be both effective and necessary.

Teacher Facilitation

· Discuss the findings of the class for Student Activity 4.5.1. Discuss students’ surroundings and the effects of prolonged exposure of high intensity to sound (gathered in #2). Brainstorm ways of reducing or eliminating loud sounds in the classroom, community, work, and home environments.

· Student Activity 4.5.2 is designed for students to consider the noise in their own environment and provides an opportunity for research into the effects of noise pollution and measures to control noise pollution. Students require time to complete their Internet research.

· Distribute graph paper and Student Activity 4.5.2.

· Two columns in Table 2 are already complete. Illustrate to the class how the number of maximum hours for these two sound levels (90 dB and 95 dB) are calculated. Students complete the rest of the worksheet.

Student Activity 4.5.2

Part A

Listening to very loud sounds over a sustained period of time can permanently damage a person’s hearing. Table 2 below shows various dB levels and the approximate corresponding maximum number of hours of exposure recommended in order to avoid hearing loss. Note that for every 5 dB increase in sound, the number of hours of maximum exposure at that sound level is reduced by one-half (in order to avoid hearing loss). Use this information to complete the following table:

Table 2 – Protecting your hearing

Sound Level (dB)	90	92	95	97	100	102	105	110
Maximum number of hours, h, of exposure per day in order to avoid hearing loss	8		4

Part B

1. Automobiles and Traffic are high contributors to noise in the environment. The following table summarizes the noise pollution of cars and trucks at various speeds.

Speed (km/h)	Noise at 20 m (dB)
	Auto	Medium Truck	Heavy Truck
10	15	20	31
48	60	70	81
64	68	80	84
80	71	83	86
96	74	86	88
112	78	87	89

a) Graph the following relationships:

i) speed versus noise at 20 m of an auto ii) speed versus noise at 20 m of a medium truck
iii) speed versus noise at 20 m of a heavy truck.

b) Use the table and your graph to describe the relationship between speed and noise level.

Teacher Facilitation

· Discuss students’ findings. As an alternative to a teacher-led discussion, students could briefly present their solutions to the class.

· For Student Activity 4.5.3, students need to attach the sound-level meter to their headphones using masking tape. Students take the sound level at equal increments on the volume control of the radio/CD player/tape player. If the volume setting does not have numbers, students can use correction fluid or some marking device to make eight to ten equal increments on the volume control. At each increment, students measure the sound level produced and record it in a table.

· Students continue to work in groups. Each group needs a radio or CD/tape player, a sound level meter, and graph paper. Distribute Student Activity 4.5.3.

Student Activity 4.5.3

Follow the instructions to complete the investigation

1. Test the sound-level meter to make sure that it is working properly and that you know how to measure the readings correctly.

2. Using the numeric settings on your volume control, determine the increments on the CD/tape player you are going to measure with the sound-level meter. If there are no settings, make eight to ten equally spaced marks on the control dial, beginning with no volume.

3. Attach the headphones to the microphone of the sound-level meter using masking tape.

4. Beginning in the off position (volume setting 0), measure the sound level at each increment on the volume control. Record the sound level in Table 3.

5. Repeat the procedure for other radio/CD/tape player sounds as additional trials.

Table 3 – Sound Level Readings

Volume Setting	Sound Level
	Trial 1	Trial 2	Trial 3
0
1
2
3
etc.

6. a) Make sure the volume is turned to a low level. Remove the sound-level meter from the headphones and place them on your head. Adjust the volume on the radio/CD/tape player until it reaches the volume at which you prefer to listen to music. Record this volume setting in Table 4.

b) In Table 4, record the estimated number of hours in a typical day that you would listen to music at this level.

Table 4 – Personal Data

Preferred volume setting for listening

Average Time spent listening to music

7. a) Use Table 3 to make a graph of sound level (y-axis) versus volume setting (x-axis).

b) Is there any obvious relationship that exists between the sound level and volume setting?

8. Use the graph in #7 to determine the sound level corresponding to your preferred listening volume setting

9. a) Use Table 2 to determine the maximum time you should be listening to your music at the volume you prefer.

b) Are your music habits potentially dangerous for your hearing? Explain.

c) If you lower the volume by two settings from your preferred volume setting, determine the maximum length of time you can now listen to your music without causing any permanent hearing damage.

10. a) For each volume setting, suggest the maximum number of hours of listening and justify your results.

b) Summarize your findings by drawing a volume line (like a number line). On it, place the decibel rating and the suggested number of hours.

Assessment & Evaluation of Student Achievement

· Thinking/Inquiry/Problem Solving can be assessed formatively by having students hand in (as groups) the investigation of Student Activity 4.5.3.

· Student Activity 4.5.1 can be assessed for Application.

· Knowledge/Understanding can be assessed using a quiz.

Extension

1. A student is considering the purchase of a new car stereo and thinks he needs 150 W amplifier producing 100 W of power. The student has asked you to determine if this is a good choice. To aid the student, you need to investigate:

a) the possible intensity of the sound from this system at a distance of r = 1 m (r represents radius or distance from the sound source);

b) the corresponding sound level. Use the equation I = , where P is the power in watts and I is the intensity measured in W/m². I₀ = 10^-12 W/m². Show all the calculations and recommend if the purchase of this stereo is a wise choice. Explain.

2. Using the Internet, research:

a) five major noise polluters in our community;

b) five effects that noise has on human health and hearing;

c) five occupations where hearing loss can be a major hazard. Research the safety measures these occupations use to attempt to protect the hearing of employees.

Accommodations

The activity needs to be adapted for students with hearing impairments.

Resources

http://www.nonoise.org/resource/trans/highway/spnoise.htm

http://www.noisesolutions.com/

http://www.lhh.org/noise/index.htm

http://www.nonoise.org

http://www.noisesolutions.com/ .

Activity 4.6: Logarithms and Applications

Time: 2 hours

Description

Students use previous knowledge of logarithms to develop and examine the laws of logarithms. Logarithms are then used to solve problems involving logarithmic scales. Authentic data relating to pH scales, decibel scales, and Richter scales is gathered and used in problem solving.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written material effectively;

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.02 - define and apply logarithmic functions;

ACV.04 - demonstrate facility in carrying out and applying key manipulation skills.

Specific Expectations

EL2.03 - simplify and evaluate expressions containing logarithms, using the laws of logarithms;

EL2.04 - solve simple problems involving logarithmic scales.

Prior Knowledge & Skills

· Understanding of the key properties of logarithmic functions and their graphs

· The decibel scale, the Richter Scale, and the pH scale

· How to calculate logarithms on a scientific calculator

· The relationship x = a^y y = log_ax

· Exponent laws

Planning Notes

· Prepare worksheets.

· Students require scientific calculators.

· Students require the Internet to research data for Student Activity 4.6.2. Students may search for their own data using search engines. However, if the teacher is recommending websites, they should be screened.

· An alternate to researching pH levels for substances may be to use the pH meter from the Science Department. In this case, simply dip the meter into the solution and the digital output shows the pH level.

Teaching/Learning Strategies

Teacher Facilitation

· In Student Activity 4.6.1, students develop an understanding of the product law of logarithms, the quotient law of logarithms, and the power law of logarithms. Students can work in pairs.

· Distribute Student Activity 4.6.1.

Student Activity 4.6.1 - Investigating the laws of logarithms

1. a) Use your knowledge of logarithms to complete the chart.

logaB	logaC	logaB + logaC	loga(B x C)
log216 =	log24 =		log264 =
log381 =	log39 =		log3729 =
log464 =	log416 =		log41024 =

b) What pattern do you notice in each row?

c) Write instructions, in words, for evaluating log_a(B x C).

d) Use the information in the chart to write the product law of logarithms.

2. Evaluate using the product law of logarithms:

a) log₆2 + log₆108 b) log₁₀5 + log₁₀20 c) log₄2 + log₄32

3. a) Use your knowledge of logarithms to complete the chart.

logaB	logaC	logaB - logaC
log5125 =	log55 =		=
log2256 =	log264 =		=
log3243 =	log327 =		=

b) What pattern do you notice in each row?

c) Write instructions, in words, for evaluating .

d) Use the information in the chart to write the quotient law of logarithms.

4. Evaluate using the quotient law of logarithms:

a) log₈1024 – log₈2 b) log₃108 – log₃4 c) log₂160 – log₂10

5. Complete the following chart

logaB	logaB^D	D(logaB)
log88 =	log88^3 =	3(log88) =
log33 =	log33^7 =	7(log33) =
log24 =	log24^5 =	5(log24) =

b) What pattern do you notice in each row?

c) Write instructions, in words, for evaluating log_aB^D.

d) Use the information in the chart to write the power law of logarithms.

6. Evaluate:

a) log₅25¹⁶ b) log₉81⁴³ c) log₄64⁹

Teacher Facilitation

· Discuss students’ findings. Summarize the laws of logarithms on the board. Ensure that the laws of logarithms are explained and clarified as needed.

· Students may need additional time to work with practice questions that use the laws of logarithms. Provide students with practice questions resembling questions #2, #4, and #6.

· Explain or review the purpose of pH, decibel, and Richter logarithmic scales. Discuss the concept of hydrogen concentration [H⁺] and its application with pH scales.

· Complete examples of simple problems involving pH, decibel, and Richter logarithmic scales together with the class. The following formulas are useful: pH = -log[H⁺]; M = log, where M is the magnitude of an earthquake, I is the intensity of the earthquake and S is the intensity of a “standard” earthquake; X =10dB log for decibels, used in an earlier activity.

· Sample examples:

a) Lemon Juice has pH 2.3. Calculate the [H⁺].

b) The [H⁺] level of tomato juice is 3.2 x 10^-4mol/L. Calculate the pH.

c) Port Hope experienced an earthquake of magnitude 1.7 on March 22, 2001. What would be the measure of an earthquake that is double the intensity?

d) How many times louder than a mosquito buzzing at 40 dB is a hair dryer at 70 dB?

Give students additional examples of similar questions to complete in pairs.

· For Student Activity 4.6.2, students use the Internet to gather authentic data relevant to logarithmic scales. Students may work in pairs. Distribute Student Activity 4.6.2.

Student Activity 4.6.2

1. The pH scale measures the acidity of a substance. Complete the chart by first researching the pH level of the given substances. Choose two additional substances to add to the last two rows. Then, calculate the associated [H⁺] for each substance.

Substance	Researched pH level	Calculated [H+].
Milk
Diet Cola
Regular Cola
Water
Orange juice
Grapefruit juice
Vinegar
Cleaning Fluid

2. The Richter Scale is used to measure the relative magnitude of earthquakes.

a) Locate information (location, date, and magnitude) for six earthquakes that are less than 8.9 on the Richter Scale and complete the first two columns in the chart:

Date and location	Magnitude on the Richter Scale	Calculate the magnitude of an earthquake twice as intense	How many times more intense is this earthquake than an earthquake measuring 8.9 on the Richter Scale?
1.
2.
3.
4.
5.
6.

b) Use logarithms to complete the third column of the chart.

c) Use logarithms to complete the fourth column of the chart.

3. The Decibel Scale is used to measure the intensity level of sound.

a) Locate the decibel level for five sounds greater than 30 dB and complete the first two columns of the chart:

Sound	dB level	How many times more intense is this sound than a soft whisper of 30 dB?	How many times more intense is this sound than a wristwatch ticking at 20 dB?
1.
2.
3.
4.
5.

b) Use logarithms to complete the third column of the chart.

c) Use logarithms to complete the fourth column of the chart.

Assessment & Evaluation of Student Achievement

· A quiz can be used to assess Knowledge/Understanding of Student Activity 4.6.1.

· For Student Activity 4.6.2, assess Teamwork if students work in pairs

· For Student Activity 4.6.2, use questions #1 and #2 as learning tasks. Collect question #3 to assess for Application, and have students write a summary of their findings for question #3 to be assessed for Communication.

Resources

http://www.nal.usda.gov:8001/Safety/SISAppen.pdf (for pH information)

http://www.gp.uwo.ca/docs/eqlist.html (for earthquake information)

http://www.pgc.nrcan.gc.ca/seismo/table.htm (for earthquake information)

http://www.equakealert.com/bc_earthquakes/intense.htm (for earthquake information)

http://www.pgc.nrcan.gc.ca/seismo/eqinfo/eq-westcan.htm (for earthquake information)

Activity 4.7: Summative Assessment: Springfield and Shelbyville’s History

Time: 1.5 hours

Description

Students demonstrate their knowledge of exponential functions and logarithms in a variety of applications.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written material effectively;

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE5g - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others;

CGE7b - a responsible citizen who accepts accountability for one’s own actions.

Strand(s): Exponential and Logarithmic Functions, Applications and Consolidation

Overall Expectations

ELV.01 - demonstrate an understanding of the nature of exponential growth and decay;

ELV.02 - define and apply logarithmic functions;

ACV.04 - demonstrate facility in carrying out and applying key manipulation skills.

Specific Expectations

EL1.04 - describe the significance of exponential growth and decay within the context of applications represented by various mathematical models;

EL1.05 - pose and solve problems related to models of exponential functions drawn from a variety of applications, and communicate the solutions with clarity and justification;

EL2.02 - express logarithmic equations in exponential form, and vice versa;

EL2.03 - simplify and evaluate expressions containing logarithms, using the laws of logarithms;

EL2.04 - solve simple problems involving logarithmic scales;

AC4.04 - demonstrate facility in using manipulation skills related to solving linear, quadratic, and polynomial equations, simplifying rational expressions, and operating with exponents.

Prior Knowledge & Skills

· Knowledge of the concepts introduced and examined throughout Activities 4.1 to 4.6

· Applications related to exponential growth/decay and logarithmic scales

Planning Notes

· Prepare worksheets.

· Students require graph paper and graphing calculators. Provide students with window settings for all graphing calculator activities.

· For Part B, each group needs a temperature probe, a calculator-based laboratory (CBL) unit with unit-to-unit link cable, and a cup of hot water.

Teaching/Learning Strategies

Teacher Facilitation

· Students can be placed in groups of two or three or they may complete the activity individually.

· If students are completing the activity individually, the teacher may allow them time to brainstorm ideas in groups of three or four at the beginning of each class.

· Distribute graphing calculators, graph paper, and Student Activity 4.7.1.

Student Activity 4.7.1

Part A

1. The neighbouring towns of Springfield and Shelbyville were both founded in the year 1900. Springfield started with 100 settlers and had an average yearly growth rate of 5%. Shelbyville started with only 40 settlers but had a growth rate of 8%. Assume a constant rate of growth for each town.

a) Determine a formula relating the population of Springfield, P, and the number of years, t.

b) Determine a formula relating the population of Shelbyville, P, and the number of years, t.

c) Determine the total population of each town in the year 1950.

d) Use a graphical model to determine the number of years it will take each town to reach a population of 75 000. Verify your solution using your algebraic model.

e) Use a graphical model to determine in what year Springfield and Shelbyville will have the same population. Verify your solution using your algebraic model.

2. Distance in kilometres above sea level is given by the formula , where P is the atmospheric pressure measured in kiloPascals, kPa.

a) At the top of the highest mountain in Shelbyville, the atmospheric pressure was recorded as being 220 kPa. Calculate the height of the mountain above sea level.

b) The town of Springfield has a mountain with a peak 4.5 km above sea level. Calculate the atmospheric pressure at the top of the mountain.

3. In the year 1980, both towns had an earthquake. Springfield’s earthquake measured 7.5 on the Richter Scale while the earthquake in Shelbyville measured a 6.4. Determine the difference in magnitude of the two earthquakes.

4. The Earthquake uncovered an archeological find in Shelbyville and a fossil was uncovered. The formula for the amount of carbon-14 remaining in a fossil is , where M(t) is the amount of carbon-14 in the fossil at time t, and M₀ is the original amount of carbon-14. Use a graphical model to calculate the age of the fossil if 20% of the original amount of carbon is remaining. Verify your solution using the formula.

5. In 1960, the city of Springfield set up a disaster relief fund based on donations. The amount of money, A, that Springfield must invest compounded annually at 8%/a in order to have B dollars in 20 years is represented by the equation 20(log1.08) + logA = logB.

a) How much money should Springfield have invested in 1960 in order to have its investment grow to $1 000 000 in 1980 when the earthquake hit?

b) How would this information be represented in an exponential equation?

6. In 1970, Shelbyville’s budget was at a surplus. City council decided to invest the $400 000 surplus at 4.25%/a compounded annually.

a) Create a table of values and graph the relationship between the amount of money, A, and the number of years, n, using a graphing calculator.

b) Describe how the graph would change if the interest rate were 7%.

c) Describe how the graph would change if $400 000 were invested initially.

d) Determine an equation relating the amount of money and the number of years.

e) Use the equation to determine the accumulated amount of money available when the earthquake hit.

f) Use a graphical model to determine how long would it take for the money in the fund to reach
$1 000 000. Verify your solution using your equation in part (d).

Part B (to be completed in groups of two or three)

1. Use a temperature probe to gather data about the temperature of your cup of hot water as it cools. Measure the temperature each minute for 10 minutes.

2. Create a graphical model of your results. Describe the features of your graph (include domain, range, intervals where increasing, intervals where decreasing, equation of asymptote, and intercepts). What type of graph does it represent?

3. Create an algebraic equation to model your data. Are there any restrictions on your equation? Explain fully.

4. Write a summary of your findings about the rate of cooling of the cup of hot water.

Assessment & Evaluation of Student Achievement

· Use a rubric for the assessment and evaluation of student achievement (see Appendix 4.2).

Appendix 4.1

Rubric for Assessment of Student Activity 4.2.3
* Use this rubric to provide formative feedback for student Activity 4.2.3

Category	Level 1 (50-59%)	Level 2 (60-69%)	Level 3 (70-79%)	Level 4 (80-100%)
Thinking/Inquiry/ Problem Solving Creation of an algebraic model to represent the data	- develops an algebraic model with limited effectiveness	- develops an algebraic model with some effectiveness	- develops an algebraic model with considerable effectiveness	- develops an algebraic model with a high degree of effectiveness
Communication Clarity of conclusions and recommendations made to the city planner	- makes conclusions and recommendations with limited clarity	- makes conclusions and recommendations with some clarity	- makes conclusions and recommendations with considerable clarity	- makes conclusions and recommendations with a high degree of clarity

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

Appendix 4.2

Rubric for Evaluation of Activity 4.7.1

Category/ Criteria	Level 1 (50-59%)	Level 2 (60-69%)	Level 3 (70-79%)	Level 4 (80-100%)
Knowledge/ Understanding Understanding Concepts	- demonstrates a limited understanding of concepts relating to exponents and logarithms	- demonstrates some understanding of concepts relating to exponents and logarithms	- demonstrates considerable understanding of concepts relating to exponents and logarithms	- demonstrates a thorough understanding of concepts relating to exponents and logarithms
Application Application of concepts or procedures	- applies concepts or procedures with limited effectiveness	- applies concepts or procedures with some effectiveness	- applies concepts or procedures effectively	- applies concepts or procedures with considerable effectiveness and efficiency
Communication Clarity of explanations	- demonstrates limited clarity in explanations	- demonstrates some clarity in explanations	- demonstrates considerable clarity in explanations	- demonstrates clear explanation consistently
Thinking/ Inquiry/Problem Solving Creation of a model (either graphical or algebraic)	- creates exponential or logarithmic models that address few aspects of the problem	- creates exponential or logarithmic models that address some aspects of the problem	- creates appropriate exponential or logarithmic models to address the problem	- creates appropriate exponential or logarithmic models that integrate all or almost all aspects of the context

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

Overview | Unit 3 | Course Profiles Main Menu