Unit #5 : Exponential Functions (10 days + 1 jazz day + 1 summative evaluation day) BIG Ideas:

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1 Unit #5 : Eponential Functions (0 days + jazz day + summative evaluation day) BIG Ideas: Students will: Collect primary data and investigate secondary data that can be modelled as eponential growth/decay functions Make connections between numeric, graphical and algebraic representations of eponential functions Identify key features of the graphs of eponential functions (e.g., domain, range, y-intercept, horizontal asymptote, increasing and decreasing) Apply an understanding of domain and range to a variety of eponential models Solve real-world applications using given graphs or equations of eponential functions Simplify and evaluate numerical epressions involving eponents DAY Lesson Title & Description 2P 2D Epectations Teaching/Assessment Notes and Curriculum Sample Problems Piles of Homework Distinguish eponential functions from linear and quadratic by eamining tables of values and graphs Lesson Included Investigating Eponential Growth Collect data that can be modelled as eponential growth functions through investigation and from secondary sources Make connections to First Differences and constant ratios N N EF.06 distinguish eponential functions from linear and quadratic functions by making comparisons in a variety of ways (e.g., comparing rates of change using finite differences in tables of values; identifying a constant ratio in a table of values; inspecting graphs; comparing equations), within the same contet when possible (e.g., simple interest and compound interest; population growth) Sample problem: Eplain in a variety of ways how you can distinguish the eponential function f () = 2 from the quadratic function f () = 2 and the linear function f () = Lesson Included Investigating Eponential Decay Collect data that can be modelled as eponential decay functions Make connections to First Differences and constant ratios Lesson Included N N EF2.0 collect data that can be modelled as an eponential function, through investigation with and without technology, from primary sources, using a variety of tools (e.g., concrete materials such as number cubes, coins; measurement tools such as electronic probes), or from secondary sources (e.g., websites such as Statistics Canada, E-STAT), and graph the data Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction. Grade U/C Unit 5: Eponential Functions

2 4 Investigating The Graphs of Eponential Functions Day Graph eponential functions in the form y = ab where b>0 and a= Identify key features (y-intercept, increasing or decreasing, domain and range, horizontal asymptotes, constant ratio) Lesson Included EF.03 EF.04 graph, with and without technology, an eponential relation, given its equation in the form y = a ( a > 0, a ), define this relation as the function f () = a, and eplain why it is a function; determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for eponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form f () = a (a > 0, a ), function machines] Sample problem: Graph f () = 2, g() = 3, and h() = 0.5 on the same set of aes. Make comparisons between the graphs, and eplain the relationship between the y-intercepts. 5 Investigating The Graphs of Eponential Functions Day 2 Graph eponential functions in the form y = ab where b>0 and a> Identify key features (y-intercept, increasing or decreasing, domain and range, horizontal asymptotes, constant ratio) N N Lesson Included 6 Domain and Range in Real World Applications Identify eponential functions that arise from real world applications involving growth and decay Determine reasonable restrictions on the domain and range N N EF2.02 identify eponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and eplain any restrictions that the contet places on the domain and range (e.g., ambient temperature limits the range for a cooling curve); Lesson Included Grade U/C Unit 5: Eponential Functions 2

3 7 How an Infectious Disease can Spread Simulate the spread of an infectious disease and analyze the results Determine restrictions that must be placed on the domain and range in order to apply an eponential model N N EF2.02 identify eponential functions, including those that arise from real-world applications involving growth and decay (e.g., radioactive decay, population growth, cooling rates, pressure in a leaking tire), given various representations (i.e., tables of values, graphs, equations), and eplain any restrictions that the contet places on the domain and range (e.g., ambient temperature limits the range for a cooling curve); This activity requires advanced preparation. Full Teacher Notes are provided in BLM Lesson Included 8,9 Developing and Applying Eponent Laws Investigate to develop eponent laws for multiplying and dividing numerical epressions involving eponents and for finding the power of a power. Investigate to find the value of a power with a rational eponent (e.g., use a graphing calculator to find the value for 2 4 or 3 27 by entering an eponential function with the given base and then using TRACE.) Evaluate numerical epressions with rational bases and integer/rational eponents. Note: Students only work with numerical epressions N N N C N C EF.05 EF.0 EF.02 determine, through investigation (e.g., by patterning with and without a Note: Students don t actually solve calculator), the eponent rules for multiplying and dividing numerical eponential equations in this course so the epressions involving eponents [e.g.,(½) 3 (½) 2 ], and the eponent main use of these eponent rules would likely rule for simplifying numerical epressions involving a power of a power be to help develop an understanding of [e.g.,(5 3 ) 2 ], and use the rules to simplify numerical epressions rational eponents (see sample problem containing integer eponents [e.g., (2 3 )(2 5 ) = 2 8 ]; below) and to understand the compound interest formula determine, through investigation using a variety of tools (e.g., calculator, paper and pencil, graphing technology) and strategies (e.g., patterning; finding values from a graph; interpreting the eponent laws), the value m n of a power with a rational eponent (i.e.,, where > 0 and m and n are integers) evaluate, with and without technology, numerical epressions containing integer and rational eponents and rational bases [e.g., 2-3, ( 6) 3, 2 4,.0 20 ]; Sample problem: The eponent laws suggest that = 4. What value would you assign to 2 4? What value would you assign 3 to 27? Eplain your reasoning. Etend your reasoning to make a generalization about the meaning of n, where > 0 and n is a natural number. Suggestion: Teachers may want to have students eplore on sketchpad or with a graphing calculator. Students can graph y = 4 and then eamine the y-value when = ½ and then graph y = 9 and eamine the y- value when = ½ and so on. and/or Grade U/C Unit 5: Eponential Functions 3

4 0 Using Graphical and Algebraic Models Students will solve problems using given graphs or equations of eponential functions Help students make connections between the algebraic model of the eponential function and the real-world application (i.e. help students understand the meanings of a and b in the contet of the problem) Note: Students are not required to generate the equation on their own, but should be encouraged to eplain the parameters in the contet of the problem. N N EF2.03 solve problems using given graphs or equations of eponential functions arising from a variety of real-world applications (e.g., radioactive decay, population growth, height of a bouncing ball, compound interest) by interpreting the graphs or by substituting values for the eponent into the equations Sample problem: The temperature of a cooling liquid over time can be modelled by the eponential function 30 T( ) = , where T() is the temperature, in degrees Celsius, and is the elapsed time, in minutes. Graph the function and determine how long it takes for the temperature to reach 28ºC. Lesson Included Review Day (Jazz Day) 2 Summative Evaluation Grade U/C Unit 5: Eponential Functions 4

5 Unit 5 : Day : Piles of Homework Minds On: 20 Action: 35 Consolidate:20 Total=75 min Minds On Description/Learning Goals Distinguish eponential functions from linear and quadratic functions in a variety of ways (eg. Comparing rates of change using finite differences; inspecting graphs, identifying a constant ratio in a table) Individual Activating Prior Knowledge Paper folding review of eponents Students receive a piece of paper. Have students fold paper in half and record the number of layers. Repeat this process and have students record the number of folds and the number of layers created by the fold. Connections can then be made to powers of 2. Assessment Opportunities Grade U/C Materials BLM 5.. BLM 5..2 BLM 5..3 Bundle of photocopy paper 6 file folders post-it notes 25 paper clips Action! Consolidate Debrief Concept Practice Whole class Guided Eploration Write the list as a sequence to look for a pattern (constant ratio). Write some other patterns that use constant ratio on the board and ask students to complete the patterns and identify how they filled in the missing parts. Jigsaw Investigation Students are first organized into home groups of 3. Each member will then be assigned a number. The home groups will then divide into 6 groups. There should be two groups of s, 2s, and 3s. In each group, one student is selected to play the role of Teacher. Provide each group a set of instructions from BLM 5... The Teacher in the group receives a file folder that has groups of paper clipped together. Each set of clipped papers represents the homework for one day. The Teacher hands out the homework one day at a time and the students will record how many pages of homework they have received in total. The students record the work on BLM After recording the data, the students return to the home group to share the data. In the home group, the students complete the remainder of BLM 5..2 Mathematical Process Focus: Problem solving (students decide on which model is being shown), connecting (students are using skills developed for selecting a type of model), representing (students represent a situation with a model). Whole Class Guided discussion Discuss with the class the questions from BLM 5.. about how much homework they would have received in 30 days from each class. Whole Class Note taking Guiding question: Can first and second differences be used to determine if a model is an eponential model? The teacher guides the class through developing a note and introduction of key term constant ratio. Begin to build the Eponential FRAME with students (BLM 5..3). Home Activity or Further Classroom Consolidation The teacher provides students with additional tables of values that include linear / quadratic/ eponential values and students are to try and determine which model the table of values represents. For the Action! component, have paper available for learners to use for representation. For the small group activity, it is recommended to try a jig-saw form of cooperative learning. Additional information on the jig-saw form can be found in Think Literacy. Crosscurricular Approaches Grades pg. 70. Literacy Strategy: Students can begin to add information to the Eponential FRAME Model (BLM 5..3). (F unction, R epresentation, A nd, M odel, Eample) The FRAME can be etended throughout the unit. Grade U/C Unit 5: Eponential Functions 5

6 5.. Piles of Homework Photocopy this sheet and cut into sections to give to groups. Scenario: You have discovered that three of your teachers give out homework in different ways. How much homework will you have on your homework pile? Class A How much homework do you get from your teacher? Record how many pieces of paper you have in total at the end of each day in your table. After 30 days, how many pieces of paper will you have been given? Scenario: You have discovered that three of your teachers give out homework in different ways. How much homework will you have on your homework pile? Class B How much homework do you get from your teacher? Record how many pieces of paper you have in total at the end of each day in your table. After 30 days, how many pieces of paper will you have been given? Scenario: You have discovered that three of your teachers give out homework in different ways. How much homework will you have on your homework pile? Class C How much homework do you get from your teacher? Record how many pieces of paper you have in total at the end of each day in your table. Note: Is there a connection between how many pieces of paper you have on your pile and how many pieces of paper you get the net day? After 30 days, how many pieces of paper will you have been given? Grade U/C Unit 5: Eponential Functions 6

7 5.. Piles of Homework (continued) Teacher notes. How to organize the paper for Class A Day Day 2 Day 3 Day 4 Day 5 sheet of paper 2 sheets of paper 2 sheets of paper 2 sheets of paper 2 sheets of paper How to organize the paper for Class B Day Day 2 Day 3 Day 4 Day 5 sheet of paper 3 sheets of paper 5 sheets of paper 7 sheets of paper 9 sheets of paper How to organize the paper for Class C Day Day 2 Day 3 Day 4 Day 5 sheet of paper 2 sheets of paper 6 sheets of paper 8 sheets of paper 54 sheets of paper It is recommended you use a post-it note to stick on the top of each pile in order for the students to know which day of homework they are receiving. Grade U/C Unit 5: Eponential Functions 7

8 5..2 Piles of Homework Scenario: You have discovered that three of your teachers give out homework in different ways. How much homework will you have on your homework pile? Record your information from your class. When you return to your home group, you will share your data with the other group members and you will receive the data from the other two classes. Class A Amount Day of work First Differences Second Difference What type of relationship is this model? Justify your answer. Class B Amount Day of work First Differences Second Difference What type of relationship is this model? Justify your answer. Grade U/C Unit 5: Eponential Functions 8

9 5..2 Piles of Homework (continued) Class C Amount Day of work First Differences Second Difference What type of relationship is this model? Is there a pattern to the first differences? Describe the pattern. Is there a pattern to the second differences? Describe the pattern. Revisiting The Paper Folding Activity Record your information from the paper-folding activity at the start of class. Complete the chart. Fold Number of layers First 0 Differences Second Difference Is there a pattern to the first differences and second differences? If yes, describe the pattern. Grade U/C Unit 5: Eponential Functions 9

10 5..3 Eponential FRAME Numerical Model Algebraic Model Description/Key words Visual/Spatial/Concrete Graphical Model Contetual Grade U/C Unit 5: Eponential Functions 0

11 Unit 5 : Day 2 : Investigating Eponential Growth Grade U/C Minds On: 0 Action: 45 Consolidate:20 Total = 75 min Minds On Action! Consolidate Debrief Reflection Description/Learning Goals Materials Collect data through investigation from primary sources that can be modelled BLM 5.2. as eponential growth functions. BLM Investigate secondary sources of data that can be modelled as eponential BLM growth functions with and without the use of technology. BLM Graph the data with and without technology. BLM Identify eponential functions that arise from real-world applications involving (Teacher) growth. 4 guitars 4 measuring tapes 4 graphing calculators Assessment Opportunities Whole Class Discussion Describe the four centres (BLM 5.2., BLM 5.2.2, BLM 5.2.3, BLM 5.2.4) as necessary to prepare the students for the investigation. Connect the activity to the previous day s work on eponential functions. Pairs Investigation Students will work in pairs and complete four different investigations related to eponential growth (BLM 5.2., BLM 5.2.2, BLM 5.2.3, BLM 5.2.4). Students will have approimately 0 minutes to work at each activity before moving to the net station. The students should be encouraged to collect all data and complete the graphs in the time that they have. They may need to take time to complete the questions related to the investigations at home. Ensure that all students record the data on their own observation sheets. Mathematical Process Focus: Representing (Students will represent applications of Eponential Growth graphically, numerically, pictorially and concretely.) Whole Class Discussion Direct the students to review the First Differences and constant ratios computed in BLM 5.2., BLM 5.2.2, BLM and BLM5.2.4 and have students look for patterns. Consider the following Guiding Questions: a) Why does an eponential model fit the data you have eamined? b) How can you determine that the data is eponential from the table of values? c) What was common in the shapes of the curves? (Sample response: They all increased over the domain, first slowly and then quickly.) Add to the FRAME Model by including various representations of Eponential Growth. Home Activity or Further Classroom Consolidation Students will reflect on the data they collected and consolidate their understanding by completing the questions provided on BLM investigation sheets. Students would benefit from a review on how to create a tree diagram and how to set up appropriate scales on aes for graphing. Set up 4 stations for each activity. This gives 6 stations in total, which will accommodate 32 students. Note: Suggested solutions for Teachers are found on BLM Literacy Strategy: Continue to add to the Eponential FRAME Model. The FRAME can be etended throughout the unit to develop the ideas further. Grade U/C Unit 5: Eponential Functions

12 5.2. Investigating Eponential Growth: Pizza Toppings INTRODUCTION In this activity, you and your partner will investigate the number of different pizzas that can be created for a given number of available toppings. INSTRUCTIONS. Fill in the chart below by drawing all the different pizzas that can be created by choosing all, some or none of the available toppings indicated. Toppings Available Different Pizza Drawings none Cheese Cheese, Pepperoni Cheese, Pepperoni, Mushrooms Cheese, Pepperoni, Mushrooms, Bacon 2. a) Use your information from question to complete the table below Number of Available Toppings 0 Number of Different Pizzas First Differences Ratio b) What do the First Differences tell us about this data? Why? 2 c) Is there a constant ratio between consecutive values in the column titled Number of Different Pizzas? 3 How does this value relate to the pattern in the First Differences? 4 c) Predict how many different pizzas could be created from 5 toppings. Justify your answer. Grade U/C Unit 5: Eponential Functions 2

13 5.2. Investigating Eponential Growth: Pizza Toppings (continued) 3. Neatly sketch a graph of your results from question 2 on the grid below. Draw a smooth curve through the points. (Note: This is DISCRETE data; however, the smooth curve assists in seeing the general shape of the graph.) NUMBER OF DIFFERENT PIZZAS 4. Using the graph comment on the shape of the curve. Use words such as the following in your description: increasing, decreasing, quickly, slowly. NUMBER OF TOPPINGS 5. Complete this statement: As the number of toppings increases by, the number of different pizza combinations 6. Predict how many different pizzas can be created if there are nine available toppings. Clearly eplain how you made your prediction. 7. If a restaurant owner would like to offer 200 different pizza combinations, what is the minimum number of available toppings she would need? Eplain your reasoning. 8. a) Your local pizza parlour offers you the choice of 5 different toppings. If you were to eat a different pizza every day, how many years would it take for you to try every possible one? (Hint: There are 365 days in a year.) b) Does this answer surprise you? Why or why not? Grade U/C Unit 5: Eponential Functions 3

14 5.2.2 Investigating Eponential Growth: Friendzy INTRODUCTION In this activity, you and your partner will use a Tree Diagram to simulate the effect of telling three friends, who each tell three friends, who each tell three friends and so on. INSTRUCTIONS. A letter is ed out to three friends. Each of these three recipients will then it to three new friends. Continue this pattern in order to complete the first four rounds in the Tree Diagram below. (Hint: Use tiny dots to represent each that is sent so that you will have enough space to draw out the entire Tree Diagram.) Round Round 2 Round 3 Round 4 2. a) Use the information from question to complete the table. Round Number of Letters ed First Differences Ratio b) Consider the Tree Diagram and the data in the first four rows of the table. Predict the Number of Letters ed in Round 5. Justify your prediction c) Comment on the patterns you see in the Number of Letters ed column and the First Differences column. d) Is there a constant ratio between consecutive values in the Number of Letters ed column? e) Comment on the value. Grade U/C Unit 5: Eponential Functions 4

15 5.2.2 Investigating Eponential Growth: Friendzy (continued) 3. Neatly sketch a graph of your results from question 2 on the grid below. Draw a smooth curve through the points. (Note: This is DISCRETE data; however, the smooth curve assists in seeing the general shape of the graph.) 4. Complete this statement: As the number of rounds increases by, the number of letters ed NUMBER OF LETTERS ED 5. Predict how many letters will be e- mailed during the 9 th round. Show how you determined this. 6. During which round will the number of letters ed eceed for the first time? Show how you determined this. NUMBER OF ROUNDS 7. Pyramid schemes work similarly to this simulation in that individuals must find others willing to invest in (or purchase a product from) a company. Consider a pyramid scheme where people are asked to invest $000 each and are required to find four more investors to do the same. How much total money will be invested in this company after three rounds? (Hint: Draw a tree diagram.) NOTE: Pyramid schemes are illegal because they usually involve fraud. Grade U/C Unit 5: Eponential Functions 5

16 5.2.3 Investigating Eponential Growth: Guitar Frets INTRODUCTION: A fret is a ridge set across the fingerboard of a stringed instrument, such as a guitar. Frets divide the neck into segments at intervals related to a musical framework. In this activity, you and your partner will measure the distances between the frets and the bridge of a guitar. (See the diagram on the accompanying page.) INSTRUCTIONS. Using a tape measure, accurately measure the distance for each indicated segment of the guitar as shown in the diagram on the accompanying page. All measurements must be taken in mm. Measurement Number Segment of Guitar 2 th fret to bridge Distance (mm) First Differences Ratio 2 th fret to bridge 3 0 th fret to bridge 4 9 th fret to bridge 5 8 th fret to bridge 6 7 th fret to bridge 7 6 th fret to bridge 8 5 th fret to bridge 9 4 th fret to bridge 0 3 rd fret to bridge 2 nd fret to bridge 2 st fret to bridge Grade U/C Unit 5: Eponential Functions 6

17 5.2.3 Investigating Eponential Growth: Guitar Frets (continued) 2. a) Is there a constant ratio between consecutive values in the column labelled Distance? If so, what is it? If not, eplain why not. b) Predict how far the 3 th fret is from the bridge and show how you made this prediction. c) Measure the distance from the 3 th fret to the bridge and compare it to your prediction. 3. Neatly sketch a graph of your results from question on the grid below.. Draw a smooth curve through the points. DISTANCE FROM FRET TO BRIDGE (mm) MEASUREMENT NUMBER 4. How would the graph change if the frets were equally spaced apart? Grade U/C Unit 5: Eponential Functions 7

18 5.2.3 Investigating Eponential Growth: Guitar Frets (continued) Diagram of Guitar Parts Source: Nut st Fret 2 th Fret Bridge Grade U/C Unit 5: Eponential Functions 8

19 5.2.4 Investigating Eponential Growth: Humpback Whale Population (Source - INTRODUCTION If a population has a constant birth rate through time, and is never limited by food, disease or threat, it has what is known as eponential growth. INSTRUCTIONS. The Humpback Whale is distributed worldwide in all ocean basins, though it is less common in Arctic waters. Barlow & Chapman (997) have estimated a population growth rate of 6.5% per year for the well-studied Humpback Whale population in the Gulf of Maine. Through various methods, the Humpback Whale population in this region was estimated to be 652 whales in 993 COMPLETE THE QUESTIONS IN THE BOX ON THE LEFT BEFORE FILLING OUT THE TABLE. THINK: a) What value do you multiply each population by if it is increasing by 6.5% per year? Year Population First Differences 995 b) Verify that your answer in (a) is correct by using it to calculate and confirm that the population for 994 is 694 (rounded to the nearest whole number) c) Complete the chart, by calculating the Population for each of the years in the table, using the growth rate of 6.5% per year. (Round answers to the nearest whole number.) Grade U/C Unit 5: Eponential Functions 9

20 5.2.4 Investigating Eponential Growth: Humpback Whale Population (continued) 2. Enter the data from the chart above into the lists L and L 2 of your graphing calculator. 3. Create a scatter plot of the data using your graphing calculator. Set your WINDOW as shown below and neatly sketch the plot as it appears on your calculator screen. Include labels and scales on your aes. 4. a) Does this data appear to be eponential from the graph? Why or why not? b) How would you convince someone that the data IS eponential? 5. Is there a relationship between the constant ratio of consecutive Population values and consecutive Finite Differences? 6. a) Predict the number of Humpback Whales that would be in the Gulf of Maine in the year Clearly eplain how you made your prediction. b) Show how you could calculate the number of whales in the Gulf of Maine in Compare this result to your prediction. Grade U/C Unit 5: Eponential Functions 20

21 5.2.5 Sample Solutions to Selected Growth Questions BLM 5.2. Pizza Toppings. Students will draw the different pizza combinations in the chart. The first row with NO toppings available will have a single pizza (plain crust). The second row with CHEESE topping will have 2 pizzas (plain, cheese). The third row with Cheese and Pepperoni will have 4 pizzas (plain, cheese, pepperoni, cheese and pepperoni). The fourth row with Cheese, Pepperoni and Mushrooms will have 8 pizzas (plain, cheese, pepperoni, mushrooms, cheese and pepperoni, cheese and mushrooms, pepperoni and mushrooms, cheese and pepperoni and mushrooms). The fifth row will have 6 different pizzas. 2. b) Students will give varied responses such as: The First Differences tell us that the data is not linear (they aren t constant). The First Differences tell us that the data is eponential because there is a constant ratio between the consecutive first differences. 2. c) There is a constant ratio of 2 in the Number of Different Pizzas column. The First Differences also double. 2. d) 32. Sample response: I made the prediction by multiplying the value in row 4 by 2 (or I doubled the value in row 4). 3. Make a scatterplot. Have students join the points with a smooth curve in order to see the shape of the graph better. Discuss the fact that this is discrete data. 4. The curve increases slowly at first and then more quickly. 5. doubles Students will probably calculate the number of toppings by successively multiplying by 2. Some students may see the pattern as 2 9 but this can be discussed in more depth in the consolidation phase of the lesson. 7. Sample approach: Students would see that the answer to the previous question was 52, so half of this is 256. Therefore, 8 toppings are needed. 8. Calculate years. 365 BLM Friendzy. The bottom row of the Tree Diagram (Round 4) will show 8 s. Encourage students to use a small dot to represent each so that all 8 will fit in the space provided. 2. b) The number of s is 243. Students may justify this by saying there is a constant ratio of 3. They may refer to the fact that there would be 3 s sent from each of the 8 individuals in Round 4, meaning there would be 83=243 s. 2. c) Students may see that both columns have a constant ratio of 3. They may comment on the fact that the First Differences are always twice as big as the values in the 2 nd column (e.g., 6 is twice as big as 3, 8 is twice as big as 9, 54 is twice as big as 27 etc.) 2. d) e) There is a constant ratio of 3. This is the same as the number of s sent by each person on each round. Grade U/C Unit 5: Eponential Functions 2

22 5.2.5 Sample Solutions to Selected Growth Questions (continued) 3. Make a scatterplot. Have students join the points with a smooth curve in order to see the shape of the graph better. Discuss the fact that this is discrete data. 4. triples Students will probably calculate the number of s by successively multiplying by 3. Some students may see the pattern as 3 9 but this can be discussed in more depth in the consolidation phase of the lesson th round. Students will likely guess and test. This will depend on their level of understanding at this point. Some may test values of 3 n while others will use successive multiplication by The total value will be $ $ $ = $ (Assuming the person at the top of the pyramid doesn t invest any money.) BLM Guitar Frets. Sample Chart showing data collected by a student. Note: Students need to be reminded to be as accurate as possible when taking the measurements. Measurement Distance (mm) (2 th fret) 32 2 ( th fret) (0 th fret) 36 4 (9 th fret) (8 th fret) (7 th fret) (6 th fret) (5 th fret) (4 th fret) 5 0 (3 rd fret) 542 (2 nd fret) ( st fret) 609 Note: The constant ratio is approimately.06. The variation is due to measurement error. 2. a) The constant ratio is close to b) To find the distance from the 3 th fret, we go the other way in the table, meaning the constant ratio is about Therefore, the 3 th fret should be = 30 mm from the bridge. 3. Make a scatterplot. Have students join the points with a smooth curve in order to see the shape of the graph better. Discuss the fact that this is discrete data. 4. The graph would be linear if the frets were equally spaced apart. Grade U/C Unit 5: Eponential Functions 22

23 5.2.5 Sample Solutions to Selected Growth Questions (continued) BLM Humpback Whales. a).065 b) Students will show is approimately 694. c) The bottom row will show there will be about 224 whales by the year a) b)the students may observe that the graph is increasing slowly at first, with a bit of an increase towards the end. Note: Some students may feel this graph looks more linear. This would lead to a good discussion as to what other information we have (besides the graph) that would help us to determine if this is linear or not. (e.g., First Differences, constant ratio) 5. Yes. They are both Students will probably use successive multiplication by.065. The number will be close to 480 depending on the rounding used by the students. Grade U/C Unit 5: Eponential Functions 23

24 Unit 5 : Day 3 : Investigating Eponential Decay Grade U/C Minds On: 0 Action: 45 Consolidate:20 Total = 75 min Minds On Description/Learning Goals Collect data through investigation from primary sources that can be modelled as eponential decay functions. Investigate secondary sources of data that can be modelled as eponential decay functions with and without the use of technology. Graph the data with and without technology. Identify eponential functions that arise from real-world applications involving decay. Whole Class Discussion Describe the four activities (BLM 5.3., BLM 5.3.2, BLM 5.3.3, BLM 5.3.4) in which the students will be engaging and connect the activity to the previous day s work on eponential functions. Assessment Opportunities Materials BLM 5.3. BLM BLM BLM BLM (Teacher) 400 dice 4 containers for shaking dice 4 bo lids Set up 4 stations for each activity. This gives 6 stations in total, which will accommodate 32 students. Action! Consolidate Debrief Reflection Pairs Investigation Students will work in pairs and complete four different investigations related to eponential growth (BLM 5.3., BLM 5.3.2, BLM 5.3.3, BLM 5.3.4). Students will have approimately 0 minutes to work at each activity before moving to the net station. The students should be encouraged to collect all data and complete the graphs in the time that they have. They may need to take time to complete the questions related to the investigations at home. Ensure that all students record the data on their own observation sheets. Mathematical Process Focus: Representing (Students will represent applications of Eponential Growth graphically, numerically, pictorially and concretely) Learning Skill/Team Work/Observation/Rating Scale: By observation, assess and record each student s level of teamwork using a four point scale (N,S,G,E). Whole Class Discussion The teacher should direct the students to review the First Differences and constant ratios computed in BLM 5.3., BLM 5.3.2, BLM and BLM and have students look for patterns. Consider the following Guiding Questions: a) Why does an eponential model fit the data you have eamined? b) How can you determine that the data is eponential, from the table of values? c) What was common in the shapes of the curves? (Sample response: They all decreased over the domain, first quickly and then slowly.) Add to the FRAME Model by including various representations of Eponential Decay. Home Activity or Further Classroom Consolidation Students will reflect on the data they collected and consolidate their understanding by completing the questions provided on BLM investigation sheets. Note: Suggested solutions for Teachers are found on BLM Suggestion: For the Radioactive Atoms (BLM 5.3.2) use a Pringles-type chip can as the container. Have students roll the dice into a bo lid to contain the dice. Literacy Strategy: Continue to add to the Eponential FRAME Model. Grade U/C Unit 5: Eponential Functions 24

25 5.3. Investigating Eponential Decay: Tiling a Floor INTRODUCTION: A lavish hotel is undergoing a renovation and has hired an interior design team to finish the very large rectangular floor in its foyer. It has been decided to use light and dark grey marble slabs to cover the 6 m by 9 m surface. (See diagram below.) Grade U/C Unit 5: Eponential Functions 25

26 5.3. Investigating Eponential Decay: Tiling a Floor (continued) INSTRUCTIONS. Using the scale cm represents m and a ruler, determine the actual length, width and area for each indicated dark grey slab and record your results in the table below. Dark Grey Slabs Slab Number Length (m) Width (m) Area (m 2 ) First Differences Ratio Repeat step for each indicated light grey slab and record your results in the table below. Light Grey Slabs Slab Number Length (m) Width (m) Area (m 2 ) First Differences Ratio Contrast these First Differences with those of eponential growth data? 4. Calculate the ratios between consecutive Areas in each table. What do these values tell us about the data? Grade U/C Unit 5: Eponential Functions 26

27 5.3. Investigating Eponential Decay: Tiling a Floor (continued) 5. Neatly sketch 2 graphs on the same grid below using your data from the tables. (Note: This is DISCRETE data; however, the smooth curve assists in seeing the general shape of the graph.) AREA OF SLAB (m 2 ) SLAB NUMBER 6. Refer back to your table and calculate the total area covered by each type of marble. 7. The light grey marble costs $ 700/m 2 and the dark grey marble costs $ 950 /m 2. Determine the amount of money saved by using light grey for the whole foyer area instead of covering the area with the combination of light and dark grey marble slabs as shown in the diagram. Grade U/C Unit 5: Eponential Functions 27

28 5.3.2 Investigating Eponential Decay: Radioactive Atoms INTRODUCTION All matter is made up of atoms. Some kinds of atoms have too much energy and are unstable. These atoms are called radioactive. It is not possible to predict eactly when a radioactive atom will release its etra energy and form a different stable atom. This process is called radioactive decay. In this eperiment you will model this process. (Source: Ministry of Education Regional Training Sessions, Grade Mathematics, Spring 2006) INSTRUCTIONS. Fill your container with 00 dice. Each die represents one atom. 2. a) Pour the dice out of the container onto a tray. The atoms with the number one facing up have decayed. Remove them and record the number of atoms remaining in the table below. Repeat the process until there are five or fewer atoms remaining. Note that each pour represents one time period (we will call each time period a day). Etend the table if needed. Number of Days 0 Number of Atoms Remaining First Differences b) How do these First Differences contrast with the First Differences of eponential growth data? c) Calculate the ratio between the first two values in the column titled Number of Atoms Remaining. What does this ratio mean in the contet of this problem? d) Calculate the ratio between the 2 nd and 3 rd values in the column titled Number of Atoms Remaining. Compare it to the ratio from (c). Account for any differences. 0 Grade U/C Unit 5: Eponential Functions 28

29 5.3.2 Investigating Eponential Decay: Radioactive Atoms (continued) 3. Neatly sketch a graph of your data. Draw a smooth curve through the points. NUMBER OF ATOMS REMAINING NUMBER OF DAYS 4. The time it takes for half of the original number of atoms to decay is called the half-life. Use your graph above to determine the half-life of your substance in days. 5. When one half-life has passed there will be 50 atoms of the original radioactive substance remaining. Use your graph to determine how long it will take for these 50 atoms to reduce to 25 atoms. 6. Use your graph to determine how long it will take for the 25 atoms to reduce to about 2 (half of 25). 7. What can you conclude about the half-life of your atoms. Grade U/C Unit 5: Eponential Functions 29

30 5.3.3 Investigating Eponential Decay: Car Depreciation INTRODUCTION: Depreciation is the decline in a car s value over the course of its useful life. It s something new-car buyers dread. Most modern domestic vehicles typically depreciate at a rate of 5%-20% per year depending on the model of the car. INSTRUCTIONS. A 2007 Ford Mustang GT convertible is valued at $ and depreciates on average at 20% per year. Since the car is depreciating at 20% per year, the remaining value at the end of the first year is % of the original value. Therefore, to find the depreciated value, multiply the previous year s value by. 2. Complete the following table to calculate the value of the car at the end of each of the first five years of ownership. Year-end Depreciated Value in $ Show Calculation in this column What would you epect the constant ratio to be for this eample. Justify your answer. Grade U/C Unit 5: Eponential Functions 30

31 5.3.3 Investigating Eponential Decay: Car Depreciation (continued) 4. Neatly sketch a graph of your data. Draw a smooth curve through the points. DEPRECIATED VALUE IN $ YEAR-END 5. If the depreciation was 5% per year, how would the constant ratio change? Grade U/C Unit 5: Eponential Functions 3

32 5.3.4 Investigating Eponential Decay: Computers in Schools INTRODUCTION: Computers have become an integral part of today s schools. However, this has not always been the case. The table below shows the average number of students per computer in public schools (elementary and secondary) since 983. (Source of data: Precalculus: A Graphing Approach, Holt, Rinehart and Winston 2002, p.398) Year Population First Differences Ratio INSTRUCTIONS. Eamine the table above. How is this data different from the eponential growth data? Grade U/C Unit 5: Eponential Functions 32

33 5.3.4 Investigating Eponential Decay: Computers in Schools (continued) 2. Neatly sketch a graph of the data from the table on the previous page. When choosing your scale for the horizontal ais, consider question 4 below. After you have plotted the points, draw a smooth curve through them. STUDENTS PER COMPUTER YEAR 3. Using the graph, comment on the shape of the curve. Use words such as the following in your description: increasing, decreasing, quickly, slowly. 4. Use your graph to predict the number of students per computer in the year Is the answer from question #4 surprising? Why or why not? Grade U/C Unit 5: Eponential Functions 33

34 5.3.5 Sample Solutions to Selected Decay Questions BLM 5.3. Tiling a Floor 3. These First Differences are negative, whereas the First Differences for eponential growth data were positive. 4. The constant ratio of 2 in the first table tells us that the areas of consecutive dark grey slabs are being cut in half. The constant ratio of 2 in the second table tells us that each consecutive light grey slab has 3 an area that is 2 of the previous slab The total dark grey area is 93 square metres and the total light grey area is 2 metres. 7. The dark grey cost is 93 $950 = $ The light grey cost is 2 $700 = $ Therefore the total is: $ $ = $ If the whole foyer is completed using the light grey, then the total area is = 304 square metres at a cost of $700 per metre is: $ The cost savings is: $ $ = $ BLM Radioactive Atoms 2. a) Values in the table will vary depending on the data collected. 2. b) The First Differences are negative, whereas the First Differences for eponential growth data were positive. 2. c) The ratio will vary depending on the data. It should be around 83% (5/6 of the atoms would theoretically remain each time.) The rate represents the percentage of atoms remaining after each day. 2. d) There will be differences due to eperimental factors. 3. Make a scatterplot. Have students join the points with a smooth curve. Discuss that this is continuous data as the atoms would theoretically be decaying all the time (not just once a day.) 4. The answers will vary depending on the data. The half life should be about 4-5 days. 5. Answers to 5,6 and 7 will also vary depending on the data. BLM Car Depreciation. The remaining value at the end of the first year is 80% of the original value. Therefore, multiply the previous year s value by The value at the end of year 5 will be $ The constant ratio will be This is the value we used to create the data. 4. Draw a smooth curve through the points. Discuss that this is continuous data. 5. The constant ratio would be Grade U/C Unit 5: Eponential Functions 34

35 5.3.5 Sample Solutions to Selected Decay Questions (continued) BLM Computers in Schools. The number of students per computer is DECREASING instead of INCREASING. 2. Teachers may wish to discuss with students why this data is continuous. 3. The curve is decreasing quickly at first and then slowly 4. The answer will depend on the accuracy of the graph. Answers may range from to 3. The answer to #5 will vary depending on the student s understanding of eponential decay and also on the connection they make to the real world with this problem (e.g., understanding that the number of computers in schools has increased dramatically over the last decade). Grade U/C Unit 5: Eponential Functions 35

36 Unit 5 : Day 4 : Investigating the Graphs of Eponential Functions Part Minds On: 0 Action: 50 Consolidate:5 Total=75 min Minds On Action! Description/Learning Goals Graph eponential functions in the form y = ab where b > 0 and a =. Identify the key features (y-intercept, increasing or decreasing, domain and range, horizontal asymptote). Use the table of values to show the pattern of the first differences and to calculate the constant ratio. Whole Class Discussion Activate prior knowledge related to eponential functions by collectively creating a class KWL chart (Think Literacy, Mathematics 0-2, p.54). Create a KWL template on the board or on chart paper. Give each student 4 post-it notes. Students individually write ideas they already KNOW related to eponential functions on 3 of the notes. On the fourth they record something they are WONDERING about. Students place their notes under the K and W on the class chart. The teacher can group the common ideas together and summarize with the class. Individual Investigation Using Technology Students will complete BLM 5.4. to identify the shapes and properties of eponential functions. Students will complete tables of values and look for patterns in the data by calculating the first differences and the constant ratio. Assessment Opportunities Grade U/C Materials Graphing calculators BLM 5.4. post-it notes It is important to distinguish between finding a difference and finding a ratio. Students will see/review that the ratio for eponential functions can be calculated from the y coordinates or finite differences. At the end of the net lesson the students will be given 2 post-it notes to record the things they have LEARNED. Consolidate Debrief Application Concept Practice Mathematical Process Focus: Connecting (Students will make the connection between the value of the base of the eponential function and the shape of the graph. Students will make the connection between the constant ratio and the base of the eponential function. Students will make connections between different representations: algebraic, graphical, numeric) Whole Class Discussion Students will share their responses to the consolidation questions at the end of the activity. Students will eplain why the relationships are functions. Summarize the similarities in the graphs (domain, range, y-intercept, regions of increase and decrease). Identify the horizontal asymptote. Students will eamine the patterns in the y-coordinates in the table of values and in particular the use of the ratio in predicting the y-coordinate when is a negative integer. Add to the FRAME model by including eamples of equations. Home Activity or Further Classroom Consolidation a) Without graphing or making a table of values predict the properties (domain, range, y-intercept, regions of increase and decrease, growth or decay, horizontal asymptote) of the graphs of y = 5 y = 5 b) To verify your predictions, make tables of values and graph on the same grid. What other observations can you make about the graphs?. This is the first time a horizontal asymptote has been identified. Refer to this pattern in the future lesson on evaluating numerical epressions containing integer eponents. Literacy Strategy: Continue to add to the Eponential FRAME Model. Grade U/C Unit 5: Eponential Functions 36

37 5.4. The Graphs of Eponential Functions Part Step : Set your graphing calculator to the following window and the TBLSET menu to the following settings. Step 2: Each of the equations is in the form: y = ab. Graph each equation using the calculator. Calculate the values in the chart for zero and positive -values. Use the TABLE feature of the calculator to verify your work. Copy the y-value for negative -values from the TABLE. Label and put the scale on each ais and carefully graph the relationship. Calculate the first differences, y, and the constant ratio.. y = 2 y-intercept is a = b = increasing, decreasing or neither (circle one) y y Ratio Domain = 3 Range = 4 2. y = 3 y-intercept is a = b = increasing, decreasing or neither (circle one) Domain = Range = y y Ratio 3. y = 4 y-intercept is a = b = increasing, decreasing or neither (circle one) Domain = Range = y y Ratio Grade U/C Unit 5: Eponential Functions 37

38 5.4. The Graphs of Eponential Functions Part (continued) 4. y = y-intercept is a = b = increasing, decreasing or neither (circle one) Domain = Range = y y Ratio 5. y = 2 y-intercept is a = b = increasing, decreasing or neither (circle one) Domain = Range = 6. y = 3 y-intercept is a = b = increasing, decreasing or neither (circle one) Domain = Range = y y y y Ratio Ratio Step 3: Consolidation. What ordered pair do these graph have in common? 2. Make a chart listing the equations that have graphs that are increasing, decreasing and neither increasing nor decreasing. Increasing Decreasing Neither What characteristic does an eponential equation have if its graph a) Increases? b) Decreases? Eplain why there is a graph that shows neither eponential growth nor decay. 3. Describe how the value of the base, b, of an eponential function determines the shape of the graph. Grade U/C Unit 5: Eponential Functions 38