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Common Core Algebra Unit 10: Exponential Functions 2/1/17Lesson 2: Exploring Exponential Patterns Objective: SWBAT identify patterns of exponential growth and decay.Do Now: A piece of paper is one layer thick. If we place another piece of paper on top of it, the stack is two layers thick. If we place another piece of paper on top, the stack is three layers thick.a) Is the relationship displayed by the table linear? Explain how you know.b) Write a function rule that models the relationship.Think About It…A piece of paper is one layer thick. If the paper is folded in half, the piece of paper is 2 layers thick. If the paper is folded again, the piece of paper becomes 4 layers thick. If the paper is folded a third time, it is 8 layers thick.4572000-635100a) Is this relationship linear? Explain how you know.b) Describe the pattern of f(x).c) Write a function rule that models the relationship.d) Graph the points on the coordinate grid.There are many things in the real world that grow faster as they grow larger or decrease slower as they get smaller. These patterns are known as exponential ____________ or _____________.Group Exploration #1:The number of people who have heard a rumor often grows exponentially. Consider a rumor that starts with 3 people and where the number of people who have heard it doubles each day that it spreads.a) Why does it make sense that the number of people who have heard a rumor would grow exponentially?b) Fill in the table below for the number of people, N, who knew the rumor after it has spread a certain number of days, d.We’d like to determine the number of people who know the rumor after 20 days, but to do that, we need to develop a formula to predict N (the number knowing the rumor) if we know d (the number of days it has been spreading).c) For the following number of days, fill in how you calculated your values based on extended products using the number 2. d) Using the pattern you developed in (a), write a formula giving the number of people who know the rumor, N, if you know the number of days, d, it has been spreading.e) How many people would know the rumor after 20 days?Group Exploration #2:Helmut (from Finland) is heading towards a lighthouse in a very peculiar way. He starts 160 feet from the lighthouse. On his first trip he walks half the distance to the light house. On his next trip he walks half of what is left. On each consecutive trip he walks half of the distance he has left. We are going to model the distance, D, that Helmut has remaining to the lighthouse after n-trips.a) Fill in the table below for the amount of distance that Helmut has left after n-trips.b) Each entry in the table could be found by multiplying the previous by what number? This is important because we always want to think about exponential functions in terms of multiplying.c) Based on (c), give a formula that predicts the distance, D, that Helmut has left after n-trips.d) How far is Helmut from the windmill after 6 trips? Provide a calculation that justifies your answer and don’t forget those units!e) Helmut believes he will reach the windmill after 10 trips. Is he correct?f) Explain why Helmut will never reach the windmill based on this pattern.Sum It Up:When multiplying by a number greater than _________, the exponential pattern is a _______________.When multiplying by a number less than _________, the exponential pattern is a ____________.Check for Understanding:In a science fiction novel, the main character found a mysterious rock that decreased in size each day. The table below shows the part of the rock that remained at noon on successive days.a) Describe the exponential pattern by identifying the number each fractional part of the rock is multiplied by from the previous day.b) Is this a growth or decay? How do you know?c) Which fractional part of the rock will remain at noon on day 7?Challenge: Write a function rule that describes the relationship between the number of days and the fractional part of the remaining rock.Problem Set:1. A piece of yarn that measures one unit is cut in half. a) Complete the table below.Number of Cuts Length of Each New Piece011?b) Which column of your table is the input? Which is the output?c) Is this a growth or a decay? Explain how you know using the number you multiplied by in the table.d) Determine a mathematical model (equation) that represents this data by examining the patterns in the table.e) Using your model, determine the length of the yarn after 10 cuts. f) Will the output ever reach zero? Explain why.2. A piece of paper is 0.01 centimeters (cm) thick. When you fold it once, it becomes 0.02 centimeters thick.If you fold it again, it doubles again to 0.04 centimeters thick. Each fold doubles the thickness of the paper.41382955461000a) How thick is the paper after: 4 Folds: 5 Folds:b) For each of the following number of folds, f, show how you can calculate the thickness, T, based on repeatedly multiplying by _____.Number of Folds(f)Thickness of Paper (T)10.0120.0230.04c) Determine a formula, based on (b), for the thickness, T, based on the number of folds, f.d) How thick would the paper be if f ?10 ? Use proper units.e) If there are 100 centimeters in a meter, how many meters thick is the paper after 20 folds? Show thework that leads to your answer.3. The Sierpinski Triangle is a type of progression where an equilateral triangle has ? of its area removed tocreate a new shape. Then ? of its remaining area is taken away. A series of these triangles is shown below, starting with an area of 64.a) If we remove ? of the area, what fraction of the area remains?b) Multiply 64 by the fraction you found in (a). What value do you get?c) Find the areas of the third and fourth pictures above by multiplying by the fraction you found in (a).d) Find a formula for the area, A, that remains after n removals of area.e) How much area remains after 10 removals?f) How much area remains after 20 removals?g) Will the area ever reach zero? Explain your thinking.4. Below is a linear table and an exponential table. Table ATable B4445806450032004007937500a) Which table is linear? Which table is exponential? Explain how you know. b) Explain the difference between a linear and an exponential pattern. 5. Explain the difference between exponential growth and exponential decay.Name ____________________Exit SlipCommon Core Algebra Unit 10: Exponential Functions 2/1/17Lesson 2: Exploring Exponential Patterns Objective: SWBAT identify patterns of exponential growth and decay.You have a very special candy bar. It is so special to you that you decide to eat only of whatever remains of the candy bar each day. a) Is this an example of exponential growth or decay? Explain how you knowb) How much of the candy is left on the 4th day? c) Will you ever finish the candy bar?Name ____________________Exit SlipCommon Core Algebra Unit 10: Exponential Functions 2/1/17Lesson 2: Exploring Exponential Patterns Objective: SWBAT identify patterns of exponential growth and decay.You have a very special candy bar. It is so special to you that you decide to eat only of whatever remains of the candy bar each day. a) Is this an example of exponential growth or decay? Explain how you knowb) How much of the candy is left on the 4th day? c) Will you ever finish the candy bar? ................
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