Radicals and Exponentials. David A Parker Secondary Mathematics

Transcription

1 Radicals and Exponentials David A Parker Secondary Mathematics

2 Page 2 Mathematics even at its simplest beginnings requires a specific vocabulary set, math is its own language. To, effectively, communicate the processes, patterns, and equations that are contained within any given mathematical sub-set, one must own this language. A sometimes-detrimental reality is that far too many of these words get commonly used for different purposes throughout everyday communications with no consideration for their mathematical definitions. With respect to the topic of Radicals, Exponentials and Logarithms, there is much of this idiomatic use of words that are also mathematical terms, thus the importance of vocabulary is very rampant and can be extremely difficult to students. It is for this that beginning with lesson one, students were required to give both explanations and diagrams. Starting students with, basically, vocabulary rather than the formulas, that they simply wish to memorize, is a fundamental difference between being able to do the work and actually understanding the mathematics. It is also essential that students are capable of reading the problems they are being required to solve. The greatest importance of vocabulary spreads across all mathematics, as the most used problem solving is the epitome of a word problem. Therefore, there is a great need for students to master mathematical vocabulary if they wish to be successful, not only in math, but in large aspects of life. Beginning with lesson one students must explain mathematical definitions for square roots, nth roots, and radicand. In addition, they must define imaginary numbers and complex numbers as a mathematical definition.

3 Assessment of Prior Knowledge Page 3 The expectation of this unit is that students have seen square roots while solving for solutions to quadratic equations. Students should be proficient at solving for x in numerous different equation formats. It is also expected that students will be familiar with systems of linear equations, factoring, and exponent rules. From this, the exponential radical form and adding, subtracting, multiplying and dividing of the unit is expected to have higher initial scores from students than the Complex numbers, exponential functions and logarithmic functions. Since many of the more complicated properties of logarithmic functions are derived from exponential rules and the fact that these rules were studied in earlier terms, it is possible that some of the higher-achieving students will perform better than expected on the initial assessments. However, it is also important to note that for some of the students it has been in excess of a year or two since they used exponent rules. Due to the previously established assessments and assumed prior knowledge of the students, the assessment will cover both sections of the unit Radicals and Exponential and Logarithmic Functions. Although both are being assessed the focus will be towards the radicals as this is the heavier weighted section of the unit and it will give imperative knowledge of what needs covered. I assessed at the beginning of the first class after Spring break in the standard classroom. This was a single student exam and had no group work assigned to it. The students had a full sixty minutes to complete the assessment. There were no given or needed accommodations for special needs students. At the end of the assessment I requested that each student fill out an index card. The card was to have: their name, the one that the computer has them listed as, any other name they would prefer to go by. Their pronoun if it was important to them. An interesting fact and a hobby. This allowed me to attempt to get to know them better, quicker, as I was coming in at the end of the year. Normally I will

4 begin my classes with something similar that will allow us to share who we are with each other. Page 4 My students only succeeded in answering 25% or less of the pre assessment. This was not overly concerning as it was mostly material they had not seen yet. Formative Assessments Sticking with what the students were used too I gave a weekly homework quiz. The quiz was used to evaluate how I would progress the following week, if I needed to revisit prior learning or if students were ready to move forward with their knowledge. Once or twice during the week, I made sure to organize the lesson around group work to make it easier to get some small group and one on one narrative evaluations of how their understanding was going. Most classes had an activity that required some form of proof or expression of written out work, another way to evaluate if they understood the material. My favorite however, was having students verbally explain to other students, the whole class, or to me how to solve a problem we were working on. End of Unit Assessments The end of unit assessments were broken into two parts. I assessed on Radicals before moving into exponential and logarithmic functions than assessed the exponential and logarithmic functions after introducing exponential growth and decay. I almost postponed the radical assessment as the feedback I was getting showed a serious lack of understanding of the radicand. However, after a blind survey of their comfort level of test I went forward with it and the class preformed higher than my expectations. Imaginary numbers were very difficult for these students to grasp. In the future I will present them sooner and dedicate more time to working with complex numbers. The post assessment on radicals was given in the standard classroom as a single student test and

5 a time frame of sixty minutes. Page 5 The second post assessment was a time crunch for my final day of teaching this class. I felt uncomfortable with the lack of time spent on the material and thought it best to give a small group assessment instead of single student work. Since this assessment was given to a group it was not only a harder assessment but I added a group collaboration to the grading aspect. The assessment was given in the standard classroom in groups of random selected teams of four. All work was to be shown and one random test from the group would be selected for the group grade. Also for group participation, all work on every test question had to be complete.

6 Page 6 Table of Contents Title Page Unit Learning Goals... 3 Calendar.. 8 Lesson 1: Radical Expressions and Functions Lesson 2: Rational Exponents. 18 Lesson 3: Multiplying and Simplifying Radical Expressions. 29 Lesson 4: Adding, Subtracting, and Dividing Radical Expressions 36 Lesson 5: Adding, Subtracting, and Dividing Radical Expressions (continued) 43 Lesson 6: Radical Equations 48 Lesson 7: Complex Numbers Lesson 8: Radical Review 66 Lesson 9: Radical Test. 74 Lesson 10: Fractals.. 79 Lesson 11: Exponential Functions 82 Lesson 12: What is an interest Rate? Lesson 13: What is an interest Rate? (Continued) Lesson 14: How is interest really calculated? Lesson 15: What is continuous compounding? Lesson 16: Inverse Functions and Logarithmic Functions Lesson 17: Exponential Growth and Decay. 114 Lesson 18: Exponential Growth and Decay Test. 120 Page

7 Page 7 Unit Learning Goals and Standards Lessons in this Unit Lesson 1: Radical Expressions and Functions Lesson 2: Rational Exponents Lesson 3: Multiplying and Simplifying Radical Expressions Lesson 4: Adding, Subtracting, and Dividing Radical Expressions Lesson 5: Adding, Subtracting, and Dividing Radical Expressions Lesson 6: Radical Equations Lesson 7: Complex Numbers Lesson 8: Radical Review Lesson 9: Radical Test Lesson 10: Fractals Lesson 11: Exponential Functions Lesson12: What is an interest Rate? Lesson13: What is an interest Rate? (Continued) Lesson14: How is interest really calculated? Lesson15: What is continuous compounding? Lesson16: Inverse Functions and Logarithmic Functions Lesson17: Exponential Growth and Decay and Review Lesson18: Exponential Growth and Decay Test Goals of this Unit Students will be able to: Manipulate and solve Radical Expressions and Functions Understand Rational Exponents Multiply and Simplify Radical Expressions Multiply multiple term Radical Expressions Rationalize numerators and denominators Simplify and solve radical equations Understand Complex numbers and simplify powers of i Evaluate exponential functions Compose inverse functions Use logarithms to solve exponential functions

8 Calendar: Based on 90 minute lessons meeting 5 times every two weeks Page 8 Monday Tuesday 04/07/15 Wednesday Thursday Friday No Class Topic: Radical Expressions and Functions No Class Topic: Rational Exponents Home Work Quiz No Class Lesson 1 Lesson 2 Objectives: SWBAT Evaluate a square root Evaluate square root functions Find the domain of square root functions Simplify expressions of the form Evaluate cube root functions Simplify expressions of the form Find even and odd roots Simplify expressions of the form Strategy: Use the recognition of the area of squares, volume of cubes to discover how to use nth radicals to an equal exponent Objectives: SWBAT Use the definition of Use the definition of Use the definition of Simplify expressions with rational exponents Simplify radical expressions using rational exponents. Strategy: Explore rational exponents as a descriptor of radicals. Explore the properties of rational exponents by proof to allow for simplification. Monday Tuesday Wednesday Thursday Friday No Class Topic: Multiplying and Simplifying Radical Expressions No Class Topic: Adding, Subtracting, and Dividing Radical Expressions Lesson 3 Lesson 4 Topic: Multiplying with More Than One Term and Rationalizing Denominators Lesson 5 Objective: SWBAT use the product rule to multiply radicals, use factoring and product rule to simplify radicals and multiply radicals and then simplify. Strategy: Use properties of exponents to understand multiplying and simplifying radical functions Objective: SWBAT add and subtract radical expressions, use the quotient rule to simplify radical expressions, and use the quotient rule to divide radical expressions. Strategy: Show how radicals can be evaluated as a base to allow for addition and subtraction. Prove the quotient rule using exponent rules Objective: SWBAT multiply radical expressions with more than one term, use polynomial special products to multiply radicals, rationalize denominators containing one term, rationalize denominators containing two terms, and rationalize numerators. Strategy: Revisit multiplication by one. Show multiplication by

9 Page 9 conjugate. Monday Tuesday Wednesday Thursday Friday No Class Topic: Radical Equations No Class Topic: Complex Numbers No Class Lesson 6 Objective: SWBAT solve radical equations and use models that are radical functions to solve problems. Lesson 7 Objective: SWBAT express square roots of negative in terms of I, add and subtract complex numbers, multiply complex numbers, divide complex numbers, and simplify powers of i. Strategy: Multiply both sides of the equation by the highest value of the radical nth root. Repeat until no radicals remain. Work through applications Strategy: Show the how it is simple to just consider i as something we need, thus we can manipulate it. This information is highly dependent on definitions Monday Tuesday Wednesday Thursday Friday No Class Topic: Radical Review No Class Topic: Radical Test Topic: Fractals Lesson 8 Objective: SWBAT manipulate radical expressions and functions, rational exponents, multiply and simplify radical expressions, add, subtract and divide radical expressions, multiply with more than one term and rationalize denominators, solve radical equations and manipulate complex numbers. Strategy: Evaluate radicands, rational exponents, simplify radical expressions including adding, subtracting, multiplying and dividing, rationalize Lesson 9 Objective: SWBAT manipulate radical expressions and functions, rational exponents, multiply and simplify radical expressions, add, subtract and divide radical expressions, multiply with more than one term and rationalize denominators, solve radical equations and manipulate complex numbers. Strategy: Test Lesson 10 Objective: SWBAT create and zoom their own fractals and explain iteration. Strategy: Video on the use of fractals and student exploration of a fractal zoomer

10 Page 10 denominators, solve radical equations and work with complex numbers. Monday Tuesday Wednesday Thursday Friday No Class Topic: Exponential Functions No Class Topic: Interest Rate Lesson 12 No Class Lesson 11 Objective: SWBAT write Objective: SWBAT an exponential growth calculate and graph or decay function to exponential functions model a given real world situation. Strategy: manipulating exponential functions Strategy: Students will explore the cost of paying varying interest rates, and differing compounding s Monday Tuesday Wednesday Thursday Friday No Class Topic: Interest rate (cont.) No Class Topic: Compound Interest Topic: Compound interest (cont) Lesson 13 Objective: SWBAT write an exponential growth or decay function to model a given real world situation. Strategy: Students will explore the cost of paying varying interest rates, and differing compounding Lesson 14 Objective: SWBAT derive the compound interest formula and apply it to solve real world problems. Strategy: Students will explore the true cost of paying for something on an interest based payment plan. Lesson 15 Objective: SWBAT explain the meaning of the mathematical constant e and use the continuous compounding interest formula to solve real world problems. Strategy: Students will explore compounding interest at higher and higher frequencies. Monday Tuesday Wednesday Thursday Friday No Class Topic: Inverse Functions and Logarithmic Functions No Class Topic: Exponential Growth and Decay: Modeling Data and Post Test. Lesson 16 Objective: SWBAT model Lesson 17 Objective: SWBAT model

11 Page 11 exponential growth and decay, choose an appropriate model for data and express an exponential model in base e. and show their knowledge on their tests. exponential growth and decay, choose an appropriate model for data and express an exponential model in base e. and show their knowledge on their tests. Strategy: show how logarithmic is simply the inverse of exponentials Strategy: Same as last lesson but in more of a review nature and then test Monday Tuesday Wednesday Thursday Friday No Class Topic: Exponential Growth No Class and Decay: Test. Lesson 18 Objective: SWBAT model exponential growth and decay, choose an appropriate model for data and express an exponential model in base e. and show their knowledge on their tests. Strategy: Group Test

12 Page 12 Lesson 1: Radical Expressions and Functions Topic: Radical Expressions and Functions Rational: Students will visually explore square and cube roots so they can use this too abstractly transfer it to nth roots that are necessary for solving higher lever equations. Objectives: Students will be able to evaluate square roots, evaluate square root functions, find the domain of square root functions, simplify expressions of the form, evaluate cube root functions, simplify expressions of the form, find even and odd roots, and simplify expressions of the form Language Objective: Students will be able to understand, use and define square root, cube root, nth root, radical sign, radicand, and radical expression. Lesson Assessments: Students will answer aloud questions about square roots, cube roots, and nth roots. Visual assessments will be done while groups and one on one students work through example and classroom work. Student will take exit card assessment at the end of class Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson (40 minutes) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and

13 Page 13 five minutes to discuss student solutions. Project four boxes on the screen that shows the area, area 16, 25, 36 and 49. Ask students to figure out what the side lengths are. Have students show their work and compare answers. Go over solutions if needed, no more than five minutes. If it takes more than five minutes for the warm up than there needs to be a lesson on the subject matter. Introduction (10 minutes) Introduce the Unit with something like this; this is another way to look at square roots that you have already been working on with quadratic equations. But first let us talk for a moment This being the introduction lesson to taking over the class I want to both give students some information about me personally and professionally and also get some information about them. I will give them some background information about my life and explain that teaching them is my class. As such I must collect data in an attempt to prove that I can teach, thus I must begin our time together gathering data about what knowledge they may already posses about what I am planning on covering. It is important to make sure they understand that there is no grade attached that this is simply data. Give Pre-Test (Up to 30 minutes, since the information is unknown I am planning on no more then 20 minutes.) Especially since no real activity is planned for today after the Pre Test have the students get up and move around, talk, etc for five minutes, breaks or some type are very necessary for a 90 minute class. Middle of lesson (35 minutes) Square root is not new to these students, as they have been working with quadratics so I will be very brief about the symbol and want to be sure they understand the definition of the principal square root. Definition of the Principal Square Root If a is a nonnegative real number, the nonnegative number b such that b 2 = a, denoted by b = a, is the principal square root of a. In different terms this means that 81 = 9, because 9 2 = 81 and 9 is positive. However - 81 = -9 because -9 2 = 81 and -9 is negative. Here discuss that the point I want them to take away is that they have been solving equations and when solving an equation you take the root of a squared number so it is plus or minus which we will get into soon. However, as for simply the square root of a number it is positive. Quick check in on evaluating functions,

14 Page 14 Solve Now what is the domain? 5x-6 greater than or equal to zero, thus set it equal to zero and solve, 5x-6 = 0 -> x = 6/5 so the domain is [6/5, inf). Can you tell me the basic shape of a square root graph? Half a sidewise parabola A quick discussion of the plus and minus, when you solve an equation like You are evaluating the square root of each side and since acknowledge that and we can check this we must What you need to take from this is that square roots are positive, but when you square an equation to solve it you must look at both values. How about cube roots, what does that mean? show a cube and discuss how volume is simply multiplication of three and in a cube they all equal, thus s cube root is a number that can be broken into the multiplication of three equal numbers. Show them the base graph of a cube root. The point is to show that = a So if there is a three here then a = x*x*x and if there is a two than a = x*x; What do you think it means if there is a five, six or seven? Finally, let us discuss nth roots and nth roots of a number to the nth power, therefore covering all possibilities. This looks like, First, what does nth root mean? The hope is that a student will get the correlation. And? If n is even;, if n is odd This covers all roots from square and cube to infinite. End (10 minutes)

15 Page 15 We have covered square roots, cubed roots, as they are the most commonly used. This led us to nth roots. We followed this idea to taking the root of an equal power i.e. Exit Ticket (5 min) Find the domain of the function The numerator is not important to the domain, as cube roots have no restrictions. The denominator is an even root thus must not equal less than zero, but since it is in the denominator it must not equal zero therefore the radical expression must be greater than zero. To solve, Set square both sides subtract 2x from both sides rewrite 30 =2x divide both sides by 2 and we get that x = 15. This tells us that the range is everything less than x = 15. Domain: If the function models the median height of boys who are x months of age, and the function models the median height of girls who are x months of age. Then at what month are they the same height? Here we set the equations equal to each other thus We know it is positive as for now we cannot go backwards in aging time. To find the actual age we plug in the x value and get months from both equations. Homework: Is taken from the book (5 th edition Blitzer Intermediate Algebra for college students) Section 7.1, problems: 29, 61,63,83,88,103 and 120

16 Page 16 Homework 29. Find the domain of Find the indicated root, or state that the expression is not a real number Simplify each expression. Include absolute value bars where necessary Using the function to estimate the speed of a car. A motorist is involved in an accident. A police officer measures the car s skid marks to be 245 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 50 miles per hour and the motorist tells the officer he was not speeding, should the officer believe him? Explain If I am given any real number, that number has exactly one odd root and two even roots. Does this statement make sense or does it not make sense explain your reasoning?

17 Page 17 Lesson Reflection This was my first lesson with this group of students. They didn t know what to expect from me and I didn t know what to expect from them. The warm up went well enough, though a little different than what they were used to. The lesson followed the template and the students were genuinely interested in learning. The one thing I will change is the lack of an activity, even with a break the 90 minutes was to long for direct instruction.

18 Page 18 Lesson 2: Rational Exponents Topic: Rational Exponents Rational: Using exponents to represent radical expressions allows for understanding how to simplify adding, subtracting, multiplying and dividing radical expressions. Objectives: Students will be able to use the definition of, use the definition of, use the definition of, simplify expressions with rational exponents, and simplify radical expressions using rational exponents. Language Objective: Students will be able to understand, use and define numerator, denominator, rational exponent, radical s index, and base. Lesson Assessments: Students will answer aloud questions about square roots, cube roots, and nth roots. Visual assessments will be done while groups and one on one students work through example and classroom work. Student will take exit card assessment at the end of class Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson; Warm up, Home Work Quiz and introduction of new material. (30 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and

19 Page 19 five minutes to discuss student solutions. Show a reasonable proof that or that Have students show their work and compare answers. Go over solutions if needed, no more than five minutes. If it takes more than five minutes for the warm up than there needs to be a lesson on the subject matter. None rigorous proof: Square both sides: Subtract a and b from both sides: such that Divide by 2: or both can be zero that solves for the equals part. However, if neither is zero and we stated that they had to be real numbers than they have to be positive and any two positive numbers multiplied together is greater than zero. Home Work Quiz (15 minutes) Introduction (5 minutes) Introduce the Unit with something like this; Now we need to take what we learned about radicals and (do stuff with and too them). For instance simplifying things like this; Middle of lesson (25 minutes) Here we take the radicals we just revisited last class and we turn them into fractional exponents to make it easier to simplify, especially radicals within radicals. Defining Rational Exponent What does mean? Lets logic it out; well if we think back to exponent rules we can cube each side and get and this becomes, and this we saw last class, it translates to. The Formal Definition of ask the students if anyone would like to take a stab at defining this. If represents a real number and is an integer, then

20 Page 20 If a is negative, n must be odd. If a is nonnegative, n can be any index. Since we want to be able to go back and forth between these two forms lets practice a couple. Rewrite each expression in radical notation Examples = = 8 = = Now that we have seen the definition how about you all walk me through the definition of. The Definition of If represents a real number, is a positive rational number reduced to lowest terms, and is an integer, then And In other words, it does not matter which we do first, but it is important to recognize that sometimes it is easier to do one before the other. This is how it looks. Examples = or =

21 Page 21 = or = = or = And one more definition, how would we define The Definition of If is a nonzero real number, then Examples I do not know how long it has been since the students have seen properties of exponents, so here is a review You must have these memorized or be able to derive them on the spot. Here is how you would derive the Thus, this is because exponents are repeated multiplication. Similarly, the rest of the exponent

22 Page 22 properties can be derived and I would recommend working these out on your own. Examples Search and Rescue Game (25 minutes) This can be either a solo activity or paired or group How to play: The ten problems listed below will be hung on the walls around the room preferably on colored paper. Each problem will have an answer on the bottom of the page but this is not the answer to the problem they are doing instead the answer to the problem you solve is what you need to find on the bottom of the sheet of the next problem you need to work out. (Students can either be given random numbers to start at different locations or better have them choose on their own just remind them to not group up on the same one.) Example, Student A starts at problem #3 they find the solution to be next they find the problem with the solution written on the bottom of the page and solve that problem until they have completed all ten problems. This will get them on their feet and moving about the room. List of Problems with answers 1. Solve send to problem 6 2. Rewrite with rational exponents send to problem 10

23 Page 23 Name Date Homework Quiz 5 3. Simplify and answer in radical form send to problem 7 4. Solve send to problem 1 5. Simplify send to problem 8 6. Simplify and answer in radical form send to problem 9 7. Solve send to problem 4 8. Solve send to problem 2 9. Solve send to problem Simplify send to problem 3 Exit Ticket (5 min) Simplify Homework: Is taken from the book (5 th edition Blitzer Intermediate Algebra for college students) Section 7.2, problems: 1,13,30,104,113,133,135, and Graph

24 Page Write a radical function whose domain is 3. If is the estimate of the speed of a car,, in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. Than if, a motorist is involved in an accident and a police officer measures the car s skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer, she was not speeding, should the officer believe her? Explain.

25 Page 25 Names Search and Rescue Problem # Work (show all work) Answer

26 Problem # Work (show all work) Answer Page 26

27 Page 27 Homework Answer on separate sheet of paper and show all work. Rewrite each expression in radical notation. Simplify, if possible Rewrite with rational exponents. 3. Use rational exponents to simplify after simplifying write your answer in radical notation. 4. Distribute and write in simplest form 5. If the qualifications for a yacht to enter the America s Cup is figured by the equation Where L is the yacht s length, in meters, S is its sail area, in square meters, and D is its displacement, in cubic meters then: 6. a. Rewrite the inequality using rational exponents. b. Determine if a yacht with length meters, sail area square meters, and displacement cubic meters is eligible for the America s Cup. 7. What is the meaning of? Give an example to support your explanation. 8. By adding the exponents, I simplified and obtained 49. Does this make sense? Explain why or why not.

28 Lesson Reflection There was some confusion with this lesson until we discussed and practiced exponential rules. Next time I will begin the class by revisiting exponent rules before introducing a new way to use them. The activity was a huge hit and had all the students on their feet walking around the classroom while doing math and collaborating with one another. Page 28

29 Page 29 Lesson 3: Multiplying and Simplifying Radical Expressions Topic: Multiplying and Simplifying Radical Expressions Rational: Understanding and manipulating radicals without converting them into rational exponents. Objectives: Students will be able to use the product rule to multiply radicals, use factoring and the product rule to simplify radicals, and multiply radicals and then simplify. Language Objective: Students will understand what it means to simplify a radicand. They will also be able to explain perfect nth powers of a radicand. Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through example and classroom work including writing on the board and describing the work to the class. Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson; warm up, and revisit last class activity (30 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. The least answered from last class activity.

30 Page 30 #2 = #5 #6 #9 If you had no problems with the activity and these are too easy simplify this Revisit last class activity (15 min) Have students show their work and compare answers. Have students express solutions than project the remaining questions from the activities and ask if any others needed worked out. Introduction (5 minutes) Verbally express the objectives of today s class. Today we are going to cover the product rule for radicals, Simplifying Radical Expressions by Factoring, and Multiplying Radicals and then Simplifying. Middle of lesson (55 minutes) The Product Rule for Radicals (10 minutes) Formal definition: The Product Rule for Radicals Examples to work out with class

31 Page 31 Product rule activity (10 minutes) Break the class into four groups give them each a product rule problem and two minutes to write out the solution. Then each group will have two minutes to present their problem to the class. Group a: Group b: Group c: Group d: Simplifying Radicals (15 minutes) Formal definition: Simplifying Radical Expressions by Factoring A radical expression whose index is n is simplified when its radicand has no factors that are perfect nth powers. To simplify, use the following procedure: 1. Write the radicand as the product of two factors, one of which is the greatest perfect nth power. 2. Use the product rule to take the nth root of each factor. 3. Find the nth root of the perfect nth power. Examples to work out with class Remind students that the square root of a square is absolute value. One last thing before you group up again. Remembering we can simplify things like:

32 Page 32 Factoring activity (10 minutes) Break the class into four groups give them each a product rule problem and two minutes to write out the solution. Then each group will have two minutes to present their problem to the class. Group a: Group b: Group c: Group d: Multiplying Radicals and Then Simplifying (10 minutes) Examples to work out with class Exit Ticket (5 min) Go over in class if time permits and they seem to struggle with it. Homework: Is taken from the book (5 th edition Blitzer Intermediate Algebra for college students) Section 7.3, problems: 10,14,28,32,38,48,52,76,80,88,92,98, and 117

33 Page 33 Homework Answer on separate sheet of paper and show all work. Use the product rule to multiply Simplify by factoring Express the function f, in simplified form. Assume that x can be any real number. 38. Simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers Multiply and simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers Simplify. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. 88. The function models the distance d(x), in miles, that a person h feet high can see to

34 Page 34 the horizon. 92. The captain of a cruise ship is on the star deck, which is 120 feet above the water. How far can the captain see? Write the answer in simplified radical form. Then use the simplified radical form and a calculator to express the answer to the nearest tenth of a mile. 98. Explain why is not simplified. What do we mean when we say a radical expression is simplified? 117. If.

35 Page 35 Lesson Reflection I followed the lesson plan and shouldn t have. I realized during the introduction that I should not have started abstractly. In the future I will begin with numerical examples and then move to abstract absolutes. Factoring is going to be a problem and the students need practice with it along with the idea of multiple things becoming one as in the cubed root of x cubed is only a single x. The activities and group work are very beneficial and enjoyed by this class.

36 Page 36 Lesson 4: Adding, Subtracting, and Dividing Radical Expressions Topic: Adding, Subtracting, and Dividing Radical Expressions Rational: Understanding and manipulating addition, subtraction and division of radicals without converting them into rational exponents. Objectives: Students will be able to add and subtract radical expressions, use the quotient rule to simplify radical expressions, and use the quotient rule to divide radical expressions. Language Objective: Students will be able to use the product rule to multiply radicals, use factoring and the product rule to simplify radicals, and multiply radicals and then simplify. Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through example and classroom work including writing on the board and describing the work to the class. Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson; warm up (10-15 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions.

37 Page 37 Introduction (5 minutes) Verbally express the objectives of today s class. Today we are going to cover adding, subtracting and dividing radical expressions. Middle of lesson (60 minutes) Adding and Subtracting Radical Expressions (20 minutes) Examples to work out with class Sometimes we can simplify the radicals first Adding and subtracting radicals activity (10 minutes) Break the class into four groups give them each a product rule problem and two minutes to write out the solution. Then each group will have two minutes to present their problem to the class. Group a: Group b:

38 Page 38 Group c: Group d: Quotient Rule to Simplify Radicals (20 minutes) Formal definition: The Quotient Rule for Radicals Examples to work out with class Factoring activity (10 minutes) Break the class into four groups give them each a product rule problem and two minutes to write out the solution. Then each group will have two minutes to present their problem to the class. This is the same activity from the last class but seemed to fit. Group a: Group b: Group c:

39 Page 39 Group d: Check Point (25 minutes) Check Point Hand out Homework: Is taken from the book (5 th edition Blitzer Intermediate Algebra for college students) Section 7.4, problems: 24,26,42,44,52,54,86, and 107

40 Page 40 Homework Answer on separate sheet of paper and show all work. Add or Subtract as indicated. You will need to simplify terms to identify the like radicals Simplify using the quotient rule Divide and if possible, simplify What does travel in space have to do with radicals? Imagine that in the future we will be able to travel in starships at velocities approaching the speed of light (approximately 186,000 miles per second). According to Einstein s theory of relativity, time would pass more quickly on Earth than it would in the moving starship. The radical expression Gives the aging rate of an astronaut relative to the aging rate of a friend,, on Earth. In the expression, v is the astronaut s velocity and c is the speed of light. a. Use the quotient rule and simplify the expression that shows your aging rate relative to a friend on Earth. Working in a step-by-step manner, express your aging rate as 107. b. You are moving at velocities approaching the speed of light. Substitute c, the speed of light, for v in the simplified expression from part (a). Simplify completely. Close to the speed of light, what is your aging rate relative to a friend on Earth? What does that mean? Check Point Name Date Simplify the given expression or perform the indicated operation(s) and, if possible, simplify. Assume that all variables represent positive real numbers

41 Page Find the domain Lesson Reflection The student had a horrific time understanding/excepting simplifying to a common base to be able

42 to add and subtract the radicals. I had to go back to examples of you can add together 3x + 5x but not 3x + 5y and work from there. I think a prior lesson on abstract addition and subtraction would be highly beneficial to this lesson. The activity was hard since they were not comfortable with the material. Page 42

43 Page 43 Lesson 5: Adding, Subtracting, and Dividing Radical Expressions Topic: Adding, Subtracting, and Dividing Radical Expressions Rational: Understanding and manipulating addition, subtraction and division of radicals without converting them into rational exponents. Objectives: Students will be able to add and subtract radical expressions, use the quotient rule to simplify radical expressions, and use the quotient rule to divide radical expressions. Language Objective: Students will be able to use the product rule to multiply radicals, use factoring and the product rule to simplify radicals, and multiply radicals and then simplify. Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through example and classroom work including writing on the board and describing the work to the class. Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson; warm up (15-20 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions.

44 Page 44 Have students show their work and compare answers. Have students express descriptive solutions on the board or document camera. These are very similar to last classes warm ups and we are going to spend another day going over these to be sure that everyone is comfortable with it. Home Work Quiz (15 minutes) Introduction (5 minutes) Verbally express the objectives of today s class activities. Today we are going to work with adding and subtracting radicals. I will break the class into groups 3 groups of 4 and two groups of 5. Each group will receive a large sheet of paper and an addiction of radicals problem to work out in two ways, by radical simplification and by exponent rules. Then the papers will be taped to the wall and we (the class) will walk around to each problem. At each problem I will call on a group member of my choosing to explain the problem and answer and take any question from the class. After all the problems have been seen and understood I will choose new groups and we will repeat the process with a subtraction problem. Middle of lesson (55 minutes) Problems for groups to work out

45 Exit Ticket (5 min) Explain in clear detail how to add and subtract radicals. Page 45

46 Page 46 Name Date Homework Quiz

47 Lesson Reflection At first there was some push back about the lesson as it was set out but when I expressed that only two students succeeded in getting the correct answers on the check point they did they agreed that more work with adding and subtracting was a good thing. There were no deviations from this lesson and I would not change it other than harder problems if I had students who could do the work easier. I walk about and talked to groups individually and between the feedback and the exit ticket I am comfortable moving forward. Page 47

48 Page 48 Lesson 6: Radical Equations Topic: Solving radical equations Rational: Now that students have seen and simplified radicands they will solve problems involving them in preparation for solving things like exponential growth and decay Objectives: Students will be able to solve radical equations and use models that are radical functions to solve problems. Language Objective: Students will be able to understand, use and define radical equations Lesson Assessments: Students will answer aloud questions about square roots, cube roots, and nth roots within functions and equations. Visual assessments will be done while groups and one on one students work through example and classroom work. Student will take exit card assessment at the end of class Standards: CCSS.Math.Content.HSF.IF.C.7.b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Beginning of Lesson (50 minutes) Warm Up (20 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Multiply: Solve:

49 Page 49 Solve: Have students show their work and compare answers. Go over solutions if needed, no more than twenty minutes. If it takes more than twenty minutes for the warm up than there needs to be a lesson on the subject matter. Introduction (10 minutes) Introduce this lesson with something like this; how many solutions are there to the expression x=4? This question will cause some confusion and it is meant to. Let the class think about it for a couple minutes and then start writing out this example. If we square both sides, we obtain standard form. Subtract 16 from both sides and write the quadratic equation in Factor. Set each factor equal to zero. Solve the resulting equations. The equation has two solutions, -4 and 4. By contrast only 4 is a solution of the original equation, x=4. For this reason, when raising both sides of an equation to an even power, always check proposed solutions in the original equation. Examples (20 minutes) Here are the steps to solving a radical equation Solve: Solution: Step 1. Isolate the radical on one side. Step 2. Raise both sides to the nth power. Because n, the index is 2, we square both sides. Square both sides to eliminate the radical.

50 Page 50 Simplify. Step 3. Solve the resulting equation. The resulting equation is a linear equation. Subtract 3 from both sides Divide both sides by 2. Step 4. Check the proposed solution in the original equation. Because both sides were raised to an even power, this check is essential. The solution is 11 and the solution set is {11}. Now do your own using the same step by step and then we will have you check with your class mates. Solve: After this example is completed and discussed in groups It is time to take a 5 minute break and when the students come back get them into 4 5 person groups. As a group showing all steps solve: Allow 15 minutes of group work than write the problem on the board and call on an individual from each group to explain each step. Now using the same idea and steps Solve: If running out of time this can be the exit ticket if you have 10 minutes left give the next two problems as exit if you are down to 5 give only the last problem. Exit Ticket (5-10 min) Solve:

51 Homework: Is taken from the book (5 th edition Blitzer Intermediate Algebra for college students) Section 7.6, problems: 7, 11,13,15,17,19 and 21 Page 51

52 Page 52 Homework

53 Page 53 Lesson Reflection Originally I was going to introduce more material in this lesson but found that the pace of understanding the given material was not what I had planned thus I increased the time spent on examples until I felt that there was enough comfort from the students to move on. The exit ticket revealed that there will need to be some one on one discussions with five of the students, the remaining 18 solved them with no problems. I think it would be beneficial to find more interactive activities for this lesson and then spread it out over two days.

54 Page 54 Lesson 7: Complex Numbers Topic: Adding, Subtracting, Multiplying, Dividing Complex numbers Rational: Understanding and manipulating the square root of negative one. In order to perform addition, subtraction, multiplication and division of Complex numbers. Objectives: Students will be able to 1. Express square roots of negative numbers in terms of i 2. Add and subtract complex numbers 3. Multiply complex numbers 4. Divide complex numbers 5. Simplify powers of i Language Objective: Students will be able to describe imaginary numbers, and complex numbers Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through examples and classroom work. Small group assessments during activity. Standards: CCSS.Math.Content.HSN.CN.A.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. CCSS.Math.Content.HSN.CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.Math.Content.HSN.CN.A.2 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Color papers with written equations

55 Page 55 Students Materials: Pencil and paper Beginning of Lesson; warm up (10-15 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Home Work Quiz (15 minutes) Introduction (5 minutes) Verbally express the objectives of today s class. Today we are going to cover imaginary and complex numbers, thus everything we have said about not being able to square root -1, was a lie. Middle of lesson (60 minutes) The Imaginary Unit i Definition The imaginary unit i is defined as Example to work out with class Using the imaginary unit i, we can express the square root of any negative number as a real multiple of i. For example, We can check that by squaring 4i and obtaining -16. Abstractly, the Square Root of a Negative Number If b is a positive real number, then These are for students to work out. a.

56 Page 56 b. c. Complex Numbers All numbers can be written as a complex number, (a) the real part and (bi) the imaginary part. This gives us the set of all numbers in the form a+bi The Real part and the Complex part Write these on the board and then probe for questions, what is the real part? What is the imaginary part? Adding and Subtracting Complex Numbers It is easiest to understand adding and subtracting complex numbers if it is first broken into grouping like this. And in general it is like this: Multiply Complex Numbers With multiplication it is still easier to break it into groupings like this. Here is an example that shows that sometimes all you can pull out is the i Now have the class walk you through how to do these.

57 Page 57 Using Conjugates to Divide Complex Numbers Although I am calling it division it is actually just writing fractions in the standard complex form of and then multiplying by the conjugate of the denominator. Example as so. Simplifying Powers of 1. Express powers of in forms of 2. Replace with -1 and simplify. Use that -1 to an even power is 1 and -1 to an odd power is -1. Exit ticket (5-10 minutes) These examples will solidify the idea of simplifying powers of I, have the students work them out. a. b. c. Group activity (30 minutes) Break the class into groups of four to five (you need as many groups as people are in a group) Each group gets a different equation pertaining to i to solve together (it works best to use different colored papers, the same at each table) Allow ample time for each group to solve their equation and ask for questions and verify answers. Rearrange groups so each group has one of each color. The task is for each (color) to explain the solution process to the rest of the group.

58 Homework: Do enough research, about the history and how imaginary or complex numbers are used, to write a short paragraph about them. Page 58

59 Page 59 Name Date Homework Quiz

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65 Lesson Reflection This lesson ended amazingly. But getting to the end was rough. As much as I tried explaining different ways to understand i there simply seemed to be a mindset block about it. As the class rounded past the mid-way point even though I knew they had a lack of understanding I decided it was time to jump into the activity, get them on their feet and talking to each other and individually to me. The activity start off was bumpy but as some of the students began to grasp the whole idea they were able to explain it to the others who were not getting it and this expedited the learning process. I would change the homework about the history of imaginary and complex numbers to the lesson prior to this one so the students came in with some understanding of what we were doing. Page 65

66 Page 66 Lesson 8: Radical Review Topic: Evaluate radicands, rational exponents, simplify radical expressions including adding, subtracting, multiplying and dividing, rationalize denominators, solve radical equations and work with complex numbers. Rational: Show proficiency in the new material covered. Objectives: Students will be able to manipulate radical expressions and functions, rational exponents, multiply and simplify radical expressions, add, subtract and divide radical expressions, multiply with more than one term and rationalize denominators, solve radical equations and manipulate complex numbers. Language Objective: Students will solidify the information they have learned Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through examples and classroom work. Small group assessments during activity. Standards: CCSS.Math.Content.HSN.CN.A.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. CCSS.Math.Content.HSN.CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.Math.Content.HSN.CN.A.2 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Time: (90 minutes) Teacher Materials: Balloons Printout of the problems cut up and stuffed into balloons Printout of secondary problems Paper clips for each group for balloon popping

67 Page 67 Bag of candy Students Materials: Pencil and paper Individual small white boards or paper Beginning of Lesson; warm up (10-15 min) Put the class into groups. Hand out the Radical Review sheet and explain the activity that will take place half way through the class. The first 30 minutes of class after we are all in groups will be calaborating in groups about how to solve, and solving, the problems on the Radical Review sheet that I will hand out in a minute. At forty-five minutes left of class I have group selections that everyone will be assigned. Each group will get five balloons. The purpose is to end with the most balloons, you cannot have more than five. How we play: I will project a problem from the overhead. The first person to hold up the correct solution on their white board wins the round. The winner gets to pop a different group s balloon and pull the equation out to start the next round or if the group who won the round has less than five balloons they can take a balloon from another group and I will pull a question out of my extras. You can only win as an individual once before the rest of you group wins one, thus everyone in your group must understand each solution. At the end each participant gets a bag of candy and the winning group gets the remainder of the bag. During the first half while students work on the Radical Review sheet check in with groups to see if you can direct the learning where needed. Remind the students that tomorrow will be a test on radical after they have a final 30 minutes to discuss the review with me. Radical Review 1. find the indicated root 2. find the indicated values 3. simplify 4. simplify 5. simplify

68 Page simplify 7. multiply 8. simplify 9. multiply and simplify 10. add 11. subtract 12. simplify 13. simplify 14. multiply and simplify 15. rationalize the denominator and simplify 16. rationalize the numerator and simplify 17. solve 18. solve 19. solve 20. write in form 21. write in form 22. write in form 23. simplify

69 Page 69 Problems for in balloons 1. simplify 2. simplify 3. simplify 4. simplify 5. simplify 6. simplify 7. simplify 8. simplify 9. simplify 10.

70 Page simplify simplify simplify 13. simplify 14. simplify 15. rationalize the numerator 16. solve =5 solve 18. in terms of

71 Page rationalize the numerator 23. rationalize the denominator 24. simplify 25. simplify Extra problems 1. simplify 2. simplify 3. simplify 4. simplify

72 Page simplify 6. simplify 7. simplify 8. simplify 9. solve 10. simplify End (5 10 minutes) Remind them that the test will be the next day and they will have review time prior to it. Ask for any final questions. Get a closed eyes thumbs up thumbs down understanding of the information covered on the review.

73 Lesson Reflection When I did this lesson I allowed the same person to answer the questions during the activity and it started to become a one man show, it was necessary to not allow him to answer any more and is way there must be something in place that ensures everyone is engaged. The students loved popping the balloons. Page 73

74 Page 74 Lesson 9: Radical Test Topic: Evaluate radicands, rational exponents, simplify radical expressions including adding, subtracting, multiplying and dividing, rationalize denominators, solve radical equations and work with complex numbers. Rational: Show proficiency in the new material covered. Objectives: Students will be able to manipulate radical expressions and functions, rational exponents, multiply and simplify radical expressions, add, subtract and divide radical expressions, multiply with more than one term and rationalize denominators, solve radical equations and manipulate complex numbers. Language Objective: Students will show their proficiency of the information they have learned Lesson Assessments: Students will answer aloud questions about radicals. Visual assessments will be done while groups and one on one students work through examples and classroom work. Individual assessments Radical Test. Standards: CCSS.Math.Content.HSN.CN.A.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. CCSS.Math.Content.HSN.CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.Math.Content.HSN.CN.A.2 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Time: (90 minutes) Teacher Materials: Printout of test Printout of review

75 Page 75 Students Materials: Pencil and paper Beginning of Lesson; warm up (30 min) Put the class into groups. They have 30 minutes to work with others on anything they don t understand from the review. I am also available for any questions. Students can be shy about asking for help so in small groups it allows them to interact with other students and also allows you to come around and ask question and help them even when they don t ask. Also pay attention to getting the same question multiple times that is an indicator that it is beneficial to do a quick run through of it on the board for the whole class. The last 60 minutes of the class is for testing, if they finish early they can write me a note on classroom feedback and then read.

76 Page find the indicated root Radical TEST 25. simplify 26. add/subtract 27. simplify 28. multiply and simplify

77 Page rationalize the denominator and simplify 30. solve 31. solve 32. write in form 33. simplify

78 Lesson Reflection The group work was fantastic and I went around helping with what was needed. I didn t do any review on the board, it wasn t needed. The test was administered. ( I wouldn t change anything) Page 78

79 Page 79 Lesson 10: Fractals Topic: Fractals Rational: Show real world applications for math they have been learning. Objectives: Strike interest in the use of mathematics Language Objective: Students will be able to explain fractal and iteration Lesson Assessments: Students will answer aloud questions about iterations. Visual assessments will be done while groups and one on one students work through examples and classroom work. Standards: CCSS.Math.Content.HSN.CN.A.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. CCSS.Math.Content.HSN.CN.A.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. CCSS.Math.Content.HSN.CN.A.2 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Time: (90 minutes) Teacher Materials: Projection capabilities for a computer, with sound Link to Mandelbrot Set Zoom Link to YouTube Fractals Hunting The Hidden Dimension A down loaded copy of xaos put on student computers for their use. Students Materials:

80 Page 80 Pencil and paper Beginning of Lesson; warm up (10-15 min) Once the students are seated play the Mandelbrot Set Zoom off of YouTube. Now that you have their attention ask them what an iteration is. The dictionary definition is repetition of a mathematical or computational procedure applied to the result of a previous application, typically as a means of obtaining successively closer approximations to the solution of a problem. You start with a function, lets pick a simple one and we are going to start at x =2, And you simply keep putting the answer back into the equation. Does anyone have any idea how this could be useful? There is a good chance you won t get any reasons on why this is useful. At this point play Fractals Hunting The Hidden Dimension, its 53 minutes long and could be watched in its entirety, I use 26:25 to about 31: Put the class into groups and let help them play with the fractal zoomer.

81 Lesson Reflection This lesson was strictly about showing a different side of mathematics. Something that hopefully some of them can connect to. Maybe even spark and interest from. The students were totally wrapped in the zoomer and seriously enjoyed both creating and zooming fractals even to the point of locking up computers because they asked it to do more than its processing capabilities. I would like to make time to show the entire video Fractals Hunting The Hidden Dimension and have students research some of the other uses of fractals like the attempt to cure blindness using fractal metals. There was no exit ticket or need for them the idea was to add excitement and relax them before entering into compound interest and that was successful. Page 81

82 Page 82 Lesson 11: Exponential Functions Topic: Exponential functions Rational: Understanding and manipulating exponential functions Objectives: Students will be able to calculate and graph exponential functions Language Objective: Students will be able to explain exponential Lesson Assessments: Students will answer aloud questions about exponential functions. Visual assessments will be done while groups and one on one students work through example and classroom work including writing on the board and describing the work to the class. Standards: CCSS.Math.Content.HSF.LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. CCSS.Math.Content.HSF.LE.A.1.a Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. CCSS.Math.Content.HSF.LE.A.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. CCSS.Math.Content.HSF.LE.A.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Print outs of If offered a million dollars Poster board for each pair of students

83 Page 83 Colored pencils and markers Students Materials: Pencil and paper Beginning of Lesson; warm up (15-20 min) Warm Up (10 minutes) Allow the first five minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Write on the board or project the question You have a pair of rabbits, every 8 weeks they produce a pair of rabbits, and so does each new pair of rabbits. How many rabbits do you have in 72 weeks? Have students show their work and compare answers. Have students express descriptive solutions on the board or document camera. It is helpful but not necessary for the students to discover exponential functions it can be done by simple iterations and the discovery can come with the activity. Introduction (5 minutes) Verbally express the objectives of today s class activities. Today we are going to work with exponential functions. I am going to hand out an information sheet with a few questions and ideas. After working out these ideas and questions, as a group, you will create a poster showing all your math and giving your answers, choice, and why. The last fifteen minutes we will gather up as a class and come to each table to have you discuss what you chose and why. It is on the pamphlet but I want to reiterate that I will choose who presents so be sure that everyone understands the material enough to present it. Now get into groups of four. And I will hand out the information. Middle of lesson (60 minutes) Hand out the if offered a million dollars paper. Today will be a lot about walking around the classroom and answering questions but not giving answers. Have poster board and markers ready. The last 15 minutes do a gallery walk and pick members from each group to discuss why they chose what they chose. Exit Ticket (5 min) What was the most interesting part of today s discoveries

84 Page 84 Would you take a million dollars if offered? What would you take if given the choice? a. $1,000,000 b. A single dollar whose value doubles each day for the month of May Example, $1 +$2 +$4 +$8 Or c. A penny whose value doubles each week for one year Example, $.01 +$.02 +$.04 +$.08 Which would you choose? Create a poster that justifies your answer by showing all equations and use words for how you came to your decision. Make sure to include a graph for all three choices, each in a different color. Be certain that all group members can explain the reasoning, as I will pick who presents. Once finished with the poster, ponder and answer the following questions. Would it change your answer if it was $1,000,000,000 instead of $1,000,000? Why? How about a $100 per day for the month of May? Why?

85 Lesson Reflection When I wrote this lesson it the activity was only going to be maybe 20 minutes. However, as the discovery took longer and longer I witnessed a lot of peer learning and self-discovery going on and decided to add the poster and presentation part allowing it to run the length of the class. A couple students didn t choose the pennies even though it was the largest amount of money they said it was too much money to comprehend, the number was simply too large. There was a lot of shock at how quickly the exponentials grew. Page 85

86 Page 86 Lesson12: What is an interest Rate? Topic: What is interest? Rational: Students will explore the cost of paying varying interest rates, and differing compounding s Objectives: Students will be able to write an exponential growth or decay function to model a given real world situation. Language Objective: Students will be able to understand and use compound, interest, growth and decay Lesson Assessments: Students will work through a packet that will be collected Visual assessments will be done while in groups and one on one students work through example and classroom work. Student will turn in self made questions and work the answers of other student s questions. Standards: CCSS.Math.Content.HSF.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Packet Warm Up (15-20 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Solve: 15% of 1000 =?

87 Page 87 If I have $100 and it is increasing at a rate of $10 per day what percentage is it increasing by? How much would you have if you started with $20 and increased by 30% three times? Have students show their work and compare answers. Go over solutions if needed, no more than five minutes. If it takes more than five minutes for the warm up than there needs to be a lesson on the subject matter. Introduction: 30 minutes In part 1 of the handout introduces calculating percent increase and decrease. It is imperative that students get comfortable with using a one-step calculation. The skill and thinking required to do so helps them to write the exponential function in general form for the next section. Be sure to not rush students through this section. It is ESSENTIAL to students' success over the next four days that they understand how to calculate a percent increase or decrease using only a one-step calculation. In groups of four you will have 30 minutes to work through part 1 of the pamphlet I will be handing out. A reminder, as group work it is imperative that all group members understand what is going on and how to solve the problems. Before moving on to part 2 have a brief download of what was learned in part 1, probe a few answers look for any confusion you might have missed. Middle of lesson 30 minutes Part 2 During this section pay close attention to look for students who get stuck while generalizing the pattern, in terms of x, at the bottom of the table. If necessary give guidance but try to let it happen through discovery. The most important is that by the end they are able to see a pattern as an operation based in time, instead of iterations. For example, if a student is trying to find the balance on year 5 with a 25% interest rate they should not rely on finding the year 4 information first. Make sure that your students understand the balance as the result of $1000 being multiplied by 1.25 five times. I expect many students will need a reminder or if not seen before and introduction to recognize the process as repeated multiplication. I want them to be fluent with determining the percent rate of change.

88 Page 88 End of Lesson This will need a solid 5-10 minutes. Spend the last of class open to discussion and questions. Hopefully you get questions like: Why would you want a credit card? Are they good or bad to have? What is interest? Why do you have to pay someone to have a credit card? Is this like a loan when you buy a car? Homework: Listen to the first 20 minutes of Terry Gross interview with Elizabeth Warren located at Take down a few notes about any interesting points you hear in the interview to share with the class the next time we meet. To find this audio clip: Go to In the search bar type and click on the article.

89 Page 89 Name: Day 1: What is an Interest Rate? Part 1: Calculating percent By definition a percent is defined as a fraction with 100 as the denominator. To find a percent of a value means to multiply the value by that fraction. Example: 22% of 80 means 22 80, which equals Therefore, 22% of 80 is If you want to find a percentage increase, you need to add the percentage back to the original value. If you want to find a percentage decrease, you subtract the percentage from the original value. This can be done in two steps, or one step. For our purposes, it will be important to understand the one-step method. Example: You need to pay 15% tax on a car that cost $12,000. To do that, first find 15% of 12,000. Then, add that result to $12,000. That is the total cost. Two steps: 15 $12,000 $1,800 (that s the tax) $12,000 + $1,800 = $13,800 (that s the total cost). 100 One step: (0.15)(12000) = ( ) = (1.15) = Price + Tax = factor out price = 1-step calculation Answer the following questions with the one-step method. Write down what you multiplied, not just the final answer. Round your answer to the nearest cent. Answers are given to nearest dime. 1) You go out to dinner at a fancy restaurant, and the bill is for $68. But, you want to leave an 18% tip for the waiter. How much do you pay in total? One-step calculation: Answer: 2) Vegan meals at Tofu Com Chay are on sale for 30% off. The Mixed Buddha Special normally costs $8.50. How much does it cost now? 3) Your current balance for your Visa card is $522. Visa charges you a percentage of the balance each year in exchange for allowing you to borrow money from them, called the Annual Percentage Rate. Your APR is 12%. If you don t make any payments this year (assuming there are no late fees), how much will you owe at the end of one year? If you continue to not make any payments, how much will you owe at the end of two years?

90 Page 90 Part 2: Skipping your payments Now that you are experts at finding percent s, let s apply that knowledge to comparing credit cards. Card A has an APR of 10%. Card B has an APR of 25%. Card C has an APR of 50% (this is realistic for illegal enterprises, and payday loans.) Let s say you start off with a $100 balance on all cards, and you are not making any payments (because you re bad like that). How much will you owe in each year? Complete the table. Year Card A Card B Card C x (type into Y1) Use the TABLE in the calculator to carefully graph the trajectory of all three balances. Label everything. (x-scale = 1 year, y-scale = $50.)

91 Page 91 1) Draw the line on the graph where the money you owe doubles. Using the graph, find the year in which the amount you owe doubles for each card. 2) It turns out Card A has a $30 application fee! Explain how this would change the graph. Now is Card A or Card B a better idea? Justify your response. 3) What would be the interest rate on a card that, on an initial balance of $100, owed $ after 5 years, and $ after 10 years?

92 Page 92 Lesson Reflection These next four lessons starting with this one was a huge learning curve for me. I had originally planned on doing a packet of information the first half of the class and then a activity to end the class. The Packet became a four day lesson. I noticed a lot more engagement as the math moved towards real life, money, problems that the students could relate to. I helped that they were mostly senior and money had all of a sudden exponentially grown in importance to them. The was a lot of self-learning in this lesson which was good, however I would like to find some more interactive activities to go with it.

93 Page 93 Lesson13: What is an interest Rate? (Continued) Topic: What is interest? Rational: Students will explore the cost of paying varying interest rates, and differing compounding s Objectives: Students will be able to write an exponential growth or decay function to model a given real world situation. Language Objective: Students will be able to understand and use compound, interest, growth and decay Lesson Assessments: Students will work through a packet that will be collected Visual assessments will be done while in groups and one on one students work through example and classroom work. Student will turn in self made questions and work the answers of other student s questions. Standards: CCSS.Math.Content.HSF.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Packet Warm Up (10 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Solve: What would be the interest rate on a card that, on an initial balance of $250, owed after 5

94 Page 94 years and after 10 years? Have students show their work and compare answers. Go over solutions if needed, no more than five minutes. If it takes more than five minutes for the warm up than there needs to be a lesson on the subject matter. Introduction: 30 minutes In this section students will explore how making a $50 payment vs. a $20 payment every month will affect the credit card balance. Emphasize to students that this isn't really how credit card interest is calculated (once a year), but we will explore more during the next class how interest is really calculated. In groups different groups of four than you were in for parts 1 and 2, you will have 30 minutes to work through part 3 of the pamphlet I will be handing out. Again remember, as group work it is imperative that all group members understand what is going on and how to solve the problems. Before moving on to part 2 have a brief download of what was learned in part 1, probe a few answers look for any confusion you might have missed. Middle of lesson 30 minutes Part 2 In this section, students will be required to apply their results and knowledge from parts 1 through 3 of the activity. This is a great check to see if students understood the material. This section could be assigned as homework if students did not finish it in class. Homework (be sure to have watched the pod cast) End of Lesson This will need a solid 5-10 minutes. Spend the last of class open to discussion and questions. Push for a class discussion on how one might check there solutions.

95 Page 95 Name: Day 2: What is an Interest Rate? (Continued) Part 3: Making your payments You ve decided to become a responsible person, and you are going to actually try to pay off your credit card bill. Let s try some different options. Assume you currently owe a balance of $1000, and your APR is 29%. 1) Your minimum payment is $20 per month. At that rate, you will have paid $ by the end of the year (12 months), leaving you with $ that you still owe. When you apply the interest rate to the amount you owe, how much will your balance be? 2) Instead of paying $20 per month, you decide to increase it to $50 per month. How much will you have paid by the end of the year? What will your new balance be after you apply the interest? Part 4: Apply Your Knowledge 1) Since 1995, the daily cost of patient care in community hospitals in the United States has increased about 4% per year. In 1995, such hospitals cost an average of $968 per day. a. Write an equation to model the cost of hospital care since Let x = years since 1995, y = cost. b. Use your equation to estimate the approximate cost per day in ) The FM school district has 4512 students this year. The student population is declining by 2.5% per year. Write an equation to model the population decline, and find the population 3 years from now. 3) The function y = 10(1.08) x models the cost of annual tuition (in thousands of dollars) at a local college x years after a. What is the percent of increase? b. How much was tuition in 1997?

96 Page 96 c. How much was tuition in 2000? d. How much will the tuition be the year you plan to graduate from high school? 4) Write an exponential function y = ab x for a graph that includes (2, 2) and (3, 4). Lesson Reflection Another full engagement day, money seems to be a strong motivator. Today there were a lot more questions from groups as I wandered around and prodded for information on what they were doing. I find myself looking forward to the next lesson and from the feedback so do the students. Today a couple groups got finished before the rest and I sent them around to help their peers.

97 Page 97 Lesson14: How is interest really calculated? Topic: How much is really charged for interest. Rational: Students will explore the true cost of paying for something on an interest based payment plan. Objectives: Students will be able to derive the compound interest formula and apply it to solve real world problems. Language Objective: Students will be able to explain compound interset Lesson Assessments: Students will work through a packet that will be collected Visual assessments will be done while in groups and one on one students work through example and classroom work. Student will turn in self made questions and work the answers of other student s questions. Standards: CCSS.Math.Content.HSF.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Packet Warm Up (10 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Have students take a moment to reflect on their homework of listening to the pod cast. Each student in every group should think of one thing they learned in the NPR podcast and then share it with their group. Each group should pick the best fact and write it up on the board. Then as a

98 Page 98 whole class we can review some of the most interesting aspects of the conversation. Introduction: 30 minutes In this part of the credit card investigation students explore how interest is really calculated. Using a table students will be asked to write a general equation to calculate compound interest. I assume this section will be difficult for students. As students are working, I will assess their progress by checking their answers in the column expression for balance after x years. It is important that students have the correct expressions in Questions 3 and 4 if they are going to obtain the interest formula. I may also give students the guidance that they can break down the middle column of the table. In groups different groups of four than you were in for days 1 and 2, you will have 30 minutes to work through part 1 of day 3 of the pamphlet I will be handing out. Again remember, as group work it is imperative that all group members understand what is going on and how to solve the problems. Before moving on to part 2 have a brief download of what was learned in part 1, probe a few answers look for any confusion you might have missed. Middle of lesson 30 minutes Part 2 In this section, students will be answering some very straightforward questions about interest calculations. I will encourage them to self-assess as their work, focusing particular attention on the reasonableness of their answers. I will also give students some parts to these answers at some point in the lesson to help with their self-checks. Maybe I will give them the whole dollar amount and have them find the change part of the answer. Something like this: 1. $ $ $ $6918. I want students to know if they have made a mistake, without providing them with the exact answers so they can check their work.

99 Page 99 End of Lesson This will need a solid 5-10 minutes. Spend the last of class open to discussion and questions. Push for a class discussion on how interest can work for you and against you. Ask questions about how much is something really going to cost if you buy it in payments Name: Day 3: How Is Interest Really Calculated? Part 1: In reality, interest on a credit card isn t only charged once a year. Let s investigate how it s really calculated. 1) Suppose you charge $1,000 on a card that charges 12% APR. Find the account balance after 5 years. Try to do this by evaluating a single expression. Write down the expression as well as the answer. Expression you evaluated: Balance: 2) If interest were charged at a 12% APR semi-annually instead of once a year, the same card would charge 6% twice a year. By evaluating a single expression, find the account balance after 5 years. 3) Complete the Table: Frequency of Calculation Annual Number of Times Interest is Calculated per Year Interest Rate per Period at 12% APR Expression for Balance after x years Account Balance After 5 Years Semi-Annual

100 Page 100 Quarterly Monthly Weekly 4) Try to write an expression using only variables, for the account balance A, after t years, on an initial charge P, at an annual percentage rate r, calculated n times a year. 5) In reality, credit cards charge interest monthly. Adjust your equation to make it credit-card specific: Part 2: Apply your knowledge 1) You owe $200 on a debt with 16% APR, compounded monthly. How much will you owe after 2 years? Try to do this by evaluating a single expression. Write down the expression as well as the answer. Expression you evaluated: Balance: 2) You deposit $1,000 in a college fund that pays 6.1% interest compounded quarterly. Find the account balance after 5 years. Write down the expression as well as the answer. 3.) Suppose you have $1,500 in a savings account that pays 4.7% annual interest. Find the account balance after 25 years with the interest compounded semi-annually. 4) You deposit $4,500 into an account earning 3.6%, compounded quarterly. How much will be in the account after 12 years?

101 Page 101 Lesson Reflection The class finished this part of the lesson quicker than I had expected. This was not a bad thing as it allowed for extra time to talk about the really cost of things when you buy them on payments. As it was I had to cut them short on their conversations as class ended. In the future I will do some sort of activity with them looking up credit card companies and what they charge for interest what the minimal payment is and how long it would take to pay off a charge at the minimum payment.

102 Page 102 Lesson15: What is continuous compounding? Topic: How does nature compound things? Rational: Students will explore compounding interest at higher and higher frequencies. Objectives: Students will be able to explain the meaning of the mathematical constant e and use the continuous compounding interest formula to solve real world problems. Language Objective: Students will be able to explain continuous interest and e Lesson Assessments: Students will work through a packet that will be collected Visual assessments will be done while in groups and one on one students work through example and classroom work. Student will turn in self made questions and work the answers of other student s questions. Standards: CCSS.Math.Content.HSF.IF.C.7.e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. CCSS.Math.Content.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Packet Warm Up (10 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions.

103 Page 103 Find the accumulated value of an investment of $5000 for 10 years at an interest rate of 6.5% if the money is; a. Compounded semiannually; b. Compounded monthly Introduction: 60 minutes Today is the final day of the compound interest investigation. In the first two days of this investigation students learned all about interest. They learned how to calculate a percent increase or a percent decrease using only a one-step calculation. Then they applied this new knowledge to quickly compare how three credit cards balances grow with three different interest rates. They also explored a simplified problem where we increase a credit card payment by only $30 a month and saw how much quicker the $1000 balance we owed decreased. In the third day of the credit card investigation, students learned how interest is actually calculated. We talked about different rates of compounding and students analyzed how compounding rates can affect the amount of money we owe on a credit card. They then derived the compound interest formula by analyzing the structure of the expression used to calculate the interest for each compounding period and the patterns they observed. Today, students will extend their learning to continuously compounding interest. In this section of the investigation, students first review how to calculate compound interest again and then continue to increase the compounding frequency in order to introduce continuously compounding interest. This then leads students to the constant e. Finally, students will apply their knowledge of calculating compound interest by solving some example problems. Don t push the learning, give the entire hour as this can be a difficult derivation and it will be time to talk to individual groups about what they are learning. In groups different groups of four than you were in for days 1, 2 and 3 you will have 60 minutes to work through this last pamphlet that I will be handing out. At the end of the 60 minutes we will wrap up class by having you all present your conclusion at the end of the hand out. End of Lesson 15 minutes Have each group present their summary, if you run out of time have the remaining groups present as the warmup to the following class, it is important not only for their remembrance but to feel like hey this meant something.

104 Page 104 Name: Day 4: A Dastardly Scheme Take yourself back, mentally, to the previous lesson, where we investigated intervals for charging 12% APR. 1) Complete the new table: Frequency of Calculation Number of Times Interest is Calculated per Year Interest Rate Each Period Annual 1 12% Expression for Balance on $1000 after 1 year Account Balance on $1000 after 1 Year (1 0.12) 1120 Semi-Annual 2 6% Quarterly Monthly ) A dastardly accountant at the credit card company notices that if they charge interest more frequently, they will earn more money. (Sure, not much, but if they do it to everybody ) How frequently will they have to charge 12% interest in order for you to owe $1128 after one year? Weekly Daily Hourly Minute-ly What?! The accountant s tiny, cold heart skips a beat. There doesn t seem to be a way to even reach $ ! He thinks, what if we could charge interest all the time? Surely, if we could compound the interest infinite times per year, we could make infinite dollars. Sounds pretty amazing. 3) To investigate, let s consider a very simple example. 100% APR, on a $1 charge over 1 year for more and more frequent compounding. Show at least four decimal places. x , ,000 1,000,000 1 Substitute: 1 x Evaluate to 4 places x Sorry accountant. Your ingenious plan has been ruined by mathematics! Too bad There is a limit to how much interest they can earn, despite very frequent compounding. Limit is a very fancy concept. We are talking calculus-fancy, here. Even if the number in the second column there was

105 Page 105 infinity, there is a limit to how much money they could make. 4) As x approaches, this value is called e. Approximate e to three decimal places: 5) That is very unfortunate for the accountant. It seems that we can t earn $. Just to be sure, grab a x 1 calculator and graph y 1. In WINDOW, the x-min and y-min should both be zero. Make the x- x max= 100 and the y-max=5. Make a sketch of what you see and describe the shape and behavior of the graph. Label everything. Use the TRACE tool to see what happens to the y value of the function when x gets bigger and bigger. 6) Just like the number π, the number e is irrational. You can find a more precise value of by pressing the x e button on your calculator. Try this now. You need to type in the 1, so you see e ^ (1) on your calculator screen. To nine decimal places, e = The constant e is used in situations that involve continuous growth such as Growth of the body or any living organism Population growth (of very large populations) Half life of an organic compound (carbon dating) Continuous compound growth is modeled with: where P represents the initial value, r represents the growth rate, t represents units of time, and A represents the ending value.

106 Page 106 Apply your knowledge: 1.) According to statistical surveys, the annual growth rate in the world population in recent years is about 1.7%. There were about 5.3 billion people living on this planet in Unlike bank accounts, the compounding of population growth does not take place annually or quarterly. It s going on all the time. Every second of every hour, people are being born and others are dying. Thus, the growth rate (the overall effect of all of those births and deaths) can be viewed as a continuous process. If the same growth rate continues, how many people will inhabit the Earth by 2012? Use A rt Pe, where P = 5.3, r = 0.017, and t = ) Suppose you invest $1,050 at an annual interest rate of 5.5% compounded continuously. How much money, to the nearest dollar, will you have in the account after five years? (Financial institutions will not, in general, offer interest rates that are compounded continuously) 3.) A bacterial culture of 10,000 is growing continuously at a rate of 16% per day. a.) Find the population of the culture at the end of one day. b.) Find the population of the culture at the end of five days. 4.) If a population started with 20,000 members an grew continuously at a rate of 10% per year, 0.1x how long would it take the population to double? To find this, graph Y e, and look for the x (time) where the y (population) has doubled in size. (Hint: If you also graph Y2 = 40000, you can just find the intersection.) It s up to you to find the appropriate graphing window. Sketch and label your graph below. 5.) Summary Time. Now imagine that an interested and somewhat intelligent person approaches you and asks, What is e? What would you say to them? Think before you write, and maybe even go back over the previous pages. No big math words. No plagiarizing. No saying anything you don t understand.

107 Page 107 Lesson Reflection When I first attempted this lesson I tried to crunch all four of these days into half a class period. It didn t work. The lesson worked itself out to be four days long and I tried to put a activity at the end of this lesson where students create their own problem tapes them to different parts of the wall and then everyone walks about solving everyone else s problems and that activity should be a lesson on its own and will be the next lesson in this work sample. I have to admit that there was a lot of engagement and learning from peer to peer, but I still stand on wanting to find some activities that get students up and moving about at least a little bit.

108 Page 108 Lesson16: Inverse Functions and Logarithmic Functions Topic: Using composition of functions to verify inverses, solving exponentials with logs. Rational: Students will work with logarithmic functions Objectives: Students will be able to solve an equation for its inverse and use logarithmic functions as an inverse to solve exponentials. Language Objective: Students will be able to explain inverse function Lesson Assessments: Students will answer aloud questions about inverses and the class solves them Visual assessments will be done while groups and one on one students work through examples and classroom work. Small group assessments during activities. Standards: CCSS.Math.Content.HSF.LE.A.4 For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Warm Up (10 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. Compose: for the following equations

109 Page 109 Introduction: 30 minutes What you just did was compose inverses of functions. The definition of the inverse of a function although sometimes difficult in practice is simple in nature. Definition of the Inverse of a Function Here is an outline of finding the inverse of a Function The equation for the inverse of a function f can be found as follows: 1. Replace f(x) with y in the equation f(x). 2. Interchange x and y. 3. Solve for y. if this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function. 4. If f has an inverse function, replace y in step 3 with. We can verify our result by showing that. Let me walk through one with you and then I you can work on a couple. Find the inverse of Step 1. Replace f(x) with y: Step 2. Interchange x and y: Step 3. Solve for y: Step 4. Replace y with : Now you solve for the inverses of: The students should not be able to solve Middle of lesson: 30 minutes To solve you need logs. Not the ones from trees but logarithmic functions. This is the inverse to exponential. Look at this definition and then see if you can solve

110 Page 110 Definition of the Logarithmic Function A couple more things to know, properties of inverses means that and. And the last thing, most calculators will only do (log) which is base 10 and ln which is, get this, base e. So you need what is called the change of base to get a number in many cases. Change of base is simple it is the log of the x divided by the log of the base. Like this Let us put all this together. First we can look at what we couldn t solve well we now know that its inverse is. But is that any better than what we had before? Let s look at it with some numbers, and while we are at it verify some of this. We know that so how about we solve. Take the log base 2 of both sides Now apply the inverse rules and finally base change in the calculator that is exactly what we expected to get. I want to show you one more thing and then I have a pair of problems for you to work out at your tables. First isolate the exponential by dividing by 9 Now take the log base e of both sides, which would be the ln or natural log. since the same inverse properties apply we get stick that in your calculator At your tables work through these and I will walk around for questions. The formula models the population of Texas, A, in millions, t years after a. What was the population of Texas in 2005? b. When will the population of Texas reach 27 million? The formula models the population of California, A, in millions, t years after a. What was the population of California in 2005? b. When will the population of California reach 40 million? End of Lesson 15 minutes Work through the two class problems and be sure everyone was able to get to a salution as this leads into exponential growth and decay Homework

111 Exponential Problems Page 111

112 Page 112 Homework

113 Page 113 Lesson Reflection The students got very excited about being able to solve for exponentials. However some had a rough time manipulating logs and trying to get a grasp on natural logs. I will need to watch closely to see if it is necessary to spend more time on this concept. Next time I might do logs one class and then natural logs another. I had no class room for this lesson and did it out side with sidewalk chalk and used the side of the building as my board.

114 Page 114 Lesson17: Exponential Growth and Decay and Review Topic: Solving exponentials with logs. Rational: Students will work with logarithmic functions Objectives: Students will be able to solve growth and decay problems Language Objective: Students will be able to explain growth and decay in the form of what the exponents do Lesson Assessments: Students will answer aloud questions about inverses and the class solves them Visual assessments will be done while groups and one on one students work through examples and classroom work. Small group assessments during activities. Standards: CCSS.Math.Content.HSF.LE.A.4 For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Warm Up (10 minutes) Allow the first ten minutes of class for students to work alone on these warm up problems, and five minutes to discuss student solutions. The formula models the population of Hungary, A, in millions, t years after a. Find Hungary s population, in millions, for 2006, 2007, 2008, and b. Is Hungary s population increasing or decreasing?

115 Page 115 Introduction: 30 minutes What you just did is called decay. Here is the connection. We have already been using well growth and decay are the same equation with different labels. Exponential Growth and Decay The mathematical model for exponential growth or decay is given by If k > 0, the functions is growing If k < 0, the function is decaying A is the amount is the initial amount e is e k is the percentage (as a decimal) of the change. (negative if it is decaying) t is time in years Here are the formulas that model the population in 4, A, in millions, t years after 2006 India Iraq Japan Russia With this information answer the following questions. 1. What was the population of Japan in 2006? 2. What was the population of Iraq in 2006? 3. Which country has the greatest growth rate? By what percentage is the population of that country increasing each year? 4. Which country has a decreasing population? By what percentage is the population of that country decreasing each year? 5. When will India s population be 1238 million? 6. When will India s population be 1416 million? Give plenty of time for all the students to work through these problems and ask question. Middle of lesson: 30 minutes Now let s practice decay. Here are some questions pertaining to how we date things. An artifact originally had 16 grams of carbon-14 present. The decay model describes the amount of carbon-14 present after t years. 1. How many grams of carbon-14 will be present in 5715 years? 2. How many grams of carbon -14 will be present in 11,430 years? A bird species in danger of extinction has a population that is decreasing exponentially. Five years ago the population was at 1400 and today only 1000 of the birds are alive. Once the

116 Page 116 population drops below 100, the situation will be irreversible. When will this happen? End of Lesson 10 minutes Spend the last of class open to questions and remind the class that the next class will be the test covering exponentials, logarithms, and inverses. Homework Review

117 Page Find the accumulated value of an investment of $3500 for 12 years at an interest rate of 3.5% if the money is a. Compounded semiannually b. Compounded monthly c. Compounded continuously 2. Suppose that you have $15,000 to invest. Which investment yields the greater return over 3 years: 7% compounded monthly or 6.85% compounded continuously? Find the inverse functions of Solve

118 Page 118 The growth model describes Mexico s population, A, in millions, t years after a. What is Mexico s growth rate? b. How long will it take Mexico to double its populations? Using the exponential decay model for carbon-14, Prehistoric cave paintings were discovered in a cave in France. The paint contained 15%

119 Page 119 Lesson Reflection Today there was a lot of work on how to put these type of problems into your calculator. Everyone was nervous about the upcoming test and that hindered the learning environment. I did go ahead and tell them that the test would be given in groups and that helped calm some of the tension. My classroom had been made unusable and I did this lesson in the hallway with a projector, projecting onto the wall.

120 Page 120 Lesson18: Exponential Growth and Decay Test Topic: Showing proficiency Rational: Students will work in a group to take a test on the logarithmic materials learned Objectives: Students will be able to solve growth and decay problems Lesson Assessments: Growth and Decay test Standards: CCSS.Math.Content.HSF.LE.A.4 For exponential models, express as a logarithm the solution to ab ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Time: (90 minutes) Teacher Materials: Dry-Erase markers Pen and blank paper Document Camera Students Materials: Pencil and paper Warm Up (5 minutes) Put students into preselected random groups of four Introduction: 25 minutes These are the groups you will take the test with, take the next 25 minutes to look over each other s reviews and ask any needed questions

121 Page 121 Wander and probe for questions if any become obvious that it is a class wide question go over it on the board. Middle of lesson: 60 minutes Some necessary instruction on the group test. First everyone will turn in their own test. To get group participation points everyone must have the same answers and similar work. I will randomly choose one test from the group that test will be everyone s grade. I will allow 3 questions to me but they will cost, the first question is free, the second question will only allow you to get a 4- and a third question a 3 (this is out of a 4 point proficiency grading) Question? Print each group set of tests on different colors. While they test watch for group participation. End of Lesson Allow students to work to the very end. Name i. Show all your work ii. You will be graded for group participation all group members work must be complete to get group points iii. Using Jupiter grades I will randomly choose one members test, this is the one that will be graded for everyone s grade Formulas and Equations: Compound interest Continues interest Growth and Decay

122 Page 122 Logarithms 13. Find the accumulated value of an investment of $2500 for 10 years at an interest rate of 6.5% if the money is d. Compounded semiannually e. Compounded monthly f. Compounded continuously 14. Suppose that you have $12,000 to invest. Which investment yields the greater return over 5 years: 7% compounded monthly or 6.75% compounded continuously? Find the inverse functions of: 15.

123 Page Solve

124 Carbon-14 Dating: The Dead Sea Scrolls a. Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14 Page 124 b. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls.

125 Page 125 Lesson Reflection This was the first group test I ever gave. I found it very refreshing to see the extent of peer collaboration that went on for an entire hour. No group asked more than one questions, every group asked about setting up the half-life problem. After this experience I will implement group tests more often.

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127 Page 127 Radicals and Exponential Growth (Pre Assessment) Radical Expressions and Functions: 1. Police use the function = 20 to estimate the speed of a car,, in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. If a motorist is involved in an accident, and a police officer measures the car s skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer, they were not speeding, should the officer believe them? Explain. 2. Explain how to find the domain of a square root function. Rational exponents: 3. Simplify ( ) 6 show all your work. 4. Define and then give two examples. Multiplying and Simplifying Radical Expressions: 5. If the function = 3 2 models the distance,, in miles, that a person feet high can see to the horizon. If a pool deck on a cruise-ship is 72 feet above the water. How far can passengers on the pool deck see? Write the answer in simplified radical form. Use the simplified radical form and a calculator to express the answer to the nearest tenth of a mile. 6. Explain why 50 is not simplified. What do we mean when we say a radical expression is simplified?

128 Page 128 Radical TEST (post test) find the indicated root simplify add/subtract simplify ( 5 12) multiply and simplify rationalize the denominator and simplify = 5 solve 8. ( 5) = 4 solve (3 + 3 ) write in + form simplify

129 Page 129 Name (post test) i. Show all your work ii. You will be graded for group participation all group members work must be complete to get group points iii. Using Jupiter grades I will randomly choose one members test, this is the one that will be graded for everyone s grade Formulas and Equations: Compound interest = 0(1 + ) Continues interest = 0 Growth and Decay = 0 = 0 = = =, > 0, < 0 = 0 Logarithms log 1 = 0 ln 1 = 0 log = 1 ln = 1 log = ln = log = ln =