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