Sample Lesson Plan Format



Lesson Plan 7.1: nth Roots and Rational Exponents468630016002000Class: Honor Algebra 2Grade Level: Sophomores and JuniorsUnit: Powers, Roots, and RadicalsTeacher: Ms. Delaney LudwigObjectives Students will be able to:Evaluate nth roots of real numbers using both radical notation and rational exponent notation.Use nth roots to solve real-life problemsEssential QuestionWhen would you need to use the nth root? How can different occupations apply the nth root and rational exponents to their everyday life?Anticipatory Set Review/Brainstorming: (15 minutes)For today’s anticipatory set I will ask students to recall how to solve square root problems. I will give students two equations: 49 (which = 7) and 100 (which = 10)I will then ask them if square roots are the only type of roots there are. Following that discussion I will ask them how many different types of roots they can come up with. After a few minutes of students answering, I will explain that there could be an infinite amount of types of roots. This is because there is an infinite amount of numbers. Teaching: Activities (35 minutes)Nth Roots:I will explain that you can extend the concept of square roots with other types of roots. For example 2 is a cube root of 8 because 23 = 8. 3 is also a fourth root of 81 because 34 = 81. I will explain that for any number greater than 1, this equation bn = a can be used to explain that b is an nth root of a. An nth root of a can be written as na, where n is the index of the radical (equation). You can also write an nth root of a as a power of a. For example, suppose that a = ak. Then you can determine the value of k as follows: a × a = a (Definition of square root)ak × ak = a (Substitute ak for a)a2k = a1 (Product of powers property)2k = 1 (Set exponents equal when bases are equal)k = 12Therefore, you can now say that a = a12. Then I will show that you can do this with any other type of root. For example, 3a = a13. So, the equation na = a1n is what you should follow when evaluating for the nth root. I will then explain to my students that there are four main properties to know about evaluating real nth roots. These properties are:If n is odd, then a has one real nth root : na = a1nIf n is even and a is greater than 0, then a has two real nth roots: ±na = ±a1n.If n is even and a = 0, then a has one nth root: n0 = a10 = 0.If n is even and a is less than 0, then a has no real nth roots. After these steps, I will ask if there are any questions this far or if anything seems to be confusing. I will then use some example problems to show students exactly how they can use this equation and these steps to evaluate nth roots. Example problems:Find the indicated real nth roots of a.n = 3, a = -125Answer: because n = 3 is odd, and a = -125 has one real cube root according to the properties, you can write 3-125 = -5 or -12513 = -5. n= 4, a=16Answer: because n = 4 is even and a = 16 is greater than 0, according to the properties, 16 has 2 real fourth roots. Because 24 = 16 and (-2)4 = 16, you can write ±416 = ±2 or ±1614 = ±2. Rational Exponents:I will then explain to students what a rational exponent is. A rational exponent does not have to be of the form 1n where n is an integer greater than 1. Other rational numbers such as 32 and - 12 can also be used as exponents. I will explain to students, just like nth roots, rational exponents have properties too. I will then list the two main properties: Let a1n be an nth root of a, and let m be a positive integer.amn = (a1n)m = (na)ma -mn = 1amn = 1(a1n)m = 1(na)m , a ≠ 0After these steps, I will ask if there are any questions this far or if anything seems to be confusing. I will then use some example problems to show students exactly how they can use these properties to evaluate expressions with rational exponents.Example problems:Evaluate the following expressions:93/2 Answer: 93/2 = (9)3 = 33= 2732-2/5Answer: 32-2/5 = 13225 = 1(532)2 = 122 = 14I would then explain that these two answers are found by the radical notation way. I could also go over the rational exponent notation as well if students seem to understand that better or want me to walk through the steps to see if their steps were correct. Using nth Roots in Real Life:For this part of the class, I would tell students that these types of functions can come up in real life situations. I would ask them if they could think of any situations this might apply to. Based on the ideas students come up with, I would give them an example. My example would be: The Olympias is a reconstruction of a Greek galley ship used over 2000 years ago. The power P (in kilowatts) needed to propel the Olympias at a desired speed S (in knots)can be modeled by this equation P=0.0289s3. A volunteer crew of the Olympias was able to generate a maximum power of about 10.5 kilowatts. What was their greatest speed?Answer: P=0.0289s3 (Write the model of power given) 10.5 = 0.0289s3 (Substitute 10.5 for P) 363.32 = s3 (Divide each side by 0.0289) 3363.32 = s (Take the cubed root of each side) 7 = s (Can use calculator to solve and round to the nearest whole number)The greatest speed attained by the Olympias was about 7knots which is about 8 miles per hour. I would then ask if there are any questions or certain areas the class would like me to go over again. If not we will then move on to group work. ClosureGroup work: (25 minutes)Once students seem to understand the material and I answered any questions there are, I would then split students up into partners and have them work on problems from the book to make sure they understand how to solve these problems. I would determine partners by what color shirt you are wearing. Students that have the same color shirt would be partners. If a student did not have the same color shirt as anyone else I would have them find a student with the same color shoes. If students still did not have a partner I would just pair them up with students who are left and if an odd number of students I would choose a group for them to join. Independent PracticeThe problems from the book would consist of page 404, #13-28 all and #29-40 odds. I would assign only the odds for the second questions because students would be able to check the back of the book’s answer key to see if they are getting the correct answer. Students would be working with a partner on these problems, this way they can help each other out if struggling. I would be walking around the room at this time to make sure students are doing problems correctly and answering any questions. If students seem to be finishing early, I would start going over the answers by asking student what answers they got for each problem. If students seem to be struggling with certain problems from the book, I would do a problem at the board with the entire class as another example on how to solve these types of equations. I would then inform the class that if they did not get these problems finished in class, it is homework. AssessmentAs formative assessments, I would have students fill out a 3-2-1 form. I would pass out the form attached to this lesson to each student and ask them to name 3 things you found out today, 2 things you found interesting in this lesson, and 1 question you still have. I would give them the last 10-15 minutes of class because they might need some extra time to think and write out their ideas. If they seem to be finishing up early I would collect the sheets and remind them of the homework. Materials Copies of exit sheetDuration This lesson will last approximately 90 minutes. Modified from Madeline Hunters Lesson Plan Design Lesson Plan 7.4: Inverse Functions468630016002000Class: Honor Algebra 2Grade Level: Sophomores and JuniorsUnit: Powers, Roots, and RadicalsTeacher: Ms. Delaney LudwigObjectives Students will be able to:Find the inverse of a linear functionTo solve real life problems, such as finding a person’s bowling averageEssential QuestionWhen would you need to use the inverse of a linear function? How can different occupations apply inverses of linear functions to their everyday life?Anticipatory Set (15 minutes)For today’s anticipatory set I will explain the topic of inverse functions. I will explain that an inverse function maps the output values back to their original input values. This means that the domain of the inverse relations is the range of the original relation and that the range of the inverse relation is the domain of the original relation. I will then relate this definition to wrapping a present and then unwrapping it. For example, take this “wrapping function” (original function) which takes the input object and 1) puts it in a box, 2) wraps paper around the box, and 3) ties a ribbon around the box. The corresponding inverse functions, the “unwrapping function”, would then 1) untie the ribbon, 2) remove the paper, and 3) take the object out of the box. Note that the inverse function steps are the opposite order of the original function steps. I would also draw this picture to have as a visual for the students: 933450108585F (original function)020000F (original function)X (domain) Y (range)933450100965G (inverse function)020000G (inverse function)Y (range) X (domain)I will then ask if any of the students can come up with other analogies for this function. If so we can discuss them a little further. If not, we can move on to different examples of the lesson.Teaching: Activities (35 minutes)Finding an Inverse of Linear Functions:I will begin to explain that because inverse functions are properties of the original functions flipped you can easily make inverse tables from original function tables. For example, I would draw a table with random inputs for x and y and ask students how I would begin to create an inverse table. Depending on their answers, I would then explain the number values for x and y simply flip spots with one another. I would then also explain that because of this flip, when graphing the functions, the lines are just a reflection of one another. The graph of the inverse relation is the reflection of the graph of the original relation. The line of the reflection is y = x. Finding an Inverse Relation:I would begin explaining that to find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y and solve for y (if possible). For example:y=2x-4 (Write the original equation)x=2y-4 (Switch the positions of x and y)x+4=2y (Add 4 to each side)12x+2=y (Divide each side by 2)12x+2=y (This is the inverse relation)In this example both the original relation and the inverse relation happen to be functions, therefore, in such cases, the two functions are called inverse functions. These functions can be show by two properties:Functions f and g are inverse of each other provided:fgx=x and gfx=xThe function f is denoted by f-1, read as “f inverse”. I will then ask if there are any questions or confusion. If so I will explain the material again or write another example problem. Verifying Inverse Functions:I will explain to students that sometimes you might be given a problem where the inverse function of an equation is already given. When that happens they will need to prove the inverse to be true. I will then explain to the how you can do this. I will ask them to verify that fx=2x-4 and f-1x= 12x+2 are inverses. I will begin to ask students if they have any idea how we would go about this. Depending on their answers, I would first tell them they are going to need to show that f(f-1x)= x and f-1f(x)= x. Do so this I would explain to students that they will need to plug in values for f-1x and fx. For example: First Equation - f(f-1x)= f(12x+2)=2 12x+2-4= x+4-4= xSecond Equation – f-1f(x)=f-1( 2x-4)=12 2x-4+2=x-2+2=xI would then explain that these steps are basically plugging in the function values for it variables. I would ask if there are any questions or confusion. If so, I will further explain in certain areas that are needed. If not, I will move on to the next example. Partner Activity:For this activity, students will be partnered up. Their partner will be determined by the month they were born in. Students will stand up and form a line by the month they were born in (January to December). While the students are ordering themselves, I will write 5 problems on the board. These are the problems I will write: Once everyone is lined up I will pair each person up by the person they are standing next to. If there is an uneven amount of student I will find a group for the one student to join. After everyone is paired up, I will explain to the students they are to find the inverse functions of each equation. As they are working on the problems, I will be walking around making sure students are doing their work and doing it correctly. I will also be answering questions if needed. If I see students are struggling with the equations, I will work one of the problems at the board with the class. This will hopefully give them a start and help each other out since they are working in groups. Once everyone has finished the problems, I will ask groups that feel comfortable to put their work/answer on the board. After all problems are up on the board, we will go over them as a class. After all answers are clear and there are no more questions about this topic, I will move on to graphing. Graphing Inverse Functions:I would begin to have students recall how they would plot points on a graph. I would explain graphing inverses are just like that with only a few exceptions. I would start by writing on the board a property that students should follow when beginning to graph inverses. The property states that id point (a, b) is on the graph of f, then the point (b, a) is on the graph of f-1. Proof of this is, if (a, b) is on the graph of f, then b = f(a). Since b and f(a) are equal, f-1(b) and f-1(fa) are also equal, and f-1b= f-1fa=a ,according to the definition of inverse functions, so the point (b, a) is on the graph f-1. As an example, I would then graph the two functions we found in the first example. I would draw a graph on the board or by using an overhead and show that you plot the points just like you would any other equation. I would point out the slope and the y-intercept for each equation and then show the similarities of the two lines. I would also explain that both lines are a reflection of one another. I would then ask again if there were any questions/confusion or if anyone wanted me to repeat anything. If so I would further explain any details. If not I would move on to the activity. ClosureDeck of Cards Activity: (25 minutes)Once students seemed comfortable with graphing I will ask them to break off into their groups again. There should be 5 groups. I will then give each group a deck of playing cards. Each student will draw two cards at a time for a total of 5 times. This will give each student 10 cards. Each two cards they draw will represent an ordered pair relation. They will then be asked to graph the relation and inverse relation on graph paper. They will also have to determine if a relation and inverse relation are functions using the horizontal line tests. The value of the cards will be as follows:1=1, 2=2, 3=3, … 10=10Jack=11, Queen=12, King=13Negative “x” numbers = spadesPositive “x” numbers = clubsNegative “y” numbers = diamondsPositive “y” numbers = heartsIndependent PracticeStudents would have the rest of the class time to work on this deck of cards activity. Every student would need to turn in their graphs, but they would be encouraged to work together or ask each other for help if needed. I would be walking around checking on students. If students seemed confused on the directions, I would draw 2 cards from a deck and do an example problem. I would then inform the class that if they did not get these graphs finished in class, it is homework and would be collected next class period. AssessmentAs formative assessments, I would give each student a post-it note and ask them to briefly write down what the most important thing they learned today was. I would give them the last 10-15 minutes of class because they might need some extra time to think and write out their ideas. If they seem to be finishing up early I would collect the post-it notes and remind them of the homework. Materials 5 decks of cardsPost-it notesGraph paperDifferent color markers for board (for graphing)Duration This lesson will last approximately 90 minutes. Modified from Madeline Hunters Lesson Plan Design ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download