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In this unit, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (NRN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (FLE.A.4). They use appropriate tools to explore the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations of a graph of a logarithmic function relate to the logarithmic properties (FBF.B.3). Students identify appropriate types of functions to model a situation (i.e. based on the context, is an: exponential, linear, quadratic or sinusoidal the most appropriate). They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of CCSSM), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations. Notes: Lessons attached in a separate document, along with Do Nows. Vocabulary words are embedded in the lessons, as well as posted on the classroom “word wall,” following instruction for the day. Technology is used through a PowerPoint in class to present the work, the use of calculators during some lessons and the doc cam to show student work. Differentiation in the class occurs through the use of a separate graphic organizer for notes, in addition to shortened assignments, strategic seating and targeted circulation in class. All students also have access to the answer key on google classroom and we will review almost all answers in class, through presentation of student work.
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Catherine Hyman: 11^{th} Grade Algebra 2/ PreCalculus
Unit Overview:
In this unit, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (NRN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (FLE.A.4). They use appropriate tools to explore the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations of a graph of a logarithmic function relate to the logarithmic properties (FBF.B.3). Students identify appropriate types of functions to model a situation (i.e. based on the context, is an: exponential, linear, quadratic or sinusoidal the most appropriate). They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of CCSSM), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
Notes: Lessons attached in a separate document, along with Do Nows. Vocabulary words are embedded in the lessons, as well as posted on the classroom “word wall,” following instruction for the day. Technology is used through a PowerPoint in class to present the work, the use of calculators during some lessons and the doc cam to show student work. Differentiation in the class occurs through the use of a separate graphic organizer for notes, in addition to shortened assignments, strategic seating and targeted circulation in class. All students also have access to the answer key on google classroom and we will review almost all answers in class, through presentation of student work.
Standards Addressed:
FLE.A.2: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table).
NRN.A.1: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents
NRN.A.2: Rewrite expressions involving radicals and rational exponents using the properties of exponents
FBF.A.1a: Determine an explicit expression, a recursive process, or steps for calculation from a context.
FIF.B.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
FLE.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.
FBF.B.4a: Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.
ASSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MP.1: Make sense of problems and persevere in solving them
MP.2: Reason abstractly and quantitatively
MP.3: Construct viable arguments and critique the reasoning of others
MP.4: Model with Mathematics
MP.5: Use appropriate tools strategically
MP.6: Attend to precision
MP.7: Look for and make use of structure
MP.8: Look for and express regularity in repeated reasoning
Day 
Lesson 


1 
Properties of Integer Exponents 
Objective(s) SWBAT · apply the properties of exponents for integer exponents. · model a realworld scenario involving exponential growth and decay. 
Do Now Review of low questions from last quiz on rational functions – focusing on operations with rational functions, as the skill is essential 
Exit Ticket Applying properties 
Criteria for Success Students can successfully use the properties of exponents to evaluate expressions. 

CCSS FLE.A.2 NRN.A.2
SMP MP.1 MP.2 MP.4 MP.7 
Key Vocabulary · Exponential Growth/ Decay · Integer 
Key Points · This lesson reviews the properties of exponents taught in previous courses.

Misconceptions · incorrectly using the properties of exponents (i.e. multiply exponents when multiplying bases) 
Instructional Lesson Type Exploratory Lesson – students experiment with the properties to solidify their understanding, as opposed to instruction 


Day 1 Details: In this lesson, students will review exponent properties, following a review of topics from their last quiz (the close of rational functions). Students have learned much of these exponential properties in their Algebra 1 class (two years ago), so this will be more of a review than anything else. However, since it was learned so long ago, with a year of Geometry in between, review is essential. With this being largely review, students will likely feel very successful throughout the lesson, giving them an early positive push for the rest of the unit. Additionally, a lot of the focus through the lesson should be on gearing and manipulating the properties to current standards and, eventually, solving equations. Showing previews of what to come is essential for the framing of the lesson, while also providing challenge problems at the close of the lesson, gearing students up for harder solving equations problems, in addition to seeing the properties within logarithms. At the close of the lesson, students will have a 6 question exit tickets simply asking students to apply the properties of the day. 


2 
Properties of Rational Exponents 
Objective(s) SWBAT · calculate quantities that involve positive and negative rational exponents. 
Do Now Review of previous day 
Exit Ticket Applying properties and rewriting 
Criteria for Success Students can rewrite expressions from radical to exponential notation and back again. 

CCSS NRN.A.1 NRN.A.2
SMP MP.3 MP.6 MP.7 MP.8 
Key Vocabulary · n^{th} root · principal root · Rational 
Key Points · Focus of lesson: connecting rational exponents to radical expressions 
Misconceptions · exponents must be integers · there is no connection between the exponential form and the radical form of a number 
Instructional Lesson Type Discovery Lesson 


Day 2 Details: In this lesson, students will focus primarily on rational (fractional) exponents, following a do now focused on the previous day’s properties as well as receiving their corrected exit tickets. Unlike the previous lesson, the majority of students have not seen this property, making this rather new for their work. As this is largely a discovery lesson, students will first have to predict and estimate the effects of rational exponents, followed by being shown a few example problems and having to predict the property themselves. Following the discovery and share out of predictions, students will then apply the property to problems increasing in difficulty, while infusing properties from previous day. The exit ticket for this class is focused on saying whether or not a simplified answer is correct or not. If it is not correct, students have to include the correct answer. *This lesson will also include many ACT sample problems as this is a power standard on the ACT, which many students will be taking in the following months. 


3 
Rewriting Expressions Using Exponent Properties 
Objective(s) SWBAT · rewrite expressions involving radicals and rational exponents using the properties of exponents. 
Do Now Review of previous day 
Exit Ticket Practice rewriting expressions 
Criteria for Success Students can find an exact value of the product of numbers with rational exponents without the use of a calculator and can justify the validity of an equation using the properties of exponents. 

CCSS NRN.A.1 NRN.A.2
SMP MP.3 MP.7 MP.8 
Key Vocabulary · rational exponent · radical expression · Nth Root · Root · Index 
Key Points · Properties of exponents hold when the exponents are any rational numbers. · Radical expressions can be rewritten using properties of exponents. · Rational numbers (exponents) can be represented in decimal form. 
Misconceptions · properties of exponents cannot be used with rational exponents 
Instructional Lesson Type Problem Set Lesson 


Day 3 Details: This lesson is largely a summary and review of properties covered in previous two days, as these are so essential for: logarithmic understanding, calculus understanding and ACT success. In this lesson, students will complete differentiated work, based on the two previous day’s exit tickets and will close with a whole class white board activity. Additionally, in this lesson, students will be asked to calculate individual exponential expressions, through property application, as opposed to simply rewriting. For this, students will have to master and access vocabulary words, such as integer. 


4 
Modeling with Exponential Functions 
Objective(s) SWBAT · write an exponential function that represents a realworld situation. · calculate the average rate of change of a function. 
Do Now Review of previous day 
Exit Ticket Application of Euler’s number 
Criteria for Success Students can write an exponential function that represents a given situation. 

CCSS FBF.A.1a FIF.B.6 SMP MP.4 MP.8 
Key Vocabulary · Euler’s number, e · average rate of change 
Key Points · e naturally occurs in many applications. · The rate of change of the exponential function base e at a value a is that same as the value of the function at a. 
Misconceptions · e is specific to the volume of water in a tank 
Instructional Lesson Type Problem Set Lesson 


Day 4 Details: A major focus of this course is application problems and solving with modeling, which is a feature of this lesson. Students will use application and modeling problems to discover Euler’s number (e), which will be applied to later lessons, using rate of change problems. Much like when students have to discover pi in Geometry, student have to discover the constant occurrence of e in everyday situations, identifying it as the “natural number.” Not only do students need to know e for lessons in this unit, students can also focus on e being an irrational number, identifying the definition vs. rational numbers and integers. Students can also use this as a review of “rate of change” which will be essential for lessons in the last third of this larger unit in a large application chunk of lessons. This lesson can also be a great time to include some history of math by introducing the mathematician, Euler. There will be a short portion of the PowerPoint focused on Euler, his work, and how it is still relevant and essential. Since this course moves quickly and it is not permitted at my school, students will not see a movie on Euler. 


5 
Solving Exponential Equations 
Objective(s) SWBAT · solve simple exponential equations numerically. 
Do Now Review of previous day 
Exit Ticket 4 solving exponential equations: 2 one to one property and 2 requiring rewriting 
Criteria for Success 

CCSS FBF.A.1a FBF.B.4a ACED.A.1
SMP MP.8 
Key Vocabulary · Inverse · Base · OnetoOne 
Key Points · When both sides of the equation can be written as exponential expressions with the same base, then the equation can be solved exactly. · When an exponential equations cannot be solved by hand, approximating the solution can be done by “squeezing”. 
Misconceptions · If the exponential expressions on both sides of an equation cannot be written with the same base, the equation cannot be solved.

Instructional Lesson Type Problem set lesson 


Day 5 Details: Following 4 days of exponential properties, students will now by tasked with solving exponential equations. While, at this point, they will only be able to solve problems with similar bases, students will be instructed following the next day’s lesson, they will be able to. For students who finish early, I will introduce the basic logarithmic property, in an effort for them to try and discover how to solve problems without the same base. Throughout the lesson, while many students will understand immediately how to deal with the bases, I will push students to continue following the proper notation, showing each step along the way, primarily for error catching and identification. Students will first attempt to create the property themselves and I will follow up by addressing whether it is correct or not. 


6 
Calculating Logarithms 
Objective(s) SWBAT · calculate a simple logarithm using the definition. 
Do Now Review of previous day 
Exit Ticket Practice rewriting and solving logarithms 
Criteria for Success Students can explain valid values for the base b in the expression . They can calculate simple logarithms using the definition. 

CCSS FLE.A.4 FBF.B.4a
SMP MP.8 
Key Vocabulary · logarithm · base for a logarithm 
Key Points · This is the introduction to logarithms, so the “What Power” language is used as a way to make connections for students moving forward. 
Misconceptions · b can equal 0 or 1, or can be a negative value 
Instructional Lesson Type Discovery Lesson 


Day 6 Details: One of the most important lessons of the year, students will be introduced to logarithms for the first time in their math careers. Since it is a strange concept and major shift in thinking, students will be introduced to it as the “whatpower” function, in a way to reinforce the idea of what a log truly is. Students will be presented with examples and be tasked with identifying the property themselves, followed by practice problems, with an answer key, and finally a share out of predicted properties. Since many exponential properties are required for this, I will make sure to remind students to reference the page of properties in their notes (i.e. what sort of exponent makes a term go to a fraction or get smaller). The exit ticket for this lesson will require students to both rewrite and solve logarithms. As with the previous lesson, while many students can easily solve without formally writing out the expression, students will still be required to, so they are not confused for later lessons. 


7 
Discover Properties of Logarithms 
Objective(s) SWBAT · construct a table of logarithms base 10 and observe patterns that indicate properties of logarithms. 
Do Now Review of previous day 
Exit Ticket Calculate logarithms from a logarithm table, applying logarithm properties 
Criteria for Success Students will be able to use a logarithm table to approximate a specified logarithm. 

CCSS ASSE.A.2
SMP MP.7 MP.8 
Key Vocabulary · common logarithm · logarithm properties 
Key Points · For integers k, · For integers m and n, · For integers k and positive real numbers x, 
Misconceptions · Multiplication and division undo one another · Addition and subtraction undo one another 
Instructional Lesson Type Discovery Lesson 


Day 7 Details: This lesson builds on the rule developed in the previous lesson and focuses on the algebraic properties required for logarithms. Using calculators, students are assigned to use a table with specific problems intended to present students with consistencies so patterns can be identified and applied. Following calculating specific logarithms with calculators and included table, students must identify the property, using correct notation. The believed properties will be reviewed as a class and subsequently applied to practice problems. Following the identification of the first two properties, the remaining 4 will be shared and applied in practice. On the exit ticket, students will be asked to condense/expand and identify which property was used. 


8 
Prove Properties of Logarithms 
Objective(s) SWBAT · justify properties of logarithms using the definition and properties already developed. 
Do Now Review of previous day 
Exit Ticket Prove the rewriting of expressions using properties 
Criteria for Success Students can use properties of logarithms established in the lesson to project their learning in order to establish an additional property. 

CCSS ASSE.A.2 FLE.A.4
SMP MP.3 
Key Vocabulary

Key Points · · · · · ·

Misconceptions · properties of logarithms are not connected to exponential laws

Instructional Lesson Type Problem Set Lesson 


Day 8 Details: Following the introduction of six logarithmic properties in the previous class, students will be asked to apply the logarithmic properties to condense or expand expressions. However, students will also be asked to prove particular expressions of condensing or expanding but will only be allowed to use specific properties for the questions. This will require students to think very critically about the path to application and fluidity in the work, since they won’t have everything at their disposal. This is a lesson requiring ample cooperation between classmates, with the need to seek help from one another, while the teacher circulates. The push to help peers will help both one students understand the work but will help reinforce the work to the students explaining . 


9 
Change of Base 
Objective(s) SWBAT · calculate logarithms with any base using a calculator that computes only logarithms in base 10 and base e. · justify properties of logarithms with any base. 
Do Now Review of previous day 
Exit Ticket Apply change of base to solve for any base logarithm 
Criteria for Success Students can rewrite logarithms as equivalent expressions. 

CCSS FLE.A.4
SMP

Key Vocabulary · natural logarithm · change of base formula 
Key Points · The properties of logarithms are reviewed in the lesson in connection to changing the base to base 10 or base e (ln).

Misconceptions · ln is misread as the word “in”

Instructional Lesson Type Problem Set Lesson 


Day 9 Details: As one of the culminating lessons on this subunit, students will start bringing a lot of their work together. Students will first learn about the ‘natural log’ (log with base e), which will require them to pair their understanding of logs with their knowledge of the natural number e, from a previous lesson. Following this, students will be introduced to logs which are impossible to calculate using mental math and/or the log properties and will use tables to prove change of base formula and master the equation for application to various log problems. In order to build context, students will see previews of the need to use change of base to solve equations in subsequent lessons. A major focus of this lesson is also calculator fluency and rounding knowledge. 


10 
Solving Equations with Logarithms 
Objective(s) SWBAT · solve simple logarithmic equations using the definition of logarithm and logarithmic properties. 
Do Now Review of previous day 
Exit Ticket 4 solving logarithmic equations: 2 one to one property and 2 requiring rewriting 
Criteria for Success Students can solve a logarithmic equation by rewriting the equation in exponential form. 

CCSS ASSE.A.2 FLE.A.4
SMP MP.3 
Key Vocabulary · Inverse · Change of Base · Extraneous Solution 
Key Points · Logarithmic equations can be rewritten as exponential equations using the properties of logarithms in order to solve. · It is important to check for extraneous solutions. 
Misconceptions · students may neglect to check for extraneous solutions 
Instructional Lesson Type Problem Set Lesson 


Day 10 Details: As the last lesson on this subunit, students will bring all of their knowledge from the previous 9 lessons together with their understanding of solving equations and the idea of extraneous solutions. Following this lesson, students will be able to solve for problems they were unable to solve in the exponential equations unit, by applying their log knowledge and understanding of change of base formula. The lesson will require students to use all properties, rewrite functions, solve equations and check their solutions, making this the most difficult lesson of the unit. With this, the exit ticket is essential to know what the focus on reteaching needs to be, if a student doesn’t fully grasp the concept. 

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