Assignment 4- N

Nermin Fialkowski

AIL 624

Dr. Jan Richards

Assignment Week 4 3-Day Unit Plan Class: Pre-Calculus (Honors & Regular) Chapter 3: Exponentials & Logarithms Section 1: Transformations & Exponentials Textbook: College Preparatory Mathematics, Pre-Calculus with Trigonometry I teach a total of three Pre-Calculus classes, two Honors and one Regular. All classes are made up of junior and seniors. This Unit Plan is intended for both levels. My Honors classes do move at a faster pace. Students work well independently and in groups. Students in my Regular class lack procedural fluency and algebraic skills. Unit Exams are slightly modified between levels. All classes have a handful of English Language Learners, and there is one Special Education student in each class. Lesson 1 3.1.1- How do kf(x) and f(kx) Transform a Graph? Standard addressed: Common Core State Standards Functions: Building Functions (F-BF) Build new functions from existing functions 3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); and the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Mathematics Content Standards for California Public Schools Algebra II 9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions. Lesson objectives: Students will explore the effect of multiplying the output and input by a number, (kf(x)) and (f(kx)), respectively on the graph of y = f(x). Lesson description: Students will explore the effect of multiplying the output and input by a number, (kf(x)) and (f(kx)), respectively on the graph of y = f(x). Students will explore these effects by using an interactive simulator (www.desmos.com/calculator/kai88dwx8z) that

RATIONALE FOR INSTRUCTIONAL AND STUDENT ACTIVITIES. Students will explore the effect of multiplying the output and input by a number, (kf(x)) and (f(kx)), respectively on the graph of y = f(x). Students will explore these effects by using an interactive simulator (www.desmos.com/calculator/kai88dwx8z) that allows students to change the output and input values of the variable k, and visually see how the graph of f(x) is transformed. • Connectivism (Burke, 2016) o Students use technology to complete the activity. o Four Steps 1.) Aggregation Students make predictions first as to what effects k will have 2.) Remixing Develop understanding of effect of multiplying the output and input by a number 3.) Repurposing Complete homework assignment 4.) Feeding Forward Students solidify effect of multiplying the output and input by a number. • Advanced Organizers (Week 4: Teaching Approaches and Models)

allows students to change the output and input values of the variable k, and visually see how the graph of f(x) is transformed. Students have three focus questions to help guide them in determining these transformations. For Question #1 (kf(x)) and #2 (f(kx)), students will select four different k values, and describe how k effects the graph of f(x). Question #3 has students narrow down their focus specifically on k=–1. Students continue to describe the transformational effects that k=–1 has on f(x). Students will then apply their findings of the transformational effects of k. Students will graph four transformations (a. 2f(x), b. f(2x), c. –f(x), d. f(–x)) for the same given graph.

Assessment plan for lesson: Students will be given about 15-20 minutes to explore the simulator and describe the transformational effects that k has on f(x). Students will be informally assessed as to whether or not they can correct identify the transformational effects on f(x). kf(x) is a vertical transformation (stretch if k>1 or compression if 0
o Desmos simulator acts as advanced organizer. - Divided into three sections, one for each focus question - #1 kf(x) - #2 f(kx) - #3 –1f(x) and f(–1x)

descriptions of the transformations and create the new graph (What are the Types of Assessment?) Students’ homework will be a type formative assessment. Grading this assignment will allow me to provide feedback to my students, as well as identify any gaps in student knowledge that will allow me to modify my instruction as needed The homework assignment includes describing three different transformations to f(x)=x2. And finding three expressions, in function notation, that generates transformed graphs, based on a given function f(x).

Lesson 2 Match My Exponential Standard addressed: RATIONALE FOR INSTRUCTIONAL AND Common Core State Standards STUDENT ACTIVITIES. Functions: Linear, Quadratics, and Exponential Models (F-LE) Students will be exploring the effects of initial value Construct and compare linear, quadratic, and (a) and growth factor (b) of an exponential function, exponential models and solve problems y = a · b , through the use of technology. Students will use an interactive Desmos Activity, Match My 2. Construct linear and exponential Exponential. The Match My Exponential Activity is functions, including arithmetic and geometric sequences, given a graph, a interactive and provides students with immediate description of a relationship, or two input feedback. Students can make adjustments to their output pairs (include reading these from a equations to fit the requirement of passing through two points. While fulfilling these requirements, they will table). 
 be able to develop an intuitive understanding of initial Mathematics Content Standards for California value (a) and growth factor (b) of exponential Public Schools functions, y = a · b . (Students will create a total of nine Algebra II exponential graphs along with their respective 12.0 Students know the laws of fractional equations). At the end of the activity students will have exponents, understand exponential the ability to plot their own two points and create their functions, and use these functions in own unique graph and equation. problems involving exponential growth and • Connectivism (Burke, 2016) decay. o Students use technology to complete the activity. Lesson objectives: o Students share knowledge by Students will explore and have an introductory completing their own unique graph and understanding of the effects of initial value (a) and equation. growth factor (b) of an exponential function, o Four Steps y = a · b. 1.) Aggregation Guess-and-check method for creating an equation that satisfies going through two points 2.) Remixing x

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Lesson description: Today’s activity will be completed with the use of technology. The entire lesson will done through a Desmos Activity, Match My Exponential. Students will be graphing and creating various exponential equations. Students’ graphs and equations must satisfy passing through two points. The Match My Exponential Activity is interactive and provides students with immediate feedback. Students can then make adjustments to their equations to fit these requirements. While fulfilling these requirements, they will be able to develop an intuitive understanding of initial value and growth factor of exponential functions.

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Students will create a total of nine exponential graphs along with their respective equations, that pass through at least two given points. At the end of the activity students will have the ability to plot their own two points and create their own unique graph and equation. Through Desmos, I am able to view all of my students’ progress through The Dashboard. Students can work at their own pace throughout this activity and receive immediate feedback. The Dashboard home page allows me to monitor student progress. I can view which question they are currently working on and whether they answered it correctly or not. With this resource, I am able to see and spend more time with the students who are struggling. An overview of the Activity questions can be found at: https://docs.google.com/document/d/1Lhd0REg7 _etFg8JBg3XM58-zVXPwNMNAXs3YRIRcJyg/ edit?usp=sharing To view the Activity, go to: student.desmos.com, and use the code JK3ZUA Assessment plan for lesson Students will have 20-40 minutes to complete the Match My Exponential Activity. During this Activity, students will learn how to find an exponential function in the form of y = a · b , whose graph passes through two given points. x

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Develop intuitive understanding of initial value and growth factor of exponential functions 3.) Repurposing Students share create their own unique graph and equation 4.) Feeding Forward Students share their own unique graph and equation with class Concept Attainment (Week 4: Teaching Approaches and Models) o “Students generalize, hypothesis, categorize, and use questioning techniques in order to logically synthesize information.” - With the use of immediate feedback, and a total of nine examples, students will be able to develop intuitive understanding of initial value (a) and growth factor (b) of exponential functions. Where (a) is the y-intercept and (b) is the amount needed to get to the next point. Motivation (Motivating Students) o Appeal of Subject - Anticipation - Surprise - Feedback Brain-Based Teaching (Jensen, 2005) o Anticipation/Curiosity - Creates a positive state of hope and vigilance. Differentiation o Students share create their own unique graph and equation

Students will develop an intuitive understanding of initial value and growth factor of exponential functions, through the use of various interactive examples. The purpose is for students to pick up on pattern relations. Pattern recognition is essential for improving and understanding recall (Doyle & Zakrajsek, 2013) By the end of the activity students should have an introductory understanding of the effects of initial value (a) and growth factor (b) of an exponential function, y = a · b . x

The Match My Exponential Activity is a type of Formative Assessment but can also be labeled as a Diagnostic Assessment. The reason it can be labeled a Diagnostic Assessment is because I did not provide students with any prior information that in an exponential function y = a · b , (a) is the initial value and (b) is the growth factor. The plan is for students to come to that conclusion on their own through pattern recognition. Diagnostic Assessments identify “students’ current knowledge of a subject, their skill sets and capabilities, and clarify misconceptions before teaching takes place (Formative and Summative Assessment).” x

As a Formative Assessment, student responses will be observed for accuracy, but will be done so informally (students’ grades would not be affected by this activity). Additionally, students’ accuracy on the Activity will guide my instruction for the next lesson.

Lesson 3 3.1.2- How Do I Find Exponential Functions? Standard addressed: RATIONALE FOR INSTRUCTIONAL AND Common Core State Standards STUDENT ACTIVITIES. Functions: Linear, Quadratics, and Exponential Models (F-LE) Students will work through 3.1.2 of their textbook, Construct and compare linear, quadratic, and using Guided Notes. Students will find exponential exponential models and solve problems functions in the form of y = a · b , that passes through two given points. Students will then apply 2. Construct linear and exponential this knowledge and skill to real-world situations. functions, including arithmetic and geometric sequences, given a graph, a • Direct Instruction (Week 4: Teaching description of a relationship, or two input Approaches and Models) output pairs (include reading these from a x

table). 
 Mathematics Content Standards for California Public Schools Algebra II 12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

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Lesson objectives: Students will learn how to algebraically find an exponential function of the form y = a · b whose graph passes through two given points. x

Lesson description: Students will work through 3.1.2 of their textbook, using Guided Notes. Students will find exponential functions in the form of y = a · b , that passes through two given points. Students will then apply this knowledge and skill to real-world situations. x

The first questions in their textbook walks them through a step-by-step process needed to algebraically find an exponential function that passes through two points. The process has students start with a system of two equations by replacing the x and y-values of the two points with x and y of y = a · b . Next is a substitution process used to solved for a and b. x

Using real-world situations, students learn about halflife of radioactive isotopes. A table of time versus amount of radioactive isotope remaining is completed using half-life. Students then select two values from the table to create an exponential function to model the amount of radioactive isotope after t minutes.

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Using the table, students will be able to identify the initial value (a) and growth factor (b) of an exponential function, y = a · b . x

Guided Notes for this lesson can be found at: (https://docs.google.com/document/d/1PsZAHATfKhb4SDIcndNMBZfTuy6YthpWCETvYe3Co/edit?usp=sharing).

o Guided Notes 1.) Introduction/Review Initial value (a) and growth factor (b) of exponential functions, y = a · b . 2.) Development Model expected learning outcomes. “If teachers’ expectations are communicated to students and those expectations affect student’s confidence, then achievement can be impacted as well” (Woolfolk, 2017, p.416). 3.) Guided Practice Workout first questions from textbook with students. “Modeling can be most effective when the teacher makes use of all elements of observational learningattention, retention, production, and especially reinforcement and practice” (Woolfolk, 2017, p.419). 4.) Closure Highlight key components of how to solve for (a) and (b) algebraically 5.) Independent Practice Students solve real-world problems using the method/procedures from the first problem in the notes 6.) Evaluation Students work on homework Motivation o Relevant curriculum (Jensen, 2005) - Real-World Situations - Surprise - Feedback o Authentic Assessment (Woolfolk, 2017) - Formative assessment Connectivism (Burke, 2016) o Take Match My Exponential activity and build on it.

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Assessment plan for lesson As a type of formative assessment, students will be assessed informally as they complete their guided notes for this lesson. The first part of the notes walks students through a step-by-step process needed to algebraically find an exponential function that passes through two points. This process will be modeled by me, while providing students the opportunity to algebraically solve for a and b.

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Solidify intuitive understanding of initial value and growth factor

Students will then be asked to repeat this algebraic process with a peer. During this time, I will be monitoring student learning. It will help me identify any gaps in my instruction (from today and the previous day); from which I can provide students with feedback and adjust my instruction (What are the Types of Assessment?). “Formative assessment measures student progress but it can also assess [my] own progress as an instructor” (Formative and Summative Assessment). After my instruction has been modified based on students’ feedback, students will complete a question on their own for homework that has them find an exponential function that models the temperature of coffee based on the amount of time that has passed since it was poured. Students will be able to apply what they have learned about modeling real-world situations with exponential functions, by selecting two data points. This homework questions is still a type of formative assessment, but students will no longer be assessed informally. I will be looking for students to be able to appropriately and independently apply their knowledge from today’s lessons to their homework. Then based on students’ accuracy, I can make a decision about my instruction, move on, or re-teach. Write a summary of your thoughts on the value of using the Models, Connectivism, Motivation, Creativity, and Brain-based Learning in your teaching. What have you already been doing? What will you change? How will this information help you to become a more effective teacher? With the implementation of Common Core State Standards in Mathematics, there has been a shift from Direct Instruction to Group Investigation. My curriculum of College Preparatory Mathematics (CPM), implements Group Roles as a way to structure Group Investigation. These group roles include: facilitator, resource manager, recorder/reporter, and task manager. These group roles to help contribute to the learning process. The purpose of working in small groups is for students to become active participants of their learning. In these groups students are able to “discuss, share ideas, and articulate their thinking” (“Using Team Roles,” n.d.). Additionally, students are able to “make connections to different ideas through their communication with [other] students who see things differently, and are encouraged by their peers to put their ideas into word” (“Using Team Roles,” n.d.). Although the shift has moved away from Direct Instruction, it is still a

valid teaching model. And that there are times in class when Direct Instruction is the most efficient way to present students with content material. Student motivation is a concept that appeals greatly to me on a personal level. After taking this class I have a better understanding of student motivation. All students are motivated, but the question is, what are they motivated to do? This is where I need to consider student learning styles to best meet their needs of being motivated to do work, learn, and participate. I now have a better understanding of how to apply strategies to help increase student motivation. Additionally, there is both Intrinsic and Extrinsic Motivation and each has its own advantages and disadvantages (Motivating Students). What I have been doing as a part of my instruction is dividing my instruction into sections of before, during, and after. The Teaching Model described by Jensen (2005) has 10% time dedicated to before instruction (pre-exposure and physical learning environment), 80% during instruction (engagement, framing, acquisition, elaboration, and memory strengthening), and 10% after instruction (settling time and rest, and review and revision). Moving on, I will place more emphasis on physical learning environment, framing, and settling time. I will continue implementing Concept Attainment activities for students, where “students generalize, hypothesis, categorize, and use questioning techniques in order to logically synthesize information” (Week 4: Teaching Approaches and Models). I have found that students actually enjoy these activities. They like being in control and having a say in creating their own categories. To become a more effective teacher, I will provide my students with more differentiated learning opportunities, where students are given a choice. This provides students with a more authentic learning experiences and allows them to deepen their understanding. In addition, students are able to demonstrate their creativity.

Resources Burke, S. (2016, Febuary 26). Connectivism [Video file]. Retrieved from: https://youtu.be/-20Oqm1GvsU Depth of Knowledge (DOK) Overview Chart [PDF document]. Retrieved from Northern Indiana Educational Services Center. Website: http://www.niesc.k12.in.us/index.cfm/ staff-development/public-consulting-group-co-teaching-session/depthofknowledgechartpdf/ Doyle, T., & Zakrajsek, T. (2013). The New Science of Learning. Sterling, VA: Stylus Publishing, LLC. Formative and Summative Assessment [PDF document]. Retrieved from Northern Illinois University. Website: https://www.niu.edu/facdev/_pdf/guide/assessment/formative%20and_summative_ assessment.pdf

Jensen, E. (2005). Teaching with the Brain in Mind. Alexandria, VA: Association for Supervision and Curriculum Development Motivating Students. Retrieved from: https://cft.vanderbilt.edu/guides-sub-pages/motivating-students/#intrinsic Using Team Roles. (n.d.). College Preparatory Mathematics. [PDF file]. Retrieved from: https://pdfs.cpm.org/studyTeam/Using_Team_Roles_with_Study_Teams.pdf Week 4: Teaching Approaches and Models. [PowerPoint Slides]. Retrieved from: https://nu.blackboard.com /webapps/blackboard/content/listContent.jsp?course_id=_87543_1&content_id=_6592947_1& mode=reset What are the Types of Assessment? Retrieved from: https://www.onlineassessmenttool.com/knowledgecenter/assessment-knowledge-center/what-are-the-types-of-assessment/item10637 Woolfolk, A. (2017). Educational Psychology. Boston, MA: Pearson