Relationships between Inputs & Outputs I can use mapping to explain the concept of composite functions and how the connection between sets operates as the rule. I can break functions down into composite pieces and find the domains for composite functions including polynomial, radical, and rational functions 2B I can analyze functions to determine if they are classified as odd functions, even functions, or neither and explain the relationship between input and outputs 2C Given a graph, I can analyze relationships between inputs and outputs (composition) and find outputs by analyzing relationships (composite, translations, absolute value) NCTM 1 (learning objective) 2A: What connects to what you learned in Algebra 2? 2B: What question do you have? 2C: What definition from Algebra 2 connects to this LT

Non-Routine GP #2 LT 2A & 2C The suggested retail price of a new car is p dollars. The dealership advertized a factory rebate of $1200 and an 8% discount. Write a function R in terms of p giving the cost of the car after receiving the factory rebate. Write a function S in terms of p giving the cost of the car after the 8% discount. Does or yield the lower cost of the car? Explain.

Did you Match or Did you take another Pathway? Non-Routine #2 LT 2A & 2C Soln The suggested retail price of a new car is p dollars. The dealership advertized a factory rebate of $1200 and an 8% discount. R = P - 1200 S = 0.92P = (0.92)(p-1200) = 0.92p-1200 0.92p-1200 < 0.92 (p-1200) < Answer Key for previous slide Did you Match or Did you take another Pathway?

Relationships between Inputs & Outputs I can use mapping to explain the concept of composite functions and how the connection between sets operates as the rule. I can break functions down into composite pieces and find the domains for composite functions including polynomial, radical, and rational functions 2B I can analyze functions to determine if they are classified as odd functions, even functions, or neither and explain the relationship between input and outputs 2C Given a graph, I can analyze relationships between inputs and outputs (composition) and find outputs by analyzing relationships (composite, translations, absolute value) NCTM 1 (learning objective)

Describe the symmetry Describe symmetry 2B I can analyze functions to determine if they are classified as odd functions, even functions, or neither and explain the relationship between input and outputs Student Actions: Independently students take 1min to describe the symmetry in the visual “While you share your descriptions of symmetry, you need to be thinking about what question comes to mind? Now write down the question you were thinking about? Describe the symmetry Describe symmetry

Describe the Relationship Approach Describe the Relationship Approach Independently Create 2 Plans that might work by communicating with partner Create Plans

II. Even and Odd Functions Definitions 1. Even functions: Let f(x) be a real-valued function. Then f(x) is even if the following relationship is true: f(-x) = f(x) for all x in the domain of f. a. Geometrically, the graph of an even function is symmetric with respect to the y- axis On your own, use the definition to sketch an example of a function that MIGHT be even. Convince your partner that your sketch satisfies the definition of an even function.

Explain how the symmetry works Visual Go back to your definition check the definition algebraically and geometrically Explain how the symmetry works

Even and Odd Functions 2. Odd functions: Let f(x) be a real-valued function. Then f(x) is odd if the following relationship is true: f(-x) = - f(x) a. Geometrically, the graph of an odd function has rotational symmetry with respect to the origin (y=x) Describe the resulting image when you place a pin at the origin and rotate the image 180 degrees y=x? example function? y=x^3? Cool odd functions examples: x^3 – kx, where k is any constant (ex. x^3 – 4x)

Explain how the symmetry works Visual Explain how the symmetry works

Use a function relationship to describe this situation Does f(-x) = -f(x)? Use a function relationship to describe this situation

Use a function relationship to describe this situation Does f(-x) = -f(x)? Does f(-x) = f(x) Use a function relationship to describe this situation

Why Eggs? An easter egg is a joke/visual gag/in-joke that a creator (typically the artist) has hidden in the artwork for viewers to find (just like an easter egg). They range from the not-so-obscure to the really obscure.

Captain America’s shield is to the left on the work bench WHERE IS THE EASTER EGG?

Who Cares about Even/Odd Functions? Purpose: To figure out symmetry relations between the pre-image/domain and image/range of a function to help sketch quickly to use as a short-cut when differentiating and integrating Calculus

Write out approach, questions and areas of confusion Goal Problems #1 (LT 2B) Routine Recall & Reproduction Given f(x)= 1/x. Use the definitions to determine if f(x) is even, odd or neither? Odd function, Even function or Neither? Explain your reasoning. Alone Write out approach, questions and areas of confusion

Active PracticeLearning Target 2B Note to Self in Margin of notes My Personal Learning Goal during active practice. My choices to ensure action levels up my learning today. How will I know if my choices are working? When will I try the Non-Routine? When will I build my concept map?

Non-Routine Problem LT 2B http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/performance-tasks/crickets.pdf Need to print student sheets (pages 5-6). Problem will not fit here.

Write out approach, questions and areas of confusion Goal Problems #2 LT 2B Recall & Reproduction Routine Determine algebraically if the function is even, odd or neither: f(x) = x3 - 5x Find the coordinates of a second point on the graph given that: Function is even, (5, -1) Function is odd, (4, 9) Write out approach, questions and areas of confusion

Non-Routine Goal #2 Problem LT 2B If f(x) is an even function, determine if g(x) is even, odd or neither. Explain. g(x) = – f(x) g(x) = f(– x) g(x) = f(x) – 2 g(x) = – f(x – 2)

Compare to Solutions Annotate your “work” by completing the following (use a different color) Answer questions Isolate and resolve areas of confusion (without copying how to do the problem) Identify areas of strength Analyze reasoning (Approach, Plan, Execution) Analyze written response

Based on Goal Problem #2, Current Level of Mastery? 4: Demonstrate recall, solve routine problems, and solve non-routine problems (levels 1, 2, 3).  Teach concept to a peer 3: Demonstrate recall, solve routine, approach and create plans that might work for non-routine problems (levels 1, 2, and part of 3). Explain how I reasoned through a routine problem 2: Demonstrate recall and solve routine problems (levels 1 & 2).  Explain how to approach and solve a problem 1: Demonstrate recall (level 1).  I do not know how to use the concepts that I know to create a plan that works NY:  Not sure what facts apply and cannot remember how to approach any of the problems.  I really don’t get it yet, but I am trying different things I know. Score goes into GRASP

Relationships between Inputs & Outputs I can use mapping to explain the concept of composite functions and how the connection between sets operates as the rule. I can break functions down into composite pieces and find the domains for composite functions including polynomial, radical, and rational functions 2B I can analyze functions to determine if they are classified as odd functions, even functions, or neither and explain the relationship between input and outputs 2C Given a graph, I can analyze relationships between inputs and outputs (composition) and find outputs by analyzing relationships (composite, translations, absolute value) NCTM 1 (learning objective)

Explain the relationship between these three pictures

Explain the relationship

Goal Problems #1 (LT 2C Translations) Routine Recall & Reproduction Sketch the graph of – f(x + 3) – 1. Compare each function to the graph of f(x) = |x|. y = |x| – 3 y = |–x| y = |0.5 x| Alone Write out approach, questions and areas of confusion

Active Practice: Learning Target 2C Translations Note to Self in Margin of notes My Personal Learning Goal during active practice. My choices to ensure action levels up my learning today. How will I know if my choices are working? When will I try the Non-Routine? When will I build my concept map?

Non-Routine Problem LT 2B Translations Determine if the following are true or false. Justify your claim. The graph of y = f(–x) is a reflection of the graph of y = f(x) over the x-axis. The graph of y = –f(x) is a reflection of the graph of y = f(x) over the y-axis. The graph of y = –f(x) is a reflection of the graph of y = f(–x) over the x-axis.

Goal Problems #2 LT 2C Translations Recall & Reproduction Routine Piecewise, use quadratic, exp, logs, abs value. Step fcn Explain the translations necessary to turn the graph of f(x) into the graph of g(x). f(x) = need a log function with all shifts g(x) = need a e to the x with all shifts Write out approach, questions and areas of confusion

Non-Routine Goal #2 Problem LT 2B The graph of f(x) has x-intercepts at x = –1 and x = 4. Use this to find the x-intercepts of the given graph. If not possible, explain why. f(–x) f(x – 2) 2f(x) f(x) – 1

Non-Routine Goal #3 Problem LT 2B

Next Steps GRASP Build Concept Map Strengthen Concept Map Complete individual active sensemaking practice problems

Requires doing more problems Build Concept Map Requires doing more problems Use Learning Targets & Notes, Goal Problems, Quick Check, and problems from the book to design concepts and categories What is in black, includes visuals (labeled diagrams, types of problems) How to Approach is green Focus on connections where you explain how to think about using information about the concepts to figure out non-routine problems Plans to use the information and why this will lead to an accurate solution Questions I ask myself as I think through my approach, plans, and execution

Strengthen Concept Map Based on Qualitative & Quantitative Evidence, add to your concept map Concepts, Definitions, Visuals, Explanation of “what” Approach for concepts Reasons the concepts are useful in creating plans (connections) Questions I pose to myself as I move through my concept map MP 2 (abstract/quantitative reasoning) & 6 (precise mathematical language); NCTM 3 (connections, representations) – S. evidence that T. is teaching this

Non-Routine, Routine, Recall & Reproduction Strengthen Concept Map with Problems From Learning Targets 2A, 2B, 2C & Goal Problems Non-Routine, Routine, Recall & Reproduction Select from learning targets…..which ones have I not done yet? Choose one of each and write out my approach, plans if needed, and questions I pose to myself as I move through my solution pathway Move 5: Action Plan / Improve; NCTM 1 (find all learning goals in map. Focus?), 8 (use evidence of S. thinking to make decisions) Recall & Reproduction is a problem not in context (skills with finding resultant vector, magnitude) Routine: Similar to problems in Learning Targets Non-Routine should be a problem similar to the spaceship problem involving forces

LT 2E: Describe the limiting behavior Approach Independently Create 2 Plans that might work by communicating with partner Approach Create Plans https://www.illustrativemathematics.org/illustrations/386 2E I can explain how to model limiting behavior (asymptotic behavior, min/max) and explain the mathematics.