1. 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

2. 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.

3. 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
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?

4. 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)

5. 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

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

7. 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.

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

9. 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)

10. Explain how the symmetry works
Visual
Explain how the symmetry works

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

12. 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

13. 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.

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

15. 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

16. 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

17. 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?

18. Non-Routine Problem LT 2B
Need to print student sheets (pages 5-6). Problem will not fit here.

19. 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

20. 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)

21. 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

22. 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

23. 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)

24. Explain the relationship between these three pictures

25. Explain the relationship

26. 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

27. 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?

28. 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.

29. 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

30. 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

31. Non-Routine Goal #3 Problem LT 2B

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

33. 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

34. 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

35. 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

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