1. Planning to Implement CCSS-M Function Cluster:
Interpret functions that arise in applications in terms of the context
Lynn Aring
OCTM VP-Secondary
Bay High School, retired
2012 OCTM Annual Conference
October 18-19
Columbus, Ohio

5. Students proficient in this standard
What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·know that solving a problem doesn't always mean solving word problems
·look at a problem verbally, numerically, graphically, and/or symbolically providing different entry points into solving a problem
·plan a solution pathway, rather than just jumping into a solution attempt
·transform algebraic expressions into a more useful form for graphing functions
·check answers to see if they make sense especially in contextual problems
·understand other solution/graphing methods and can see similarities and differences in the methods of others to their own solution/graphing method

6. What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·have the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents
·have the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved
·pay attention to the units involved, checking that the final answer is not only numerically reasonable but that the algebraic operations used in solving an equation produce the correct units for the final answer
·know and flexibly use different properties of operations and objects

7. What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·can listen to or read arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments
·can compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in the argument, explain what it is
·can understand and use stated assumptions, definitions, properties, postulates and previously established results in constructing solution methods, arguments, and proofs

8. ·Interpreting Functions 4,5,6,7 ·Building Functions 1,2
What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·can create functions that model situations in everyday life, society, and the workplace
·routinely interpret functions in the context of the situation and reflect on whether the function make sense
The following standards in the conceptual category of Functions have been marked as modeling standards:
·Interpreting Functions 4,5,6,7
·Building Functions 1,2
·Linear, Quadratic, and Exponential Models-All
·Trigonometric Functions 1

9. ·use estimation skills to estimate the output of a function(e.g. f(3))
What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·use estimation skills to estimate the output of a function(e.g. f(3))
·use mental math or paper and pencil for simpler calculations
·use paper and pencil for simpler algebraic manipulations and when writing a function in an alternative form.
·use graph paper and pencil to graph basic functions
·use a graphing calculator to graph more complex equations and to solve equations numerically or graphically
·use spreadsheets to look at both explicit and recursive functions numerically and graphically
·Use dynamic graphing software with sliders to explore transformations of functions(e.g. f(x+k) versus f(x)+K)

10. Students proficient in this standard ·communicate precisely to others
What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·communicate precisely to others
·use mathematical terminology such as, functions, relations, inverses, properly
·read and write symbolic expressions correctly(e.g. f(x) and f-1(x))
·label axes with appropriate scales and units
·round appropriately based on the context of the problem

11. What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·look closely to discern a pattern or structure
·can see complicated things, such as algebraic expressions, as single objects or as being composed of several objects
·can see the value of different forms of a function or expression(e.g.,quadratic equations in standard form, factored form, or vertex form will reveal different properties of the graphs of the functions)

12. What do the Standards for Mathematical Practice mean in the context of the conceptual category: Functions?
Students proficient in this standard
·notice if calculations are repeated, and look for both general methods and for shortcuts
For example, after performing long division of polynomials multiple times, students may develop or at least understand synthetic division

13. From the Common Core State Standards Document
One variable is dependent on the other
Numerical inputs and outputs
Domain
Multiple Representations
Modeling
Linear
Exponential
Technology

14. From the Common Core State Standards Document
Recursion

15. From the Common Core State Standards Document
Conceptual Category
Domain
Cluster

16. From the Common Core State Standards Document
Conceptual Category: Functions
Standard
Cluster
Domain
Interpreting
Functions
Function

17. From the Common Core State Standards Document
Related Grade 8 Standards

18. Ohio Department of Education Website
Model Curriculum

19. Ohio Department of Education Website
Model Curriculum

20. Ohio Department of Education Website
Model Curriculum

21. Ohio Department of Education Website
Model Curriculum

22. Ohio Department of Education Website
Model Curriculum

23. Ohio Department of Education Website
Resources

24. Ohio Department of Education Website
Resources

25. Ohio Department of Education Website
Resources

26. Ohio Department of Education Website
Resources

27. Ohio Department of Education Website
Resources

29. From Appendix A

30. From Appendix A
Traditional Pathway

31. From Appendix A
Integrated Pathway

33. From the PARCC Model Content Frameworks

34. From the PARCC Model Content Frameworks

35. From the PARCC Model Content Frameworks
In Algebra 1, this cluster is classified as major. Standards 4, 5, and 6 are taught/tested in this course.
Assessment tasks
i. have a real world context
II. are limited to linear functions, square root functions, cube root functions. piecewise-defined functions(including step functions and absolute value functions) and exponential functions with domains in the integers

36. From the PARCC Model Content Frameworks
In Algebra 2, this cluster is also classified as major. Only standards 4 and 6 are taught/tested in this course.
Assessment tasks
i. have a real world context
II. tasks may involve polynomial, exponential, logarithmic, and trigonometric functions

37. From the PARCC Model Content Frameworks

38. From the PARCC Model Content Frameworks
In Mathematics 1, this cluster is classified as major. All 3 standards are taught/tested in this course
For Standard 4
i. Tasks have a real-world context
II. Tasks are limited to linear functions, square root functions, cube root functions, piecewise-defined functions(including step functions and absolute value functions), and exponential functions with domains in the integers.
For Standard 5
i. Tasks have a real-world context
II. Tasks are limited to linear functions, square root functions, cube root functions, piecewise-defined functions(including step functions and absolute value functions), and exponential functions with domains in the integers.
For Standard 6
i. Tasks have a real-world context
II. Tasks are limited to linear functions, square root functions, cube root functions, piecewise-defined functions(including step functions and absolute value functions), and exponential functions with domains in the integers.

39. From the PARCC Model Content Frameworks
In Mathematics 2 this cluster is classified as major. All three standards are taught/tested in this course.
For standard 4
i. Tasks have a real-world context
II. Tasks are limited to quadratic and exponential functions.
For standard 5
i. Tasks have a real-world context
II. Tasks are limited to quadratic functions.
For standard 6
i. Tasks have a real-world context
II. Tasks are limited to quadratic and exponential functions.

40. From the PARCC Model Content Frameworks
In Mathematics 3, this cluster is also classified as major. Only standards 4 and 6 are tested in this course
For Standard 4
I. Tasks have a real-world context
II. Tasks may involve polynomial, logarithmic, and trigonometric functions
For Standard 6
I. Tasks have a real-world context
II. Tasks may involve polynomial, logarithmic, and trigonometric functions

53. Domain: all reals between 0 and 7 inclusive
Degrees
Time(secs) 7
120
Domain: all reals between 0 and 7 inclusive
Range: All reals between 0 and 120 inclusive

55. Quadratic

59. Door is opening or the number of degrees is increasing
Door is closing or the number of degrees is decreasing

61. The graph appears to have a horizontal asymptote, so the door would never close

63. 100 75 50
31.3
18.8
10.9
6.3
3.5
2.0
Opening, graph and table indicate number of degrees is increasing
Opening, Rate of change is positive indicating # of degrees is increasing

65. Cubic polynomial
You would need 4 points
(0,0)
(7,0)
(time, max degree)
one other point