VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010

1. VDOE Mathematics Institute Grade Band 9-12 Functions K-12 Mathematics Institutes Fall 2010 Opening Slide Opening Slide

2. Placemat ConsensusFunctions 2 The focus of our afternoon session today is to discuss the vertical progression of functions in grades 9-12. This presentation does not intend to cover all concepts included in the mathematics curriculum grade 9-12, but only to highlight best practices in developing the idea of functions across the grade levels. To begin this session we will be using an activity called placemat consensus. This is a structure that allows a small group to think individually about a problem or topic and then to collaborate and come to a consensus about the solution or important components of the problem. The directions are as follows: 1. Get into groups of no more than 4 people. Each group will need a large post-it poster and a marker for each person. Divide the poster into four quadrants as shown in the figure, with a circle in the center, large enough to write inside. Each person will be given 2 minutes to write down what they feel are the most important concepts related to functions in one of the quadrants. This is done silently and individually. Then each group is given 3-4 minutes to discuss their individual responses and come to consensus about the three most important concepts related to functions in the circle in the center. The focus of our afternoon session today is to discuss the vertical progression of functions in grades 9-12. This presentation does not intend to cover all concepts included in the mathematics curriculum grade 9-12, but only to highlight best practices in developing the idea of functions across the grade levels. To begin this session we will be using an activity called placemat consensus. This is a structure that allows a small group to think individually about a problem or topic and then to collaborate and come to a consensus about the solution or important components of the problem. The directions are as follows: 1. Get into groups of no more than 4 people. Each group will need a large post-it poster and a marker for each person. Divide the poster into four quadrants as shown in the figure, with a circle in the center, large enough to write inside. Each person will be given 2 minutes to write down what they feel are the most important concepts related to functions in one of the quadrants. This is done silently and individually. Then each group is given 3-4 minutes to discuss their individual responses and come to consensus about the three most important concepts related to functions in the circle in the center.

3. Overview of Vertical Progression Middle School (Function Analysis) 7.12 � represent relationships with tables, graphs, rules and words 8.14 � make connections between any two representations (tables, graphs, words, rules) 3 This morning you were introduced to a vertical articulation document of the 2009 mathematics SOLs. We would like for you to turn to that section of the document that includes the row titled �FUNCTION ANALYSIS.� The column headings include Content from Earlier Grades, Algebra I, Algebra, Functions and Data Analysis and Algebra 2. The vertical articulation of function analysis has content introduced in grades 7 and 8. In the grade 7, Standard 7.12 has students represent relationships with tables, graphs, rules and words. Standard 8.17 has students identify domain, range, independent variable and dependent variable. Standard 8.14 has students make connections between any two representations (tables, graphs, words, rules).This morning you were introduced to a vertical articulation document of the 2009 mathematics SOLs. We would like for you to turn to that section of the document that includes the row titled �FUNCTION ANALYSIS.� The column headings include Content from Earlier Grades, Algebra I, Algebra, Functions and Data Analysis and Algebra 2. The vertical articulation of function analysis has content introduced in grades 7 and 8. In the grade 7, Standard 7.12 has students represent relationships with tables, graphs, rules and words. Standard 8.17 has students identify domain, range, independent variable and dependent variable. Standard 8.14 has students make connections between any two representations (tables, graphs, words, rules).

4. Overview of Vertical Progression Algebra I (Function Analysis) A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic. 4 In Algebra I the vertical progression of function analysis continues with students investigating and analyzing linear and quadratic functions, through both graphical and algebraic approaches. In standard A.7, students will determine whether a relation is a function, continuing the study of patterns and relationships from middle school. In addition, students will be able to identify domain and range, zeros, and intercepts of linear and quadratic functions, both algebraically and graphically. Students will also explore the idea of finding values of a function for elements in its domain. Making connections between multiple representations is vital for students to create deeper understanding about functions.In Algebra I the vertical progression of function analysis continues with students investigating and analyzing linear and quadratic functions, through both graphical and algebraic approaches. In standard A.7, students will determine whether a relation is a function, continuing the study of patterns and relationships from middle school. In addition, students will be able to identify domain and range, zeros, and intercepts of linear and quadratic functions, both algebraically and graphically. Students will also explore the idea of finding values of a function for elements in its domain. Making connections between multiple representations is vital for students to create deeper understanding about functions.

5. Overview of Vertical Progression Algebra, Functions and Data Analysis (Function Analysis) AFDA.1 The student will investigate and analyze function (linear, quadratic, exponential, and logarithmic) families and their characteristics. Key concepts include a) continuity; b) local and absolute maxima and minima; c) domain and range; d) zeros; e) intercepts; f) intervals in which the function is increasing/decreasing; g) end behaviors; and h) asymptotes. 5 In Algebra, Functions and Data Analysis, students investigate a multitude of function families, including linear, quadratic, exponential and logarithmic. In standard AFDA.1 students explore continuity, local absolute maxima and minima, domain and range, zeros., intercepts and intervals in which functions are increasing and decreasing. Standard AFDA.1 also includes the analysis of the end behavior of functions and asymptotes.In Algebra, Functions and Data Analysis, students investigate a multitude of function families, including linear, quadratic, exponential and logarithmic. In standard AFDA.1 students explore continuity, local absolute maxima and minima, domain and range, zeros., intercepts and intervals in which functions are increasing and decreasing. Standard AFDA.1 also includes the analysis of the end behavior of functions and asymptotes.

6. Overview of Vertical Progression Algebra, Functions and Data Analysis (Function Analysis) AFDA.4 The student will transfer between and analyze multiple representations of functions, including algebraic formulas, graphs, tables, and words. Students will select and use appropriate representations for analysis, interpretation, and prediction. 6 In AFDA.4, the students continue to make connections between multiple representations of functions (algebraic formulas, graphs, tables and words). This course lends itself well to be a curriculum in which the application of concepts is the focus. Students are given the opportunity to decide which representations of functions best represent a situation in which a student may need to analyze, interpret or predict using functions.In AFDA.4, the students continue to make connections between multiple representations of functions (algebraic formulas, graphs, tables and words). This course lends itself well to be a curriculum in which the application of concepts is the focus. Students are given the opportunity to decide which representations of functions best represent a situation in which a student may need to analyze, interpret or predict using functions.

7. Overview of Vertical Progression Algebra 2 (Function Analysis) AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions. 7 In Algebra 2, students will be exposed to many of the concepts of the Algebra, Functions and Data Analysis course, but in a more theoretical context and in greater depth. In Standard AII.7 Students investigate and analyze functions algebraically and graphically. Students explore domain and range, including limited and discontinuous domains and ranges, zeros, intercepts, and intervals in which a function is increasing or decreasing. In addition, the Algebra II standard includes a study of the end behavior of functions, inverse of a function, and composition of functions. Today, we will be providing an overview of the vertical progression of functions in grades 9-12. We will begin by discussing common vocabulary in the grade levels. We will continue by reviewing the function families that extend across the grade levels and their basic characteristics. We will then focus on each function family and some real world applications that extend from the study of each. We will discuss some of the important features of the new 2009 mathematics SOLs in grades 9-12. We will end by comparing your placemat consensus responses with what has been discussed today.In Algebra 2, students will be exposed to many of the concepts of the Algebra, Functions and Data Analysis course, but in a more theoretical context and in greater depth. In Standard AII.7 Students investigate and analyze functions algebraically and graphically. Students explore domain and range, including limited and discontinuous domains and ranges, zeros, intercepts, and intervals in which a function is increasing or decreasing. In addition, the Algebra II standard includes a study of the end behavior of functions, inverse of a function, and composition of functions. Today, we will be providing an overview of the vertical progression of functions in grades 9-12. We will begin by discussing common vocabulary in the grade levels. We will continue by reviewing the function families that extend across the grade levels and their basic characteristics. We will then focus on each function family and some real world applications that extend from the study of each. We will discuss some of the important features of the new 2009 mathematics SOLs in grades 9-12. We will end by comparing your placemat consensus responses with what has been discussed today.

8. Vocabulary The new 2009 SOL mathematics standards focus on the use of appropriate and accurate mathematics vocabulary. 8 Mathematics educators recognize the importance of emphasizing vocabulary and the 2009 standards focus on the appropriate and accurate use of mathematics vocabulary.Mathematics educators recognize the importance of emphasizing vocabulary and the 2009 standards focus on the appropriate and accurate use of mathematics vocabulary.

9. �Function� Vocabulary Across Grade Levels 9 In the table shown, we have identified some of the common vocabulary that extends across the 9-12 grade level continuum pertaining to functions. Some of the vocabulary, such as increasing/decreasing intervals for example, have not been focused upon in the standards of learning at the Algebra I level before the 2009 standards. Mathematics educators in Virginia need to become very familiar with the new standards and the Curriculum Framework that describes them.In the table shown, we have identified some of the common vocabulary that extends across the 9-12 grade level continuum pertaining to functions. Some of the vocabulary, such as increasing/decreasing intervals for example, have not been focused upon in the standards of learning at the Algebra I level before the 2009 standards. Mathematics educators in Virginia need to become very familiar with the new standards and the Curriculum Framework that describes them.

10. Vocabulary Across Grade Levels 10 This table includes some of the �action verbs� that are used commonly across grade levels as we discuss functions, their attributes and applications. It is important to be sure that students have a good understanding of this vocabulary. For example, do students recognize the differences between evaluate, simplify and solve? In what situations do we use each? And � teachers should be encouraged to use appropriate vocabulary when teaching content. A common phrase that should be avoided is �plug and chug� � the better alternative is substitution. Appropriate terminology provides opportunity for writing in the mathematics classroom, such as providing rationales for the various steps in solving an equation.This table includes some of the �action verbs� that are used commonly across grade levels as we discuss functions, their attributes and applications. It is important to be sure that students have a good understanding of this vocabulary. For example, do students recognize the differences between evaluate, simplify and solve? In what situations do we use each? And � teachers should be encouraged to use appropriate vocabulary when teaching content. A common phrase that should be avoided is �plug and chug� � the better alternative is substitution. Appropriate terminology provides opportunity for writing in the mathematics classroom, such as providing rationales for the various steps in solving an equation.

11. Wordle � Algebra I 2009 VA SOLswww.wordle.net 11 A web based application called �Wordle� was used to create the graphic above. The new 2009 Algebra I SOLs were inputted into the application. Depending upon how many times a word appears in the document dictates how large or small the word appears in the graphic. For example, solving, equations and linear appear often in the 2009 Algebra I Virginia Standards of Learning, as would be expected.A web based application called �Wordle� was used to create the graphic above. The new 2009 Algebra I SOLs were inputted into the application. Depending upon how many times a word appears in the document dictates how large or small the word appears in the graphic. For example, solving, equations and linear appear often in the 2009 Algebra I Virginia Standards of Learning, as would be expected.

12. Wordle � Algebra, Functions and Data Analysis 2009 VA SOLs 12 This shows the Wordle graphic when the Algebra, Functions and Data Analysis 2009 standards of learning are inputted. Notice the prominence of the words data, along with linear, quadratic, exponential and logarithmic.This shows the Wordle graphic when the Algebra, Functions and Data Analysis 2009 standards of learning are inputted. Notice the prominence of the words data, along with linear, quadratic, exponential and logarithmic.

13. Wordle � Algebra II 2009 VA SOLs 13 In the Algebra II Virginia Standards of Learning, functions is very prominent as a commonly used word in the standards. We notice the prominence of rational, expressions, equations and graphing as well.In the Algebra II Virginia Standards of Learning, functions is very prominent as a commonly used word in the standards. We notice the prominence of rational, expressions, equations and graphing as well.

14. Wordle � Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry 14 When all of the Virginia Standards of Learning are put into Wordle at one time, the prominence of the word function and functions is clearly seen. Functions are an important concept in Algebra I, Algebra, Functions and Data Analysis and Algebra II and will be the focus of our time together this afternoon.When all of the Virginia Standards of Learning are put into Wordle at one time, the prominence of the word function and functions is clearly seen. Functions are an important concept in Algebra I, Algebra, Functions and Data Analysis and Algebra II and will be the focus of our time together this afternoon.

15. Reasoning with Functions Key elements of reasoning and sense making with functions include: Using multiple representations of functions Modeling by using families of functions Analyzing the effects of different parameters Adapted from Focus in High School Mathematics: Reasoning and Sense Making, NCTM, 2009 15 The National Council of Teachers of Mathematics, in its high school focus on reasoning and sense making, recognizes the importance of creating a focus on multiple representation of functions as a way to promote deeper understanding. Student understanding of the function families and being able to analyze the effects of different parameters on function behavior are also recognized as important concepts in the high school study of functions.The National Council of Teachers of Mathematics, in its high school focus on reasoning and sense making, recognizes the importance of creating a focus on multiple representation of functions as a way to promote deeper understanding. Student understanding of the function families and being able to analyze the effects of different parameters on function behavior are also recognized as important concepts in the high school study of functions.

16. Using Multiple Representations of Functions Tables Graphs or diagrams Symbolic representations Verbal descriptions 16 Different representations of a function exhibit different properties. Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone. Students need to establish connections among different representations, for example, the relationship among the zeros of a function, the solution of an equation, and the x-intercepts of graphs.Different representations of a function exhibit different properties. Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with symbolic representations alone. Students need to establish connections among different representations, for example, the relationship among the zeros of a function, the solution of an equation, and the x-intercepts of graphs.

17. Algebra Tiles ~ Adding Add the polynomials. (x � 2) + (x + 1) Algebra tiles are another way to represent polynomial relationships. Students are given both a visual and kinesthetic representation of the various terms. A quadratic term, x2, is represented by a green square whose side lengths are defined as x. A linear term, x, is represented by a blue rectangle whose length is x and whose width is 1. A constant term of 1 is represented by a gold square, whose length and width are both 1. Negative values usually have the same shapes, but are usually a different color, such as black in this example. In this example, we are adding two binomials together (x � 2) and (x + 1). The manipulatives allow students to recognize that a positive and negative value will add up to be zero. The resulting polynomial is modeled by 2 x-bars and negative one tile. This concrete representation of like terms can be a very powerful way to promote student learning. Algebra tiles are another way to represent polynomial relationships. Students are given both a visual and kinesthetic representation of the various terms. A quadratic term, x2, is represented by a green square whose side lengths are defined as x. A linear term, x, is represented by a blue rectangle whose length is x and whose width is 1. A constant term of 1 is represented by a gold square, whose length and width are both 1. Negative values usually have the same shapes, but are usually a different color, such as black in this example. In this example, we are adding two binomials together (x � 2) and (x + 1). The manipulatives allow students to recognize that a positive and negative value will add up to be zero. The resulting polynomial is modeled by 2 x-bars and negative one tile. This concrete representation of like terms can be a very powerful way to promote student learning.

18. Algebra Tiles ~ Multiplying The tiles can also be used to model multiplication of polynomials. Here we will multiply (x + 2) and (x + 3).The tiles can also be used to model multiplication of polynomials. Here we will multiply (x + 2) and (x + 3).

19. Multiply the polynomials using tiles. An array can be used and easily compared to an area model, in which each side of the area has a dimension. The x2 term (green square) comes into play when we multiply x times x. Students can see that if one side of the array is x and the other side is 1, then we need to fill in with positive x-bars. Finally, the 1x1 areas fill in the remainder of the space. The resulting polynomial modeled inside of the array represents the product x2 + 5x + 6.An array can be used and easily compared to an area model, in which each side of the area has a dimension. The x2 term (green square) comes into play when we multiply x times x. Students can see that if one side of the array is x and the other side is 1, then we need to fill in with positive x-bars. Finally, the 1x1 areas fill in the remainder of the space. The resulting polynomial modeled inside of the array represents the product x2 + 5x + 6.

20. Algebra Tiles ~ Factoring Work backwards from the array. The algebra tiles are an ideal way to model factoring, since students can work backward. In this example, the trinomial x2 � 3x + 2 can be modeled inside of the array. The dimensions of the array then compose the two binomials that make up the factors of the polynomial.The algebra tiles are an ideal way to model factoring, since students can work backward. In this example, the trinomial x2 � 3x + 2 can be modeled inside of the array. The dimensions of the array then compose the two binomials that make up the factors of the polynomial.

21. Polynomial Division A.2 The student will perform operations on polynomials, including a) applying the laws of exponents to perform operations on expressions; b) adding, subtracting, multiplying, and dividing polynomials; and c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations. 21 In Algebra I, SOL A.2 includes the division of polynomials. In the 2001 SOL standards, students were only expected to divide polynomials by monomials. However, in the 2009 standards, students will be expected to divide polynomials by binomials.In Algebra I, SOL A.2 includes the division of polynomials. In the 2001 SOL standards, students were only expected to divide polynomials by monomials. However, in the 2009 standards, students will be expected to divide polynomials by binomials.

22. Polynomial Division Divide (x2 + 5x + 6) by (x + 3) Common factors only will be used��no long division! 22 This division will be limited to common factors only (no remainders).This division will be limited to common factors only (no remainders).

23. Represent the polynomials using tiles. 23 We can use algebra tiles to model division of polynomials as well. In this example we will model dividing (x2 + 5x + 6) by (x + 3). The tiles can be used to model each polynomial, as shown.We can use algebra tiles to model division of polynomials as well. In this example we will model dividing (x2 + 5x + 6) by (x + 3). The tiles can be used to model each polynomial, as shown.

24. Factor the numerator and denominator. 24 Students must then know that in order to divide, they need to find common factors in the numerator and denominator. In order to do so, the numerator must be factored. We work backward from the trinomial to find the factors of x2 + 5x + 6 as (x + 2)(x + 3).Students must then know that in order to divide, they need to find common factors in the numerator and denominator. In order to do so, the numerator must be factored. We work backward from the trinomial to find the factors of x2 + 5x + 6 as (x + 2)(x + 3).

25. Represent the polynomials using tiles. 25 Students can then recognize that one of the factors in the numerator is the same as the (x + 3) factor in the denominator. Reducing like terms to equal 1 results in the answer, (x + 2).Students can then recognize that one of the factors in the numerator is the same as the (x + 3) factor in the denominator. Reducing like terms to equal 1 results in the answer, (x + 2).

26. Points of Interest for A.2 from the Curriculum Framework Operations with polynomials can be represented concretely, pictorially, and symbolically. VDOE Algeblocks Training Video http://www.vdoe.whro.org/A_Blocks05/index.html 26 One of the essential understandings for SOL A.2 is that operations with polynomials can be represented concretely (with manipulatives such as Algeblocks), pictorially (drawings), and symbolically. Students should be able to model the polynomial operation with concrete and pictorial manipulations and relate it to the corresponding symbolic representation. VDOE provides Algeblocks training through streaming video from their website. One of the essential understandings for SOL A.2 is that operations with polynomials can be represented concretely (with manipulatives such as Algeblocks), pictorially (drawings), and symbolically. Students should be able to model the polynomial operation with concrete and pictorial manipulations and relate it to the corresponding symbolic representation. VDOE provides Algeblocks training through streaming video from their website.

27. (2x + 5) + (x � 4) = 3x + 1 27 For example, in the problem (2x + 5) + (x � 4). Students should begin working with the concrete manipulatives. Once students have the image of the manipulatives in their mind, they can begin working pictorially, drawing pictures of the concrete manipulatives. **click to animate** When students understand the underlying mathematics and have the image of the manipulatives in their mind, the connections can be made between the concrete, pictorial, and symbolic representations. For example, in the problem (2x + 5) + (x � 4). Students should begin working with the concrete manipulatives. Once students have the image of the manipulatives in their mind, they can begin working pictorially, drawing pictures of the concrete manipulatives. **click to animate** When students understand the underlying mathematics and have the image of the manipulatives in their mind, the connections can be made between the concrete, pictorial, and symbolic representations.

28. Modeling by Using Families of Functions Recognize the characteristics of different families of functions Recognize the common features of each function family Recognize how different data patterns can be modeled using each family 28 Establishing an understanding of specific families of functions in high school helps students to find similarities and differences between functions. Students can use transformational approaches when graphing functions, for example, that can extend across different families. Eventually, the goal is to have students be able to recognize how different data patterns can be modeled using the various function families.Establishing an understanding of specific families of functions in high school helps students to find similarities and differences between functions. Students can use transformational approaches when graphing functions, for example, that can extend across different families. Eventually, the goal is to have students be able to recognize how different data patterns can be modeled using the various function families.

29. Analyzing the Effects of Parameters Different, but equivalent algebraic expressions can be used to define the same function Writing functions in different forms helps identify features of the function Graphical transformations can be observed by changes in parameters 29 Students who learn to recognize the basic characteristics of the parent functions, such as the linear parent function f(x) = x, or the quadratic parent function of f(x) = x2, can begin to observe how changes in parameters result in changes to the parent function. A focus on equivalent algebraic expressions is an important concept in algebraic manipulation of function equations. Students can begin to recognize how changes in parameters affect the key features of each function family. For example, students who recognize that the function f(x) = x2 results in a parabolic type graph with a vertex at (0, 0) can then begin to analyze what happens when the function transforms to f(x) = x2 + 2.Students who learn to recognize the basic characteristics of the parent functions, such as the linear parent function f(x) = x, or the quadratic parent function of f(x) = x2, can begin to observe how changes in parameters result in changes to the parent function. A focus on equivalent algebraic expressions is an important concept in algebraic manipulation of function equations. Students can begin to recognize how changes in parameters affect the key features of each function family. For example, students who recognize that the function f(x) = x2 results in a parabolic type graph with a vertex at (0, 0) can then begin to analyze what happens when the function transforms to f(x) = x2 + 2.

30. Overview of Functions Looking at Patterns Time vs. Distance Graphs allow students to relate observable patterns in one real world variable (distance) in terms of another real world variable (time). 30 A convenient starting point for having students begin to think about functional relationships is to talk about time vs. distance graphs. Students can begin this discussion very informally in the middle grades which can provide a very good jumping off point for a beginning discussion of functions in Algebra I. The discussion of the independent variable (time) versus the dependent variable (distance) relationship within a real world context adds a foundation for the more abstract representations of functions that are to come in Algebra I and beyond.A convenient starting point for having students begin to think about functional relationships is to talk about time vs. distance graphs. Students can begin this discussion very informally in the middle grades which can provide a very good jumping off point for a beginning discussion of functions in Algebra I. The discussion of the independent variable (time) versus the dependent variable (distance) relationship within a real world context adds a foundation for the more abstract representations of functions that are to come in Algebra I and beyond.

31. Time vs. Distance Graphs 31 Students can use time vs. distance graphs to begin to think about functions and how one variable relates to another without symbols. Here we could ask students, in which graph is� 1. The car is stopped. (C) 2. The car is traveling at a constant speed. (A or B) 3. The speed of the car is decreasing. (D) 4. The car is coming back. (B)Students can use time vs. distance graphs to begin to think about functions and how one variable relates to another without symbols. Here we could ask students, in which graph is� 1. The car is stopped. (C) 2. The car is traveling at a constant speed. (A or B) 3. The speed of the car is decreasing. (D) 4. The car is coming back. (B)

32. 32 Students can begin making comparisons of graphs. Which runner stopped for a rest? (Charlie) How long did he stop? (5 seconds) Which runner won the race? (Albert)Students can begin making comparisons of graphs. Which runner stopped for a rest? (Charlie) How long did he stop? (5 seconds)

33. Slope and Linear Functions Students can begin to conceptualize slope and look at multiple representations of the same relationship given real world data, tables and graphs. 33 As students begin to think about the contextualized ideas of constant vs. non-constant rates of change by observing time-distance graphs, the idea of thinking about slope as a constant rate of change can begin to be developed. Students begin to make connections between multiple representations of slope by first exploring it through a less formalized real world context and then having that knowledge formalized and extended.As students begin to think about the contextualized ideas of constant vs. non-constant rates of change by observing time-distance graphs, the idea of thinking about slope as a constant rate of change can begin to be developed. Students begin to make connections between multiple representations of slope by first exploring it through a less formalized real world context and then having that knowledge formalized and extended.

34. Exploring Slope using Graphs & Tables 34 Students must be given the opportunity to make meaning about the concept of functions prior to the actual formalization of the concepts. For example, we can provide students the opportunity to explore a real world situation, such as the one shown about electricity costs, without having formally introduced them to the concept of slope. What is the relationship between the number of kilowatts used and the total cost of a bill? Exploring data patterns prior to formalization of content provides students the opportunity to create a deeper understanding of the concepts of linear relationships.Students must be given the opportunity to make meaning about the concept of functions prior to the actual formalization of the concepts. For example, we can provide students the opportunity to explore a real world situation, such as the one shown about electricity costs, without having formally introduced them to the concept of slope. What is the relationship between the number of kilowatts used and the total cost of a bill? Exploring data patterns prior to formalization of content provides students the opportunity to create a deeper understanding of the concepts of linear relationships.

35. Exploring Functions As students progress through high school mathematics, the concept of a function and its characteristics become more complex. Exploring families of functions allow students to compare and contrast the attributes of various functions. 35 As the vertical articulation documents highlight, the Virginia Standards of Learning are building upon the concepts of function analysis introduced in earlier grades . The types of functions that students encounter through grades 9-12 develop in complexity and allow students to build upon prior knowledge.As the vertical articulation documents highlight, the Virginia Standards of Learning are building upon the concepts of function analysis introduced in earlier grades . The types of functions that students encounter through grades 9-12 develop in complexity and allow students to build upon prior knowledge.

36. Function Families Linear: Absolute Value: 36 Students begin an informal analysis of linear functions in the middle grades and begin formalization of these concepts in grade 8 and Algebra I. Absolute value functions, while not formally addressed in the standards of learning, can be a great segue between linear and quadratic relationships, by highlighting the characteristics of both types of functions. Students begin an informal analysis of linear functions in the middle grades and begin formalization of these concepts in grade 8 and Algebra I. Absolute value functions, while not formally addressed in the standards of learning, can be a great segue between linear and quadratic relationships, by highlighting the characteristics of both types of functions.

37. Function Families Quadratic Square Root 37 Students formally begin a study of quadratic functions in Algebra I and exploration continues in Algebra, Functions and Data Analysis and into Algebra II. The exploration of square root functions is a natural extension of quadratics in Algebra II as students are introduced to the idea of inverse functions.Students formally begin a study of quadratic functions in Algebra I and exploration continues in Algebra, Functions and Data Analysis and into Algebra II. The exploration of square root functions is a natural extension of quadratics in Algebra II as students are introduced to the idea of inverse functions.

38. Function Families 38 The cube root function is a natural extension of the square root function and the discussion of limited and continuous domains. Rational functions begin the exploration of excluded values and asymptotes.The cube root function is a natural extension of the square root function and the discussion of limited and continuous domains. Rational functions begin the exploration of excluded values and asymptotes.

39. Function Families Polynomial: Exponential: 39 The introduction of polynomial functions in Algebra II provides an opportunity for students to expand their conceptual understanding of zeros and intercepts along with a discussion of end behavior. Exponential functions provide a model for many real world situations and begin to set the stage for a discussion of logarithms.The introduction of polynomial functions in Algebra II provides an opportunity for students to expand their conceptual understanding of zeros and intercepts along with a discussion of end behavior. Exponential functions provide a model for many real world situations and begin to set the stage for a discussion of logarithms.

40. Function Families Logarithmic: 40 Logarithmic functions, which are explored in both Algebra, Functions and Data Analysis and Algebra II, expand the notion of inverse functions that were introduced when discussing quadratic and square root functions. Student exploration of the concept of logarithms builds upon many of the concepts of high school mathematics, including powers and exponents.Logarithmic functions, which are explored in both Algebra, Functions and Data Analysis and Algebra II, expand the notion of inverse functions that were introduced when discussing quadratic and square root functions. Student exploration of the concept of logarithms builds upon many of the concepts of high school mathematics, including powers and exponents.

41. 41 Linear Functions As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior. As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior.

42. 42 Linear Functions As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior. As we explore each function in more detail, we will be using multiple representations to discuss each parent function of the function families. We will develop the characteristics of each function as they are addressed in Algebra I and Algebra II and view a graphical and tabular representation of each parent function. Linear functions have many algebraic representations, as shown in the upper left-hand box. In Algebra I, students begin the exploration of domain and range and zeros. The relationship between the zero of a linear function and the solution of a linear equation is an important concept for students to understand. An exploration of intercepts allows students to obtain a deeper recognition of the characteristics of linear functions. In Algebra II, students will discuss the intervals over which functions are increasing and/or decreasing and discuss end behavior.

43. Absolute Value Functions 43 Review the absolute value function and consider how students in Algebra II might attempt to identify the characteristics of this function given what they know about linear functions. Take a moment to jot down the characteristics of the absolute value function and complete the table of values shown and then compare your answers with someone seated near you. How do the ideas deemed necessary about this parent function compare to your discussion of functions during the Placemat consensus activity?Review the absolute value function and consider how students in Algebra II might attempt to identify the characteristics of this function given what they know about linear functions. Take a moment to jot down the characteristics of the absolute value function and complete the table of values shown and then compare your answers with someone seated near you. How do the ideas deemed necessary about this parent function compare to your discussion of functions during the Placemat consensus activity?

44. Absolute Value Functions 44 Compare your answers about the characteristics of the absolute value function to those shown. Why is it important for students to recognize these characteristics about functions?Compare your answers about the characteristics of the absolute value function to those shown. Why is it important for students to recognize these characteristics about functions?

45. Function Transformations A transformational approach to graphing should be encouraged early along in the study of functions. Activities that allow students to explore the transformations of the graphs as they relate to the parametric changes in the function equation are important. Allowing students to discover the patterns for vertical and horizontal transformations of graphs, along with dilations of the graph will create a stronger connection for students between an graphical and algebraic representation of a function.A transformational approach to graphing should be encouraged early along in the study of functions. Activities that allow students to explore the transformations of the graphs as they relate to the parametric changes in the function equation are important. Allowing students to discover the patterns for vertical and horizontal transformations of graphs, along with dilations of the graph will create a stronger connection for students between an graphical and algebraic representation of a function.

46. Function Transformations

47. Quadratic Functions 47 Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas. Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas.

48. Quadratic Functions Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas. Quadratic functions can be the first formal introduction to non-linear relationships that students see in mathematics. Students should be given the opportunity to explore many real-world situations that model quadratic functions such as projectile motion, in order to have concrete examples that relate to these more abstract ideas.

49. Exploring Quadratic Relationships through data tables and graphs 49 Exploring the relationship of a set of data, such as the one shown which compares the weight of water at various temperatures, allows students to link concrete representations that can then be formalized algebraically. A big idea about quadratic functions is the idea of a maximum or minimum value for a set of data. Once students recognize the idea that quadratic functions have a max or min value, the parameters of the function can be examined further.Exploring the relationship of a set of data, such as the one shown which compares the weight of water at various temperatures, allows students to link concrete representations that can then be formalized algebraically. A big idea about quadratic functions is the idea of a maximum or minimum value for a set of data. Once students recognize the idea that quadratic functions have a max or min value, the parameters of the function can be examined further.

50. TAKE a BREAK Please take a 5 minute break. We will reconvene at _______.Please take a 5 minute break. We will reconvene at _______.

51. Square Root Functions 51 The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph. The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph.

52. Square Root Functions 52 The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph. The square root function allows students to begin to think about graphical inverses. A simple activity using patty paper and tables of values can assist student s in recognizing the inverse relationship between the quadratic and square root function. This is the first function students may encounter where the domain is restricted. For the parent function, x is restricted to be greater than or equal to zero, which in turn creates a restriction on the range of the function. Students may informally note that the square root function operates similarly to a ray in Geometry, in that it has an endpoint and goes off to infinity in only one direction (be sure to point out that the square root function is not linear, like a ray). Similarly, we can only discuss end behavior of the square root function for only one part of the graph.

53. Square Root Function Real World Application The speed of a tsunami is a function of ocean depth: SPEED = g = acceleration due to gravity (9.81 m/s2) d = depth of the ocean in meters Understanding the speed of tsunamis is useful in issuing warnings to coastal regions. Knowing the speed can help predict when the tsunami will arrive at a particular location. 53 An interesting application of the square root function is associated with the height of a tsunami wave. Often caused by underwater earthquakes, a tsunami wave can be created when water is placed in motion by the force of the earthquake. The speed of the wave will depend upon the depth of the ocean as the wave moves. Thus, as tsunamis reach shallow water, they tend to slow down and diminish.An interesting application of the square root function is associated with the height of a tsunami wave. Often caused by underwater earthquakes, a tsunami wave can be created when water is placed in motion by the force of the earthquake. The speed of the wave will depend upon the depth of the ocean as the wave moves. Thus, as tsunamis reach shallow water, they tend to slow down and diminish.

54. Cube Root Functions 54 Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens.Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens.

55. Cube Root Functions Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens.Students should be encouraged to compare the characteristics of the cube root function to the square root function. Why is the domain of a square root function restricted, but not the domain of a cube root function? Why is the range also the set of all real numbers? As students explore these ideas, their prior knowledge of evaluating square and cube roots will be activated. When we naturally encourage students to make these connections, the understanding of algebraic functions deepens.

56. Cube Root Function Real World Application Kepler�s Law of Planetary Motion: The distance, d, of a planet from the Sun in millions of miles is equal to the cube root of 6 times the number of Earth days it takes for the planet to orbit the sun, squared. For example, the length of a year on Mars is 687 Earth-days. Thus, d = 141.478 million miles from the Sun 56 An interesting and authentic application of cube root functions is Kepler�s Law of Planetary Motion. This law states that it is possible to determine the distance that another planet in our solar system is from the sun by calculating the number Earth days that it takes for the planet to orbit the sun. The function is a cube root function of 6 times the number of Earth days it takes for the planet to orbit the sun, squared. For example, Mars takes 687 Earth days to orbit the sun. Thus, it can be calculated to be approximately 141.478 million miles from the Sun.An interesting and authentic application of cube root functions is Kepler�s Law of Planetary Motion. This law states that it is possible to determine the distance that another planet in our solar system is from the sun by calculating the number Earth days that it takes for the planet to orbit the sun. The function is a cube root function of 6 times the number of Earth days it takes for the planet to orbit the sun, squared. For example, Mars takes 687 Earth days to orbit the sun. Thus, it can be calculated to be approximately 141.478 million miles from the Sun.

57. Rational Functions The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes.The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes.

58. Rational Functions The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes.The parent function y = 1/x possesses two asymptotes. A vertical asymptote at x = 0 and a horizontal asymptote at y = 0. The parameters of the parent function can be adjusted to allow students to explore the affect on asymptotes.

59. Rational Expressions Real World Application A real world application of rational expressions is shown, in which total time needed to travel up and down stream can be modeled using rational expressions. A simple distance = rate x time problem becomes a rational expression problem as the speed of the tugboat is increased or decreased based upon the speed of the current of the river. We can discuss with students what would happen to the tugboat as it traveled upstream if the speed of the river current was 10 mph. This real world example of an excluded value implies that there may be values for which a variable cannot be equal. In this case, if the current is 10 mph, the tugboat would be still in the water, neither moving up nor down stream.A real world application of rational expressions is shown, in which total time needed to travel up and down stream can be modeled using rational expressions. A simple distance = rate x time problem becomes a rational expression problem as the speed of the tugboat is increased or decreased based upon the speed of the current of the river. We can discuss with students what would happen to the tugboat as it traveled upstream if the speed of the river current was 10 mph. This real world example of an excluded value implies that there may be values for which a variable cannot be equal. In this case, if the current is 10 mph, the tugboat would be still in the water, neither moving up nor down stream.

60. Rational Expressions Real World Application Solving rational equations involves multiplying both sides of the equation by the common denominator, which in this case creates a quadratic function for which we can solve for the speed of the current. Solving rational equations involves multiplying both sides of the equation by the common denominator, which in this case creates a quadratic function for which we can solve for the speed of the current.

61. Applying Solving Equations and Graphing Related Functions As mentioned previously, a big idea in algebra involves the relationship between the solutions of an equation and the zeros of the graph of its related function. In this problem if we wish to solve 5c2 - 20 = 0 we can graph the related function f(c) = 5c2 - 20 and find the zeros. In the problem, a current cannot be a negative value, so we dismiss the negative value of c. As mentioned previously, a big idea in algebra involves the relationship between the solutions of an equation and the zeros of the graph of its related function. In this problem if we wish to solve 5c2 - 20 = 0 we can graph the related function f(c) = 5c2 - 20 and find the zeros. In the problem, a current cannot be a negative value, so we dismiss the negative value of c.

62. Solving Equations & Functions A.4 The student will solve multistep linear and quadratic equations in two variables�.. Framework Identify the root(s) or zero(s) of a �.. function over the real number system as the solution(s) to the �.. equation that is formed by setting the given �� expression equal to zero. 62 The concept of relating the solution(s) of an equation to the root(s) or zero(s) of its related function is part of the A.4 standard, but this idea can be built upon in AFDA and Algebra II.The concept of relating the solution(s) of an equation to the root(s) or zero(s) of its related function is part of the A.4 standard, but this idea can be built upon in AFDA and Algebra II.

63. Exponential Functions Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. � this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly. Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. � this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly.

64. Exponential Functions Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. � this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly. Exponential Functions become important as the variable moves from the base to the exponent. This is where asymptotes become obvious as students work through values that cannot exist. � this topic requires preparation in the handling of negative exponents and their use in fraction form (and not negative numbers) Students will also experience ranges that become very large very fast (or very small) as x-values increase slowly.

65. Exponential Function Real World Application Homemade chocolate chip cookies can lose their freshness over time. When the cookies are fresh, the taste quality is 1. The taste quality decreases according to the function: f(x) = 0.8x, where x represents the number of days since the cookies were baked and f(x) measures the taste quality. When will the cookies taste half as good as when they were fresh? 65 In this application, the goal is to approximate the value of x such that f(x) = 0.5 (one-half the taste quality) The solution can be found algebraically or graphically. A quick algebraic method which uses students� skills from Algebra 1 helps them understand how the numbers are reacting in this situation. This problem is a �natural� for continuing the discussion on how to obtain a more exact answer. In this application, the goal is to approximate the value of x such that f(x) = 0.5 (one-half the taste quality) The solution can be found algebraically or graphically. A quick algebraic method which uses students� skills from Algebra 1 helps them understand how the numbers are reacting in this situation. This problem is a �natural� for continuing the discussion on how to obtain a more exact answer.

66. Logarithmic Functions 66 Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it�s use as an inverse to the exponential function, and thus, teachers should continually help students� recall their prior work in this context. Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it�s use as an inverse to the exponential function, and thus, teachers should continually help students� recall their prior work in this context.

67. Logarithmic Functions 67 Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it�s use as an inverse to the exponential function, and thus, teachers should continually help students� recall their prior work in this context. Logarithmic functions are the inverses of exponential functions and can best be introduced through this relationship. Students should notice that x- and y-values changes in the table of values and likewise domain and range characteristics change in the same manner. Caution needs to occur to avoid students from thinking that all the characteristics of the exponential function are switched on the logarithmic functions which is why students need to be prepared for the analysis of these functions through their experiences with the polynomial and square root functions. Continued work on logarithms may distance the student from the graphical connection to it�s use as an inverse to the exponential function, and thus, teachers should continually help students� recall their prior work in this context.

68. Logarithmic Function Real World Application The wind speed, s (in miles per hour), near the center of a tornado can be modeled by s = 93 log d + 65 Where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the tornado�s center. s = 93 log d + 65 s = 93 log 220 + 65 s = 93(2.342) + 65 s = 282.806 miles/hour 68 In this real world example, the wind speed at a tornado�s center can be estimated using a logarithmic model based upon the distance that the tornado travels. In this problem, a large tornado in 1925 spanned three states and 220 miles. The function can be used to estimate the wind speed near the center of the tornado, which is approximately 283 miles per hour.In this real world example, the wind speed at a tornado�s center can be estimated using a logarithmic model based upon the distance that the tornado travels. In this problem, a large tornado in 1925 spanned three states and 220 miles. The function can be used to estimate the wind speed near the center of the tornado, which is approximately 283 miles per hour.

69. Inverse Functions:Exponentials and Logarithms 69 This graphic displays the inverse relationship between y=2x and y=log2x. And when comparing the (x, y) coordinates of points located on the graphs, the switching of the values is apparent. Other compare and contrast opportunities exist when discussing major attributes such as domain and range analysis, asymptotes, end behavior, etc. And y = x, as the line of reflection clearly indicates the graphical nature of this inverse relationship. The rationale for understanding logarithms arises from it�s inverse nature with exponentials and students may need to be reminded as they journey into the logarithms that this is �from whence� we needed to know about them!This graphic displays the inverse relationship between y=2x and y=log2x. And when comparing the (x, y) coordinates of points located on the graphs, the switching of the values is apparent. Other compare and contrast opportunities exist when discussing major attributes such as domain and range analysis, asymptotes, end behavior, etc. And y = x, as the line of reflection clearly indicates the graphical nature of this inverse relationship. The rationale for understanding logarithms arises from it�s inverse nature with exponentials and students may need to be reminded as they journey into the logarithms that this is �from whence� we needed to know about them!

70. Functions and Inverses Every function has an inverse relation, but not every inverse relation is a function. When is a function invertible? A function is invertible if its inverse relation is also a function. 70 The vertical line test is commonly used to test if a function exists. This is Algebra 1 content which also is found in Alg 2 and AFDA. The Horizontal Line test can be introduced as the test to determine if an inverse function exists. This rationale is based on prior knowledge: Inverse relationships exist by switching the x- and y-values. The vertical line test is used to test if a function exists. Thus, the horizontal line test is used to test if a function�s inverse exists. The vertical line test is commonly used to test if a function exists. This is Algebra 1 content which also is found in Alg 2 and AFDA. The Horizontal Line test can be introduced as the test to determine if an inverse function exists.

71. Quadratic Functions Require Restricted Domains in order to be Invertible Function: Inverse Function: 71 Thus, the need to restrict domains so that inverse functions can occur!Thus, the need to restrict domains so that inverse functions can occur!

72. Inverse Functions 72 Graphical depiction of the reflection across y = x for a cubic function and it�s inverse. Very similar to the exponential and logarithmic inverse function slide shown earlier.Graphical depiction of the reflection across y = x for a cubic function and it�s inverse. Very similar to the exponential and logarithmic inverse function slide shown earlier.

73. Polynomial Functions End behavior ~ direction of the ends of the graph 73 Discussion of end behavior regarding the even and odd degree nature of polynomial functions.Discussion of end behavior regarding the even and odd degree nature of polynomial functions.

74. Real World ApplicationPolynomial Function Suppose an object moves in a straight line so that its distance s(t) after t seconds, is represented by s(t)= t3 + t2 + 6t feet from its starting point. Determine the distance traveled in the first 4 seconds. 74 We introduce a cubic model for a time-distance relationship in this problem and encourage students to think about the function that can model this behavior.We introduce a cubic model for a time-distance relationship in this problem and encourage students to think about the function that can model this behavior.

75. 75 Analyzing the graph of this cubic function, we expect the ends of the graph to extend in opposite directions since the function is an odd-degreed function. As we are teaching students about functions we connect ideas to prior knowledge, but we can also provide a glimpse of some of the content that students might see in math analysis when again studying polynomial function more in depth, and perhaps even in a calculus course where students can study velocity and acceleration by calculating the first and second derivatives of the function and evaluating each function at a given time.Analyzing the graph of this cubic function, we expect the ends of the graph to extend in opposite directions since the function is an odd-degreed function. As we are teaching students about functions we connect ideas to prior knowledge, but we can also provide a glimpse of some of the content that students might see in math analysis when again studying polynomial function more in depth, and perhaps even in a calculus course where students can study velocity and acceleration by calculating the first and second derivatives of the function and evaluating each function at a given time.

76. Time is our constraint, so we are only concerned with the positive domain 76 To answer the question about how far the object has traveled after 4 seconds, we find the value of the function at time = 4 seconds. When students begin to use functions to model data, we should be sure to have conversations about how the function may only be a good model over a specific domain of the function. To answer the question about how far the object has traveled after 4 seconds, we find the value of the function at time = 4 seconds. When students begin to use functions to model data, we should be sure to have conversations about how the function may only be a good model over a specific domain of the function.

77. Analyzing Functions 77 The determination of the vertical asymptote is done by analyzing the denominator for values of x that makes the value of the denominator equal to 0. Those values are then eliminated from the domain since division by 0 is not possible. The determination of the vertical asymptote is done by analyzing the denominator for values of x that makes the value of the denominator equal to 0. Those values are then eliminated from the domain since division by 0 is not possible.

78. Analyzing Functions 78 The determination of the vertical asymptote is done by analyzing the denominator for values of x that makes the value of the denominator equal to 0. Those values are then eliminated from the domain since division by 0 is not possible. The determination of the vertical asymptote is done by analyzing the denominator for values of x that makes the value of the denominator equal to 0. Those values are then eliminated from the domain since division by 0 is not possible.

79. Asymptotes f(x) = 3(x � 2) 79 What is the domain of this function? {all real numbers} What is the range? {f(x)| f(x) > 0} Does this function have an asymptote and if so, what is it? Yes, y = 0What is the domain of this function? {all real numbers} What is the range? {f(x)| f(x) > 0} Does this function have an asymptote and if so, what is it? Yes, y = 0

80. Asymptotes 3xy = 12 80 Rewrite to y = form to assist with evaluation of asymptotes. How does this function compare to its parent function y = 1/x? How does changing the parameters of the function (namely multiplying by a factor of 4) transform this function as compared to its parent? What are the asymptotes?Rewrite to y = form to assist with evaluation of asymptotes. How does this function compare to its parent function y = 1/x? How does changing the parameters of the function (namely multiplying by a factor of 4) transform this function as compared to its parent? What are the asymptotes?

81. What do you know about this rational function? 81 The conversation teachers have with students regarding discontinuity and how to find it algebraically leads to the discovery of graphs with �holes�. Such as in this example, where a quadratic is divided by a line and a line results with a �hole� in it. For quadratics that factor, the process to finding this line is done by factoring out equivalent values of �1� ( ) , which results in the line f(x) = x + 2, but students need to understand that in the original function, because division by 0 is undefined. The conversation teachers have with students regarding discontinuity and how to find it algebraically leads to the discovery of graphs with �holes�. Such as in this example, where a quadratic is divided by a line and a line results with a �hole� in it. For quadratics that factor, the process to finding this line is done by factoring out equivalent values of �1� ( ) , which results in the line f(x) = x + 2, but students need to understand that in the original function, because division by 0 is undefined.

82. Discontinuity (Holes) 82 This is the graphical representation of This is the graphical representation of

83. Function Development 9-12 83 This slide shows how the content covered in the functions of Algebra 1 is essential as a building block for the AFDA and Alg2 function content. AFDA and Alg2 teachers will use their students� prior knowledge from Algebra 1 to increase their understanding of functions and behavior through the addition of content that can be more analytical in nature. This slide shows how the content covered in the functions of Algebra 1 is essential as a building block for the AFDA and Alg2 function content. AFDA and Alg2 teachers will use their students� prior knowledge from Algebra 1 to increase their understanding of functions and behavior through the addition of content that can be more analytical in nature.

84. Draw a function that has the following characteristics Domain: {all real numbers} Range: {f(x)| f(x)>0} Increasing: {x| -2<x<2 U x>5} Decreasing: {x| 2<x<5} Relative maximum(turning point): (2, 4) Relative minimum(turning point): (-2, 1) End Behavior: As x approaches 8, f(x) approaches 8. As x approaches - 8, f(x) approaches 8. Asymptotes: y=0 84 When you click, �Is it possible� will pop up. Suggest some changes in the values�for example�would it be possible to draw this function if as x approached infinity, f(x) approaches 0? NO. Not possible b/c we say that it is increasing on x>5. Change other values.When you click, �Is it possible� will pop up. Suggest some changes in the values�for example�would it be possible to draw this function if as x approached infinity, f(x) approaches 0? NO. Not possible b/c we say that it is increasing on x>5. Change other values.

85. Revisit Placemat ConsensusFunctions 85 As we end our session, let�s reflect back on our placemats and compare what we felt were the most important concepts of functions in grades 9-12. Did we identify the concepts that were addressed in this presentation? Do the teachers in your school divisions recognize the importance of functions in the study of high school algebra courses?As we end our session, let�s reflect back on our placemats and compare what we felt were the most important concepts of functions in grades 9-12. Did we identify the concepts that were addressed in this presentation? Do the teachers in your school divisions recognize the importance of functions in the study of high school algebra courses?