Mathematics Task Arcs

Transcription

1 Overview of Mathematics Task Arcs: Mathematics Task Arcs A task arc is a set of related lessons which consists of eight tasks and their associated lesson guides. The lessons are focused on a small number of standards within a domain of the Common Core State Standards for Mathematics. In some cases, a small number of related standards from more than one domain may be addressed. A unique aspect of the task arc is the identification of essential understandings of mathematics. An essential understanding is the underlying mathematical truth in the lesson. The essential understandings are critical later in the lesson guides, because of the solution paths and the discussion questions outlined in the share, discuss, and analyze phase of the lesson are driven by the essential understandings. The Lesson Progression Chart found in each task arc outlines the growing focus of content to be studied and the strategies and representations students may use. The lessons are sequenced in deliberate and intentional ways and are designed to be implemented in their entirety. It is possible for students to develop a deep understanding of concepts because a small number of standards are targeted. Lesson concepts remain the same as the lessons progress; however the context or representations change. Bias and sensitivity: Social, ethnic, racial, religious, and gender bias is best determined at the local level where educators have in-depth knowledge of the culture and values of the community in which students live. The TDOE asks local districts to review these curricular units for social, ethnic, racial, religious, and gender bias before use in local schools. Copyright: These task arcs have been purchased and licensed indefinitely for the exclusive use of Tennessee educators.

2 mathematics Algebra 2 Building Polynomial Functions A SET OF RELATED S UNIVERSITY OF PITTSBURGH

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4 Table of Contents 3 Table of Contents Introduction Overview... 7 Identified CCSSM and Essential Understandings... 9 Tasks CCSSM Alignment Lesson Progression Chart Tasks and Lesson Guides TASK 1: Income and Expenses Lesson Guide TASK 2: Sums and Differences Lesson Guide TASK 3: Adding Functions Lesson Guide TASK 4: Missing Function Lesson Guide TASK 5: Comparing Products Lesson Guide TASK 6: Three Points Lesson Guide TASK 7: Triple Trouble Lesson Guide TASK 8: Factors and Products Lesson Guide... 60

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6 mathematics Algebra 2 Introduction Building Polynomial Functions A SET OF RELATED S

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8 Introduction 7 Overview In this set of related lessons, students use arithmetic to combine polynomial functions and explore the relationship between the original and new functions. Particular attention is focused on tabular and algebraic strategies for combining functions using arithmetic and the relationship between factors and x-intercepts of polynomial functions. There are a total of eight tasks in this set of related lessons. Six of the tasks are developing understanding tasks and two are solidifying understanding tasks. The related lessons address the Common Core State Standards A-APR.B.3 and F-BF.A.1b and require students to make use of all of the Mathematical Practice Standards. The first three tasks focus on adding and subtracting functions and understanding the relationship between the degree of the addends and the degree of the sum. In Tasks 1 and 2, students develop methods of determining the sum and difference of polynomial functions using the graph, table, and algebraic representation. They also explore the relationship between the degree of the addends and the degree of the sum. Students solidify their understanding of what it means to sum functions and what they can tell about the resulting function from the addends in Task 3. In Tasks 4-7, students develop understanding of the relationship between factors and zeros of polynomial functions. Task 4 asks students to determine characteristics of a missing factor function given a graph of the quadratic product and a table representing one of the factors. Task 5 further explores the linear factorization of quadratic functions by asking students to investigate the product of parallel lines, perpendicular lines, intersecting/non-perpendicular lines, and the product of a line and itself. Tasks 6 and 7 introduce higher order polynomials and continue to develop understanding of the relationship between factors and zeros of polynomial functions. Task 8 solidifies understanding that all polynomial functions can be written as the product of linear factors and that the zeros of the factors are the zeros of the product. The prerequisite knowledge necessary to enter these lessons is a developing understanding of linear functions including movement between the table, graph, and algebraic representations. Through engaging in the lessons in this set of related tasks, students will: deepen understanding of how multiple representations are used to represent the same relationship between two variable quantities; learn how to combine polynomial functions arithmetically using multiple representations; and understand the relationship between factors and zeros of a polynomial function. By the end of these lessons, students will be able to answer the following overarching questions: How is the graph of the sum or difference of two functions related to the addends? How is the graph of the product of two polynomial functions related to the factor functions? How is the degree of the sum (or product) function related to the degrees of the addend (or factor) functions? The questions provided in the guide will make it possible for students to work in ways consistent with the Standards for Mathematical Practice. It is not the Institute for Learning s expectation that students will name the Standards for Mathematical Practice. Instead, the teacher can mark agreement and disagreement of mathematical reasoning or identify characteristics of a good explanation (MP3). The teacher can note and mark times when students independently provide an equation and then re-contextualize the equation in the context of the situational problem (MP2). The teacher might also ask students to reflect on the benefit of using repeated reasoning, as this may help them understand the value of this mathematical practice

9 8 Introduction in helping them see patterns and relationships (MP8). In study groups, topics such as these should be discussed regularly because the lesson guides have been designed with these ideas in mind. You and your colleagues may consider labeling the questions in the guide with the Standards for Mathematical Practice.

10 Introduction 9 Identified CCSSM and Essential Understandings CCSS for Mathematical Content: Algebra and Functions Essential Understandings Understand the relationship between zeros and factors of polynomials. A-APR.B.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Polynomial functions can be written as a product of two or more linear factors. The value of a polynomial function is equal to zero if and only if at least one factor of the polynomial is equal to zero. Therefore, a polynomial function will have the same x-intercepts as its factor functions. The product of two or more polynomial functions is a polynomial function. The product function will have the same x-intercepts as the original functions because the original functions are factors of the polynomial.

11 10 Introduction CCSS for Mathematical Content: Algebra and Functions Essential Understandings Build a function that models a relationship between two quantities. F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Two or more functions can be multiplied using the algebraic representation by applying the distributive property and combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x 1 ) and g(x 1 ) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. The CCSS for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. *Common Core State Standards, 2010, NGA Center/CCSSO

12 Introduction 11 Tasks CCSSM Alignment Task A-APR.B.3 F-BF.A.1B MP 1 MP 2 MP 3 MP 4 MP 5 MP 6 MP 7 MP 8 Task 1 Income and Expenses Developing Understanding Task 2 Sums and Differences Developing Understanding Task 3 Adding Functions Solidifying Understanding Task 4 Missing Function Developing Understanding Task 5 Comparing Products Developing Understanding Task 6 Three Points Developing Understanding Task 7 Triple Trouble Developing Understanding Task 8 Factors and Prodcucts Solidifying Understanding

13 12 Introduction Lesson Progression Chart Overarching Questions How is the graph of the sum or difference of two functions related to the addends? How is the graph of the product of two polynomial functions related to the factor functions? How is the degree of the sum (or product) function related to the degrees of the addend (or factor) functions? TASK 1 Income and Expenses Developing Understanding TASK 2 Sums and Differences Developing Understanding TASK 3 Adding Functions Solidifying Understanding TASK 4 Missing Function Developing Understanding Content Polynomial functions can be subtracted. The sum or difference of two linear functions will be a linear function. Polynomial functions can be added and subtracted. The sum or difference of a linear function and a quadratic function is quadratic. Functions can be added or subtracted using graphs and tables or using the algebraic representations. Solidify understanding that polynomial functions can be added and subtracted using tables, graphs, and equations and that the degree of the sum function is dependent on the degree of the addends developed in Tasks 1 2. The product of two non-constant linear functions is always a quadratic function. The x-intercepts of the linear function are the x-intercepts of the quadratic. Functions can be multiplied using graphs and tables or using their algebraic representations. Strategy Calculating and representing the difference of two functions using multiple representations. Determining the characteristics of the sum and difference of a linear function and a quadratic function. Analyzing the truth of always, sometimes, never statements. Identifying characteristics of a missing factor function given graphs of the product and one factor. Representations Starts with context and asks students to create a graph. Starts with two functions represented in tables. Students will incorporate other representations into their solution path. Propositions are made in written language. Students explore those propositions using tables, graphs, and equations. Two functions are represented one graphically and one in a table. Students will use multiple representations to determine characteristics of a third function.

14 Introduction 13 TASK 5 Comparing Products Developing Understanding TASK 6 Three Points Developing Understanding TASK 7 Triple Trouble Developing Understanding TASK 8 Factors and Products Solidifying Understanding Content The product of two non-constant linear functions is always a quadratic function. The graph of the quadratic function has predictable characteristics determined by whether the linear factors are parallel, perpendicular, intersecting/nonperpendicular, or represent the same line. Regardless of the relationship between the linear factors, however, the x-intercepts of the linear factors are the x-intercepts of the quadratic function. All polynomial functions can be written as a product of linear factors. The value of the polynomial function is equal to zero when one or more of the linear factors is equal to zero. Polynomials can be multiplied algebraically by applying the distributive property and collecting like terms. The product of two polynomial functions is a polynomial function. The x-intercepts of the original functions are the x-intercepts of the product function. Polynomial functions can be multiplied using graphs and tables or using the algebraic representations. Solidify understanding of the relationships between factors of a polynomial function and its zeros developed in Tasks 4 7. Strategy Exploring products of parallel lines, perpendicular lines, intersecting/nonperpendicular lines, and coinciding lines. Determining possible factor functions when the x-intercepts of the product are known. Students are asked to consider the product of two functions represented graphically. They respond to reasoning put forth by four hypothetical students about properties of the product. Given x-intercepts of a product function, students determine possible linear factors. Representations Students construct graphs. Starts with written language and equations. Students construct graphs and equations. Starts with graph of a linear and quadratic function and student predictions in written language about the product function. Starts with graphs, equations, and written language.

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16 mathematics Algebra 2 Tasks and Lesson Guides Building Polynomial Functions A SET OF RELATED S

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18 Tasks and Lesson Guides 17 Name Income and Expenses TASK 1 Santiago recently started working at his first job. He has already earned $400 and will continue to earn at a rate of $250 per week. He is spending some of his money on a gym membership. The membership costs $15 per week and has a one-time enrollment fee of $50. The rest of the money he earns is going into a savings account for college. On the grid below, graph: the function representing Santiago s earnings; the function representing Santiago s gym expenses; and the function representing Santiago s savings Explain how you determined the function representing Santiago s savings.

19 18 Tasks and Lesson Guides 1 Income and Expenses Rationale for Lesson: Introduce the concept of subtracting functions using multiple representations. Students are presented with a context that can be modeled using linear functions. Task: Income and Expenses Santiago recently started working at his first job. He has already earned $400 and will continue to earn at a rate of $250 per week. He is spending some of his money on a gym membership. The membership costs $15 per week and has a one-time enrollment fee of $50. The rest of the money he earns is going into a savings account for college. On the grid below, graph: the function representing Santiago s earnings; the function representing Santiago s gym expenses; and the function representing Santiago s savings. Common Core Content Standards F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP6 Attend to precision. MP8 Look for and express regularity in repeated reasoning.

20 Tasks and Lesson Guides 19 Essential Understandings Materials Needed Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Task. Straight edge. Calculator. 1

21 20 Tasks and Lesson Guides 1 SET-UP PHASE Please read the task out loud. You will have about 5 minutes of private think time to begin working on the task independently. I will let you know when it is time to work with your group. EXPLORE PHASE Possible Student Pathways Group can t get started. Uses algebraic expressions. Earnings: x - Expenses: - ( x) Savings: x Uses tables. Weeks Income Expesnses Savings Assessing Questions What do you know about Santiago s earnings and his expenses? Why did you subtract the expressions? Tell me about the numbers in your table. Where do they come from? Advancing Questions How is the amount of money Santiago saves related to his earnings and expenses? Are these expressions functions? How can you write and graph functions representing earnings, expenses, and savings? What patterns do you see in the table? What do those patterns tell you about the graphs of these functions? Relates savings to distance between income and expenses functions. The savings is the difference. It keeps going up because the earnings keep getting bigger faster than the expenses. What do you mean when you say the savings is the difference? You say that the savings keeps going up. How can you determine how quickly it is increasing?

22 Tasks and Lesson Guides 21 SHARE, DISCUSS, AND ANALYZE PHASE EU: Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) Tell us about how you used the table to reason about this problem. Who can tell us about how this group used the table to determine points on the graph of the savings function? (They subtracted y values for each week. Income minus gym membership cost each week to get the total savings.) Why did they use the operation of subtraction? (Santiago pays his expenses from his income, so we have to subtract his expenses from the money he has coming in.) Why is the savings function increasing? (He makes more than he spends, so he adds money to his savings every week.) I am hearing that we can use tables to represent the total income over time and the total expenses over time. For any given week, we can subtract the expenses from the income to find Santiago s savings. Keeping track of this in a table, let us see how his savings are changing over time. (Revoicing) How much money is Santiago adding to his savings account every week? Who can show us where we see the rate of change in the table? On the graph? In the context? 1 EU: Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Tell us about how you used equations to think about this problem. Who can restate in their own words how this group wrote equations to represent Santiago s income and his expenses? Who can explain how this group used those equations to find the function representing Santiago s savings? (They just subtracted the things that are the same. The x s from the x s and the constants from the constants.) So, are you saying that if we know the algebraic representation of two functions, we can subtract them by combining like terms? (Marking) How does this group s work compare to the table we looked at? Where do we see the rate of change from the table in this group s equation?

23 22 Tasks and Lesson Guides 1 EU: The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Why is the difference of these two linear functions a linear function? Will the difference of two linear functions always be linear? Why or why not? (I don t know, because the x-term might go away and it would be just a number, like (2x + 3) (2x) = 3.) What do others think of that? Is y = 3 a linear function? Why or why not? I hear many of you saying yes. So, y = 3 is a constant function and it does form a straight line. A constant function is a linear function. Could the difference of two linear functions ever have a higher degree? Could it be quadratic, for example? (Challenging) (That isn t possible. How could some x minus some other x give you x 2?) Who would like to respond to (student name) s question? (I agree. It isn t possible. You have to multiply x by x to get x 2. Subtracting won t do it.) What about the context tells us that Santiago s savings will be linear and not some other kind of function? (Since he is making $250 every week and spending $15 every week, he is saving the same amount every week, $235.) What do others think about this? Who can add on? The savings increase at a constant amount. (Marking) Can we see this in the table as well? Let s look at what we talked about today. We considered subtracting two linear functions using a table and saw that the difference of corresponding y-values will be on the difference function. We also considered the algebraic equations and saw that we can subtract two linear functions by combining like terms. Finally, we asked ourselves if the difference of two linear functions will always be a linear function and we figured out that if we subtract two linear functions, the x coefficient may change and it may even change to zero, but that these changes will still always result in a linear equation. (Recapping) As we continue to explore how we can use arithmetic on functions, we will see if we can discover similar truths about other polynomial functions. Application Summary Quick Write Jaime has $20 in his piggy bank and earns $5 each week in allowance. Any money he doesn t spend on baseball cards, he will put in his piggy bank. He spends $5 each week on baseball cards. Describe the graph of the function representing the amount of money in his piggy bank over time. How can we determine the difference of two functions using their graphs, tables, and equations? What kind of function results when you subtract two linear functions? Explain how you know.

24 Tasks and Lesson Guides 23 Support for students who are English Learners (EL): 1. Private think time allows students to organize their thoughts and struggle individually. 2. Small group collaboration allows students to express their thinking to a few peers before sharing with the whole group. 3. Use of multiple representations and teacher prompts to connect the multiple representations helps all students make sense of the mathematical relationships. 4. Create an interactive word wall or have students record new vocabulary terms in a special section of their notebook. As they progress through the set of related lessons, add terms to the word wall and refer to it often. 1

25 24 Tasks and Lesson Guides TASK 2 Name Sums and Differences Consider the two functions k(x) and g(x) shown in the table of values. x k(x) x g(x) We can define two new functions, h(x) = k(x) + g(x) and m(x) = k(x) g(x). What can you determine about h(x) and m(x)? Use at least two representations in your response.

26 Tasks and Lesson Guides 25 Sums and Differences Rationale for Lesson: Continue developing the concept of combining functions using arithmetic. Extend reasoning to include non-linear functions. 2 Task: Sums and Differences Consider the two functions k(x) and g(x) shown in the table of values. x k(x) x g(x) We can define two new functions, h(x) = k(x) + g(x) and m(x) = k(x) g(x). What can you determine about h(x) and m(x)? Use at least two representations in your response. Common Core Content Standards F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.

27 26 Tasks and Lesson Guides 2 Essential Understandings Materials Needed Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Task. Graph paper. Calculator (optional).

28 Tasks and Lesson Guides 27 SET-UP PHASE Please read the task out loud. What is the relationship between h(x) and the functions described in the tables? So, h(x) is the sum of the two functions and m(x) is the difference. We will take 5 minutes to work on the task individually before you begin working in your groups. 2 EXPLORE PHASE Possible Student Pathways Group can t get started. Uses tables. x k(x) g(x) h(x) Assessing Questions What do you know about k(x) and g(x)? Why did you add the values in the k(x) column by the values in the g(x) column to fill the h(x) column? Advancing Questions What does it mean to add two functions together? Would a different representation be helpful? What patterns do you see in the h(x) column? What do these patterns tell you about the function? Can you use a similar method to learn about m(x)? Why or why not? Uses equations. k(x) + g(x) = (-2x + 4) + (x 2 +1) h(x) = x 2 2x+ 5 k(x) g(x) = (-2x + 4) (x 2 +1) m(x) = - x 2 2x+ 3 Conjectures that the sum will be cubic and the difference will be linear. How did you generate the equations to represent k(x) and g(x)? Tell me about your conjectures. What led you to these ideas? How are the sum and difference similar? How are they different? How can you prove or disprove your conjectures? How can you represent the sum using tables, graphs, or equations?

29 28 Tasks and Lesson Guides 2 SHARE, DISCUSS, AND ANALYZE PHASE EU: Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) Tell us about how you used the table to find the sum and difference. Who can explain this group s method in their own words? Show us on the table how they determined points on the sum function and the difference function. What did you notice when you plotted the points from your table? (Student name) said they were both parabolas. Did other groups find this, too? Hmm I wonder why that happened. Let s look at another group s work and keep thinking about this question. EU: Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Tell us about these equations. How did you determine equations for k(x) and g(x)? Do this group s equations represent the data in the tables? Why or why not? Tell us about how you found the sum and difference using equations. Who can tell us about how they used the equations to find the sum and difference? (They wrote the equations for both of the functions and then they just added the terms that were alike.) They found the sum and difference by collecting (or combining) like terms. (Revoicing) Do their sum and difference functions represent the relationships found by the first group? How do you know? So, we have explored two strategies for adding or subtracting two functions that appear to get us the same results. What are those two methods and what are the strengths and weaknesses of each? (We used the table to just add up specific points. This is a nice method, because it is easy to do and I don t have to deal with variables. It tells me points on the graph and then I can graph them to see what it looks like. The other method is to write the equations and then add or subtract them by combining like terms. This is good because you get the whole equation at the end so you can graph it using a graphing calculator and can figure out any points on the graph if you need to.) So far, we have explored two methods of adding or subtracting equations. We can add their corresponding y-values using a table or we can collect like terms using their algebraic representations. These result in either a table representation or an algebraic representation, either of which can be used to create a graph. (Recapping)

30 Tasks and Lesson Guides 29 EU: The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Let s return to this idea that the sum and difference are both quadratic. Why did you classify the sum and difference as quadratic? What is your evidence? Who heard what (student name) said about the shape of the graph and can add on? How do we know from the equations that these functions are quadratic? Listen to this group s conjectures and be prepared to indicate why you agree or disagree. Do you agree or disagree that the sum of a linear and a quadratic will always be quadratic? Why? Is there any way we could end up with an x 3 term or any term with a power greater than 2? Why or why not? Could the sum or difference ever be linear? How can you support your position using the algebraic representation? (Just like the problem we did yesterday with Santiago, if we have a quadratic and a linear, then when we subtract, the x 2 term can t go anywhere, and the exponent can t get any bigger either because we aren t multiplying. The sum and difference have to be quadratic.) Yesterday we saw that the difference of two linear functions will always be linear. Now, we are finding that for similar reasons, the sum and difference of a linear and a quadratic will always be quadratic. Looking at the algebraic representations, we see that combining like terms will never result in the x 2 term being eliminated and will never result in a variable being raised to a higher power. (Recapping) How is the difference function similar to the sum function? How is it different? Why does this parabola open downwards? Will the difference of a linear and a quadratic always be a parabola opening downwards? Why or why not? We know that addition is commutative, but subtraction is not. How would the difference of g(x) f(x) be similar to and different from f(x) g(x)? (Challenging) (g(x) f(x) is still quadratic, but the parabola would open downward because the x 2 term will be negative.) 2 Application Describe the sum and difference of p(x) = 2x 5 and r(x) = 3x 2 + 2x 5. Summary What do we know about the sum of a linear and a quadratic function? Explain. What do we know about the difference of a linear and a quadratic function? Explain. Quick Write What kind of function results when you add a linear and a quadratic function? Explain how you know. Support for students who are English Learners (EL): 1. Take time to discuss and define the terms function, sum, and difference. Each of these words has multiple meanings or has a homonym (sum, some). 2. Add new terms to the word wall. Refer to existing word wall terms throughout the lesson.

31 30 Tasks and Lesson Guides TASK 3 Name Adding Functions 1. Sheila, Thomas, and Zenia are examining functions together. They make the following claims: The graph of the sum of two linear functions is sometimes a parabola. The graph of the sum of a linear function and a quadratic function is always a parabola. The graph of the sum of two quadratic functions is always a parabola. The graph of the sum of a quadratic function and a cubic function is never a parabola. Do you agree or disagree with their claims? Use mathematics to support your position. 2. When adding functions of different degrees, what will be true about the degree of the sum? 3. Write an example of each of the following sums of functions: Cubic + cubic = cubic Cubic + cubic = quadratic Cubic + cubic = linear Why isn t the sum of two cubic functions always a cubic function?

32 Tasks and Lesson Guides 31 Adding Functions 3 Rationale for Lesson: Solidify understanding of adding polynomial functions using tables, graphs, and equations. Generalize rules to determine the degree of the sum of two polynomial functions. Task: Adding Functions 1. Sheila, Thomas, and Zenia are examining functions together. They make the following claims: The graph of the sum of two linear functions is sometimes a parabola. The graph of the sum of a linear function and a quadratic function is always a parabola. The graph of the sum of two quadratic functions is always a parabola. The graph of the sum of a quadratic function and a cubic function is never a parabola. Do you agree or disagree with their claims? Use mathematics to support your position. 2. When adding functions of different degrees, what will be true about the degree of the sum? 3. Write an example of each of the following sums of functions: Cubic + cubic = cubic Cubic + cubic = quadratic Cubic + cubic = linear Why isn t the sum of two cubic functions always a cubic function? Common Core Content Standards F-BF.A.1b Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.

33 32 Tasks and Lesson Guides 3 Essential Understandings Materials Needed Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Task. Graph paper. Graphing calculators (recommended).

34 Tasks and Lesson Guides 33 SET-UP PHASE Does the result of the sum of two polynomials always have equal degree to the polynomial of highest degree? Please read the task out loud. Consider the result of adding different polynomials. We will take 5 minutes to work on the task individually before you begin working in your groups. 3 EXPLORE PHASE Possible Student Pathways Group can t get started. Uses graph. Creates sketches of possible pairs of functions and their sums. Uses equations. Creates equations of possible pairs of functions and their sums. Students finish early. Assessing Questions What does it mean to add two functions? What representations have we used to add two functions? Tell me about your graph. How did you add these functions? Tell me about the equations you wrote. How do you know that this equation (point to one) represents a quadratic function? What generalizations did you make based on the examples you looked at? Advancing Questions Come up with an example of two linear functions and find their sum. Would the graph of this sum be a parabola? Why or why not? I see that the linear functions you sketched have a sum that is linear. Does that mean that the sum of two linear functions is always linear? Why or why not? I see that the quadratic functions you chose do sum to a quadratic. How could we change the functions so that their sum is not quadratic? Do the generalizations you made apply to higher order polynomials as well? Why or why not?

35 34 Tasks and Lesson Guides 3 SHARE, DISCUSS, AND ANALYZE PHASE EU: Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) We have explored two strategies for adding or subtracting two functions. We can add their corresponding y-values using a table or we can collect like terms using their algebraic representations. Did either of those methods help you think about this task? There are no specific functions to add or subtract in this task. This group created functions to use to test the conjectures. Tell us about how you used the graph and table to find the sum of the functions you chose. Who can explain this group s method in their own words? Show us on the table how they determined points on the sum function. What did you notice when you graphed the sum of two quadratic functions? Someone said it was a parabola. Did other groups find this too? (Not us. Ours was a line, because the x 2 terms disappeared.) I am glad you mentioned the x 2 terms, because we need to talk about equations. (Revoicing) As we move on to talking about equations, keep thinking about how we can predict the degree of the sum or difference of two polynomial functions. EU: Two or more functions can be added or subtracted using their algebraic representations by combining like terms. Tell us about how you found the sum of your functions using equations. Who can repeat what this group said about like terms? How is this method similar to or different from the method using tables? You mentioned that the x 2 terms disappeared. How is that possible? How can you add two things together and get zero? (Challenging) (They were opposites, 3x 2 and -3x 2.) Opposites sum to zero. (Marking) Let s look at what we figured out about how we can make predictions about the degree of the sum of two functions.

36 Tasks and Lesson Guides 35 EU: The degree of the sum of two polynomial functions is dependent upon the degree of the addends. When two polynomial functions are added using their algebraic representations by combining like terms, the coefficient of the highest order terms may change, but the exponent will not. Therefore, if the degree of the addends is unequal, the sum will have the degree of the addend with the higher degree. If the degree of the addends is equal, the degree of the sum is less than or equal to the degree of the addends. Let s consider the sum of two quadratic functions. We heard from a group who summed two quadratic functions and got a quadratic function. We heard from a group that summed two quadratic functions and got a linear function. Are there any other possible outcomes? Could a quadratic plus a quadratic equal a cubic? (I don t think so. I think that only the number in front of x 2 can change, like x 2 + 2x 2 = 3x 2, but it doesn t change the exponent.) S/he said that adding like terms affects the coefficient, but does not change the exponent. (Revoicing) What do others think about that? What does that mean about the sum of two polynomial functions with the same degree? (It means that the sum has the same degree.) If the two polynomial functions have the same degree, their sum will also have that degree. (Marking) Another group made a conjecture about the sum of two functions that don t have the same degree. Listen to their conjecture and be prepared to say why you agree or disagree. (The sum is the same as the one with the higher exponent, because there is nothing to cancel out that term and it can t get a higher exponent because there is no multiplying.) Who agrees that the sum of two functions can never have a greater degree than the degree of the highest addend? Why? (Revoicing) Can the sum have a degree that is less than the degree of the highest addend? Can it be less than the degree of the lowest addend? Explain. (The sum always has the same degree as the higher exponent. It can never be less. I know this from the equations. If I add a quadratic plus a linear, the x 2 term can t be canceled out, because there is no other x 2 terms to combine it with.) Yesterday we heard some convincing arguments for why the sum of a linear and a quadratic will always be quadratic. Who can recall and restate one of those arguments? (Student name) reminded us that the x 2 term will never be eliminated when we add or subtract a linear and a quadratic. What about adding a cubic and a quadratic? Or a cubic and a linear? Show us an example to support your thinking. I am hearing the argument that when the addends have different degrees, the sum ALWAYS has the same degree as the higher addend. (Marking) Who can state for us why this is true? When the degree of the addends is the same, like in #3, there are different possible degrees for the sum. Why? Who can repeat what (student name) just said about combining like terms and can add on? 3 Application Write two quadratic functions whose sum is linear. Explain how you know the sum of these functions will be linear. Summary What methods have we explored for adding two polynomial functions? How is the degree of the sum related to the degree of the addends? Quick Write Why is the sum of a quadratic function plus a quadratic function not always quadratic? Be specific and use examples in your response. Support for students who are English Learners (EL): 1. During the set-up of the task, ask students to provide non-mathematical examples of sometimes, always, and never statements.

37 36 Tasks and Lesson Guides TASK 4 Name Missing Function If h(x) = f(x) g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. x f(x)

38 Tasks and Lesson Guides 37 Missing Function Rationale for Lesson: Consider what it means to multiply two polynomial functions. Develop understanding of the relationship between factors and zeros of a polynomial function by observing that the x-intercepts of the linear factors coincide with the x-intercepts of the product function. 4 Task: Missing Function If h(x) = f(x) g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. x f(x) Common Core Content Standards A-APR.B.3 F-BF.A.1b Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Standards for Mathematical Practice MP1 Make sense of problems and persevere in solving them. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning.

39 38 Tasks and Lesson Guides 4 Essential Understandings Materials Needed Polynomial functions can be written as a product of two or more linear factors. The value of a polynomial function is equal to zero if and only if at least one factor of the polynomial is equal to zero. Therefore, a polynomial function will have the same x-intercepts as its factor functions. Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) Task. Calculator (optional).

40 Tasks and Lesson Guides 39 SET-UP PHASE Please read the task out loud. What is the relationship between g(x) and the functions in the table and graph? So, the graphed function, h(x), is the product of two functions. The function in the table, f(x) is one of the functions being multiplied to get h(x). We will take 5 minutes to work on the task individually before you begin working in your groups. 4 EXPLORE PHASE Possible Student Pathways Group can t get started. Uses tables. x f(x) g(x) h(x) /1= /2= /3= /4=1 4 Assessing Questions What can you tell about the shape of f(x) from the data in the table? Explain. Why did you divide the values in the last column by the values in the f(x) column? Advancing Questions What type of function will g(x) be? Why? What patterns do you see in the g(x) column? What do these patterns tell you about the function? Guess and check. f(x) = x + 2 (x + 2)*x = x 2 + 2x check at 0 g0 2 +2(0) = 0 (NO) (x + 2) (x + 1) = x 2 + 3x + 2 check at 0 g (0) + 2 = 2 (NO) Conjectures that g(x) will have negative y-values for -2 < x < 1. Why did you choose x and x + 1 to multiply by f(x)? What will you choose next? Tell me about your conjecture. Why do you think g(x) will be negative over that interval? What do you know about the product that can help you make an educated guess about what g(x) is? What about x < -2 and x > 1? Is g(x) positive or negative over those intervals? How do you know?

41 40 Tasks and Lesson Guides 4 SHARE, DISCUSS, AND ANALYZE PHASE EU: Two or more polynomial functions can be added using their graphs or tables of values because given two functions f(x) and g(x) and a specific x-value, x 1, the point (x 1, (f(x 1 ) + g(x 1 )) will be on the graph and in the table of the sum f(x) + g(x). (This is true for subtraction and multiplication as well.) Tell us about how you used the table to reason about this problem. Who can tell us how this group used the table to find points on the graph of y = g(x)? (Just like the table method for adding and subtracting, they multiplied the y-terms that go with the same x to get a point on the new graph.) Are you saying that points on the product of f(x)*g(x) can be found by multiplying corresponding y-values of f(x) and g(x)? (Revoicing) Use this relationship to explain how you know that the y-intercept of g(x) in the table is correct. (The y-intercept of h(x) is -2 and the y-intercept of f(x) is * -1= 2, so to get the y-intercept of h(x) to be 2, the y-intercept of g(x) has to be -1.) Show us where we see this in the table. What do we know about g(x) based on the points in the table? Who agrees that g(x) is linear? What about the data in the table indicates that g(x) is linear? (The y-values go up by 1 when the x-values go up by 1.) EU: Polynomial functions can be written as a product of two or more linear factors. This group made conjectures about g(x) before they found any specific points on g(x). Listen to their conjectures and be prepared to explain why you agree or disagree. What was their reason for conjecturing that g(x) is a linear function? Do you agree or disagree with their reasoning? Why is a quadratic function the product of two linear functions? What was their reason for conjecturing that g(x) is negative for x < 1? Do you agree or disagree with their reasoning? (A negative times a negative is a positive and a negative times a positive is a negative.) (f(x) is negative for x < 1 and h(x) is positive until -2 and then negative from -2 to 1. So, before x = -2, f(x) is negative, so g(x) has to be negative to multiply together to make h(x) positive. From -2 to -1, f(x) is positive and h(x) is negative, so g(x) has to be negative.) Can you show us these regions on the graph to help us understand? Do the points in the other group s table support these conjectures? Why or why not? (If we add more x-values to the table, we can see where f(x) is negative and then we can see the negative times a negative and the positive times a negative that makes h(x) positive before x = -2 and negative from -2 to 1.) Who can explain how to determine an equation for g(x) from the points in the table?