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Supporting Rigorous Mathematics Teaching and Learning Deepening Our Understanding of CCSS Via A Constructed Response Assessment Tennessee Department of Education High School Mathematics Algebra 2 © 2013 UNIVERSITY OF PITTSBURGH Forms of Assessment Assessment as Learning Assessment of Learning © 2013 UNIVERSITY OF PITTSBURGH Assessment for Learning Session Goals Participants will: • deepen understanding of the Common Core State Standards (CCSS) for Mathematical Practice and Mathematical Content; • understand how Constructed Response Assessments (CRAs) assess the CCSS for both Mathematical Content and Practice; and • understand the ways in which CRAs assess students’ conceptual understanding. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: • analyze Constructed Response Assessments (CRAs) in order to determine the way they are assessing the CCSSM; • analyze and discuss the CCSS for Mathematical Content and Mathematical Practice; • discuss what it means to develop and assess conceptual understanding; and • discuss the CCSS related to the tasks and the implications for instruction and learning. © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards The standards consist of: The CCSS for Mathematical Content The CCSS for Mathematical Practice © 2013 UNIVERSITY OF PITTSBURGH Analyzing a Performance-Based Assessment © 2013 UNIVERSITY OF PITTSBURGH Tennessee Focus Clusters Algebra 2 Extend the properties of exponents to rational exponents. Write expressions in equivalent forms to solve problems. Understand the relationship between zeros and factors of polynomials. Build a function that models a relationship between two quantities. © 2013 UNIVERSITY OF PITTSBURGH 7 Analyzing Assessment Items (Private Think Time) Four assessment items have been provided: Car Depreciation Writing a Polynomial Patterns in Patterns One Rocket, Three Equations For each assessment item: • solve the assessment item; and • make connections between the standard(s) and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH 1. Car Depreciation After you purchase a new car, it begins to lose value. As the years pass, more and more value is lost. This process is known as depreciation. a. Carmen buys a new car for $24,500. Carmen’s new car loses 14% of its value each year through depreciation. Write a function that can be used to model the value of the car at the end of each year that Carmen owns the car. Justify your equation mathematically. b. After Carmen buys the car, he adds an audio and speaker system worth $500. The audio system loses 15% of its value the minute it is installed into the car, and 7% of the remaining value each year through depreciation. Write a function that can be used to model the value of the car with the audio system and speakers at the end of each year that Carmen owns the car. © 2013 UNIVERSITY OF PITTSBURGH 2. Writing a Polynomial Recall that polynomial functions with only real number zeros can be written in factored form as follows: y a x z1 1 x z 2 2 ... x z n p p pn where each zn represents some real root of the function, and each pn is a whole number exponent greater than or equal to 1. Consider the graph of the polynomial function below. 14 a. Lisa claims that, since the point (0, 6) 13 12 is on the graph, (x – 6) is a factor of 11 10 this polynomial. Explain why you 9 8 agree or disagree with Lisa’s claim. 7 Identify all the zeroes of the function 6 5 and use that information in your 4 3 explanation. 2 1 1 b. Suppose a = . Write a function in 0 12 -5 -4 -3 -2 -1 -1 0 1 2 3 4 factored form to represent this graph. -2 Justify your equation mathematically. © 2013 UNIVERSITY OF PITTSBURGH 5 3. Patterns in Patterns Laura creates a design of circles embedded in each other for a poster. The largest circle has a diameter of 28 inches, and each successive circle has a diameter of 3 the previous circle. 4 a. Write a function that can be used to determine the diameter of any circle drawn in the poster in this way. Explain the meaning of each term in your expression in the context of the problem. b. Laura eventually draws 10 circles. Write and use a formula for the sum of a series to find the sum of the circumferences of the 10 circles, accurate to two decimal places. Show your work. © 2013 UNIVERSITY OF PITTSBURGH 28 inches 4. One Rocket, Three Equations For his science project, Grady designs and launches a model rocket from the rooftop of an empty building. He places a sensor on the rocket to provide him with data about the height of the rocket (from ground level in feet) over time (in seconds) during the flight. From the data returned by the sensor, Grady was able to write and graph a function modeling the height of the rocket over time. = −16 2 + 96 + 112 a. Rewrite the function in vertex form. Then describe what this form of the function reveals in the context of the situation. b. Rewrite the function by factoring completely. Then describe what this form of the function reveals in the context of the situation. © 2013 UNIVERSITY OF PITTSBURGH Discussing Content Standards (Small Group Time) For each assessment item: With your small group, discuss the connections between the content standard(s) and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH Deepening Understanding of the Content Standards via the Assessment Items (Whole Group) As a result of looking at the assessment items, what do you better understand about the specifics of the content standards? What are you still wondering about? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Number and Quantity The Real Number System (N-RN) Extend the properties of exponents to rational exponents. N-RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N-RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Seeing Structure in Expressions (A-SSE) Write expressions in equivalent forms to solve problems A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression. ★ A-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Arithmetic with Polynomials and Rational Expressions (A-APR) Understand the relationship between zeros and factors of polynomials A-APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). 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. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Functions Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1 Write a function that describes a relationship between two quantities.★ F-BF.A.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 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. F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 Determining the Standards for Mathematical Practice Associated with the Constructed Response Assessment © 2013 UNIVERSITY OF PITTSBURGH Getting Familiar with the CCSS for Mathematical Practice (Private Think Time) • Count off by 8. Each person reads one of the CCSS for Mathematical Practice. • Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical Practice. © 2013 UNIVERSITY OF PITTSBURGH 20 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 for Mathematics, 2010, NGA Center/CCSSO 21 Discussing Practice Standards (Small Group Time) Each person has 2 minutes to share important information about his/her assigned Mathematical Practice. © 2013 UNIVERSITY OF PITTSBURGH 22 Discussing Practice Standards (Small Group Time) For each assessment item: With your small group, discuss the connections between the practice standards and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH 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 Deepening Understanding of the Practice Standards via the Assessment Items (Whole Group) Which Standards for Mathematical Practice do you better understand? What are you still wondering about? © 2013 UNIVERSITY OF PITTSBURGH Assessing Conceptual Understanding © 2013 UNIVERSITY OF PITTSBURGH Rationale We have now examined assessment items and discussed their connection to the CCSS for Mathematical Content and Practice. A question that needs considering, however, is if and how these assessments will give us a good means of measuring the conceptual understandings our students have acquired. In this activity, you will have an opportunity to consider what it means to develop conceptual understanding, as described in the CCSS for Mathematics, and what it takes to assess for it. © 2013 UNIVERSITY OF PITTSBURGH Assessing for Conceptual Understanding The set of CRA items are designed to assess student understanding of expressions and equations. Look across the set of related items. What might a teacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole? What is varying from one item to the next? © 2013 UNIVERSITY OF PITTSBURGH Conceptual Understanding • What do the authors mean by conceptual understanding? • How might analyzing student performance on this set of assessments help us determine if students have a deep understanding of the assessed standards? © 2013 UNIVERSITY OF PITTSBURGH Developing Conceptual Understanding Knowledge that has been learned with understanding provides the basis of generating new knowledge and for solving new and unfamiliar problems. When students have acquired conceptual understanding in an area of mathematics, they see connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press The CCSS on Conceptual Understanding In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO Assessing Concept Image Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties. A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations which then provides a base from which they can move to another level of understanding. Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars. http://mathematicallysane.com/analysis/trenches.asp Developing and Assessing Understanding Why is it important, when assessing a student’s conceptual understanding, to vary items in these ways? © 2013 UNIVERSITY OF PITTSBURGH Using the Assessment to Think About Instruction In order for students to perform well on the CRA, what are the implications for instruction? • What kinds of instructional tasks will need to be used in the classroom? • What will teaching and learning look like and sound like in the classroom? © 2013 UNIVERSITY OF PITTSBURGH Step Back • What have you learned about the CCSS for Mathematical Content that surprised you? • What is the difference between the CCSS for Mathematical Content and the CCSS for Mathematical Practice? • Why do we say that students must work on both Standards for Mathematical Content and the Standards for Mathematical Practice? © 2013 UNIVERSITY OF PITTSBURGH Seeing Structure in Expressions A container of ice cream is taken from the freezer and sits in a room for t minutes. Its temperature in degrees Fahrenheit is − 2 − + , where and are positive constants. Write this expression in a form that shows that the temperature is always 1. Less than + 2. Greater than A.SSE.3 http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5 Seeing Structure in Expressions Susan is told to take 250 mg doses of an antibiotic for an ear infection regularly every 12 hours for 10 days. Susan does some research online and finds out that at the end of 12 hours, about 4% of the drug is still in the body. 1. What quantity of the drug is in the body right after the first dose, the second dose, the third dose? 2. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount? A.SSE.B.4 http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5 Seeing Structure in Expressions 1. What quantity of the drug is in the body right after the first dose, the second dose, the third dose? Let 1 be the amount of the drug in the body after the first dose 1 = 250 To find 2 , the amount of the drug in the body after the second dose, we have to find out how much of the first dose is still present and add the new dose. Since after 12 hours about 4% of the drug is still present in the body we get: 2 = 0.041 + 250 = 0.04 250 + 250 = 260 We use the same reasoning to find 3 : 3 = 0.042 + 250 = 0.04 0.04 250 + 250 + 250 = 260.4 Or 3 = 0.042 1 + 0.041 + 1 http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, pg. 5 Seeing Structure in Expressions 2. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount? Since the amount of antibiotic in Susan’s body increases every time she takes a dose, we need to find the amount in her body after the last (20th) dose: 20 = 0.0419 1 + 0.0418 1 + ⋯ + 0.041 + 1 20 = 1 0.0419 + 0.0418 + ⋯ + 0.04 + 1 20 = 1 1−0.0420 1−0.04 or 20 = 1 −1 0.04 20 1−0.04 Therefore, the amount of antibiotic in Susan’s system after 20 does of 250 mg is: 250−250 0.04 20 20 = =260.4167 mg ≈ 260.4 mg pg. 5 http://commoncoretools.files.wordpress.com/2011/09/ccss_progression_rp_67_2011_11_121.pdf, 1−0.04 1. Car Depreciation Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1 Write a function that describes 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. ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 2. Writing a Polynomial Arithmetic with Polynomials and Rational Expressions (A-APR) 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. Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.1 Write a function that describes a relationship between two quantities.★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 3. Patterns in Patterns Task Building Functions (F-BF) Build a function that models a relationship between two quantities F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★ ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star ( ★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 4. One Rocket, Three Equations Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by an expression. ★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010