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Study Group 3 - High School Math (Algebra 1 & 2, Geometry) Welcome Back! Let’s spend some quality time discussing what we learned from our Bridge to Practice exercises. © 2013 UNIVERSITY OF PITTSBURGH Let’s Go Over Bridge to Practice #2: Time to Reflect on Our Learning Part 1: For Algebra 1, Using the Bike and Truck Task: For Algebra 2, Using the Missing Function Task: For Geometry, Using the Building a New Playground Task: a. Choose the Content Standards from the relevant pages in your module 2 handout (or view the standards on the following slides for each subject area Alg 1: 6-9, Alg 2: 16-19, Geometry: 25-28) b. Choose the Practice Standards students will have the opportunity to use while solving this task and find evidence to support them. © 2013 UNIVERSITY OF PITTSBURGH For Algebra 1: Bike and Truck Task Distance from start of road (in feet) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance. Time (in seconds) © 2013 UNIVERSITY OF PITTSBURGH For Algebra 1 - Bike and Truck Task 1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. 2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. 3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. 4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack and why. © 2013 UNIVERSITY OF PITTSBURGH Algebra 1 - Reflecting on Our Learning • Which CCSS for Mathematical Content did we discuss? • Which CCSS for Mathematical Practice did you use when solving the task? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content Algebra 1 Task CCSS Conceptual Category – Algebra Creating Equations* (A–CED) Create equations that describe numbers or relationships. A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. *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 a star,OFeach standard in that domain is a modeling standard. © 2013with UNIVERSITY PITTSBURGH Common Core State Standards, 2010, p. 65, NGA Center/CCSSO The CCSS for Mathematical Content Algebra 1 Task CCSS Conceptual Category – Algebra Reasoning with Equations and Inequalities (A–REI) Solve equations and inequalities in one variable. A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-REI.B.4 Solve quadratic equations in one variable. A-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), © 2013 UNIVERSITY OF PITTSBURGH taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Common Core State Standards, 2010, p. 65, NGA Center/CCSSO The CCSS for Mathematical Content Algebra 1 Task CCSS Conceptual Category – Algebra Reasoning with Equations and Inequalities (A–REI) Represent and solve equations and inequalities graphically. A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ A-REI.D.12 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. ★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. © 2013 UNIVERSITY OF PITTSBURGH Common Core State Standards, 2010, p. 65, NGA Center/CCSSO The CCSS for Mathematical Content Algebra 1 Task CCSS Conceptual Category – Functions Interpreting Functions (F–IF) Interpret functions that arise in applications in terms of the context. F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific © 2013 UNIVERSITY OF PITTSBURGH 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 Common a modeling standard. Core State Standards, 2010, p. 69, NGA Center/CCSSO For Algebra 1 Task: What standards for mathematical practice made it possible for us to learn? 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 Common Core State Standards for Mathematics, 2010 reasoning. © 2013 UNIVERSITY OF PITTSBURGH Part 3 - Underlying Mathematical Ideas Related to the Lesson – For Algebra 1 (Essential Understandings) • The language of change and rate of change (increasing, decreasing, constant, relative maximum or minimum) can be used to describe how two quantities vary together over a range of possible values. • A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables (NCTM, EU 2b). • The average rate of change is the change in the dependent variable over a specified interval in the domain. Linear functions are the only family of functions for which the average rate of change is the same on every interval in the domain. © 2013 UNIVERSITY OF PITTSBURGH Essential Understandings – Algebra 1 EU #1a: Functions are single-valued mappings from one set—the domain of the function—to another—its range. EU #1b: Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions. EU #1c: The domain and range of functions do not have to be numbers. For example, 2-by-2 matrices can be viewed as representing functions whose domain and range are a two-dimensional vector space. EU #2a: For functions that map real numbers to real numbers, certain patterns of covariation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has. EU #2b: A rate of change describes how one variable quantity changes with respect to another—in other words, a rate of change describes the covariation between variables. EU #2c: A function’s rate of change is one of the main characteristics that determine what kinds of real-world phenomena the function can model. © 2013 UNIVERSITY OF PITTSBURGH Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics. Essential Understandings – Algebra 1 EU #3a: Members of a family of functions share the same type of rate of change. This characteristic rate of change determines the kinds of real-world phenomena that the function can model. EU #3c: Quadratic functions are characterized by a linear rate of change, so the rate of change of the rate of change (the second derivative) of a quadratic function is constant. Reasoning about the vertex form of a quadratic allows deducing that the quadratic has a maximum or minimum value and that if the zeroes of the quadratic are real, they are symmetric about the x-coordinate of the maximum or minimum point. EU #5a: Functions can be represented in various ways, including through algebraic means (e.g., equations), graphs, word descriptions, and tables. EU #5b: Changing the way that a function is represented (e.g., algebraically, with a graph, in words or with a table) does not change the function, although different representations highlight different characteristics, and some may only show part of the function. EU #5c: Some representations of a function may be more useful than others, depending on the context. EU #5d: Links between algebraic and graphical representations of functions are especially important in studying relationships and change. © 2013 UNIVERSITY OF PITTSBURGH Cooney, T.J., Beckmann, S., & Lloyd, G.M., & Wilson, P.S. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12 (p. 8-10). Reston, VA: National Council of Teachers of Mathematics. For Algebra 2: Missing Function Task If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. x -2 -1 0 1 2 f(x) 0 1 2 3 4 © 2013 UNIVERSITY OF PITTSBURGH Algebra 2 - Reflecting on Our Learning • Which CCSS for Mathematical Content did we discuss? • Which CCSS for Mathematical Practice did you use when solving the task? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content – Alg 2 Task 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, p. 60, NGA Center/CCSSO The CCSS for Mathematical Content - Alg 2 Task 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 the 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, p. 64, NGA Center/CCSSO The CCSS for Mathematical Content – Alg 2 Task 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, p. 64, NGA Center/CCSSO The CCSS for Mathematical Content – Alg 2 Task 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, p. 70, NGA Center/CCSSO For Algebra 2 Task: What math practices made it possible for us to learn? 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 Part 3 - Underlying Mathematical Ideas Related to the Lesson – For Algebra 2 (Essential Understandings) • The product of two or more linear functions is a polynomial function. The resulting function will have the same xintercepts as the original functions because the original functions are factors of the polynomial. • Two or more polynomial functions can be multiplied using the algebraic representations 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) and g(x) and a specific x-value, x1, the point (x1, f(x1)+g(x1)) will be on the graph of the sum f(x)+g(x). (This is true for subtraction and multiplication as well.) © 2013 UNIVERSITY OF PITTSBURGH For Geometry: Building a New Playground Task The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown. © 2013 UNIVERSITY OF PITTSBURGH For Geometry: Building a New Playground PART A 1. Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. 2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B 3. The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. 4. Describe a strategy for determining a point equidistant from any three points. © 2013 UNIVERSITY OF PITTSBURGH Geometry - Reflecting on Our Learning • Which CCSS for Mathematical Content did we discuss? • Which CCSS for Mathematical Practice did you use when solving the task? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Congruence (G-CO) Understand congruence in terms of rigid motions. G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. © 2013 UNIVERSITY OF PITTSBURGH Common Core State Standards, 2010, p. 76, NGA Center/CCSSO The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Congruence (G-CO) Prove geometric theorems. G-CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. © 2013 UNIVERSITY OF PITTSBURGH G-CO.C.11 Common Core State Standards, 2010, p. 76, NGA Center/CCSSO The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Similarity, Right Triangles, and Trigonometry (G-SRT) Define trigonometric ratios and solve problems involving right triangles. G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ ★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. © 2013 UNIVERSITY OF PITTSBURGH Common Core State Standards, 2010, p. 77, NGA Center/CCSSO The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically. G-GPE.B.4 G-GPE.B.5 G-GPE.B.6 G-GPE.B.7 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ ★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,UNIVERSITY each standard in that © 2013 OF PITTSBURGH domain is a modeling standard. Common Core State Standards, 2010, p. 78, NGA Center/CCSSO For Geometry Task: Which Standards for Mathematical Practice made it possible for us to learn? 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 © 2013 UNIVERSITY OF PITTSBURGH Part 3 - Underlying Mathematical Ideas Related to the Lesson - For Geometry (Essential Understandings) • Coordinate Geometry can be used to form and test conjectures about geometric properties of lines, angles and assorted polygons. • Coordinate Geometry can be used to prove geometric theorems by replacing specific coordinates with variables, thereby showing that a relationship remains true regardless of the coordinates. • The set of points that are equidistant from two points A and B lie on the perpendicular bisector of line segment AB, because every point on the perpendicular bisector can be used to construct two triangles that are congruent by reflection and/or Side-Angle-Side; corresponding parts of congruent triangles are congruent. • It is sometimes necessary to prove both 'If A, then B' and 'If B, then A' in order to fully prove a theorem; this situation is referred to as an "if and only if" situation; notations for such situations include <=> and iff. © 2013 UNIVERSITY OF PITTSBURGH Part 2 - Research Connection: Findings by Tharp and Gallimore This slide pertains to Alg 1, Alg 2, & Geometry • For teaching to have occurred - Teachers must “be aware of the students’ ever-changing relationships to the subject matter.” • They [teachers] can assist because, while the learning process is alive and unfolding, they see and feel the student's progression through the zone, as well as the stumbles and errors that call for support. • For the development of thinking skills—the [students’] ability to form, express, and exchange ideas in speech and writing—the critical form of assisting learners is dialogue -- the questioning and sharing of ideas and knowledge that Tharp & Gallimore, 1991 happen in conversation. End of review of Bridge to Practice #2 Now we will move into our new Study Group Module 3 which is divided into two parts: 1. The impact of teacher implementation of a high level task on student learning 2. Using assessing and advancing questions to support student learning © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Part 1 Enacting Instructional Tasks: Maintaining the Demands of the Tasks Tennessee Department of Education High School Mathematics © 2013 UNIVERSITY OF PITTSBURGH Using the Assessment to Think About Instruction In order for students to perform well on the Constructed Response Assessments (CRAs), 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 Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). By analyzing the classroom actions and interactions of six teachers enacting the same high-level task, teachers will begin to identify classroom-based factors that are associated with supporting or inhibiting students’ high-level engagement during instruction. © 2013 UNIVERSITY OF PITTSBURGH Session Goals Participants will: • learn about characteristics of the written task that impact students’ opportunities to think and reason about mathematics • learn about the factors of implementation that contribute to the maintenance and decline of thinking and reasoning • analyze student work to determine what students know and can do • develop assessing and advancing questions based on student work (this will be part of the Bridge to Practice #3) © 2013 UNIVERSITY OF PITTSBURGH The Mathematical Tasks Framework TASKS TASKS TASKS as they appear in curricular/ instructional materials as set up by the teachers as implemented by students Student Learning Stein M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development, p. 4. New York: Teachers College Press. © 2013 UNIVERSITY OF PITTSBURGH The Enactment of the Task (Private Think Time) • Read the vignettes. • Consider the following question: What are students learning in each classroom? Scenario A – Mrs. Fox Scenario B – Mr. Chambers Scenario C – Ms. Fagan Scenario D – Ms. Jackson Scenario E – Mr. Cooper Scenario F – Ms. Gorman © 2013 UNIVERSITY OF PITTSBURGH The Enactment of the Task (Small Group Discussion) Discuss the following questions and cite evidence from the cases: What are students learning in each classroom? What made it possible for them to learn? © 2013 UNIVERSITY OF PITTSBURGH The Enactment of the Task (Whole Group Discussion) What opportunities did students have to think and reason in each of the classes? © 2013 UNIVERSITY OF PITTSBURGH Research Findings: The Fate of Tasks TASKS TASKS TASKS as they appear in curricular/ instructional materials as set up by the teachers as implemented by students © 2013 UNIVERSITY OF PITTSBURGH Student Learning Linking to Research/Literature: The QUASAR Project How High-Level Tasks Can Evolve During a Lesson: • Maintenance of high-level demands. • Decline into procedures without connection to meaning. • Decline into unsystematic and nonproductive exploration. • Decline into no mathematical activity. © 2013 UNIVERSITY OF PITTSBURGH Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Decline • Problematic aspects of the task become routinized. • Understanding shifts to correctness, completeness. • Insufficient time to wrestle with the demanding aspects of the task. • Classroom management problems. • Inappropriate task for a given group of students. • Accountability for high-level products or processes not expected. © 2013 UNIVERSITY OF PITTSBURGH Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands Maintenance • Scaffolds of student thinking and reasoning provided. • A means by which students can monitor their own progress is provided. • High-level performance is modeled. • A press for justifications, explanations through questioning and feedback. • Tasks build on students’ prior knowledge. • Frequent conceptual connections are made. • Sufficient time to explore. © 2013 UNIVERSITY OF PITTSBURGH Linking to Research/Literature: The QUASAR Project Task Set-Up Task Implementation Student Learning A. High High High B. Low Low Low C. High Low Moderate Stein & Lane, 1996 © 2013 UNIVERSITY OF PITTSBURGH Mathematical Tasks: A Critical Starting Point for Instruction There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995 © 2013 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Part 2 Illuminating Student Thinking: Assessing and Advancing Questions Tennessee Department of Education High School Mathematics © 2013 UNIVERSITY OF PITTSBURGH Rationale Effective teaching requires being able to support students as they work on challenging tasks without taking over the process of thinking for them (NCTM, 2000). • Asking questions that assess student understanding of mathematical ideas, strategies or representations provides teachers with insights into what students know and can do. • The insights gained from these questions prepare teachers to then ask questions that advance student understanding of mathematical ideas, strategies or connections to representations. © 2013 UNIVERSITY OF PITTSBURGH The Structures and Routines of a Lesson Set Up of the Task The Explore Phase/Private Work Time Generate Solutions The Explore Phase/Small Group Problem Solving 1. Generate and Compare Solutions 2. Assess and Advance Student Learning Share, Discuss, and Analyze Phase of the Lesson 1. Share and Model 2. Compare Solutions 3. Focus the Discussion on Key Mathematical Ideas 4. Engage in a Quick Write © 2013 UNIVERSITY OF PITTSBURGH MONITOR: Teacher selects examples for the Share, Discuss, and Analyze Phase based on: • Different solution paths to the same task • Different representations • Errors • Misconceptions SHARE: Students explain their methods, repeat others’ ideas, put ideas into their own words, add on to ideas and ask for clarification. REPEAT THE CYCLE FOR EACH SOLUTION PATH COMPARE: Students discuss similarities and difference between solution paths. FOCUS: Discuss the meaning of mathematical ideas in each representation REFLECT: By engaging students in a quick write or a discussion of the process. Small Groups based on subject area (Algebra 1, Algebra 2, or Geometry) Participants will: • analyze given student work for their subject area to determine what the students know and what they can do based only on the evidence from student work © 2013 UNIVERSITY OF PITTSBURGH Bike and Truck Task – Algebra 1 Distance from start of road (in feet) A bicycle traveling at a steady rate and a truck are moving along a road in the same direction. The graph below shows their positions as a function of time. Let B(t) represent the bicycle’s distance and K(t) represent the truck’s distance. Time (in seconds) © 2013 UNIVERSITY OF PITTSBURGH Bike and Truck Task - Algebra 1 1. Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. 2. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. 3. Which vehicle was first to reach 300 feet from the start of the road? How can you use the domain and/or range to determine which vehicle was the first to reach 300 feet? Explain your reasoning in words. 4. Jack claims that the average rate of change for both the bicycle and the truck was the same in the first 17 seconds of travel. Explain why you agree or disagree with Jack. © 2013 UNIVERSITY OF PITTSBURGH What Does Each Student Know? Algebra 1 Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Bike and Truck Task in your participant handout. What does each student know? Be prepared to share and justify your conclusions. © 2013 UNIVERSITY OF PITTSBURGH Response A - Algebra 1 54 Response B - Algebra 1 55 Response C - Algebra 1 56 Missing Function Task – Algebra 2 If h(x) = f(x) · g(x), what can you determine about g(x) from the given table and graph? Explain your reasoning. x -2 -1 0 1 2 f(x) 0 1 2 3 4 © 2013 UNIVERSITY OF PITTSBURGH What Does Each Student Know? Algebra 2 Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Missing Function Task in your Participant Handout. What does each student know? Be prepared to share and justify your conclusions. © 2013 UNIVERSITY OF PITTSBURGH Response A – Algebra 2 59 Response B – Algebra 2 60 Response C – Algebra 2 61 Building a New Playground Task Geometry The City Planning Commission is considering building a new playground. They would like the playground to be equidistant from the two elementary schools, represented by points A and B in the coordinate grid that is shown. © 2013 UNIVERSITY OF PITTSBURGH Building a New Playground Geometry PART A 1. Determine at least three possible locations for the park that are equidistant from points A and B. Explain how you know that all three possible locations are equidistant from the elementary schools. 2. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B 3. The City Planning Commission is planning to build a third elementary school located at (8, -6) on the coordinate grid. Determine a location for the park that is equidistant from all three schools. Explain how you know that all three schools are equidistant from the park. 4. Describe a strategy for determining a point equidistant from any three points. © 2013 UNIVERSITY OF PITTSBURGH What Does Each Student Know? Geometry Now we will focus on three pieces of student work. Individually examine the three pieces of student work A, B, and C for the Building a New Playground Task in your Participant Handout. What does each student know? Be prepared to share and justify your conclusions. © 2013 UNIVERSITY OF PITTSBURGH Response A - Geometry 65 Response B - Geometry 66 Response C - Geometry 67 Group D - Cannot Get Started Imagine that you are walking around the room, observing your students as they work on the task for either Algebra 1 or 2, or Geometry. Group D has little or nothing on their papers. Consider an assessing question and an advancing question for Group D. Be prepared to share and justify your conclusions. Reminder: You cannot TELL Group D how to start. What questions can you ask them? © 2013 UNIVERSITY OF PITTSBURGH Before Beginning Bridge to Practice #3: As you complete your next Bridge to Practice, reflect on the Content Standards and Essential Understandings as needed to help focus our discussion • For Algebra 1, Using the Bike and Truck Task: F-IF.B.4; F-IF.B.5; F-IF.B.6 • For Algebra 2, Using the Missing Function Task: A-APR.A.1; A-APR (cluster); F-BF.A.1b • For Geometry, Using the Building a New Playground Task: G-GPE.B.4; G-GPE.B.5; G-GPE.B.6 © 2013 UNIVERSITY OF PITTSBURGH Bridge to Practice #3 Part A: Use the list developed of what the students know and what they can do from the Student Work A-D to develop questions to be asked during the Explore Phase of the lesson – Develop at least one assessing question for Students A-D for your subject area – Develop at least one advancing question for Students A-D for your subject area © 2013 UNIVERSITY OF PITTSBURGH Bridge to Practice #3 Part B: Now that you have solved the task, examined some student work, and developed your assessing and advancing questions, facilitate this task with your students and record your assessing and advancing questions during the small group explore phase of the lesson. Note: Record could be audio or video using a device such as your phone, or have a colleague script your questions for you. Come prepared to share the questioning from your lesson. © 2013 UNIVERSITY OF PITTSBURGH