# Study Group 3 - High School Math (Algebra 1 & 2, Geometry)

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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. Our Bridge to Practice through TNCore Training is linking our classroom instruction to the CCSS!

Let’s Go Over Bridge to Practice #2: Time to Reflect on Our Learning

For Algebra 1: Bike and Truck Task
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. Distance from start of road (in feet) Time (in seconds) Facilitator Note: This slide is here for you to reference during the group discussion.

For Algebra 1 - Bike and Truck Task
Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. 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. 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. Facilitator Note: This slide is here for you to reference during the group discussion.

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? (SAY) Share the standards for mathematical content you found that students had an opportunity to make sense of on this task. The focus cluster standards are on pages 10–11 in the handout from Module 2. Facilitator Note: These are the standards that relate to the Bike and Truck task! Discuss why using the slides that follow with the standards on them. F-IF.B.4 F-IF.B.5 F-IF.B.6 Note: Students might demonstrate other content standards when solving the task, but we are determining which content standards the task is specifically designed to assess AS THE TASK IS WRITTEN (not our view of all of the ways it can be interpreted). The focus here is also on rich conversation to get more familiar with the content standards and see how they could be addressed in a problem.

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. Facilitator Note: These standards do NOT align to the Bike and Truck task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: What do the standards on the slide say? How do they differ from each other? (Give participants a minute to read the standards.) One standard references creating equations and inequalities in one variable; while another references two variables. How does the work we did on this task give students insight into creating and solving equations? We used a graphical approach. We moved from graph to equation. We made sense of the average rate of change. *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. 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), 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. Facilitator Note: These standards do NOT align to the Bike and Truck task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: What do the standards on the slide say? How do they differ from each other? (Give participants a minute to read the standards.) They reference solving equations, but the type of equation is different. They reference solving in one variable. How does the work we did on this task give students insight into solving equations? Students make connections between representations. Students make comparisons between two different function. 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 half-plane (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. Facilitator Note: These standards do NOT align to the Bike and Truck task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: What do the standards on the slide say? How do they differ from each other? (Give participants a minute to read the standards.) How does A.REI.D.10 relate to solving equations? How is A.REI.D.10 related to A.REI.D.11? Each point on the graph represents a solution. Equations may be solved graphically. 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 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. These are the standards that relate to the Bike and Truck task! Discuss why. Reminder: Students might demonstrate other content standards when solving the task, but we are determining which content standards the task is specifically designed to assess AS THE TASK IS WRITTEN (not our view of all of the ways it can be interpreted). The focus here is also on rich conversation to get more familiar with the content standards and see how they could be addressed in a problem. Probing Facilitator Questions and Possible Responses:. What key features of the graph did we interpret? We interpreted rate of change and intervals where the function is increasing or decreasing. We used the y-intercept as we interpreted the graphs. How did we address F-IF.B.5? When we determined which vehicle reached 300 feet first, we had to realize that the function with the “smaller” domain was the one that reached 300 feet first. This standard doesn’t talk about range. But similarly, we realized when calculating the average rate of change, one function has a more restricted range than the other. The CCSS Key Shift of Rigor indicates that conceptual understanding, procedural skill and fluency, and application should be pursued with “equal intensity.” How does this cluster of standards support this Key Shift? These standards ask students to interpret, relate, and calculate. The verbs imply all three components of the key shift. Common 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?
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Directions: Show the list of standards for mathematical practice. Ask participants which of these standards for mathematical practice students will have an opportunity to use when solving the task. Probing Questions and Possible Responses: What does it mean to reason abstractly and quantitatively? This standard means that you have to abstract information from the problem context, use the information to complete calculations, and then interpret the solutions in the context of the problem. In the Bike and Truck task, we had to abstract values from the graph, calculate the average rate of change and then interpret the average rate of change in the context of the problem. Attend to precision means more than just getting the correct answer. What would it mean in this problem? It means performing calculations correctly, but also attending to the meaning of words and being able to talk about the domain (and range), and where the average rate of change is visible in the graph. Common Core State Standards for Mathematics, 2010

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. (SAY) When I was working with you I was watching and listening for your understanding of the content standards. When I heard essential understandings, such as these related to rate of change, then I knew you understood the concept. Read just one of the bullets, bullet #2. Why do I need to know this level of detail about rate of change in order to determine if you understand the structure behind rate of change?

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. (SAY) We have identified standards that align to the Bike and Truck task, but sometimes standards are not specific enough to design instruction around. Essential understandings are more specific statements that identify the underlying mathematical truths that are the goals of our instruction. These essential understandings are written and published by NCTM. Which of these essential understandings did we explore today as we engaged in the task? 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. 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 f(x) -2 -1 1 2 3 4 Facilitator Information: This slide is here for you to reference during the group discussion.

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? (Say) Share the standards for mathematical content you found that students had an opportunity to make sense of on this task. Facilitator Information: These are the standards that relate to the Missing Function Task! Discuss why using the slides that follow with the standards on them. A-APR.A.1 A-APR (cluster) F-BF.A.1b NOTE: Students might demonstrate other content standards when solving the task, but we are determining which content standards the task is specifically designed to assess AS THE TASK IS WRITTEN (not our view of all of the ways it can be interpreted). The focus here is also on rich conversation to get more familiar with the content standards and see how they could be addressed in a problem.

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. Facilitator Information: These standards do NOT align to the Missing Function task. Briefly discuss these standards, only if time permits. 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 ͌ t 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. Facilitator Information: These standards do NOT align to Missing Function task. Briefly discuss these standards only if time permits. 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. Probing Facilitator Questions and Possible Responses: What do the standards on the slide say? How do they differ from each other? (Give participants a minute to read the standards.) They both are concerned with characteristics of polynomials, but the one standard specifically references the Remainder Theorem while the other relates the zeros of a polynomial to its graph. How does the work we did in this task give students insight into the relationship between the zeros of a function and its graph? The zeros are one of the key characteristics that the students are identifying. We connected the algebraic representation to the graphical representation and this included the zeros. 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. Probing Facilitator Questions and Possible Responses: How did we address F-BF.A.1? We determined the algebraic representation of the functions. How did this task specifically help students develop conceptual understanding of this standard? We used multiple representations. Students connected the graph to the table to the equation. Students were asked to write characteristics of a function and not simply write the equation, so this will help bring out the connections. Reminder: Students might demonstrate other content standards when solving the task, but we are determining which content standards the task is specifically designed to assess AS THE TASK IS WRITTEN (not our view of all of the ways it can be interpreted). The focus here is also on rich conversation to get more familiar with the content standards and see how they could be addressed in a problem. Common Core State Standards, 2010, p. 70, NGA Center/CCSSO

For Algebra 2 Task: What math practices made it possible for us to learn?
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Directions: Show the list of standards for mathematical practice. Ask participants which of these standards for mathematical practice students will have an opportunity to use when solving the task. Probing Questions and Possible Responses: What does it mean to look for and make use of structure with respect to this task? In this problem, students make use of structure when they relate the factored form of h(x) to the linear functions f(x) and g(x) that are factors of h(x). The parabola has two zeros, so this structure determines the algebraic representation. Attend to precision means more than just getting the correct answer. What would it mean in this problem? It means performing calculations correctly, but also attending to the meaning of words and being able to talk about the intercepts and rate of change, and use the underlying mathematical meanings of these words to approach this problem. 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 x- intercepts 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.) (SAY) When I was working with you I was watching and listening for your understanding of the content standards. When I heard essential understandings such as these related to the function, then I knew you understood the concept. Read just one of the bullets, bullet #3. Why do I need to know this level of detail about a quadratic function in order to determine if you understand the structure behind quadratics?

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. Facilitator Information: This slide is here for you to reference during the group discussion.

For Geometry: Building a New Playground
PART A  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. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B 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. Describe a strategy for determining a point equidistant from any three points. Facilitator Information: This slide is here for you to reference during the group discussion.

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? (SAY) Which of the standards for mathematical content did students have an opportunity to make sense of? Facilitator Information: These are the standards that relate to the Build a New Playground task! Discuss why using the slides that follow with the standards on them. G-GPE.B.4 G-GPE.B.5 G-GPE.B.6 Note: Students might demonstrate other content standards when solving the task, but we are determining which content standards the task is specifically designed to assess AS THE TASK IS WRITTEN (not our view of all of the ways it can be interpreted). The focus here is also on rich conversation to get more familiar with the content standards and see how they could be addressed in a problem.

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. Facilitator Information: These standards do NOT align to the Playground task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: Many groups observed or constructed congruent triangles. Did any of you use the relationships in these standards to determine the triangles were congruent. Not really. I used SAS to determine that the triangles I drew were congruent, but I was not thinking in terms of rigid motions. This will be a shift for me in terms of how I think about and how I teach congruence. 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. G-CO.C.11 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. Facilitator Information: These standards do NOT align to the Playground task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: What do you notice about the standards on this slide? What are the implications for your classrooms? Prove, prove, prove… Students are expected to prove these relationships. The implication for my learning and my classroom is that I need to figure out what the CCSS defines as proof and provide opportunities for all of my students to construct proofs. 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. Facilitator Information: These standards do NOT align to the Playground task. Briefly discuss these standards, only if time permits. Probing Facilitator Questions and Possible Responses: The CCSS Key Shift of Rigor indicates that conceptual understanding, procedural skill and fluency, and application should be pursued with “equal intensity.” How does this cluster of standards support this Key Shift? These standards ask students to understand, explain and use. The verbs imply all three components of the key shift. Common Core State Standards, 2010, p. 77, NGA Center/CCSSO

The CCSS for Mathematical Content CCSS Conceptual Category – Geometry

For Geometry Task: Which Standards for Mathematical Practice made it possible for us to learn?
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Probing Questions and Possible Responses: What does it mean to reason abstractly and quantitatively with respect to this problem? In this situation, students might abstract information from the diagram to determine the coordinates of A and B. They then work with that information, without reference to the actual problem to determine the distance of a third point from A and B, then return to the diagram to consider the situation as a whole and make some generalizations. Attend to precision means more than just getting the correct answer. What would it mean in this problem? In this case, students must attend to precision of language when describing the relationships they observe. Even if they do not know the vocabulary term “perpendicular bisector,” they should be able to describe lines, right angles, segments, midpoints and endpoints in such a way that it is clear that they are describing the perpendicular bisector of segment AB. How do students look for and express regularity in repeated reasoning in this problem? Students first find several locations for the park using a method repeatedly. Then they observe patterns to see that these points all lie on the same line. Finally, they express regularity by making a conjecture that the points all lie on the perpendicular bisector of segment AB. Common Core State Standards for Mathematics, 2010

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. (SAY) When I was working with you I was watching and listening for your understanding of the content standards. When I heard essential understandings such as these related to the use of coordinate geometry, then I knew you understood the concept. Read just one of the bullets, bullet #3. Why do I need to know this level of detail about coordinate geometry in order to determine if you understand the structure behind the relationships?

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 happen in conversation. (SAY) Read the Tharp and Gallimore Quotes. What are Tharp and Gallimore saying? Probing Facilitator Questions and Possible Responses: What does it mean that teachers must “be aware of the students’ ever changing relationships to the subject matter”? In order to say, “I was teaching,” you need to know if you were monitoring and advancing learning. Why is talk critical in this process? Who ever talks the most learns the most. If you don’t hear from students, then you don’t now what they know or don’t know. Tharp & Gallimore, 1991

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: The impact of teacher implementation of a high level task on student learning Using assessing and advancing questions to support student learning

Supporting Rigorous Mathematics Teaching and Learning
Part 1 Enacting Instructional Tasks: Maintaining the Demands of the Tasks Tennessee Department of Education High School Mathematics Overview of the Module: (SAY) Students often struggle to solve high-level tasks. Their discomfort with the struggle when they are solving the high-level task causes them to ask the teacher for assistance. In turn, the teacher provides assistance, often by doing the problem solving for the student. The students, as a result, do not have the opportunity to learn to think, reason, and to practice engaging in problem solving and communicating mathematically. This part of the module is designed to help teachers recognize when a high level task declines into the teacher doing too much thinking for the student.

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? Directions: Give participants a moment to read the slide before you ask the probing questions. (SAY) We will discuss the answers to these questions as we engage in the module.

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. Directions: Read or paraphrase the rationale.

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) Directions: Give participants a minute to read the slide. Emphasize that there are two aspects of a task to consider: The task as it is written The task as it is implemented by the teacher

as they appear in curricular/ instructional materials as set up by the teachers as implemented by students Student Learning (SAY) Research from the QUASAR Project, noted in this tool called the Mathematical Tasks Framework, indicates there are three major influences on student learning. Consider these 3 phases in our next activity. 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.

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 (SAY) These cases are examples of six ‘episodes of teaching’ – a snapshot of what occurred during classroom instruction. Norms: When analyzing episodes of teaching, which may be narrative cases, video clips, etc., keep in mind that this is only a snapshot in time. A lens for analyzing such episodes is always established prior to reading or viewing it. Although there might be interest in discussing a variety of things that are read or seen in this case, a focus, or lens, is established so that conversations can occur around a particular aspect of teaching mathematics. For this case, our lens – what we are focusing on – is what we think students were learning in each classroom. Remember these particular situations are ones that could happen in any one of our classrooms. Use a highlighter or take notes on the case so that, during our discussion, we can all be referring to the same passage. You will have about 10 minutes to read the six cases. Be respectful of others if you finish and others are still reading. Tasks are to be compared as written, in terms of the opportunities they provide students to engage in thinking and reasoning about mathematics.

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? (SAY) Remember, when contributing to a discussion, identify the case to which you are referring. Since any episode of teaching is only a snapshot in time, no judgments or evaluations about the teacher can be made. The goal is not to “fix” the teacher or the teaching – it is to learn from what occurred during the episode. Facilitator Note: Make certain the discussion focuses on students’ opportunities to learn in the classrooms and not on what teachers are doing in the vignettes. If participants begin to make judgments about the teacher's work, rephrase participants’ responses asking, "Are you saying the opportunities differed in each room?" Ask participants to identify specifics in the paragraph that make them claim certain things.

The Enactment of the Task (Whole Group Discussion)
What opportunities did students have to think and reason in each of the classes? Probing Questions and Possible Responses: Each teacher used the same task. Which classroom had the greatest opportunity to think and reason? Mrs. Fox gave students opportunities to work in small groups. Mrs. Fox did not interrupt students if they were on the right path. Mrs. Fox asked pressing questions. Facilitator Note: Press participants to elaborate on the kinds of questions asked. You might have participants read the questions out loud so others can hear them. Ask participants if they notice how specific the questions are, and if this surprises them. Marking: Note that the questions help students to make conceptual connections. Ask participants what this means. What do you think of the use of the hexagons in Mrs. Fox’s classroom? Marking: We refer to this as a means of scaffolding.) How do the learning opportunities in Mrs. Fagan's classroom differ from the learning opportunities in Mrs. Fox's classroom? Seems like they are similar? How many people think they are alike? I think they are different because Mrs. Fagan guides students a little more. Does this matter? Yes, because students do not have an opportunity to do the figuring out. She gives them the pathway. Do the students have to do any thinking and reasoning in Mrs. Fagan's classroom? Yes, students still have to think about how and why the perimeters are changing. What is going on in Mr. Cooper's classroom? The teacher gives students a procedure. There is no press for mathematical meaning. What happens in Ms. Jackson’s classroom? Students are off-task. They get the markers and focus on drawing. Students do not do any mathematics. What is going on in Mr. Chambers’ classroom? Students are off-task and the teacher does not intervene..

Research Findings: The Fate of Tasks

Linking to Research/Literature: The QUASAR Project

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands
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. (SAY) These factors were apparent in Mr. Cooper's, Mrs. Jackson’s, Mr. Chambers’ room and/or, to some degree, even Mrs. Fagan's classroom. What might have happened in Mrs. Fagan's room? CHALLENGES BECOME NON-PROBLEMS: Mrs. Fagan was not comfortable with the struggle her students were experiencing. Therefore, she gave in to their pressure and removed the challenge by telling them how to do the problem. FOCUS SHIFTS TO CORRECT ANSWER: Mr. Cooper was focused on students arriving at the correct solution so that they would feel successful. NOT ENOUGH TIME: Mrs. Fagan was also uncomfortable with the amount of time it was taking for students to wrestle with the challenges of the task.

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands
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. (SAY) Do you recognize these factors in the cases that we read?

Linking to Research/Literature: The QUASAR Project

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 Directions: Give participants a minute to read the slide.

Supporting Rigorous Mathematics Teaching and Learning
Part 2 Illuminating Student Thinking: Assessing and Advancing Questions Tennessee Department of Education High School Mathematics Overview of the Module: This part of the module is designed to help teachers learn an alternative method of assisting student learning—asking assessing and advancing questions instead of telling students the steps and procedures that they can follow.

The Structures and Routines of a Lesson

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 Directions: Work in subject area groups to analyze student work as described on the slide

Bike and Truck Task – Algebra 1
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. Distance from start of road (in feet) Time (in seconds) Directions: Remind participants that they have themselves engaged in this task; refer to any work in the room that was done with the task. Tell participants that we will be examining student work from this task.

Bike and Truck Task - Algebra 1
Label the graphs appropriately with B(t) and K(t). Explain how you made your decision. Describe the movement of the truck. Explain how you used the values of B(t) and K(t) to make decisions about your description. 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. 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.

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. Give the groups time to discuss what each student knows and what they can do for Students A-C in their subject area. Then facilitate a discussion using the notes on slides 54-56: Facilitator Note: Display student work A, B, and C in the room. Hang one piece of poster paper under each piece of student work with the label “The student knows…” If participants make inferences about what the student knows, then press participants to give you evidence from the student work to support their claims. Directions: Small Group Work: Take 10 minutes to identify what students know and can do. Group Discussion: What does each student know and what can s/he do? Let us know how you decided that the student knows x or y or z.

Response A - Algebra 1 Facilitator Note:
The goal of this part of the session is to help participants become aware of the difference between evidence-based statements and inferences. Directions: Lead a group discussion of evidence-based observations related to each piece of student work. Record participants’ responses to the question, “What does the student know and what can he do?” Probing Facilitator Questions and Possible Responses Related to Response A’s Work: What does the student know and what can he do? The student has labeled B(t) and K(t) correctly on the graph. The student associates the x-axis with the “domain.” The student associates the y-axis with the “range.’” The student associates “steady speed” with “constant.’” 54

Response B - Algebra 1 Probing Facilitator Questions and Possible Responses Related to Response B’s Work: What does the student know and what can he do? The student has labeled B(t) and K(t) correctly on the graph. The student indicates that the truck was the first to reach 300 feet, and there is a vertical line at 18 seconds that may be associated with this decision, though nothing in the student writing specifically says this. 55

Response C - Algebra 1 Probing Facilitator Questions and Possible Responses Related to Response C’s Work: What does the student know and what can he do? The student indicates that the truck “will stop at lights.” The student indicates that the bike is going ‘”straight up,” a possible reference to the straight line. There is, however, a question mark that indicates uncertainty about the issue. The student indicates that the truck was the first to reach 300 feet because “truck got there at 18 sec.” 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 f(x) -2 -1 1 2 3 4 Directions: Remind participants that they have themselves engaged in this task; refer to any work in the room that was done with the task. Tell participants that we will be examining student work from this task.

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. Give the groups time to discuss what each student knows and what they can do for Students A-C in their subject area. Then facilitate a discussion using the notes on slides 59-6. Facilitator Note: Display student work A, B, and C in the room. Hang one piece of poster paper under each piece of student work with the label “The student knows…” If participants make inferences about what the student knows, then press participants to give you evidence from the student work to support their claims. Directions: Small Group Work: Take 10 minutes to identify what students know and can do. Group Discussion: What does each student know and what can s/he do? Let us know how you decided that the student knows x or y or z.

Response A – Algebra 2 Facilitator Note: The goal of this part of the session is to help participants become aware of the difference between evidence-based statements and inferences. Directions: Lead a group discussion of evidence-based observations related to each piece of student work. Record participants’ responses to the question, “What does the student know and what can he do?”. Probing Facilitator Questions and Possible Responses Related to Response A’s Work: What does the student know and what can he do? The student recognizes the values in the table as points on the graph. The student can graph the points. The student can write a linear equation, y = x + 2, describing the values, but does not explain the thinking behind the equation. The student describes x-values where the graph crosses the x-axis. 59

Response B – Algebra 2 Probing Facilitator Questions and Possible Responses Related to Response B’s Work: What does the student know and what can he do? The student can graph the points noted in the table, and connects them with a line segment. The student can write a linear expression, x + 2, describing the values, but does not explain the thinking behind the expression. The student labels the points where the graph cross the x-axis. The student correctly extends the table to include values for g(x) and h(x). 60

Response C – Algebra 2 Probing Facilitator Questions and Possible Responses Related to Response C’s Work: What does the student know and what can he do? The student can graph the points noted in the table, and connects them with a line. The student includes the linear expression x + 2 in the product (x + 2)(x + 1). The student explains some of the thinking behind the expression x + 1, e.g., line, positive, increases by 1 and graph is infinite The student notes the x-intercepts of the parabola, but does not describe them as being on the graph. The student can graph y = x + 1. The student can expand and simplify the product (x + 2)(x + 1). 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. Directions: Remind participants that they have themselves engaged in this task; refer to any work in the room that was done with the task. Tell participants that we will be examining student work from this task.

Building a New Playground - Geometry
PART A  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. Make a conjecture about the location of all points that are equidistant from A and B. Prove this conjecture. PART B 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. Describe a strategy for determining a point equidistant from any three points.

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. Give the groups time to discuss what each student knows and what they can do for Students A-C in their subject area. Then facilitate a discussion using the notes on slides 65-67 Facilitator Notes: Display student work A, B, and C in the room. Hang one piece of poster paper under each piece of student work with the label “The student knows…” If participants make inferences about what the student knows, then press participants to give you evidence from the student work to support their claims. Directions: Small-Group Work: Take 10 minutes to identify what students know and can do. Group Discussion: What does each student know and what can s/he do? Let us know how you decided that the student knows x or y or z.

Response A - Geometry 65 Facilitator Notes:
The goal of this part of the session is to help participants become aware of the difference between evidence-based statements and inferences. Directions: Lead a group discussion of evidence-based observations related to each piece of student work. Record participants’ responses to the question, “What does the student know and what can he do?”. Probing Facilitator Questions and Possible Responses Related to Response A’s Work: What does the student know and what can he do? The student can find the midpoint of a line segment. The student can find two points equidistant from A and B by forming a square. The student can label points on his/her diagram. The student knows the equidistant points are on the perpendicular bisector. How do you know the student is aware that other points on that line drawn are equidistant from A and B? 65

Response B - Geometry Probing Facilitator Questions and Possible Responses Related to Response B’s Work: What does the student know and what can he do? The student can find three points equidistant from A and B. The student is forming right triangles with legs equal in measurement to find equidistant points. The student recognizes that the lengths of the segments can be found using the Pythagorean Theorem. How do you know if the student is aware of those facts? How can we be sure? The student knows segments QM and AB are perpendicular and proves they are using slope. 66

Response C - Geometry Probing Facilitator Questions and Possible Responses Related to Response C’s Work: What does the student know and what can he do? The student can use a compass with radius AB to find several points equidistant from A and B. The student connects the points s/he found. The student can write about his/her thinking, but some of the writing lacks clarity. 67

Group D - Cannot Get Started

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 a Bridge to Practice #3: Remind participants that we analyzed which content standards and essential understandings apply to these tasks as the previous Bridge to Practice #2 and this should help in focusing the discussion over assessing and advancing questions. Questions teachers develop should press students toward demonstrating their understanding of the content standards.

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 You will write questions that you might ask students if you want to assess and advance student learning. After everyone has written questions, the whole group will analyze the set of questions in the next Study Group session. Our ultimate goal, which will be completed in the next Study Group session, is to identify a set of characteristics that describe the assessing and advancing questions—characteristics we can then use to guide us as we plan questions for other tasks.

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.

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