Building Students’ Deep Understanding through a Common Core-Aligned Unit Course Goals: Participants will complete this course with  A revised unit outline that it is Common Core-aligned, coherent, and rigorous  Experience planning and teaching a lesson with a focus on opportunities for mathematical talk 1

Building Students’ Deep Understanding through a Common Core-Aligned Unit Session 1 – Setting the Stage  Reflect on planning and pedagogical practices in context of the CCLS instructional shifts in mathematics  Use a protocol to analyze the alignment of a unit’s culminating task with the targeted standards of the unit Session 2 – Rigorous Tasks as Unit Drivers Session 3 – Planning with the Instructional Shifts in Mind Session 4 – Supporting the Instructional Shifts through Mathematical Talk Session 5 – Bringing it all Together 2

Building Students’ Deep Understanding through a Common Core-Aligned Unit Session 1 – Setting the Stage 1.Course Overview, Logistics, and Introduction 2.Reflecting on Common Core Work 3.Activity 1: Understanding how the Instructional Shifts in Mathematics impact Our Work 5.Activity 2: Aligning a Unit’s Culminating Task to the CCLS 6.Reflection, Strategies for Sharing, Bridge to Practice, and Feedback Forms 3

The Common Core State Standards Initiative  Goal: Prepare students to graduate from high school prepared for success in college and careers  45 states and the District of Columbia have fully adopted the Common Core Standards for ELA & Mathematics 4

Transitioning to the Common Core is a Multi-Year Process  NYS adopts Common Core standards 5 5  NYS integrates Common Core into State tests; NYS releases curriculum materials; all schools implement broader and deeper citywide instructional expectations *New York is part of a consortium of states, the Partnership for the Assessment of Readiness for College and Careers (PARCC), working together to develop new state assessments. SY SY SY SY SY SY  NYC students take State PARCC* assessments  Common Core Pilots (100 schools)  All schools implement citywide instructional expectations  All schools implement citywide instructional expectations as final part of transition to the Common Core

Citywide Instructional Expectations These expectations build on the inquiry work of the last several years. Strengthening student work by examining and refining curriculum, assessment, and classroom instruction; Strengthening teacher practice by examining and refining the feedback teachers receive. 6

Teacher teams analyze student work samples in relation to selected Common Core standards Schools understand current state of teacher and student work and determine how to strengthen in relation to Common Core Fall STRENGTHENING STUDENT WORK ~ EXPECTATIONS

Engage all students in a rigorous, Common Core- aligned literacy and math task embedded within a well-sequenced curricular unit: >Literacy: Read and analyze informational texts and write opinions and arguments in response >Math: Engage in a cognitively demanding task that requires students to demonstrate their ability to model with mathematics and/or construct and explore the reasoning behind arguments to arrive at a viable solution In teams, look closely at resulting student work to continue the cycle of inquiry, making future instructional adjustments and communicating lessons learned to other school staff 8 Winter/Spring 2012

 Every school adopts a research-based rubric of teacher practice  School leaders >Engage in short, frequent cycles of classroom observation and collaborative examination of student work >Provide feedback that teachers can act on to increase the effectiveness of their instruction  Teachers >Understand what is expected >Engage in ongoing reflection on their practice >Receive support to continually develop STRENGTHENING TEACHER PRACTICE ~ EXPECTATIONS

Reflection What worked well about implementing the Citywide Instructional Expectations? What was challenging? What did you learn about your students in the process? What did you learn about your own practice in the process? What lessons learned do you want to apply to your work going forward? How did your experience with the Common Core relate to your inquiry work? 10

ACTIVITY: UNDERSTANDING HOW THE MATHEMATICS INSTRUCTIONAL SHIFTS IMPACT OUR WORK 11

12 Priority shifts

Standards for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. 13

Format of K-8 Mathematics Standards 14 Each grade includes an overview of cross-cutting themes and critical domains of study… … as well as the Standards for Mathematical Practice.

Format of K-8 Mathematics Standards 15 > Domains: overarching ideas that connect topics across the grades > Clusters: illustrate progression of increasing complexity from grade to grade > Standards: define what students should know and be able to do at each grade level

Format of High School Mathematics Standards Each conceptual category includes an overview of its domains of study… 16 …as well as the Standards for Mathematical Practice.

Reflection  What connections do you notice between the selected instructional shifts in mathematics, the selected standards for mathematical practice, and the content standard you examined? 17

18 Implications for Students and Teachers 18 If students have to… (know and be able to) Then teachers have to… Solve words problems and reflect on the reasonableness of responses Design tasks that have multiple entry points and opportunity for reflection

Implications for Students and Teachers  If Students Have To… 19  Then Teachers Have To…

Reflection  What are your key learnings from this activity?  What structures and supports must teachers have in place to support students in meeting the demands of the Common Core?  What structures and supports must schools have in place to support teachers in meeting the demands of the Common Core 20

ACTIVITY #2: ALIGNING A UNIT’S CULMINATING TASK WITH THE CCLS 21

What do you consider to be the important components of a unit plan in math? 22

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Unit Outline Components Overarching Questions: The big inquiries of a unit. These questions reach across and connect content and skill in the unit. Each experience allows students to deepen responses to the overarching questions. Primary CC Content Standards addressed in the Unit: What content, skills, and performances will be the major focus for this unit? Primary CC Mathematical Practices addressed in the Unit: Which Practices lend themselves to deepening conceptual understanding and fluency in math for this unit? 24

Unit Outline Components  Sequenced Activities: Activities cohere around the same content/performances as assessment task. Activities a) allow for CCLS-based work that parallels assessments in smaller chunks; b) are sequenced to prompt retrospective work; and c) provide a model of the kind of writing, skills, and performances required by the culminating assessment.  Instructional Tasks: Provide opportunities for students to a) learn/experience/build their content knowledge b) develop skills embedded in the Math Practices; c) are sequenced to move learners from basic procedural knowledge to higher level thinking about math; and d) mirror the assessment tasks in order to provide support for students to learn the content, habits, and skills they need to successfully and independently complete the unit's culminating task. 25

Unit Outline Components  Embedded Assessments: List the ways in which you will assess student learning during and after the instructional task.  Culminating Assessment Task: How can the content, skills, and performances in the unit be combined to elicit student responses that demonstrate a deep understanding of the unit standards? 26

Performance Tasks  What comes to mind when you hear the term “performance task?” 27

Step 1. Work the task thoroughly. Step 2. Compare your work with the answer key/rubric and other instructional support materials. Step 3. Identify the content and performances required. Step 4. Match the content and performances to the CCLS. Step 5. Rate the alignment of content. Step 6. Rate the alignment of performances. Step 7. Additional considerations. 28 SEVEN STEPS FOR ALIGNING MATHEMATICS TASKS TO THE CCLS

Step 5: CCLS Mathematics Content Standards: Content vs. Performance 29 CCLSContentPerformance 4.NF.3c Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

Sample Task Alignment  Grade 2 – Carol’s Numbers Grade 2 – Carol’s Numbers  Grade 8 – Cell Phone Plans Grade 8 – Cell Phone Plans  HS Algebra – Aussie Fir Tree HS Algebra – Aussie Fir Tree 30

Common Practice Task: Carol’s Numbers – Grade 2 31

DEBRIEF CAROL’S NUMBERS What was your solution? What content was required to solve the task? What performances were required to solve the task? 32

DEBRIEF CAROL’S NUMBERS: SOLUTION AND CONTENT AND PERFORMANCES Solution: 1) 742 2) 247; Put the smallest number on the left, the next smallest number in the middle and the largest number last. 3-5) See number line and discuss. Content: Place value, three-digit numbers, number line Performances: Represent, compare, read and write, understand 33

DEBRIEF CAROL’S: ALIGNMENT WITH MATH PRACTICES  MP.1 Make sense of problems and persevere in solving them. (Rating – Performance: 3)  MP.3 Construct viable arguments and critique the reasoning of others. (Rating – Performance: 2)  MP.6 Attend to precision. (Rating – Performance: 3) 34

DEBRIEF CAROL’S NUMBERS: ALIGNMENT WITH THE CCLS CONTENT STANDARDS  2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. (Rating – Content: 3, Performance: 3)  2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. (Rating – Content: 3, Performance: 2)  2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. (Rating – Content: 3, Performance: 2)  2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100 on a number line diagram. (Rating – Content: 3, Performance: 2) 35

Performance Task Characteristics Analysis 36 StrengthsAreas for Improvement Suggestions for Modification

Reflection  How can you update your current units and tasks to better align with the Common Core Learning Standards? 37

Bridge to Practice Choose a unit to upgrade during the course that focuses on a piece of the major work of the grade. Evaluate and modify the unit’s culminating task to ensure that it meets the characteristics of a performance task and is aligned to the targeted standards. >Use the blank Task Alignment Grids provided. We will review them at the beginning of Session 2. 38

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Common Practice Task: Cell Phone Plans – Grade 8 40

DEBRIEF CELL PHONE PLANS  Jumel and Ashley have two of the most popular phones on the market, a Droid and an iPhone. Jumel’s monthly cell phone plan is shown below, where c stands for the cost in dollars, and t stands for the number of texts sent each month.  Jumel: c = t  Ashley’s plan costs $.35 per text, in addition to a monthly fee of $45.  a. Whose plan, Jumel’s or Ashley’s, costs less if each of them sends 30 texts in a month?  Explain how you determined your answer.  How much will Ashley’s plan cost for the same number of texts as when Jumel’s costs $75.00?  Explain in writing how you know if there is a number of texts for which both plans cost the same amount. 41

DEBRIEF CELL PHONE PLANS What was your solution? What content was required to solve the task? What performances were required to solve the task? 42

DEBRIEF CELL PHONE PLANS: SOLUTION AND CONTENT AND PERFORMANCES A: C = t; J: C = t  For 30 texts, Ashley’s plan costs $55.50 and Jumel’s costs $ Ashley’s is cheaper.  b) $150  c) If they both sent 50 text messages their plans would cost the same. Content: linear equations in two variables, simultaneous equations, real word mathematical problems Performances: solve, analyze, explain, reason 43

DEBRIEF CELL PHONE PLANS: ALIGNMENT WITH MATH PRACTICES  MP.1 Make sense of problems and persevere in solving them. (Rating – Performance: 3)  MP.2 Reason abstractly and quantitatively. (Rating – Performance: 3)  MP.3 Construct viable arguments and critique the reasoning of others. (Rating – Performance: 2)  MP.4 Model with mathematics. (Rating – Performance: 3)  MP.6 Attend to precision. (Rating – Performance: 3) 44

DEBRIEF CELL PHONE PLANS: ALIGNMENT WITH CCLS CONTENT STANDARDS  8.EE.8 Analyze and solve pairs of simultaneous linear equations. (Rating – Content: 3, Performance: 2)  8.EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. (Rating – Content: 2, Performance: 3)  8.EE.8c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. (Rating – Content: 3, Performance: 3) 45

DEBRIEF CELL PHONE PLANS: ALIGNMENT WITH CCLS CONTENT STANDARDS  8.EE.7 Solve linear equations in one variable. (Rating – Content: 3, Performance: 3)  8.EE.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. (Rating – Content: 2, Performance: 3)  8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Rating – Content: 2, Performance: 3)  8.F.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. (Rating – Content: 2, Performance: 3) 46

Performance Task Characteristics Analysis 47 StrengthsAreas for Improvement Suggestions for Modification

Bridge to Practice Choose a unit to upgrade during the course that focuses on a piece of the major work of the grade. Evaluate and modify the unit’s culminating task to ensure that it meets the characteristics of a performance task and is aligned to the targeted standards. >Use the blank Task Alignment Grids provided. We will review them at the beginning of Session 2. 48

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Common Practice Task: Aussie Fir Tree – HS Algebra 50

DEBRIEF THE AUSSIE FIR TREE  Consider the following function that generates the geometric pattern of a reverse growing fir tree.  Stage unit squares  Stage unit squares  Stage unit squares  Stage 4 - ?? 1.Draw and describe Stage 5 of the pattern in terms of its shape and number of unit squares needed to construct the fir tree. 51

DEBRIEF THE AUSSIE FIR TREE 2. Describe how the pattern is growing. 3. How many unit squares are needed to build a Stage 10 Aussie Fir Tree? Show your work. 4. Given any stage number n, determine a closed form equation to determine the amount of unit squares needed to build the tree. 5. Your mate tells you that exactly 274 unit squares will make an Aussie Fir Tree. He is wrong. Explain to him why his statement is false. 52

DEBRIEF THE AUSSIE FIR TREE What was your solution? What content was required to solve the task? What performances were required to solve the task? 53

DEBRIEF THE AUSSIE FIR TREE: SOLUTION 1) 30 unit squares 2) It grows by consecutive even numbers 3) 110 unit squares 4) n(n + 1) or n 2 + n 5) Because the quadratic equation n 2 + n = 274 cannot have rational solutions (by quadratic formula, factoring, etc.) 54

DEBRIEF THE AUSSIE FIR TREE: CONTENT & PERFORMANCE Content: Sequences, quadratic equations, functions Performances: Draw the next stage, describe the pattern’s growth, find a partial sum (for stages 5 and 10), find general formula to generate the pattern, present argument 55

DEBRIEF THE AUSSIE FIR TREE: Practices aligned with the task  MP.1 Make sense of problems and persevere in solving them.  MP.3 Construct viable arguments and critique the reasoning of others.  MP.4 Model with mathematics.  MP.6 Attend to precision.  MP.7 Look for and make use of structure. 56

DEBRIEF THE AUSSIE FIR TREE: ALIGNMENT WITH THE CCLS CONTENT STANDARDS  F.BF.1 Write a function that describes a relationship between two quantities.  F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context.  F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.  A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions. 57

DEBRIEF THE AUSSIE FIR TREE: ALIGNMENT WITH THE CCLS CONTENT STANDARDS  A.REI.4 Solve quadratic equations in one variable.  A.REI.4b Solve quadratic equations by inspection (e.g., for x^2 = 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. 58

Aussie Fir Tree – Rating Scale for Content 3 = Excellent: The content of the task is clearly consistent with the content of the identified CCLS. 2 = Good: This rating is used for a partial match. Content addressed in the task is consistent with the most critical content of the identified CCLS. However, supporting content of the CCLS may not be addressed (possibly by design). 1 = Weak: This rating is used for a partial match when the most critical content addressed in the identified CCLS is NOT addressed in the task. However, supporting content of the CCLS is addressed. 0 = No Alignment: None of the content addressed in the task matches the content of the identified CCLS. (Delete this CCLS from the list of standards identified as aligned with the task). 59

Aussie Fir Tree – Rating Scale for Performance 3 = Excellent: The performances of the task are clearly consistent with the performances of the identified CCLS. 2 = Good: This rating is used for a partial match. Performances addressed in the task are consistent with the most critical performances of the identified CCLS. However, supporting performances of the CCLS may not be addressed (possibly by design). 1 = Weak: This rating is used for a partial match when the most critical performances addressed in the identified CCLS are NOT addressed in the task. However, supporting content of the CCLS is addressed. 0 = No Alignment: None of the performance addressed in the task matches the performances of the identified CCLS. (Delete this CCLS from the list of standards identified as aligned with the task). 60

Aussie Fir Tree – ALIGNMENT WITH THE CCLS MATH PRACTICES 61 CCLSCPAlignment Comments MP.1 Make sense of problems and persevere in solving them. NA 3 = Excellent: The (content/performance) of the task is clearly consistent with the (content/performance) of the identified CCLS. 2 = Good: This rating is used for a partial match. (Content/performance) addressed in the task is consistent with the most critical content of the identified CCLS. However, supporting (content/performance) of the CCLS may not be addressed (possibly by design). 1 = Weak: This rating is used for a partial match when the most critical (content/performance) addressed in the identified CCLS is NOT addressed in the task. However, supporting (content/performance) of the CCLS is addressed. 0 = No Alignment: None of the (content/performance) addressed in the task matches the (content/performance) of the identified CCLS. (Delete this CCLS from the list of standards identified as aligned with the task).

Aussie Fir Tree – ALIGNMENT WITH THE CCLS MATH PRACTICES 62 CCLSCPAlignment Comments MP.1 Make sense of problems and persevere in solving them. NA 3 Finding the equation requires perseverance on the student’s part. 3 = Excellent: The (content/performance) of the task is clearly consistent with the (content/performance) of the identified CCLS. 2 = Good: This rating is used for a partial match. (Content/performance) addressed in the task is consistent with the most critical content of the identified CCLS. However, supporting (content/performance) of the CCLS may not be addressed (possibly by design). 1 = Weak: This rating is used for a partial match when the most critical (content/performance) addressed in the identified CCLS is NOT addressed in the task. However, supporting (content/performance) of the CCLS is addressed. 0 = No Alignment: None of the (content/performance) addressed in the task matches the (content/performance) of the identified CCLS.

Aussie Fir Tree – ALIGNMENT WITH THE CCLS MATH PRACTICES 63 CCLSCPAlignment Comments MP.3 Construct viable arguments and critique the reasoning of others N/A MP.4 Model with mathematics. N/A MP.6 Attend to precision. N/A MP.7 Look for and make use of structure. N/A

Aussie Fir Tree – ALIGNMENT WITH THE CCLS MATH PRACTICES – Expert Ratings 64 CCLSCPAlignment Comments MP.3 Construct viable arguments and critique the reasoning of others N/A3 Descriptions, explanations, and demonstrations are required in parts 1 to 3 and 5. MP.4 Model with mathematics. N/A3 The equation (function) found in part 4 is a mathematical model that generates the pattern. MP.6 Attend to precision. N/A3 Precise terms and notation are required when presenting a mathematical explanation. MP.7 Look for and make use of structure. N/A3 Finding and using patterns is a key component of this task.

Aussie Fir Tree – ALIGNMENT WITH THE CCLS CONTENT STANDARDS 65 CCLSCP Alignment Comments F.BF.1 Write a function that describes a relationship between two quantities. 3 = Excellent: The (content/performance) of the task is clearly consistent with the (content/performance) of the identified CCLS. 2 = Good: This rating is used for a partial match. (Content/performance) addressed in the task is consistent with the most critical content of the identified CCLS. However, supporting (content/performance) of the CCLS may not be addressed (possibly by design). 1 = Weak: This rating is used for a partial match when the most critical (content/performance) addressed in the identified CCLS is NOT addressed in the task. However, supporting (content/performance) of the CCLS is addressed. 0 = No Alignment: None of the (content/performance) addressed in the task matches the (content/performance) of the identified CCLS.

AUSSIE FIR TREE – ALIGNMENT WITH THE CCLS CONTENT STANDARDS 66 CCLSCPAlignment Comments F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions. A.REI.4 Solve quadratic equations in one variable. A.REI.4b Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. as a ± bi for real numbers a and b. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b.

AUSSIE FIR TREE – ALIGNMENT WITH THE CCLS CONTENT STANDARDS 67 CCLSCPAlignment Comments F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 33 Determination of the recursive and explicit processes are required in parts 1 to 3, and 4. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 33 This CCSS connects the sequence to the equation that must be created from the patterning used in the task. A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions. 33 The equation is created in part 4. A.REI.4 Solve quadratic equations in one variable. 33 In part 5 the explanation requires finding a solution (or not finding one) for the equation found in part 4. A.REI.4b Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. as a ± bi for real numbers a and b. Recognize when the quadratic formula gives complex solutions and write them as a +/- bi for real numbers a and b. 32 In part 5 the quadratic equation found in part 4 must be solved. Students may use any method and may even stop short of solving if, by inspection, they are able to determine that the solutions are irrational.

Performance Task Characteristics Analysis 68 StrengthsAreas for Improvement Suggestions for Modification

Bridge to Practice Choose a unit to upgrade during the course that focuses on a piece of the major work of the grade. Evaluate and modify the unit’s culminating task to ensure that it meets the characteristics of a performance task and is aligned to the targeted standards. >Use the blank Task Alignment Grids provided. We will review them at the beginning of Session 2. 69

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