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Albert Shanker School for Visual and Performing Arts - I.S. 126 Alexander Angueira, Principal

CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Daniel J. Purus, Network Leader. Albert Shanker School for Visual and Performing Arts - I.S. 126 Alexander Angueira, Principal Thursday, October 13, 2011.

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Albert Shanker School for Visual and Performing Arts - I.S. 126 Alexander Angueira, Principal

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  1. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network MeetingDaniel J. Purus, Network Leader Albert Shanker School for Visual and Performing Arts - I.S. 126 Alexander Angueira, Principal Thursday, October 13, 2011

  2. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 AGENDAIntroductions and Review of Folder ContentCommon Core State Standards InitiativeNYS Department of Education Information LinksNYC Department of Education Citywide Instructions Expectations for 2011-2012 Areas of Focus – Selected Common Core Standards in MathematicsCommon Core Library – “See Student Work” – Instructional TasksSeven Steps for Aligning Mathematics Tasks to the CCLS Instructional Tasks: “75 Points” – Grade 4: Share Solution, Match to CCLS Content and Performance Standards and Rate CCLS Content and Performance Standards “Cell Phone Plans” – Grade 8 “Carol’s Numbers” – Grade 2 “The Aussie Fir Tree” – High School AlgebraReflections of Alignment ProcessClosing

  3. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 Common Core State Standards Initiative – corestandards.orghttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf (Mathematics Standards)The K- 8 Mathematics standards:The K-5 standards provide students with a solid foundation in whole numbers, addition, subtraction, multiplication, division, fractions and decimalsThe 6-8 standards describe robust learning in geometry, algebra, and probability and statistics Modeled after the focus of standards from high-performing nations, the standards for grades 7 and 8 include significant algebra and geometry contentStudents who have completed 7th grade and mastered the content and skills will be prepared for algebra, in 8th grade or after (College and Career Readiness)

  4. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 New York State Department of Education Common Core Information LinksNew York State Department of Education http://www.p12.nysed.gov/ciai/common_core_standards/P-12 Common Core Learning Standards Toolkit http://www.p12.nysed.gov/ciai/common_core_standards/toolkit.htmlEngageNY - http://engageny.org/ * Common Core State Standards * Math Practice Standards and Content Standards * Key Points/Instructional Shifts * Common Core State Standards Implementation Timeline

  5. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 Common Core Six Instructional Shifts in Math Shift 1 - Focus Teachers use the power of the eraser and significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards so that students reach strong foundational knowledge and deep conceptual understanding and are able to transfer mathematical skills and understanding across concepts and grades. Shift 2 - Coherence Principals and teachers carefully connect the learning within and across grades so that, for example, fractions or multiplication spiral across grade levels and students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. Shift 3 - Fluency Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions (found in the attached list of fluencies) such as multiplication tables so that they are more able to understand and manipulate more complex concepts.

  6. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 Common Core Six Instructional Shifts in Math Shift 4 - Deep Understanding Teachers teach more than “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying them to new situations. As well as writing and speaking about their understanding. Shift 5 – Applications Students are expected to use math and choose the appropriate concept for application even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly science, ensure that students are using math – at all grade levels – to make meaning of and access content. Shift 6 - Dual Intensity Students are practicing and understanding. There is more than a balance between these two things in the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in “drills” and make use of those skills through extended application of math concepts. The amount of time and energy spent practicing and understanding learning environments is driven by the specific mathematical concept and therefore, varies throughout the given school year

  7. NYSED Common Core State Standards (CCSS) for ELA/ Literacy, Math, and the Arts Implementation Timeline SY 2011- 2012 • Recommend initial phase of CCSS implementation: every teacher is delivering at least one CCSS aligned unit each semester. • Math and ELA tests continue to be aligned with 2005 Standards • Vendors who will collaborate with NYS educators to develop curricular modules in ELA, Math, and the Arts are chosen between September and January and their submissions (several exemplary units) are immediately made available to the field Summer 2012 • The first 50% of ELA, Math, and Arts exemplary modules are in the field from the vendor. The nature and number of these modules is linked w/the common core-aligned tests to come in 12-13. • Additional training on CCSS occurs • Intensive training on the nature of the transition in the SY 12-13 tests occurs SY 2012-2013 • Phase II of CCSS implementation happens in schools • NYS tests aligned to CCSS in ELA and Math begin Summer 2013 • Full menu of ELA, Math, and Arts exemplary units are available • Ongoing training on curriculum and testing SY 2013-2014 • Phase III of CCSS implementation happens in schools • NYS Tests aligned to CCSS in ELA and Math Summer 2014 • Ongoing training on CCSS implementation • Intensive training on PARCC assessments (if adopted by the Board of Regents) SY 2014-15 • Full implementation of CCSS • PARCC assessments (if adopted by the Board of Regents)

  8. CITYWIDE INSTRUCTIONAL EXPECTATIONS for 2011 -2012FINAL June 1, 2011*Strengthening student work by examining and refining curriculum, assessment, and classroom instruction; and*Strengthening teacher practice by examining and refining the feedback teachers receive. *Rigorous Curriculum and Tasks for All Students *In teams, look closely at current student work to understand the steps needed to reach the level of performance that the Common Core demands (spring/fall 2011). *Engage all students in at least one literacy task and one math task aligned to strategically selected Common Core standards. These tasks should be embedded in Common Core-aligned curricula and include multiple entry points for all learners, including students with disabilities and English language learners (winter 2011-12). – In literacy, students will complete a task that asks them to read and analyze informational texts and write opinions and arguments in response. – In math, students will engage in a cognitively demanding mathematics task that requires them 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 (spring 2012).

  9. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  10. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 Rationale for Selected Common Core Standards in MathematicsThe Standards for Mathematical Practices are one of the most distinct portions of the Common Core standards. Taken together, these standards paint a picture of a mathematically proficient student. We understand that the Standards for Mathematical Practice are interconnected and that a rich task may require students to demonstrate many of the practices. However, for the winter task, we ask that schools focus on one or both of the high-leverage practices we’ve selected—Model with Mathematics (#4) and Construct Viable Arguments and Critique the Reasoning of Others (#3)—and the domains of focus representing key content in each grade. Modeling is what real mathematicians do: using mathematics to represent and solve authentic problems in our world. Key to this process of problem solving is the ability to clearly construct and analyze mathematical arguments, evaluating and articulating the reasoning behind claims. Facility with these two practices, and solid knowledge of key concepts, will enable students to demonstrate their mathematical thinking by successfully analyzing authentic problems and constructing and defending logical paths to solving problems.

  11. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 New York City Department of Education - Common Core Library http://schools.nyc.gov/Academics/CommonCoreLibrary/default.htm“See Student Work” - Instructional Tasks Each Common Core-Aligned Task is accompanied by a variety of instructional supports, including: Culminating performance tasks Guidance around use of Universal Design for Learning (UDL) principles for each task Rubrics and/or scoring guides Annotated student work ranging across levels Unit outlines developed using the Understanding by Design (UBD) framework, aligned to CCLS Standards Supports for helping students access complex texts Additional assessments and learning activities (e.g. initial, formative assessments) Task-specific supports for ELLs and Students with Disabilities Steps for Aligning Mathematics Tasks to the Common Core Learning Standards (CCLS)

  12. SEVEN STEPS FOR ALIGNING MATHEMATICS TASKS TO THE CCLS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 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. 12

  13. SEVEN STEPS FOR ALIGNING MATHEMATICS TASKS TO THE CCLS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • Step 7.In addition to the alignment of the task with the CCCS, the following considerations should be part of your overall review. • Task Improvement • Teacher Support Materials • Student Support Materials • Appropriateness of Age and Content • Source of Challenge • Effective Instruction/Assessment 13

  14. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 Common Practice Task: 75 POINTS – Grade 4 The diagram below shows the plan for some empty lots in a neighborhood. They are formed with the same properties as your pattern blocks. (This means that the trapezoids are congruent and the triangle, parallelograms, and hexagons are equilateral.) Mike and Juan are going to clean the lots by removing papers, bottles, and cans. Each lot they clean will earn them points in a contest. If they earn a total of at least 75 points they will win a prize. a) If Lot D is worth 20 points to clean up, what is the fair value of the other lots? Show all your mathematical thinking. b) If Mike and Juan clean up all the lots, will they have earned enough points to win the prize? Justify your answer using your mathematical thinking. 14

  15. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 POINTS • What was your solution? • What content was required to solve the task? • What performances were required to solve the task?

  16. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – Solution a) Lot A = Lot D = 20 points Lot B = 1/6 of 20 = 20/6 or 10/3 or 3 1/3 points Lot C = Lot F = 1/3 of 20 or 20/3 or 6 2/3 points Lot E = Lot G = ½ of 20 or 10 points b) YES – Lots A and D together = 40 points Lot B = 3 1/3 points Lots C and F together = 12 4/3 or 13 1/3 points Lots E and G together = 20 points TOTAL = 76 2/3 points 76 2/3 > 75 points 16

  17. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – Match Content and Performance to CCLS Which of the Standards of Mathematical Practice (processes) align with the requirements of the task? Which CCLS standards align with the requirements of the task? 17

  18. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – CCLS 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.6 Attend to precision. MP.7 Look for and make use of structure. 3.G.2 Partition shapes into parts with equal areas. … 4.G.2 Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, … 4.NF.3c Add and subtract mixed numbers with like denominators… 4.NF.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators … 4.NF.2 Compare two fractions with different numerators and different denominators, … 4.NF.4c Solve word problems involving multiplication of a fraction by a whole number, … 18

  19. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – 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). 19

  20. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – 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). 20

  21. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – CCLS Math Practices (processes) for Demonstration: MP.1 Make sense of problems and persevere in solving them. MP.3 Construct viable arguments and critique the reasoning of others. MP.6 Attend to precision. 21

  22. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 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).

  23. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 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).

  24. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  25. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  26. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 75 Points – Grade Level Standards for Demonstration: 4.NF.3c Add and subtract mixed numbers with like denominators… 4.NF.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators … 4.NF.4c Solve word problems involving multiplication of a fraction by a whole number, … 26

  27. 75 POINTS – ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 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).

  28. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  29. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  30. 75 POINTS ALIGNMENT WITH THE CCLS MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  31. PRACTICING WITH A COMMON TASK CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • Cell Phone Plans– Grade 8 • Carol’s Numbers– Grade 2 • The Aussie Fir Tree – High School

  32. CELL PHONE PLANS – 8th Grade CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 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 = 60 + 0.05t 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. b. How much will Ashley’s plan cost for the same number of texts as when Jumel’s costs $75.00? c. Explain in writing how you know if there is a number of texts for which both plans cost the same amount.

  33. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 CELL PHONE PLANS • What was your solution? • What content was required to solve the task? • What performances were required to solve the task?

  34. CELL PHONE PLANSSOLUTION AND CONTENT AND PERFORMANCES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • A: C = 45+.35t; J: C = 60+.05t • For 30 texts, Ashley’s plan costs $55.50 and Jumel’s costs $61.50. 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

  35. CELL PHONE PLANSALIGNMENT WITH MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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)

  36. CELL PHONE PLANSALIGNMENT WITH CCLS CONTENT STANDARDS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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)

  37. CELL PHONE PLANSALIGNMENT WITH CCLS CONTENT STANDARDS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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)

  38. CAROL’S NUMBERS - Grade 2 CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011

  39. CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 CAROL’S NUMBERS • What was your solution? • What content was required to solve the task? • What performances were required to solve the task?

  40. CAROL’S NUMBERSSOLUTION AND CONTENT AND PERFORMANCES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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. 3) 85? 4) 21? 5) 31? 0 213142 85 • Content: Place value, three-digit numbers, number line • Performances: Represent, compare, read and write, understand

  41. CAROL’S NUMBERSALIGNMENT WITH MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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)

  42. CAROL’S NUMBERS ALIGNMENT WITH THE CCLS CONTENT STANDARDS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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)

  43. THE AUSSIE FIR TREE – High School Algebra CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • Consider the following function that generates the geometric pattern of a reverse growing fir tree. • Stage 1 - 2 unit squares • Stage 2 - 6 unit squares • Stage 3 - 12 unit squares • Stage 4 - ?? • Draw and describe Stage 5 of the pattern in terms of its shape and number of unit squares needed to construct the fir tree. • Describe how the pattern is growing. • How many unit squares are needed to build a Stage 10 Aussie Fir Tree? Show your work. • Given any stage number n, determine a closed form equation to determine the amount of unit squares needed to build the tree. • 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.

  44. THE AUSSIE FIR TREE CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • What was your solution? • What content was required to solve the task? • What performances were required to solve the task?

  45. THE AUSSIE FIR TREESOLUTION CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 1) 30 unit squares • 2) It grows by consecutive even numbers • 3) 110 unit squares • 4) n(n + 1) or n2 + n • 5) Because the quadratic equation n2 + n = 274 cannot have rational solutions (by quadratic formula, factoring, etc.)

  46. THE AUSSIE FIR TREECONTENT & PERFORMANCE CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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

  47. THE AUSSIE FIR TREEALIGNMENT WITH MATH PRACTICES CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • 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: 3) • MP.4 Model with mathematics. (Rating – Performance: 3) • MP.6 Attend to precision. (Rating – Performance: 3) • MP.7 Look for and make use of structure. (Rating – Performance: 3)

  48. THE AUSSIE FIR TREE ALIGNMENT WITH THE CCLS CONTENT STANDARDS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • F.BF.1 Write a function that describes a relationship between two quantities.★ (Rating – Content: 3, Performance: 3) • F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. ★ (Rating – Content: 3, Performance: 3) • F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (Rating – Content: 3, Performance: 3) • A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions.(Rating – Content: 3, Performance: 3)

  49. THE AUSSIE FIR TREEALIGNMENT WITH THE CCLS CONTENT STANDARDS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • A.REI.4 Solve quadratic equations in one variable.(Rating – Content: 3, Performance: 3) • 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.Recognizewhen the quadratic formula givescomplex solutions andwritethem as a ± bi for real numbers a and b. (Rating – Content: 3, Performance: 2)

  50. REFLECTIONS ON THE ALIGNMENT PROCESS CFN 208 Coach/Assistant Principal/Teacher-Leader Mathematics Network Meeting Thursday, October 13, 2011 • Turn and Talk • How do content and performance alignment contribute to your understanding of the CCLS? • How does this activity influence how you will work within your schools to develop teacher practice? • How does the alignment process apply to your work within your schools? • How does the alignment process inform how you think about improving student work through strengthening teacher practice? 50

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