NYC Schools Task Alignment Project The Common Core State Standards Aligning and Rating Math Tasks April 28, 2011

Alignment and Rating Protocol Consider these steps for the task, The Aussie Fir Tree 1.Work the task. 2.Compare your work with the key/rubric and other instructional support materials. 3.Identify the content, skills, and performances required. 4.Match those content, skills, and performances to the CCSS. 5.Rate Content Centrality. 6.Rate Performance Centrality. 7.Record your comments regarding any partial matches or possible task adaptations.

Step 1: Work the task. 1.Carefully work through the task from start to finish, keeping a record of your work and your reasoning. 2.Think about how children at the target grade level would solve the problem. 3.Make notes about issues you see in the task and about required skills. 4.Identify the subjects (concepts/content) and the actions (performances) needed to answer all the questions.

Step 1: The Aussie Fir Tree – Work the Task The solutions for the five separate parts of the task include drawing the next stage, description of the pattern, determination of the 10 th term, finding a closed form equation for the pattern, applying the equation to answer a question. Students in high school are likely to use the progression in the task to solve it and answer the questions. Discuss solutions and notes. The solutions for the five separate parts of the task include drawing the next stage, description of the pattern, determination of the 10 th term, finding a closed form equation for the pattern, applying the equation to answer a question. Students in high school are likely to use the progression in the task to solve it and answer the questions. Discuss solutions and notes.

Step 2: Use Instructional Support Materials After working the task compare your results to the scoring rubric or other support materials (e.g. lesson overview, unit guide, guidelines for administration, etc). Make note of any alternative strategies that may be present in the rubric along with any additional content and performances required.

Step 2: The Aussie Fir Tree – Support Materials Unfortunately there are no support materials included with this task. Having, at the minimum, an answer key and scoring rubric is essential for this process. Without supporting documents the rater can only guess at the intention (level of rigor, strategies, solution path) of the task designer was. Unfortunately there are no support materials included with this task. Having, at the minimum, an answer key and scoring rubric is essential for this process. Without supporting documents the rater can only guess at the intention (level of rigor, strategies, solution path) of the task designer was.

Step 3a: Identify Content Consider the larger mathematical topics addressed in the task (e.g. number systems, functions, probability, geometry, etc). Also consider the other “nouns,” some of which may relate to the larger topic (e.g. rational numbers, proportional relationships, counting techniques, similarity, etc).

Step 3a: The Aussie Fir Tree - Identify Content Sequences is the larger mathematical topic for this task. Other sub-topics include the “nouns” of the task: patterns* and closed form equation. Other content can be found in the working of the task. For example: – The closed form equation represents a function. – The equation is a quadratic, making quadratic equations part of the content. (*Note: In the prompt there may be an issue with using the words “geometric pattern.” Students with experience with sequences and series may be led to believe this sequence is geometric, when it is not.) Sequences is the larger mathematical topic for this task. Other sub-topics include the “nouns” of the task: patterns* and closed form equation. Other content can be found in the working of the task. For example: – The closed form equation represents a function. – The equation is a quadratic, making quadratic equations part of the content. (*Note: In the prompt there may be an issue with using the words “geometric pattern.” Students with experience with sequences and series may be led to believe this sequence is geometric, when it is not.)

Step 3b: Identify the Performances Ask yourself what does the student need to be able to DO to correctly answer the questions and explain or justify the answers? Identify those “verbs,” or actions required in the task. Also identify which Mathematical Practices relate to the task.

Step 3b: The Aussie Fir Tree – Identify Performances The visible “verbs” of the task include draw, describe, determine, and explain. Imbedded performances include create or write an equation and solve a quadratic equation. (This particular equation is not factorable.) In addition there are the imbedded processes that relate to this task: problem solving, seeing and using patterns, justifying solutions, modeling with an equation, communicating precisely. The visible “verbs” of the task include draw, describe, determine, and explain. Imbedded performances include create or write an equation and solve a quadratic equation. (This particular equation is not factorable.) In addition there are the imbedded processes that relate to this task: problem solving, seeing and using patterns, justifying solutions, modeling with an equation, communicating precisely.

Step 4: Match to the Appropriate CCSS Search the CCSS for the content and performances required in the task.* At first limit yourself to the grade level assigned to the task. Find the standard(s) that address all the content, skills, and processes needed to complete the task successfully. * When searching the CCSS use an electronic version so you can do word searches. Use key words to find your concepts/skills. For example when looking for standards to match a problem involving rates, you could search for “rate,” but also try “slope,” or “proportion.” Also look for key performance words like “graph,” “solve,” or “compare” to make sure you find all possible matches.

Step 4: The Aussie Fir Tree – Matching Content CCSS matching content: o Functions is one the domains of the high school CCSS. o Supporting standards under Functions include: Interpreting Functions; Building Functions; Linear, Quadratic, and Exponential Models; Trigonometric Functions. o In each of those sections search for key content words such as: patterns or sequence; closed-form, explicit, or recursive; quadratic. CCSS matching performance: o In the same domain and supporting standards sections of the CCSS search for key performance terms such as: explain or describe; determine; create or write; solve. CCSS matching content: o Functions is one the domains of the high school CCSS. o Supporting standards under Functions include: Interpreting Functions; Building Functions; Linear, Quadratic, and Exponential Models; Trigonometric Functions. o In each of those sections search for key content words such as: patterns or sequence; closed-form, explicit, or recursive; quadratic. CCSS matching performance: o In the same domain and supporting standards sections of the CCSS search for key performance terms such as: explain or describe; determine; create or write; solve.

Step 4 (cont.) : The Aussie Fir Tree - CCSS Aligned Content Standards CCSS 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. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 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.

Step 4 (cont.): The Aussie Fir Tree - CCSS Aligned Mathematical Practices * Modeling is defined in the CCSS in the High School Mathematics section in the following way: “Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. … A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object like a coin.” CCSS 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.

Step 5: Rate Content Centrality Compare the identified “nouns” of the task to each of the matched CC standards individually. Do the standard and the task clearly address the same content? Use this scale to rate Content Centrality 2 = There is clear evidence of content of the standard in the task. 1 = There is partial evidence of the content of the standard in the task. 0 = There is no evidence of the content of the standard in the task.

Step 5: The Aussie Fir Tree - Content Centrality CCSSCCComment F.BF.1 Write a function that describes a relationship between two quantities. ★ 2 The equation required in part 4 is the function that models the design. F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 2 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. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 2 This CCSS connects the sequence to the equation that must be created from the pattern. A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions. 2 The equation is created in part 4. A.REI.4. Solve quadratic equations in one variable. 2 In part 5 the explanation requires finding a solution (or not finding one) for the equation found in part 4. MP.1 Make sense of problems and persevere in solving them. NA MP. 3 Construct viable arguments and critique the reasoning of others. NA MP. 4 Model* with mathematics. NA MP. 6 Attend to precision. NA MP. 7 Look for and make use of structure. NA

Step 6: Rate Performance Centrality Compare the identified “verbs/actions” of the task to each of the matched CC standards individually. Do the standard and the task clearly address the same content? Use this scale to rate Performance Centrality 2 = There is clear evidence that the performances of the standard are required in the task. 1 = There is partial evidence that the performances of the standard are required in the task. 0 = There is no evidence of that the performances of the standard are required in the task.

Step 6: The Aussie Fir Tree - Performance Centrality CCSS PC Comment F.BF.1 Write a function that describes a relationship between two quantities. ★ 2 The equation required in part 4 is the function that models (describes) the relationship. F.BF.1a Determine an explicit expression, a recursive process, or steps for calculation from a context. 2 The recursive process is required in parts 1 to 3, the explicit expression is required in part 4. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 2 This CCSS connects the sequence to the equation that must be created from the pattern. A.CED.1 Create equations in one variable and use them to solve problem. Include equations arising from linear and quadratic functions. 2 Creation of an equation is required in part 4. A.REI.4. Solve quadratic equations in one variable. 2 In part 5 the explanation requires finding a solution (or not finding one) for when the expression found in part 4 is set equal to MP.1 Make sense of problems and persevere in solving them. 2 Finding the equation requires some perseverance on the student’s part. MP. 3 Construct viable arguments and critique the reasoning of others. 2 Descriptions, explanations, and demonstrations are required in parts 1 to 3 and 5. MP. 4 Model* with mathematics. 2 The equation (function) found in part 4 is a mathematical model that generates the pattern. MP. 6 Attend to precision. 2 Precise terms and notation are required when presenting a mathematical explanation. MP. 7 Look for and make use of structure. 2 Finding and using patterns is a key component of this MP

Step 7: Comments If any of your standards earned two 0’s, you will likely want to delete the CCSS from your group. Before you delete look at the CC standard again and remember why you selected it. Are you forgetting some component of the task when you worked through it? Look back at your work and notes for the task before you delete. Record your reasons for each rating, making notes of specific contents and performances important to the task and/or the matched CCSS.

Step 7: The Aussie Fir Tree – Comments and Issues Comments specific to content and performance centrality are included in the right column of the slides for steps 5 and 6. In the original submission there were standards matched to the task, not all of which were good matches. In your alignment work you may find that some of your ratings are 0 for either content or performance. It is also appropriate in the alignment process to make comments regarding the task itself: the interest level of the task, possible adaptations for special needs students, etc. Always make sure the CCSS are accurately stated. Comments specific to content and performance centrality are included in the right column of the slides for steps 5 and 6. In the original submission there were standards matched to the task, not all of which were good matches. In your alignment work you may find that some of your ratings are 0 for either content or performance. It is also appropriate in the alignment process to make comments regarding the task itself: the interest level of the task, possible adaptations for special needs students, etc. Always make sure the CCSS are accurately stated.