CLAIMS 1-Concept and Procedures 2-Problem Solving 3-Communicating 4-Modeling and Data Analysis Unit 3 Learning Outcomes Grade 4 MATHEMATICAL PRACTICES 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 CLUSTERS (SBAC Targets) TARGET F Major: Extend understanding of fraction equivalence and ordering. TARGET G Major: Build fractions from unit fractions by applying and extending previous understanding on whole numbers. STANDARDS 4.NF.1 Explain why a fraction a b is equivalent to a fraction (n x a) (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g. by creating common denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparison with symbols >, =, or <, justify the conclusions, e.g., by using a visual fraction model. 4.NF.3 Understand a fraction a b with a > 1 as a sum of fractions 1 b. 3a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 3b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: ⅜ = ⅛ + ⅛ + ⅛; ⅜ = ⅛ + 2 / 8 ; 2 ⅛ = 1 + 1 + ⅛ = 8 / 8 + 8 / 8 + ⅛. 3c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, 1 Benchmark Blueprint 2 SR 1 CR PT-Part B Notes In grade four, fractions include those with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. The value of n cannot equal zero. Refer to frameworks for visual representation of concept. 4 SR Key tools of comparison in 4 th grade include benchmark fractions, common denominators, and common numerators. NCTM provides an equivalent fraction tool: http://illuminations.nctm.org/activity.aspx?id=3 510 4 SR PT-Part A PT-Part C Use 3b skills to develop conceptual understanding of fractional subtraction e.g., ⅞ - ⅜ = ⅞ - ⅛ - ⅛ - ⅛ = 4 / 8 = ½ Student should develop understanding of mixed number as a sum of a whole number and a fraction (a b c = a + b c ).
TARGET B Additional/Supporting Gain familiarity with factors and multiples. (Supports Major Target E: 4.NBT.4-6 and Major Target F: 4.NF.1-2) TARGET C Additional/Supporting Generate and analyze patterns. (Supporting Major Target E: 4.NBT.4-6) and/or by using properties of operations and the relationship between addition and subtraction. 3d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 4.OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is prime or composite. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule Add 3 and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. 3 SR Supports students work with multi-digit arithmetic as well as their work with fraction equivalence. Can support using place value and properties to perform multi-digit arithmetic (TARGET E: Major) and understanding of fraction equivalence and ordering (TARGET F: Major). 4 SR PT-Part D Visual examples of pattern activities: Double Plus One and Multiples of Nine : https://www.illustrativemathematics.org/4 Can support using place value and properties to perform multi-digit arithmetic. (TARGET E: Major) Major- Areas of intensive focus where students need fluent understanding and application of the core concepts. These clusters require greater emphasis than the others based on the depth of the ideas, the time that they take to master, and/or their importance to future mathematics or the demands of college and career readiness Supporting- Rethinking and linking; areas where some material is being covered, but in a way that applies core understanding; designed to support and strengthen Additional- Expose students to other subjects; may not connect tightly or explicitly to the major work of the grade. Benchmark Item Types and points 17 Selected Responses (1 point) 1 Constructed Responses (2 points) 1 Performance Task with 4 parts (6 points) Updated 11/2015 2
Mathematical Performance Task Development Purpose: At the end of the unit, students should be able to make connections with the learning of concepts and skills, problem solving, modeling, analysis, communication and reasoning. Students should be able to perform at higher levels of complexity in their thinking and application of the math content. Mathematical tasks should Integrate knowledge and skills across multiple claims and targets Measure depth of understanding, research skills, complex analysis Take age appropriate development into consideration Task Development Professional Learning Communities will Step 1: Consider the learning targets needed to be mastered throughout the unit. Step 2: Consider the Depth of Knowledge or level of complexity that students will need to perform. (DOK 1) Recall and Reproduction (DOK 2) Skills and Concepts/ Basic Reasoning Engage students in relevant and interesting topics Have an authentic purpose and connected components Accessible to all learners (DOK 3) Strategic Thinking/ Complex Reasoning Performance Task Expectations (DOK 4) Extended Thinking/ Reasoning Recall of a fact, information or procedure Recall or recognize fact Recall or recognize definition Recall or recognize term Recall and use a simple procedure Perform a simple algorithm. Follow a set procedure Apply a formula A one-step, well-defined, and straight algorithm procedure. Perform a clearly defined series of steps Identify Recognize Use appropriate tools Measure Students make some decisions as to how to approach the problem Skill/Concept Basic Application of a skill or concept Classify Organize Estimate Make observations Collect and display data Compare data Imply more than one step Visualization Skills Probability Skills Explain purpose and use of experimental procedures. Carry out experimental procedures Requires reasoning, planning using evidence and a higher level of thinking Strategic Thinking Freedom to make choices Explain your thinking Make conjectures Cognitive demands are complex and abstract Conjecture, plan, abstract, explain Justify Draw conclusions from observations Cite evidence and develop logical arguments for concepts Explain phenomena in terms of concepts Performance tasks Authentic writing Project-based assessment Complex, reasoning, planning, developing and thinking Cognitive demands of the tasks are high Work is very complex Students make connections within the content area or among content areas Select one approach among alternatives Design and conduct experiments Relate findings to concepts and phenomena Definitions To view full list of DOK descriptors, visit http://education.ky.gov/curriculum/docs/documents/cca_dok_support_808_mathematics.pdf 3
Mathematics Depth of Knowledge Levels by Norman L. Webb Level 1 (Recall) includes the recall of information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. That is, in mathematics a one-step, well-defined, and straight algorithmic procedure should be included at this lowest level. Other key words that signify a Level 1 include identify, recall, recognize, use, and measure. Verbs such as describe and explain could be classified at different levels depending on what is to be described and explained. Level 2 (Skill/Concept) includes the engagement of some mental processing beyond a habitual response. A Level 2 assessment item requires students to make some decisions as to how to approach the problem or activity, whereas Level 1 requires students to demonstrate a rote response, perform a well-known algorithm, follow a set procedure (like a recipe), or perform a clearly defined series of steps. Keywords that generally distinguish a Level 2 item include classify, organize, estimate, make observations, collect and display data, and compare data. These actions imply more than one step. For example, to compare data requires first identifying characteristics of the objects or phenomenon and then grouping or ordering the objects. Some action verbs, such as explain, describe, or interpret could be classified at different levels depending on the object of the action. For example, if an item required students to explain how light affects mass by indicating there is a relationship between light and heat, this is considered a Level 2. Interpreting information from a simple graph, requiring reading information from the graph, also is a Level 2. Interpreting information from a complex graph that requires some decisions on what features of the graph need to be considered and how information from the graph can be aggregated is a Level 3. Caution is warranted in interpreting Level 2 as only skills because some reviewers will interpret skills very narrowly, as primarily numerical skills, and such interpretation excludes from this level other skills such as visualization skills and probability skills, which may be more complex simply because they are less common. Other Level 2 activities include explaining the purpose and use of experimental procedures; carrying out experimental procedures; making observations and collecting data; classifying, organizing, and comparing data; and organizing and displaying data in tables, graphs, and charts. Level 3 (Strategic Thinking) requires reasoning, planning, using evidence, and a higher level of thinking than the previous two levels. In most instances, requiring students to explain their thinking is a Level 3. Activities that require students to make conjectures are also at this level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple answers, a possibility for both Levels 1 and 2, but because the task requires more demanding reasoning. An activity, however, that has more than one possible answer and requires students to justify the response they give would most likely be a Level 3. Other Level 3 activities include drawing conclusions from observations; citing evidence and developing a logical argument for concepts; explaining phenomena in terms of concepts; and using concepts to solve problems. Level 4 (Extended Thinking) requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. The extended time period is not a distinguishing factor if the required work is only repetitive and does not require applying significant conceptual understanding and higher-order thinking. For example, if a student has to take the water temperature from a river each day for a month and then construct a graph, this would be classified as a Level 2. However, if the student is to conduct a river study that requires taking into consideration a number of variables, this would be a Level 4. At Level 4, the cognitive demands of the task should be high and the work should be very complex. Students should be required to make several connections relate ideas within the content area or among content areas and have to select one approach among many alternatives on how the situation should be solved, in order to be at this highest level. Level 4 activities include designing and conducting experiments; making connections between a finding and related concepts and phenomena; combining and synthesizing ideas into new concepts; and critiquing experimental designs. 4
Collaborative Team Planning Guide SEGMENT 1: Properties of Equivalent Fractions What do I want my students to know and be able to do? Questions to Consider Application of Mathematical Practices (Behaviors/Actions) How can framework examples help make sense of mathematical tasks? How will we use number talks along with conceptual strategies to check for reasonable responses and build fluency? What tools/models are appropriate in demonstrating mastery towards the skills addressed by the standard/cluster? What misconceptions/errors do the frameworks anticipate with this particular skill set? Which correlating lessons are appropriate in rigor in alignment to clusters and standards as presented in the frameworks? What does the framework say about grade-level specific mathematical practices and implementation? What previous skills/concepts were addressed in previous grade(s) in order to make connections to current learning? Standard Content Objective Language Objective Social Objective What will my students learn? What language will my students use? What will my students do as they learn? 4.NF.1 4.NF.1 4.OA.4 4.AO.4 5
Language Functions and Considerations Vocabulary (specialized, technical) I can use the following math terms: Factor pairs, prime number, composite number, numerator, denominator, equivalent fractions, simplest form, greatest common factor, equivalent (4.NF.1) Structure/Syntax (The way words and vocabulary are used to express ideas) (4.NF.1) I can use the following sentence frame: Equivalent fractions name the part of a by showing the same in different ways. Equivalent fractions name the same part of a whole by showing the same amount in different ways. 1/2 2/4 4/8 Function (The intended use of language): illustrate to make clear or intelligible, as by examples or analogies( 4.NF.1) I can illustrate fractions by drawing a model. 1/3 I can illustrate equivalent fractions by drawing a model. 2/3 = 4/6 6
Structure/Syntax (The way words and vocabulary are used to express ideas) 4.OA.4 I can use the following sentence frame: A whole number is a of each of its factors. A whole number is a multiple of each of its factors. Simon is organizing his 36 toy cars into equal-sized piles. Which list shows all of the possible numbers of cars that could be in each pile? A 2. 3, 4, 6 B 1, 2, 3, 4, 6 C 2, 3, 4, 5, 9, 12, 18 D 1, 2, 3, 4, 6, 9, 12, 18, 36 Function (The intended use of language): Apply to make use of as relevant, suitable or pertinent 4.OA.4 I can apply prior knowledge of and to solving the math. I can apply prior knowledge of factors and multiples to solving the math. I know that different factor pairs result in the same product (i.e. 1 x 12 = 12; 2 x 6 = 12; 3 x 4 = 12;) Curricular Connections: Chapter 8 Lesson 1 Factors and Multiples Lesson 2 Prime and Composite Numbers Lesson 3 Hands On: Model Equivalent Fractions Lesson 4 Equivalent Fractions Lesson 5 Simplest Form Unit Connections: Ready Common Core Unit 2-7: Multiples and Factors Unit 4-13: Understand Equivalent Fractions Unit Connections: iready Finding Multiples Level D Equivalent Fractions Level C Find Equivalent Fractions Level C Content Specific Vocabulary: Factor pairs, prime number, composite number, numerator, denominator, equivalent fractions, simplest form, greatest common factor, equivalent 7
How will we know if they have learned it? Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom. Grade level teams will develop and implement common formative assessment. Claim Claim Descriptor MP s **Achievement Level Descriptor Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 3 Communicating Claim 4 Modeling and Data Analysis Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve, Evaluate, Extend, Apply Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze *5, 6, 7, 8 *1, 5, 7, 8 *3, 6 *2, 4, 5 *All practices may be integrated in each claim, however certain practices are emphasized above Target B Additional/Supporting: Gain familiarity with factors and multiples. (4.OA.4) Target F Major: Extend understanding of fraction equivalence and ordering. (4.NF.1-2) Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target B: Select and use appropriate tools strategically. Target C: Interpret results in the context of a situation. Target A: Test propositions or conjectures with specific examples. Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target B: Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. 8
How will we know if they have learned it? 9
Collaborative Team Planning Guide SEGMENT 2: Comparing and Ordering Fractions What do I want my students to know and be able to do? Questions to Consider Application of Mathematical Practices (Behaviors/Actions) How can framework examples help make sense of mathematical tasks? How will we use number talks along with conceptual strategies to check for reasonable responses and build fluency? What tools/models are appropriate in demonstrating mastery towards the skills addressed by the standard/cluster? What misconceptions/errors do the frameworks anticipate with this particular skill set? Which correlating lessons are appropriate in rigor in alignment to clusters and standards as presented in the frameworks? What does the framework say about grade-level specific mathematical practices and implementation? What previous skills/concepts were addressed in previous grade(s) in order to make connections to current learning? 4.NF.2 4.OA.4 Standard Content Objective Language Objective Social Objective What will my students learn? What language will my students use? What will my students do as they learn? 4.NF.2 4.OA.4 10
Language Functions and Considerations Vocabulary (specialized, technical): I can use the following terms: Mixed number, improper fraction, like fractions (4.NF.3a) Structure/Syntax (The way words and vocabulary are used to express ideas) (4.NF.3a) I can use the following sentence frame: When adding like fractions, we need to add and subtract the. Sometimes the answer could be an fraction (i.e.3/10+9/10= 12/10.) We need to convert the fraction to a mixed number (12/10 = 1 and 2/10.) When adding like fractions, we need to add and subtract the numerators. Sometimes the answer could be an improper fraction (i.e.3/10+9/10= 12/10.) We need to convert the improper fraction to a mixed number (12/10 = 1 and 2/10.) Function (The intended use of language): (4.NF.3a) Language Function: Explain- phrases or sentences to express the rationale, reasons, causes or relationships related to one or more actions or events ideas or processes. I can explain how to and fractions with like denominators. I can explain how to add and subtract fractions with like denominators. I can transfer my prior knowledge of adding whole to adding like. I can transfer my prior knowledge of adding whole numbers for adding like fractions (Example 3/10 + 6/10= 9/10) I added the numerators. Curricular Connections: Chapter 8 Lesson 6 Compare and Order Fractions Lesson 7 Use Benchmark Fractions to Compare and Order Lesson 8 Problem Solving Investigation: Use Logical Reasoning Unit Connections: Ready Common Core Unit 4-14: Compare Fractions Unit Connections: iready: Comparing and Ordering Three Unlike Fractions Level C Vocabulary: Least Common Multiple, benchmark fractions 11
How will we know if they have learned it? Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom. Grade level teams will develop and implement common formative assessment. Claim Claim Descriptor MP s **Achievement Level Descriptor Students can explain and apply mathematical concepts and interpret and Target B carry out mathematical procedures with precision and fluency. *5, 6, 7, 8 Additional/Supporting: Gain familiarity with factors and multiples. (4.OA.4) Target F Major: Extend understanding of fraction equivalence and ordering. (4.NF.1-2) Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 3 Communicating Claim 4 Modeling and Data Analysis Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve, Evaluate, Extend, Apply Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze *1, 5, 7, 8 *3, 6 *2, 4, 5 *All practices may be integrated in each claim, however certain practices are emphasized above Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target B: Select and use appropriate tools strategically. Target C: Interpret results in the context of a situation. Target A: Test propositions or conjectures with specific examples. Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target B: Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. 12
How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. How will we know if they have learned it? 13
Collaborative Team Planning Guide SEGMENT 3: Adding and Subtracting Fractions with Like Denominators What do I want my students to know and be able to do? Questions to Consider Application of Mathematical Practices (Behaviors/Actions) How can framework examples help make sense of mathematical tasks? 4.NF.3a How will we use number talks along with conceptual strategies to check for reasonable responses and build fluency? What tools/models are appropriate in demonstrating mastery towards the skills addressed by the standard/cluster? What misconceptions/errors do the frameworks anticipate with this particular skill set? Which correlating lessons are appropriate in rigor in alignment to clusters 4.NF.3d and standards as presented in the frameworks? What does the framework say about grade-level specific mathematical practices and implementation? What previous skills/concepts were addressed in previous grade(s) in order to make connections to current learning? Standard Content Objective Language Objective Social Objective What will my students learn? What language will my students use? What will my students do as they learn? 4.NF.3a 4.NF.3d 14
Language Functions and Considerations Vocabulary (specialized, technical): I can use the following terms: Mixed number, improper fraction, like fractions (4.NF.3a) Structure/Syntax (The way words and vocabulary are used to express ideas) (4.NF.3a) I can use the following sentence frame: When adding like fractions, we need to add and subtract the. Sometimes the answer could be an fraction (i.e.3/10+9/10= 12/10.) We need to convert the fraction to a mixed number (12/10 = 1 and 2/10.) When adding like fractions, we need to add and subtract the numerators. Sometimes the answer could be an improper fraction (i.e.3/10+9/10= 12/10.) We need to convert the improper fraction to a mixed number (12/10 = 1 and 2/10.) Function (The intended use of language): (4.NF.3a) Language Function: Explain- phrases or sentences to express the rationale, reasons, causes or relationships related to one or more actions or events ideas or processes. I can explain how to and fractions with like denominators. I can explain how to add and subtract fractions with like denominators. I can transfer my prior knowledge of adding whole to adding like. I can transfer my prior knowledge of adding whole numbers for adding like fractions (Example 3/10 + 6/10= 9/10) I added the numerators. Curricular Connections: Chapter 8 Lesson 9 Mixed Numbers Lesson 10 Mixed Numbers and Improper Fractions Chapter 9 Lesson 1 Hands On: Use Models to Add Like Fractions Lesson 2 Add Like Fractions Lesson 3 Hands On: Use Models to Subtract Like Fractions Lesson 4 Subtract Like Fractions Unit Connections: Ready Common Core Unit 4-15: Understand Fraction Addition and Subtraction Unit 4-16: Add and Subtract Fractions Unit Connections: iready Understand Adding and Subtracting Fractions Level D Vocabulary: Mixed number, improper fraction, like fractions How will we know if they have learned it? 15
Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom. Grade level teams will develop and implement common formative assessment. Claim Claim Descriptor MP s **Achievement Level Descriptor Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 3 Communicating Claim 4 Modeling and Data Analysis Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve, Evaluate, Extend, Apply Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze *5, 6, 7, 8 *1, 5, 7, 8 *3, 6 *2, 4, 5 *All practices may be integrated in each claim, however certain practices are emphasized above Target F Major: Extend understanding of fraction equivalence and ordering. (4.NF.1-2) Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target B: Select and use appropriate tools strategically. Target C: Interpret results in the context of a situation. Target A: Test propositions or conjectures with specific examples. Target B: Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. How will we know if they have learned it? 16
Collaborative Team Planning Guide SEGMENT 4: Combining Mixed Numbers with Like Denominators What do I want my students to know and be able to do? Questions to Consider Application of Mathematical Practices (Behaviors/Actions) How can framework examples help make sense of mathematical tasks? 4.NF.3b How will we use number talks along with conceptual strategies to check for reasonable responses and build fluency? What tools/models are appropriate in demonstrating mastery towards the skills addressed by the standard/cluster? What misconceptions/errors do the frameworks anticipate with this particular skill set? Which correlating lessons are appropriate in rigor in alignment to 4.NF.3c clusters and standards as presented in the frameworks? What does the framework say about grade-level specific mathematical practices and implementation? What previous skills/concepts were addressed in previous grade(s) in order to make connections to current learning? Standard Content Objective Language Objective Social Objective What will my students learn? What language will my students use? What will my students do as they learn? 4.NF.3b 4.NF.3c 17
Language Functions and Considerations Vocabulary (specialized, technical): I can use the following terms: equivalent fraction, associative property, decompose, mixed number; (4.NF.3b) Structure/Syntax (The way words and vocabulary are used to express ideas) (4.NF.3b) I can use the following sentence frame: I can fractions by decomposing the like a whole number ( 5=2+3; 5/8=2/8+3/8.) I can decompose fractions by decomposing the numerator like a whole number ( 5=2+3; 5/8=2/8+3/8.) Function (The intended use of language): (4.NF.3b) Language Function: Model- To make or construct a descriptive or representational shape or pattern I can model how to a fraction into a sum of fractions with the same in more than one way. I can model how to decompose a fraction into a sum of fractions with the same denominator in more than one way. Example : 3/6 = 1/6 + 1/6 + 1/6; Curricular Connections: Chapter 9 Lesson 5 Problem Solving Investigation: Work Backward Lesson 6 Add Mixed Numbers Lesson 7 Subtract Mixed Numbers Unit Connections: Ready Common Core Unit 4-17: Add and Subtract Mixed Numbers Unit Connections: iready: Understand Mixed Numbers Level D Content Specific Vocabulary: Equivalent fraction, associative property, decompose, mixed number 18
How will we know if they have learned it? Teachers will plan and implement daily formative assessments in order to provide specific immediate feedback in the classroom. Grade level teams will develop and implement common formative assessment. Claim Claim Descriptor MP s **Achievement Level Descriptor Students can explain and apply mathematical concepts and interpret and Target F carry out mathematical procedures with precision and fluency. *5, 6, 7, 8 Major: Extend understanding of fraction equivalence and ordering. (4.NF.1-2) Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 3 Communicating Claim 4 Modeling and Data Analysis Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve, Evaluate, Extend, Apply Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze *1, 5, 7, 8 *3, 6 *2, 4, 5 *All practices may be integrated in each claim, however certain practices are emphasized above Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target A: Test propositions or conjectures with specific examples. Target B: Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. How will we know if they have learned it? 19
Collaborative Team Planning Guide SEGMENT 5: Following Rules to Extend Patterns What do I want my students to know and be able to do? Questions to Consider Application of Mathematical Practices (Behaviors/Actions) How can framework examples help make sense of mathematical tasks? 4.OA.5 How will we use number talks along with conceptual strategies to check for reasonable responses and build fluency? What tools/models are appropriate in demonstrating mastery towards the skills addressed by the standard/cluster? What misconceptions/errors do the frameworks anticipate with this particular skill set? Which correlating lessons are appropriate in rigor in alignment to clusters and standards as presented in the frameworks? What does the framework say about grade-level specific mathematical practices and implementation? What previous skills/concepts were addressed in previous grade(s) in order to make connections to current learning? Standard Content Objective Language Objective Social Objective What will my students learn? What language will my students use? What will my students do as they learn? 4.OA.5 20
Language Functions and Considerations Vocabulary (specialized, technical): I can use the following math terms: dividend, multiples, compatible numbers, remainder (4.OA.5) Structure/Syntax (The way words and vocabulary are used to express ideas) (4.OA.5) I can use the following sentence frame: Another way to a would be by drawing shapes. Another way to represent a pattern would be by drawing shapes. Example: 1 2 3 4 5 6 A pattern can follow more than a rule. Possible answer: The shapes go back and forth between and, and the odd-numbered spots have, whereas, the even-numbered spots have. The shapes go back and forth between triangles and circles, and the odd-numbered spots have triangles, whereas, the even-numbered spots have circles. Function (The intended use of language): I can analyze a process for solving the math. (4.OA.5) Language Function Analyze phrases or sentences to indicate parts of a whole and/ or he relationship between/ among parts of an action, event, idea or process. I can and extend in numbers and shapes. I can analyze and extend patterns in numbers and shapes. When given the numbers 20, 24, 28, 32, 36, I can determine that the is add. If I were to extend the, I could predict that the following numbers in the will all be. When given the numbers 20, 24, 28, 32, 36, I can determine that the rule is add 4. If I were to extend the pattern, I could predict that the following numbers in the pattern will all be even. Curricular Connections: Chapter 7 Lesson 1 Nonnumeric Patterns Lesson 2 Numeric Patterns Lesson 3 Sequences Lesson 4 Problem-Solving Investigation: Look for a Pattern Lesson 5 Addition and Subtraction Rules Lesson 6 Multiplication and Division Rules Unit Connections: Ready Common Core Lesson 2-8: Number and Shape Patterns Unit Connections: iready: Describing Numerical Relationships Level C Content Specific Vocabulary: Pattern, nonnumeric patterns, numeric patterns, rule, term, sequence, input, output 21
How will we know if they have learned it? Claim Claim Descriptor MP s **Achievement Level Descriptor Students can explain and apply mathematical concepts and interpret and Target B carry out mathematical procedures with precision and fluency. *5, 6, 7, 8 Additional/Supporting: Gain familiarity with factors and multiples. (4.OA.4) Claim 1 Concepts and Procedures Claim 2 Problem Solving Claim 3 Communicating Claim 4 Modeling and Data Analysis Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Relevant Verbs: Understand, Solve, Apply, Describe, Illustrate, Interpret, Analyze Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Relevant Verbs: Model, Construct, Compare, Investigate, Build, Interpret, Estimate, Analyze, Summarize, Represent, Solve, Evaluate, Extend, Apply Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Relevant Verbs: Understand, Explain, Justify, Prove, Derive, Assess, Illustrate, Analyze *1, 5, 7, 8 *3, 6 *2, 4, 5 *All practices may be integrated in each claim, however certain practices are emphasized above Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. Target E: Distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in the argument explain what it is. Target A: Apply mathematics to solve well-posed problems in pure mathematics and those arising in everyday life, society, and the workplace. http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ How will we respond when learning has not occurred? Professional Learning Communities will develop and implement Response to Intervention. How will we respond when learning has already occurred? Professional Learning Communities will develop and implement Enrichment. How will we know if they have learned it? 22
G.R.R. (Gradual Release of Responsibility) Lesson Delivery Model You do it together collaborative component is infused throughout the Focus Lesson, Guided Instruction and Independent delivery stages. GRR Claims & Targets MP s Instructional Lesson Sequence: Focus- Coherence- Rigor Step 1: Introduce Content, Language and Social Objective: as determined by PLC Modeling Focused Lesson Infuse Claims and Targets throughout lesson delivery/ learning Infuse Mathematical Practices throughout lesson delivery/ learning Step 2: Connect to real world problems and/or prior learning (i.e. concepts that link across grade spans, future learning, or construction justification of a concept) : Teacher presents the problem Teacher overtly explains purpose of strategies, tools/models, etc. for the day s learning Engage in Collaborative Conversations and use Language Functions (Develop within PLC and embedded within steps) : The learning objective for today is. I think that we will learn about. reminds me of. One strategy for is. Step 1: Introduce and Contextualize Vocabulary (Throughout the remaining phases): Step 2: Model how and why the math works using Skills and Concepts along with Think A Louds: Step 3: Deconstruct the math problem and state the operations needed to solve the problem(s): Teacher and students share thinking Teacher asks students what is their understanding of the problem Teacher asks students the skills needed to get to the solution A way of thinking about solving this problem is. The most important thing to remember in this problem is. I believe the question is asking us to. Engage in Collaborative Conversations and use Language Functions ( Develop within PLC and embedded within steps): In order to add two numbers you need to. and are accomplished by. This first step in is to, followed by. 23
Step 1: Revisit or introduce a new real world math problems that are connected to Content, Language and Social Objective: Step 2: Provide students with opportunities to engage in the learning process: Solving real-world problems Describing and illustrating their understanding (speaking and writing) Justifying and explaining their reasoning in solving problems (speaking and writing) Asking questions to generate mathematical thinking We Do You Do Together Infuse Claims and Targets throughout lesson delivery/ learning Infuse Mathematical Practices throughout lesson delivery/ learning Step 3: Differentiation and Feedback Provide differentiated problems and Think A Louds Specific feedback and scaffolding Multiple explanations for solving particular problems Aide in the processing of the content for students (Questions, Prompts and Cues) Assessing students progress and adjusting teaching Determine student grouping or pairing Intervene as needed (responsibility begins to shift) Engage in Collaborative Conversations and use Language Functions ( Develop within PLC and embedded within steps): Based on, I determined that. Given that we can deduce that. I agree/disagree with that. Given that we can deduce that. I agree/disagree with that. Step 1: Revisit Objectives. Clearly define expectations and structures for collaborative conversations. Provide appropriate Mathematical Tasks. Step 2: Teacher Role Teacher facilitates Teacher Questions, Prompts, Cues appropriately Teacher provides Corrective Feedback Provide Enrichment Opportunities as needed Student Role Students communicate thinking and understanding Students will problem solve and reason Students will process, justify, explain, prove, critique, etc. (Utilize mathematical practices) Students will use and discuss strategies to extend/deepen understanding of learning Engage in Collaborative Conversations and use Language Functions ( Develop within PLC and embedded within steps): 24
Step 1: Provide students time to work individually or in pairs in order to assess mastery of the skills and concepts presented to them: Do it Alone Teacher facilitates learning with specific immediate feedback Teacher provides differentiation if needed Teacher provides students with the opportunity to write, justify, and explain their reasoning (math journal) Students practice and apply skills using curriculum/resources Students apply problem solving strategies Step 2: Assess and Close Assess the students on the learning for the lesson Revisit the Content, Language and Social Objectives Extend Connect the concepts to future lessons 25