Unit 3 Learning Outcomes Grade 5 CLAIMS 1-Concept and Procedures 2-Problem Solving 3-Communicating 4-Modeling and Data Analysis 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 STANDARDS Benchmark Blueprint TARGET E 5.NF.1 Add and subtract fractions with unlike Major: Use equivalent denominators (including mixed numbers) by replacing fractions as a strategy given fractions with equivalent fractions in such a way to add and subtract as to produce an equivalent sum or difference of fractions. fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd) 2 SR 1 CR Notes In grade four, students learned to calculate sums of fractions with different denominators, where one denominator is a divisor of the other, so that only one fraction has to be changed. Framework pg. 19-20 Textbook Lessons: 9.4-9.8, 9.10-9.13 Ready Common Core: Unit 2 Lesson 10 Greatest Common Factor (GCF) and understand how to use the GCF when simplifying fractions Lowest Common Multiple (LCM) and understand how to use the LCM when finding a common denominator for equivalent fractions Understand the difference between a factor and a multiple Know what an equivalent fraction is Know how to change an improper fraction to a mixed number and vice versa Understand the process of regrouping in adding and subtracting mixed numbers Subtraction with renaming (Textbook lesson 9.13) 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using 1 3 SR PT-Part B Students use summative understanding of fractions from grades 3-5 and synthesize adding and subtracting from grades K-4 when they arrive at this standard.
visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result ⅖ + ½ = 3 / 7, by observing that 3 / 7 < ½. Framework pg. 20 Textbook Lessons: 9.2-9.13 Ready Common Core: Unit 2 Lesson 11 Understand how fraction tiles are broken into pieces of one whole Understand how fraction tiles with different denominators when stacked can be the equivalent (Textbook Lesson 9.4) Find the LCM of two unlike fractions using fraction tiles Estimate fractions to the nearest whole number in order to check for reasonableness (Textbook Lesson 9.8) TARGET F Major: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 3 SR In 4 th grade, students multiply fractions by whole numbers. Students examine the relationship between fractions and division. Framework pg. 21 Textbook Lessons: 8.1 Ready Common Core: Unit 2 Lesson 12 Understand that a fraction represents division of the numerator by the denominator The numerator is the number of parts represented The denominator represents the number of parts in the whole Have a clear understanding of which number represents the divisor (denominator) and which number represents the dividend (numerator) in a word problem. Understand how to plot fractions on a number line Use models to represent the numerator (dividend) and show how to use it to divide by the denominator (divisor) TARGET F Major: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 5.NF.4 Understand a fraction a/b with a>1 as a sum of fractions 1/b. 4a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) = 8/15. (In general, (a/b) (c/d) = ac/bd.) 4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate 1 SR PT-Part A PT-Part D Extension of multiplication will bring students closer to division of fractions in grade six. Framework pg. 22-24 Textbook Lessons NF4, NF 4a: 10.1-10.6, 10.12 Textbook Lessons NF 4b: 10.5, 10.6 Ready Common Core 4a: Unit 2 Lesson 13 Ready Common Core 4b: Unit 2 Lesson 14 2
unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.5 Interpret multiplication as scaling (resizing), by: 5a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a) / (n b) to the effect of multiplying a/b by 1. 5.NF.6 Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 5a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. 5b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4. 5c. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual 3 3 SR 2 SR PT-Part B 3 SR Standard serves as preparation for ratio and proportional reasoning. Framework calls for extensive exploration and explanation by students. Framework pg. 25 Textbook Lessons NF 5, NF 5b: 8.3, 8.6-8.8, 10.6, 10.8 Textbook Lessons NF 5a: 10.8 Ready Common Core: Unit 2 Lesson 15 Framework pg. 25 Textbook Lessons: 10.1-10.4, 10.6-10.8, 10.12 Ready Common Core: Unit 2 Lesson 16 Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. Division of a fraction by a fraction is not a requirement at 5 th grade. Framework pg. 26-27 Textbook Lessons NF 7, NF 7c: 10.9-10.12 Textbook Lessons NF 7a: 10.9, 10.11 Textbook Lessons NF 7b: 10.9, 10.10, 10.12 Ready Common Core 7a & 7b: Unit 2 Lesson 17 Ready Common Core 7c: Unit 2 Lesson 18
fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Benchmark Item Types and points 17 Selected Responses (1 point) 1 Constructed Responses (2 points) 1 Performance Task with 4 parts (6 points) 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. 4
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. Task Development Mathematical tasks should Integrate knowledge and skills across multiple claims and targets Engage students in relevant and interesting topics Measure depth of understanding, research skills, complex analysis Have an authentic purpose and connected components Take age appropriate development into consideration Accessible to all learners 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 (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 Definitions 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 To view full list of DOK descriptors, visit http://education.ky.gov/curriculum/docs/documents/cca_dok_support_808_mathematics.pdf 5
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. 6
Collaborative Team Planning Guide SEGMENT 1: Concepts of Multiplication with 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? 5.NF.3 5.NF.3 5.NF.5b 5.NF.5b 7
Language Functions and Considerations Vocabulary (specialized, technical): Structure/Syntax (The way words and vocabulary are used to express ideas): Function (The intended use of language: Curricular Connections: Chapter 8 Lesson 1 Fractions and Division Lesson 3 Simplest Form Lesson 6 Compare Fractions Lesson 7 Hands On: Use Models to Write Fractions as Decimals Lesson 8 Write Fractions as Decimals Unit Connections: Ready Common Core Unit 2-12: Fractions as Division Unit 2-15: Understand Multiplication as Scaling Unit Connections: iready Fractions as Division: Level E Understand Multiplication as Scaling Level E Vocabulary: Fraction, numerator, denominator, simplest form, equivalent fractions, least common denominator 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: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. (5.NF.3-7) 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 8 *1, 5, 7, 8 *3, 6 *2, 4, 5 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 B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. 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. *All practices may be integrated in each claim, however certain practices are emphasized above http://www.cde.ca.gov/ci/ma/cf/draft2mathfwchapters.asp http://bcsd.com/cipd/sbac-item-specification-tools-2/ 9
Collaborative Team Planning Guide SEGMENT 2: Add and Subtract Fractions What do I want my students to know and be able to do? Questions to Consider 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? Application of Mathematical Practices (Behaviors/Actions) 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? 5.NF.1 5.NF.1 5.NF.2 5.NF.2 Vocabulary (specialized, technical): Language Functions and Considerations Structure/Syntax (The way words and vocabulary are used to express ideas): Function (The intended use of language): 10
Curricular Connections: Chapter 9 Lesson 1 Round Fractions Lesson 2 Add Like Fractions Lesson 3 Subtract Like Fractions Lesson 4 Hands-On: Use Models to Add Unlike Fractions Lesson 5 Add Unlike Fractions Lesson 6 Hands-On: Use Models to Subtract Unlike Fractions Lesson 7 Subtract Unlike Fractions Unit Connections: Ready Common Core Unit 2-10: Add and Subtract Fractions Unit 2-11: Add and Subtract Fractions in Word Problems Unit Connections: iready: Understand Mixed Numbers Level D Add and Subtract Fractions Level E Add and Subtract Fractions in Word Problems Level E Content Specific Vocabulary: Like fractions, unlike fractions 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 E carry out mathematical procedures with precision and fluency. *5, 6, 7, 8 Major: Use equivalent fractions as a strategy to add and subtract fractions. (5.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 B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. 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. 11
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? 12
Collaborative Team Planning Guide SEGMENT 3: Addition and Subtraction of Mixed Numbers 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? 5.NF.2 5.NF.2 Vocabulary (specialized, technical): I can use the following math terms: Language Functions and Considerations Structure/Syntax (The way words and vocabulary are used to express ideas): Function (The intended use of language): 13
Curricular Connections: Chapter 9 Lesson 8 Problem Solving Investigation: Determine Reasonable Answers Lesson 9 Estimate Sums and Differences Lesson 10 Hands On: Use Models to Add Mixed Numbers Lesson 11 Add Mixed Numbers Lesson 12 Subtract Mixed Numbers Lesson 13 Subtract with Renaming Unit Connections: Ready Common Core Unit 2-11: Add and Subtract Fractions in Word Problems Unit Connections: iready: Understand Mixed Numbers Level D Add and Subtract Fractions in Word Problems Level E Content Specific Vocabulary: Mixed number 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 E carry out mathematical procedures with precision and fluency. *5, 6, 7, 8 Major: Use equivalent fractions as a strategy to add and subtract fractions. (5.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 B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. 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. 14
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? 15
Collaborative Team Planning Guide SEGMENT 4: Multiply 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? 5.NF.4 5.NF.4 5.NF.6 5.NF.6 Language Functions and Considerations Vocabulary (specialized, technical): Structure/Syntax (The way words and vocabulary are used to express ideas): Function (The intended use of language): 16
Curricular Connections: Chapter 10 Lesson 1 Hands On: Part of a Number Lesson 2 Estimate Products of Fractions Lesson 3 Hands On: Model Fraction Multiplication Lesson 4 Multiply Whole Numbers and Fractions Lesson 5 Hands On: Use Models to Multiply Fractions Lesson 6 Multiply Fractions Lesson 7 Multiply Mixed Numbers Lesson 8 Hands On: Multiplication as Scaling Unit Connections: Ready Common Core Unit 2-13: Understand Products of Fractions Unit 2-14: Multiply Fractions Using an Area Model Unit 2-16: Multiply Fractions in Word Problems Unit Connections: iready: Understand Products of Fractions Level E Multiplying a Whole Number and a Fraction Level E Content Specific Vocabulary: Factor, product, simplify, simplest form, scaling 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: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. (5.NF.3-7) 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 B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. 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. 17
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? 18
SEGMENT 5: Divide Fractions and Whole Numbers 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? 5.NF.6 5.NF.6 5.NF.7 5.NF.7 Language Functions and Considerations 19
Vocabulary (specialized, technical): Structure/Syntax (The way words and vocabulary are used to express ideas): Function (The intended use of language): Curricular Connections: Chapter 10 Lesson 9 Hands On: Division with Unit Fractions Lesson 10 Divide Whole Numbers by Unit Fractions Lesson 11 Divide Unit Fractions by Whole Numbers Lesson 12 Problem Solving Investigation: Draw a Diagram Unit Connections: Ready Common Core Unit 2-16: Multiply Fractions in Word Problems Unit 2-17: Understand Division with Unit Fractions Unit 2-18: Divide Unit Fractions in Word Problems Unit Connections: iready: Multiplying a Whole Number and a Fraction Level E Understand Division with Unit Fractions Level E Divide Unit Fractions in Word Problems Level E 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 1 Concepts and Procedures Content Specific Vocabulary: Unit fraction 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: Apply and extend previous understandings of multiplication and division to multiply and divide fractions. (5.NF.3-7) Claim 2 Problem Solving Claim 3 Communicating 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 *1, 5, 7, 8 *3, 6 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 B: Construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures. 20
Claim 4 Modeling and Data Analysis 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 *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. 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? 21
Modeling Focused Lesson Infuse Claims and Targets throughout lesson delivery/ learning Infuse Mathematical Practices throughout lesson delivery/ learning 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 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. 22
You Do Together We Do Infuse Claims and Targets throughout lesson delivery/ learning Infuse Mathematical Practices throughout lesson delivery/ learning 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 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): 23
Do it Alone Step 1: Provide students time to work individually or in pairs in order to assess mastery of the skills and concepts presented to them: 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 24