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## Module 6B for Middle/High School Teachers

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**Module 6B for Middle/High School Teachers**Florida Standards for Mathematics: Focus on Content Standards**Professional Development Session Alignment Set 1 –**Completed 2013 Data Use Governing Board Data Use ELA Math School Leaders Data Use ELA Math Teachers Leadership Teams Session 1 Session 2**Professional Development Session Alignment**Set 2 - August, 2013 to May, 2014 Governing Board Florida Standards School Leaders Assessments Data Analysis VAM Data ELA Math Data & ELA Data & Math Teachers Leadership Teams Session 3 Session 4 Session 5 Session 6 Module 7 ELA & Data Use Module 8 Math & Data Use**You Are Here**Module 3 Math Module 4 Data Use Module 1 Data Use Module 2 ELA Module 6 Math Module 7 ELA & Data Use Module 8 Math & Data Use Module 5 ELA**Travel Notes**• Mileage to/from the trainings will be reimbursed to the school at $.445/mile (documentation with map and mileage required) • Parking and tolls will also be reimbursed with receipt • Reimbursement is limited to two cars per school • Forms and directions to request reimbursement are available under “Resources” on www.flcharterccrstandards.org • There are specific instructions included with the form to help fill it out correctly • Reimbursements for substitutes are NOT an eligible expense**Focus on Content Standards Outcomes**• Learn more about the Practice Standards • Examine the language and structure of the Florida Standards for Math Content • Create and solve standards-based tasks • Observe Florida Standards for Math-aligned instruction • Share implementation successes and challenges and plan next steps**Today’s Agenda**Welcome and Introductions • Pre-Assessment • Sharing Implementation Experiences • The Language of the Content Standards • High Level Tasks and Assessment Lunch • Creating High Level Tasks • The Progression of Concepts • Teaching the Content Standards Through Problem Solving • Next Steps • Post-Assessment Wrap Up**Introductory Activity**Pre-Assessment Guide Page 4**Section 1**Sharing Implementation Experiences**Coherence**Instructional Shifts for Mathematics • The Standards for Mathematical Content • The Standards for Mathematical Practice Rigor Focus Two Areas**Fewer standards allow for focusing on the major work for**each grade Focus**Coherence**The Standards are designed around coherent progressions and conceptual connections. Use place value understanding and properties of operations to add and subtract Use place value understanding and properties of operations to add and subtract fluently Use place value understanding and properties of operations to perform multi-digit arithmetic**Coherence**The Florida Standards for Math are designed around coherent progressions and conceptual connections. All Roads Lead to Algebra…… Math Concept Progression K-12**The major topics at each grade level focus equally on:**Rigor • CONCEPTUAL UNDERSTANDING • More than getting answers • Not just procedures • Accessing concepts to solve problems • PROCEDURAL SKILL • AND FLUENCY • Speed and accuracy • Used in solving more complex problems • Comes after conceptual understanding • APPLICATION OF MATHEMATICS • Using math in real-world scenarios • Choosing concepts without prompting**Developing Mathematical Expertise**The Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning**Activity 1: Sharing Experiences Implementing Math Practice**Standards Positive Highlights Challenges Lessons Learned Guide Pages 6-7 Guide Pages 8-9**Section 2**The Language of the Mathematical Content Standards**Activity 2: What Do These Students Understand?**1 2 2 412 X 412 Guide Pages 9-10 Guide Pages 8-9**The major topics at each grade level focus equally on:**Rigor • CONCEPTUAL UNDERSTANDING • More than getting answers • Not just procedures • Accessing concepts to solve problems • PROCEDURAL SKILL • AND FLUENCY • Speed and accuracy • Used in solving more complex problems • Comes after conceptual understanding • APPLICATION OF MATHEMATICS • Using math in real-world scenarios • Choosing concepts without prompting**Conceptual Understanding**“Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.” (Adding it Up: Helping Children Learn Mathematics. 2001) Guide Page 11**Conceptual Understanding**Example How is multiplying 8x36 like multiplying 7(x+y)? Could you use the procedures you use for one multiplication to do the other? Explain how you know. Guide Page 11**Conceptual Understanding**Possible Response: 8x36 could be written as 8(30+6) and done the same way as the algebraic multiplication. Both use the distributive property. The usual arithmetic algorithm is just another way of writing the same thing. 36 X8 48 240 288 8(30+6) = 8x30 + 8x6 = 240 + 48 = 288**Conceptual Understanding**• Expressions and Equations • 6th Grade, Cluster1, Standards 3 & 4: Apply and extend previous understandings of arithmetic to algebraic expressions. • 7thGrade, Cluster1, Standards 1 & 2: Use properties of operations to generate equivalent expressions.**Procedural Skill and Fluency**“Procedural skill and fluency is demonstrated when students can perform calculations with speed and accuracy.” (Achieve the Core) “Fluency promotes automaticity, a critical capacity that allows students to reserve their cognitive resources for higher-level thinking.” (Engage NY)**Procedural Skill and Fluency**Simplify the following 12a3b-4c9 18a-1b5c12 Guide Page 12**Procedural Skill and Fluency**Answer: 2a4 . 3b9c3 Algebra Standard – Arithmetic with Polynomials and Rational Expressions MAFS.912.A-APR.4.6: Rewrite rational expressions.**Application of Mathematics**• The Standards call for students to use math flexibly for applications. • Teachers provide opportunities for students to apply math in context. • Teachers in content areas outside of math, particularly science, ensure that students are using math to make meaning of and access content. (ASCD and Achieve the Core. 2012)**Application of Mathematics**At the T-Shirt store you get a 20% discount but you must pay a 15% sales tax. Which would you prefer to have calculated first, discount or tax? Explain how you know what’s best. Guide Page 13 Adapted from: Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. England: Addison-Wesley Publishers Limited.**Application of Mathematics**Answer:It doesn’t matter. The result is the same. Possible Reasoning: (let x = original price) Discount First 80% of x = .8x (new price). 15% of .8x = .12x (tax) .8x + (.15)(.8x) = .8x(1 + .15) = (.8x)(1.15) = .92x Tax First 15% of x = .15x (tax). x + .15x = 1.15x (price w/tax) 80% of 1.15x = (.8)(1.15)x = .92x (price after discount)**Application of Mathematics**• Ratios and Proportional Relationships • 7th Grade, Cluster 1: Analyze proportional relationships and use them to solve real-world and mathematical problems.**Think About It…**How does the approach of the Florida Standards for Math Content differ from previous approaches to mathematics teaching and learning?**Let’s Take A Break…**Be back in 15 minutes…**Section 3**High Level Tasks and Assessment**Math Class Needs a MakeoverDan Meyer**Watch Video**New Standards and New Tasks will be Assessed in a New Way**Middle School Task – TV Sales High School Task – Golf Balls in Water Guide Pages 15-18**Activity 3: What Makes These Tasks “High Level”?**Guide Page 19**High Level Tasks**• Don’t have a predictable, well-rehearsed approach or pathway to the solution. • Require students to explore and understand the nature of mathematical concepts, processes, or relationships. • Demand self-monitoring or self-regulation of one’s own cognitive processes. • Require students to access relevant knowledge and experiences and make appropriate use of them in working through the task.**High Level Tasks**• Require students to analyze the task and actively examine task constraints that may limit possible solution strategies and solutions. • Require considerable cognitive effort and may involve some level of anxiety for the student due to the unpredictable nature of the solution process required. Adapted from (Stein, Smith, et al (2000). Implementing Standard-Based Mathematics Instruction)**Section 4**Creating High Level Tasks**Strategies for Differentiating High Level Tasks**Open Questions Parallel Tasks**What are Open Questions?**• An open question is framed in such as way that multiple responses and approaches can correctly answer the question. • An open question allows students at varying developmental and readiness levels to equally participate in and grow from thought provoking tasks. • An open question provides multiple pathways into the mathematics.**Creating Open Questions**Strategy: The answer is the question • Solve: x2 - 4x + 3 = 0 • One of the solutions to a quadratic equation is 3, what might be the equation. Can you find more than one solution? Guide Page 21**Parallel Tasks**Parallel Tasks are sets of tasks, usually two or three, that get at the same big idea and are close enough in context that they can be discussed simultaneously. (Small. 2012, 10)**Creating High Level Tasks**Strategy: Parallel Tasks – You Choose • The point (9,-2) is the top right vertex of a parallelogram? What might the coordinates of the other vertices be? • The points (9,2) and (1,1) are two vertices of a parallelogram. What might the coordinates of the other vertices be? Guide Page 21 (Small & Lin, 2010)**Principles to Keep in Mind**All open questions must allow for correct responses at a variety of levels. Parallel tasks need to be created with variations that allow struggling students to be successful and proficient students to be challenged. Questions and tasks should be constructed in such a way that will allow all students to participate together in follow-up discussions.**Activity 4: Open Questions and Parallel Tasks**Guide Pages 22-26**Reflect**• How does teaching with high level tasks engage students in rigorous work - conceptual understanding, procedural skill and fluency, and application of mathematics? 2. How does teaching with high level tasks support Florida’s ‘New Way of Work’? Guide Page 27**Section 5**The Progression Of Mathematical Concepts