Module 6B for Middle/High School Teachers Florida Standards for Mathematics: Focus on Content Standards

Professional Development Session Alignment Set 1 – Completed 2013 Governing Board School Leaders Teachers Math Leadership Teams Session 2 Session 2 Session 1 Session 1 ELA Data Use ELA Math Data Use 2

Professional Development Session Alignment Set 2 - August, 2013 to May, 2014 Governing Board School Leaders Module 7 ELA & Data Use Module 8 Math & Data Use Teachers Math Leadership Teams Session 4 Session 4 Session 3 Session 3 ELA Assessments Data Analysis VAM Data Analysis VAM Florida Standards Data & ELA Data & ELA Data & Math Data & Math Session 5 Session 5 Session 6 Session 6 3 Data

Module 5 ELA Module 6 Math Module 7 ELA & Data Use Module 8 Math & Data Use You Are Here Module 2 ELA Module 1 Data Use Module 4 Data Use Module 3 Math

5 8 Components of Full Florida Standards Implementation

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 There are specific instructions included with the form to help fill it out correctly Reimbursements for substitutes are NOT an eligible expense 6

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 Focus on Content Standards Outcomes 7

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 Today’s Agenda 8

Pre-Assessment Introductory Activity 9 Guide Page 4

Sharing Implementation Experiences Section 1 10

Instructional Shifts for Mathematics 11 The Standards for Mathematical Content The Standards for Mathematical Practice Focus Coherence Rigor Two Areas

Fewer standards allow for focusing on the major work for each grade Focus 12

The Standards are designed around coherent progressions and conceptual connections. Coherence Grade 1 Grade 2Grade 3 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 13

The Florida Standards for Math are designed around coherent progressions and conceptual connections. Coherence Math Concept Progression K-12 All Roads Lead to Algebra…… 14

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 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 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 APPLICATION OF MATHEMATICS Using math in real- world scenarios Choosing concepts without prompting 15

The Standards for Mathematical Practice Developing Mathematical Expertise 16 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

Activity 1: Sharing Experiences Implementing Math Practice Standards 17 Sharing Implementation Experiences 1.Each participant will discuss with their table group one positive highlight, one challenge, and one lesson learned from their personal implementation of the Practice Standards thus far. 2.Each table group will then determine two positive highlights, one common challenge, and one common lesson learned that they will present to the larger group. 3.Participants will record notes and “New Ideas” generated from the discussion. Guide Pages 8-9 Positive Highlights Challenges Lessons Learned Guide Pages 6-7

The Language of the Mathematical Content Standards Section 2 18

Activity 2: What Do These Students Understand? 19 What Do These Students Understand? 1.Read and analyze the “Who Knows Math” handout on page 9 in the Participant Guide. Record your observations on page 10 in the Participant Guide. 2.Answer the following: What does each student know? What does each student not know? What don’t we know about what each student knows? Guide Pages 8-9 Guide Pages 9-10

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 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 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 APPLICATION OF MATHEMATICS Using math in real- world scenarios Choosing concepts without prompting 20

21 Guide Page 11 “ Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.” (Adding it Up: Helping Children Learn Mathematics. 2001) Conceptual Understanding

22 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. Conceptual Understanding Guide Page 11

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. 23 Conceptual Understanding 8(30+6) = 8x30 + 8x6 = = X

Expressions and Equations 6 th Grade, Cluster1, Standards 3 & 4: Apply and extend previous understandings of arithmetic to algebraic expressions. 7 th Grade, Cluster1, Standards 1 & 2: Use properties of operations to generate equivalent expressions. 24 Conceptual Understanding

25 “ 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

26 Guide Page 12 Simplify the following 12a 3 b -4 c 9 18a -1 b 5 c 12 Procedural Skill and Fluency

Answer: 2a 4. 3b 9 c 3 Algebra Standard – Arithmetic with Polynomials and Rational Expressions MAFS.912.A-APR.4.6: Rewrite rational expressions. 27 Procedural Skill and Fluency

28 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. 29 Application of Mathematics Guide Page 13 Adapted from: Mason, J., Burton, L., & Stacey, K. (1985). Thinking mathematically. England: Addison-Wesley Publishers Limited.

30 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 7 th Grade, Cluster 1: Analyze proportional relationships and use them to solve real-world and mathematical problems. 31 Application of Mathematics

How does the approach of the Florida Standards for Math Content differ from previous approaches to mathematics teaching and learning? Think About It… 32

Let’s Take A Break… 33 Be back in 15 minutes …

High Level Tasks and Assessment Section 3 34

Math Class Needs a Makeover Dan Meyer 35 Watch Video

Click to edit Master title style Middle School Task – TV Sales High School Task – Golf Balls in Water 36 New Standards and New Tasks will be Assessed in a New Way Guide Pages 15-18

Click to edit Master title style 37 Activity 3: What Makes These Tasks “High Level”? Guide Page 19 High Level Tasks 1.Discuss with your table group what makes the sample tasks “high level”. 2.Record the characteristics of high level tasks on page 19 in the Participant Guide.

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 38

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) High Level Tasks 39

40 Lunch

Creating High Level Tasks Section 4 41

Open Questions Strategies for Differentiating High Level Tasks 42 Parallel Tasks

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. 43 What are Open Questions?

Strategy: The answer is the question Solve: x 2 - 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? Creating Open Questions 44 Guide Page 21

45 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) Parallel 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? Creating High Level Tasks 46 (Small & Lin, 2010) Guide Page 21

Principles to Keep in Mind 47 1.All open questions must allow for correct responses at a variety of levels. 2.Parallel tasks need to be created with variations that allow struggling students to be successful and proficient students to be challenged. 3.Questions and tasks should be constructed in such a way that will allow all students to participate together in follow-up discussions.

48 Activity 4: Open Questions and Parallel Tasks Create Your Own Higher Level Questions and Tasks 1.Review the information about Open Questions and Parallel Tasks on pages in the Participant Guide. 2.As a table group, use these strategies to convert the problems on pages into higher level questions and tasks. Try to compose questions that require students to think more deeply about concepts and to exercise mathematical practices. Generate possible solutions for each problem and be prepared to discuss what different students might do. 3.Make a poster of your favorite problem and share it with the larger group. Guide Pages 22-26

1.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’? Reflect 49 Guide Page 27

The Progression Of Mathematical Concepts Section 5 50

Progressions came first, before the standards. K-8 has developmental progressions for each domain by grade. High school has progressions of conceptual categories without grades. Each grade K-8 has major, supporting and additional standards. Grade level introductions list grade emphases. 51 Math Handouts Progressions of Standards

The Organization of the Standards 52 Math Handouts

High School Pathways 53 Math Handouts

Domain Distribution 54 (

Activity 5: Exploring the Content Standards 55 Exploring the Content Standards – Part 1 Which standards focus on Conceptual Understanding? Which standards focus on Procedural Skill and Fluency? Which standards focus on Applications? Guide Pages 29-71

Activity 5: Exploring the Content Standards 56 Exploring the Content Standards – Part 2 How do standards progress? What is the interaction of Conceptual Understanding, Procedures, and Applications over time? How do the standards progress in complexity over time? Guide Pages 29-71

Activity 5: Exploring the Content Standards 57 Exploring the Content Standards – Part 3 How can the practice standards be infused into working on the content? Guide Pages 29-71

Teaching the Content Standards Section 6 Through Problem Solving 58

Sorting & Classifying Equations - Part 1: Overview Sorting & Classifying Equations - Part 2: Discussion Putting it All Together 59 Guide Pages 73-75

Next Steps Section 7 60

1.What do we think should happen at school to promote implementation of the Florida Standards for Math? 2.What can we do now in our classrooms and in the school to promote implementation of the Florida Standards for Math? 3.What are some expected challenges? 4.How can we work around the challenges? What's Your Plan? 61 Guide Page 77

Closing Activities 62

Don’t Forget Your Resources… flcharterccrstandards.org 63 cpalms.org/project/cpalmscharter.aspx

Learned more about the Practice Standards Examined the language and structure of the Florida Standards for Math Practice Created and solved standards-based tasks Observed Florida Standards for Math-aligned instruction Shared implementation successes and challenges and planned next steps Focus on Content Standards Outcomes 64

Click to edit Master title style Where Are You Now? Assessing Your Learning 65 Post-Assessment and Session Evaluation Guide Page 79

Module 2 ELA Module 1 Data Use Module 3 Math Module 4 Data Use Module 5 ELA Module 6 Math Module 7 ELA & Data Use Module 8 Math & Data Use What’s Next? Module 7 ELA & Data Use Module 4 Data Use

Thanks and see you next time! 67