3/28/2017 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices.

3/28/2017 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices

Overall Outcomes Recognize the interconnectedness of the Standards for Mathematical Practice and content standards in developing student understanding and reasoning. Illuminate practices that establish a culture where mistakes are a springboard for learning, risk-taking is the norm, and there is a belief that all students can learn. Deeping content knowledge and pedagogy within an important focus area for our grade band: Number and Operations - Fractions Share that whenever possible our examples and activities come from this domain. Number and Operations - Fractions

Effective Classrooms

What research says about effective classrooms
The activity centers on mathematical under-standing, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session. Handout – Classroom Culture Template Document.

The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long- lasting student understanding. Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

Effective implies: Students are engaged with important mathematics.
Lessons are very likely to enhance student understanding and to develop students’ capacity to do math successfully. Students are engaged in ways of knowing and ways of working consistent with the nature of mathematicians ways of knowing and working. Teachers Development Group-Best Practices Workshop Donovan and Bransford, eds., 2005; Weiss et al, 2003; Kilpatrick et al, 2001; Glenn et al, 2000;

Reflection What is your current reality around classroom culture?
What can you do to enhance your current reality? Handout Handout – Classroom Culture Template Document.

Outcomes: Day 1 Reflect on teaching practices that support the shifts in the Standards for Mathematical Practice and content standards. Understand how to analyze student work with the Standards for Mathematical Practice and content standards. Analyze, adapt and implement a task with the integrity of the Common Core State Standards.

A message from OSPI Link to Greta’s Video

Outcomes Norms slide here

WA CCSS Implementation Timeline
Phase 1: CCSS Exploration Phase 2: Build Awareness & Begin Building Statewide Capacity Phase 3: Build State & District Capacity and Classroom Transitions Phase 4: Statewide Application and Assessment Ongoing: Statewide Coordination and Collaboration to Support Implementation Drop if everywhere else

Theory of Practice for CCSS Implementation in WA
2-Prongs: 1. The What: Content Shifts (for students and educators) Belief that past standards implementation efforts have provided a strong foundation on which to build for CCSS; HOWEVER there are shifts that need to be attended to in the content. 2. The How: System “Remodeling” Belief that successful CCSS implementation will not take place top down or bottom up – it must be “both, and…” Belief that districts across the state have the conditions and commitment present to engage wholly in this work. Professional learning systems are critical

Transition Plan for Washington State
K-2 3-5 6-8 High School Year 1- 2 School districts that can, should consider adopting the CCSS for K-2 in total. K – Counting and Cardinality (CC); Operations and Algebraic Thinking (OA); Measurement and Data (MD) 1 – Operations and Algebraic Thinking (OA); Number and Operations in Base Ten (NBT); 2 – Operations and Algebraic Thinking (OA); Number and Operations in Base Ten (NBT); and remaining 2008 WA Standards 3 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) 4 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) 5 – Number and Operations – Fractions (NF); Operations and Algebraic Thinking (OA) and remaining 2008 WA Standard 6 – Ratio and Proportion Relationships (RP); The Number System (NS); Expressions and Equations (EE) 7 – Ratio and Proportion Relationships (RP); The Number System (NS); Expressions and Equations (EE) 8 – Expressions and Equations (EE); The Number System (NS); Functions (F) Algebra 1- Unit 2: Linear and Exponential Relationships; Unit 1: Relationship Between Quantities and Reasoning with Equations and Unit 4: Expressions and Equations Geometry- Unit 1: Congruence, Proof and Constructions and Unit 4: Connecting Algebra and Geometry through Coordinates; Unit 2: Similarity, Proof, and Trigonometry and Unit 3:Extending to Three Dimensions

Why Shift? Almost half of eighth-graders in high achieving countries showed they could reach the “advanced” level in math, meaning they could relate fractions, decimals and percent to each other; understand algebra; and solve simple probability problems. In the U.S., 7 percent met that standard. Results from the 2011 TIMMS Taiwan, Singapore, and South Korea

The Three Shifts in Mathematics
Focus: Strongly where the standards focus Coherence: Think across grades and link to major topics within grades Rigor: Require conceptual understanding, fluency, and application This is a reminder of the three shifts that are required by the Common Core State Standards for Mathematics. Read slide

Focus on the Major Work of the Grade
Two levels of focus: What’s in/What’s out The standards at each grade level are interconnected allowing for coherence and rigor There are two levels of focus. The first level is the focus of what is in versus what is out; what is being taught at each grade level compared to what is not. It is because of this level of focus that teachers will have the time to go deeper with the math that is most important. Compared to the typical state standards of the past (which in some cases were literally volumes of standards that would have taken years to “cover,” even one grade’s worth of math), the Common Core State Standards for Mathematics have fewer standards which are manageable and it is clear what is expected of the teachers and students at each grade level. That is the 1st level of focus. The other level of focus is the shape of the content that is in each grade or course. What that means is that if you look at the “focused” list, say for Kindergarten, you can see the list in terms of shades. There are things that are really sharp and focused in the middle, that are the major content for that grade. The other topics are there in a supporting way and help to support that major work. So, even within the list that exists, there is focus. That is the 2nd level of focus.

Focus in International Comparisons
TIMSS and other international comparisons suggest that the U.S. curriculum is ‘a mile wide and an inch deep.’ “…On average, the U.S. curriculum omits only 17 percent of the TIMSS grade 4 topics compared with an average omission rate of 40 percent for the 11 comparison countries. The United States covers all but 2 percent of the TIMSS topics through grade 8 compared with a 25 percent noncoverage rate in the other countries. High-scoring Hong Kong’s curriculum omits 48 percent of the TIMSS items through grade 4, and 18 percent through grade 8.” Coverage does not equal knowledge. Knowing a few topics well helps students apply their knowledge to areas they haven’t studied. Note: find out what topics Hong Kong drops or a resource to give participants – Ginsburg et al., 2005

The domain and clusters are important

Content Emphasis by Cluster—Grade 3
Achieve the Core has created these documents to help teachers emphasize content. 75% of instruction should be based on the areas of focus. Not that the other areas aren’t taught but they support the major areas of focus.

Grade 3 (supporting cluster)
Linking supporting clusters to the major work of the grade

Have participants look at their supporting clusters to see how they can support fractions.

Focus on Major Work In any single grade, students and teachers spend the majority of their time, approximately 75% on the major work of the grade. The major work should also predominate the first half of the year. This is from the Achieve the Core document Publisher’s Criteria p. 7 Find areas where minor clusters support the major cluster in fractions from PARCC

Engaging with the 3-5 Content
How would you summarize the major work of 3-5? What would you have expected to be a part of the major work that is not? Give an example of how you would approach something differently in your teaching if you thought of it as supporting the major work, instead of being a separate, discrete topic. Let’s take some time to closely review 3-5 grade band. After reviewing, please discuss with your groups the questions on this page.

Focus on Fractions One of the Major Works of the 3-5 Grade Band
Deeping Content Knowledge and

Shifts - Implications for Fractions
gression Grade 3: Developing an understanding of fractions as numbers is essential for future work with the number system. It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related component of their ever expanding number system. An example of Focus: grade 3 from the PARCC document p. 16 Optional - video from Illustrative math project focusing on the shifts within CCSS-M for Fractions

Shift Two: Coherence Think across grades, and link to major topics within grades
Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.

Coherence Across and Within Grades
It’s about math making sense. The power and elegance of math comes out through carefully laid progressions and connections within grades. Coherence is about math making sense. Just like there are two levels of focus, there are two types of coherence found in the math standards. One is the coherence of topics across grades and the other is the coherence within a grade. You will see coherence across grades as the Standards direct us to have students apply learning from a previous grade to learn a new topic. You will see coherence across grades as you see the thoughtfully laid out progressions of mathematics that are meaningful and make sense. You will see coherence within a grade in the Standards as they direct us to have students reinforce a major topic in a grade by utilizing a supporting topic. You will see coherence within a grade as you see the meaningful introduction to topics in the same grade that complement each other.

Coherence Think across grades, and link to major topics within grades
Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards. ~Excerpt from The Structure is the Standards Phil Daro, Bill McCallum, Jason Zimba

How will it look different?
Varied problem structures that build on the student’s work with whole numbers 5 = builds to 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 and 5/3 = 5 x 1/3 Conceptual development before procedural Use of rich tasks-applying mathematics to real world problems Effective use of group work Precision in the use of mathematical vocabulary You will see evidence of these throughout the session Learning Progressions

Coherence -Think Across Grades
Here is an example of the coherence for fractions across the grade levels

Coherence -Think Across Domains
Grade 4: Operations and Algebraic Thinking: Students use four operations with whole numbers to solve problems. Students gain familiarity with factors and multiples which supports student work with fraction equivalency. Number and Operations Fractions: Students build fractions from unit fractions by applying and extending previous understandings of operations with whole numbers.

The Structure is the Standards
You have just purchased an expensive Grecian urn and asked the dealer to ship it to your house. He picks up a hammer, shatters it into pieces, and explains that he will send one piece a day in an envelope for the next year. You object; he says “don’t worry, I’ll make sure that you get every single piece, and the markings are clear, so you’ll be able to glue them all back together. I’ve got it covered.” Absurd, no? But this is the way many school systems require teachers to deliver mathematics to their students; one piece (i.e. one standard) at a time. They promise their customers (the taxpayers) that by the end of the year they will have “covered” the standards. ~Excerpt from The Structure is the Standards Phil Daro, Bill McCallum, Jason Zimba

Rigor: Illustrations of Conceptual Understanding, Fluency, and Application
Here rigor does not mean “hard problems.” It’s a balance of three fundamental components that result in deep mathematical understanding. There must be variety in what students are asked to produce. Rigor, as defined here, does not mean hard problems. It doesn’t mean more difficult. Rigor, here, means something very specific. We are talking about the balance of these components of conceptual understanding, fluency, and application. We are going to look at a set of problems; some assess fluency, some require conceptual understanding, and some are examples of application. By working through these problems, we can start seeing what this looks like.

Some Old Ways of Doing Business
Lack of rigor Reliance on rote learning at expense of concepts Severe restriction to stereotyped problems lending themselves to mnemonics or tricks Aversion to (or overuse) of repetitious practice Lack of quality applied problems and real-world contexts Lack of variety in what students produce E.g., overwhelmingly only answers are produced, not arguments, diagrams, models, etc. Either teachers don’t allow for repetitious practice or spend too much on repetitious practice Greta will add a slide about new ways of doing business

Some Old Ways of Doing Business
Concrete Semi Concrete Abstract Unfortunately this model (Jerome Bruner, 1964) was interpreted as giving hierarchal value to the symbolic above the concrete or semi concrete… Which lead to: Abstract Semi Concrete (used to “prove” or show why the abstract worked) and an implication that the concrete was only for those who didn’t “get it”

Conceptual and Procedural Understanding
Desired outcome is a balance that leads to flexible thinking about concepts and an ability to apply knowledge in novel situations Concrete Semi-Concrete Abstract Conceptual and Procedural Understanding We used to teach lower grade students with concrete manipulative objects and then take them away as they matured. Mathematics uses concrete objects at all levels to build understanding and justify reasoning. Going from the linear model to a strong math foundation built on a circular model

How do students currently perceive mathematics?
Doing mathematics means following the rules laid down by the teacher. Knowing mathematics means remembering and applying the correct rule when the teacher asks a question. Mathematical truth is determined when the answer is ratified by the teacher. -Mathematical Education of Teachers report (2012) Coherence is about making sense but currently this is how students perceive mathematics

How do students currently perceive mathematics?
Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less. Ordinary students cannot expect to understand mathematics: they expect simply to memorize it and apply what they have learned mechanically and without understanding. -Mathematical Education of Teachers report (2012)

Redefining what it means to be “good at math”
Expect math to make sense wonder about relationships between numbers, shapes, functions check their answers for reasonableness make connections want to know why try to extend and generalize their results Are persistent and resilient are willing to try things out, experiment, take risks contribute to group intelligence by asking good questions Value mistakes as a learning tool (not something to be ashamed of) We need to redefine what it means to be “good at math” or a good math student

Mathematical Practices
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. The Mathematical Practices support Claims 2-4 the combination of the content with the mathematical practice is how they will operationalize the standards with the assessment.

The foundation for student learning is built using the CCSS for Mathematical Practices. It is important that we as educators provide students with opportunities to experience mathematics in a way that builds understanding. The 8 standards have been grouped into the following categories: 1 and 6 Over arching Habits of Mind of a productive mathematical thinker, Reasoning and Explaining, Modeling and using tools and seeing structure and generalizing.

Standards for Mathematical Practices
Poster Activity Poster Activity: We are going to giver you time to dig deeper into these standards for mathematical practice. Divide teachers into groups of 2 to 4. Assign each group a pair of standards using the visual graphic from the previous slide. They fill out their own 4 quadrant chart and then match up with 1 or 2 other teachers and share. Summarize their discussion and have each group prepare a poster to share with the rest of the group. What will students be doing? What does the teacher need to provide in order to foster these practices. Teachers-opportunities for students and culture of the classroom Standards for Mathematical Practices

Sharing posters, can be done in several ways
Sharing posters, can be done in several ways. Each group can share out orally or you can give them post it notes and proceed through a gallery walk and add notes and comments to the posters. Teachers will often collect ideas from each other so they might like to have time to make some personal notes about what they learned from each other. You can have them talk with an elbow partner and share what they learned during the share out.

Shifts in Focus, Coherence and Rigor in the assessment
Let’s look at the Assessment Shifts in Focus, Coherence and Rigor in the assessment

A Balanced Assessment System
English Language Arts/Literacy and Mathematics, Grades 3-8 and High School School Year Last 12 weeks of the year* DIGITAL CLEARINGHOUSE of formative tools, processes and exemplars; released items and tasks; model curriculum units; educator training; professional development tools and resources; scorer training modules; and teacher collaboration tools. Optional Interim Assessment Optional Interim Assessment PERFORMANCE TASKS ELA/Literacy Mathematics COMPUTER ADAPTIVE TESTS ELA/Literacy Mathematics Computer Adaptive Assessment and Performance Tasks Computer Adaptive Assessment and Performance Tasks Assessment system that balances summative, interim, and formative components for ELA and mathematics: Summative Assessment (Computer Adaptive) Mandatory comprehensive assessment in grades 3–8 and 11 (testing window within the last 12 weeks of the instructional year) that supports accountability and measures growth Selected response, short constructed response, extended constructed response, technology enhanced, and performance tasks Interim Assessment (Computer Adaptive) Optional comprehensive and content-cluster assessment Learning progressions Available for administration throughout the year Formative Processes and Tools Optional resources for improving instructional learning Assessment literacy Scope, sequence, number and timing of interim assessments locally determined Re-take option *Time windows may be adjusted based on results from the research agenda and final implementation decisions.

Time and format Summative: For each content area - ELA & Math
Computer Adaptive Testing (CAT) Selected response (SR), Constructed Response (open-ended—CR, ECR), Technology enhanced (e.g., drag and drop, video clips, limited web- interface) Performance Tasks (like our CBAs) (PT) 1 per content area in grades 3-8 Up to 3 per content area in High School Performance tasks: Performance tasks are scenario-based sets of materials and items/tasks that cohere around a single theme, or real-world problem. The items/tasks comprising each PT require greater amounts of time to administer than more traditional selected- and constructed-response items. The points allocated for these significant events must be commensurate with the time students will spend on them. The content of PTs focus on those aspects of the CCSS that rely on research, problem solving, and application and transfer of knowledge, including higher DOK levels (3 and 4). – Computer adaptive testing: As each student takes a computer adaptive test the test delivery system adjusts the difficulty of the items to be tailored to how the student is performing on the assessment. This means that students do not receive the same items, unlike a “fixed-form” test where all students see the same items. Thus, blueprints for assessments with a CAT component must provide acceptable lower and upper bounds for the number of items to be presented for each assessment target. Our CAT blueprints ensure sufficient breadth and depth at the student level to obtain a reliable score across our claims and full breadth and depth across students at the classroom level. Therefore, the full breadth of content must be taught; however, the burden of testing the entire breadth of content will not be placed on each student.

Assessment Claims for Mathematics
“Students can demonstrate progress toward college and career readiness in mathematics.” Overall Claim (Gr. 3-8) Overall Claim (High School) “Students can demonstrate college and career readiness in mathematics.” Claim 1 Concepts and Procedures “Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.” Claim 2 Problem Solving “Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.” Overall Claims (gr 3-8) refers to the content standards Claim 3 Communicating Reasoning “Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.” 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.”

Claim 1 Concepts and Procedures
Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency. Grade Level Number of Assessment Targets 3 11 4 12 5 6 10 7 9 8 17 Foreshadow that we will be looking at all four claims during the course of the instruction. The number of Claim 1 assessment targets varies across grade levels. The assessment targets for each grade level are presented in detail in the Content Specifications and are explored in greater detail in the Grade Level Considerations training modules. For now, note that careful thought went into examining the progression of mathematical knowledge and skills as students progress from early elementary grades through high school and as students develop college and career readiness. This progression informed the development of each grade level assessment target. Also note that for high school, assessment targets are established for grade 11 only and reflect the skills and knowledge students are expected to demonstrate in order to be college and career ready. To help use assessment targets to inform the development and review of items and tasks, let’s take a closer look at the structure of an assessment target. The number of assessment targets are the same as the number of clusters for each grade level. If you are interested in more information or detail about these assessment targets we have that available on handouts.

Cognitive Rigor and Depth of Knowledge
The level of complexity of the cognitive demand. Level 1: Recall and Reproduction Requires eliciting information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. Level 2: Basic Skills and Concepts Requires the engagement of some mental processing beyond a recall of information. Level 3: Strategic Thinking and Reasoning Requires reasoning, planning, using evidence, and explanations of thinking. Level 4: Extended Thinking Requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. Smarter Balanced items and tasks will elicit evidence that students have the ability to integrate knowledge and skills across multiple assessment targets and are ready to meet the challenges of college and careers. {+} Items and tasks must be constructed at various levels of cognitive rigor. Smarter Balanced has defined four levels of depth of knowledge. The first level focuses on recall and reproduction of facts and other types of information. The second level focuses on basic skills and concepts that require cognitive processes that extend beyond the recall of information. The third level focuses on strategic thinking and reasoning. The fourth and final level requires extended thinking that includes complex reasoning, planning, development, and cognition that occurs over an extended period of time. Let’s take a look at a sample item for each of the four levels of depth of knowledge.

DOK Distribution on SBAC
Grade 4 25% 40% 26% 9% Grade 8 18% 43% 27% 12% High School 41% 23%

What is the depth of knowledge of these tasks?
Looking at SBAC Tasks: Depth of Knowledge and Mathematical Practices, two lenses What is the depth of knowledge of these tasks? Which mathematical practices do they promote? When we look at the SBAC sample items think about these two areas? Use the top part of the deep Task Analysis template

3.NF.A.3a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. What is the depth of knowledge of this task? DOK 1 Recall and Reproduction Which mathematical practices do they promote? MP 3 Construct Viable Arguments and Critique the Reasoning of others and MP 6 Attend to precision 3.NF.A.3a Grade 3 Number and Operations Fraction

5.NF.C.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions What is the depth of knowledge of this task? DOK 2 Basic Skills and Concepts Which mathematical practices do they promote? MP 1 Make Sense of Problems and Persevere in Solving Them , MP 2 Reason abstractly and quantitatively, and MP 4 Model with mathematics

Claim 2 – Problem Solving
Claim 2: Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace Select and use tools strategically Interpret results in the context of the situation Identify important quantities in a practical situation and map their relationships. Claim 2 focuses on Problem Solving. The purpose of this claim is to elicit evidence that students can solve a range of complex well-posed problems in pure and applied mathematics and can make productive use of knowledge and problem solving strategies. Items and tasks written to assessment targets for this claim will ask students to: {+} Apply mathematics to solve well-posed problems arising in everyday life; select and use tools strategically; Interpret results in the context of a situation; And identify important quantities in practical situations and to map their relationships. Items and tasks written for Claim 2 will provide evidence for several of the Claim 2 assessment targets. Each target should not lead to a separate task: it is in using content from different areas, including work studied in earlier grades, that students demonstrate their problem solving proficiency.”

4.NF.B.4c Solve word problems involving multiplication of a fraction by a whole number
Maybe have them use the one on the website so they can interact with it (the image is hyperlinked) Which mathematical practices do they promote? MP 1 Make sense of problems and persevere in solving them. MP 2 Reason abstractly and quantitatively. and MP4 Model with mathematics What is the Depth of Knowledge of this task? DOK 2 Basic Skills and Concepts

Take a Break!

Learning Progression Number and Operations Fractions
The authors of the Common Core created Learning Progressions which describe the mathematics content with it’s focus and coherence

Domain Cluster Heading Cluster Standard
Each standard includes a cluster heading and cluster We are going to dig into resources from the Learning Progression to better understand the progression of the standards related to number and operations of fractions

Sorting Standards You are going to be working in groups to sort mathematical concepts and tasks in a progression, to better understand the Coherence of the standards and the focus at each grade level

Sort the clusters under the standards using the CCSS document
Work in groups of 2 or 3 to sort the “fractions cards” into a progression of concepts. Sort the clusters under the standards using the CCSS document Handouts: Fractions Sorting Cards Fractions Fractions Progression Area to Number for Card Sort Particpants use this document to sort the progression of examples Debief: Which mathematical practices did you use during this activity? Major clusters and supporting clusters

Learning Progression Number and Operations Fractions Unit Fractions
The authors of the Common Core created Learning Progressions which describe the mathematics content with it’s focus and coherence Unit Fractions

What is a unit fraction? Discuss at your table – we will predict and adjust as we move through the materials

Exploring Unit Fractions
atics.org/pages/fractions_pro gression Facilitator’s discretion about whether to include this section

Grade 3 number line The next few slides are ones to high light important concepts at each grade level. 3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (3rd grade spends much time on unit fractions that have 1 as a numerator to compare fractions based on the denominator: 1/6 is smaller than ½ because it takes more of them to make one whole). 3.NF.3 a Understand that two fractions are equivalent if they are the same size, or the same point on a number line

4.NF.3 4.NF.3 4.NF.4 Equivalent to mixed numbers with understanding and not with a procedure 4.NF.3 Add and subtract mixed numbers with like denominators,e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Note that students are not just changing mixed numbers to a fraction but reasoning that the fraction is made of more than one whole . 4. NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.

This leads to 4.NF.4 This leads to 4 NF.4 They are reasoning about how many ¼ are in the whole. Notice that 2 ¾ they don’t use the standard process to make 2 and ¾ into an improper fraction they know that 4/4 is one whole so 2 wholes are 8/4 and ¾ is 11/4.

Note this from the learning progression. No longer do we have to go through Least common denominator in fact it is.

Halves How many ways can you show halves on a Geoboard? (the whole is determined by you) What is the most creative design you can create? How did you determine that it was a half? Which of your halves are equivalent? Why? Using a Geoboard cut it in half using a rubber band. Allow participants to create other ways they can divide the area in half, they can record their ideas on Geo dot paper. After they spend some time exploring have participants share their ideas. The handout has them divide their geoboard into 1/4s and 1/8ths but the video only shows students working with ½. This is an opportunity to develop and discuss the culture of safety being developed within the classroom

Mathematical Practices - looking at video from the lens of SMP # 3
Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. - Read full standard and highlight important ideas 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. We are going to focus on one mathematical practice more deeply. Have participants read the paragraph in the Standards for Mathematical Practice #3 in their standards document

Protocol: Construct viable arguments and critique the reasoning of others.
Review the text and underline the parts that pertain to constructing arguments argumentation and critiquing. Brainstorm the actions you would expect to see if students were critiquing the reasoning of others. Protocol Handout Construct viable arguments and critique the reasoning of others in preparation to watch the video on this Geoboard lesson

Fractions with Geoboards
Fourth- and fifth-graders investigate the concept of halves using the geoboard as an area model. They learn that one-half means two equal-sized parts with equal areas, but that are not necessarily congruent. #37 start at 2:29 and end at 10:00 includes NCTM analysis and identifies Mathematical Practice 3 as the main focus Watching for evidence of critiquing the reasoning of others. Note the time of the activity or interaction so the group can easily find it to analyze. Notes: put link to video resource onto PPT slides when presenting #37 start at 2:29 and end at 10:00 l

Finding Evidence of: Construct viable arguments and critique the reasoning of others.
What makes this activity evidence of critiquing the reasoning of others? What observable conditions, supported critiquing the reasoning of others? What, observable conditions, constrained critiquing the reasoning of others? Share out notes teachers have from their protocol “Construct Viable Arguments and Critiquing the Reasoning of Others” . Bring out that critiquing the reasoning of others is also happening in small groups as students talk to each other Look at slide for reflection from grades 6-8

How did this activity support the big ideas of unit fractions
Determining what represents “whole” Reasoning about the size of a unit fraction based on the meaning of a unit fraction If the whole is represented by a region in the plane (area model) it is possible for identical fractions to be represented by different shapes Turn and talk with your partners and them we will share out with the group

Implications for your students
Consider your lessons over the next few weeks Develop an instructional action intended to improve your instructional practice for constructing viable arguments and critiquing the reasoning of others. Pair share Small group talk Whole group discussion

Can You See? Lets work on some
I am going to put a model on the overhead, and I want you to think yourself what it might be a model of. And after you think of one thing an have it on your paper, see if you can think of what else it might model. Ask them to first work silently on their own. Ask, “Can you see 4/7 of something?” “Could someone come up and show us?” “What is the part?” “What is the whole?” Participants may talk about 4/7, 3/7. Point out that the Singapore materials do something that our textbooks don’t – that is, point out not only that 4/7 Is shaded but also that 3/7 is not shaded and that 7/7 is the whole. Could anyone ask a question, “Can you see …?” The participant said, “Can you see 3 parts to 4 parts?” Could someone come up and explain this?” Can you see 1 and 1/3? 1 and 1/3 of what? What is the whole and what is the part? Can you see ¾? ¾ of what? What is the whole? What is the part? What is ¾ of 4/7? Can anyone thing of another question? Can you see 1 divided by 4/7? Can you see 7/4? Of what? What is 7/4 of 4/7? What summary statements can you make about these relationships?

Can You See? I am going to put a model on the overhead, and I want you to think yourself what it might be a model of. And after you think of one thing an have it on your paper, see if you can think of what else it might model. Ask them to first work silently on their own. Ask, “Can you see 3/5 of something?” “Could someone come up and show us?” “What is the part?” “What is the whole?” Participants may talk about 3/5, 2/5. Point out that the Singapore materials do something that our textbooks don’t – that is, point out not only that 3/5 Is shaded but also that 2/5 is not shaded and that 5/5 is the whole. Could anyone ask a question, “Can you see …?” The participant said, “Can you see 2 parts to 3 parts?” Could someone come up and explain this?” Can you see 1 and 1/2? 1 and 1/2 of what? What is the whole and what is the part? Can you see 5/2? 5/2 of what? What is the whole? What is the part? What is 2/3 of 3/5? Can anyone thing of another question? Can you see 1 divided by 3/5? Can you see 5/3? Of what? What is 5/3 of 3/5? What summary statements can you make about these relationships?

Mathematical Practices
What are the content standards can be addressed at your grade level in this task? What mathematical practices does it promote? 3.NF.A.1, 4.NF.B.4a, 5.NF.B.7b MP 3 Consider using the Deep Task Analysis sheet with this activity.

Using Rich Tasks in the Classroom

What makes a rich task? Is the task interesting to students?
Does the task involve meaningful mathematics? Does the task provide an opportunity for students to apply and extend mathematics? Is the task challenging to all students? Does the task support the use of multiple strategies and entry points? Will students’ conversation and collaboration about the task reveal information about students’ mathematics understanding? Rich tasks are opportunities to increase the cognitive complexity of the instruction in your classroom…but on its own a rich task is not rich unless it is supported by the environment in which it is presented Talk through each of the 6 questions – provide them with a handout Discussion around mathematical endeavors Should address more than one standard Potential to broaden skills and content knowledge Potential for revealing patterns or lead to generalizations Involves the learner in testing, proving, explaining, justifying, reflecting and interpreting Adapted from: Common Core Mathematics in a PLC at Work Larson,, et al

Environment for Rich Tasks
Learners not passive recipients of mathematical knowledge Learners are active participants in creating understanding and challenge and reflect on their own and others understandings Instructors provide support and assistance through questioning and supports as needed This slide is animated, don’t pull up the talking points until participants have had a chance to come up with their own ideas and share them.

Let’s Try a Rich Task Using a task card with students Who Got What?
With your table group engage in this task and predict what sort of entry points and strategies students might use Create a list of misconceptions that might arise Task adapted from Young Mathematicians at Work-Sub Sandwich

Homework Before our next meeting
Use the task “Who Got What?” with your students. Bring back one or two student artifacts - ready to discuss student generated strategies, etc. (Please remove student names) Use the Standards for Mathematical Practice Matrix to reflect on where your classroom falls on the continuum and be ready to discuss any activities you used to move your classroom forward on this scale Please bring your instructional materials for fractions to our next class. The next slide has a planning sheet for them to fill out.

Standards for Mathematical Practice Matrix
Where are the opportunities for students in most of your classroom lessons

Looking at Student Work – Next Session
Protocol Form small groups of 3 or 4 Each person selects 1 or 2 work samples to share with the group You will follow a protocol to review student work which focuses on student understanding Please remove student names from any papers

Reflection and quick write
What are the instructional shifts needed to make these practices a reality? From the work so far today, What are the instructional shifts needed to make these practices a reality?

3/28/2017 Day 2 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices

Outcomes: Day 2 and 3 Analyze, adapt and implement a task with the integrity of the Common Core State Standards. Understand how to analyze student work with the Standards for Mathematical Practice and content standards. Deepen understanding of the progression of learning around Number and Operations - Fractions

The activity centers on mathematical understanding, invention, and sense-making by all students. The culture is one in which inquiry, wrong answers, personal challenge, collaboration, and disequilibrium provide opportunities for mathematics learning by all students. The tasks in which students engage are mathematically worthwhile for all students. A teacher’s deep knowledge of the mathematics content she/he teaches and the trajectory of that content enables the teacher to support important, long-lasting student understanding. Begin session with research-based best practices…we would like to encourage you to engage in personal challenge throughout the sessions, collaborate with your colleagues, welcome disequilibrium and wrong answers, and be open to questions…take them as a challenge to your ideas not to you personally. if this is what needs to occur in the classroom, we are going to try to model this in the workshop. offering you suggestions and techniques to establish a classroom culture that is conducive to learning mathematics. You will notice that these best practices encompass the Standards for Mathematical practice. Could link with the shifts in classroom practice (7) if you are using that in your session.

Looking at Student Work
Form small groups of 3 or 4 Each person selects 1 or 2 works samples to share with the group Follow the Collaboration Protocol to review student work samples and record information observations Inferences implications Handouts for protocol and recording sheet – screen shots are on the next two pages

Collaboration Protocol-Looking at Student Work (55 minutes)
1. Individual review of student work samples (10 min) All participants observe or read student work samples in silence, making brief notes on the form “Looking at Student Work” 2. Sharing observations (15 min) The facilitator asks the group What do students appear to understand based on evidence? Which mathematical practices are evident in their work? Each person takes a turn sharing their observations about student work without making interpretations, evaluations of the quality of the work, or statements of personal reference. 3. Discuss inferences -student understanding (15 min) Participants, drawing on their observation of the student work, make suggestions about the problems or issues of student’s content misunderstandings or use of the mathematical practices. Adapted from: Steps in the Collaborative Assessment Conference developed by Steve Seidel and Project Zero Colleagues Select one group member to be today’s facilitator to help move the group through the steps of the protocol. Teachers bring student work samples with student names removed. 4. Discussing implications-teaching & learning (10 min) The facilitator invites all participants to share any thoughts they have about their own teaching, students learning, or ways to support the students in the future. How might this task be adapted to further elicit student’s use of Standards for Mathematical Practice or mathematical content. 5. Debrief collaborative process (5 min) The group reflects together on their experiences using this protocol. Please remind teachers that they will first focus on evidence for students use of the Mathematical Practices and understanding of content. Example of Evidence: Students drew pictures to represent the problem. Students could … Inference: Students didn’t know how to find a common denominator

Looking at student work
This form is adapted from the book Looking Together at Student Work

What instructional strategies did you use with this lesson?
What Standards for Mathematical Practices did you notice your students engaging in during this task? Using the SMP Matrix how would you describe your classroom – with examples to support

“Students can demonstrate progress toward college and career readiness in mathematics.” Overall Claim (Gr. 3-8) Overall Claim (High School) “Students can demonstrate college and career readiness in mathematics.” Claim 1 Concepts and Procedures “Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.” Claim 2 Problem Solving “Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.” Overall Claims (gr 3-8) refers to the content standards “Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.” Claim 3 Communicating Reasoning 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.”

Revisit the practices - The foundation for student learning is built using the CCSS for Mathematical Practices. It is important that we as educators provide students with opportunities to experience mathematics in a way that builds understanding. The 8 standards have been grouped into the following categories: 1 and 6 Over arching Habits of Mind of a productive mathematical thinker, Reasoning and Explaining, Modeling and using tools and seeing structure and generalizing.

Cognitive Rigor and Depth of Knowledge (DOK)
The level of complexity of the cognitive demand. Level 1: Recall and Reproduction Requires eliciting information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. Level 2: Basic Skills and Concepts Requires the engagement of some mental processing beyond a recall of information. Level 3: Strategic Thinking and Reasoning Requires reasoning, planning, using evidence, and explanations of thinking. Level 4: Extended Thinking Requires complex reasoning, planning, developing, and thinking most likely over an extended period of time. Smarter Balanced items and tasks will elicit evidence that students have the ability to integrate knowledge and skills across multiple assessment targets and are ready to meet the challenges of college and careers. {+} Items and tasks must be constructed at various levels of cognitive rigor. Smarter Balanced has defined four levels of depth of knowledge. The first level focuses on recall and reproduction of facts and other types of information. The second level focuses on basic skills and concepts that require cognitive processes that extend beyond the recall of information. The third level focuses on strategic thinking and reasoning. The fourth and final level requires extended thinking that includes complex reasoning, planning, development, and cognition that occurs over an extended period of time. Let’s take a look at a sample item for each of the four levels of depth of knowledge.

Claim 2 – Problem Solving
Claim 2: Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies. Apply mathematics to solve well-posed problems arising in everyday life, society, and the workplace Select and use tools strategically Interpret results in the context of the situation Identify important quantities in a practical situation and map their relationships. Claim 2 focuses on Problem Solving. The purpose of this claim is to elicit evidence that students can solve a range of complex well-posed problems in pure and applied mathematics and can make productive use of knowledge and problem solving strategies. Items and tasks written to assessment targets for this claim will ask students to: {+} Apply mathematics to solve well-posed problems arising in everyday life; select and use tools strategically; Interpret results in the context of a situation; And identify important quantities in practical situations and to map their relationships. Items and tasks written for Claim 2 will provide evidence for several of the Claim 2 assessment targets. Each target should not lead to a separate task: it is in using content from different areas, including work studied in earlier grades, that students demonstrate their problem solving proficiency.”

4.NF.B.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b
This comes back to the work on day one with unit fractions MP 1? DOK 2?

Claim 3 – Communicating Reason
Claim 3: Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Test propositions or conjectures with specific examples. Construct, autonomously, chains of reasoning that justify or refute propositions or conjectures. State logical assumptions being used. Use the technique of breaking an argument into cases. Distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in the argument—explain what it is. Base arguments on concrete referents such as objects, drawings, diagrams, and actions. Determine conditions under which an argument does and does not apply. Claim 3 focuses on Communicating Reasoning. The purpose of this claim is to elicit evidence that students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Items and tasks written for this claim will ask students to explain his or her reasoning, justify a conjecture, and assess the validity of a claim. The Claim 3 targets require students to: {+} Test propositions or conjectures; Construct chains of reasoning that justify or refute propositions or conjectures; State logical assumptions that are made; Use techniques of breaking arguments into cases; Distinguish correct logic and flawed reasoning and explain what it is Base arguments on concrete referents And determine conditions under which an argument does and does not apply. Items and tasks written for Claim 3 will provide evidence for several of the Claim 3 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 3 in a given grade will draw upon knowledge and skills articulated in the standards in that same grade, with strong emphasis on the major work of the grade.”

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size MP 3 and 6 DOK 3

Shift Two: Coherence Think across grades, and link to major topics within grades
Carefully connect the learning within and across grades so that students can build new understanding onto foundations built in previous years. Begin to count on solid conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning. Refer back to the Grecian Urn article (could have them read this)

Fix this slide.

Focus and Coherence through the Major and Supporting Clusters of the CCSS-M
Discuss with a partner which: Major, Supporting, and/or Additional Clusters are involved in the task Major Numbers and Operations with Fractions—developing understanding of fractions as numbers Supporting Geometry—reasoning with shapes and their attributes

Exploring Equivalent Fractions
atics.org/pages/fractions_pro gression Facilitator discretion about the inclusion of these ideas

Justify the truth of this statement in at least two different ways.
See if they solve it based on number sense i.e. 2/5 is less than ½ so ½+1/3 must be greater than 2/5 Strategies: Reasoning based on Numerators Denominators Benchmark fractions Reaso The clock can be useful for this.—if they don’t come up with alternatives, we will return to it after string work, after the work ask them about how their responses have changed and what might have affected their understanding.

Strings How does this support the development of efficient computational strategies and overall number sense? What mathematical practices does it support? Use the clock to work through this string and the context of exercising or some other context dealing with time (see notes slide following this one) 1/2 + 1/3 1/2 + 3/4 2/3 + 1/4 5/12 + 1/4 35/60 + 5/12 3/4 + 5/6 The clock allows children to convert common fractions into whole numbers (minutes), operate with them, and then convert them back to fractions. This model inherently provides a common denominator of 60. 5.NF.A MP 3 and 5

Presenter Notes for String
List one problem at a time: 1/2 + 1/3 “I am exercising for the New Year and have decided that I am going run for 1/2 hour and walk for 1/3 hour. How long did I exercise? What fraction of a hour is that?” Have participants use thumbs up when they know an answer and can tell how they arrived at it. Model it on the clock “I knew that 1/2 hour was 30 min (draw a circle and cut in half) I knew that 1/3 hour was 20 min (add the 20 min onto the 30 so that your minute hand is pointing to the ‘10’ on the clock) My answer is 10/12 of an hour (or 5/6 or 50/60) Continue with the scenario of exercising and modeling for the remaining problems as you finish the string

Exploring Adding Fractions
atics.org/pages/fractions_pro gression

Reflect back on 1/2 + 1/3, do you have another way of proving that it is not equal to 2/5?
What might have affected the way that you changed your response?

Justify the truth of this statement in at least two different ways.
Return to the 1/3 + 1/2 asking How might their responses change, what might affect that change

Does the task involve meaningful mathematics? Does the task provide an opportunity for students to apply and extend mathematics? Is the task challenging to all students? Does the task support the use of multiple strategies and entry points? Will students’ conversation and collaboration about the task reveal information about students’ mathematics understanding? Talk through each of the 6 questions – provide them with a handout Discussion around mathematical endeavors Should address more than one standard Potential to broaden skills and content knowledge Potential for revealing patterns or lead to generalizations Involves the learner in testing, proving, explaining, justifying, reflecting and interpreting Adapted from: Common Core Mathematics in a PLC at Work Larson,, et al

Learners not passive recipients of mathematical knowledge Learners are active participants in creating understanding and challenge and reflect on their own and others understandings Instructors provide support and assistance through questioning and supports as needed

Task Analysis Protocol Sheet

Fraction Tracks Play Fraction Tracks (access online)
Video Add protocol for video review NCTM Illuminations has Fraction Tracks Encourage games for practice of these skills Fraction Tracks is available to play online. Ann has cards to use for hard copies

How did this activity support the big idea of equivalent fractions
Reasoning about the size of a unit fraction based on the meaning of a unit fraction Turn and talk with your partners and then we will share out with the group

Using your materials modify or create a rich task We will review the homework briefly on Day Three

Homework Before our next meeting
Use the task you modified with your students Bring back one or two student artifacts - ready to discuss student generated strategies, etc. (Please remove student names) Use the Standards for Mathematical Practice Matrix to reflect on where your classroom falls on the continuum and be ready to discuss any activities you used to move your classroom forward on this scale The next slide has a planning sheet for them to fill out.

3/28/2017 Day 3 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices

Outcomes: Day 2 and 3 Analyze, adapt and implement a task with the integrity of the Common Core State Standards. Understand how to analyze student work with the Standards for Mathematical Practice and content standards. Deepen understanding of the progression of learning around Numbers and Operations - Fractions

Looking at Student Work
From small groups of 3 or 4 Each person selects 1 or 2 works samples to share with the group Follow the Collaboration Protocol to review student work samples and record information observations Inferences implications Slides 132 – 134 Only if sessions 1 and 2 are separate days

Collaboration Protocol-Looking at Student Work (55 minutes)
1. Individual review of student work samples (10 min) All participants observe or read student work samples in silence, making brief notes on the form “Looking at Student Work” 2. Sharing observations (15 min) The facilitator asks the group What do students appear to understand based on evidence? Which mathematical practices are evident in their work? Each person takes a turn sharing their observations about student work without making interpretations, evaluations of the quality of the work, or statements of personal reference. 3. Discuss inferences -student understanding (15 min) Participants, drawing on their observation of the student work, make suggestions about the problems or issues of student’s content misunderstandings or use of the mathematical practices. Adapted from: Steps in the Collaborative Assessment Conference developed by Steve Seidel and Project Zero Colleagues Select one group member to be today’s facilitator to help move the group through the steps of the protocol. Teachers bring student work samples with student names removed. 4. Discussing implications-teaching & learning (10 min) The facilitator invites all participants to share any thoughts they have about their own teaching, students learning, or ways to support the students in the future. How might this task be adapted to further elicit student’s use of Standards for Mathematical Practice or mathematical content. 5. Debrief collaborative process (5 min) The group reflects together on their experiences using this protocol. Please remind teachers that they will first focus on evidence for students use of the Mathematical Practices and understanding of content.

Looking at student work

Which Fraction Pair is Greater?
Think like a fourth grader Give one or more reasons for the comparison Try not to use models or drawings DO NOT USE cross multiplication Elementary and Middle School Mathematics VandeWalle, p. 311 Alternate slides for Session 3 following in the same day as Session 2

Create a poster with comparison method and problem
Create a poster that shows which method they used and which problems they used that method with.

Ways In Which To Compare
Same size whole (same denominator) - B&G Same number of parts (same numerator) but different sized wholes – A,D,& H More than/less than one-half or one – A,D,F,G, and H Closeness to one-half or one – C,E,I,J,K, and L

More or Less Find a partner Place the cards face down
Take turns drawing a card and deciding which fraction is more or less Sort the cards by strategies used to solve them 9:55-10:05 4.NF.2 Compare two fractions with different numerators and different denominators 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. Justify by using a visual model.

Read a story and explore the math
Read Beasts of Burden from The Man who Counted by Tahan 1993 Pass out the camel cards and have students model the story Rusty Bresser in Math and Literature describes three days of activities with fifth graders, based on this story

Exploring Multiplying Fractions
atics.org/pages/fractions_pro gression Semi-concrete explanation Multiplying part 2 Abstract explanation Multiplying part 1 Facilitator discretion

Paper folding I planted half my garden with vegetables this summer. One third of the half that is vegetables is planted with green beans. What fraction of the whole garden is planted with green beans? Math Matters- use a piece of paper to model this action. Multiplication with whole numbers and a fraction follows the repeated addition using whole numbers When two fractions are multiplied it is based on a fraction as an operator. A fraction is operating on another number and changes the other number. The use of the word “of” to multiply is based on the operator interpretation of fraction: when we multiply ½ and 8 we are taking one half of eight.

Math Matters When two fractions are multiplied it is based on a fraction as an operator. A fraction is operating on another number and changes the other number. The use of the word “of” to multiply is based on the operator interpretation of fraction: when we multiply ½ and 8 we are taking one half of eight. p. 125 Math Matters Chapin and Johnson

Finding a fraction of a fraction
Conceptually – with no subdivisions Pictures – YES Algorithms - NO

Finding a fraction of a fraction

Exploring Dividing Fractions
atics.org/pages/fractions_pro gression This three minutes covers a huge conceptual leap for most elementary teachers Fraction as a division problem ie Two whole numbers written as a fraction give the same number as the result of taking the numerator divided by the denominator And why the invert and multiply rule works

Dividing by one –half Illustrative Math Example
Shauna buys a three-foot-long sandwich for a party. She then cuts the sandwich into pieces, with each piece being 1/ 2 foot long. How many pieces does she get? Phil makes 3 quarts of soup for dinner. His family eats half of the soup for dinner. How many quarts of soup does Phil's family eat for dinner? A pirate finds three pounds of gold. In order to protect his riches, he hides the gold in two treasure chests, with an equal amount of gold in each chest. How many pounds of gold are in each chest? Leo used half of a bag of flour to make bread. If he used 3 cups of flour, how many cups were in the bag to start? Illustrative Math Example

Video—Cathy Humphreys
Defending Reasonableness: Division of Fractions 1 ÷ 2/3 Use the Standards for Mathematical Practice Matrix to reflect on where this classroom falls on the continuum and be ready to discuss any activities you could use to move this classroom forward on the scale Defending Reasonableness Division of Fractions Chapter 4 Connecting Mathematical Ideas by Jo Boaler and Cathy Humphreys Can use Video Reflection Recording Sheet if desired

Standards for Mathematical Practice Matrix
Where are the opportunities for students in most of your classroom lessons

Claim 4 – Modeling and Data Analysis
Claim 4: Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Apply mathematics to solve problems arising in everyday life, society, and the workplace. Construct, autonomously, chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for a complex problem. State logical assumptions being used. Interpret results in the context of a situation. Analyze the adequacy of and make improvement to an existing model or develop a mathematical model of a real phenomenon. Identify important quantities in a practical situation and map their relationships. Identify, analyze, and synthesize relevant external resources to pose or solve problems. Claim 4 focuses on modeling and data analysis. Claim four requires extended response items and performance tasks that elicit evidence that students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Items and tasks written for this claim will ask students to investigate problems that have more than one solution pathway, summarize the results within the context of the problem, and evaluate the solution within the context of the problem. The assessment targets associated with Claim 4 require students to: {+} Apply mathematics to solve problems arising in everyday life; Construct chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for complex problems; State logical assumptions that are made; Interpret results in the context of a situation; Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon; Identify important quantities in a practical situation and map their relationships; And identify, analyze, and synthesize relevant resources to pose or solve problems. Items and tasks written for Claim 4 will provide evidence for several of the Claim 4 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 4 in a given grade will draw upon knowledge and skills articulated in the progression of standards up to that grade, with strong emphasis on the “major” work of the grades. Now, let’s shift our focus from the claims and assessment targets specified in the Content Specifications to the additional information presented in the Item Specifications.

“Students can demonstrate progress toward college and career readiness in mathematics.” Overall Claim (Gr. 3-8) Overall Claim (High School) “Students can demonstrate college and career readiness in mathematics.” Claim 1 Concepts and Procedures “Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.” Claim 2 Problem Solving “Students can solve a range of complex well-posed problems in pure and applied mathematics, making productive use of knowledge and problem solving strategies.” Overall Claims (gr 3-8) refers to the content standards Claim 3 Communicating Reasoning “Students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others.” 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.”

Looking at SBAC Performance Task: Depth of Knowledge and Mathematical Practices, three lenses
What are the specific content standards in this performance task? What is the depth of knowledge of these tasks? Which mathematical practices do they promote? When we look at the SBAC sample items think about these two areas? Use the top part of the deep Task Analysis template

Planting Tulips DOMAINS: Operations and Algebraic Thinking, Number and Operations—Fractions, Measurement and Data

Does the task involve meaningful mathematics? Does the task provide an opportunity for students to apply and extend mathematics? Is the task challenging to all students? Does the task support the use of multiple strategies and entry points? Will students’ conversation and collaboration about the task reveal information about students’ mathematics understanding? Rich tasks are opportunities to increase the cognitive complexity of the instruction in your classroom…but on its own a rich task is not rich unless it is supported by the environment in which it is presented Talk through each of the 6 questions – provide them with a handout Discussion around mathematical endeavors Should address more than one standard Potential to broaden skills and content knowledge Potential for revealing patterns or lead to generalizations Involves the learner in testing, proving, explaining, justifying, reflecting and interpreting Adapted from: Common Core Mathematics in a PLC at Work Larson,, et al

Learners not passive recipients of mathematical knowledge Learners are active participants in creating understanding and challenge and reflect on their own and others understandings Instructors provide support and assistance through questioning and supports as needed This slide is animated, don’t pull up the talking points until participants have had a chance to come up with their own ideas and share them.

Using your materials modify or create a rich task or one that is more conceptually focused for your students

Top Resources for Math Educators
RMC website: OPSI website:

3/28/2017 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices.

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3/28/2017 CCSS-M in the Classroom: Grades 3-5 Number and Operations Fractions Weaving Content and Standards for Mathematical Practices.

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