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Grades 1 and 2 Mathematical Standards for Operations and Algebraic Thinking. June 2014. Introduction.

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## Grades 1 and 2 Mathematical Standards for Operations and Algebraic Thinking

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**Grades 1 and 2Mathematical Standards forOperations and**Algebraic Thinking June 2014**Introduction**“Teachers need to understand the progressions in the standards, so they can see where individual students and groups of students are coming from and where they are going. Progressions disappear when standards are torn out of context and taught as isolated events.” -Achievethecore.org**Session Goals**Participants will: • Develop an understanding of trajectories to deepen content knowledge of Operations and Algebraic Thinking. • Engage in lessons; and • Reflect on the learning in relationship to the CCSS in mathematics and the Math Practices.**Rationale**Key Shifts of the Common Core State Standards Focus Coherence Rigor achievethecore.org achievethecore.org**Rationale**Focus Teachers focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom.**Rationale**Coherence The standards are designed around coherent progressions from grade to grade. Principals and teachers carefully connect the learning across grades so that students can build new understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual understanding of core content and build on it. Each standard is not a new event, but an extension of previous learning.**Rationale**Rigor Conceptual understanding: Teachers support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures.**Rationale**Rigor (continued) Procedural skill and fluency: The standards call for speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions such as single-digit multiplication so that students have access to more complex concepts and procedures.**Rationale**Rigor (continued) Application- 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.**Rationale**Mathematical proficiency…has five strands: • Conceptual understanding- comprehension of mathematical concepts, operations, and relations • Procedural fluency • Strategic competence • Adaptive reasoning • Productive disposition (Kilpatrick, Swafford, & Findell, 2001, p.5)**Overview of Activities:**The Standards for Mathematical Practice The Trajectory- Understanding Parts of Numbers The Trajectory- Understanding Place- Value Situations and Strategies-**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.**Mathematical Practice 1**MP1 – Make sense of problems and persevere in solving them • “Does this make sense?” • Explain the meaning of the problem and look for entry points to its solution.**Mathematical Practice 2**MP2- Reason abstractly and quantitatively. • The ability to contextualize and decontextualize. • Make sense of quantities and their relationships in problem situations. • Know and flexibly use different properties of operations and objects.**Mathematical Practice 3**MP3- Construct viable arguments and critique the reasoning of others. • Construct arguments using objects, drawings, diagrams, and action. • Justify conclusions, communicate them to others, and respond to the arguments of others.**Mathematical Practice 4**MP4- Model with mathematics. • Apply the math you know to solve problems in everyday life. • Decide which operation applies to a particular context, and why.**The Trajectory- Understanding Parts of Numbers**• Identifying parts of numbers (1.OA.C.6) (2.OA.B.2) • Combining parts of numbers (1.OA.C.6) (2.OA.B.2) • Decomposing numbers (1.OA.C.6) (2.OA.B.2) • Using symbols (1.OA.D.7) (2.OA.A.1)**The Trajectory- Understanding Parts of Numbers**• Identifying parts of numbers (1.OA.C.6) (2.OA.B.2) • Combining parts of numbers (1.OA.C.6) (2.OA.B.2) • Decomposing numbers (1.OA.C.6) (2.OA.B.2) • Using symbols (1.OA.D.7) (2.OA.A.1)**Preparing for Addition and Subtraction**• Trajectory for Understanding Number Relationships • 1. Identifying parts of numbers • * Recognizes and describes parts contained in larger numbers (conceptual subitizing) • Subitizing is the ability to 'see' a small amount of objects and know how many there are without counting. Play Snapshots**Identifying parts of numbers (1.OA.C.6) (2.OA.B.2)**Title- Snapshots Trajectory- How is this lesson/activity situated in the bigger mathematical picture? Math Goal- What will students know and understand after today’s lesson? Evidence: How will you know that learners understand? What evidence will you collect? Relevant standards and math practices. Adapted form teachersdg.org**Identifying parts of numbers (1.OA.C.6) (2.OA.B.2)**Title- Snapshots Launch/Set Up (grouping, time, materials, actions) Investigate (materials, actions) Summarize/Close (materials, actions) Exit Task (what will be your next steps?)**Identifying parts of numbers (1.OA.C.6) (2.OA.B.2)**Title- Snapshots Do the math. The point is to examine your mathematical thinking about the math. • What are ways the task can be approached/solved correctly? • What strategies/ideas can or cannot be generalized and why or why not?**Identifying parts of numbers (1.OA.C.6) (2.OA.B.2)**Title- Snapshots Anticipate student struggles. • What correct and incorrect student strategies and responses do you anticipate? Why? • What do you think students will struggle with?**Preparing for Addition and Subtraction**• Trajectory for Understanding Number Relationships 2. Combining Parts of Numbers • *Counts on to determine total • *Knows the amount is not changed when a number is broken apart and recombined in various ways. • *Combines parts using related combinations. • Game Double Compare**The Trajectory- Understanding Parts of Numbers**• Identifying parts of numbers (1.OA.C.6) (2.OA.B.2) • Combining parts of numbers (1.OA.C.6) (2.OA.B.2) • Decomposing numbers (1.OA.C.6) (2.OA.B.2) • Using symbols (1.OA.D.7) (2.OA.A.1)**Preparing for Addition and Subtraction**• Trajectory for Understanding Number Relationships • 3. Decomposing Numbers • *Identifies the missing parts of numbers (develops more slowly) by using related combinations • Play Start with/Get to and Drop the Chips**Preparing for Addition and Subtraction**• Trajectory for Understanding Number Relationships • 4. Using Symbols • * Uses equations to record combining and taking away parts • (2+3=5 and 5=2+3)**Reflections**• What was your biggest insight or learning in this module? • What one thing will you do differently based on your understanding of this module’s content and meeting the rigor of the TN-ELDS or the CCSS for Math? • What questions do you still have about this content?**The Trajectory- Place Value**• Understanding ten as a unit (1.NBT.B.2a, 1.NBT.B.2c) • Understanding the structure of 1 ten and some ones (1.NBT.B.2b) • Understanding the structure of tens and ones (2.NBT.B.5)**Understanding the structure of tens and ones**(1.NBT.A.1,1.NBT.B.2c) (2.NBT.B.5) Title- Cover the Square Trajectory- How is this lesson/activity situated in the bigger mathematical picture? Math Goal- What will students know and understand after today’s lesson? Evidence: How will you know that learners understand? What evidence will you collect? Relevant standards and math practices. Adapted form teachersdg.org**Reflections**• What was your biggest insight or learning in this module? • What one thing will you do differently based on your understanding of this module’s content and meeting the rigor of the TN-ELDS or the CCSS for Math? • What questions do you still have about this content?**Situations and Strategies**• Methods used for solving single-digit addition and subtraction problems. • Count All- Level 1 • Count On- Level 2 • *Students should see the first addend embedded in the total-counting words rather than objects • Recompose- Level 3- Convert to an easier problem • How can I use what I know to solve what I don’t know?**Strategies**Methods used for solving single-digit addition & subtraction problems.**Situations and Strategies**Situations: For Grade 1: • Experience with all the problem situations, including all subtypes and language variants (master all but unshaded) • These problems involve single-digit numbers • Solve easier problem subtypes with Level 3 methods (most important- involving making a ten)**Situations and Strategies**Situations: For Grade 2: • Master all problem situations and their subtypes and language variants • Addition and subtraction within 20- Level 3 methods or “just know” • Numbers involved are within 100 represented with diagrams and/or equations • Problems within 100, use developing place value skills and understandings • Solve two-step problems, especially with single-digit addends- No two step problems with the most difficult subtypes and variants**Situations and Strategies**Grade 1 1.OA.A.1; 1.OA.A.2 With connections to 1.0A.B.3 and 1.OA.C.6 MP 1, MP 2, MP 3, MP 4, MP 5, MP 8 • What this is showing is that in grade 1- using addition and subtraction w/in 20 to solve word problems involving situations and adding 3 whole #s = or less than 20 is related to applying properties of operations and demonstrating fluency. ex. 8+6= 8+2+4=10+4=14**Situations and Strategies**Grade 2 2.OA.A.1 With connections to 2.NBT.B.5 MP 1, MP 2, MP 3, MP 4, MP 5, MP 8 Using addition and subtraction w/in 100 to solve 1 or 2 step word problems connects to fluently adding and subtracting w/in 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.**Properties**• Identity property of addition (e.g., 6 = 6 + 0) • Identity property of subtraction (e.g., 9 – 0 = 9) • Commutative property of addition (e.g., 4 + 5 = 5 + 4) • Associative property of addition (e.g., 3 + 9 + 1 = 3 + 10 = 13)**Meaning of Fluency in each grade…**• Just knowing some answers • Knowing some answers from pattern (e.g., “adding 0 yields the same number”) • Knowing some answers from the use of strategies. • The important press toward fluency should also allow students to fall back on earlier strategies when needed. (Relating addition to subtraction).Our goal for students is to be accurate, efficient, flexible problem solvers.**Progressions….**• The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics..**Looking at Operations and Algebraic Thinking in the**Progressions K-2 The addition and subtraction situations present problems in context. These contexts, and the models, representations, and equations used to solve them: • Help students visualize the structure of the problem as a whole. • Help students “see” relationships between addition and subtraction • Help students begin to develop more flexible strategies to solve addition and subtraction problems**Representations- Ways to capture an abstract math concept or**relationship • Representations can help students keep track of the steps used to solve the problems. • Representations help students have a sense of whether their answer is or isn’t accurate. • *Can be visible, such as a number sentence, manipulative, or graph**Preparing to Pose a Story Problem**• Tell students that you are going to tell them a story. Encourage them to imagine the story in their minds as you tell it. Read the problem out loud. • Imagine what is happening. Don’t solve it yet. Turn and talk. Explain what they think is happening in the problem. • Ask for volunteers to retell the story in their own words. What information do we know? What are we trying to find out? Ask students to think about what happens with the two quantities. • Will the answer to this problem be more or less than ___ (an amount in the story)? Why?**The Lesson: Add To with Result Unknown**Title- Add To with Result Unknown Trajectory- How is this lesson/activity situated in the bigger mathematical picture? Math Goal- What will students know and understand after today’s lesson? Evidence: How will you know that learners understand? What evidence will you collect? Relevant standards and math practices. Adapted form teachersdg.org

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