Pathways to Teacher Leadership in Mathematics Monday, July 7, 2014 Laying the Foundation for Fluency: Single-Digit Addition and Subtraction Common Core State Standards for Mathematics Pathways to Teacher Leadership in Mathematics Monday, July 7, 2014

Learning Intentions & Success Criteria We are learning to… deepen our understanding of the meaning of fluency of single-digit addition and subtraction. We will be successful when we can … use visual models to support the development of fluency with single-digit addition and subtraction. verbalize CCSSM fluency expectations for addition and subtraction for Grades K-6. clarify OA connections to fluency for addition and subtraction basic facts.

K-2 Standards Progression: Fluency

A Content Standards Progression Domain: Operations and Algebraic Thinking (OA) Clusters: K: Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. 1: Understand and apply properties of operations and the relationship between addition and subtraction. 1& 2: Add and subtract within 20. Standards: K.OA.3; K.OA.4; 1.OA.4; 1.OA.6; 2.OA.2 Clusters phrases highlight the “big ideas” that groups together a set of standards within and across grade levels. 4

Let’s start with the end in mind… Standard 2.OA.2: Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. “Acquiring proficiency in single-digit arithmetic involves much more than memorizing.” (Adding It Up, NRC, 2001, p. 6) What is that “more” and how do we help students get there? We end with this and the question at the end is rhetorical and hopefully they see the connection.

Laying the Foundation in Kindergarten K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

Prompting quantitative reasoning a.k.a “number relationships” Dot Patterns & Ten Frames Play “Flash” How many dots did you see? How did you see it? What’s the math? What’s the math? (decomposing quantities 0 – 10, subitizing, cardinality, part/whole, combinations of 10, making sense of number relationships.)

Dot Pattern How many dots? How did you see it?

How many dots? How did you see it?

How many dots? How did you see it?

Ten Frames Ten frames show relationships of small numbers to five and ten.

How many dots? How did you see it?

How many dots? How did you see it?

How many dots? How did you see it?

Laying a foundation for understanding Morgan, Dot Plates, & Ten Frames Enjoy! Command-F to get to full screen. Select chapter; click on “menu” when ready to switch, don’t click stop.

Through the lens of the Math Content K.OA.3: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings… K.OA.4: For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings... What is Morgan understanding and what is she able to do?

Through the lens of the Math Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Use appropriate tools strategically. How is Morgan engaged in these practices while working with the content of decomposing numbers?

Developing Conceptual Understanding for Addition and Subtraction Strategies 1.OA.6

Grade 1: 1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Optional slide….. Point being, not all standards are created equal and this one standard might require attention all school year in first grade and be a major component of your instruction and assessment.

Strategies for Single Digit Addition Make a ten. Use an easier “equivalent” problem. Use fives Use a helping fact Use doubles Transform the problem in some way

8 + 6 Strategies: Make a ten. Use a double. Use fives. Put 8 counters on your first frame & 6 counters on your second frame. Strategies: Make a ten. Use a double. Use fives. Use some other equivalent problem.

Make a ten: 8 + 6 How could you make a ten?

Make a ten: 8 + 6 How could you make a ten? Move 2 counters to the top frame. Then you have 10 and 4 more counters. Write an equation. 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14

Use a double: 8 + 6 What doubles might you use? Reason 6 + 6 = 1 Then add 2 more. Reason 6 + 6 = 12. Write an equation. 8 + 6 = 6 + 6 + 2 = 12 + 2 = 14

Use fives: 8 + 6 Can you see some fives? Where? Reason: 5 + 5 is 10; need to add 3 more and 1 more. Write an equation. 8 + 6 = 5 + 3 + 5 + 1 = 5 + 5 + 3 + 1 = 10 + 4 = 14

7 + 9 6 + 7 Select a problem. Draw a strategy card for the group. 7 + 9 6 + 7 Select a problem. Draw a strategy card for the group. Everyone uses ten frames and counters to reason through the strategy and writes an equation(s) that shows the reasoning. Share, compare, and discuss as a group. Repeat with another strategy card. Strategy Cards: Counting on, Make a ten, Use a double, Use fives Reflect: Which strategies seem to work best for each problem?

Decompose to ten: 15 – 6 Place 15 counters on the double ten frame. Completely fill one frame, place 5 on the other frame. CCSS language: “decomposing a number leading to a ten”

Decompose to ten: 15 – 6 How can you remove 6 counters in parts by decomposing it in a way that gets you to or “leads to a ten”? Remove 5 counters to get to ten. CCSS language: “decomposing a number leading to a ten” Remove 1 more. Write an equation. 15 – 5 – 1 = 9 or 15 – 5 = 10; 10 – 1 = 9

---------------------------- Try it: 13 – 5 16 – 7 Use ten frames and counters to reason through the “Decompose to Ten” strategy. Write an equation(s) to show the reasoning. Share and discuss in your small group. ---------------------------- Brainstorm: What other subtraction facts would lend themselves well to this strategy? Make a list of facts and try them out.

Building Procedural Fluency from Conceptual Understanding

Homework Reflection -- Fluency • One key idea related to “fluency” and one question or wondering about developing fluency with your students. • One key message from the Thornton article on basic facts.

Fluency Expectations Complete the stem. Being fluent means… Turn and share your thoughts.

CCSSM K-6: Key Grade Level Fluency Expectations Source: www.achievethecore.org

Launching from the OA Domain Why is fluency for single digit operations found in the Operations and Algebraic Thinking Domain? What does this clarify for us as teachers?

A Press Toward Fluency Fluency in each grade involves a mixture of just knowing some answers, knowing some answers from patterns (e.g., “adding 0 yields the same number”), and knowing some answers from strategies. It is important to push sensitively and encouragingly toward fluency…recognizing that fluency will be a mixture of these kinds of thinking which may differ across students. –OA Progressions p. 18 p. 18 of progressions document - Silently read. What does this mean Kindergarten through 2nd grade? Talk to each other – in what way in K, 1st and 2nd will they tap into the mixture above. Kindergarten – dot patterns, part whole, story problems – connected to the milestones 1st grade – beginning of 1st grade to end of 1st grade 1OA6 – may know their doubles, they may +1, +2, partners of 10. Based on our work so far: What ways might this “mixture” evolve in Kindergarten and 1st Grade students? Justify your thinking for K and 1st grade with examples from class activities and discussions

Computational Fluency Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently. --Principles to Actions, 2014, p. 42 When fluent, students may use visual images or objects to direct or constrain their method or self-correct errors. --Focus in Grade 2: Teaching with Curriculum Focal Points, 2011, p. 61

Reflect How will a focus on MP7 and MP8 support fluency in addition and subtraction basic facts? What implications does this knowledge have for all students K-8 in regard to achieving fluency for grade level expectations?

Learning Intentions & Success Criteria We are learning to… deepen our understanding of the meaning of fluency of single-digit addition and subtraction. We will be successful when we can … use visual models to support the development of fluency with single-digit addition and subtraction. verbalize CCSSM fluency expectations for addition and subtraction for Grades K-6. clarify OA connections to fluency for addition and subtraction basic facts.

Disclaimer Pathways to Teacher Leadership in Mathematics Project University of Wisconsin-Milwaukee, 2014-2017   This material was developed for the Pathways to Teacher Leadership in Mathematics project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II Improving Teacher Quality Program.