Focus Whole Number Computation Strategies for Computing Progression of Addition and Subtraction Decomposition Partial Product and Partial Quotient Leveraging Partial Understanding

Whole Number Computation Big ideas Flexible methods of computation involve taking apart and combining numbers in a wide variety of ways. Most are based on place value or “compatible” numbers. Invented strategies are flexible methods of computing that vary with the numbers and the situation. Flexible methods for computation require a good understanding of the operations and properties of the operations, especially the commutative property and the distributive property. The traditional algorithms are clever strategies for computing that have been developed over time. Each is based on performing the operation on one place value at a time with transitions (trades, regroupings) to adjacent positions.

Strategies for Computing Direct Modeling Counts by ones Use of base-ten models Traditional Algorithms Usually require guided development Strategies Supported by written recordings Mental Methods when appropriate

The POWER of DOUBLES!

Multiplication Facts Conceptually

Decomposition and Partial Products 10 5 100 100 50 100 50 10 + 5 165 10 10 5 1

Decomposition and Multiplication 28 x 36 Using partial products, how many products will you have? Model your thinking.

Multiplication with Partial Products 57 x 42 57 X 42 14 (7x2) 100 (50x2) 280 (40x7) 2000 (40x50) 2394 40 2 50 50 x 40 = 2000 50 x 2 = 100 7 7 x 40 = 280 7 x 2 = 14

Traditional Algorithm For Multiplication Using your base 10 pieces show and record: 24 x 32 Discuss: How does all the conceptual work support what you are showing and recording? How does this affect decisions about teaching the traditional algorithm in your classroom?

Strategies for Computing Multiplication Computation strategies for multiplication are more complex Students need to think about NUMBERS not DIGITS The ability to decompose numbers in flexible ways is even more important The Distributive Property deepens conceptual understanding Students need ample opportunities to make sense of multiplication

Choose a Model – Choose a Strategy Partitioning (decomposing) Strategy for Multiplication Partial Products Area Model 84 x 5 68 x 6 12 x 28 72 x 14

Progression of Multiplication https://gfletchy.com/progression-videos/

Strategies for Division Even though many adults think division is the most onerous of the computational operations, it can be considerably easier than multiplication! Partition or Fair Shares The bag has 783 jelly beans, and Aidan and her four friends want to share them equally. How many jelly beans will Aidan and each of her friends get? Measurement or Repeated Subtraction Jumbo the elephant loves peanuts. His trainer has 625 peanuts. If he gives Jumbo 20 peanuts each day, how many days will the peanuts last? Discuss the difference between these problems. What makes them difficult? What makes them accessible?

Area Model with Decomposition 32 ÷ 4 224 ÷ 14 5 3 10 5 1 +10 12 224 84 30 2 14 -140 84 - 70 14 - 20 10 - 14 4 14 5 + 3 8 10 + 5 + 1 16

Decomposition and Division

Examples

Try It! Where do you see the connections? Using Partial Quotients: 2862 ÷ 6 Try Using the Traditional Algorithm: 2862 ÷ 6 Where do you see the connections?

Progression of Division

Traditional Algorithm for Division Turn and Talk: How does this model set the stage for the traditional algorithm?

Traditional Algorithm for Division Using your base 10 pieces to show and record: 2862 ÷ 7 Discuss: How does all the conceptual work support what you are showing and recording? How does this affect decisions about teaching the traditional algorithm in your classroom?

Turn and Talk How does decomposition support conceptual understanding? How does using multiple strategies support conceptual understanding?

Mastery vs. Partial Understanding What is something you have truly mastered?

Leveraging Partial Understanding Dr. Megan Franke, UCLA

The Idea of Leveraging Partial Understanding Builds on partial understandings Allows for innovation – try and try again Includes all students’ learning Allows students to show what they DO know in order to build on it Allows students to do tasks again before interventions Creates a broader view of what counts as knowing and understanding Students see themselves as competent

Analyzing student work for partial understanding Look at the student work as a group. Discuss what you notice about how the student approached the problem. Discuss the following: What do they know? What strategies are they using? How can they grow? What instructional decisions could be next steps? Select one analysis to share with the whole group.

Ways to initiate conversation about student work Tell me about your model. Show me what you have tried so far. Tell me about what you have done. Do you know where you are stuck? If so, show me what is confusing about it. Where are you confused? Does your answer make sense? How do you know? I notice you did this (something in their work). Explain that to me.

What Partial Understandings Do? Create spaces to help students to see themselves (and for others to see them) as competent Create openings for supporting students to build on their current understandings Create broader views of what counts as knowing and understanding – and values those Disrupt the idea that you either know or don’t know

What Does This Mean For Us? As a PLC have conversations about how to: Use student work to have discussions about partial understanding and what types of questions to ask. Use partial understanding as a starting place to build on in order to deepen understanding. Find ways to capitalize on student’s partial understanding to meet them where they are to take them further.

Discussion … With a partner, discuss 1-2 of the following questions: What does this mean for my instruction tomorrow? What will I do differently in my next lesson having this new knowledge? What types of questions might I ask students to gain insight into their understanding? What could be an action step for tomorrow related to partial understanding?

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