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Applying the Distributive Property to Large Number Math Alliance Tuesday, June 8, 2010

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Learning Intention (WALT) & Success Criteria We are learning to… Understand how and why the partial product algorithm works for multiplication of large numbers. We will know we are successful when… We can apply and explain the partial products algorithm for multiplication utilizing modes of representation.

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Extending Our Learning: Homework Sharing Each person shares the following: The “focus fact.” Strategies used from class to help their student learn that fact. Why you chose to use each strategy attempted How you used each strategy with your student Concept-based language used to support your selected strategy. As a table group, keep track of each strategy and concept-based language used.

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Surfacing Strategies Used Review the list of strategies created at your table Pick 2 strategies and place each on a separate large post-it. Be sure to provide a quick sketch, if needed, to further illustrate the strategy. Provide a heading or title for each post-it Place your large post-its on the white board at the front of the room.

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Generalizing The Experience As you attempted teaching a strategy (or strategies) for multiplication basic facts: What did you learn about yourself as a teacher of mathematics? What did you learn about your case study student that can be applied to future students or future similar experiences?

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As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and understood more deeply. Manipulative models Pictures Real-world situations Oral language Written symbols Modes of representation of a mathematical idea Lesh, Post & Behr (1987)

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Puzzled Penguin Needs Our Help! Dear 4 th grade math student, Today I had to find 8×7. I didn’t know the answer so I used two multiplications I did know: 5 × 3 = 15 3 × 4 = 12 8 × 7 = 27 Is my answer right? If not, please help me understand why it is wrong. Thank you, Puzzled Penguin 4 th grade Expressions Curriculum Unit 1 Lesson 11 Take a minute on your own and think about what Puzzled Penguin is attempting to do. Which mode of representation might help you “see” his thinking? ???

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Helping Puzzled Penguin Share the mode of representation you found yourself working with to better understand Puzzled Penguins thinking. How does that representation help surface Puzzled Penguin’s misconception? Why might an array (made with tiles or graph paper) or an open array be a good choice? 8 × 7 = ? 5 × 3 = 15 3 × 4 = 12 8 × 7 = 27

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What does the array model reveal? Where are 5 × 3 and 3 × 4 in this array? Why do his beginning steps make sense? How does conceptual-based language support this work? 5 × 3 3×4

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Building Arrays for Larger Dimensions: A Scaffold Approach First Problem: 27 x 34 Step 1: 20 x 30 Talk: What does 20 x 30 mean? (Hands in your lap, must talk only) Build: Build array for 20 x 30 with place value blocks. Draw: Record your 20 x 30 using grid paper. Color in the rectangle.

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20 × 30 Array Conceptual-based language: 20 rows of 30 objects 20 groups of 30 objects 20 sets of 30 objects How does 20 × 30 relate to the original problem 27 x 34?

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Building Arrays for Larger Dimensions: A Scaffold Approach 27 × 34 Step 2: 20 x 34 Talk: What does 20 × 34 mean? How would you modify your model to show this problem? Build: Use the place value models to change your 20 × 30 array to a 20 × 34 array. Draw: Add to your 20 × 30 array to show the 20 × 34 array Color: Use another color to show what you added.

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20 × 30 Open Array 20 × 34 Open Array 20 x × 30 What does 20 × 34 mean? What conceptual-based language helps us connect the array to the meaning of multiplication? How does 20×34 relate to the original problem of 27×34?

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Building Arrays for Larger Dimensions: A Scaffold Approach 27 × 34 Step 3: 27 x 34 Talk: How would you modify your current model for 20 × 34 to show 27 x 34? What conceptual-based language are you using? Build: Using the place value blocks First, model to show 7 x 30, 7 rows of 30; Then, modify to show 7 x 4, 7 rows of 4. Draw: Use another color to show 7 x 30; then a fourth color to show 7 x 4.

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7 x 307 x 4 Write the partial product for each array and calculate the total. 600 = 20 x 30 (Step 1) 80 = 20 x 4 (Step 2) 210 = 7 x 30 (Step 3) 28 = 7 x 4 (Step 3) x x 4 27 x This is commonly call the Partial Product Algorithm. Why?

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Time to practice Try the scaffold approach for the partial product algorithm with the following: 14 × 26

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14 × 26 Step 1: 10 × 20 Build the model Draw Color Step 2: 10 × 26 Modify the model Modify your drawing Color Step 3: 14 × 26 Modify the model Modify your drawing Color Write out equations that match the arrays 200 = 10 × = 10 × 6 80 = 4 × = 4 × 6 364

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Try it again! 28 × 31 Talk over your steps to scaffold this equation using the partial product method.

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As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and understood more deeply. Manipulative models Pictures Real-world situations Oral language Written symbols Modes of representation of a mathematical idea Lesh, Post & Behr (1987)

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Homework Assignment Read Section 5.7 of Beckmann (pp ) Do problems 5, 6, & 7 (p. 258) using the grid paper provided in class. Please follow and complete all instructions for each problem. Do problem #10 using an open array. Problems 2 & 4 on p. 254 are recommended for further practice.

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Learning Intention (WALT) & Success Criteria We are learning to… Understand how and why the partial product algorithm works for multiplication of large numbers. We will know we are successful when… We can apply and explain the partial products algorithm for multiplication utilizing modes of representation.

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