Applying the Distributive Property to Large Number

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

2. 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.

3. 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.

4. 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.

5. 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?

6. Modes of representation of a mathematical idea Pictures 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. Written symbols Manipulative models Real-world situations Oral language Lesh, Post & Behr (1987)

7. Puzzled Penguin Needs Our Help! Dear 4th 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 4th 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? ???

8. 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

9. What does the array model reveal? 7 3 4 5 8 3 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

10. 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.

11. 20 × 30 Array How does 20 × 30 relate to the original problem 27 x 34? 30 20 • Conceptual-based language: • 20 rows of 30 objects • 20 groups of 30 objects • 20 sets of 30 objects

12. 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.

13. 20 × 30 Open Array 20 × 34 Open Array How does 20×34 relate to the original problem of27×34? 30 4 20 x 4 20 × 30 20 • What does 20 × 34 mean? • What conceptual-based language helps us connect the array to the meaning of multiplication?

14. 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.

15. 27 x 34 30 4 20 x 30 20 x 4 This is commonly call the Partial Product Algorithm. Why? 20 7 • 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) • 918 7 x 30 7 x 4

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

17. 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 × 20 60 = 10 × 6 80 = 4 × 20 24 = 4 × 6 364 14 × 26

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

19. Modes of representation of a mathematical idea Pictures 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. Written symbols Manipulative models Real-world situations Oral language Lesh, Post & Behr (1987)

20. Homework Assignment Read Section 5.7 of Beckmann (pp. 249-254) 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.

21. 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.