Applying the Distributive Property to Large Number

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

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### Applying the Distributive Property to Large Number

Math Alliance

Tuesday, June 8, 2010

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.
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.
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.
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?
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)

Puzzled Penguin Needs Our Help!

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

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.

???

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

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

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.
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

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.
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?

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.
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

Time to practice
• Try the scaffold approach for the partial product algorithm with the following:
• 14 × 26
Step 1: 10 × 20

Build the model

Draw

Color

Step 2: 10 × 26

Modify the model

Color

Step 3: 14 × 26

Modify the model

Color

Write out equations that match the arrays

200 = 10 × 20

60 = 10 × 6

80 = 4 × 20

24 = 4 × 6

364

14 × 26
Try it again!
• 28 × 31
• Talk over your steps to scaffold this equation using the partial product method.
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)

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.

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.