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## PowerPoint Slideshow about 'Applying the Distributive Property to Large Number' - iokina

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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.

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!

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?

???

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

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.

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?

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.

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

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 × 26Try 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.

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