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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

Applying the Distributive Property to Large Number

Math Alliance

Tuesday, June 8, 2010

learning intention walt success criteria
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
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
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
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?
slide6
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
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
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
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
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
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
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
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 27 341
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.
slide15
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
Time to practice
  • Try the scaffold approach for the partial product algorithm with the following:
    • 14 × 26
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 × 26
try it again
Try it again!
  • 28 × 31
  • Talk over your steps to scaffold this equation using the partial product method.
slide19
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)

slide20
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 criteria1
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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