Applying the Distributive Property to Large Number

Math Alliance

Tuesday, June 8, 2010

Learning Intention (WALT) &Success CriteriaWe 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?

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.

Manipulativemodels

Pictures

Real-worldsituations

Orallanguage

Written symbols

Modes of representation of a mathematical idea

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 = 153 × 4 = 128 × 7 = 27Is 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 = 153 × 4 = 128 × 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 ApproachFirst 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

Conceptual-based language:

•20 rows of 30 objects

•20 groups of 30 objects

•20 sets of 30 objects

How does 20 × 30relate to the originalproblem 27 x 34?

Building Arrays for Larger Dimensions: A Scaffold Approach 27 × 34Step 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 x 4

30 4

20 20 × 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?

Building Arrays for Larger Dimensions: A Scaffold Approach 27 × 34Step 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.

7 x 30 7 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)918

20 x 3020 x 4

27 x 34

20

7

30 4

This is commonly call the Partial Product Algorithm. Why?

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 arrays200 = 10 × 20 60 = 10 × 6 80 = 4 × 20

24 = 4 × 6 364

Try it again!

28 × 31 Talk over your steps to scaffold this equation

using the partial product method.

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.

Manipulativemodels

Pictures

Real-worldsituations

Orallanguage

Written symbols

Modes of representation of a mathematical idea

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 CriteriaWe 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.