Applications of the Distributive Property

Math AllianceJune 22, 2010

Beth Schefelker, DeAnn Huinker, Melissa Hedges, Chris Guthrie

Learning Intention (WALT) & Success Criteria We are learning to…

Make connections between the distributive property, the use of arrays, and the area model for multiplication.

We will be successful when… We can explain how splitting arrays, “start

with facts,” and the partial products algorithm are grounded in the distributive property.

Our Journey with Multiplication

Grounded ourselves in the foundations of a conceptual understanding of multiplication

Viewed multiplication as more than basic facts

Learned flexible strategies for multiplication

Expanded our understanding of representations

Learned new vocabulary

Factor

Product

Partial product

Array

___ groups of ____

___ rows of _____

____ sets of ______

The Distributive Property of Multiplication Over Addition

The distributive property is the most important and computationally powerful tool in all of arithmetic.

Beckmann (2005)

For all real numbers, A, B, and C,

A × (B + C) = (A × B) + (A × C)

“A times the quantity (B + C), is the same as A times B plus A times C.”

… or conceptually “partitioning & distributing.”

NOTE: Real numbers are all the numbers that have a spot on the number line.

Revisiting Splitting Arrays and “Start-with facts”

9 × 7 = ___

Use concept-based language to describe the meaning of this equation.

Write down two different “start with” facts that could help you solve 9x7.

Visualize how your start-with fact works on this array.

Think about 9x7 and the use of the distributive property.What is being partitioned?What is being distributed?

9

5 + 2

9 x 5 9 x 2

Make a quick sketch of a 9×7 open array. Start with 9×5 to partition it. Label all dimensions and each partial product.

Use the Notetaking Guide….

The Distributive PropertyA × (B + C) = A × B + A × C

A × (B+C) = A × B + A ×C

Write equations to show the partitioning and distributing.

A

B + C

A x B A x C

Visualize 9x7, starting with 9x5.

Think, then turn to your neighbor:

In the above equation,

what is the value of A? B? C?

On your Guide, make the second open array using letters to label dimensions and partial products.

9 x 7 = 9 x (5+2) = 9 x 5 + 9 x 2

Variation 1: Splitting the Array (A + B) × C = A × C + B × C

Use concept-based language to describe the relationship between the expressions. 9 x 7 5 x 7

On your guide, sketch a 9x7 array, partition usingthe 5x7 “start with” fact.

Shade and label the array to represent the dimensions and the partitions.

5 x 7

4 x 7

5

+

4

7

Variation 1: The Distributive Property(A+B) × C = A × C + B × C

9 ×7 = (5+4) × 7 = 5 × 7 + 4 × 7

(A+B) × C = A × C + B × C

C

A

B

A x C

B x C

Visualize 9x7, start with 9x5.

On your recording sheet, draw the second array with letters to label dimensions and partial products.

Write equations to show the partitioning and distributing.

Variation 2A × (B+C+D) = A × B + A × C + A × D

Complete Variation 2 on your guide sketch and label both arrays write the equations.

Consider 9 x 7• What is being partitioned?• What is being distributed?

9x7 9 groups of 7

9 groups of 2 is 189 groups of 2 is 189 groups of 3 is 27

9 x 7 = 9 x (2+2+3)

= 9x2 + 9x2 + 9x3

= 18 + 18 + 27

= 63

A × (B + C + D)

= A × B + A × C + A × D

A

B + C + D

A x B A x C A x D

Variation 2: The Distributive PropertyA × (B+C+D) = A × B + A × C + A × D

Variation 3: A x (B–C) = A x B – A x C

or (B–C) x A = B x A – C x A

“9 groups of 7”… Too hard!Think…10x7…10 groups of 7. Much better!

Complete Variation 3.• sketch and label both arrays• write the equations

Consider 9 x 7

Variation 3: A x (B–C) = A x B – A x C

or (B–C) x A = B x A – C x A

9 groups of 7

“10 groups of 7 less 1 group of 7.”

9 x 7 = (10 – 1) x 7 = 10 x 7 – 1 x 7

= 70 – 7 = 63

Quick Quiz 7× 8

Match the algebraic notation of the distributive property and each of its variations to corresponding number sentences and arrays.

A × (B + C) = A × B + A × C7 × 8 = 7 × (5 + 3) = 7×5 + 7×3

(A+B)× C= A×C +BxC7 × 8 = (5+2) × 8 = 5×8 + 2×8

A × (B+C+D) = A×B + A×C + A×D

7× 8 =7 × (2+2+4) = 7×2 + 7×2 +7×4

(B-C)×A = B×A – C×A 7 × 8 = (8-1) × 8 =(8×8) - (8-1)

Learning Intention (WALT) & Success Criteria We are learning to…

Make connections between the distributive property, the use of arrays, and the area model for multiplication.

We will be successful when… We can explain how splitting arrays, “start

with facts,” and the partial products algorithm are grounded in the distributive property.

Exam Next Week!

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