Transcript Applications of the Distributive Property

Applications of the
Distributive Property
Math Alliance
June 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
 ___ groups of ____
 Product
 ___ rows of _____
 Partial product
 ____ sets of ______
 Array
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.
The Distributive Property
A × (B + C) = A × B + A × C
5
Use the Notetaking Guide….
Think about 9x7 and the use
of the distributive property.
What is being partitioned?
What is being distributed?
Make a quick sketch of a 9×7
open array. Start with 9×5 to
partition it. Label all dimensions
and each partial product.
9
9x5
+
2
9x2
A × (B+C) = A × B + A ×C
B
Visualize 9x7, starting with 9x5.
+
C
Think, then turn to your neighbor:
In the above equation,
what is the value of A? B? C?
AxB
On your Guide, make the second
open array using letters to label
dimensions and partial products.
A
Write equations to show the
partitioning and distributing.
9 x 7 = 9 x (5+2) = 9 x 5 + 9 x 2
AxC
Variation 1: Splitting the Array
(A + B) × C = A × C + B × C
 Use concept-based language
7
to describe the relationship
between the expressions.
9x7
5x7

 On your guide, sketch
5
+
a 9x7 array, partition using
the 5x7 “start with” fact.
4
Shade and label the array
to represent the dimensions
and the partitions.
5x7
4x7
Variation 1: The Distributive Property
(A+B) × C = A × C + B × C
C
Visualize 9x7, start with 9x5.
On your recording sheet,
draw the second array with
letters to label dimensions
and partial products.
A
AxC
Write equations to show the
partitioning and distributing.
B
9 ×7 = (5+4) × 7 = 5 × 7 + 4
×7
(A+B) × C = A × C + B
×C
BxC
Variation 2
A × (B+C+D) = A × B + A × C + A ×
D
Consider 9 x 7
•What is being partitioned?
•What is being distributed?
Complete Variation 2
on your guide
sketch and label both arrays
write the equations.
9x7
9 groups of 7
9 groups of 2 is 18
9 groups of 2 is 18
9 groups of 3 is 27
Variation 2: The Distributive Property
A × (B+C+D) = A × B + A × C + A ×
D
B
9 x 7 = 9 x (2+2+3)
= 9x2 + 9x2 + 9x3
= 18 + 18 + 27
= 63
AxB
A
A × (B + C + D)
=A×B+A×C+A×
D
+
C
AxC
+
D
AxD
Variation 3:
or
A x (B–C) = A x B – A x C
(B–C) x A = B x A – C x A
Consider 9 x 7
“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
Variation 3:
or
A x (B–C) = A x B – A x C
(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 × C
7 × 8 = 7 × (5 + 3) = 7×5 + 7×3
(A+B)× C= A×C +BxC
7 × 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
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!
 Review study guide