Transcript The Distributive Property

Where We’ve Been…….
Properties, Part I……..
commutative
If I say “order,” you say………
associative
If I say “grouping,” you say………
Value stays the same
If I say “identity,” you say………
Where We Are Going…….
Just listen.
Today, we are going to investigate one of the
most important properties you will use this
year and in future classes.
Algebra
Associative Properties
Commutative Properties
Distributive Property
Identity Properties
Order of Operations
Translating Expressions
What Is Our Objective?
• Use the distributive property to rewrite and
simplify equivalent multiplication problems
Notes.
Distributive Property
What property will we use?
What will we do with this property?
We will simplify multiplication problems
with this property!
Naming What You Know
You already use this property when you work
with mental math.
Let’s say you bought 23 CD’s for $6.00 each.
$6 each
Is there a way you could
mentally rearrange these values
to find your total without a pencil
and paper?
If you have ever tried this mental math,
Many of you would mentally multiply the $6
you have used the distributive property!!
by 20, multiply the $6 by 3, and add the
products. 6(20 + 3)
Let’s try this mentally….
120 + 18 = 138
6(20)
$6 each
6(3)
What does this look like?
Let’s take this basic multiplication problem:
6 x 23
We will rewrite 23 as an addition problem.
6 x (20 + 3)
Now we will multiply the 6 by EACH of the values that add up to 23.
6 x 20 + 6 x 3
120
138
+ 18
23
First?????
x 6
Next???
What it looks like…….
Notes.
Algebraically, the distributive property is defined
with variables.
a(b + c) = a(b) + a(c)
Think about our CD example.
In expanded form, how would we write 23?
20 + 3 Now, let’s use our price of 6
6(20 + 3) = 6(20) + 6(3)
= 120
+ 18
= 138
Guided Practice:
5 x 27
5 (20 + 7)
Let’s take the larger factor and write it in expanded form.
Remember, no sign means to multiply!
Let’s “ distribute “ the 5. It is the factor used on both
the 20 and the 7.
(5 x 20) + (5 x 7) =
(100) + (35) =
135
4 x 28
4 (20 + 8)
Take the larger factor and write it in expanded form.
Let’s “ distribute “ the 4 . It is the value used on both
the 20 and the 8.
(4 x 20) + (4 x 8) =
(80) + (32) = 112
7 x 108 = 7 (100 + 8 )
(7 x 100) + (7 x 8) = 700 + 56 =
756
Small Group Work
In your groups, practice this process. Talk to each other as you complete each of
these steps:
First rewrite the problem, breaking up the larger value.
Next, show how the single multiplier is “distributed” to both parts of the
expanded number.
WE ARE ONLY BREAKING UP THE LARGER VALUE AND “DISTRIBUTING “
THE SINGLE-MULTIPLIER!
After completing the first four problems, move on to the next three. Try to
find a value (evaluate) the expression after you rewrite it!
Give me a sign when
you finish this section.
1) 8(43) = 8( 40 + 3)
344
8( 40 ) + 8 ( 3 )
320 + 24
2) 5(67) = 5( 60 + 7)
335
5(60 ) + 5 ( 7 )
300 + 35
3) 9(42) = 9( 40 + 2)
___(40)
9
+
378
9 (2)
__
360 + 18
4) 3(53 ) = 3( 50 + 3)
159
3( 50 ) + 3 ( 3 )
150 + 9
5) 7(52) = 7( 50 + 2 )
7(50) + 7(2)
350 + 14 = 364
6) 9(57) = ___
9 ( 50 + 7
9 (50) + 9(7)=
450 + 63 = 513
)
Application
Big Group Menu
Food
Cost
Organic Chili
$6.00
Chicken Tacos
$4.00
Fruit Salad
$5.00
Organic Salad
$5.00
Veggie Plate
$6.00
Discuss with your group……..
Thirty-three 6th graders ordered from
the Big Group Menu. They ordered
25 organic chilis and 8 veggie plates.
Create a problem using the distributive
property to represent this situation.
Both of these items cost the same…….$6.00. This will be our common
multiplier.
6 x (33) =
6 x (25 + 8) =
6 x (30 + 3) =
6(30) + 6(3) = 180 + 18 = $198.00
Take a moment to look at the
operations we used with the
Distributive property.
Did we leave something out??????
We rewrote all problems as addition.
Let’s look at two problems and change
them to subtraction.
Our first problem in the Guided Practice was
5 x 27
Could we use a subtraction problem
to create a value of 27????
30 – 3 = 27
5 x 27 =
5 ( 30 - 3 )
5(30) - 5(3)
150 - 15 = 135
Could we use a subtraction problem
to create a value of 8(78)????
80 – 2 = 78
8 x 78 =
8 ( 80 - 2
)
8(80) - 8(2)
640 - 16 = 624
Think Critically
Look at the expressions represented
by the properties we have studied…..
What is the one distinct
14 x 3 = 3 x 14
difference that separates the
distributive property from the
commutative and associative
14 + 4 = 4 + 14
properties?
(4 x 5) x 12 = 4 x (5 x 12)
(4 + 5) + 12 = 4 + (5 + 12)
12 X 6 = (6 x 10) + (6 x 2)
Distributive Property, Part I
We have defined the distributive property.
We have used the distributive property to
rewrite multiplication problems.
Distributive Property
+
+
Models
+
You can apply the Distributive Property to unknown values (models).
You are looking at a model that represents 2x + 5 that is written
three times.
Instead of writing 2x + 5 + 2x + 5 + 2x + 5, we could write
3(2x + 5)
We have to use the distributive property to “pull” the values
out of the parentheses……
3(2x) + 3(5)
= 1
We use the associative property to multiply 3(2)x = 6x
and we multiply 3(5).
6x + 15 represents our simplified value. We can do no more.
= x
You can apply the Distributive Property to unknown values (models).
What expression represents this model?
We have 3x + 7 that is written two times.
2(3x + 7) =
2(3)x + 2(7)
6x + 14
= 1
= x
The Distributive Property
Words: To multiply a sum by a number, multiply each addend by
the number outside the parentheses.
2(74) =
2(70 + 4) = 2 x 70 + 2 x 4
140 + 8 = 148
9x
1
4
3
Ask yourself, “What is one-third of 9?”
=
9(4 +
9(4) +
1
)
3
1
9( )
3
=
36 + 3 = 39
SHOW YOUR STEPS as you evaluate each of these.
3
5x2 =
5
5(2 +
12 x
3
)
5
1
2
4
12 ( 2 +
3
5( )
5
Think of 15 ÷ 5
= 5(2) +
10 + 3 = 13
=
1
)
4
2 x 3.6 =
= 12(2) + 12(1)
4
24 + 3 = 27
+
2(.6)
2(3)
2 ( 3 + .6) =
6 + 1.2 = 7.2
Think Critically
Faye is making a pair of earrings and a bracelet for
four friends. Each pair of earrings uses 4.5 cm of wire and each
bracelet uses 13 cm of wire. Write two equivalent expressions and then
find out how much wire is used.
4(4.5 + 13) and 4(4.5) + 4(13) are equivalent.
18 +
52 = 70 cm
4(17.5) = 70 cm
Mark all of the information you have:
What do you need for one set of stuff????
Earrings: 4.5 cm
How many are being made? 4
bracelet: 13 cm
Think Critically
Each day, Martin lifts weights for 10 minutes and runs on
the treadmill for 25 minutes. Write two equivalent
and find the total minutes that Martin exercises in 7 days.
7(10 + 25)
7(35)
245 minutes
and
7(10) + 7 ( 25)
70 + 175
245 minutes
7(10) + 7(20) + 7(5) ???????
70 + 140 + 35 105 + 140 = 245
Reasoning…..Subtraction???
A coyote can run up to 43 mph while a rabbit can run up to
35 mph.
We will write two equivalent expressions to
Show the difference in how far they would run after 6 hours.
Choose one to solve.
6( 43 – 35)
6( 8) = 48
or
or
6(43) – 6(35)
258 – 210 = 48
Factoring Expressions
Factoring the Expression: We will take a simple addition problem and rewrite
it using the distributive property.
Let’s factor the numerical expression 12 + 8
Ask yourself, “What the largest number 12 and 8
share?” Some find it helpful to write this out.
(4 x 3) + (4 x 2)
4( 3 + 2 )
We know it is 4. OK…..if we divide each of the numbers by 4 –
factor the 4 out of each one – what’s left??? Think of the
values as multiplication problems with 4’s.
4 is the shared factor…..What do we have left?
Let’s look at this process again.
14  21  7  2  7  3
(2  3)
7
9 + 21
3x3 + 3x7
3
3 ( 3 + 7)
7 x 2 + 7 x 4 ?????
14 + 28
14 x 1 + 14 x 2
14
14 ( 1 + 2)
80 + 56
8 x 10 + 8 x 7
8
8( 10 + 7)
Let’s add some variables!
3 x  15
3x  3  5
( x  5)
3
7n  35 
7n  7(5) 
9n  9(3) 
(n  5)
7
(n  3)
9
9n  27 
Let’s add some variables!
16 + 4x
4 · 4 + 4x
7x + 42
7x + 7 · 6
4(4 + x)
7(x + 6)
36x + 30
6·6x +6·5
6(6x + 5)
Factor Each Expression
8 + 16
8(1) + 8(2)
54 + 24
6(9) + 6(4)
6(9 + 4)
63 + 81
9(7) + 9(9)
11x + 55
32 + 16x
11x + 11(5)
16(2) + 16x
77x + 21
7(11)x + 7(3)
11(x + 5)
16(2 + x)
7(11x + 3)
8(1 + 2)
9(7 + 9)
3 ( 2x +5 )
Use your imagination!
What just happened here?
(3 ∙ 2x) + ( 3 ∙ 5 ) =
6x
+ 15
6x + 15
Product Island
3( a  4) 
3a
 3(4)
3a  12
5(b  6) 
5b  5(6)
5b  30
3
(4n  6)
3(4n  6) 
3(4)n  3(6)
12n  18
Lesson Practice
2
8
3
9x
2
9(8 + )
3
2
3
3(x + 1)
3x + 3(1)
9(8) + 9( )
3x + 3
4) 4(x + 6)
4x + 4(6)
4x + 24
5) 25 + 60
5(5 + 12)
5(x + 8)
5x + 5(8)
5x + 40
6) 4x + 40
4( x + 10)
Six friends: Admission is $9.50. One ride is $1.50
6(9.5 + 1.5) =
6(9.5) + 6(1.5) =
6(11) = $66
What have we done?
• We have used the distributive property to
solve multiplication problems.
• We have written problems with variables
using the distributive property.