Math 6A (Periods 2 & 5)

Distributive Property 3.4 

Definition in words: To multiply a sum of difference by a number, multiply each number in the sum of difference by the number outside the parentheses. Then evaluate.

Distributive Property of Multiplication with Respect to Addition:

For any real  numbers a, b, and c,

a( b + c) = ab + ac

and

(b +c)a = ba +bc

Distributive Property of Multiplication with Respect to Subtraction:

For any real  numbers a, b, and c,

a( b - c) = ab - ac

and

(b - c)a = ba – bc

If you are given the problem 23 x 6 normally you would stack them and multiply but what if you consider

6(23) = 6(20+ 3) = 6 (20) + 6(3) 120

How about 8(26) = 8(20 + 6) = 8(20) + 8(6) = 160 + 48 = 208

But you could have changed it to 8 ( 25 + 1) = 8(25)  = 8(1) = 200 + 8 = 208

What should we do with 20(19)? well… 20(15 + 4) or 20(10 + 9) work but are they easy to multiply—especially in your head?

Why not try 20(20-1) = 20(20) – 20(1) = 400 – 20 = 380

20(39) = 20(40 – 1) = 20(40) – 20(1) = 800 – 20 = 780

25(39) has all sorts of possibilities

25(40 -1) = 25(40) – 25(1) = 1000 – 25 = 975

25( 36 + 3) = 25(36) + 25(3) = 900 + 75 = 975

We talked about the “trick” of multiplying any number by 25… if you forgot… come in and ask again. I would love to go over this. Hint: get out quarters and figure out how many dollars you have with all those quarters…

15(47) ?  again lots of possibilities to make the multiplication easier

15(50-3) = 15(50) – 15(3)

or change 15(47) = 47(15) { what property allows you to change the order?}

Then 47(10 + 5) = 47(10) + 47(5)

How do you multiply by 5 quickly?  Multiply by 10 and take half. Why does that work?

Now the book gives the following example:

3(7 + 2) = 3(7) + 3(2) = 21 + 6 = 27

My question, why would we even want to do that? It does not make it easier—in fact it is just too many steps. So you need to start to realize WHEN the DP (Distributive Property) helps you to make the multiplication easier!

8(53) could be an example when it makes sense because 8(50 + 3) = 8(50) + 8(3) and both of those are easy to multiply—even in your head.. and then add them! 424

What about mixed numbers? We were taught that if you had

 

You needed to change the mixed number to an improper fraction

But watch the Distributive Property in action:

We then reviewed some of the problems similar to our Apples & Bananas WS

4(n + 5) = 4n +4(5) = 4n + 20

9( 6 + x + 2) = 9(6) + 9x + 9(2) = 54 + 9x + 18

but then we combined like terms to get 9x + 72

Why not combine like terms inside the HUGS first?

9( 6 + x + 2) = 9(x + 8) = 9x + 72

Combining LIKE TERMS

Like terms are terms that have the same variable(s) raised to the exact same sxponent(s)

Constants are like terms

5x+ 19 + 2x + 2 becomes

7x + 21

Jose is x years old. His brother is 2 years older than Jose. Their aunt, Maria is three times as old as Felipe. Write and simplify an expression that represents Maria’s age

Jose x

Felipe x + 2

Maria 3( x + 2)

3(x + 2) = 3x + 6

Art Museum question:

	Museum	Exhibit
Child (under 5)	Free	Free
Student	$8	$x
Regular	$12	$4
Senior	$10	$3

A class of 30 students visit an art museum and a special exhibit. Use the DP to write and simplify an expression for the cost.

each student would be 8 + x so for 30 students you would have

30(8 + x) = 240 + 30x

We estimate reasonable values for x—and had a lively discussion about why we picked the price for the student’s exhibit fee.

Then we evaluated for the price of $2

30(8 + 2) = 30(10) = 300

and we also substituted in for 240 + 30x  = 240 + 30(2) = 260 + 60 = 300

We found that the values were the same $300

Like Terms are terms that have the SAME VARIABLES RAISED TO THE SAME EXPONENTS
Constants terms are like terms:
5x + 19 + 2x + 2
Combine like terms
7x + 21

12x + y + 3 + y - 5x + y
Careful
we really have 12x -5x --> which is 7x
and we have y + y + y --> that's really 3y
so
7x + 3y + 3
1y + 1y + 1y = y ( 1+1+1) = y(3) which must be written as 3y

7z +2(z-5y)
We must use the Distribute Property first! 
7z + 2z +2(-5y)
7z + 2z -10y
9z -10y

Extension:

Factoring Expressions:

When you factor an expression, you can factor out any common factor!

Factor 20 -12

Find the GCF of 20 and 12  … the book says by listing the factors… we use inverted division or the box method

12 = 2²∙3

20 = 2²∙5

So the GCF(20,12) = 4

Write each term of the expression as the product of the GCF and the remaining factor(s)

20 – 12 = 4(5) – 4(3)

=4(5 -3)

14x -98

What is the GCF?  Look only at the numbers this time

14 = 2∙7

98 = 2∙7∙7

The GCF( 14, 98) = 2∙7 = 14

so 14x -98 = 14 (x) – 14(7)

14(x -7)