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)

Math 6A ( Periods 2 & 5)

Properties of Addition & Multiplication 3.4 

Math 6A (Periods 2 & 5)

Writing Expressions 3.2 (Continued) 

 Math Operations  vocab of what means addition, subtraction, multiplication, and division
Make sure to glue the '1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

Use a "LET STATEMENT"
Let n = the number
That is the simplest "LET STATEMENT.  You are defining your variable!


We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n - 5

The product of a number and 3   3n
4 times a number  4n
The quotient of 3 and a number  3/n
The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.
10 less than a number is n-10
a number increased by 9  n + 9
9 more than a number is also n + 9



In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

8w would be our expression


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
So the next consecutive whole number after w is w + 1.

A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

So the number which precedes x would be x - 1.
The next number after n is n + 1



What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2



What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2


What if I asked what is the next consecutive EVEN number after the even number "m"
It would be m + 2


What would it be if I asked what was the next consecutive odd number, after the odd number x?
x + 2
Let's look at those relationships like the workdays from Monday's lesson... We are going to find another strategy to use for some of the more complicated expressions...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply by 8
so the mathematical expression in this case is "8w."

What about writing an expression for how many dimes are in x dollars?

Use your T- chart
x dollars    dimes
1..................10
2..................20
3..................30
x...................10x

Put 3 known relationships and then ask yourself "What did I do to the left side to get the right side"
What did I do to 1 to get 10? ... answer multiply by 10
What did I do to 2 to get 20? ... answer multiply by 10
What did I do to 3 to get 30?  ... answer multiply by 10
So What must I do to x? it must be the same  ... answer multiply by 10  so 10x is what I am looking for!

How many dollars do we have in n nickles?
set up-- unknown on the left!
n nickles      dollars
20....................1
40....................2
60....................3
n......................?
Again, Ask yourself  "What did I do to the left side to get the right side"
What did I do to 20 to get 1? ... divide by 20
What did I do to 40 to get 2?  ... divide by 20
What did I do to 60 to get 3?  ... divide by 20
So what must I do to n? it must be the same  ... divide by 20 so n/20

The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
"What do you do to the left side to get the right side?"

and in this case, specifically...

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12

Algebra ( Periods 1 & 4)

Chapter 4-4 Parallel and Perpendicular Lines 

Parallel Lines:

2 lines that are parallel to each other have the SAME SLOPE

Example:

y = 2x – 10

y = 2x + 4

Those two lines are  parallel because they both have a slope of 2

BE CAREFUL:
To be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS

If they have the same m and the same b then they are COLLINEAR
(or they COINCIDE) they are the same line and NOT parallel

Perpendicular Lines

2 lines that are perpendicular to each other have SLOPES that are:

Opposite signs

and

Reciprocals

y = 2x – 10 is perpendicular to y = (-1/2)x + 4

This is another twist to our Mystery Line Puzzles…

If you know that the mystery line is parallel  or perpendicular to another given line they you know the mystery line’s slope!!

EXAMPLE 1:

Your mystery line has a point of ( 2, -5) and is PARALLEL to the line y = 2x + 4

So you know that the mystery line’s slope
because it is the same as the given line  (m = 2)

Substitute the slope and the point given for the mystery line and solve for b
OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)

y + 5 = 2x – 4

y = 2x – 9

If I used Slope Intercept form or y = mx + b

-5 = 2(2) + b

-5 = 4 + b

b = -9

Plug everything back in

y = 2x – 9

Same results—but I think that it takes longer ( students also tend to FORGET to put everything together)

EXAMPLE 2:

Your mystery line has a point of ( 2, -5) and is perpendicular to the line y = 2x + 4

So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )

Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!  

y + 5 = (-½)(x -2)

y + 5 =  -½x + 1

y =  -½x - 4

If I used Slope Intercept form or y = mx + b

-5 = (-½)(2) + b

-5 = -1 + b

-4 = b

Plug everything back

 y = -½x - 4

Algebra Honors ( Period 6)

Chapter 4-4 Parallel and Perpendicular Lines 

Parallel Lines:

2 lines that are parallel to each other have the SAME SLOPE

Example:

y = 2x – 10

y = 2x + 4

Those two lines are  parallel because they both have a slope of 2

BE CAREFUL:
To be parallel they must have the SAME SLOPE BUT DIFFERENCE y-INTERCEPTS

If they have the same m and the same b then they are COLLINEAR
(or they COINCIDE) they are the same line and NOT parallel

Perpendicular Lines

2 lines that are perpendicular to each other have SLOPES that are:

Opposite signs

and

Reciprocals

y = 2x – 10 is perpendicular to y = (-1/2)x + 4

This is another twist to our Mystery Line Puzzles…

If you know that the mystery line is parallel  or perpendicular to another given line they you know the mystery line’s slope!!

EXAMPLE 1:

Your mystery line has a point of ( 2, -5) and is PARALLEL to the line y = 2x + 4

So you know that the mystery line’s slope
because it is the same as the given line  (m = 2)

Substitute the slope and the point given for the mystery line and solve for b
OR use Point Slope Form and solve easily!
This is why I like Point Slope: y + 5 = 2(x – 2)

y + 5 = 2x – 4

y = 2x – 9

If I used Slope Intercept form or y = mx + b

-5 = 2(2) + b

-5 = 4 + b

b = -9

Plug everything back in

y = 2x – 9

Same results—but I think that it takes longer ( students also tend to FORGET to put everything together)

EXAMPLE 2:

Your mystery line has a point of ( 2, -5) and is perpendicular to the line y = 2x + 4

So you know that the mystery line’s slope is the opposite sign and the reciprocal of the given line
(since the given line’s slope is 2, the mystery line’s slope must be –½ )

Substitute the slope and the point given on the mystery line and solve for b OR use Point Slope Form and solve easily!  

y + 5 = (-½)(x -2)

y + 5 =  -½x + 1

y =  -½x - 4

If I used Slope Intercept form or y = mx + b

-5 = (-½)(2) + b

-5 = -1 + b

-4 = b

Plug everything back

 y = -½x - 4