Math 6H (Period 6 & 7)

Distributive Property 1-4
The Distributive Property of Multiplication with Respect to addition
For any whole numbers a, b, c,

a(b + c) = ab + ac
and (b + c)s = ba + ca

The Distributive Property of Multiplication with Respect to Subtraction
For any whole numbers a, b, c
a(b-c) = ab- ac
and
(b-c)a= ba - ca

Since multiplication is distributive with respect to both addition and subtraction, we refer to both properties as the DISTRIBUTIVE PROPERTY

The Distributive Property is used to simplify expressions.
for example
(8 × 6) + (2 × 6) can be solved quickly when you use the distributive property
(8 +2)6 = (10)6 = 60
The following were taken from the Class Exercises on Page 14... Check them out
4(3 + 8) = 4 × 3 + 4 × 8
6(5 + 9) = 6 × 5 + 6 × 9
9(7 - 4) = 9× 7 - 9 × 4
7( 12 + 15) = 7× 12 + 7×15
5(11 - 3) = 5 × 11 - 5 × 3

(9 ⋅ 7) + (9 ⋅ 13) could be solved by multiplying 9× 7 = 63 and adding it to 9× 13= 117
BUT... using the distributive property
(9 ⋅ 7) + (9 ⋅ 13)= 9( 7 + 13) = 9(20) = 180

and
(8 × 9) + (8 × 1) could have been solved by first doing
8 × 9 = 72 and then 8 × 1= 8 and then adding them BUT..
with the distributive property
(8 × 9) + (8 × 1) = 8(9 + 1) = 8(10) = 80 :)
(54 × 11) - (24 × 11) simplifies to
(54 - 24)11 =
30(11) = 330

11(88 - 42) ...
now with this one it doesn't help to separate
that is making 11(88 - 42 ) = 11(88) -11(42) is NOT going to make it easier
so
11(46) = 506... using the trick of 11's that we learned today... do you remember it???

(93 × 5) - (23 × 5) =
(93-23)5 =
(70)5 = 350

Pre Algebra (Period 2 & 4)

Coordinate Graphing 1-10

Review of the xy coordinate plane.


Cartesian plane:
Named after French mathematician Descartes.

plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.



quadrant: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4

quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)

They are named I, II, III, IV in Roman Numerals


coordinate - A coordinate is the position of a point in the Cartesian plane


coordinate = "co" means goes along with (coworker, co-president, co-champions)

"ordinate" means in order

So coordinate means numbers that go along with each other in a certain order

The numbers are the x and y values and the order is that the x always comes first


Also called an ordered pair (x y order and pair of numbers)

Ordered pairs are recognized by the use of ( x , y ) format



origin = (0, 0) the center of the graph (its beginning or origin)

When you count the coordinate' s position, you count from the origin.


x comes before y in the alphabet so the order is (x, y) ....
always go right or left first, then up or down


the x axis is the horizontal axis (goes across)


Remember that because the number line also is horizontal and you learn that first 
(the pattern to remember is x is always first and the number line is before going up and down)


NOW LET'S GET TO WHAT YOU ACTUALLY DO TO GRAPH!!!

1) Count your x value:

positive x, count right from origin (positive numbers are to the right of zero on number line)

negative x, count left from origin

2) Count your y value:

positive y value, count up from where your x value was (up is the positive direction)

negative y value, count down from where your x value was (down is the negative direction)


EXAMPLES:

(3, 5) Count 3 to the right from the origin, then 5 up

(3, -5) Still count 2 to the right, but now count 5 down

(-3, 5) Count 3 to the left from the origin, then count 5 up

(-3, -5) Again count 3 to the left, but now count 5 down



BUT WHAT HAPPENS WHEN 
ONE OF THE VALUES IS ZERO?

If the y value is zero it means that you move right or left, but don't go up or down:

SO YOUR POINT WILL BE ON THE x AXIS........
x axis is where y = 0 

Example: (3, 0) is a point on the x axis, 3 places to the RIGHT

Example: (-3, 0) is a point on the x axis, 3 places to the LEFT



If the x value is zero it means that you don't move right or left, you just go up or down.

SO YOUR POINT WILL BE ON THE y AXIS...........
y axis is where x = 0 

Example: (0, 3) is a point on the y axis, 3 places UP

Example: (0, -3) is a point on the y axis, 3 places DOWN
Now when the question is
if x is positive and y is negative, what quadrant is it? IV
If x is positive and y is positive, what quadrant is it? I
If x is negative and y is positive, what quadrant is it? II
If x is positive and y is negative, what quadrant is it? III
These questions could be stated like this:
if x >0 and y <0 what quadrant is it? Now think: IF x > 0 it must be a positive number and if y <0 it must be a negative number.. and then use the rules above!!

Pre Algebra (Period 2 & 4)

Evaluating Expressions 1-3 (continued)


STRATEGY #1: MAKE A T-CHART
To translate known relationships to algebra, it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with 3 lines of numbers and look for the relationship between the 2 columns.
Then, you use that mathematical relationship with a variable.

EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:

d days ---> hours
1 --->24
2 ---> 48
3 --->72

Now look at the relationship between the left column and the right column.
You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d

d days ---> hours
1 --->24
2 --->48
3 --->72
d --->24d

EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:

h hours -> days
24 --->1
48 --->2
72 --->3



(Why did I start with 24 and not 1 hour this time?)


Now look at the relationship between the left column and the right column.
You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h

h hours -> days
24 --->1
48 ---> 2
72 ---> 3
h --->h/24


The number of inches in y yards
You know that there are 36 inches in 1 yard.
y yards is the unknown so that must go on the left

y yards -> inches
1 ---> 36
2 --->72
3 ---> 108

Ask yourself
"What did I do to 1 to get 36?"
"What did I do to 2 to get 72?"
"What did I do to 3 to get 108?"
Answer for all three multiplied by 36

so what do you do to y ?
multiply by 36

y yards -> inches
1 ---> 36
2 --->72
3 ---> 108
y ---> 36y

36y is the expression you are looking for!!

How about the number of yards in i inches.
This time i inches is the unknown
i inches -> y
36 ---> 1
72 --->2
108 ---> 3

This time when you ask the question it is
"What did I do to 36 to get 1?"
What did I do to 72 to get 2?"
"What did I do to 108 to get 3?"

With each of these your answer is you divided by 36.

So what must you do to i... divide by 36
i inches -> y
36 ---> 1
72 --->2
108 ---> 3
i ---> i/36

so i/36 is your expression


We drew a grocery bag with the following info:
Food -> Calories
Bread ---> 55
Apple ---> 70
Banana --->100
Egg ---> 110


3 apples + 2 bananas
3(70) + 2 (100)
210 + 200 = 410 calories
a apples + b bananas
becomes 70a + 100b

e eggs + 1 slice of bread?
110e + 55

Algebra (Period 1)

Chapter 1 Review ... continued...
CHAPTER 1-2: COMMUTATIVE PROPERTY

PROPERTIES ARE OUR FRIENDS! (mathematically speaking)

YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!


COMMUTATIVE PROPERTY (works for all multiplication or all addition)

You can SWITCH THE ORDER and still get the same sum or product.

This is the property YOU CAN HEAR because you've switched the order.

a + b = b + a [you can abbreviate as C+]
OR
ab = ba [ you can abbreviate as Cx]

Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS

SO WHY SHOULD YOU CARE????

Because it makes the math easier sometimes!

Which would you rather multiply:

(2)(543)(5) OR
(2)(5)(543) ???



TWO MORE FRIENDS: 
THE IDENTITY PROPERTY OF
 ADDITION AND
THE IDENTITY PROPERTY OF MULTIPLICATION

For addition, we know that adding zero to anything will not change the IDENTITY of what you started with:
a + 0 = a (what you started with)

0 is known as the ADDITIVE IDENTITY.


For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with:
(1)(a) = a (what you started with)

1 is known as the MULTIPLICATIVE IDENTITY.

Sometimes 1 is "incognito" (disguised!)

We use this concept all the time to get EQUIVALENT FRACTIONS.

Say we have 3/4 but we want the denominator to be 12

We multiply both the numerator and the denominator by 3 and get 9/12


We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3

ANYTHING OVER ITSELF = 1 (except zero because dividing by zero is UNDEFINED!)

a + b - c = 1

a + b - c



We also use this property to SIMPLIFY fractions.
We "cross cancel" all the parts on the top and the bottom that equal 1
(your parents would say that we are reducing the fraction)

6abc/2a = 3bc since both the numerator and denominator can be divided by 2a.
                                 

CHAPTER 1-4: ASSOCIATIVE PROPERTY

ANOTHER FRIEND!

This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!

a + (b + c) = (a + b) + c
 [ you can abbreviate as A+]
a(bc) = (ab)c
 [you can abbreviate as Ax]
Why? TO MAKE THE MATH EASIER OF COURSE!

This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.

EXAMPLE: [(543)(5)](2)

Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]

But our friend the Associative Property allows us to simply move the [ ]

[(543)(5)](2) = (543)[(5)(2)] which is so much easier to multiply in your head!!!



CHAPTER 1-5: DISTRIBUTIVE PROPERTY

A new friend! It's a property. It has NO counterexamples.

a( b + c) = ab + ac

or

(b + c)a = ba + ca (What other property justifies this corollary?)

You don't need to show the "middle step" once you understand this property.

EXAMPLE: -5(3y - 4) = (-5)(3y) + (-5)(-4) = - 15y + 20


You can't combine the y term with the 20 because they are UNLIKE TERMS.

LIKE TERMS:

1. Same variable
(s)
2. Same exponent(s)

Constants are like terms because they all have no variables

You can only combine (add or subtract) like terms.


BUT YOU CAN MULTIPLY UNLIKE TERMS!

3a(7y) = 21ay

BUT

3a + 7y cannot be simplified


Math 6H (Period 6 & 7)

Inverse Operations 1-3

Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication & division are inverse operations

We can use these relationships between inverse operations to simplify some numerical and variable expressions

17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17

9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...

108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108

3n ÷ 3 = 3

Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0

Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1

Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0

Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'

Algebra (Period 1)

Chapter 1 Review

Exponential Notation

exponent ---> little number that is raised called superscript on your computer.
4³
the 4 is the base and the 3 is the exponent.
the power is how many times the repeated multiplication will occur... in this case the power is 3!!
Some terms to know:
exponentiation notation : written with an exponent (8k)⁴
expanded notation: 8k⋅ 8k⋅ 8k⋅ 8k
(10c)³ 10⋅10⋅10⋅c⋅c⋅c
standard notation: the number value
3³ = 27
(3x)³ = 27x³
but 3x³ is just 3⋅x⋅x⋅x or just the same as you started with!! 3x³
Watch your HUGS.. they are very important in life.. and in math!!
(3x)³ where x = 2
is [(3)(2)]³ or 6 ³ = 6⋅6⋅6 = 216
but 3x³ is 3 (2)³ or 3⋅2⋅2⋅2⋅ = 24

Powers of negative numbers and the odd/even rule

If there is a negative inside the (hugs)
odd number of negative signs or an odd power ---> its negative

even # of negative signs or an even power ---> positive

(-2) ⁵ really means (-2)(-2)(-2)(-2)(-2) = -32
(-2) ⁴ really means (-2)(-2)(-2)(-2) = 16

If there is a negative but NO (hugs)... it is ALWAYS negative
-2⁵ = -32 and
-2 ⁴ = -16
because these really read
"Take the opposite of 2 to the 5th power" and
"Take the opposite of 2 to the 4th power"

Equation Vocab

solution = answer

replacement set: the set of answers that you have to choose from for the answer.
solution set: all the answers that make a statement true

{ } is set symbol
For example
solve y + 4 = 10 if the replacement set is {2, 6, 10}
you discover that
6 + 4 = 10 makes the statement true so 6 is the solution set and you write it
{6}

But what if you have y + 4 = 10
but the replacement set is { 2, 7, 10}
when you plug in each of the values none of them are true
2 + 4 ≠ 10
7 + 4 ≠ 10
10 + 4 ≠ 10
so the solution set is the NULL SET. which you write
{ } or ∅

Equations can be true, false or open!!
If it is numerical they are either true or false...
If equation is Algebraic ( variables) it is always OPEN

Math 6H (Periods 6 & 7)

Properties of Addition & Multiplication 1-2

1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers

Some important properties are found using whole numbers:

Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a

Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba

You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!

However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18

Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)

Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)

You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!

Addition Property of Zero from the book
also known as the Identity Property of Addition ( Id+)
for any whole number a
a + 0 = a and 0 + a = a

Multiplication Property of One
also know as the Identity Property of Multiplication (Idx)
For any whole number a
a(1) = a and 1(a) = a

Multiplication Property of Zero
for any whole number a
0(a) = 0 and 0(a) = 0

So how do we justify?

16 + 19 + 34 ---> given
16 + 34 + 19 ---> C+
(16 + 34) + 19---> A+
50 + 19---> simplify
69---> simplify


2 (14)(15)---> given
2 (15) (14) ---> Cx
[2(15)] (14) ---> Ax
30 (14)---> simplify
420---> simplify

how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 ---> given
14 ⋅ 8 ⋅ 5⋅ 25 ---> Cx
14 ⋅ (8 ⋅ 5)⋅ 25 ---> Ax
14 ⋅ 40⋅ 25--->simplify
14 ⋅ (40⋅ 25) ---> Ax
14 ⋅ 1000 ---> simplify
14000 ---> simplify

Make sure you justify each step of tonight's homework!!

Algebra (Period 1)

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Pre Algebra (Periods 2 & 4)

Our Class Blog
Welcome to our class blog... where you can earn extra credit by adding your own relevant comments about our class notes for the day.. or where you can find answers from others in your class. Check here often, especially if you have been absent. You might just find out the math strategy that works for you!!

Email me if you are interested in adding notes and/or comments to this blog-- for extra credit!!

Expressions 1-1
Expression are NOT equations. An Expression does NOT have an equal sign (=).
Words you need to know:
Addition: sum, increased by, more than
Subtraction: difference, less THAN, subtracted from
Be careful with this one... "3 less than a number n" is n-3. It isn't in the order that you read. Think about "3 less than 10" What would you write ? 10-3...

Multiplication: product, times
Division: quotient or Per... When we say " miles per hour" it is really the MILES DIVIDED by the HOURS!!


The Order Of Operations 1-2

O³ = Order Of Operations

It’s Aunt Sally’s Rule—
Please Excuse My Dear Aunt Sally

Please ( ) parentheses

Excuse E² exponents

My Dear (or even Dear My) × ÷ or ÷ ×

Aunt Sally (Sally Aunt) + - or - +

also called ... PEMDAS

The most common mistake of O³

Student forget that when you see multiplication and division—you do whichever comes first in order!!

So if you see division then multiplication, you do division first then multiplication
When you see addition and subtraction, you do whichever comes first in order!
So if you see subtraction then addition, do the subtraction first!

When you see more than one set of parentheses, do the innermost ones first, then work outwards.
They're called nested parentheses because a bird builds his nest from the inside out.


Evaluating Expressions 1-3

First substitute in for the variable..
Follow AUNT SALLY's RULE or PEMDAS

4y - 15 where y = 9
4(9) - 15
36 - 15
21

3ab + c/2 where a = 2 b = 5 and c = 10
3(2)(5) + 10/2
30 + 5
35

Be careful when substituting in negatives...