Algebra Period 3

Exponents: 5-1

Review the odd/even rule
IF THERE IS A NEGATIVE INSIDE PARENTHESES:
Odd number of negative signs or odd power = negative
Even number of negative signs or even power = positive

EXAMPLES:
(-2)^5 = -32
(-2)^4 = +16

IF THERE IS A NEGATIVE BUT NO PARENTHESES:
ALWAYS NEGATIVE!!!!
-2^5 = -32
-2^4 = -16

MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m^5m^3 = m^8
You can check this by EXPANDING:
(mmmmm)(mmm) = m^8

DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m^8/ m^5 = m^3

Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm

ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
By the transitive property of equality : 1 = m^0

NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!

EXAMPLE:
m^3/m^5 = m^-2


m^3/m^5 = m^-2
mmm/mmmmm 1/mm

Again, by transitive property of equality:
m^3/m^5 = m^-2 = 1/m^2
m2

EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER

EXAMPLE:
(-2)^-5 = 1/(-2)^5 = 1/-32 OR -1/32

RESTATE A FRACTION INTO A NEGATIVE POWER:
1) Restate the denominator into a power
2) Move to the numerator by turning the power negative

EXAMPLE:
1/32
1/(2)^5
(2)^-5


More on Exponents: 5-2

POWER to another POWER

Multiply the POWERS
(m^5)^3 = m^15
to check EXPAND it out (m^5) (m^5) (m^5) = m^15

PRODUCT to a POWER
DISTRIBUTE the power to EACH FACTOR
(m^5n^4)^3 = m^15n^12

RAISING a QUOTIENT to a POWER
distribute the power to the numerator AND to the denominator

Math 6 H Periods 1, 6 & 7

Review of Sections 5.1- 5.3
You know that 60 can be written as the product of 5 and 12. 5 and 12 are whole number FACTORS of 60.
A number is said to be divisible by its whole number factors if the remainder is 0.
A multiple of a whole number is the product of that number and any other whole number ( including zero-- because zero is a whole number)
You can find the multiples of any given whole number by multiplying that number by 0, 1, 2, 3, 4...

Any multiple of 2 is called an even number. A whole number that is not an even number is called an odd number.
Zero is an even number!!

Tests for Divisibility

A Number is Divisible by:
2 if is even
3 if the SUM of its digits is a multiple of 3
4 if the last two digits in the number represent a multiple of 4
5 if its last digit is either a 0 or a 5
6 if the number is divisible by 2 AND 3
8 if the last three digits in the number represent a multiple of 8
9 if the SUM of its digits is a multiple of 9
0 if the last digit is a 0.

A perfect number is one that is the sum of all its factors except itself. The smallest perfect number is 6 , since 6 = 1 + 2 + 4. What would be the next perfect number?
Try this and let me know for extra credit... email me your response.

Square Numbers and Square Roots 5-3

Numbers such as 1, 4, 9, 16, 25 are called perfect squares or square numbers.
1 = 1 times 1
4 = 2 times 2
9 = 3 times 3 and so on
One of the two equal factors of a square number is called the square root of that number. To denote the square root of a number we use a radical sign. √ it looks something like a check mark with a line extending. See our textbook for better examples. ( Page 157)
We can use our knowledge of perfect squares to find the lengths of sides of squares. For example IF the Area of a square is 49 cm squared. We can find the length of each side using the formula for the area of a square. A = lw but in this case it is also A = s^2, which is read "Area equals side squared."
so we know 49 = s^2. so the square root (SQRT) of 49... hmmm.. we know 7 times 7 = 49 so s = 7 cm. Notice the label isn't squared- we are talking about linear measurements with each side-- only the AREA is squared.
Positive SQ RT are also called principal square roots.
negative square roots exist also. For example the SQRT of 81 is 9 but it could also be -9 because (-9) times (-9) also = 81
We are focusing on positive SQ RTs in this chapter.
Memorize the Perfect Squares and corresponding SQ RTs up to 30.
KNOW the rule for squaring any number that ends in 5.
See me if you forgotten the rule!!
Check out these websites for how to calculate a SQ RT without a calculator

http://mathforum.org/library/drmath/sets/mid_square_roots.html

http://www.homeschoolmath.net/teaching/square-root-algorithm.php

http://www.nist.gov/dads/HTML/squareRoot.html

Algebra Period 3

CHAPTER 4-1
Introduction to INEQUALITIES and graphing them:
Writing an inequality
Graphing an inequality: open dot is < or >

Closed dot mean “less than or EQUAL” or “ greater than or EQUAL”
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
The closed dot INCLUDES the number as part of the solution set.

For example, x ≥ -5 is any number greater than -5 but it also includes -5.

What’s the difference of inequalities from equations? Inequalities have many answers (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution

GRAPHING INEQUALITIES:
First, graphing an equation's solution is easy
1) Say you found out that y = 5, you would just put a dot on 5 on the number line

2) But now you have the y ≥ 5

You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions

3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!

TRANSLATING WORDS:
Some key words to know: so make FLASH CARDS or do whatever you do to MEMORIZE these words!!
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!
I want at most 15 minutes of homework means that I can have 15 minutes,
but I'm sure hoping for even less!

CHAPTER 4-2 Solving Inequalities with adding and subtracting:
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get:
y > 5
Your answer is infinite! Any real number bigger than 5 will work!

See Graphing above to review how to graph the solution!!

CHAPTER 4-3 Solving inequalities with multiplication or division:
Again, you will use your equation skills,
but this time with the Multiplicative Inverse Property.

ONE MAJOR DIFFERENCE FROM EQUATIONS!
When you multiply or divide by a NEGATIVE to BALANCE,
you must SWITCH the inequality SYMBOL!
(Does not apply to adding or subtracting negatives.)

EXAMPLE: -3y > 9
You need to divide both sides by NEGATIVE 3 so the symbol
will switch from > to < in the solution
y < -3 is the answer

If you want to understand why:
3 < 10
Now multiply both sides by -1 (multiplication prop of equality)
You get -3 < -10, but THAT'S NOT TRUE!!!
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10

REMEMBER: when you MULTIPLY or DIVIDE by a NEGATIVE,
the symbol SWITCHES

Chapter 4-4; DOING 2 STEPS WITH INEQUALITY SIGNS -
Same as equations except make sure you switch the symbol
if you multiply or divide by a negative!
Always finish with the variable on the left.
Check with whatever solution is easiest in the solution set!

With two steps:

Before you start, you may want to clear fractions or decimals, but if you don't mind using them, just get started with the checklist below. If you want to clear them, you should do that right after you distribute (between steps 1 and 2 below)

1. Do distributive property first (if necessary)
2 Combine like terms on each side of the wall (equal sign)
3. Jump the variables to one side of the wall (get all the variables on one side of the equation) by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term)

4. Add or subtract
5. Multiply or divide
6. Make sure the variable is on the LEFT side when finished.

By putting the variable on the left side of the inequality, you will be able to graph the solutions much easier. The arrow you will draw will follow the same direction as the inequality sign.

Pre Algebra Period 2

Statistics Unit

There are many ways to ORGANIZE and REPRESENT data.
The whole point is to make the data more UNDERSTANDABLE.

There are 3 ways to express the CENTRAL TENDENCY of data:
Mean or average (add up all the values and divide by the number of items)
Mode or the most often seen value
Median or middle of data from least to greatest (if even number of items, average the middle two)

The Range can be expressed in two ways:
Showing the lowest value to the highest value
Calculating the difference between those two values (highest - lowest)

Tally Tables help you put COUNT the data and begin to ORGANIZE it.
You can tally numeric data (like grades) or non-numeric data (like names for the giraffe)

Frequency Tables take the tallies and SUMMARIZE them in an easy to read table.
Can be numeric or non-numeric (like the Tally Table)

Line Plots are graphs that make the Frequency Table summary more VISUAL.
It's really easy to see the MODE in the Line Plot.
For numeric data, you can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
Can be numeric or non-numeric (Like Tally and Frequency Tallies)

Stem and Leaf Plots are graphs that CLUSTER the data in INTERVALS.
This helps make the graph more UNDERSTANDABLE because you can see PATTERNS.
Data must be NUMERIC.
You can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
You can also see the MODE but not as quickly as the Line Plot because it will be in an interval with other data items. On the other hand, if you're looking for the interval that has the most data (not the individual item), the Stem and Leaf will show the "Interval Mode" very quickly.
You can read every data item on the Stem and Leaf.

Scatter Plots are graphs that plot TWO DIFFERENT TYPES OF DATA against each other, one one the x axis and one on the y axis to determine if there is a CORRELATION between the two types of data. ("co" meaning together and "relation" meaning a pattern)
If the two types of data have a correlation, you can draw a LINE OF BEST FIT through the center of the data points.
POSITIVE CORRELATION: the data moves in the SAME DIRECTION
(the more you study, the higher your grade....or the less you study, the lower your grade)
The line of best fit looks like it is going UP from left to right
NEGATIVE CORRELATION: the data moves in OPPOSITE DIRECTIONS.
(the higher the temperature, the lower the number of people using their heat....or the lower the temperature, the higher the number of people using their heat)
The line of best fit looks like it is going DOWN from left to right
NO CORRELATION: the two types of data has no relationship and so the data points are scattered everywhere with no pattern
(temperature changes should have no relationship with how well a student does)

TODAY:
LINE GRAPHS: p. 98-99
Data that takes place OVER TIME is represented well on a Line Graph.
For example, a student's math grade over the course of a year, or over middle school, or even over a single month. The length of the time interval depends on what you're trying to show.
If you have more than one set of data over the same time period,
you can make MULTIPLE LINES on the graph.
For example, a student's math grade, along with his science grade.

BAR GRAPHS:
Data over time can also be shown as a Bar Graph, but Bar Graphs are especially useful to show data that compares either non-numeric data or comparing multiple types of the same data at the same time.
Non-numeric data: Really a Line Plot but without the "X's" Allows you to represent very large data sets easily because you can label the y axis (vertical axis) in any way that you need to.
Numeric data: Going back to a student's grades on the Line Graph...Say you wanted to compare multiple students' math grades and science grades in 8th grade, you could make a DOUBLE BAR GRAPH with one bar color representing math grades and another bar color representing their science grades

HISTOGRAMS
Isn't this just a Bar Graph???
Yes! It's a specific type of bar graph that shows the FREQUENCY of data items

Isn't that what a Line Plot and a Stem and Leaf Plot show???
Yes!
If you made Bar Graph out of a Line Plot, it would be called a HISTOGRAM
or
If you turn a Stem and Leaf Plot on its side, outline the data listed in the leaves and then erase the data numbers and instead color in the bars, you will also have a HISTOGRAM.

What you lose:
The actual data items (you'll still know 12 students scored in the 80's, but not exactly what score in the 80's that each of them received)
You cannot determine the exact MODE or MEDIAN, but can make an analysis of the intervals.
What you gain:
For large amount of data items, a HISTOGRAM is much less cluttered than either a Stem and Leaf or Line Plot, making the patterns easier to understand.
For large amount of data items, because you can adjust the labels on the y axis, you can fit all the data in in whatever

Math 6 Honors Periods 6 & 7 (Tuesday )

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10 10 ∙ 10 = 100 10 ∙ 100 = 1000

10 ∙ 0.1 = 1 10 ∙ 0.01 = 0.1 10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620 = 7620

10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 2.63874 = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the nth power of ten, move the decimal point n places to the right.

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of a number greater than or equal to 1, but less than 10, AND a power of ten are said to be written in scientific notation.

To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the umber of places we moved the decimal point.

4,592,000,000 in scientific notation

First move the decimal point to the left to get a number between 1 and 10

4,592,000,000 4.592

Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation

4.592 x 10⁹

When we move a decimal point to the left, we are actually dividing by a power of ten.

Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

3.1 ÷ 10⁴ = 0003.1 = 0.00031

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

Math 6 Honors Periods 1, 6 & 7 ( Yosemite Week in review)

Comparing Decimals 3-4

We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.

In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:

1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.

2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .

3. 0.83 and 0.833

To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833

0.83 = 0.830 Then compare

0.830 and 0.833 since 0 <3 Then 0.830 < 0.833.

Write in order from least to greatest

4.164, 4.16, 4.163, 4.1

First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100

The order of the numbers from least to greatest is

4.1, 4.16, 4.163, 4.164

Rounding 3-5

A method for rounding may be stated as follows: Find the place to which you wish to round, mark it with an underline ___ Look at the digit to the right. If the digit to the right is 5 or greater, add 1 to the marked digit. If the digit to the right is less than 5, leave the marked digit unchanged. Replace each digit to the right of the marked place with a 0

Round 32,567 to (a) the nearest ten thousand, (b) the nearest thousand, (c) the nearest hundred, and (d) the nearest ten

(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros

30,000

(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros

33,000

(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros

32,600

(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero

32,570

A similar method of rounding can be used with decimals. The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.

Round 4.8637 to (a) the nearest thousandth, (b) the nearest hundredth, (c) the nearest tenth, and (d) the nearest unit

a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7

4.864

b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7

4.86

c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7

4.9

d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7

5

Adding and Subtracting Decimals 3-6

Decimals may be added or subtracted using the same rules as whole numbers

Write the given numbers one above the other with the decimal points in line.

Annex zeros to get the same number of decimal places and then add or subtract as if the numbers were whole numbers.

Place a decimal point in the number for the sum or difference in position under the decimal points in the given numbers.

Add 6.47 + 340.8 + 73.523

STEP 1	STEP 2	STEP 3
0006.47	006.470	006.470
0340.8	340.800	340.800
+ 073.523	+ 73.52	+ 73.523
	420 793	420.793

The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. Use estimates as a habit to check if your answer is reasonable. To find an estimate, first round the highest place value of the smallest number, then compare.

Add 8.574 + 81.03 + 59.432. Then estimate to check your answer. What is the highest place value of the smallest number? 9 + 81 + 59 = 149

Algebra Period 3

REVIEW OF SIMPLE EQUATIONS 3-1 through 3-3

1 and 2-step equations
What is the goal? ... to determine the value of the variable
What does isolate the variable mean? ... getting the variable alone on one side of the equation
How? Use inverse operations ( the opposites) to 'get rid' of everything on the side with the variable.
FOCUS? ... on the variable
Addition/ Subtraction Properties of Equality
For all rational numbers a, b, and c if you begin with a= b then
a+c = b+c
as long as you add/subtract the same value to BOTH SIDES you are still equal or in balance
Sometimes, you combine these two properties becasue we say you never subtract-- you always ADD the Opposite

Multiplication/ DIvision Proeprties of Equality
For all rational numbers a, b, and c if you begin with a = b then
ac= bc
or a/c = b/c

Again, as long as you multiply/ divide the same valu to BOTH SIDES you are still equal or in balance

ALso works for (3/4) x = 12 because you multiply by the reciprocal....
You are using the Multiplication Property of Equality with the Multiplicative Inverse Property
that's 2 friends at once!!
(3/4)x = 12
(4/3)(3/4)x = 12 (4/3)
x = 16

Using both properties together 2-step equations

Use the Addition Property of Equality first ( get rid of addition or subtraction)
Use the Multiplication Property of Equality next ( get rid of mult/division)

Collecting Terms- sometimes you need to collect LIKE TERMS on the SMAE SIDE of the equation before balancing

FORMAL CHECK- 3 steps
1) REWRITE the original equation
2) SUBSTITUTE your solution and put a "?" over the equal sign
3) DO THE MATH and check it!!


2- Steps with distributive Property
USUALLY you want to distribut first!!
(Unless the factor outside the ( ) can be dividied out of BOTH SIDES PERFECTLY!!

5y -2(2y +8) = 16
5y -4y -16 = 16 {distibute accurately}
y - 16 = 16 { collect like terms}
y = 32 { solve by adding 16 to both sides!!}

-3(4 +3x) = -9
Wait you could distribute but if you divide both sides by -3
{the -3 goes into BOTH SIDES PERFECTLY}
you have 4 + 3x = 3
3x = -1 { subtract 4 from both sides}
x = -1/3 { Divide both sides by 3}

What if you have variables on BOTH SIDES of the equation?

Simplify each side of the equation first
THEN use the additive inverse property to move the variables to the other side, Usually we try to move the smaller coefficient to the larger because somethimes that avoids negative coefficients. But it isnot always the case.

3y - 10 - y = -10y + 12
combine like terms
2y - 10 = -10y +12
In this case I would add 10y to both sides to get rid of a negative coefficient
12y - 10 = 12
Now add 10 to both sides
12y = 22
Now divide both sides by 12
y = 22/12 or y = 11/6

Pre Algebra Period 2( Week of October 6)

Two Step Equations 7-1

Use the Addition Property of Equality or the Subtraction Property of Equality to get rid of addition or subtraction
+prop =
-prop =

Then use the Multiplication Property of Equality or the Division Property of Equality to get rid of multiplication or division
xprop =
÷prop=

This is the opposite of Aunt Sally because for solving equations you must work backwards-- or go in reverse!!

3x + 7 = 28
Subtract seven from both sides
-7 = -7 you subtract 7 form both sides so you used the -prop =
3x = 21
NOW Divide both sides by 3
3x/3 = 21/3
x = 7

CHECKING a 2 step equation

You need 3 steps to check also

1) rewrite the equation
3x + 7 = 28
2) substitute your answers and put a "?" over the "="
3) DO THE MATH!! use order of operations
3(7) + 7 ? 28
21 + 7 ? 28
28 = 28

Math 6 Honors Periods 1, 6 & 7

One Step Equations with all 4 Operations 2-4 and 2-5

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

FORMAL CHECK OF YOUR ANSWER

There are three (3) steps to a formal check:

1. REWRITE the original equation from the original source-- this is just in case you find you copied the problem wrong!!

2. SUBSTITUTE your answer where the variable is and QUESTION your answer by placing a "?" over the =.

3. DO THE MATH-- that is check your answer by doing the math and finally putting a check mark at the end.

Pre Algebra Period 2 (Monday)

One Step Equations with all 4 Operations 2-5 and 2-6

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one? that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of

x = and box this in.

FORMAL CHECK OF YOUR ANSWER

There are thre (3) steps to a formal check:

1. REWRITE the original equation from the original source-- this is just in case you find you copied the problem wrong!!

2. SUBSTITUTE your answer where the variable is and QUESTION your answer by placing a "?" over the =.

3. DO THE MATH-- that is check your answer by doing the math and finally putting a check mark at the end.

Math 6 Honors Periods 1, 6 & 7

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

The point of the number line that is paired with a number is called the graph of that number.
Check out the graph on page 39 of our textbook.

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 would be 6 < n < 12 or 12 > n > 6

Pre Algebra Period 2 (Thursday)

Simplifying Variable Expressions 2-3
Review of Algebraic terminology:
In the expression, 3y + 5
3 is the coefficient (number attached to variable –
remember, "co" means to go along with)
y is the variable
5 is the constant (number not attached to variable)
terms are separated by ADDITION ONLY!

COMBINING LIKE TERMS:
1) Same variable (or no variable)
2) Same power

You can combine by addition or subtraction LIKE TERMS.
You cannot combine UNLIKE TERMS.

EX: 3a + 4a = 7a
but
3a + 4b = 3a + 4b
3a + 4a² = 3a + 4a²

YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!
IT'S MUCH SIMPLER!
-25a + 5a - (-10a) when a = -14
First combine like terms: -10a
Then plug in for a = -14: -10(-14) = 140

Algebra Period 3 (Wednesday)

Number properties and Proofs 2-10
MORE NEW FRIENDS! (PROPERTIES)
There are 2 types of Properties: Axioms and Theorems
Axioms = properties we accept as obvious and so we don't need to prove them
Theorems = properties that need to be proved USING THE AXIOMS WE ACCEPT AS FACT!

EXAMPLES OF AXIOMS:
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse

EXAMPLE OF A THEOREM:
Distributive Property in REVERSE (a + b)c = ac + bc


PROPERTIES OF EQUALITY
(these are AXIOMS)
” Prop = “
REFLEXIVE:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.


SYMMETRIC:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5
In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12

TRANSITIVE:
a = b and b = c then a = c
3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.

I've got a pattern that will help you recognize the difference between these 3 properties specifically.
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations

Algebra Period 3 (Monday)

Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac

Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN

INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES

EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b

Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]

Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]

Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]

Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21

Simplify by combining like terms:
12x - 6y -17

Pre Algebra Period 2 (Monday)

Properties of Numbers 2-1
WHAT ARE PROPERTIES? (Why are they your friends?)
You can count on properties. They always work. There are no COUNTEREXAMPLES!

COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)

An example from Math: You can't switch the order of subtraction because it's not the same value.
10 - 8 DOES NOT EQUAL 8 - 10
That's a COUNTEREXAMPLE to saying that you can switch subtraction
(We'll say that it's a COUNTEREXAMPLE to the existence of a
COMMUTATIVE PROPERTY OF SUBTRACTION
so that property does not exist!)

An example from Math: You switch the order of addition to make the adding easier.
20 + 547 + 80 = 20 + 80 + 547
(both equal 647, but the right side is much easier!)
What allowed you to switch the order?
A property called the Commutative Property of Addition says you can!
You'll always get the same value!

Now Aunt Sally doesn't like some of the properties because they allow us to do things that are exception to the Order of Operations!

Commutative Property
You can switch the order of all addition or all multiplication
a + b = b + a
ab = ba
3 + 5 = 5 + 3
3 (5) = 5 (3)
(you can HEAR the change in order!)
Aunt Sally says that you always need to go left to right, but Commutative says not necessary if
you have all multiplication or all addition.

Associative Property
You can group all addition or all multiplication any way you want
a + b + c = a + (b + c)
abc = a(bc)
(3 + 2) + 8 = 3 + (2 + 8)
(Why would you want to? Sometimes it's easier!)
[57 x 5] (2) = (57) [ 5 (2) ]
(you can't hear this property! but you can SEE it!)
Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if
you have all multiplication or all addition.
These properties give you a choice when it's all multiplication OR all addition
There are no counterexamples for these two operations.

BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples! 10 - 2 does not equal 2 - 10
15 ÷ 5 does not equal 5 ÷ 15)

SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???
Commutative allows you to choose!

ANOTHER 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!!!

TWO MORE FRIENDS:
THE IDENTITY PROPERTIES
OF ADDITION AND 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/a + b - c = 1


We also use this property to SIMPLIFY fractions.
We "simplify" all the parts on the top (the numerator) and the bottom (the denominator) that equal 1
(your parents would say that we are reducing the fraction)
6abc/10a =3bc/5 since both the numerator and denominator can be divided by
2a/2a



WE LOVE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!

Math 6 H Periods 1, 6 & 7 (Monday)

Writing Mathematical Expressions 2-1

Make sure to glue the 'pink 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

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 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.

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

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

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 b 8
so the mathematical expression in this case is "8w."

What about writing an expression for
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

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


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
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.

Math 6 Honors Periods 1, 6 & 7

Order of Operations 1-5
PEMDAS in my math class... PEMDAS in my math class... just follow the song and make sure you use the rules

Always perform the operation enclosed in the inner pair of group symbols ( such as parentheses and bracket [ ]) FIRST.
Then do all the exponents
Do all multiplication and division in order from left to right
Do all addition and subtractions in order from left to right.

A Problem Solving Model 1-6

make sure to glue in the "Plan for Solving Word Problems" into your spiral notebook

Read the problem carefully. Make sure that you understand what it says. You may need to read it more than once... That's okay!! It is great to re read things!!

Use questions like these in planning the solution:
What is asked for?
What facts are given?
Are there enough facts? Are there some unnecessary facts?
Determine which operation or operations can be used to solve the problem

Carry out the operations CAREFULLY!!
Check your results with the facts given in the problem.

Math 6 Honors Periods 1, 6 & 7

The Distributive Property 1-4

Distributive Property of Multiplication
(with Respect to Addition)

For any whole numbers a, b, and c,

a x (b + c) = (a x b) + (a x c)
and (b + c ) x a = (b x a ) + (c x a)

or written without the multiplication operator symbol
a(b + c) = ab + ac
(b+c)a = ba + bc
Remember that the parentheses indicate which operation to do first.


How could we simplify the following using the distributive property?

a. 13 x 15
13(10 + 5) = 13(10) + 13(5)


b. (11 x 4) + (11 x 6)
11( 4 + 6) = 11(10)

Distributive Property of Multiplication
(with Respect to Subtraction)

For any whole numbers a, b, and c,

a x (b - c) = (a x b) - (a x c)
and (b - c ) x a = (b x a ) - (c x a)

or written without the multiplication operator symbol

a(b -c) = ab - ac
(b-c)a = ba - ca which is better written ab- ac

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

Pre Algebra Period 2 (Monday)

CHAPTER 1-9: MULTIPLYING AND DIVIDING INTEGERS

They have the SAME rules!!!!

Math Book Rules:
If you have 2 signs that are the SAME -
answer is POSITIVE
If you have 2 signs that are DIFFERENT -
answer is NEGATIVE

Good Guy/Bad Guy Rules:
GOOD thing happens to GOOD person= GOOD = POSITIVE

BAD thing happens to BAD person = GOOD = POSITIVE
(they got what they deserved!)

GOOD thing happens to a BAD person = BAD = NEGATIVE
(we hate when good things happen to people who don't deserve it!)

BAD thing happens to a GOOD person = BAD = NEGATIVE
(we hate when that happens because it's so UNFAIR!)

Finger Rules:
If your index finger represents the negative sign, then if you have 2 negatives, you have the index fingers of both your left hand and your right hand and they make a plus sign!
If you just have one negative, it just stays negative because you don't have another finger to cross it!
If you have more than 2 negatives, you just keep using your index fingers to determine the sign.
We did this in class. It's fun! But once you get it, you won't need to keep doing it! (unless you want to keep having fun!!!)

WHAT IF THERE IS MORE THAN 2 SIGNS?
Use Aunt Sally Rules and go left to right
or
BE A SIGN COUNTER:
an ODD number of NEGATIVES = NEGATIVE
an EVEN number of NEGATIVES = POSITIVE

EXAMPLE: (-2) (-5) (-3) = -30 (odd number of negatives)
(2) (-5) (-3) = +30 (even number of negatives)
(2) (5) (-3) = -30 (odd number of negatives)

Averages ( or the Mean): Adding up all the numbers and dividing by the number of numbers

Algebra Period 3 (Monday)

CHAPTER 2-1: ABSOLUTE VALUE
Note:
It is difficult to show the symbol for absolute value here so l n l should be read as “the absolute value of n”

l n l = 5 has 2 possible answers: {-5, 5}
l n l= -5 is impossible! It's the null set and that symbol is either { } or a 0 with a slanted line through it

FORMAL DEFINITION OF ABSOLUTE VALUE:
absolute value of n is n if n was a positive number or zero
absolute value is the opposite of n if n was a negative number
(since absolute value is always positive)

The absolute value is a distance concept- the absolute value is the distance a number is from zero on a number line.

2 words often misunderstood:
withdrawing money is actually considered negative, while depositing is considered positive
The focus is not the money in your wallet, but the money in your bank account!

CHAPTER 2-2: RATIONAL NUMBERS:
Counting Numbers = natural numbers = 1, 2, 3, 4 …..
Whole Numbers = the natural numbers + 0… so 0, 1, 2, 3, 4, ….
Integers = the whole numbers and their opposites… so …-4, -3, -2, -1, 0, 1, 2, 3, 4….
Rational number = any number that can be expressed as the ratio (fraction) of two integers
a/b, where a and b are both integers and b cannot be zero
b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
Proof of this was done in class

There are positive and negative rational numbers.
One way to put them in order from least to greatest is to simply place them on the REAL number line. Numbers will be least to greatest if read from the left to the right.
If you have both fractions and decimals, it's often easier to change the fractions into decimals by simply dividing the numerator by the denominator.
REMEMBER: When numbers are NEGATIVE, the closer they are to zero, the bigger they are.
Example: -1/2 is greater than -3/4
-2.3 is greater than -2.5

You can also find the absolute value of rational numbers -
They are ALWAYS POSITIVE unless you're talking about zero (which is neutral)

Pre Algebra Period 2 (Tuesday)

Absolute Value 1-4
ABSOLUTE VALUE IS ALWAYS POSITIVE!
(except zero because zero has no sign)
Absolute value is a DISTANCE concept and that is why it can't be negative.

On the other hand, every integer has an ADDITIVE INVERSE which will be its OPPOSITE SIGN (except zero which has no sign - neutral)

Adding Integers 1-5
Three ways to understand adding integers:
1. positive negative sketch - make positive numbers positive signs and make negative numbers negative signs. Then match up all the positives with the negatives by box them in. Whatever is left, is the answer and the sign of the answer.

2. number line - draw the arrows and see where you end up

3. Who wins and by how much?
Different signs: Put the winner on top and take the difference
Take the DIFFERENCE (subtract) and keep the BIGGER (absolute value)number's SIGN
I say: 2 teams came to play: the positive team and the negative team
If you have 10 + (-15) then the positive team scored 10 while the negative team scored 15
Who won?
The negative team (so keep that sign)
By how much?
5 points
Answer: -5

Same sign: Just add and keep the sign you have
I say: only one team came to play so, of course, they won!
You would just add up the scores because all the players are on the same team!
Example: -5 + (-10), so just the negative team came to play
Therefore, the negative team won and you just add up their scores
Answer: -15

Additive inverses: the sum of additive inverses (same number with opposite signs) is always zero.
I say: it's a tie and no one wins! The answer would be zero!
Example: -5 + 5 = 0
-x + x = 0

Again, the rules are:
SAME sign: Just ADD them and KEEP the sign
DIFFERENT signs: Take the difference (SUBTRACT) and take the BIGGER number's SIGN.

MORE THAN 2 ADDENDS:
When adding a lot of addends, here's a good strategy:
1) SEE IF ANY ARE ADDITIVE INVERSES AND JUST CROSS THEM OUT BECAUSE ADDING INVERSES ALWAYS GIVES YOU ZERO!
What justifies crossing them out? The Identity Property of Addition
2) Add the positives to the positives
Add the negatives to the negatives
What justifies changing the order and grouping the addends this way?
The Commutative (order) and Associative (grouping) Properties of Addition

3) Finally, add the positive sum to the negative sum at the end and
see who wins and by how much
Usually, you will make less silly mistakes this way than just going left to right!

Adding integers with variable expressions:
Just substitute in for the variable, putting the substituted number into
( ),then evaluate using the integer rules.
(I say plug and chug!)
y + 5 where y = -12
(-12) + 5 = -7

Algebra Period 3 (Tuesday)

CHAPTER 1-5: FACTORING (part 2 starting on p. 25)
First, I'll review what Like Terms are and clear up the difference between
"combining" vs multiplying/dividing terms
LIKE TERMS:
1. Same variable
2. Same exponent
Constants are like terms because they all have no variables
(you can actually say that they have a variable raised to the 0 power,
which is 1 times the coefficient)
Example: 3x + 4 can be thought of as 3x + 4x0.= 3x + 4(1).
You can only combine (add or subtract) like terms.

BUT YOU CAN MULTIPLY/DIVIDE UNLIKE TERMS!
3a(7y) = 21ay
BUT
3a + 7y cannot be simplified

48xy/6x = 8y
BUT
48xy - 6x cannot be simplified

FACTORING
THE MAJOR "WATERSHED" MOMENT IN ALGEBRA I!
Undoing the Distributive Property
5 ( 6y + 11 )
This simplifies to 30y + 55

Now you can WORK BACKWARDS to the way the expression began before you used the distributive property.
In 7th grade, we called this using the distributive property backwards.
Now, in Algebra, we call this FACTORING.
FACTORING IS A KEY CONCEPT TO UNDERSTAND IN ALGEBRA.
WITHOUT IT, YOU WILL STRUGGLE THE ENTIRE YEAR!

Let's use the same problem as above and work backwards (FACTOR)
EXAMPLE: FACTOR 30y + 55

THINK: What do 30y and 55 have in common (What is their GCF?)
They both divide by 5

FACTOR out the GCF of 5
(factoring out is really dividing each term by the 5)

MIDDLE STEP TO UNDERSTAND FACTORING:
5 ( 30y + 55 )
5 5
we haven't changed the value because we've divided each piece by 5 and multiplied by 5 as well
(multiply a number by 5 and then divide it by 5 and you'll see have your starting number!)

FINAL FACTORED FORM: 5 ( 6y + 11 )
Wait a minute...Isn't that what I started with before I used the Distributive Property????
Of course it is! This is the key concept in factoring!!!!!

FACTORING DOES NOT CHANGE THE VALUE OF WHAT YOU STARTED WITH!
It's incognito the same thing!
Therefore, you can always check your factored form by multiplying back to the original simplified form.
If it doesn't get back to that,
YOU HAVEN'T FACTORED PROPERLY!

We say that the factored form and simplified form are EQUIVALENT EXPRESSIONS..
(I say they are INCOGNITO the same thing!)

Pre Algebra Period 2 (Monday)

Variables & Expressions 1-1 (continued)

You can translate words into Algebra word by word
just like you translate English to Spanish or French.
5 more than a number
5 + n
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: BECAUSE MULTIPLICATION & ADDITION IS COMMUTATIVE, YOU DON'T NEED TO WORRY ABOUT THE ORDER
BUT
FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER
GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS
EXCEPT (counterexample!)
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it.
For word problems like someone's age or the amount of money you have, you should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: T = 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
T = 3 + 5(4) = 23
In your check, Tom is 23. Is Tom 3 years older than 5 times Julie's age?


STRATEGY #2: 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 number of 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 number of 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 number of 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 number of days
24 1
48 2
72 3
h h/24