Math Notes for Miss

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Absolute Value
ABSOLUTE VALUE-- A number’s distance from zero. The absolute value of
any number is always positive.
1. Absolute value is represented like this: | -4| = 4 and
is read “the absolute value of negative four is four”
meaning that negative four is four spaces away
from zero.
2. For example:
a.
| -27| = 27 b. | -50| = 50 c. | 10| = 10
3. So…if you put the absolute value of these numbers is
order from least to greatest you would get: 10, 27, 50
BUT if you put the numbers themselves in order it would
be quite different:
-50, -27, 10
Integers and Intervals
o
o
o
o
o
INTEGER – positive and negative whole numbers
and zero. In other words: the counting numbers.
This does not include decimals or fractions!
(ie: is 14.5 an integer?)
INTERVAL – a “piece” of the set of integers (show
how to graph and write integers, what numbers
the interval includes)
My interval – timeline example (ie: [6 years old, 12
years old] I played softball)
Notecard 1 Multiplication and Division of Integers
Math notes: quick review of rules for multiplication and division of
integers. Go over in detail adding and subtracting integers.
-MULTIPLICATION AND DIVISION – for these two operations the rules are the
same!!!!
-Positive x/ positive = positive
-Negative x/ negative = positive
-Negative x/ positive = negative
-ADDITION AND SUBTRACTION
-The rules for these are more complicated…but lets begin by always
reading these equations like a sentence (left to right)
-First remember: a positive/plus sign indicates moving to the right on the
number line-a negative/minus sign indicates moving to the left on the
number line
Notecard 2 on the Multiplication and Division of Integers
HOWEVER--- when you are reading your sentence and you see a
negative number to the right of your operation sign, it is like a “u turn” sign
telling you to move the opposite way on the number line.
5+3=8
But---5 + -3 = 2
TRICK: with subtraction you can change the subtraction sign to an
addition sign and change the number to the right of the operation sign to
its opposite.
Ie: 5 + -3 = 2 is the same as 5 – 3 = 2
Subtraction of POSITIVE AND NEGATIVE INTEGERS!!!!!!!!
**Subtraction of positive and negative integers no longer exists!!!!
1.
2.
change the subtraction sign into addition
change the number to the right of the operation sign (the
subtraction sign you turned into an addition sign) into its’
opposite.
Ex: 5 – 4 = 1 is the same as 5 + (-4) = 1
Addition of Positive and Negative Integers
If the numbers are the same sign, keep that sign for your answer
and add the numbers.
Ex: -6 + (-3) = - 9 or
8+3 = 11
1.
2.
If the signs are different keep the sign of the number with the
larger absolute value for your answer and subtract.
Ex: -6 + 12 = 6
or
-20 + 10 = -10
**A little song to help you remember!! Sing this to the tune of “Row Row
Row Your Boat”:
“Same sign add and keep, different signs subtract. Take the sign of the
higher number then it will be exact!”
Absolute Value
ABSOLUTE VALUE-- A number’s distance from zero. The absolute value of
any number is always positive.
-Absolute value is represented like this: | -4| = 4 and is read “the absolute
value of negative four is four” meaning that negative four is four spaces
away from zero.
For example:
b. | -27| = 27
c.
| -50| = 50
d. | 10| = 10
So…if you put the absolute value of these numbers is order from least to
greatest you would get: 10, 27, 50 BUT if you put the numbers themselves
in order it would be quite different: -50, -27, 10
PEMDAS
This is an acronym for the order of operations. Order of operations is the
order in which you complete problems with more than one operation.
We will go deeper into this next week but for now know what each letter
stands for:
o
P – parenthesis
o
E –exponents
o
M – multiplication
o
D – division
o
A – addition
o
S – subtraction

Exponents
Exponents are used to represent REPEATED MULTIPLICATION
Here is how to write in exponential form: 3^5
3 is the base
5 is the exponent/ power
Expanded form—writing it out
4^3 = 4 x4 x 4 = 64
Special case!!! **anything raised to the zero power is one
Word Problem Strategies
1.
2.
3.
4.
5.
6.
Read through at least once to get
the main idea. What is the problem
about? What is it asking?
Read it a second time, underline
important numbers and units.
Identify what you are looking for
Write out the key information you
need to use in order to solve
Determine what operation to use.
How do we determine which
operation to use?
Set up an equation/expression to
solve the problem
Patterns and Sequences








Example 1: Find the pattern: 1, 3, 6, 10, 15….
What is the rule? (add 2, add 3, add 4, add 5…etc.) What
comes next? 21, 28…
Example 2: Find the pattern: -20, -15, -10, -5, 0…
What is the rule? (Add 5) What comes next? 5, 10…
A sequence is an ordered set of numbers.
When these numbers change by the SAME AMOUNT EACH TIME
it is called an arithmetic sequence.
Example 1 is NOT an arithmetic sequence while Example 2 is.
Each number in a sequence is called a term.
Variables and Algebraic Expressions

Variable -- a letter or symbol that represents a quantity.
Ex: x, y, z, #, b, etc.

Algebraic Expression – and expression that contains one or more
variables
Ex:
a+b
3d + 4x
Expression vs. Equation
An expression does not have an equal sign.
An equation has an equal sign.




Translating Words into Math
Addition (combine)
o
Add, Plus, Sum, Total, Increased by, More than

Subtraction (less)
o
Minus, Difference, Subtract, Less than, Decreased by,
Take away

Multiplication (put together groups of equal parts)
o
Product, Times, Multiply

Division (separate into equal groups)
o
Quotient, Divide
Mean, Mode, Median, Range
Mean -- the average of a set of numbers
1. add al l the numbers in the set
2. divide the total by the amount of numbers in the set
Mode – the number in the set that appears most often
Range – the difference between the highest number and the lowest
number in the set
Median -- the number in the middle of the set
1. Order the numbers in the set from least to greatest
2. Take the number in the middle as your median
3. When the amount of numbers in the set is odd, this is easy.
Ex. 4, 5, 7, 8, 19 the median of this set is 7
However, when the amount of numbers is even you must follow these
rules:
a. Order the numbers in the set from least to greatest
b. Take the two numbers in the middle and find their
average. The average of these two numbers in the
median.
Ex. 4, 5, 7, 9, 10, 20
7+9=16 16/2=8 the median of this set is 8
Integers: positive and negative whole numbers and zero
Rational numbers: any number that can be expressed as a ratio of two
integers.
Ratio: comparing two quantities (we haven’t talked too much about this
so don’t worry)
So your thinking…what does that definition of rational numbers mean?!
Basically---we’re talking about fractions and decimals people!
2 ,4 1 3 ,3 1 2, 3 4 5 . 6 7 8 9


Rounding Review
When rounding we look at the number to the RIGHT of the
number we are rounding to. If it is greater than or equal to 5, we
round up. If if is less than 5, round down!
Ex. 145.1 round to the nearest ten…… 150.0
Adding and Subtracting Decimals
ALWAYS REMEMBER:

When adding and subttacting decimals always line up the
decimal points!! If there is a blank spot, insert a zero as a place
holder.
Money Baby!
*Dealing with money is dealing with decimals!
$money
decimal
5 cents……………………………….0.05
20 dollars……………………………20.00
50 cents……………………………….0.50
2 dollars and 50 cents………………2.50
Multiplying Decimals



Multiply the same as when you are using whole numbers!
You do not have to line up the decimal points
To place the decimal point add the number of decimal places
in each number you are multiplying and move the decimal point
that many places to the left in your answer.
o
Ex:
0.17- 2 decimal places
X
3  0 decimal places
0.51 2 decimal places
Dividing Decimals
Dividend: the number to be DIVIDED
Divisor: the number you are dividing by (the evil villain that chops
numbers into smaller pieces)
Step 1: write out the problem in long division form
Step2: move the decimal point of the divisor right until you have a whole
number
Step3: move the decimal point of the dividend right the same number of
positions!
Step4: perform long division as usual, bringing the decimal straight up!
Scientific Notation
Scientific notation: a method for writing very LARGE or very SMALL
numbers.
**Scientific notation is used by both mathematicians AND scientists! (think
of how big numbers get when talking about astronomy or cell biology for
example)
Scientific notation uses exponents, namely: powers of ten.
What are powers of 10?
101 = 10
102 =100
103 = 1,000
104 = 10,000


Multiplying and Dividing by powers of 10
When multiplying by a power of ten, move the decimal point to
the RIGHT the same number of places as there are zeros in the
power of ten.
When dividing by a power of ten, move the decimal point to the
LEFT the same number of places as there are zeros in the power
of ten.
Synthesizing info for Scientific Notation
Exponential Form
102
10-2


Expanded Form
10X10
1
= 1
102
100
Solution
100
0.01
Positive exponents make the number BIGGER. The decimal
point moves to the RIGHT.
Negative exponents make the number SMALLER. The
decimal point moves to the LEFT.
Let’s put what we know together so solve some problems in Scientific
Notation: 6.0 x 103 = 6.0 x 1000 = 6,000.0
Positive exponents.
The number being multiplied by the power of ten gets LARGER. The
decimal point moves to the RIGHT. The exponent dictates how many
zeros will be in the solution to the exponential form. The exponent also
dictates how many places to the RIGHT the decimal point will move.
6.0 x 10-3 =6.0 x 0.001 = 0.006
Note the negative exponent. The number being multiplied by the power
of ten gets SMALLER. The decimal point moves to the LEFT. The exponent
dictates how many spaces behind the decimal there will be. The
exponent also dictates how many places to the LEFT the decimal point will
move (multiplying with negative exponents is like dividing with powers of
ten!)
Divisibility



A number is divisible by another number if their quotient is a
whole number with no remainder.
o
Ex. 46 is divisibly by 2 and 7 because:

42 ÷ 2 = 21 (whole number, no remainder)

46 ÷ 7 = 6 (whole number, no remainder)
Any number greater than one is divisible by at least two
numbers: one and itself. Numbers that are divisible by more
than two numbers are called composite numbers.
A prime number is divisible by only one and itself.
Divisibility Rules
A number is divisible by:

2 if the last digit is even

3 if the sum of the digits is divisibly by 3

4 if the last two digits form a number that is divisibly by 4

5 if the last digit is 0 or 5

6 if the number is divisible by both 2 and 3

9 if the sum of the digits is divisibly by 9

10 if the last digit is 0


Factors
Whole numbers that are multiplied to find a product are called
FACTORS of that product. (all the numbers that can be
multiplied together to get a certain number)
o
Ex. 2 x 3 = 6
o
2 and 3 are factors of the number 6
o
All the factors of 6 are: 1,2,3,6
A number is DIVISIBLE by its factors.
o
6÷3=2
o
6÷2=3
o
o
o
o
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





PRIME FACTORIZATION –
a number written as a product of its prime factors.
You can use factors to write a number in different ways.
The factorization of 12 =
o
12 x 1
o
2x6
o
3x2x2
o
3x4

Note that 3 x 2 x 2 is the PRIME FACTORIZATION
of 12 because it uses all prime factors to
represent 12
1.
2.
3.
4.

FACTOR TREES!
Choose any two factors of the number you are factoring.
Keep finding factors until each branch ends in a prime number
The prime numbers you are left with are called the roots of the
number you were factoring.
Using exponents, write the number you were factoring as a
product of its prime factors or roots.
Fractions and Decimals




fractions and decimals are known as RATIONAL NUMBERS.
Decimals and fractions are often used to represent the same
number.

Ex. ½ = 0.5

Ex. 5/100 = 0.05
Decimals can be written as fractions OR as MIXED NUMBERS.
MIXED NUMBER -- a number that contains and whole number
and a fraction. Ex: 3 ½
Writing Decimals as Fractions Note Card 1
Identify the place value of the digit farthest to the right.
Use that place value to determine the denominator.
Decimal
Fraction
0.23…………………..23/100 (twenty-three hundredths)
1.7…………………..1 7/10 (one and seven tenths)
0.004…………………..4/1000 (four thousandths)
1.
2.
Writing Fractions as Decimals
1. It’s simply a division problem!!
Ex. 1/5 = one divided by five = 1 ÷ 5
The numerator is the dividend
The denominator is the divisor
Writing Decimals as Fractions Note Card 2
Numerator
Denominator
Dividend
Divisor
Fraction
Decimal
¾………..3 ÷ 4 =
0.75
5 2/3……2 ÷ 3 =
5.66 (with mixed numbers keep the whole
number the same)
TERMINATING DECIMAL – has a terminating (ending) number of decimal
places.
REPEATING DECIMAL – has a block of one or more digits that repeat
continuously.
Steps for REDUCING fractions:

If the numerator and the denominator are both
divisible by a common number, the fraction can
be reduced.

Find a common factor and divide both numerator
and denominator by that factor

Continue this process until the numerator and
denominator have no common factors or until you
get a prime number for the numerator or
denominator
Improper Fractions
Improper Fraction-- a fraction where the numerator is bigger then the
denominator.
It is often best to convert improper fractions into mixed numbers!
Step 1: Divide! Divide the numerator by the denominator. Use the answer
for the whole number in the mixed number
Step 2: Use the remainder as the numerator in the fraction of your mixed
number.
Step 3: Use the same denominator as the improper fraction
Ex: Convert the improper fraction 16/5 into a mixed number= 3 1/5
Sometimes you’ll have to convert mixed numbers back into improper
fractions as well!!! Ex: 5 1/3 = 16/3
LCM = Lowest Common Multiple
The multiples of a number are what you get when you multiple it by other
numbers. Just like the multiplication table.
Multiples of 4: 4, 8,12, 20, 24
Multiples of 5: 5, 10, 15, 20, 25
The LCM of 4 and 5 would be 20, the smallest common multiple.
The LCM becomes your common denominator when doing operations
with fractions. Once you have a common denominator you can
combine fractions! Yes!
2/3 + ¼ = 3/12
Converting Fractions
To convert the fractions you must find what number you multiply the
denominator by to get the denominator and multiply the numerator by
that same number, giving you your new numerator.
Step 3: Add or subtract the numerators.
Step4: Reduce!
Ordering Fractions and Percents
Ordering Fractions: Strategies

Draw a picture, picture it in your mind…if this works for you:
GREAT!

Find a common denominator

Convert the fractions into decimals

Convert the fractions into percents
Percents
Percent means “per one hundred”
Ex. 8% means “8 per 100” or “8 out of 100”
Percents
*Because percent means “out of 100” 100% is 100 out of 100. That’s
why 100% means “all” or “the whole thing.” If 24% means “24 out of
100”… As a fraction it would be: 24/100 As a decimal it would be:
0.24
Writing Decimals as Percents
Method 1:

Write the decimal as a fraction

Convert the fraction to have 100 as the denominator

Write the numerator with a percent sign

Ex. 0.3= 30/100  30%
Method 2:

Multiply by 100

Ex. 0.3 x 100= 30  30%
Writing Fractions as Percents
Method 1

Convert the fraction to have 100 as the denominator

Write the numerator with a percent sign

Ex. 4/5 = 80/100  80%
Method 2

Convert the fraction into a decimal

Multiply by 100

Ex. 4/5 = 0.8; 0.8 x 100= 80  80%
Ratios
Intruments in the Baltimore Symphony Orchestra
Violins 29
Violas
12
Cellos 10
Flutes
5
Total number of instruments: 56

You can copare the different groups using RATIOS.

RATIO: a comparison of two quantities* using division. Think:
ratios are like “fancy fractions”
*quantity: amount
EQUIVALENT RATIOS:
Ratios that name the same comparison. You can find equivalent ratios by
multiplying OR dividing both terms of a ratio by the SAME number (think:
EQUIVALENT FRACTIONS)
Proportions and Rates
What is the scientific formula for water?
H2O
-This equation means there are two atoms of Hydrogen and one atom of
Oxygen…when chemically bonded they make one molecule of water.
What is the RATIO of hydrogen to oxygen?
2:1, 2 to 1, 2/1, two hydrogen atoms to one oxygen atom
*No mater how many water molecules of water----hydrogen and oxygen
will always be in the same RATIO---- 2:1
Using your knowledge of ratios and equivalent fractions, fill out the
following chart:
#of Water
1
2
3
4
Molecules
Ratio:
Hydrogen
2
Oxygen
1
*notice that all four of these fractions or RATIOS are equivalent!
Proportion: an equation that shows two equivalent ratios.
**Just like EQUIVALENT FRACTIONS!
Using Percents!
Using percents is a snap—PERCENTS ARE PROPORTIONS!
% = IS
100
OF
Ex: What is 12% of 50?
What is 12% is 30?
12% of what is 10?
12% of 16 is what?
Multiplying and Dividing Fractions!!
The formula for multiplying fractions is easy:

Multiply the numerators

Multiply the denominators

Remember to reduce!
EX: 1/3 × 3/5
Multiply the numerators: 1 × 3 = 3 (3 is your numerator)
Multiply the denominators: 3 × 5 = 15 (15 is your denominator)
So… 3/15 is your answer…reduced, that’s 1/5
*if you are multiplying with mixed numbers….
you must convert them to improper fractions before multiplying!!!!!!!!!!!
Dividing Fractions Note Card 1
With dividing fractions we have to know what a reciprocal is:
Two numbers are reciprocals if their product is 1. To find the reciprocal of
any fraction, switch the numerator and the denominator…FLIP IT!
EX: the reciprocal of ¾ is 4/3 and ¾ × 4/3 = 1



Just like with multiplying fractions, you must first convert any
mixed numbers into improper fractions!!!!
Change the problem into a multiplication problem
Find the reciprocal of the second number in the equation and
multiply by the reciprocal
Dividing Fractions- Note Card 2
Ex: ¾ ÷ ½

First rewrite the problem as a multiplication problem: ¾ × ½

Now find the reciprocal of the second number: the reciprocal of
½ is 2/1

Use the reciprocal to multiply: ¾ × 2/1 =

Multiply the numerators: 3 × 2 = 6 (6 is the numerator)

Multiply the denominators: 4 × 1 = 4 (4 is the denominator)

The answer is 6/4, reduced is 1 ½
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