Date:___________
Created by: Loren L. Spencer
Main Idea
Determining whether a
number is positive or
negative
Through
Integer Rules
Integer definition—Whole numbers and their
opposites-- (Positive and Negative numbers)
Rule 1: The sign in front of the
number tells us whether or not the
number is positive or negative.
a. If the sign is a (+), the number is
positive.
Example:
b. If the sign is a (-), the number is
negative.
**(Hint: Always circle the sign and
its number.)
Rule 2: If the number does not
have a sign, the number is always
positive.
Rule 3: Positive numbers are
always greater than negative
numbers.
**(Hint: As you move to the left on the number line
the numbers get smaller. As you move to the right the
numbers get larger.)
Example: -6×7=
Multiplying and dividing with
integers
Rule 1: If the signs of both
numbers are the same, the answer
is positive.
Example: -2×-2=
Example: -6÷-3=
Rule 2: If the signs of both
numbers are different, the answer
is negative.
Example: 20÷-4=
1
Date:___________
Created by: Loren L. Spencer
Main Idea
Rules for adding and
subtracting integers
Through
More Integer Rules
Example: 3+4=
Rule 1: If the signs of both
numbers are the same, add the
numbers and keep the sign.
Example: -6-2=
Rule 2: If the signs of both
numbers are different, subtract
the numbers and keep the sign of
the number with the larger
absolute value.
Steps to use when two signs
are side by side
Step 1: Circle the two signs that
are next to each other like this.
Step 2: Now turn the (2) two
signs that are next to each other
into (1) one sign.
Step 3: Rewrite the equation.
Example: -2+4=
Example: 6-9=
***To solve this type of problem: Refer to the
multiplication and division rules for integers. You
must turn the (2) two signs into (1) one sign.
-
Example: -6+ 2=
-
Example: 3- 5=
Step 4: Solve using the addition
and subtraction rules for integers.
2
Date:___________
Main Idea
Adding and Subtracting
Created by: Loren L. Spencer
Through
Decimal Operations
Example: .06+2.375=
Step 1: Rewrite the Problem so
that the numbers stack. (Always
line up the decimals when you add
and subtract) Sometimes it helps
to draw a vertical line through the
decimals.
Step 2: Add or subtract like
normal.
Example: 8.25-6.375=
Multiplying Decimals
Example: .22×.675=
Step 1: Multiply the problem like
there are no decimals in the
problem.
Step 2: Count the number of
places “cheeks” to the right of the
decimal(s).
Example: 3.14×15=
Step 3: Now go back to your
answer in Step 1. Starting from the
left and moving towards the right
count the same number of places as
you had in Step 2.
3
Date:___________
Main Idea
Loops & Mixed Numbers
Created by: Loren L. Spencer
Through
Adding and Subtracting Mixed Numbers
Step 1: Add or subtract the
whole numbers.
Step 2: Circle the denominator on
the left and draw a loop to the
right and place the denominator
over itself.
Step 3: Circle the denominator on
the right and draw a loop to the
left and place the denominator
over itself.
Step 4: Multiply straight across.
(The denominators should be equal)
Step 5: Add or subtract the
numerators and keep the
denominators the same.
Step 6: If the answer is an
improper fraction simplify by
division.
Step 7: Reduce
4
Date:___________
Main Idea
Multiplying Mixed Numbers
Created by: Loren L. Spencer
Through
Multiplying and Dividing Mixed Numbers
Step 1: To multiply or divide
mixed numbers you need to turn
the mixed numbers into improper
fractions.
A. For the left Mixed number,
Multiply the denominator and the
whole number and add the
numerator—this total becomes the
numerator for the improper
fraction.
B. Follow the above step for the
right Mixed number.
Step 2: Multiply straight across.
Step 3: If the answer is an
improper fraction simplify by
division.
Step 4: Reduce
5
Date:___________
Main Idea
Dividing Mixed Numbers
Created by: Loren L. Spencer
Through
Multiplying and Dividing Mixed Numbers
Step 1: To multiply or divide
mixed numbers you need to turn
the mixed numbers into improper
fractions.
A. For the left Mixed number,
Multiply the denominator and the
whole number and add the
numerator—This becomes the
numerator for the improper
fraction.
B. Follow the above step for the
right Mixed number.
Now Remember--Dividing
Fractions is as easy as pie,
Flip the 2nd and multiply.
Step 2: Take the reciprocal of
the 2nd fraction. Flip it
Step 3: Multiply straight across.
Step 4: If the answer is an
improper fraction simplify by
division.
Step 5: Reduce
6
Date:___________
Main Idea
Dividing Decimals
Created by: Loren L. Spencer
Through
Decimal Operations
Example: .0675÷2.5=
Step 1: Place the 1st number in
“the house” (the division symbol.)
and place the 2nd outside.
Step 2: Move the decimal on the
divisor (the number outside of
the house) to the right to make a
whole number.
Step 3: Move the decimal on the
divisor (the number inside the
house) to the right the same
number of places “cheeks.”
Example: 15.7÷3.14=
Step 4: Draw a vertical line
through the decimal and bring the
decimal on top of the house. (the
line eliminates sloppy mistakes)
Step 5: Now you can divide like a
normal division problem.
7
Date:___________
Created by: Loren L. Spencer
Main Idea
Converting Decimals to
Percents
Rule 1: If there is no decimal
shown then the decimal is on the
right side.
Through
Converting between Decimals, Fractions, and
Percents
Percent—a ratio/fraction comparing a number to 100
Remember: the % sign has two zeros, so you must
Step 1: Move the decimal place 2
“cheeks” to the right.
Remember: D
P
Step 2: If there are no
digits/numbers in a cheek, fill with
zeros—No empty cheeks
Step 3: Place a percent sign to
the right of the last number.
move the decimal 2 places
Example: Change 22 to a percent
Decimal
D
Percent
P
Example: Change .0735 to a percent
Decimal
D
Percent
P
Example: Change .0003 to a percent
Converting Percents to
Decimals
Decimal
D
Percent
P
Rule 1: It is only a percent if
there is a percent sign.
Rule 2: If there is no decimal
shown then the decimal is on the
right side.
Step 1: Move the decimal place 2
“cheeks” to the Left.
Remember: D
P
Step 2: If there are no
digits/numbers in a cheek, fill with
zeros—No empty cheeks
Example: Change 125% to a decimal
Decimal
D
Percent
P
Example: Change .0034% to a decimal
Decimal
D
Percent
P
Example: Change 7.21% to a decimal
Decimal
D
Percent
P
8
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Converting Fractions to
Decimals
Converting between Decimals, Fractions, and
Percents
Cowboy
𝟑
Example:
𝟖
Step 1: Divide the numerator by
the denominator. The numerator
goes in the house.
Example:
Horse
𝟐
𝟓
or
𝟑
𝟏
Remember: The cowboy sits on
the saddle on top of the horse.
When he gets home the cowboy
goes inside, and the horse stays
outside.
Memorize:
=.10 so that 𝟏𝟎 =3×.10=.30
𝟏𝟎
& multiply 𝟏
𝟓
=
.
125
so
that
=5×.125=.625
by the
𝟖
𝟏
numerator
𝟓
𝟏
𝟒
𝟏
Step 2: Move the decimal place 2
“cheeks” to the right.
Remember: D
=.25
so that
=7×.25=1.75
𝟒
𝟓
=
.333 so that
=5×.333=1.666
𝟑
𝟑
𝟏
𝟗
=
.50 so that
=9×.50 =4.50
𝟐
𝟐
Converting Fractions to
Percents
Step 1: Change the fraction into
a decimal by dividing the numerator
by the denominator. The
numerator goes in the house.
=.20
𝟖
𝟒
so that
=4×.20=.80
𝟓
𝟕
Example:
Example:
𝟑
=
𝟏𝟓
𝟐
𝟑
=
P
Step 3: If there are no
digits/numbers in a cheek, fill with
zeros—No empty cheeks
Step 3: Place a percent sign to
the right of the last number.
Or by
ball bat
𝟑
=
𝟏𝟓
𝟐
𝟑
𝟏𝟎𝟎%
=
𝟏𝟎𝟎%
9
Date:___________
Created by: Loren L. Spencer
Main Idea
Converting Decimals to
Fractions
Through
Converting between Decimals, Fractions, and
Percents
Fraction: a part of a whole, in the form of
𝑵
𝑫
Denominator: The bottom number of a fraction
which represents the number of parts in the whole
Numerator: The top number of a fraction
Step 1: If there is a whole
number (the number to the left of
the decimal bring) it down. It will
stay to the left of the fraction.
Example: .375
Step 2: The numbers to the right
of the decimal will become the
numerator. Place them over 1.
Step 3: Count the places “cheeks”
to the right of the decimal in the
original problem and place that
many zeros to the right of the one
in the denominator.
Example: 2.56
Step 4: Reduce
10
Date:___________
Created by: Loren L. Spencer
Main Idea
Ordering Rational Numbers
3 major ways to order rational
Numbers:
1. Make everything a Decimal
2. Make everything a Percent
3. Make everything a Fraction
(Least preferred)
Through
Ordering Decimals, Fractions, and Percents
Rational Number: Any number that can be
expressed as the ratio/fraction of two integers.
Example:
1/3
Step 1: Using the rules above
change everything into a decimal
Step 2: Stack the numbers by
lining up the decimals. (You should
be able to draw a vertical line
through all of the decimals)
20%
.3
3%
Example:
1/5
5%
.35
3.5%
Step 3: Now add zeros to the
ends to make every number the
same length.
Step 4: Add a dollar sign in front
of each number and determine
which is the largest. Write it
down. Then choose the next
largest. (Remember to mark out
each one as you go.)
Example:
62.5%
2/3
75%
1/2
7/10
Note 1: You can convert all the
numbers to percents and follow the
above steps. This is an equally
good method and is often
preferred by my students
Note 2: I do not recommend
converting to fractions as it is
difficult to find a common
denominator.
Remember: A positive number is
always greater than a negative one.
11
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Multiplying fractions
Fraction Operations
Improper Fraction: A fraction which has a
numerator greater than or equal to the
denominator.
Step 1: Multiply straight across
Example:
Multiply the denominator times the
denominator and multiply the
numerator times the numerator.
Step 2: If the answer is an
improper fraction, simplify by
division
Step 3: Reduce
Example:
Example:
𝟑
𝟓
𝟓
𝟑
𝟖
𝟓
𝟒
× =
𝟕
𝟏
× 𝟏𝟎=
𝟒
× 𝟔=
Hint: If you are multiplying or
dividing fractions and a whole
number, always place the whole
number over 1.
Example:
𝟕
𝟖
× 𝟑=
12
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Dividing Fractions
More Fraction Operations
Dividing Fractions is as easy as
pie,
Flip the 2nd and multiply.
Example:
𝟒
𝟓
𝟔
÷ 𝟕=
Step 1: Take the reciprocal of
the 2nd fraction. Flip it
Step 2: Multiply straight across
Multiply the denominator times the
denominator and multiply the
numerator times the numerator.
Step 3: If the answer is an
improper fraction, simplify by
division
Example:
Example:
𝟒
𝟓
𝟑
𝟖
𝟔
÷ 𝟕=
÷ 𝟓=
Step 4: Reduce
Example:
𝟓
𝟔
𝟐
÷ 𝟑=
Hint: If you are multiplying or
dividing a fraction and a whole
number, always place the whole
number over 1.
13
Date:___________
Created by: Loren L. Spencer
Main Idea
Loops
Rule 1: To add & subtract
fractions, you must have like
denominators.
Step 1: Circle the denominator on
the left and draw a loop to the
right and place the denominator
over itself.
Through
Adding and Subtracting Fractions
Example:
Example:
𝟑
𝟖
𝟐
𝟑
𝟐
+𝟕
𝟓
+𝟕
Step 2: Circle the denominator on
the right and draw a loop to the
left and place the denominator
over itself.
Example:
𝟓
𝟔
-
𝟑
-
𝟕
𝟓
Step 3: Multiply straight across.
(The denominators should be equal)
Step 4: Add or subtract the
numerators and keep the
denominators the same.
Example:
𝟑
𝟒
𝟗
Step 5: If the answer is improper
simplify by division.
Step 6: Reduce
14
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Exponents with Positive
Bases
Exponents and Square Roots
4
Definition: Exponent/power
Example: 5
Step 1: Re-write in Expanded
form
Example: (6)
3
Step 2: Multiply
Exponents with Negative
Bases
Example: (-4)
3
Example: (-3)
4
Rule 1: If the base is negative
and the exponent is an even number
the answer will be positive.
Rule 2: If the base is positive and
the exponent is an odd number the
answer will be negative.
Square Roots
Definition: Square Root
Step 1: Make the Perfect
squares chart
Step 2: Determine where the
number is located. Be careful The
answer is in the first column.
This the
square root sign it is not a
division symbol
Example: What two integers is the
200
between?
Example: Which is the closest to
90
a) 9.3
b) 9.4
c) 9.5
d) 9.6
15
Date:___________
Created by: Loren L. Spencer
Main Idea
Scientific Notation
Rule 1: To be in Scientific
Notation, you must have one and
only one number not equal to zero
to the left of the decimal.
Rule 2: To be in Scientific
Notation it will always be written
as some number x10
Power
Converting from Standard
Form to Scientific Notation
Step 1: Draw a line where the
decimal is going to go.
Don’t Forget Rule 1.
Step 2: Write the number with its
new decimal location x10. (Zeros
at the end of the number are not
needed.) Refer to example
Step 3: Determine the sign of the
exponent negative or positive and
write it above and to the right of
the 10.
(Hint: If there is a zero to the
right of the decimal, the exponent
will be negative)
Step 4: Find the value of the
exponent by counting the number
of cheeks from the decimal to the
line you drew. Place the exponent
next to the sign you wrote above
base 10.
Through
Scientific Notation
Correct Examples:
a) 3.1234 x 108
b) 2.36 x 10-7
Non-examples---These are WRONG!!!
1. 14.256 x 105
2.
0.2345 x 10-8
3.
1.23 x 1-9
4.
8.756
Example 1: The area of an ecological reserve is
450,000 square acres. How is this area expressed
in scientific notation?
e) 4.5 x 10-5 square acres
f) 45 x 10-5 square acres
g) 450 x 103 square acres
h) 4.5 x 105 square acres
Example 2: A certain bacterium measures
approximately 0.000015 millimeters in length. How
is this expressed in scientific notation?
a) 1.5 x 10 5 mm
b) 1.5 x 10-5 mm
c) 1.5 x 10-4 mm
d) 15 x 106 mm
16
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Converting from Scientific
Notation to Standard Form
Negative Exponents
Hint: If the exponent is negative
the resulting standard form
number will be less than 1. There
will be no whole numbers to the
left of the decimal
Step 1: Move the decimal the
number of places “cheeks” to the
left indicated by the exponent. Fill
all empty cheeks with a zero.
Scientific Notation
Example 3: The diameter of a human red blood
cell is 7.65 x 10-3 millimeters. Which represents
this number in standard form?
a) 0.000765 mm
b) 7.65000 mm
c) 0.00765 mm
d) 7,650mm
Step 2: Re-write the new number
without the X 10
Positive Exponents
Step 1: Move the decimal the
number of places “cheeks” to the
right indicated by the exponent.
Fill all empty cheeks with a zero.
Step 2: Re-write the new number
without the X 10
Hint1: If the exponent is positive
the resulting standard form
number will be greater than 1.
There will be at least 1 whole
numbers to the left of the decimal.
Example 4: The sun’s core temperature reaches
close to 2.7 x 107 degrees Fahrenheit. Which of
the following represents this temperature in
standard notation?
º
a) 270,000 F
º
b) 2,700,000 F
º
c) 27,000,000 F
º
d) 270,000,000 F
Hint2: If the exponent is
negative, there will be no whole
number to the right of the decimal.
17
Date:___________
Created by: Loren L. Spencer
Main Idea
next to the sign you wrote above base 10.
Through
Scientific Notation
Rule 1: To be in Scientific
Notation, you must have one and
only one number not equal to zero
to the left of the decimal.
Rule 2: To be in Scientific
Notation it will always be written
as some number x10
Power
Converting from Standard
Form to Scientific Notation
Step 1: Draw a line where the
decimal is going to go.
Don’t Forget Rule 1.
Scientific Notation
Correct Examples:
a) 3.1234 x 108
b) 2.36 x 10-7
Not Correct
5. 14.256 x 105
6.
0.2345 x 10-8
7. 1.23 x 1-9
8. 8.756
Example:
Step 2: Write the number with its
new decimal location x10. (Zeros
at the end of the number are not
needed.) Refer to example
Step 3: Determine the sign of the
exponent negative or positive and
write it above and to the right of
the 10.
(Hint: Think about money. Line the
standard form numbers decimal up
with $1.00 and determine which is
greater. If you would prefer to
have a $1.00, then the exponent is
negative. If you want the standard
form number then the exponent is
positive.
Step 4: Find the value of the
exponent by counting the number
of cheeks from the decimal to the
line you drew. Place the exponent
18
Date:___________
Created by: Loren L. Spencer
Main Idea
Through
Converting from Scientific
Notation to Standard Form
Scientific Notation
Negative Exponents
Hint: If the exponent is negative
the resulting standard form
number will be less than 1. There
will be no whole numbers to the
left of the decimal
Example:
Step 1: Move the decimal the
number of places “cheeks” to the
left indicated by the exponent. Fill
all empty cheeks with a zero.
Step 2: Re-write the new number
without the X 10
Positive Exponents
Hint: If the exponent is positive
the resulting standard form
number will be greater than 1.
There will be at least 1 whole
numbers to the left of the decimal.
Example:
Step 1: Move the decimal the
number of places “cheeks” to the
right indicated by the exponent.
Fill all empty cheeks with a zero.
Step 2: Re-write the new number
without the X 10
19
Date:___________
Main Idea
Loops & Mixed Numbers
Created by: Loren L. Spencer
Through
Adding and Subtracting Mixed Numbers
Step 1: Add or subtract the
whole numbers.
Step 2: Circle the denominator on
the left and draw a loop to the
right and place the denominator
over itself.
Step 3: Circle the denominator on
the right and draw a loop to the
left and place the denominator
over itself.
Step 4: Multiply straight across.
(The denominators should be equal)
Step 5: Add or subtract the
numerators and keep the
denominators the same.
Step 6: If the answer is an
improper fraction simplify by
division.
Step 7: Reduce
20
Date:___________
Main Idea
Multiplying Mixed Numbers
Created by: Loren L. Spencer
Through
Multiplying and Dividing Mixed Numbers
Step 1: To multiply or divide
mixed numbers you need to turn
the mixed numbers into improper
fractions.
A. For the left Mixed number,
Multiply the denominator and the
whole number and add the
numerator—this total becomes the
numerator for the improper
fraction.
B. Follow the above step for the
right Mixed number.
Step 2: Multiply straight across.
Step 3: If the answer is an
improper fraction simplify by
division.
Step 4: Reduce
21
Date:___________
Main Idea
Dividing Mixed Numbers
Created by: Loren L. Spencer
Through
Multiplying and Dividing Mixed Numbers
Step 1: To multiply or divide
mixed numbers you need to turn
the mixed numbers into improper
fractions.
A. For the left Mixed number,
Multiply the denominator and the
whole number and add the
numerator—This becomes the
numerator for the improper
fraction.
B. Follow the above step for the
right Mixed number.
Now Remember--Dividing
Fractions is as easy as pie,
Flip the 2nd and multiply.
Step 2: Take the reciprocal of
the 2nd fraction. Flip it
Step 3: Multiply straight across.
Step 4: If the answer is an
improper fraction simplify by
division.
Step 5: Reduce
22
Date:___________
Created by: Loren L. Spencer
23