SOL 7.1, 7.3, 7.16
SOL 7.3: The student will
a) Model addition, subtraction, multiplication, and division of integers, and
b) Add, subtract, multiply and divide integers
Integers include the set of whole numbers and their opposites (…-3, -2, -1, 0, 1, 2, 3, …)
Integers are used in practical situations such as temperature changes, balancing a checkbook, and changes in altitude.
Integer Rules (negative numbers)
Adding/Addition
 If both numbers have the same sign (both positive or both negative), add the numbers and keep the sign
o Ex) 17 + 5 = 22
Ex) -23 + -4 = -27
 If the numbers have different signs, subtract and take the sign of the larger number.
o Ex) 9 + -2 = 7
Ex) -18 + 5 = -13
Ex) 3 + -7 = -4
Subtracting/Subtraction
Don’t bother subtracting. Make everything an addition problem by following KFC!
When you see subtraction, think “chicken”: KFC
 KEEP—keep the sign of the first number
 FLIP—flip the subtraction sign to an addition sign
 CHANGE—change the sign of the second number (from + to -, or – to +)
o Ex) -5 – 6 = -5 + -6 = -11
Ex) 24 - -7 = 24 + 7 = 31
o Ex) 3 – 9 = 3 + -9 = -6
Ex) -11 - -3 = -11 + 3 = -8
Multiplication and Division
If both numbers have the same sign, then the answer will be positive.
Ex) 6 x 7 = 42
Ex) -4 x -4 = 16
Ex) 45 ÷ 9 = 5
Ex) -49 ÷ -7 = 7
If the two numbers have different signs, then the answer will be negative.
Ex) 4 x -3 = -12
Ex) -2 x 8 = -16
Ex) 56 ÷ -8 = -7
Ex) -24 ÷ 6 = -4
SO for multiplication and division: Same sign = Positive; Different sign = Negative
SOL 7.1: The student will
a) Investigate and describe the concept of negative exponents for powers of ten
b) Determine scientific notation for numbers greater than zero
c) Compare and order fractions, decimals, percents, and numbers written in scientific notation
d) Determine square roots
e) Identify and describe absolute value for rational numbers
An exponent is a little number high and to the right of a number. It tells how many times a number is to be multiplied
BY ITSELF. Ex) 45 = 4 x 4 x 4 x 4 x 4 = 1024
Negative Exponent Property states: For any integer n and any nonzero integer b, b-n is the reciprocal (numerator and
denominators switch places) of of bn.
1
Ex) b-n = 1/bn
Ex) 10-2 = 1/102 = 100
Negative exponents for powers of 10 can be investigated through patterns such as:
102 = 100
101 = 10
100 = 1
10-1 = 1/101 = 1/10 = 0.1
10-2 = 1/102 = 1/100 = 0.01
A base of 10 raised to a negative exponent represents a number between 0 and 1.
SOL 7.b: Scientific Notation is used to represent very large or very small numbers. A number written in scientific
notation is the product of two factors—a decimal greater than or equal to 1 but less than 10, and a power of 10. (Move
the decimal the number of times the exponent tells you—If it is a positive exponent you will move the decimal to the
right to make it a large number. If it is a negative exponent, you will move the decimal to the left that number of times
to make it a very small number.)
Ex) 3.1 x 105 = 310,000
2.85 x 10-4 = 0.000285
SOL 7.c: To compare and order fractions, decimals, percents and numbers written in scientific notation put them all in
decimal format!
Fraction to Decimal: Divide the bottom number into the top number; adding a decimal and a zero. No remainders.
Decimal to Fraction: count the number of digits behind the decimal; put it over a fraction bar with a 1; put the number
of digits that you counted and give the 1 that many zeroes.
Decimal to Percent: Move the decimal 2 places to the left.
Percent to Decimal: Move the decimal 2 places to the right.
Fraction to Percent: Change the fraction to a decimal first and then change the decimal to a percent.
Percent to Fraction: Put the number over 100 and reduce.
To compare them after you have changed them all to decimals:
1) Line up the decimals
2) Fill zeros in the missing blanks
3) Starting left to right, compare each number to find the bigger number
Ex) Compare and order least to greatest:
15
38%
2.1 x 10-2
25
15
25
= 0.6
38% = 0.38
2.1 x 10-2 = 0.021
0.002
0.002 = 0.002
0.600
0.380
0.021
0.002
0.002 < 2.1 x 10-2 < 38% <
15
25
7.1d: A square root of a number is a number which, when multiplied by itself, produces the given number
Ex) √121 = 11 because 11 x 11 = 121
The square root of a number can be represented as the length of a side of the square.
√100 = x
The absolute value of a number is the distance from 0 on the number line regardless of direction.
Ex) |−8| = 8
Ex) |8| = 8
The absolute value of a number represents distance from zero on a number line regardless of direction. Distance is
always positive so answers to an absolute value is always a positive number.
SOL 7.16: The student will apply the following properties of operations with real numbers:
a) The commutative and associative properties for addition and multiplication
b) The distributive properties
c) The additive and multiplicative identities properties
d) The additive and multiplicative inverse properties
e) The multiplicative property of zero
The commutative property of multiplication states that we can multiply numbers in any order without changing the
result.
Ex) 5 x 6 = 6 x 5
Ex) (4 x 5) + 2 = (5 x 4) + 2
The commutative property of addition states that we can add numbers in any order without changing the result.
Ex) 4 + 3 = 3 + 4
Ex) 7 + 6 + 8 = 6 + 8 + 7
COmmutative—Change Order
The associative property of multiplication states that we can group numbers in any order without changing the result.
Ex) 4 x (5 x 6) = (4 x 5) x 6
The associative property of addition states that we can group numbers in any order withouth changing the result.
Ex) (2 + 3) + 6 = 2 + (3 + 6)
ASSOciative—Always Stay Same Order
The distributive property ties addition and multiplication together.
Ex) 4(9 + 2) = 4(9) + 4(2)
Distribute the number on the outside of the ( ) to all the numbers inside the ( ).
The additive identity property states that the sum of any number and 0 is that number.
Ex) 4 + 0 = 4
Ex) 0 + -3 = -3
The multiplicative identity property states that the product of one and any number is that number.
Ex) 5 x 1 = 5
Ex) 1 x -9 = -9
IDENTITY properties keep their identities.
The additive inverse property states that the sum of any number and its inverse (opposite) always equals zero.
Ex) 4 + -4 = 0
Ex) -6 + 6 = 0
The multiplicative inverse property states that the product of any number and its inverse (reciprocal) always equals one.
1
1
6
Ex) 7 x 7 = 1
Ex) 6 x 1 = 1
INVERSE – REVERSE (opposite)
The multiplicative property of zero states that the product of any number and zero is zero.
Ex) 5 x 0 = 0
Ex) 0 x -6 = 0