STANDARD 1 & 2:
NUMBER PROPERTIES, PEMDAS, EXPONENTS & ROOTS
Standard 1: Number Properties & Simple Operations
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as "a(b + c) = ab + ac". In numbers, this means, that 2(3 + 4) = 2×3 + 2×4.
The Associative Property
The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. Any time they refer to the Associative Property, they want you to regroup things.
Commutative Property
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around.
Additive & Multiplicative Identities
"The identity" is whatever doesn't change your number at all. For addition, "the identity" is zero, because adding zero to anything doesn't change anything. With numbers, that is 5 + 0 = 5. For multiplication, "the identity" is 1. For example 6 x 1 = 6.
Additive & Multiplicative Inverses
"The inverse" is whatever turns your number into the identity.
The additive inverse is the same number with the opposite sign. For instance 3 + -3 = 0. The additive inverse of 3 is -3.
The multiplicative inverse is the reciprocal. For example, 2 x 1/2 = 1. The multiplicative inverse of 2 is 1/2.
Order of Operations
A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance,15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first.
The Associative Property
The word "associative" comes from "associate" or "group";the Associative Property is the rule that refers to grouping. For addition, the rule is "a + (b + c) = (a + b) + c"; in numbers, this means 2 + (3 + 4) = (2 + 3) + 4. Any time they refer to the Associative Property, they want you to regroup things.
Commutative Property
The word "commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around.
Additive & Multiplicative Identities
"The identity" is whatever doesn't change your number at all. For addition, "the identity" is zero, because adding zero to anything doesn't change anything. With numbers, that is 5 + 0 = 5. For multiplication, "the identity" is 1. For example 6 x 1 = 6.
Additive & Multiplicative Inverses
"The inverse" is whatever turns your number into the identity.
The additive inverse is the same number with the opposite sign. For instance 3 + -3 = 0. The additive inverse of 3 is -3.
The multiplicative inverse is the reciprocal. For example, 2 x 1/2 = 1. The multiplicative inverse of 2 is 1/2.
Order of Operations
A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance,15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first.
Standard 2: Opposites, Reciprocals, Exponents & Roots
The opposite of a number is literally the same number, with the opposite sign. For example, the opposite of 4 is -4. Or the opposite of -100 is 100.
The reciprocal of a number is the fraction flipped over. Think: ReCIProcal means to "FLIP" it over (it kind of rhymes). For example, the reciprocal of 1/6 is 6/1 or just 6. And the reciprocal of -2 is -1/2. Keep in mind that the sign stays the same for a number's reciprocal.
That is, however, unless you are asked for the "negative reciprocal" or "opposite reciprocal." In this case, you would use the opposite sign and flip the fraction. (We use the opposite reciprocals when we are talking about the slopes of perpendicular lines- Standard 8!).
The reciprocal of a number is the fraction flipped over. Think: ReCIProcal means to "FLIP" it over (it kind of rhymes). For example, the reciprocal of 1/6 is 6/1 or just 6. And the reciprocal of -2 is -1/2. Keep in mind that the sign stays the same for a number's reciprocal.
That is, however, unless you are asked for the "negative reciprocal" or "opposite reciprocal." In this case, you would use the opposite sign and flip the fraction. (We use the opposite reciprocals when we are talking about the slopes of perpendicular lines- Standard 8!).
Exponent Properties
Most often used square roots
Complete All | |
File Size: | 133 kb |
File Type: | docx |