|Intersection of Sets||The intersection of sets A and B, written A (upside down U) B, is the set of elements that are in both A and B.|
|Union of Sets||The union of sets A and B, written A U B, is the set of elements in either A or B (or in both).|
|Order of Operation in Evaluating Expressions|| 1. Do operations within parentheses or other grouping|
2. Within grouping symbols, or if there are no grouping symbols:
a. Do all powers from left to right
b. Do all multiplications and divisions from left to right.
c. Do all additions and subtractions from left to right.
|Square of the Square Root Property||For any nonnegative number n, √n • √n = n.|
|Pythagorean Theorem|| In a right triangle with legs of lengths a and b and hypotenuse of length c,|
a2 + b2 = c2.
|Area Model for Multiplication||The area A of a rectangle with length L and width w is Ew.|
|Commutative Property of Multiplication||For any real numbers a and b, ab = ba.|
|Area Model for Multiplication (discrete form)||The number of elements in a rectangular array with r rows and c columns is rc.|
|Associative Property of Multiplication||For any real numbers a, b, and c, (ab)c = a(bc).|
|Multiplicative Identity Property of 1||For any real number a, a • 1 = 1 • a = a.|
|Property of Reciprocals||Suppose a ≠ 0. The reciprocal of a is 1/a. That is, a • 1/a = 1/a • a = 1.|
|Reciprocal of a Fraction Property||Suppose a ≠ 0 and b ≠ 0. The reciprocal of a/b is b/a.|
|Multiplication Property of Zero||For any real number a, a • 0 = 0 • a = 0.|
|Multiplying Fractions Property|| For all real numbers a, b, c, and d, with b and d not zero,|
a/b • c/d = ac/bd.
|Equal Fractions Property|| If b ≠ 0 and k ≠ 0, then |
ak/bk = a/b.
|Rate FActor Model for Multiplication||When a rate r is multiplied by another quantity x, the product is rx. So the unit of rx is the product of the units for r and x.|
|Multiplication Property of -1||For any real number a, a • -1 = -1 • a = -a.|
|Rules for Multiplying Positive and Negative Numbers||If two numbers have the same sign, their product is positive. If two numbers have different signs, their product is negative.|
|Properties of Multiplication of Positive and Negative Numbers|| 1. The product of an odd number of negative numbers is negative.|
2. The product of an even number of negative numbers is positive.
|Multiplication Property of Equality||For all real numbers a, b, and c, if a = b, then ca = cb.|
|Multiplication Property of Inequality (Part 1)||If x < y and a is positive, then ax < ay.|
|Multiplication Property of Inequality (Part 2)||If x < y and a is negative, then ax > ay.|
|Multiplication Counting Principle||If one choice can be made in m ways and a second choice can be made in n ways, then there are mn ways of making the first choice followed by the second choice.|
|Permutation Theorem||There are n! possible permutations of n different items when each item is used exactly once.|
|Putting-Together Model for Addition||If a quantity x is put together with a quantity y with the same units, and there is no overlap, then the result is the quantity x + y.|
|Slide Model for Addition||If a slide x is followed by a slide y, the result is the slide x + y.|
|Commutative Property of Addition||For any real numbers a and b, a + b = b + a.|
|Associative Property of Addition||For any real number a, b, and c, (a + b) + c = a + (b + c).|
|Additive Identity Property||For any real number a, a + 0 = 0 + a = a.|
|Property of Opposites||For any real number a, a + -a = -a + a = 0.|
|Opposite of Opposites (Op-op) Property||For any real number a, -(-a) = a.|
|Addition Property of Equality||For all real numbers a, b, and c, if a = b, then a + c = b + c.|
|Distributive Property: Adding or Subtracting Like Terms|| For any real numbers a, b, and c,|
ac + bc = (a + b)c and
ac - bc = (a - b)c.
|The Distributive Property: Removing Parentheses|| For all real numbers a, b, and c, |
c(a + b) = ca + cb and
c(a - b) = ca - cb.
|Distributive Property: Adding Fractions||For all real numbers a, b, and c, with c ≠ 0, a/c + b/c = a + b/c.|
|Addition Property of Inequality|| For all real numbers a, b, and c, |
if a < b,
then a + c < b + c.
|Algebraic Definition of Subtraction|| For all real numbers a and b, |
a - b = a + -b.
|Take-Away Model for Subtraction||If a quantity y is taken away from an original quantity x, the quantity left is x - y.|
|Comparison Model for Subtraction||The quantity x - y tells how much the quantity x differs from the quantity y.|
|Opposite of a Sum Property|| For all real numbers a and b,|
-(a + b) = -a + -b = -a - -b.
|Opposite of a Difference Property|| For all real number a and b, |
-(a - b) = -a + b.
|Triangle Sum Theorem|| If any triangle with angle measures a, b, and c in degrees, |
a + b + c = 180.
|Triangle Inequality|| Part 1: If A, B, and C are any three points, then AB + BC ≥ AC.|
Part 2: If A, B, and C are vertices of a triangle, then AB + BC > AC.
|Third Side Property|| If x and y are the lengths of two sides of a triangle, and x > y, then the length z of the third side must satisfy the inequality|
x - y < z < x + y.