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353 Cards in this Set
- Front
- Back
- 3rd side (hint)
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What are real numbers?
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Both rational and irrational numbers.
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These arethe numbers that you encounter each day and are classified into various sets and subsets.
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Rational Numbers
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Numbers that can be expressed as a fraction in the form a/b, where a and b are integers and b does not = 0. Rational numbers can be classified into subsets.
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These are the numbers we actually use daily, and always have a known ending.
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Natural Numbers
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Also known as the counting numbers. {1,2,3,4,5,6,7...} Positive, whole, without 0. These are rational numbers.
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These are what little kids would think of as numbers.
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Whole Numbers
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Natural numbers including 0. {0,1,2,3,4,5,6,7...} These are rational numbers. All positive except 0, which is neither positive or negative.
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Whole = 0.
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Integers
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Whole numbers and their opposits. {...-3,-2,-1,0,1,2,3...} These are rational numbers. Negative, positive, include 0 (neither negative or positive)
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Add negatives in.
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Rules of integers
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All numbers >0 are positive. All numbers <0 are negative.
Zero is neither positive or negative. Even intergers are divisible by 2 and includes 0. Odd integers are not divisible by 2. |
Everything that we generally think of as numbers.
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Irrational numbers
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Numbers that cannot be expressed as fractions such as sqrt 2, sqrt 3, sqrt 5, and pi. When expressed as decimals, they are non-terminating and non-repeating.
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Numbers that we can't predict the end of.
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How do you define an even integer?
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Divisible by 2
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count even numbers by what?
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How do you define an odd integer?
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Not divisible by 2
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everything that isn't even
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Is Zero positive or negative?
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Neither
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Imagine a numberline with 0 in it.
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What is the order of operations used to evaluate mathematical expressions?
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PEMDAS
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Please Excuse My Dear Aunt Sally
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What does PEMDAS stand for?
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Parentheses (or other grouping symbols)-Exponents-Multipication & Division (left to right)-Addition & Subtraction (left to right)
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Please Excuse My Dear Aunt Sally
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What are grouping symbols?
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parenthesis, brackets, absolute value symbol, and fraction bar
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What groups things together in an equation?
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Exponents include what 2 symbols?
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Any exponents and any radiacals.
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What is the opposite of an exponent?
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What is the commutative property of addition?
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It states that changing the order of the addends in a sum does not change the sum.
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a+b = b+a, where a and b are any real numbers
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What property does this equation represent?
a+b = b+a |
commutative property of addition
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does it matter who hangs out with whom in a commune?
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What is the commutative property of multiplication
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It states that changing the order of the factors in a product does not change the product.
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a*b = b*a, where a and b are any real numbers
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What property does this equation represent?
a*b = b*a |
Commutative Property of Multiplication
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Does it matter who multiplies whom in a commune?
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Associative Property of Addition
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It states that changing the grouping (parenthesis or brackets) of addends in a sum does not change the resulting sum.
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a+(b+c) = (a+b)+c, where a, b, and c are any real numbers
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What property does this equation represent?
a+(b+c) = (a+b)+c |
Associative Property of Addition
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Does it matter who adds to whom in an association?
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Associative Property of Multipication
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It states that changing the grouping (parenthesis or brackets) of factors in a product does not change the resulting product.
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a*(b*c) = (a*b)*c, where a, b, and c are any real numbers
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What property does this equation represent?
a*(b*c) = (a*b)*c |
Associative Property of Multiplication
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Does it matter who multiplies whom in an association?
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The Distributive Property
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It states that multiplication distributes over addition and subtraction
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a*(b+c) = a*b+a*c, where a,b, and c are real numbers
a*(b-c) = a*b-a*c, where a, b, and c are real numbers |
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What does this equation represent?
a*(b+c) = a*b+a*c |
The distributive property of multiplication over addition
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What would distribute over what?
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What does this equation represent?
a*(b-c) = a*b-a*c |
The distributive property of multiplication over division
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What would distribute over what?
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What is the relationship between integers and the number line?
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Each integer has a location on a real number line, where the sign of the number determines to which side of 0 the number is located.
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Integers are {...,-3,-2,-1,0,1,2,3,...}
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What is the absolute value of a number?
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It is the number of units a number is away from 0 on a number line.
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Consider position on number line.
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How do you order integers?
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When comparing integers, a number farther to the left of 0 on the number line will be less in value than a number to the right of 0 on the number line.
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The value of an integer is determined by its location on the real number line, where negative numbers appear to the left of 0 and positive numbers are located to the right of 0.
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How do you add integers?
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If the signs are the same add and keep the sign. If the signs are different subtract and take the sign of the number with the larger absolute value.
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Consider comparing the number of positives and negatives in a cloud.
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How do you subtract integers?
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add the opposite of the number being subtracted and apply the sign of the greater number
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switch signs
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How do you multiply negative integers?
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If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
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evens vs odds
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How do you divide negative integers?
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If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
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evens vs odds
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neg * neg = ?
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pos
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even
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neg * pos = ?
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neg
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odd
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neg / neg = ?
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pos
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even
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neg / pos = ?
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neg
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odd
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neg + neg = ?
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neg
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abs + sign
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neg + pos = ?
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which ever sign is larger
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abs sub + sign
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pos + pos = ?
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pos
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abs + sign
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neg - pos = ?
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neg
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add abs + neg
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pos - neg = ?
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larger value
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- turns pos and switches signs
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neg - neg = ?
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larger value
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- turns pos and switches signs
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How do you know if a number is divisible by 2?
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If the number is even.
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24, 50, and 66 are all divisible by 2 by being even
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How do you know if a number is divisible by 3?
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If the sum of the individual digits of the number is divisible by 3.
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312 is (3+1+2) = 6 which is divisible by 3 and 9,021 is (9+0+2+1) = 12 which is divisible by 3.
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How do you know if a number is divisible by 4?
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If the last two digits, taken as a two-digit number, are divisible by 4.
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736 is because 36 is divisible by 4. 12,716 is because 16 is divisible by 4.
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How do you know if a number is divisible by 5?
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If the last digit of the number is a 5 or a 0.
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10, 25, 30, and 75 are divisible by 5 because they end in 5 or 0.
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How do you know if a number is divisible by 8?
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If the last three digits, taken as a three-digit number, are divisible by 8.
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3,024 is because 024 is divisible by 8. 79,128 is because 128 is divisible by 8 (128/8 = 16).
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How do you know if a number is divisible by 9?
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If the sum of the individual digits of the number is divisible by 9.
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9,135 is (9+1+3+5) = 18 which is divisible by 9 and 414,972 is (4+1+4+9+7+2) = 27 which is divisible by 9
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How do you know if a number is divisible by 6?
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If it is divisible by both 2 and 3.
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336 is divisible by both 2 and by 3 (3+3+6) = 12
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How do you know if a number is divisible by 12?
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If it is divisible by both 3 and 4.
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How do you know if a number is divisible by 10?
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If the number ends in a 0.
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1100, 70, 970 all end in 0.
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How do you know if a number is divisible by 7?
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If you remove the last digit of the number, double it, and subtract it from the remaining digits and the result is divisible by 7.
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357 becomes 35-(2*7) = 21 which is divisible by 7. 2,016 becomes 201-(2*6) = 189 which is divisible by 7.
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What is a factor?
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A factor of a number X is a whole number that divides into X evenly without remainder.
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Parts of a number in terms of multipication.
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What is a prime factor?
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The most basic factors of a number, which are always prime.
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The bare elements.
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What is the Greatest Common Factor (GCF)?
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It is the largest common factor of 2 or more numbers. Determine this by finding all the prime factors of both numbers, then taking the prime factors that are in both (match them up) and multiply them.
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Think of venn diagrams of prime factors for both numbers.
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What is the Least Common Multiple (LCM)?
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It is the smallest number that two or more numbers will divide into evenly. Find this by multiplying the prime factors of both or all numbers together, however the Greatest Common Factors should only be multiplied once.
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Use the venn diagram and multiply all the numbers in the diagram.
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What is a fraction?
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It is a rational number in the form a/b. a is the numerator and b is the denominator, which cannot be 0.
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Must be a rational number.
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What is a numerator?
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It is the number on the top half of a fraction.
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top
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What is a denominator?
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It is the number on the bottom half of a fraction.
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bottom
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What is an equivalent fraction?
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It is a fraction that is equal to another fraction, by a factor of a/a.
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things that equal each other
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How do you find an equivalent fraction?
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Multiply or divide the fraction by the number one, in the special form of a/a.
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For example, (5/35) / (5/5) = 1/7
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How do you multiply fractions?
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Multiply the numerators with each other and the denominators with each other.
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(4/5) * (1/7) = (4/35)
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How can you make a fraction multiplication simpler?
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Eliminate common factors between numerators and denominators.
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(7/9)*(3/16) = (7/3)*(1/16) = 7/48. (factored out 3)
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How do you find x/y of t/v?
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Multiply.
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x*t/y*v and simplify
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How do you divide fractions?
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Multiply the first fraction by the recipricol of the second fration.
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(x/y) / (t/v) = (x/y)*(v/t) =xv/yt
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What is a reciprical fraction?
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It is the inverse of a fraction.
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The recipricol of x/y is y/x
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What must you do first to add or subtract fractions?
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You must find equivalent fractions for each fraction that have common denominators.
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The bottoms must be the same.
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What is the least common denominator?
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It is the least common multiple of the denominator of 2 fractions, used for addition or subraction of fractions.
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Find the least common multiple for 2 or more denominators.
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How do you add fractions?
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Convert the fractions so they have an equivalent least common denominator, add numerators while keeping same denominator, and simplify the results.
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add tops with same bottoms
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How do you subtract fractions?
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Convert the fractions so they have an equivalent least common denominator, subtract numerators while keeping same denominator, and simplify the results.
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subtract tops with same bottoms.
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What is a mixed number?
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It is a fraction that includes a whole number as in 2(2/5)
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It is also a simplified fraction.
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What is an improper fraction?
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A fraction where the numerator is greater than the denominator and which can be expressed as a mixed fration.
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22/17 or 690/13
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How do you change a mixed number to an improper fraction?
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A*(B/C) = (A*C+B)/C
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whole times denominator plus numerator all over denominator
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What is a complex fraction?
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Any fraction where either the numerator or the denominator is also a fraction.
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(a/b)/c or 17/(15/6)
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What is the place value for decimals.
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If the number is abc.def then a is hundreds, b is tens, c is ones, d is tenths, e is hundredths, f is thousandths.
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ends in s in front of decimal, ends in ths behind decimals, and no 'onths' place.
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How do you convert a decimal to a mixed fraction?
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Keep the whole number and place the numbers past the decimal over the appropraite place value number.
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9.5 is 9(5/10) or 9(1/2)
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How do you convert a fraction to a decimal?
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Divide the numerator by the denominator.
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4/5 = 4 / 5 = .80 = .8
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What must you do to add or subtract decimals?
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Line up the decimals, add trailing 0s if you need to maintain spacing.
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line up what?
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How do you multiply decimals?
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Multiply the entire numbers, then count the numbers to the right of both factors, then move the decimal over to the left of the same number as counted in the previous step.
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move the decimal to equal the number of spaces of both factors.
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How do you divide decimals?
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Set up as long division, count how many digits are to the right of the decimal point in the divisor (number divding with), move decimal point in the dividend (number dividing into) by the amount in the previous step, raise the nely placed decimal point up to the quotient, divide as usual around decimal point
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alter decimal by both factors and raise into new number
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What is a ratio?
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A comparison of two or more different quantities.
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compare
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How can ratios be written?
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a:b or a/b or a to b
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3 ways
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How can you simplify ratios?
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Remove common factors between the quantities and in the case of fractions, make improper, then divide.
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common factors or fractions
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What do you have to consider when looking at the actual numbers of a ratio?
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What the numbers represent in terms of whole quantities or relationships between quantities. ie the ratio of girls to boys in the school is 3:2, but the ratio of girls to students in the school is 3:5 and the ratio of girls to students in the school is 2:5.
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whole, part, etc
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What is a rate?
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A rate is a comparison of two types of units.
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miles per hour, feet per second, miles per gallon,
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What is a unit rate?
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A rate with a denominator of 1.
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50 miles per hour, 6 feet per second
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What is a proportion?
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It is a comparison of two ratios.
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matching ratios
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What is an extreme when referring to a proportion?
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When a/b = c/d, a and d are extremes.
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The farthest limits of the ratio
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What is a mean when referring to a proportion?
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When a/b = c/d, b and c are means.
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The middle ground of the ratio.
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How do you solve a proportion problem?
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Multiply the numerator from the first ratio by the denominator from teh second ratio and then the denominator from teh first ratio by the numerator of the second ratio, and set the values equal to each other.
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Multiply the means and set it equal to the product of the extremes then solve.
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What categories of proportions are there?
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part to part or part to whole
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what are the pieces
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What is a scale?
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It is where a smaller number is used to represent a larger unit.
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1 inch = 1 mile
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What does it mean when figures are called similar?
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It means figures have corresponding sides that are in proprotion with one another and corresponding angles withe the same measure.
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enlarging or shrinking a shape
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If two figures are known to be similar, how can you determine sides or angles?
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By setting up proportions.
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If thier similar, how are they related?
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How can you determine the perimeter of a figure similar to another figure?
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The ratio of the sides of two similar figures will be the same as the ratio of the perimeters of the same two similar figures.
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perimeters are proportional like sides.
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What is a percent?
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It is a special ratio that compares a numerical quantity to 100.
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it is part of what
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How do you change a percent to a decimal?
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Remove the percent symbol and divide by 100
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Compare to 100
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How do you change a decimal to a percent?
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Multiply by 100
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Multiply
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How do you change a fraction to a percent?
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Use the proportion x/111 = a/b and cross multiply to solve for the variable x.
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set it up for cross multiplication
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How do you convert a percent into a fraction?
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Put the given percent over 100, remove the percent symbol (%) and simplify if needed.
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over 100
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What type of a relationship is a percentage?
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part - whole
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a percentage is a ___ of a ____
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The word 'of' in mathematics usually means what?
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Multiply
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45% of 200 = 90
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The word 'is' in mathematics usually means what?
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Equals
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252 is what percent of 600? 42%
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How is percent increase or decrease calculated?
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change/original amount = percent/100
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compare change to original for percent to 100
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How is simple interest calculated?
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I = prt where I is the interest charged or paid out, p is the principle amouth that is saved or borrowed, r is the percentage rate written as a decimal, and t is the time in years
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principle, rate, time
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How do you calculate the total amount of money that is either earned or owed using simple interest?
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Total = p + prt
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what do you need to add to interest?
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What are the common measures of central tendency?
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mean, median, mode, range
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what can you tell about a set of numbers?
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What is the definition of the measures of central tendency?
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They are values that are examined from a set of data in order to make predictions and draw conclusions about that set of data as a whole
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Why use statistics scientifically
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What is the definition of the mean of a set of numbers?
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It is the sum of the numbers in the set divided by the total number of values in the set.
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Average
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What is the definition of the median of a set of numbers?
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It is the middle number when the values are listed in order from smallest to largest.
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middle
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How do you determine the median of an odd number of values?
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It will be the middle number if the values are listed from smallest to largest
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actual median
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How do you determine the median of an even number of values?
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It will be the average of the two middle values if the values are listed from smallest to largest
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middle numbers/2
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What is the definition of the mode of a set of numbers?
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It is the number that occurse the most in a data set
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frequency
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What does it mean if a set is 'no mode'?
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In a set of numbers there is not a number that occurs more than any other in the list
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lack of frequency
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What does it mean if a set is 'bimodal'?
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In a set of numbers there is more than one value that occurs as much as another number. It is also possible to have more than two modes.
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2 modes
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What is the definition of the range of a set of numbers?
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It is the difference between the largest value in the list and the smallest value in the list.
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largest value - smallest value
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What are bar graphs for?
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They are used to compare data and can be drawn using horizontal or vertical bars
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compare
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What are line graphs for?
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They are used to show a trend that occurs over time
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time
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What are circle graphs for?
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They are used to show parts of a whole and percents are often used; always make sure that any percents used in one always add up to 100%
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part/whole
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What are scatter plots for?
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They are used to show trends in data. The data points are not connected and the general trend of the data is examined
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science data collection
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What are the three most common conclusions that can be made based on the graph?
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positive correlation, negative correlation, and no correlation
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positive, neutral, negative
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What does a positive correlation look like on a scatter plot?
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As the x-values increase there is also an increase in the y-values. The points look like a line rising from the lower left to the upper right.
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increases together
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What does a negative correlation look like on a scatter plot?
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As the x-values increase there is a decrease in the y-values. The points look like a line descending from upper left to lower right.
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increase vs decrease
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What does a no correlation look like on a scatter plot?
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There is no trend in the values across the y-values or x-values. The points look random or cloud-like.
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nothing
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What is a stem and leaf plot?
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It is a way to organize a list of numbers. There are typically two columns. the first column, known as the stem, often represents the first digits in the number and the secnod column, known as the leaf, represents the last digit of the number
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tens are stems, ones are leafs.
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In a stem and leaf plot, how can you tell the total number of values in the list?
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Count the leaves.
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what represents a number
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In a stem and leaf plot, how can you tell if a number repeats?
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If the stem has a leaf of a particular number that repeats
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similar leaves on the same stem
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A stem and leaf plot can make it easy to determine what measures?
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mode, median, and range
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all but mean
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What is a histogram used for?
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The bars represent number values or intervals of values. Visually the bars are touching each other. Stem and leaf plots are particularly helpful for these.
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represents number units
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What is probability used for?
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Calculating the chance that these and other events could happen.
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probably
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What is the definition for the probability of an event (E)?
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P(E) = (the number of ways event E can occur)/(the total number of possible outcomes)
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# of a way/# of total ways
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What is the outcome set?
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A list of all possible outcomes
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sample space
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What is the sample space?
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A list of all possible outcomes
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outcome set
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What is the probability of an event that can't happen or an impossible event?
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0
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null
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What is the probability of an event that is certain to happen?
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1
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single whole
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What are the probability of an event that can possibly happen?
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between 0 and 1
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In between impossible and certain
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What is simple probability?
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The likelihood of a single event
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single option
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What is compound probability?
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The likelihood of more than one outcome to occur. The key word is often 'or'.
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multiple options
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What is the formula for calculating the compound probability of event A and event B?
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P(A or B) =
P(A) + P(B) - P(A and B) |
either - both
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What are mutually exclusive events?
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Situations that cannot occur at the same instance
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seperate
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If an event is mutually exclusive then P(A and B) are equal to what?
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0
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seperate
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In probability, what are independent events?
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Two or more events that will occur without the outcome of one affecting the outcome of any other.
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not connected
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How do you calculate the probability of two or more independent events occuring is succession?
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Multiply the probability of each event happening independently.
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P(A) x P(B) x...
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In probability, the phrase 'replace' refers to what type of event?
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Independent
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The original 'set' is being restored
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In probability, what are dependent events?
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Situations where the outcome of the first event does affect teh probability of the second event.
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connected
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In probability, the phrase 'without replacement' refers to what type of event?
|
Dependent
|
The original 'set' is not restored or is altered.
|
|
|
How do you calculate the probability of two or more dependent events occuring is succession?
|
Calculate the probability of the first event and multiply it by the probability of the second event, remembering that the original set might decrease.
|
decreasing set
|
|
|
In probability, what is the counting principle?
|
A strategy to determine the number of combinations created from the number of choices.
|
ways to calculate the total number of variations
|
|
|
In probability, what is a permutation
|
It is the number of possible arrangements for that set of objects.
|
also a factorial
|
|
|
What is a factorial?
|
For a whole number it is the product of that whole number and each of the natural numbers less than the number.
n! = n*(n-1)*(n-2)*...*1 |
descending multiplication
|
|
|
What does n! mean?
|
Factorial.
n! = n*(n-1)*(n-2)*...*1 |
dexcending multiplication
|
|
|
How do you calculate the number of permutations of n objects taken n at a time?
|
nPn = n!
|
factorial
|
|
|
How do you calculate the number of permutations of n objects you have to choose from and r is the number of objects actually selected to be arranged? (also when not all of the objects are considered for each different arrangement)
|
nPr = (n!)/(n-r)!
|
factorial over possabilities or decreasing the numbers available for the position
|
|
|
In probability, what is a combination?
|
It is the total number of groupings of a set of objects
|
grouping of objects changes the number of combinations that exist
|
|
|
What is the formula for calculating a combination?
|
nCr = (n!)/[r!(n-r)!]
Note: nCn will always be equal to 1 since there is only one way to select all the components to a combination. |
factorial over possabilities times variations
|
|
|
An exponent is also known as a what?
|
Power
|
PE
|
|
|
A power is also known as a what?
|
Exponent
|
PE
|
|
|
What is an exponent?
|
Total number of times a base is used as a factor.
|
^
|
|
|
What is the base number?
|
The number that will be the factor in the exponent
|
the big number
|
|
|
What is the exponent in an equation?
|
It tells you how many times to use the base number as a factor.
|
The smaller font, raised number.
|
|
|
If a number has no apparent exponent, what is the exponent?
|
1
|
It increases only on its own
|
|
|
If a base is raised to the 0 power, what is it equal to?
|
1
|
a whole
|
|
|
How do you multiply exponents with like bases?
|
Add the exponents and maintain the same base.
a^m * a^n = a^(m+n) |
x^2 * x^8 = x^10
|
|
|
How do you divide exponents with like bases?
|
Subtract the exponents and maintain the same base.
a^m / a^n = a(m-n) |
x^8 / x^2 = x^6
|
|
|
How do you raise an exponent to another exponent?
|
Multiply the exponents.
(a^m)^n = a^(m*n) |
(x^2)^4 = x^8
|
|
|
How do you raise a product to a power?
|
multiply the exponents for each base
((a^m)*(b^n))^t = (a^(m*t))*(b^(n*t)) |
((x^3)*(y^4))^2 = (x^6)*(y^8)
|
|
|
How do you raise a fraction to a power?
|
apply the exponent to both the numerator and denominator
(a/b)^m = (a^m)/(b^m) |
(x/y)^4 = (x^4)/(y^4)
|
|
|
How do you simplify an expression with a negative exponent?
|
Turn it into a fraction with the numerator as 1 and the base number as the denominator to a positive power.
a^(-m) = 1/(a^m) |
x^(-3) = 1/(x^3)
|
|
|
What is scientific notation?
|
It is used to write very large or very small numbers more efficiently.
|
15 x 10^5
|
|
|
How do you write a number in scientific notation?
|
Place a decimal within the non-zero digits of the number to create a number between 1 and 10. Multiply this number by a factor of 10, where the exponent to the 10 is the number of decimal places the decimal point has moved.
|
15 x 10^5
|
|
|
What are perfect squares?
|
Numbers that are the result of multiplying two of the same integers together.
|
1, 4, 9, 16, 25, ....
|
|
|
In a square root, the number within the radical sign is a what?
|
radicund
|
-al +und
|
|
|
How can you write a square root as an exponent?
|
the exponent must be a fraction where the denominator is the root or index
|
25^(1/2) = sqr rt (25) = 5
|
|
|
What is the equation for how fractional exponents can be written in terms of square roots?
|
a^(x/y) = (y sqr root (a))^x
|
the denominator is the root or index of the square root
|
|
|
How do you simplify square roots that are not perfect squares?
|
Look for the perfect square factors of the number in the radical, and factor out the perfect square.
|
sqr rt (8) = sqr rt (4) * sqr rt (2) = 2 * sqr rt (2)
|
|
|
How do you add or subtract with radicals?
|
The radicals must have the same radicand for all terms. If the randicands are not the same and all are in the simplest form, the radicals cannot be combined.
|
4sqrrt(3) + 7sqrrt(3) = 11sqrrt(3) and sqrrt(2) + sqrrt(5) cannot be combined
|
|
|
How do you multiply or divide radicals?
|
Multiply or divide coefficients with coefficients and radicals with radicals.
|
2sqrrt(5) * 3sqrrt(3) = 6sqrrt(15) and 15sqrrt(10)/5sqrrt(2) = 3sqrrt(5)
|
|
|
How can you simplify an expression with a radical in the denominator?
|
Rationalize the denominator by multiplying both the numerator and the denominator by the same radical that is in the denominator to eliminate the denominator.
|
3/sqrrt(2) = (3*sqrrt(2))/(sqrrt(2)*sqrrt(2)) = (3*sqrrt(2))/2
|
|
|
How do you know if a radical is simplified?
|
There are no perfect square factors of the radicand other than 1, there are no fractions under the radical sign, there are no radicals in the denominator of a fraction.
|
No squares, or radical fractions
|
|
|
What is a point?
|
A position in space, it has no length or width.
|
without size
|
|
|
What is a line?
|
The shortest distance between two points. It has no width, and an infinite length.
|
no width, infinite length
|
|
|
What is a line segment?
|
A piece of a line and includes two points called endpoints and all points in between
|
line with points
|
|
|
What is a plane?
|
Any three non-colinear points determine a plane and has an infinite width and length
|
never ending flat space with at least 3 points
|
|
|
What is a ray?
|
One half of a line. It includes an endpoint and all of the collinear points on one side of this endpoint.
|
endpoint in one direction
|
|
|
What is congruent in geometry terms?
|
When two geometric figures have the same measure.
|
the same
|
|
|
What is a segment bisector?
|
It is a line or line segment that divides the segment into two congruent segments.
|
divides in half
|
|
|
What is the midpoint of a segment?
|
It is where the bisector intersects the segment.
|
center point
|
|
|
What are parallel lines?
|
Lines in the same planes that do not intersect. Also applies to line segments, rays, and planes
|
equal distance apart at all points
|
|
|
What are perpendicular lines?
|
Lines in the same plane that intersect at one point and form four 90 degree angles. Also applies to line segments, rays, and planes
|
meet with right angles
|
|
|
What are skew lines?
|
Lines in different planes that do not intersect.
|
Ske lines exist in three dimensions.
|
|
|
What is an angle?
|
It is defined by two distinct rays that have the same endpoint.
|
shared point of two rays or line segments
|
|
|
What is the vertex of an angle?
|
The common endpoint of the two rays that form an angle
|
point of angle
|
|
|
What is an angle bisector?
|
It is a ray in the interior of the angle that divides the angle into congruent angles with the same measure
|
half angles
|
|
|
What is an acute angle?
|
It is an angle whose measure is between 0 degrees and 90 degrees
|
0 < angle < 90
|
|
|
What is a right angle?
|
It measures exactly 90 degrees.
|
angle = 90
|
|
|
What is an obtuse angle?
|
It measure is between 90 degrees and 180 degrees.
|
90 < angle < 180
|
|
|
What is a straight angles?
|
It measures exactly 180 degrees.
|
angle = 180
|
|
|
What are complementary angles?
|
Any two angles whose combined measures equal 90 degrees. It is not necessary for these angles to share a side.
|
degrees sum to 90
|
|
|
What are supplementary angles?
|
Any two angles whose combined measures equal 180 degrees. It is not necessary for these angles to share a side.
|
degrees sum to 180
|
|
|
What is a linear pair?
|
Two supplementary angles that share a common side and no common interior points. They will form a straight angle or a line.
|
joined supplementary angles
|
|
|
What is a transversal?
|
A line that cuts through two parallel lines, eight angles are formed.
|
line across parallel lines
|
|
|
What are vertical angles?
|
They are formed when two line intersect and share a common vertex but no common sides and are congruent.
|
opposite angle when line interesect
|
|
|
What are corresponding angles?
|
Two angles on the same side of the transversal, but one is in the interior of the parallel lines and one is in the exterior of the parallel lines and are congruent.
|
Matching angles an both parallel lines where another line cuts it.
|
|
|
What are altenate interior angles?
|
Two angles on different sides of the transversal, both in the interior of the parallel lines and are congruent.
|
Matching angles between the parallel lines on opposing sides of the transversal.
|
|
|
What are alternate exterior angles?
|
Two angles on different sides of the transversal, boh in the exterior of the parallel lines and are congruent.
|
Matching angles outside of the parallel lines on opposing sides of the transversal.
|
|
|
What are polygons?
|
Closed geometric figures made up of line segments and angles.
|
geometrical shapes
|
|
|
What is a triangle?
|
A 3 sided, 3 angled polygon.
|
3 angles
|
|
|
What is an acute triangle?
|
A triangle with three acute angles.
|
3 acute
|
|
|
What is a quadrilateral?
|
A four sided polygon.
|
4 lines
|
|
|
What is a pentagon?
|
A 5 sided polygon.
|
5 angles
|
|
|
What is a hexagon?
|
A 6 sided polygon.
|
6 angles
|
|
|
What is an octagon?
|
An 8 sided polygon.
|
8 angles
|
|
|
What is an obtuse triangle?
|
A triangle with one obtuse angle and two acute angles.
|
1 obtuse, 2 acute
|
|
|
What is a right triangle?
|
A triangle with exactly one right angle and two acute angles.
|
1 right, 2 acute
|
|
|
What is a scalene triangle?
|
A triangle with 3 sides of different lengths.
|
3 different lengths
|
|
|
What is an isosceles triangle?
|
A triangle with two sides of the same length. Two angles are also the same; the angles opposit the sides of the same length.
|
2 sides one length
|
|
|
What is an equilateral triangle?
|
A triangle with all three sides the same length. Three angles are also the same and always equal 60 degrees
|
3 sides one length
|
|
|
What do the 3 angles in a triangle add up to?
|
180 degrees
|
180
|
|
|
What is equal to the measure of an exterior angle of a triangle?
|
The sum of the two remote interior angles.
|
opposits equal
|
|
|
In a right triangle, what do the two acute angles equal up to?
|
90 degrees, they are complementary.
|
90
|
|
|
In a triangle, the length of the longest side must be what?
|
Less than the sum of the two shorter sides.
|
less
|
|
|
What is a trapezoid?
|
A quadrilateral with one pair of parallel sides.
|
1 parallel
|
|
|
What is a parallelogram?
|
A quadrialteral with two pairs of parallel sides.
|
2 parallel
|
|
|
What is the sum of the angles of a quadrilateral?
|
360 degrees
|
360
|
|
|
In a parallelogram, the opposits are what?
|
The opposit sides are parallel, and the opposit angles are congruent.
|
the same
|
|
|
What is an isosceles trapezoid?
|
A trapezoid with non-parallel sides that are equal in length.
|
short and long parallel with other sides equal
|
|
|
In an isosceles trapezoid, the angles are what?
|
2 pairs of congruent angles, and also 2 diagonals.
|
same
|
|
|
In a rectangle, what are the angles?
|
All are 90 degrees, and opposite sides are congruent as are diaganols.
|
90
|
|
|
What is a rhombus?
|
A parallelogram with four congruient sides and perpendicular diaganols.
|
stilted rectangle
|
|
|
What is a rectangle?
|
A parallelogram with four right angles.
|
four right angles
|
|
|
In a rhombus, what are the angles?
|
Opposite angles are congruent as are opposite sides. The diaganol is perpendicular.
|
congruent opposites
|
|
|
What is a square?
|
It is both a rhombus and a rectangle and a parallelogram with four 90 degree angles and all sides are congruent. The diagonals of a square are both congruent and perpendicular.
|
equals everywhere
|
|
|
What is a perimeter?
|
It is the distance around a polygon.
|
around
|
|
|
How do you calculate the perimeter of a square?
|
P = 4x where x is the length of one side
|
4 congruent sides
|
|
|
How do you calculate the perimeter of a parallelogram?
|
P = 2y + 2x where y and x are non-congruent sides.
|
around
|
|
|
What is the area of a geometric shape?
|
It is the amount of square units needed to cover the shape.
|
area units are always squared
|
|
|
How do you determine the height of a polygon?
|
It is the length of the segment that is perpendicular to a side of the polygon (called the base)
|
base determines where height is...highest point
|
|
|
What is the radius of a circle?
|
It is the segment whose endpoints are the center of the circle and any point on the circle
|
half a diameter
|
|
|
What is the diameter of a circle?
|
It is a segment that passes through the center of the circle and whose endpoints are both on the circle.
|
Twice the radius
|
|
|
How do you calculate the circumference of a circle?
|
C = 2*pi*r = pi*d
|
perimeter, single units
|
|
|
What is the perimeter of a circle called?
|
A circumference.
|
around the circle
|
|
|
How do you calculate the area of a circle?
|
A = pi*r^2
|
units squared
|
|
|
What is a prism?
|
It is a three-dimensional solid with two congruent bases, and any number of other faces that are all rectangles.
|
2 bases + other faces
|
|
|
What is a cylinder?
|
It has two congruent circular bases, with one rectangular face wrapped around the bases.
|
canister
|
|
|
How do you calculate surface area for three-dimensional solids?
|
It is the sum of the areas of its faces.
|
add areas
|
|
|
How do you calculate the volume of a three-dimensional solid?
|
It is the area of one of the bases multiplies by the height h.
|
V = A*h
|
|
|
What is the volume of a rectangular solid?
|
V = l*w*h
|
base * height
|
|
|
What is the volume of a cylinder?
|
V = pi*r^2h
|
base * height
|
|
|
What is the volume of a pyramid?
|
V = (1/3)*l*w*h
|
(base * height) / 3
|
|
|
What is the volume of a cone?
|
V = (1/3)*pi*r^2*h
|
(base * height) / 3
|
|
|
What is the hypotenuse of a right triangle?
|
The side that is opposite from the right angle
|
opposite right
|
|
|
What are the legs of a right triangle?
|
The sides that make up the right angle
|
touching right
|
|
|
What is the Pythagorean theorem?
|
A theorem states that in every right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse
|
a^2 + b^2 = c^2
|
|
|
What is the sine of an acute angle in a right triangle?
|
length of the opposite side/length of the hypotenuse
|
O/H
|
|
|
What is the cosine of an acute angle in a right triangle?
|
length of the adjacent side/length of the hypotenuse
|
A/H
|
|
|
What is the tangent of an acute angle in a right triangle?
|
length of the opposite side/length of the adjacent side
|
O/A
|
|
|
What is the coordinate plane?
|
It is formed by the intersection of two perpendicular number lines, the x and y axis, and consists of 4 quadrants
|
2D
|
|
|
What is the origin of a coordinate plane?
|
0,0
|
0 units up or down and 0 units right or left, or the center
|
|
|
Where is quadrant I?
|
Upper Right, both x and y axis are positive
|
I is all positive
|
|
|
Where is quadrant II?
|
Upper Left, x axis is negative, y axis is positive
|
II is positive Y
|
|
|
Where is quadrant III?
|
Lower Left, both x and y axis are negative
|
III is all negative
|
|
|
Where is quadrant IV?
|
Lower Right, X axis is positive, y axis is negative
|
IV is negative y
|
|
|
How is a point in the coordinate system named?
|
With 2 numbers (X,Y) which may be positive or negative and indicate how far in each direction from the origin and which direction.
|
side to side then up or down
|
|
|
What is the abscissa?
|
The X value in the (X,Y) of the coordinate plane
|
sci = x
|
|
|
What is the ordinate?
|
The Y value in the (X,Y) of the coordinate plane?
|
It's 'ord'inary to think of numbers as going up or down, as in the y axis
|
|
|
What is the formulat to determine the midpoint between two endpints of a line segment between (X1,Y1) and (X2,Y2)?
|
( [(X1+X2)/2],[(Y1+Y2)/2] )
|
Add each point together, in order, and divide by 2
|
|
|
What is the formula to calculate the distance between any tow points in the coordinate plane?
|
d = SqrRt[(X1-X2)^2+(Y1-Y2)^2]
|
Pythagorean Formula using coordinates
|
|
|
What information does the slope of a line tell us?
|
The rate of change and pattern of a graph
|
rise over run
|
|
|
What is the formulat for calculating slope from two points?
|
m = (Y1-Y2)/(X1-X2)
|
change in y over change in x
|
|
|
What are the four major categories of slope?
|
positive, negative, 0, or no slope
|
How many directions can a line go?
|
|
|
What do positive sloped lines look like?
|
They go up to the right as you move from left to right.
|
Uphill
|
|
|
What do negative sloped lines look like?
|
They go up to the left as you move from right to left
|
Downhill
|
|
|
What do lines with a slope of 0 look like?
|
A straight horizontal line represented by y = k where k is a constant value
|
0 rise means what?
|
|
|
What do lines with no slope look like?
|
A straight vertical line represented by x = k where k is a constant value, since 0 would be the divisor, it is an impossible calculation
|
DNE slopes are straight up
|
|
|
What are intercepts on the coordinate plane?
|
Where a line crosses the X or Y axis.
|
Where it intercepts known axis.
|
|
|
What will the point look like for a y intercept?
|
(0,Y)
|
The value of the other axis must be 0.
|
|
|
What will the point look like for a x intercept
|
(X,0)
|
The value of the other axis must be 0.
|
|
|
What is the equation of the slope-intercept form of a linear equation?
|
y = mx+b where m is the slope and b is the y intercept
|
both points plust slope and a y intercept
|
|
|
How is a linear inequality (< or >) represented on a coordinate plane?
|
The line should be dashed with the side of the line (or half plane) is shaded to show all the points in the solution set
|
How would you draw a line that represented not being included?
|
|
|
How is a linear inequality represented if the symbol includes =?
|
The line should be solid withe the side of the line (or half plane) is shaded to show all the points in the solution set.
|
How would you show that all points in a direction would be included.
|
|
|
How are the slopes of parallel lines related?
|
They have the same slope.
|
What would match in the equation?
|
|
|
How are the slopes of perpendicular lines related?
|
The slopes are negative reciprocals of one another.
|
What would be the same but different in the equation?
|
|
|
What are the four basic types of transformations in the coordinate plane?
|
reflections, translations, rotations, and dilations
|
A transformation is a change in some way, what are 4 ways you could change on the coordinate plane?
|
|
|
How do you solve a system of equations?
|
Graph each equation on the same coordinate plane, the point where the two lines intersect is the solution to the system of equations.
|
Where would they meet?
|
|
|
What is a system of equations in relation to a coordinate plane?
|
When more than one equation (line) is graphed on the same set of axes.
|
lines
|
|
|
What are the special systems of equations?
|
parallel lines, perpendicular lines, and coincident lines
|
three ways lines can relate in a plane
|
|
|
What are coincident lines?
|
Lines that have the same equation and are essentially on top of each other and cover the same identical points
|
CO - incident, same thing, same time
|
|
|
What are the possibles solutions for systems of equations?
|
one solution, infinite solutions, no solutions
|
how can lines meet (or not)?
|
|
|
What is the solution for parallel lines?
|
No solution
|
They never meet
|
|
|
What is the solution for perpendicular lines?
|
One solution
|
Where do they meet?
|
|
|
What is the solution for conincident lines?
|
Infinite solutions
|
What turf do they both cover
|
|
|
In systems of equations in coordinate planes, when is there infinite solutions?
|
When the lines are coincident
|
What is exactly the same?
|
|
|
In systems of equations in coordinate planes, when is there no solution?
|
When the lines are parallel
|
What lines never meet?
|
|
|
What is a reflection transformation?
|
The object is reflected over a line of relection, or flipped over, or mirrored across a given relection line.
|
What is a reflection?
|
|
|
If relection is across the y axis, what will be the reflections coordinates?
|
The Y coordinates remain the smae while the x coordinates become their opposites
|
x changes
|
|
|
If relection is across the x axis, what will be the reflections coordinates?
|
The X coordinates remain the smae while the Y coordinates become their opposites
|
y changes
|
|
|
If relection is across the y = x lines, what will the reflections coordinates be?
|
All coordiantes will be the opposites
|
Pos to neg or vice versa
|
|
|
In a reflection over the x-axis, the point (x,y) will be?
|
(x,-y)
|
y changes
|
|
|
In a reflection over the y-axis, the point (x,y) will be?
|
(-x,y)
|
x changes
|
|
|
In a reflection over the line y = x, the point (x,y) changes to?
|
(y,x)
|
swap
|
|
|
What is a translation tranformation?
|
Where an object is moved a certain number of unites either left or right and then either up or down.
|
to translate is to have the same thing in a different place
|
|
|
How is a translation expressed?
|
As: Ta,b where a is the number of horizontal units and b is the number of vertical units.
|
T what?
|
|
|
What does Ta,b mean?
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The point (x,y) moves to (x+a,y+b)
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units to be moved
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How is a reflection expressed?
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As:R?-axis where the ? is the axis of reflection (y,x, y=x, or others)
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R what?
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How are transformations notated?
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Over an arrow drawn from the original object coordinates to the new object coordinates
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point
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What is a rotation transformation?
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Object is roatated or turned a particular number of degrees around a central point, often the origin.
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move in circles
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How is a rotation transformation notated?
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Rot? or just R? where ? is the degrees it is rotated.
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degrees are important
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What direction is a positive rotation?
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counter-clockwise
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against time
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What direction is a negative rotation?
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clockwise
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with time
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A rotation in the positive or negative is the same as how many degrees in the opposite?
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360-the rotation degrees
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circle - rot
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A rotation of 90degrees transforms the point (x,y) to?
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(-y,x)
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swap with front neg
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A rotation of 180degrees transforms the point (x,y) to?
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(-x,-y)
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same but opposite signs
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A rotation of -90degrees transforms the point (x,y) to?
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(y,-x)
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swap with rear neg
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What is a dilation transformation?
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A change in the size (but not the proportions) and possibly the origin of the object using a center and a scale factor.
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enlarge or shrink
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What is the only type of transformation that does not preserve the size of the object?
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Dilation
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increase or decrease
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A dilation D with scale factor c changes the pint (x,y) to?
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(c*x,c*y)
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Just adjust the points by multiplying
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What is the notation for a dilation tranformation?
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D? where ? is the scale factor
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multiply
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What does an algebraic expression contain?
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Numbers, variables, and operations to state a relationship
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What 3 things are in every single expression
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What is an equation?
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Two algebraic expressions set equat to each other.
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=
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What is a defined operation?
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An operation in which symbols (any ones, even odd ones) are used to represent algebraic expressions. Essentially you just sub in values into the script to spit out a number
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Sub in for characters.
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What are numbers called in algebra?
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Constants
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what doesn't change?
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What is a algebraic constant?
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Numbers
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what doesn't change?
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What are letters representing unknown quantities called in algebra?
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Variables
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What changes?
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What is an algebraic variable?
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Letters representing unknown quantities.
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What does change?
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What is the number that appears in front of a variable called?
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Coefficient
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helping efficiency
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What is an algebraic coefficient?
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A numbers that appears in front of a variable.
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attaches to variables
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What is a polynomial?
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Algebraic expressions containing terms created by the product of constants and variables.
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poly = many, nomial = expressions
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What is a monomial?
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A single term in algebra
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examples: 3, 4x, -6ab, 24(x^2)y
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What is a bi or tri nomial?
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algebraic terms sperated by addition and subtraction
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examples bi: a+b, 3x-4yz
examples tri: x^2+3x-4 |
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What does the phrase like terms mean?
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It refers to polynomial terms that contain exactly the same variables and exponents
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things that can be matched/combined
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How do you multiply a monomial by a monmial?
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Multiply the coefficients and then follow the rules for multiplying the exponents of like bases (add exponents.)
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Example: 3(x^2)+4(x^3)=12(x^5)
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How do you multiply a monomial by a polynomial?
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Use the distributive property then simplify.
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Example:6(y^2)*(2y+3) = 12(y^4)+18(y^3)
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How do you multiply a binomial by a binomial?
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Use the distributive property known as FOIL
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first, outer, inner, last
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What does FOIL mean?
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First: multiply the first term in each binomial
Outer: multiply the outer term in each binomial Inner: multiply the inner terms in each binomial Last: multiply the last term in each binomial. |
first, outer, inner, last
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What is the factor or a number or expression?
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The pieces of the number or expression that multiply together to form the expression.
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parts
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What is the factor of the difference between two perfect squares?
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x^2-a^2=(x-a)(x+a)
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each square root plus/minus
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How do you factor a factorable polynomial?
a(x^2)+bx+c |
If a = 1, then the constant terms of the factors have a sum of b and a product of c, if a isn't 1, use the factors of c with trial and error to find the factors. Remember that not all trinomials are factorable.
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plus and product
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What is a perfect square trinomial?
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It has two equal binomial factors and are either: (x^2)+2ax+(a^2) or (x^2)-2ax+(a^2)
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all positive or positive negative
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What is the factor of (x^2)+2ax+(a^2)?
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(x+a)^2
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positive square
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What is the factor of (x^2)-2ax+(a^2)?
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(x-a)^2
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negative square
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What must you do to factor an expression completly?
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Factor out the GCF, if it exists. Factor the difference between two squares. Factor the trinomial into two binomials (FOIL)
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all the way to base elements
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What is a rational expression?
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An expression that may involve constants and variables in the form a/b, where b cannot equal 0.
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fraction
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How do you simplify rational expressions?
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Factor both numerator and denominator and then cancel like factors.
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factor to cancel
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What is the factor of (x-a)/(a-x)?
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-1
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cross simplifying
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What is the goal when solving an equation?
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Isolate the variable
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need only one
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What should you do to solve an equation?
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Simplify each side of the equation separately by applying the distributive property and combining like terms. Move the variable to one side of the equation. Perform the inverse operations of either addition or subtraction. Perform the inverse operations of multiplication or division. Check your answer by substituting the value of the variable into the original equation.
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Simplify, Isolate, PEMDAS, Check
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What is the substitution method for solving a system of equations?
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Solve one equation for one variable (ie x) the substitute into the other equation to solve for the other variable (ie y), then substitute the answer back into the first equation to solve for the first variable.
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put one equation into the other
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What is the elimination method for solving a system of equations?
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Multiply one equation by a factor that will allow like terms to be eleminated across both equations when the equations are added together, isolating one variable for solving, then put that variable back into the original equation.
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multiply then add the equations together.
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What is a quadratic equation?
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ax^2 + bx + c = 0 where a is not zero
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square, times, plus
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What does ax^2 + bx + c = 0 represent?
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A quadratic equation
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square, times, plus
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What type of solutions do quadratic equations have?
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Two solutions that are unequal, or two solutation that are equal, or no solution
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3 possabilities, uneq, eq, ns
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What is the quadratic formula?
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x = [-b+-(sqrt(b^2-4ac))]/2a
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free falling or pop goes the weasel
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What is this formula:
x = [-b+-(sqrt(b^2-4ac))]/2a ? |
Quadratic equation
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free falling or pop goes the weasel
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