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“Factorials” are written with an exclamation point (!), and mean that you should multiply that number by every counting number less than it. For example: 3! = 3 × 2 × 1 = 6 If you are dividing two factorials, you can save time by canceling first. For example: 5! 5 × 4 × 3 × 2 × 1 = = 5 × 4 = 20 3! 3×2×1 “Exponents” are written as a small number to the upper right of a number (the “base”), and mean that you should multiply the regularsized number by itself the small number of times. E.g. 25 = 2 × 2 × 2 × 2 × 2 = 8 × 4 = 32 If you are dividing two exponents with the same base, you can save time by canceling first. E.g. 25 2 × 2 × 2 × 2 × 2 = =2×2=4 3 2 2×2×2 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒 If you bicycle at ten miles per hour for three hours, your total mileage will be 10 × 3 = 30 miles. You can also rearrange the equation as 𝑠𝑝𝑒𝑒𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 or 𝑡𝑖𝑚𝑒 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑠𝑝𝑒𝑒𝑑 So if you go 100 miles in four 4, you must 100 have gone an average speed of = 25 4 miles per hour. If you went 60 miles at 5 60 miles per hour, it would take = 12 5 hours. In a right triangle, the two sides next to the right angle are called “legs”, and the side opposite the right angle is called the “hypotenuse”. If we call the lengths of the legs “a” and “b”, and the length of the hypotenuse “c”, the equation a leg 𝑎2 + 𝑏 2 = 𝑐 2 is true. hypotenuse c E.g., if the legs are 3 and 4, 𝑐 2 = 32 + 42 = 9 + 16 = 25 = 𝑐 2 , so the hypotenuse is 5. leg b The “triangle inequality” says that any two sides of a triangle have to add up to more than the other side. For example, if two sides of a triangle are 3 m and 7m, the other side can’t be 1 m, because 1 + 3 is not more than 7. The other side also can’t be 20 m, because 3 + 7 is not more than 20. We normally write numbers in “base 10”, where the rightmost digit is 100 = 1, the next one is 101 = 10, then 102 = 100, etc. Any counting number can be a base, however. For example, computers think in “base 2”, where the rightmost digit is 20 = 1, the next one is 21 = 2, 2 then 2 = 4, etc. If a computer is thinking about the number 1012 , 0 that number is the same as 1 × 2 + 0 × 21 + 1 × 22 = 1 × 1 + 0 + 1 × 4 = 1 + 4 = 5 in base 10. Instead of adding over and over, it is frequently easier to multiply, so long as you can turn the addition problem into addition of the same number. For example, if you’re trying to add the numbers from 1 to 10, you might notice that 1 + 10 = 11, 2 + 9 = 11, etc. You might also realize that there were 10 numbers to start with, so there must be 5 pairs that each add to 11, for a total of 5 × 11 = 55. In a data set, the mode is the number that appears the most often. The median is the number that appears in the center if the numbers are arranged in increasing order. The mean is the average of the numbers, which you get by adding all the numbers and dividing by how many numbers there are. The range is the difference between the largest and smallest numbers in the set. When a sequence is not obvious, looking at the differences can help. If you don’t see a pattern there, try it again, etc... For example, in the sequence 9, 12, 17, 24, 33, 44, … the differences are 3, 5, 7, 9, 11, … If you still don’t see it, the differences of the differences are 2, 2, 2, 2, … so the next double difference is probably 2, the next difference is probably 13, and the next number in the sequence is probably 57. For many problems, making an organized list can often help if you don’t know the “best” way to do it. For example, if you want to find all of the ways to add three different single-digit numbers and get 15, start with 1 & 2. What can be added to 1 & 2 to get 15? 12, which isn’t a single-digit number. 1 & 3? No. 1 & 4? No. 1 & 5? Yes, 9 works, so we write 159, then 168, but not 177. Then we try 2 & 3 (no), then 2 & 4, which works, so we write 249, 258, & 267, then 348, 357, & 456. A prime number is a number with exactly two factors, 1 and itself. 1 is NOT prime, because it has only one factor, 1. 2 is prime, because it has exactly two factors, 1 and 2. 3 is prime, because it has exactly two factors, 1 and 3. 4 is NOT prime, because it has three factors: 1, 2, and 4. To turn in a circle, you must turn 360° (degrees). If you turn only halfway around, you turn 180°, which is a “straight angle”. When two lines intersect at a vertex, angles that are opposite one another are equal, and are called “vertical angles”. When a line intersects two parallel lines, corresponding angles at each intersection will be equal. Any convex n-gon (n-sided polygon) can be divided into 𝑛 − 2 triangles by drawing all of the diagonals from one vertex. For example, a pentagon (a 5-gon) can be divided into 5 − 2 = 3 triangles. Every triangle has angles that add up to 180°. So every n-gon has angles that add up to (𝑛 − 2) × 180°. The perimeter of a 2D figure is the distance around its edge. The area of a 2D figure is the number of 1x1 squares it would take to fill the figure, even if they might need to be chopped up. The volume of a 3D solid is the number of 1x1x1 cubes it would take to fill the space (perhaps with chopping). The surface area of a 3D solid is the sum of the areas of its surfaces. Generally, “sum” means “add”, “difference” means “subtract”, “product” means “multiply”, and “quotient” means “divide”. When discussing fractions, “of” often means “multiply”. When discussing probability, “and” frequently means “multiply” and “or” often means “add”. In the metric system, there is a set of prefixes that indicate how many of something you’re talking about. “kilo” means 1000, “hecta” means 100, “deka” means 10, “deci” means 1 1 , “centi” means , and “milli” 10 1 100 means , but there are many 1000 others. For example, a kilometer is 1000 1 meters, and a deciliter is of a liter. 10 When naming things, mathematicians frequently use prefixes to indicate numbers. Some common ones are “uni” for 1, “duo” for 2, “tri” for 3, “tetra” for four, “pent” for five, “hex” for six, “sept” for seven, “oct” for eight, “non” for nine, “dec” for ten, but there are many more. For example, a tetrahedron is a solid with four faces, and a hexagon is a two-dimensional figure with six sides. To convert a repeating decimal to a fraction, you must find a way to eliminate the repeating part of the decimal. ̅̅̅, we will For example, to convert . ̅63 consider that 100 of that number ̅̅̅. If we subtracted would be 63. ̅63 them, we’d get 99 of that number, which would be 63. Thus, one of 63 21 7 that number would be = = . 99 33 11 Whenever you’re doing complicated arithmetic, remember the made-up word PEMDAS, which is supposed to help you remember to do anything in parentheses first, then any exponents, then all multiplication & division (which are sort-of the same thing) from left to right, then all addition & subtraction (also sort-of the same) from left to right. For example, 4 + 5 × 6 = 4 + 30 = 34 and 3 × (7 − 1)2 ÷ 4 = 3 × 36 ÷ 4 = 27. In algebra, a letter or other symbol might be used to represent a number we don’t know. For example, 2𝑤 + 3 = 45 means if double some number (math people avoid writing × if they can) and add 3, we get 45. To figure out what 𝑤 is, we need to work backwards; 2𝑤 must be 3 less than 45, so we write 2𝑤 = 45 − 3 = 42. Then 𝑤 must be half of 42, so we write 𝑤 = 42 ÷ 2 = 21. Guess and check is a totally valid solution strategy, but should probably be your last resort, and is better if you have some reasons for guessing what you’re guessing (e.g. “My guess gave me an answer that was WAY to big; I should guess a lot less this time.”). Every counting number other than 1 can be written as the product of prime numbers in exactly one way; this is called the “prime factorization” of the number. For example, 12 = 2 × 3 × 3 = 21 × 32 and 120 = 2 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 = 22 ⋅ 32 ⋅ 51 . One way to find prime factorizations is to find numbers that multiply to the number, e.g. 120 = 10 × 12, then do the same for those numbers until you only have prime numbers. Probability is a measure of how likely something is. If something is certain, it has a probability of 1. If it is impossible, it has a probability of 0. In between, probabilities are fractions, and represent the number of ways the thing could happen divided by the total number of ways 𝑔𝑜𝑜𝑑 anything could happen ( 𝑡𝑜𝑡𝑎𝑙 ). For example, if a bag has three red marbles and two blue marbles, the probability of drawing a red marble 𝑔𝑜𝑜𝑑 𝑟𝑒𝑑 3 is = = . 𝑡𝑜𝑡𝑎𝑙 𝑎𝑙𝑙 5 The “counting principle” says that if you have two (or more) independent sets of choices, you can multiply the numbers of choices to figure out the overall number of choices. If you have three pairs of pants and five shirts, you have 3 × 5 = 15 outfits. If you are flipping four coins, and each of them could land two different ways, there are 2 × 2 × 2 × 2 = 16 total ways they could flip. An “arithmetic” sequence is one where the same number is added (or subtracted) between terms, such as 1, 3, 5, 7, … or 100, 93, 86, 79, … A “geometric” sequence is one where the same number is multiplied (or divided) between terms, such as 1, 3, 9, 27, … or 3, 6, 12, 18, … Sometimes two sequences are interspersed, such as 1, 1, 3, 3, 5, 9, 7, 27, … Any number can be written in “scientific notation”, which is a number between 1 and 10 multiplied by a power of 10. For example, 0.098 = 9.8 × 10−2 and 76543 = 7.6543 × 104 . The exponent of the 10 is the number of decimal places you moved the decimal point to the left. It will be a negative number if you move the decimal point to the right. When two angles add up to a “straight angle” (180°), they are “supplementary”, and each is the “supplement” of the other. When two angles add up to a “right angle” (90°) they are “complementary”, and each is the “complement” of the other. A “permutation” is when you are picking items from a group and the order in which you pick them matters. For example, if five people run a race and you’ve got a firstplace trophy, a second-place certificate, and a third-place handshake, the order matters. There are 5 × 4 × 3 = 20 × 3 = 60 ways you could give the awards. A “combination” is when you’re picking items from a group, but order doesn’t matter. For example, if you’re electing three kids from your five-person math club to be part of the student council, the order doesn’t matter. The math works out just like a “permutation”, except that you divide by the “factorial” of the number of things you’re picking (in this case 3). So, 5×4×3 5×4×3 there are = =5×2= 3! 3×2×1 10 ways to pick them. In a circle, the diameter (𝑑 ) is the distance across the circle through the center. The radius (𝑟) is the distance from the center to the edge, so it is half the diameter (𝑑 = 2𝑟). The circumference (𝐶 ) is the perimeter, the distance around the circle, and is equal to the diameter times 𝜋 (pi) (𝐶 = 𝜋𝑑 ). The area (𝐴) is equal to 𝜋 times the square of the radius (𝐴 = 𝜋𝑟 2 ). 𝜋 (pi) is a really awesome number that comes up in circle problems, and in a lot of other places as you learn more math. 𝜋 is an irrational number, which means that it is not equal to any fraction or mixed number, and that its decimal representation goes on forever without repeating. 𝜋 is commonly approximated as 1 22 3.14, 3 , or , but you should only 7 7 use these if a problem tells you to. A palindrome is a number that reads the same forward and backward, such as 11, 202, 9999, or 87678. Polygons are two-dimensional figures who sides are all line segments. Their corners are called vertices. Polyhedrons are three-dimensional solids whose faces are all polygons. Their corners are called vertices, their “corner sides” are called edges, and their “flat sides” are called faces. Factors are numbers that can divide into another number. For example, 2 and 8 are both factors of 24. Multiples are the opposite of factors. For example, 24 and 38 are both multiples of 2. The greatest common factor of two numbers is the largest number that is a factor of both of them. The least common multiple of two numbers is the smallest number that is a multiple of all of them.