PEMDAS This is an acronym for the order of operations. Order of operations is the order in which you complete problems with more than one operation. o P parenthesis o E exponents o M multiplication OR D division o A addition OR S subtraction Rounding Review When rounding, we look at the number to the RIGHT of the number we are rounding (because he is always right.) If the number to the right is greater than or equal to 5, we round up. If it is less than 5, leave the number the same and make everything afterwards zeroes! Divisibility Rules o Ex. 145.1 round to the nearest ten 150.0 o Ex. 3261 round to the nearest thousand..3000 A number is divisible by: (Can be divided leaving NO remainder) 2 if the last digit is even 3 if the sum of the digits is divisibly by 3 4 if the last two digits form a number that is divisibly by 4 5 if the last digit is 0 or 5 6 if the number is divisible by both 2 and 3 9 if the sum of the digits is divisibly by 9 10 if the last digit is 0 Fraction-Decimal-Percent Conversion Decimals to Fractions 1. Identify the place value of the digit farthest to the right. 2. Use that place value to determine the denominator. 3. The numerator is the entire number (without the decimal). Decimal Fraction 0.23. (twenty-three hundredths) 1.7.1 (one and seven tenths) 0.004.. (four thousandths) Fractions to Decimals Divide the denominator INTO the numerator (bottom into top) Percent to Decimals Move the decimal two places to the right. Decimals to Percents Move the decimal two places to the left. Fraction to Percents Change the fraction to a decimal (see above) and then change the decimal to a percent (see above). Percents to Fraction Change the percent to a decimal (see above) and then change the decimal to a fraction (see above). Ordering Decimals, Fractions, and Percents Change everything into decimals Stack the decimal numbers on top of each other lining up the decimals. Fill in any spaces with zeroes. Compare from left to right.
Decimals Adding and Subtracting Decimals ALWAYS REMEMBER: When adding and subtracting decimals, always line up the decimal points!! If there is a blank spot, insert a zero as a place holder. *Dealing with money is dealing with decimals! $money decimal 5 cents.0.05 20 dollars 20.00 50 cents.0.50 2 dollars and 50 cents 2.50 Multiplying Decimals Multiply the same as when you are using whole numbers! You do not have to line up the decimal points. To place the decimal point in the answer, count how many numbers are past the decimal in your entire problem and move the decimal point that many places to the left in your answer. o Ex: 0.17-2 decimal places X.3 1 decimal places.051 3 decimal places Dividing Decimals Dividend: the number to be DIVIDED Divisor: the number you are dividing by Quotient: the answer Step 1: write out the problem in long division form Step2: move the decimal point of the divisor right until you have a whole number Step3: move the decimal point of the dividend right the same number of positions! Step4: perform long division as usual, bringing the decimal straight up! Fractions Steps for REDUCING fractions: o If the numerator and the denominator are both divisible by a common number, the fraction can be reduced. o Find a common factor and divide both numerator and denominator by that factor o Continue this process until the numerator and denominator have no common factors or until you get a prime number for the numerator or denominator Improper Fractions-a fraction where the numerator is bigger than the denominator. To convert improper fractions into mixed numbers! Step 1: Divide! Divide the numerator by the denominator. Use the answer for the whole number part of your mixed number. Step 2: Use the remainder as the numerator in the fraction part of your mixed number. Step 3: Use the same denominator as the improper fraction. Sometimes you ll have to convert mixed numbers back into improper fractions as well. Step 1: Multiply the denominator by the whole number and then add the numerator. Step 2: Put the answer to step 1 over the original denominator. LCM = Lowest Common Multiple The multiples of a number are what you get when you multiply it by other numbers. Just like the multiplication table. o Multiples of 4: 4, 8,12, 20, 24 o Multiples of 5: 5, 10, 15, 20, 25 o The LCM of 4 and 5 would be 20, the smallest common multiple. The LCM becomes your common denominator when adding or subtracting fractions. Once you have a common denominator you can combine fractions!
Adding or Subtracting Fractions YOU MUST GET A COMMON DENOMINATOR! Step 1: Find a number that both denominators will go into: 1/3 2/9 BOTH 3 and 9 will go into 9 Step 2: Whatever you multiply the OLD denominator by to get the new denominator, YOU MUST MULTIPLY the OLD numerator by that same number to get the new numerator! Step 3: Add or subtract the numerators. LEAVE THE DENOMINATORS the SAME!!!!!! Step4: ALWAYS reduce if possible! Multiplying Fractions!! Multiply the numerators Multiply the denominators Remember to reduce! *If you are multiplying with mixed numbers.you must convert them to improper fractions before multiplying!!!!!!!!!!! Dividing Fractions Keep change flip the second fraction upside down. Clue Words Adding: sum, increased, more, altogether, plus, total, combined, perimeter Subtracting : difference, elapsed, deduct, used, less, loss, passed, left, remain, exceed, decreased Multiplying: product, area, double (x2), 7n, 7(6), (per/each) Dividing: quotient, division, distribute, half ( 2), 12/2, (per/each) Answer: solve, find, express, evaluate, calculate, model, solution Equal: is, was, were, have, has, had Reverse the order: than, from Real Numbers Rational numbers: any number that can be expressed as a fraction. Irrational numbers: nonrepeating and nonterminating decimals (Numbers that go on forever without a repeating pattern.) Integers positive and negative whole numbers and zero. This does not include decimals or fractions! Integers Absolute Value - A number s distance from zero. The absolute value of any number is always positive. -4 = 4 the absolute value of negative four is four meaning that negative four is four spaces away from zero. For example: a. -27 = 27 b. -50 = 50 c. 10 = 10 So if you put the absolute value of these numbers in order from least to greatest you would get: 10, 27, 50 BUT if you put the numbers themselves in order it would be quite different: -50, -27, 10 Addition of Integers 1. If the numbers are the same sign, keep that sign and add the numbers. Ex: -6 + (-3) = - 9 or 8+3 = 11 2. If the signs are different, keep the sign of the number with the larger absolute value and subtract. Ex: -6 + 12 = 6 or -20 + 10 = -10 **A song to help you remember!! Sing this to the tune of Row Row Row Your Boat Same sign add and keep, different signs subtract. Take the sign of the higher number then it will be exact!
Subtraction of Integers Keep- change-change!!! 1. Keep the first number. 2. Change the operation to adding. 3. Change the second integer to the opposite sign. 4. Use integer adding rules. Ex: 5 4 = 1 is the same as 5 + (-4) = 1 **A song to help you remember!! Sing this to the tune of Mary Had a Little Lamb When you subtract, keep change change, keep change change, keep change change, When you subtract, keep change change, keep change change, keep change change, Then use your adding rules. Multiplication and Division of Integers for these two operations the rules are the same!!!! 1. Multiply or divide like normal. 2. If the signs are the same, the sign of the answer will be positive even if the signs of the two numbers are negative. 3. If the signs are different, the sign of the answer will be negative. Variables and Algebraic Expressions Variable -- a letter or symbol that represents a quantity. Ex: a, b, d Algebraic Expression and expression that contains one or more variables Ex: a + b 3d + 4x Expression vs. Equation An expression does not have an equal sign. An equation has an equal sign. To Solve an Equation: 1. What number is attached to the variable? 2. What operation is attaching them? 3. Do the opposite operation with that same number to both sides of the equation. 2n + 5 = 19 2n = 14 2 2 n = 7 Distribute and Solve Equations You must distribute and combine like terms BEFORE trying to solve the equation. In the example below: FIRST, you must distribute the -3 to the x and to the -5 NEXT, you will combine like terms FINALLY, solve the equation. 12-3(x-5) = 21 (distribute the -3 to the x and to the -5.) 12-3x + 15 = 21 (combine like terms: 12 + 15.) 27 3x = 21-27 -27 (SUBTRACT 27 because you look at the sign BEFORE the number and do -3x = -6 the opposite operation to both sides.) -3-3 x = 2 Solving Equations with Variables on Both Sides - You must get the variables together before trying to solve the equation. In order to get the variables together, you must do the opposite operation to BOTH sides to move the variable from one side to the other. 3n + 4 = 2n +16 OR 3n + 4 = 2n + 16-2n -2n -3n -3n n + 4 = 16 4 = -1n +16-4 -4-16 -16 n = 12-12 = -1n -1-1 12 = n Word Problem Strategies o Read through at least once to get the main idea. What is the problem about? What is it asking? o Read it a second time to identify what you are looking for. o Look for clue words. o Pick out the key information needed to solve the problem. o READ THE QUESTION AGAIN TO MAKE SURE YOU ANSWERED WHAT THE QUESTION ASKED!!!!!!!!!!!!!!!!!!!!!!!!!
Inequalities Inequalities are solved just like equations using inverse operations EXCEPT when you multiply or divide by a negative number, you flip the inequality sign. -2n + 5 = 19-2n = 14-2 -2 n = -7-2n + 5 < 19-2n < 14-2 -2 n > 7 Clue Words At most means < At least means >