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EIGHTH GRADE “YOU SHOULD KNOW” Study Guide and Practice Sets of Numbers: 1) Natural or Counting Numbers: 1, 2, 3, 4… 2) Whole Numbers: 0, 1, 2, 3, 4… 3) Integers: …-3, -2, -1, 0, 1, 2, 3, … 4) Rational: Terminating decimals (4.56), Repeating decimals (.333333…), fractions, perfect square roots 5) Irrational: non-terminating decimals (3.14157…), nonperfect square roots 6) Real numbers: all the sets → ↑ RATIONAL ↑ INTEGERS ↑ WHOLE ↑ NATURAL REAL ← ↑ IRRATIONAL Identify the set(s) of numbers that each belongs to: 1) 9.43243…_____________________________________ 2) -4 __________________________________________ 3) 0___________________________________________ 4) 5.12 _________________________________________ 5) 16 ________________________________________ Properties: 1) Commutative Property – when adding or multiplying, the numbers can be switched around and you still get the same answer. EX: 2 + 4 = 4 + 2 (5)(2) = (2)(5) 2) Associative property – when adding or multiplying numbers that are in and out of parenthesis, you can switch the numbers around. EX: 5 + (2 + 3) = 3 + (2 + 5) 8(9 × 7) = 9(8 × 7) 3) Distributive Property – when multiplying a number by a sum or difference, you can distribute the number through the parenthesis by multiplying. EX: 3(5 + 2) = 3 × 5 + 3 × 2 or 15 + 6 7(x – 5) = 7x – 35 4) Additive Identity: add zero to any number to obtain the original number. EX: 7 + 0 = 7 0 = identity element of addition 5) Multiplicative Identity: multiply one by any number to obtain the original number. EX: 8 × 1 = 8 1 = identity element of multiplication 6) Additive Inverse: the opposite of a number 7+-7=0 7) Multiplicative Inverse: reciprocal or “flip” of a fraction 2×½=1 Identify the property: 1) 8 + 0 = 8 ______________________________________ 2) 5(x – 6) = 5x – 30 _______________________________ 3) 18 + (-18) = 0 __________________________________ 4) 9 + (4 + 5) = 5 + (4 + 9) ___________________________ 5) What is the additive inverse of 8 ___________ 6) What is the multiplicative inverse of 2 5 ______________ 7 Basic Operations with Decimals and Fractions: Decimal: 1) When ADDING and SUBTRACTING decimals, you must line up the decimal points and bring the decimal straight down into your sum or difference. 18.2 - 6.008 = 4,785 + 9 + 2.307 18.200 - 6.008 12.192 4,785 9 + 2 .307 4,796.307 2) When MULTIPLYING with decimals, the decimal points are irrelevant until the very end. Multiply the numbers like normal and count the number of digits that are behind the decimal point in each number. This is how many places you move the decimal point in the product beginning at the end of the number. .2 × 6.03 = 1.206 3) When DIVIDING with decimals, move the decimal point to the end of the divisor (the outside number) and move the decimal point the same number of places in the dividend (the inside number). Divide like normal and bring the decimal point straight up in the quotient. Fractions 1) When ADDING and SUBTRACTING fractions, you must find a common denominator and make equivalent fractions to the ones given. Example with Borrowing: 2) When MULTIPLYING fractions, change all mixed numbers into improper fractions, cross-cancel if possible, and then multiply straight across. 3) When DIVIDING fractions, KEEP it, CHANGE it, FLIP it. 3 2 3 5 15 ÷ = × = 4 5 4 2 8 = 7 8 Converting Fractions ↔ Decimals Fraction to Decimal – divide numerator by denominator Change 5/8 into a decimal: Decimal to Fraction – put the decimal number over the ending place value Change .8 to a fraction in lowest terms. Read it: .8 (eight tenths) Write it: 8/10 Reduce it: 4/5 Change .09 to a fraction in lowest terms. Read it: .09 (nine hundredths) Write it: 9/100 ***Any Repeating Decimal must be placed over 9!!*** Practice: On your Own!! 1) 5.36 + 9.1 = 2) 5 2 1 +3 = 5 4 3) 98.01 – 16.482 = 4) 9 1 2 – 6 = 8 3 5) 4.1 (2.3) = 6) 7 1 1 × 1 = 2 4 7) 62.4 ÷ 0.04 = 8) 12 ÷ 4 ½ = Change each fraction into a decimal or each decimal into a fraction: 9) 5 = 8 11) 1.12 = 10) 9 8 = 9 12) 0.034 = Ratios: Ratio: a comparison of two numbers Example: 6 boys and 8 girls The ratio of boys to girls = 6:8 or 3:4 Rate: a comparison of different units Ex: 150 miles in 3 hours Unit Rate: when changing a rate into a unit rate, just divide!! Ex: 15 inches per 5 hours = 3 inches per hour Proportion: two ratios that are set equal to each other. Practice: On your Own!! Reduce each ratio (remember to have the same units): 1) 9 boys to 12 girls 2) 300 cars to 500 trucks 3) 8 inches to 3 feet 4) 1 pound to 20 ounces Convert each to a UNIT RATE: 5) $81 for 3 shirts 6) 500 people in 4 rooms 7) 672 miles in 10 hours Solve each proportion: 8) 3 x = 8 12 9) 9 10 = x 12 INTEGER RULES Adding with Same Signs – add and keep the sign Adding with Different Signs– subtract and keep the sign of largest number Subtraction – Keep, Multiply or Divide with Same Signs = Positive Multiply or Divide with Different Signs = Negative Absolute Value – distance from zero (always positive) Change, Change (make it an addition problems and follow the addition rules) Practice: Perform the operation 1) - 4 + 3 = ____ 9) 300 ÷ (-100) = _______ 2) -16 – (-8) = ______ 10) |-2 × 4 - 3| = ______ 3) 5 – (-2) = ______ 11) -32 = ________ 4) 16 + (-4) = ______ 12) (-3)2 = ______ 5) |-6| - |10| = ______ 13) 4│-2 + 4│ = _____ 6) -3 × -2 = _____ 14) -2 × 3 – 6 = ____ 7) 16 × (-2) = ______ 8) -14 ÷ (-2) = ______