7th Grade SCReady Study Guide Operations with Integers: ❖ Multiplying and Dividing integers – o Same signs when multiplying or dividing always equal a positive. Neg and Neg = Pos or Pos and Pos = Pos o Different signs when multiplying or dividing always equal a negative Pos and Neg = Neg or Neg and Pos = Neg ❖ Adding Integers – o Same signs: keep signs and add and answer keeps the sign of the addends. ▪ Ex -5 + -3 = -8 or 5 + 3 = 8 o Different signs: subtract and answer keeps the sign of the number with the largest absolute value. ▪ Ex -5 + 6 = 1 or -15 + 8 = -7 ❖ Subtracting Integers – o Keep first number, change subtraction to addition, and then change the 2nd number to the opposite sign. o Follow rules for adding integers. ▪ Ex. -12 - 8 Ex. 15 - (-8) Ex. -5 – (-8) -12 + (-8) - 20 15 + 8 23 -5 + 8 3 ❖ Operations with Fractions Adding or Subtracting Fractions: ❖ Must have like denominator; add or subtract numerator Multiplying fractions: ❖ Multiply numerators and multiply denominators. Dividing fractions: ❖ Invert last fraction, then multiply the fractions. 1 2 1 2 1 2 2 3 3 6 − = 2 1×2 3 2×3 × = 2 1 3 2 ÷ = 4 −1 6 6 − = 2 1 6 3 = = 3 3 2 4 × = FDP - Fraction Decimal Percent ❖ Convert Fractions to Decimals – Divide numerator by denominator. ❖ Convert Decimals to Fractions – Look at the last digit of your decimal number and determine its place value. This place value number is the denominator of the fraction, then rewrite the number without a decimal as the numerator. Make sure your final answer is in simplest fraction form. Ex: 0. 35 = 35 = 7 EX. 2.5 = 2 5 = 2 1 10 2 100 20 DP (Doctor Pepper) ❖ Convert decimals to percent: move decimal two places to right Example: 0.56 = 56% or 6 = 600% ❖ Convert percent to decimals: move decimals two places to the left. Example: 84% = 0.84 or 135% = 1.35 ❖ Convert percent to fraction - convert to decimal and then follow directions for converting decimals to fractions. ❖ Convert fractions to percent – divide numerator by denominator (convert fraction to decimal) then follow directions for converting decimals to percent. Order of operations: PEMDAS 1) Grouping symbols – includes parentheses and brackets, absolute value, fraction bar, and square roots 2) Exponents 3) Multiplication or Division LEFT TO RIGHT – whichever comes first is the operation you do first. 4) Addition or Subtraction LEFT TO RIGHT - – whichever comes first is the operation you do first. **CHECK YOUR ANSWERS USING YOUR CALCULATORS!** Solving Equations: 1) 2) 3) 4) Distribute if necessary Combine any like terms if necessary Add or subtract like terms off each side using inverse operations. Multiply or Divide to isolate variables using inverse operations. 5x + 4 = 34 -4 -4 5x = 30 /5 /5 x = 6 Solving Inequalities ❖ Follow steps for solving equations. ❖ JUST REMEMBER WHEN YOU MULTIPLY OR DIVIDE BOTH SIDES BY A NEGATIVE NUMBER THE INEQUALITY SIGN FIIPS! ❖ You can flip an inequality if you change the sign. 7 > x is the same as x < 7 ❖ when graphing, < and > is a closed circle, < and > is open circle. ❖ x > 4 - open circle at 4 and arrow to the right Proportions ❖ Unit Rates, Constant of Proportionality Unit rate is rate per one. If I can fill 20 jars in 4 hours, unit rate is 5 jars per 1 hour. Constant of Proportionality is same as Unit Rate. ❖ To solve a proportion, set up 2 ratios, cross multiply, set up as an equation and solve. 𝑥 28 = 5.6x = 1.4(28) 1.4 5.6 Your Turn: 5 is to 8 as 15 is to w. Sales tax, discount, tip, part of whole ❖ Always put your percent over 100 and part over whole, then solve 14 is 25% of what number? 25 Set up as:100 = 14 𝑥 25x = 1400 𝑃𝑎𝑟𝑡 𝑤ℎ𝑜𝑙𝑒 % = 100 x = 56 Finding missing lengths with scale factors Ex: Wyheim is building a model airplane that is 1/80 the size of the real airplane. If the real one has a wingspan of 120 feet, what will be the wingspan of the model? 𝑚𝑜𝑑𝑒𝑙 1 𝑥 MAKE YOURSELF A KEY!!! 𝑟𝑒𝑎𝑙 𝑡ℎ𝑒𝑛 𝑠𝑒𝑡 𝑢𝑝 ∶ 80 = 120 Fundamental counting principle: (FCP) ❖ Multiply the number of choices together in each category to find the total possible arrangements. This is better and quicker at finding just the TOTAL number of possibilities than tree diagrams. Example 1: Tom is going to subway to buy lunch. If he chooses one item from each category, how many lunch combinations are possible if he can choose from 12 different sandwiches, 6 different types of chips, 4 types of cookies, and 9 different types of drinks? Example 2: James bought a new lock that allows him to create his own 4 digit code. If he can chose any number from 0 thru 9 and numbers can repeat, how many different possible codes can be created? Probability: DO NOT FORGET TO SIMIPLIFY YOUR ANSWERS INTO SIMPLEST FRACTION FORM!!!! 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒘𝒂𝒚 𝒆𝒗𝒆𝒏𝒕 𝑨 𝒐𝒄𝒄𝒖𝒓𝒔 P(A) = 𝒐𝒗𝒆𝒓 𝒕𝒐𝒕𝒂𝒍 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆 𝒆𝒗𝒆𝒏𝒕𝒔 ❖ Theoretical probability: Probability that is based on what SHOULD happen! ❖ Experimental probability: Probability that is based on what DID happen…not what should! Compound Probability (2 probabilities multiplied together) ❖ Independent Probability: (one event can NOT effect following events) P(A and B) = P(A) ● P(B) ❖ Dependent Probability: The event of one even CAN effect the following events. P(A and B) = P(A) ● P(B after A) Laws of Exponents ▪ Product Rule – add exponents (must have same base) 34 ∙ 37 = ▪ Quotient Rule – subtract exponents ( must have same base) 49 ▪ 45 = 44 Power to a Power Rule – multiply exponents (83 )4 ▪ 311 = 812 Product to a Power Rule – distribute exponent (8 ∙ 17)4 = 84 ∙ 174 ▪ Quotient to a Power Rule – distribute exponent to numerator and denominator 4 3 43 (11) ▪ = 113 Zero Power Property – always equals 1 210 = 1 Properties ❖ Commutative – terms change order ❖ Associative – always stays same order, but parentheses move ❖ Identity o Adding 0 or multiplying by 1 ❖ Inverse o Adding or multiplying the opposite 2 3 o Ex: 4 + -4 = 0 and 3 ° 2 = 1 ❖ Distributive 5 (x + 4) = 5x + 20 ❖ Substitution: If x = 3, then x – 2 is 3 – 2 ❖ Zero Product Property or Multiplicative Property of Zero o Anything times 0 is 0 Verbal Translations ❖ Remember order switchers are “more than”, “less than”, “subtracted from”, “added to”, etc. ❖ Ex: six less than x is nineteen x – 6 = 19 Geometric Constructions ❖ Any two sides of a triangle will always add up to be greater than the third side. o 2”, 7.5” and 5” cannot make a triangle because 2 + 5 is less than 7.5 o Can 8, 5 and 2.5 be the sides of a triangle? o Strategies: Draw pictures to help understand the questions. ❖ Quadrilaterals o What quadrilateral has all congruent sides and no right angles? o What quadrilateral has right angles and opposite sides parallel ? Statistics ❖ ❖ ❖ ❖ Mean is average (add all numbers and divide by how many there are) Median is the middle number when arranged least to greatest Mode is number appearing most often MAD (Mean Absolute Deviation) o Subtract each number from the mean o Take absolute value of each (make positive) o Find the average of those numbers Circles Radius is from center to edge. Diameter is across the circle. Circumference is around the circle. r = ½d d = 2r C = πd or C = 2πr Area = πr2 Volume and Surface Area V = lwh SA = 2lw + 2lh + 2wh Volume is how much it holds. Surface area is found by adding together the areas of all the sides. V = ½ blh SA = bl + ah + ch + bh Volume is how much it holds. Surface area is found by adding together the areas of all the sides. Cross Sections ❖ A cross section is created when a figure is sliced. ❖ What 2D shapes are created by slicing the 3D figure? Angles ❖ ❖ ❖ ❖ ❖ Supplementary angles add up to 180° Complementary angles add up to 90° Vertical angles are across from each other Adjacent angles are beside each other A linear pair is two angles that are supplementary and adjacent Your Turn: Find each missing angle!