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MATH RESOURCE GUIDE Increase/decrease % = subtract the two #s and divide by 1st # given D = RT distance = rate or speed x time CELSIUS ℃ = ( ℉ − 𝟑𝟐 ) 𝟓/𝟗 Unit or Each or per unit = $ / amount I = PRT (simple): interest $ = $ put in x % x time (yrs) I = PRT (compounded): interest $ = (1 + rate %)™ x $ Put in c² * b² = c * b = z ² 𝒃² ∗ 𝒃³ = 𝒃 ³ + ² = 𝒃 ⁵ (𝒂)³ (b) 𝒂³ b³ (𝑪² )³ = 𝑪 ² ∗ ³ = 𝑪 ⁶ b⁵ b ² (𝒂 𝒃)³ = 𝒂³ 𝒃³ b⁵ - ² =b ³ (𝒇 𝑶𝑭 𝒙) 𝒇 (𝒙) = 𝟑 𝒙 + 𝟗 𝒇(𝒙) = 5 𝒚 = 𝟑 (𝟓) + 𝟗 … 𝒚 = 𝟏𝟓 + 𝟗 𝒚 = 𝟐𝟒 Coordinate Pair would be ( 𝟓 , 𝟐𝟒 ) Scientific Notation: MUST have one digit __. X 10 to a power ? ; The power is the movement of the decimal to the first digit. 345, 000 = 3.45 x 10 ⁵; positive power moves to right; negative power moves to left .000078 = 7.8 x 10 Absolute Value: distance from zero; MUST be positive when leaving the | | 5 |= 5 | -7 + 5 | = |-2 | = 2 | 9 + -23| = |-14 | = 14 | REAL NUMBERS: all numbers that exist. Rational Numbers: Integers, fractions, or decimals that Terminate (end) OR a Repeat decimal… 12, ½, .363636.. , 2 ¾ , -15, 0 Integers: Positive and Negative Whole Numbers Integer Rules: Multi or Divide: − 𝑥 − = + + 𝑥 += + − 𝑥 + = − + 𝑥 − = − Add or Subtract: Change to addition and PUT ON the correct side of Alge mat and Zero pair off; record what is left on the mat on which side. Irrational Numbers: Anything that doesn’t end or repeat… √2 , √3 , √ 5 , √7 , √ 8 , π, etc………. ORDER OF OPERATIONS (Please Excuse My Dear Aunt Sally) Step 1 Do Parentheses ( ) Step 2 Do Exponents / Powers / Square Roots Step 3 Do Multiplication or Division MUST DO In order from left to right Step 4 Do Addition or Subtraction MUST DO In order from left to right SOLVING EQUATIONS: 1. Change ALL subtraction to addition (if need) 2. Do the Distributive Property (If needed) 3. Combine Like TERMS on EACH side 4. Get the VARIABLE on one side (do the INVERSE to remove) 5. SOLVE 6. CHECK your answer (substitute in problem) SOLVING RATIOS/ PROPORTIONS/ FRACTIONS & PROBABILITY 1. Look for a RULE BETWEEN Fractions/Ratios/Proportions 2. REDUCE fraction/ratio/proportion to LOWER TERMS to get RULE 3. CROSS MULTIPLY and solve using steps above for SOLVING EQUATIONS FRACTION RULES: Addition and Subtraction: May Change to same denominator or Butterfly Method ⅚ + or - ⅞ = Multiplication: multiply straight across Division: Flip / Reciprocal/Inverse of 2nd fraction THEN multiply across or ZIG/ZAG by multiplying. Improper fraction: 24/4 = 6; 36/7 = 5 1/5 (divide) Mixed Fraction: 5 ¾ = 5.75 = 575% = 23/4 SLOPE Line rises to Rt = positive slope Line horizontal = 0 slope ** m is the slope Line rises to Lt = negative slope Line vertical = undefined slope b is the y- intercept (hits the y axis)** Standard Equation: Ax + By = C 𝐲 = 𝐦𝐱 + 𝐛 Slope-Intercept Equation: Slope of a line using points: Point/Slope form: 𝒚₂ − 𝒚 ₁ x₂ − 𝒙₁ m= y - y1 = m( x – x1) Slope of a Line: rise (y) run (x) Table: In (x) Out (y) -2 -4 -1 -2 0 0 Slope = 2 (Y) 1 (X) To find the y-intercept (b): (x , y) put a zero in for x (0, y) and solve the equation given; will get y = To find the x-intercept: (x , y) put a zero in for y (x, 0) and solve the equation given; will get x = Perpendicular Slope Equations MUST have OPPOSITE AND FLIPPED slope and may have the same y- intercept or b Parallel Slope Equations MUST have SAME SLOPE, BUT different y-intercept or b (𝒇 𝑶𝑭 𝒙) 𝒇 (𝒙) = 𝟑 𝒙 + 𝟗 𝒇(𝒙) = 5 𝒚 = 𝟑 (𝟓) + 𝟗 … 𝒚 = 𝟏𝟓 + 𝟗 𝒚 = 𝟐𝟒 Coordinate Pair would be ( 𝟓 , 𝟐𝟒 ) Arithmetic Sequence Geometric Sequence f(x) = mx +b f(x) = a(b) or f(x) = a² PROPERTIES OF OPERATIONS Commutative Property: Numbers can change order 3 +* 4 = 4 +* 3 and problem answer doesn’t change. Associative Property: Parentheses change order or one set leaves: ( 3+*6) +* ( 4 +*9) = 3 +*( 6 +* 4 ) + * 9 = Distributive Property: a letter or # is given to the inside (): 6 ( 4a + 5b) = 24a + 30b OR a letter or # is removed to the outside of ( ): 6x + 8x = x (6+8) RANGE: Maximum - Minimum MEAN: (Average) Add up data and divide by # of data MODE: Most repeated data MEDIAN: Put data in order and find the MIDDLE < is less than or equal . > is greater than or equal . < is less than o > is greater than o CAPACITY OR FLUID OUNCE 8 fl oz = 1c 32 fl oz = 4c = 2 pt = 1 qt 128 fl oz = 16c = 8pt = 4 qt = 1 gal WEIGHT 16 oz = 1 pound (lb) 2,000 lb = 1 ton DISTANCE 12 in. = 1 ft 36 in. = 3 ft = 1 yd 5,280 ft = 1 mi TIME 60 sec = 1 min; 60 min = 1 hr; 24 hrs = 1 day 365 days = 1 year 10 years = 1 decade 100 years = 1 century TEMPERATURE F = Fahrenheit C = Celsius Water Freezes 32 ⁰ 0⁰ Normal Body 98.6 ⁰ 37 ⁰ Water Boils 212 ⁰ 100 ⁰ CONVERSIONS 1 oz ≈ 28g 1.06 qt ≈ 1 L 1 gal ≈ 3.79 L 1 in. ≈ 2.54 cm 1 K ≈ .62 mi 1 tsp ≈ 5mL 2.2 lb ≈ 1 Kg 1 mi ≈ 1.6 Km METRIC k = kilo = 1,000 meters, grams, Liters h = hecto = 100 meters, grams, Liters da = deca = 10 meters, grams, Liters d = deci = 0.1 meters, grams, Liters c = centi = 0.01 meters, grams, Liters m = milli = 0.001 meters, grams, Liters GEOMETRY POLYGONS Line = 180° Triangle 3 SIDES = 180 ⁰ Quad 4 SIDES= 360 ⁰ Perpendicular Lines make a 90◦ Total Degrees in Polygon RULE = 180 ( n – 2) n = number of sides Scalene = NO sides same Isosceles = 2 sides same Equilateral = 3 sides same Acute ∆ = ALL angles acute Obtuse ∆ = 1 angle obtuse Rt ∆ = 1 angle right ANGLES Supplementary Angles = 180° Complementary Angles = 90° Adjacent Angles = share a side Vertical Angles = are diagonal from each other and have = degrees PERIMETER (unit) = ADD all the sides of polygon; CIRCUMFERENCE (Circle perimeter) C = d 𝜋 or 2r𝜋 (d=diameter: across full circle through the center; r=radius: from center to one side of circle) AREA (sq unit or unit 2) Triangle A = 1/2hb or hb/2 Rectangle / Parallelogram / Square A = bh Trapezoid A = ½ h (b + b) Circle 𝐴 = 𝜋𝑟 2 SURFACE AREA (sq. unit or unit 2) Find the AREA of EACH Shape …. then ADD together Cylinder SA = (𝜋𝑟 2 ∗ 2) + (d𝜋 ∗ ℎ) VOLUME (cubic or unit 3) Cube 𝑉 = 𝑠³ 1 Triangular Prism 𝑉 = 𝐵 ℎ (B = lw) 2 Rectangular Prism 𝑉 = 𝐵ℎ (B=lw) 2 Sphere V = 𝜋 𝑟 4/3 Cone 𝑉 = 𝜋 𝑟 2 ℎ 1/3 Cylinder V = 𝜋 𝑟 2 h Pyramid 𝑉 = 𝐵ℎ1/3 (B=lw) Pythagorean Theorem: RIGHT TRIANGLE ONLY; Also, DISTANCE of two points on a GRAPH 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 a & b Must be the legs/sides that form RT ANGLE!