8th Grade Math Cheat Sheets PDF

REAL NUMBER SYSTEM Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ CLASSIFYING REAL NUMBERS A number means what square...

REAL NUMBER SYSTEM Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ CLASSIFYING REAL NUMBERS A number means what square number times itself will give you the number inside roots REAL NUMBERS the square root? D E Estimate non-perfect squares by using C a number line. B For example, to find 80, find the two A closest perfect squares. 64 81 A. NATURAL: positive counting numbers starting with 1 8 9 B. WHOLE: positive counting numbers starting with 0 The number line above shows it would be slightly less than 9, or 8.9. C. INTEGER: whole numbers and their opposites D. RATIONAL: fractions, repeating decimals and If given the area of a square, the side length of the square can be found by terminating decimals taking the square root of the area. E. IRRATIONAL: non-terminating or non-repeating AREA  AREA decimals, and non-perfect squares FRACTIONS TO DECIMALS: FRACTIONS & DECIMALS 2 13 4 1 9 = 0.2… 99 = 0.13… 33 = 0.12… 11 = 0.09 DECIMALS TO FRACTIONS: scientific 0. 4 4= 9 0.47 = 47 99 5 0.15 = 33 6 0.54 = 11 notation Put numbers in the same form first! ordering numbers For example, convert numbers to decimals before ordering. LEAST TO GREATEST: INCREASING, ASCENDING GREATEST TO LEAST: DECREASING, DESCENDING ©Maneuvering the Middle LLC, 2017  Name __________________________________ EXPONENTS AND SCIENTIFIC NOTATION CHEAT SHEET - A Date __________________________Pd______ EXPONENT rules MULTIPLYING LIKE DIVIDING LIKE BASES A POWER TO A POWER INTEGER EXPONENTS BASES xa (xa)b = xa · b 1 a–b x-n = xa · xb = xa+b xb = x xn 76 1 1 EX: 22 · 23 = 25 or 32 EX: 74 = 72 or 49 EX: (22)3 = 26 or 64 EX: 7-2 = 72 or 49 SQUARE: Raise a number to a power of 2 CUBE: Raise a number to a power of 3 SQUARE & CUBE SQUARE ROOT: Inverse of squaring a number CUBE ROOT: Inverse of cubing a number SQUARES SQUARE ROOTS CUBES CUBE ROOTS ROOTS 43 = 64 3 92 = 81 81 = 9 64 = 4 (-103) = -1,000 3 (-112) = 121 121 = 11 1,000 = 10 2 4 4 2 2 8 3 8 2 (3)2 = = (3)3 = = 9 9 3 27 27 3 ADDING/SUBTRACTING: Numbers must have the Scientific notation same power of 10. Add or subtract the numbers in front and keep the power of 10. A value written as the product of a number between 1-10 and a power of ten. (5.3 x 104) + (2.6 x 104) 8.52 x 106 (5.3 + 2.6) x 104 Between Power of 7.9 x 104 1-10 10 MULTIPLYING/DIVIDING: Rearrange the problem SCIENTIFIC TO STANDARD and apply your exponent rules. 4.235 x 107 = 42,350,000 (1.2 x 103)(3 x 106) *Remember to check 6.88 x 10-7 =.000000688 the sign of (1.2 x 3)(103 x 106) the STANDARD TO SCIENTIFIC exponent! 3.6 x 109 129,200 = 1.292 x 105.000097 = 9.7 x 10-5 Operations with sci. not. ©Maneuvering the Middle LLC, 2017 FUNCTIONS AND SLOPE Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ Distance vs. time graphs functions Moving away at a FUNCTION: A relationship in which every input (x) constant speed has exactly one output (y). TO CHECK IF IT’S A FUNCTION: Moving closer at a ORDERED PAIRS & TABLES: Each x-value must constant speed correspond with exactly one y-value. Check for repeating x-values. EQUATIONS: See if any input would result in At rest; not changing more than one output. For example, y2 = x could distance result in ± y. GRAPHS: Must pass the “vertical line test”, where any vertical line touches the graph at only Moving away and one point. increasing speed slope Also called the “RATE OF CHANGE” Moving away and decreasing speed POSITIVE: Increases left to right NEGATIVE: Decreases left to right ZERO: A horizontal line UNDEFINED: A vertical line 5 x y 4 0 -2 THE 3 FORMULA: 2 1.4 2 RISE y2 – y1 1 4 4.8 x2 – x1 1 2 3 4 5 6 8.2 or RUN y2 – y1 The slope of the graph The slope of the table RISE is -3/2. is 1.7. RUN x2 – x1 (2, 6) and (-2, 7) The slope between the ordered pairs is -0.25. Triangles on the same line have the same SLOPE and are SIMILAR triangles. The ratios of their corresponding sides are EQUAL. FINDING slope ©Maneuvering the Middle LLC, 2017 LINEAR RELATIONSHIPS Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ Writing equations y −y Slope-intercept form 1. Find the SLOPE by using x2 −x1 2 1 2. Find the Y-INTERCEPT by finding the y = mx + b value of y when x = 0, or where a graphed line crosses the y-axis. 3. Write an equation in SLOPE-INTERCEPT SLOPE Y-INTERCEPT FORM. (y = mx + b) EXAMPLE 1: A line with a slope of 8 and a y-intercept of 5 -10 would have an equation of y = 8x – 10. 1 4 SLOPE: 2 An equation of y = -3x + 5 means the slope of 3 Y-INTERCEPT: 1 the line is -3 and the y-intercept is 5. 2 1 EQUATION: y = 2x + 1 1 Linear relationships can be represented verbally, 1 2 3 4 5 with an equation, with a graph and with a table. View the example below. EXAMPLE 2: x y VERBAL: A puppy weighs 2 pounds at SLOPE: 4 0 -9 birth and gains half a pound each week. Y-INTERCEPT: -9 2 -1 EQUATION: y = 0.5x + 2 EQUATION: y = 4x – 9 4 7 Multiple representations TABLE: Weeks (x) Weight (y) 0 2 If an equation is linear, it will be written in 1 2.5 the form of Y = MX + B. 2 3 LINEAR: y =.75x – 11 3 3.5 NON-LINEAR: y = 3x2 + 2 GRAPH: If a graph is linear, it will look like a 5 STRAIGHT LINE. 4 3 LINEAR NON-LINEAR 2 1 Is it 1 2 3 4 5 linear? ©Maneuvering the Middle LLC, 2017 LINEAR EQUATIONS Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ Parts of an equation Multi-step equations -12x + 4 = 40 STEPS TO SOLVE: 1 Distribute (if necessary) 2 Combine LIKE TERMS (if necessary) COEFFICIENT CONSTANT 3 Collect VARIABLES on the same of the equal VARIABLE sign 4 Collect CONSTANTS on the same side of the COEFFICIENT: the number in front of a variable equal sign (multiplying the variable) 5 Isolate the VARIABLE with inverse operations CONSTANT: a fixed value, or a number on its own VARIABLE: a letter used to represent an unknown Check your answer by PLUGGING IT IN! value NO SOLUTION: An equation where NO value for x will make the equation true. SPECIAL Work ends in a false statement, such as “7 = 5”. CASES Graph shows two PARALLEL lines. ALL REAL NUMBERS: An equation where ANY value for x will make the equation true. Work ends in a TRUE statement, such as “7 = 7”. Graph shows the SAME line. To solve a system SOLVING SYSTEMS 1 Solve one of the equations for BY SUBSTITUTION “y = ”, if necessary. of equations by graphing, graph 2 Substitute the value for “y” in the second equation. both linear equations and find 3 Solve the new equation to find “x”. the point of 4 Substitute the value for “x” in either INTERSECTION. of the original equations to find “y”. SOLVING Systems by graphing ©Maneuvering the Middle LLC, 2017 ANGLE RELATIONSHIPS Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ Parallel lines and transversals When parallel lines are cut by a transversal, 8 different angles are formed. An angle labeled “A” is CONGRUENT to any other angle labeled “A”. A B B A An angle labeled “B” is CONGRUENT to any other angle labeled “B”. A B An angle labeled “A” is SUPPLEMENTARY to any angle labeled “B”. B A Types of Angle pairs INTERIOR ANGLES: The sum of the three interior angles in any CORRESPONDING ANGLES triangle will add up to 180°. A -In the same relative ∠A + ∠B + ∠C = 180° position A ∠B -CONGRUENT angles ∠A ∠C ALTERNATE EXTERIOR ANGLES EXTERIOR ANGLES: Any exterior angle of a triangle is equal to the -Opposite sides of A sum of its two remote interior angles. transversal and outside parallel lines ∠D = ∠B + ∠C ∠B A -CONGRUENT angles ∠D ∠C ALTERNATE INTERIOR ANGLES If two triangles have TWO pairs of -Opposite sides of A corresponding angles that are CONGRUENT, the transversal and inside parallel lines triangles are SIMILAR. D A B 80° -CONGRUENT angles 80° 32° VERTICAL ANGLES A 32° C E F A -Opposite angles formed A TRIANGLE ABC ~ TRIANGLE DEF by intersecting lines -CONGRUENT angles Angle-angle criterion ©Maneuvering the Middle LLC, 2017  PYTHAGOREAN THEOREM Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ Parts of right triangles The Pythagorean theorem c a In any RIGHT triangle, the SUM of the SQUARES of the shorter sides (a and b) will equal the SQUARE of the longest side, c. In other words… b LEGS: the two sides touching a2 + b 2 = c 2 the right angle, known as “a” and “b”. To visualize the theorem, picture “a2”, “b2” and “c2” as actual squares with side lengths equal to the side lengths HYPOTENUSE: the side opposite of a, b and c: the right angle, known as “c”. The AREAS of the two The hypotenuse is always the smaller squares (a2 + b2) 3 25 units2 longest of the three sides. will always equal the AREA of the largest 5 9 units2 3 5 square (c2). 4 The converse states that IF Pythagorean converse 32 + 42 = 52 a2 + b2 = c2, then the triangle 9 + 16 = 25 4 16 units2 is a RIGHT triangle. 25 = 25 EXAMPLE 1: Can the lengths 16, 30 and 34 make a right triangle? Plug it into the theorem: The coordinate plane The Pythagorean Theorem can be used to find the 162 + 302 = 342 diagonal distance between points on a graph. Create a 256 + 900 = 1156 RIGHT triangle where the diagonal is c, the HYPOTENUSE. 1156 = 1156 Yes! 102 + 72 = c2 A EXAMPLE 2: 100 + 49 = c2 149 = c2 Can the lengths 12, 12 and 24 make a right triangle? Plug it 10 c 149 = c into the theorem: 12.2 = c The distance from 122 + 122 = 242 B point A to point B is 7 about 12.2 units. 144 + 144 = 576 288 ≠ 576 No! ©Maneuvering the Middle LLC, 2017 VOLUME Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ 1 Volume of cylinders FORMULA: V = Bh 3 h Multiply the AREA OF THE BASE by the HEIGHT of the cylinder. BASE *If given the diameter of the base, divide by 2 to find the radius. B = Area of the base (𝜋𝜋r2) Volume Volume of of r cones h spheres BASE B = Area of the base (𝜋𝜋r2) 1 4 3 FORMULA: V = Bh FORMULA: V = 𝜋𝜋r 3 3 Multiply the AREA OF THE BASE by the HEIGHT 4 Multiply times PI by times the RADIUS to the 1 3 of the cone. Then multiply by 3 , or divide by 3. THIRD power. *If given the diameter of the base, divide by 2 *If given the diameter of the sphere, divide by 2 to find the radius. to find the radius. When given the volume of a 3D figure, use the formula to find missing pieces of information. Working backwards Ex. 1: A cylinder has a volume of 5,024 inches3. Ex. 2: A cone has a volume of 4,710 inches3. If If the height of the cylinder is 16 inches, find the the height of the cone is 20 inches, find the radius of the cylinder. radius of the cone. V = 3.14(r2)(h) V = (1/3)(3.14)(r2)(h) 5,024 = 3.14(r2)(16) 4,710 = (1/3)(3.14)(r2)(20) 314 = 3.14(r2) 14,130 = (r2)(20) 100 = r2 4,500 = (r2)(20) 10 = r 225 = r2 15 = r ©Maneuvering the Middle LLC, 2017 TRANSFORMATIONS Name __________________________________ CHEAT SHEET - A Date __________________________Pd______ TRANSFORMATIONS PRE-IMAGE: ORIGINAL (A) IMAGE: NEW (A’) TRANSLATIONS REFLECTIONS ROTATIONS DILATIONS A B B’ C’ A’ D’ C d To SLIDE each point of a To FLIP a figure over a To TURN a figure around To ENLARGE or REDUCE a figure the same distance line of reflection, a point of rotation figure and same direction creating a mirror image (clockwise or counterclockwise) Algebraic representations TRANSLATION (RIGHT/UP) (x + n, y + n) Dilations & scale factor TO FIND SCALE FACTOR: TRANSLATION (LEFT/DOWN) (x – n, y – n) NEW IMAGE ORIGINAL OR PRE−IMAGE RELFECTION (OVER X-AXIS) (x, -y) REFLECTION (OVER Y-AXIS) (-x, y) SCALE FACTOR >1: Enlarges figure ROTATION (90° CLOCKWISE) (y, -x) SCALE FACTOR