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Math Review Pg. # 1 of 10 Math Review (8th) 1. Exponents: Example 1.) Example 2.) Example 3.) Example 4.) Example 5.) Example 6.) Base →32← Exponent 31 = 3 (any number to the 1st power equals itself) 32 = 3 • 3 = 9 33 = 3 • 3 • 3 = 27 30 = 1 (any number to the zero power equal 1) 3-1 = 1/3 (any number to a negative power equals 1 over the number to the positive power) -2 3 = 1/32 = 1/9 2. Square Root: One of the two equal factors of a number The √ symbol is called a radical sign. Perfect squares are the squares of whole numbers. o Example: The numbers 4, 9, 16, 25, 36, 49, and 64 are examples of perfect squares (This is because 22, 32, 42, 52, 62, 72, 82, etc.) Negative numbers cannot have a square root because there is no number that can be multiplied by itself to produce a negative number. (A negative times a negative is a positive.) √9 =3, √169 = 13, √30.25 = 5.5, √2 = 1.414213562 3. Absolute Value: always gives a positive value of the number Example 1. ) │-3│ = 3 and │3│ = 3 Example 2.) │5│ = 5 and │-5│ = 5 Example 3.) │-9 + 3│ = │-6│ = 6 4. Additive Inverse: Example 1.) Example 2.) Example 3.) the opposite of a number the additive inverse of -1 is 1 the additive inverse of ¼ is -¼ the additive inverse of 5 is -5 5. Multiplicative Inverse: (means reciprocal) Example 1.) the multiplicative inverse of 5 (or 5/1) is 1/5 Example 2.) the multiplicative inverse of 3/8 is 8/3 Example 3.) the multiplicative inverse if -¼ is -4/1 or simply -4 Math Review Pg. # 2 of 10 6. Sets of Numbers: Natural Numbers – the counting numbers {1, 2, 3, 4, 5, 6, 7, 8,…} Whole Numbers – zero plus the counting numbers {0, 1, 2, 3, 4, 5, 6, 7, 8,…} Integers – the positive and negative whole numbers {…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…} Rational Numbers - A number that can be put into fraction form. Example. { ½, 5 = 5/1, 0, -20} A decimal that ends (terminates) or repeats in a pattern. Example. { -4.1, 3.5, 4.1212, .25, √25 = 5} Irrational Numbers - Numbers which never end and never repeat in a pattern. Example. { √2 = 1.414213562… or π = 3.141592654…} 6. Roman Numerals: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000 7. Prime or Composite: Prime – a whole number greater than one whose only factors are 1 and itself. The first seven prime numbers are {2, 3, 5, 7, 11, 13, 17} Composite – a whole number greater than one that is not prime. The first seven composite numbers are {4, 6, 8, 9, 10, 12, 14} 8. Greatest Common Factor (GCF): Greatest # that can fit into both #’s. Example 1.) factors of 32: 1, 2, 4, 8, 16, 32 factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The GCF of 32 and 24 is 8. 9. Least Common Multiple (LCM): Smallest # that all #’s have in common if you skip count. Example 1.) multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 Multiples of 12: 12, 24, 36, 48, 60, 72 The LCM of 5 and 12 is 60 10. Prime Factorization: Example 1.) Answer: 22 x 3 x 5 60 6 2 10 3 2 5 ← Factor until you get all primes Math Review Pg. # 3 of 10 11. Properties of numbers: Commutative Property: a+b=b+a or a•b=b•a (the order of addition or multiplication does not affect the answer) Associative Property: (a + b) + c = a + (b + c) or (a • b) • c = a • (b • c) (the grouping of addition or multiplication does not affect the answer) Distributive Property: a(b + c) = ab + ac or a(b - c) = ab – ac (individual multiplication by a group of items in a set of parentheses) Additive Identity Property: a + 0 = 0 + a = a (Any real number plus 0 is the original number) Multiplicative Identity Property: a • 1 = 1 • a = a (Any real number times 1 is the original number) Additive Inverse Property: a + (-a) = -a + a = 0 (The sum of a real number and its opposite is zero) Multiplicative Inverse Property: a • 1/a = 1/a • a = 1 (The product of a nonzero real number and its reciprocal is 1) Multiplication Property of Zero: a • 0 = 0 (any number times zero equals zero) 12. Rules for Operations on Signed Numbers Addition: 1. If the signs are the same – keep the sign and add the numbers. 2. If the signs are different – subtract the smaller number from the larger number and keep the sign of the larger number. Subtraction: 1. Add the opposite of the second number (Change the sign of the number following the minus sign, then follow the rules of addition.) Multiplication and Division: 1. If multiplying or dividing two numbers of the same sign, the answer is positive. 2. If multiplying or dividing two numbers of different signs, the answer is negative. Math Review Pg. # 4 of 10 13. Geometry: Angles: Supplementary angels – 2 angles whose sum is 180° Complementary angels – 2 angles whose sum is 90° Acute angle – an angle which is greater than 0° but less than 90° Right angle – a 90° angle Obtuse angle – an angle which is greater than 90° but less than 180° Straight angle – a 180° angle Angles formed by parallel lines and a transversal Vertical angles are congruent (2 and 3) (1 and 4) 1 2 3 4 5 7 6 8 Alternate interior angles are congruent: (3 and 6) or (4 and 5) Alternate exterior angles are congruent: (1 and 8) or (2 and 7) Corresponding angles are congruent: (2 and 6) or (4 and 8) (1 and 5) or (3 and 7) Parallel lines: Lines that extend infinitely without intersecting {symbol ║} Example: AB ║ CD A B C D Perpendicular lines: Two lines that form 90° angles to each other. {symbol ┴} JK ┴ LM K L M J Bisect: To cut into 2 equal pieces Similar Figures: Same shape, different size. (the sides can change, but the angles remain the same) (the sides of similar triangles are always in proportion to each other.) Similar Triangles (Not Congruent) Math Review Pg. # 5 of 10 Congruent: Same exact size and shape. Congruent Trapezoids 14. Classifying Polygons: Polygon – a closed figure formed by three or more line segments Regular Polygons – a polygon whose sides and angles are all equal Naming Polygons Triangle – 3 sides Pentagon – 5 sides Heptagon – 7 sides Nonagon – 9 sides Quadrilateral – 4 sides Hexagon – 6 sides Octagon – 8 sides Decagon – 10 sides 15. Triangles: Triangle – a polygon with exactly 3 sides The sum of the angles of a triangle equals 180° A triangle cannot have more than one 90° angle A triangle cannot have more than one obtuse angle Classification of triangles by angles Acute triangle: has three acute angles Right triangle: has 1 right angle Obtuse triangle: has 1 obtuse angle Classification of triangles by sides Scalene triangle: no congruent sides Isosceles triangle: has 2 congruent sides Equilateral triangle: has 3 congruent sides 16. Classifying Quadrilaterals Quadrilateral: A polygon with 4 sides and 4 angles Parallelogram Opposite sides are parallel Opposite sides are congruent Trapezoid Exactly 1 pair of parallel sides Rectangle Opposite sides are parallel Opposite sides are congruent All 4 angles are right angles Opposite sides are parallel All 4 sides are congruent Rhombus Square Opposite sides are parallel All 4 sides are congruent All 4 angles are right angles Math Review Pg. # 6 of 10 Quadrilateral Parallelogram Rectangle Trapezoid Rhombus Square 17. Area and Perimeter Notes Perimeter: The total distance around the outside edge of a geometric figure. To find the perimeter of any shape, add up the lengths of all of its sides. Area: The total number of square units enclosed by a geometric figure. Area Formulas: Rectangle A = bh base x height or Square A = bh base x height or Parallelogram A = bh base x height Rhombus Triangle A = lw length x width A = s2 side squared A = bh base x height A = ½ bh ½ x base x height or A = bh 2 (base x height) ÷ 2 b1 Trapezoid b2 a = (b1 + b2)h 2 (base 1 x base 2 x height) ÷ 2 Math Review Pg. # 7 of 10 18. Circles Circumference (C) – the distance around a circle Diameter (D) – the distance across a circle through its center Radius (r) - the distance from the center to the edge of a circle Pi (π) – the ratio of the circumference to the diameter (the value of π is approximately 3.14 or 22/7) Chord – a line segment with endpoints on the circle Arc – a part of the circumference of a circle Formulas for Area and Circumference of a circle: Circumference (C) C = πd Remember: Cherry Pie’s Delicious (Circumference equals pi x diameter) Area (A) A = πr2 Remember: Apple Pies aRe 2 (Area equals Pi x radius squared) 19. Coordinate Plane: 1. Coordinate System – a system used to precisely locate points on a geometric plane. It is formed by two intersecting number lines. 2. x-axis – the horizontal number line 3. y-axis – the vertical number line 4. Origin – the place where the two axes intersect. (0,0) 5. Quadrants – the four sections of the coordinate plane formed by the two intersecting axes. Quadrant II Quadrant III Quadrant I Quadrant IV Math Review Pg. # 8 of 10 6. Ordered Pair – any location on the coordinate plane can be represented by an ordered pair of numbers called coordinates. The coordinates are always in the order (x,y). 7. x-coordinate (abscissa) – The first number in an ordered pair 8. y-coordinate (ordinate) - The second number in an ordered pair 20. Reflections and Symmetry Transformation: A movement of a geometric figure Reflection: The image is flipped (mirrored) over a line Translation: The sliding of an image Dilation: The enlargement or reduction of an image Rotation: Turn Reflection in the x-axis Reflection in the y-axis Translation: slide Dilation (enlargement) {the other type of dilation is a reduction} Lines of symmetry through a geometric figure A Vert. B Horiz. C Horiz. D Horiz. E F Horiz. None X Both Z None Rotational Symmetry – if a figure can be rotated less than 360° and still look like the original figure. Math Review Pg. # 9 of 10 21. Scientific Notation: A way of expressing a number as the product of a number that is at least 1 but less than 10 and a power of 10. 1.) 678,000 2.) .000543 3.) 4,805,000 = = = 6.78 x 105 5.43 x 10-4 4.805 x 106 4.) 90,302,000,000,000 5.) .00000002 6.) .000702 *** When writing a number in scientific notation: The decimal point always goes after the first digit The answer should always be in the format: → The exponent is positive for numbers greater than 1 (ex. 324,000,000) The exponent is negative for numbers less than 1. (decimals ex. .0000243) = = = 9.0302 x 1013 2.0x10-8 7.02 x 10-4 __.__ __ x 10# 22. Order of Operations: 1 2 3 4 Rules for Order of Operations (P E MD AS) FIRST: Do all operations within parentheses. SECOND: Do all powers or exponents. THIRD: Working from left to right, do all multiplication and division. FOURTH: Working from left to right, do all addition and subtraction. 23. Pythagorean Theorem: (used to find the missing side of a right triangle when 2 sides are given) a2 + b2 = c2 a and b are legs c is always the hypotenuse c a b 24. Trigonometry of a Right Triangle: (can be used to find a missing angle or side of a right triangle) SOH CAH TOA SIN = opposite hypotenuse COS= adjacent hypotenuse TAN= opposite adjacent 25. Percent Problems: Part (is) = Percent (%) Whole (of) 100 1.) 42 is 50 percent of what number? 2.) 3.75 is what percent of 75? 3.) What is 12 ½ percent of 88? 4.) 12 is 30% of what number? 5.) 19 is what percent of 95? 6.) What percent of the area of the square is shaded? Math Review Pg. # 10 of 10 26. Estimation of percent 25% = ¼ 50% = ½ 20% = 1/5 40% = 2/5 33 1/3% = 1/3 66 2/3% = 2/3 75% = ¾ 60% = 3/5 10% = 1/10 80% = 4/5 27. Central Tendency Mean – the average (add up items and divide by the # of items) Median – the middle number when arranged in size order. If there is an even number of elements you must average the two middle elements. Mode – the item that occurs the most. 28. Proportions: An equation where the two ratios are equal. Example 1.) 2 = 8 In a proportion the cross products are equal (2 x 12 = 8 x 3) 3 12 29. Graphing inequalities An inequality is a mathematical sentence that contains the symbols {<, > ,≤, or ≥} When graphing inequalities use the following notation: ● Solid Endpoint means included example {≤ or ≥} ○ Hollow Endpoint means not included example {< or >} → Arrow endpoints extend infinitely Example 1. x≤3 Example 2. x>2 Example 3. -5 ≤ x < 2 ←|----|---|---|---|---|---|---|---|---●---|---|---|→ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ←|----|---|---|---|---|---|---|---○---|---|---|---|→ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 ←|---●---|---|---|---|---|---|---○---|---|---|---|→ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 30. Solving inequalities: To solve an inequality follow the same procedures as used in solving equations, with the following exception: When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality. Example 1 Example 2 Example 3 3x + 2 > 20 -5x – 4 > 21 -7x ≤ 28 -2 -2 +4 +4 3x 3 x > 18 3 > 6 -5x -5 x > 25 -5 < -5 -7x ≤ 28 -7 -7 x ≥ -4