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7th Grade Math – Semester 2 Exam Study Guide Chapter 6 -- Percents Percent ratio that compares a # to 100 o percent is a relative value not an absolute value same % of different values gives different absolute #s o % expresses parts per 100 or “/100” Model a representation that helps you visualize the “real thing” Converting % to decimal move decimal point 2 places to left Converting decimal to % move decimal point 2 places to right % to fraction put % value over 100, then simplify. o If % < 1, write it over 100, then X by how much it takes to make numerator a whole # 2 ratios are proportional if their cross-products are equal model a representation that helps you visualize the real thing o models help to identify how to write proportion (p311) formula: part / whole = n% / 100% o hint in a statement…e.g. “what is 39% of 120”…what comes after the “of” represents the WHOLE to multiply by a %, turn it into a decimal first Sales tax = tax rate * purchase price per cent means per 100 or divide by 100…to do this, move decimal 2 places to left Chapter 7 -- Geometry Geometry Basic Concepts (7-1 to 7-2) Point in spot or location. Has no size. Named with capital letter. Line series of points extending to ∞ in both directions. Named by two points on it. Ray one endpoint and extends to ∞ in other direction. Named by endpoint and another point. Line segment part of a line with 2 endpoints. Plane flat surface extending to ∞ in all directions Intersecting lines 2 or more lines with exactly 1 point in common. Parallel lines lines on same plane that never intersect. Perpendicular lines lines that intersect and form right angle (90°). Skew lines lines on different planes…never intersect nor are || Acute between 0 and 90°. Right 90°. Obtuse between 90° and 180° Straight angle 180°. Complementary ∑ of angles is 90° Supplementary ∑ of angles is 180° Adjacent angles 2 angles next to each other, share vertex…supplementary. Vertical angles formed by 2 intersecting lines and are opposite each other…have = measures. Congruent angles angles with equal measures Triangles Scalene no congruent sides Isosceles at least 2 congruent sides Equilateral 3 congruent sides Right 1 right angle Acute 3 acute angles Obtuse 1 obtuse angle ∑ of all angles in a triangle = 180° Polygons (7-5 to 7-7) Polygon closed plane figure formed by 3 or more line segments Regular polygon all sides and angles congruent Irregular polygon all sides and angles not congruent Quad 4 Lateral lines Penta 5 Hexa 6 Octa 8 Deca 10 Trapezoid quadrilateralwith exactly 1 pair of || sides Parallelogram quadrilateral w/ 2 pairs of || sides Rhombus || gram with 4 congruent sides Rectangle || gram with 4 right angles Square || gram w/4 right angles and 4 congruent sides congruent polygons have same size and shape…corresponding parts (angles and sides) are congruent # triangles in a polygon = # sides of polygon – 2 ∑ of all angles in the polygon can be found by multiplying # of triangles that can be made in the polygon by 180° Circles (7-8) circle set of points in a plane that are all the same distance from a given point (center). Diameter segment that passes through center and has both endpoints on circle Radius segment connecting center to the circle o radius = ½ diameter Central angle angle with its vertex at the center of a circle Chord segment w/bot end points on circle Arc curve connecting 2 points on a circle Semi-circle arc that is ½ circle 7th Grade Math – Semester 2 Exam Study Guide page 4 of 8 Chapter 8 -- Geometry and Measurement Area of parallelograms and triangles area there are 2 dimensions in measuring area…length and width or base and height o area is always stated in units squared (units2) because of the two dimensions Area of parallelogram = bh … the base x height Area of triangle = ½ bh … the area of a triangle is exactly one-half of a parallelogram Area of trapezoid = ½ h (b1 + b2) Circumference and Area of Circles (8-4) …first understood by Archimedes as ratio of circumference and the radius of a circle Circumference of a circle C = d Diameter of a circle d = 2r Area of a circle A = r2 Square roots and irrational #s (8-5) area of a square = side x side (s2) perfect square product of the square of an integer square root inverse of squaring a #… when we use the sq root symbol, we are referring to the non-neg sq root of a number (remember… 2 negatives multiplied together are positive) sq root of fraction take sq root of numerator and denominator Irrational #s non-terminating, non-repeating #s…cannot be written as ratio of two integers. Pythagorean Theorem (8-6) Right triangle sides that form right angle are called legs. Side opposite right angle is hypotenuse. Pythagorean Theorem – sum of the squares of the lengths of the legs is = to the square of the length of the hypotenuse. o Formula a2 + b2 = c2 o a and b are the legs of the right triangle, c is the hypotenuse Three-dimensional figures (8-7) 3-dimensional figure a solid figure with length, width, height that does not lie on a plane height measured as perpendicular length from one base to another face flat surface on 3D figure prism 3D figure w/2 parallel and congruent faces, called bases… prism is named based on its bases cylinder 3D w/2 || bases that are circles pyramid 3D w/triangular faces that meet at a point (vertex) and a base that is a polygon 7th Grade Math – Semester 2 Exam Study Guide cone circular base and one vertex sphere set of all points in space equidistant from a center. page 5 of 8 7th Grade Math – Semester 2 Exam Study Guide page 6 of 8 Volume of Rectangular Prisms and Cylinders (volume = area of base x height) volume number of cubic units (units3) needed to fill the space inside a 3-dimensional figure volume of rectangular prism lwh … or area of base (lw) x height volume of cylinder r2 h … or area of base ( r2) x height Chapter 9 graphs used to visually represent relationship between 2 or more sets of numbers / data scale ratio usually label the dimensions of a graph (axes) as X and Y interval pattern between values on a scale sequence set of numbers that follow a pattern term each # in a sequence arithmetic sequence sequence created by adding a fixed # to the previous one geometric sequence sequence created by multiplying a fixed # to the previous one Functions (9-4 to 9-6) function a relationship that assigns exactly 1 output value for each input Graphing functions (9-5) graphs typically use Y as vertical and X as horizontal axes points on graph labeled as (x,y) origin pt is (0,0) to graph, create X-Y table and calculate at least 2 points linear function (variable X is power of 1) – graphs a straight line parabolic function (X2) 7th Grade Math – Semester 2 Exam Study Guide page 7 of 8 Chapter 10 – Graphing in the Coordinate Plane points on graph labeled as (x,y)…this is called an ordered pair 4 quadrants in coordinate plane labeled I, II, III, IV Y is vertical axis and X is horizontal axis Linear equations o y = mx + b b where the line crosses the y axis m slope (steepness of the line) rise over run (rise / run) ∆y / ∆x o you can calculate slope from 2 points by: (y2 – y1) / (x2 – x1) if m is positive line is ascending if m is negative line is descending you can graph a line 2 ways: 1. make an x y table and plot points 2. use y = mx + b non-linear equations o some form of y = x2 graph is a parabola o some form of y = | x | graph is a “V” o translation “slide”…moves every point the same distance and direction reflection “flip”…flips a figure over a line of reflection…creates a mirror image of the original… you can fold a figure on itself symmetry a figure has line symmetry if a line can be drawn through the figure so that one side is a mirror image of the other rotation transformation that turns a figure about a fixed point rotational symmetry figure can be rotated < 180° to get the figure look as it did before 7th Grade Math – Semester 2 Exam Study Guide page 8 of 8 Chapter 12 – Probability Probability o Theoretical and experimental probability Counting Principle o m ways of making a first choice and n ways of making a second choice, then there are m x n ways of making a first choice followed by a specific second choice Compound Events o Independent events the occurrence of one event does not affect the probability of the occurrence of the other event o Dependent events the occurrence of one event affects the probability of the occurrence of the other event Permutations o Factorial 5! = 5 x 4 x 3 x 2 x 1 o Arrangement of objects in a particular order…e.g., there are 16 teams competing in Olympic Hockey. What are the number of ways that gold, silver, and bronze medals can be given out the number of ways k objects can be taken out of a set of n total objects in order Combinations o Grouping of objects in which the order does not matter o o the number of ways k objects can be taken out of a set of n total objects