Contemporary Mathematics GEO , ELEM ALGE , STAT PROB.docx

Contemporary Mathematics Plane Geometry Show a working knowledge of basic terms and concepts in Plane Geometry Lines and curves, perpendicular and parallel lines Angles, angle properties Special triangles and quadrilaterals Solve problems involving the basic terms and concepts in Plane Geometry...

Contemporary Mathematics Plane Geometry Show a working knowledge of basic terms and concepts in Plane Geometry Lines and curves, perpendicular and parallel lines Angles, angle properties Special triangles and quadrilaterals Solve problems involving the basic terms and concepts in Plane Geometry Statistics and Probability Show mastery and knowledge of basic terms and concepts in statistics and probability Counting techniques Probability of an event Measure of central tendency Measure of variability Solve, evaluate, and manipulate symbolic and numerical problems in elementary algebra by applying fundamental rules, principles and processes. 139 Introduction: This part focuses on the concepts identified in the LET competencies for Gen ED Mathematics. It provides a review of the definitions, formulas, operations, postulates, and theorems in Geometry and Statistics. However, in an effort to effectively guide students to a deeper understanding of the concepts involved, it begins with the foundations ang gives a wider, more comprehensive discussion than those specifically identified by the competencies. GEOMETRY BASIC DEFINITIONS Undefined Terms: the basic geometric concepts for which no definitions are given. These are the points, lines and planes. Collinear points: points that lie on the same line. Coplanar points: points that lie on the same plane. Space: the set of all points. Line Segment: a part of a line consisting of two endpoints and all the points between them. Ray: a part of a line having one endpoint and extending infinitely in one direction. Opposite rays: rays with a common endpoint but extending in opposite directions. Congruent segments: two segments having the same measure or length. Angle: formed by two non-collinear rays with a common endpoint. The two rays are the sides of the angle. The common endpoint of the two rays is the vertex of the angle. POSTULATES Space contains at least 4 noncoplanar points. Every plane contains at least three noncollinear points. Every line contains at least two points. Two points determine a line. Three noncollinear points determine a plane. If two points are in a plane, then the line containing the points are in the same plane. If two lines intersect, then their intersection is a point. If two planes intersect, then their intersection is a line. POLYGONS Polygon: a closed plane figure formed by fitting together segments end to end with each segment intersecting exactly two others. Diagonal of a polygon: a line segment that connects two non-consecutive vertices. Convex polygon: no diagonal is in the exterior of the polygon. Concave polygon: at least one diagonal is in the exterior of the polygon. Equilateral polygon: all the sides have equal lengths. 140 Equiangular polygon: all the angles have equal measure. Regular polygon: it is both equilateral and equiangular. Interior angles of a polygon: the angles formed by the sides of a polygon Exterior angles of a polygon: the angles formed in the exterior of the polygon when its sides are extended. THEOREMS Angle-Sum Theorem for Triangles: The sum of the measures of the three angles of any triangle is 180˚. Angle-Sum Theorem for Quadrilaterals: The sum of the measures of the four angles of any quadrilateral is 360˚. Angle-Sum Theorem for Polygons: The sum(s) of the measures of the interior angles of any polygon with n sides is given by s = (n – 2)180˚. Exterior Angle Theorem for Polygons: The sum of the measures of the exterior angles of a convex polygon (one at each vertex) is 180˚. TRIANGLES: Classifying Triangles According to Sides Scalene: No two of its sides are congruent. Isosceles: At least two of its sides are congruent. The two congruent sides are the legs. The third side is called the base. The angle opposite the base is the vertex. The angles adjacent to the base are the base angles. Equilateral: All of its sides are congruent. Classifying Triangles According to Angles Acute: all of its angles are acute Right: has one right angle. The hypotenuse is the longest side. The legs are the other two sides. Obtuse: One of its angles is obtuse. Equiangular: all of its angles are congruent. Secondary Parts of a Triangle: Altitude of a triangle: a segment from a vertex perpendicular to the line that contains the opposite side. Median of a triangle: a segment from one vertex to the midpoint of the opposite side. Concurrent lines: Three or more lines that meet at the same point. QUADRILATERALS – a quadrilateral is a four-sided polygon. Kinds of Quadrilaterals A quadrilateral is a parallelogram if and only if one of the following is satisfied: Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. 141 Both pairs of opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Each diagonal divides the quadrilateral into two congruent triangles. A quadrilateral is a rectangle if and only if it is a parallelogram and one of the following is satisfied: It has four right angles. Its diagonals are congruent. A quadrilateral is a rhombus if and only if it is a parallelogram and one of the following is satisfied: Its sides are all congruent. Its diagonals are perpendicular. Its diagonals bisect the angles of the quadrilateral. A quadrilateral is a square if it both a rhombus and a rectangle. A quadrilateral is a trapezoid if it has a pair of parallel opposite sides. The parallel sides are the bases and the non-parallel sides are its legs. If the legs are congruent, the trapezoid is said to be an isosceles trapezoid. A trapezoid is isosceles if it satisfies one of the following: Its legs are congruent. Each pair of base angle is congruent. Diagonals are congruent. CIRCLES Circle: the set of all points in a plane at a given distance (radius) from a given point (center) in the plane. Radius: the line segment from the center to any point of the circle. Chord: a line segment whose endpoints lie on the circle. Diameter: a chord containing the center. Secant: a line that intersects the circle in two points. Tangent: a line lying on the same plane as the circle that intersects the circle in exactly one point. Point of Tangency: The point where the tangent touches the circle. Congruent circles: two or more circles having the same radius. Concentric circles: two or more coplanar circles sharing the same center. RELATIONS INVOLVING SEGMENTS AND ANGLES Segment-Addition Postulate: Point Y is between X and Z if and only if X, Y, and Z are collinear and XY + YZ = XZ. Midpoint of the segment: A point that bisects a segment, or divides a segment into two congruent segments. Bisector of the line segment: A ray, line or line segment that contains the midpoint. Angle bisector: A ray that contains the vertex and divides the angle into two congruent parts. 142 PERPENDICULAR LINES AND BISECTOR OF A LINE Perpendicular lines: two lines that intersect at right angles. Perpendicular bisector or a segment: the line which is perpendicular to the segment at its midpoint. Theorem: The shortest segment from a point to the line is the perpendicular segment. ANGLE PAIRS Adjacent angles: two angles with a common vertex, a common side, and no common interior points. Supplementary angles: two angles whose measures have a sum of 180˚. Complementary angles: two angles whose measures have a sum of 90˚. Vertical angles: Two angles are vertical if and only if their sides form two pairs of opposite rays and their angles are nonadjacent formed by two intersecting lines. Linear pair: two angles which are adjacent and supplementary. Theorems: Supplements of congruent angles are congruent. Complements of congruent angles are congruent. Vertical angles are congruent. ANGLES AND SIDES OF A TRIANGLE Exterior angle of a triangle: an angle which forms a linear pair with one of the triangle’s interior angles. Remote interior angles: two interior angles of the triangle not adjacent to the exterior angle. Theorems: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In a triangle, if one side is longer than the other side, the angle opposite the longer side is the larger angle. In a triangle, if one angle is larger than the other angle, the side opposite the larger angle is the longer side. PARALLEL LINES AND TRANSVERSALS Intersecting lines: coplanar lines having a point in common. Perpendicular lines: lines that intersect at right angles. Paralell lines: coplanar lines that do not intersect at do not intersect 1 2 ting two or more coplanar lines at different points 3 4 Interior angles: <3, <4, <5, <6 5 6 143 7 8 Rizjoe villacruz ([email protected]) <8 and <5 and <7 and <6 Exterior angles: <1, <2, <7, <8 Corresponding angles: <1 and <5, <2 and <6, <3 and <7, <4 and Alternate interior angles: <3 and <6, <4 Alternate exterior angles: <1 and <8, <2 Same-side interior angles: <3 and <5, <4 Postulates: If parallel lines are cut by a transversal, then: The alternate interior angles are congruent. The corresponding angles are congruent. The alternate exterior angles are congruent. The same-side interior angles are supplementary. COUNTING TECHNIQUES Experiment: any activity that can be done repeatedly (e.g. tossing a coin, rolling a die) Sample space: the set of all possible outcomes in an experiment. Example: In a rolling die, the sample space is S = {1, 2, 3, 4, 5, 6}. Sample point: an element of the sample space. Example: In a rolling die, there are six sample points. Counting Sample Points Fundamental Principle of Counting (FPC) If a choice consists of k steps, of which the steps can be performed in n1 ways, for each of these the second can be performed in n2 ways, for each of these the third can be performed in n3 ways . . . , and for each these the kth can be made in nk ways, then the whole choice can be made in n1n2n3…nk ways. Example: In how many ways can two dice fall? Ans.: 6 • 6 = 36 ways Permutation Permutation is an arrangement of objects wherein the order is important. Linear Permutation If n objects are to be arranged r objects at a time the number of distinct arrangements is given by formula  Example: In how many can the first, second and third winners may be chosen in a beauty pageant with 10 contestants? 10P3 = = 10 • 9 • 8 = 720 ways Circular Permutation 144 If n objects are to be arranged in a circular manner, then the number of distinct arrangements is (n-1)! Answer: (7-1)! = 6! Permutation with Repetitions The nuber of distinct permutations of n things of which p are of one kind, q are of a second kind,… r of the kth kind is Example: How many different permutations are there in the word COMMITTEE if all letters are to be taken? Answer: Combination Combination is the arrangement of objects regardless of order. In other words, the order of arranging the objects is not important. If n objects are to be arranged r at a time, the number of distinct combinations is given by the formula: nCr Example: In how many ways can a committee of 4 be chosen from 6 persons? Answer: 6C4 PROBABILITY Probability: the likelihood of the occurrence of an event. If E is any event, then the probability of an event denoted by P(E) has a value between 0 and 1, inclusive. In symbol, 0 ≤ P(E) ≤ 1 If P(E) = 1, then E is sure to happen If P(E) = 0, then E is impossible to happen. Moreover, the probability that E will not happen is P(E’), then P(E) + P(E’) = 1. Theoretical Probability Theoretically, the probability of an event E, denoted by P(E), is defined as where n(E) = number favourable outcomes n (S) = number of possible outcomes Experimental Probability The probability of an event may also be obtained experimentally. Suppose we want to find out the probability of obtaining a tail in a toss of coin. We can perform an experiment by tossing the coin 50 times and record the number of occurrences of tail. Suppose that tail occurred 24 times, then the probability of getting a tail based on this experiment is P (tail) = 24/50 STATISTICS Statistics is the branch of mathematics used to summarize quantities of data and help investigators draw sound conclusions. Its two main branches are descriptive statistics and inferential statistics. A sample is a specified set of measurements or data, which is drawn from a much larger body of measurements or data called the population. 145 Kinds of sampling Random sampling techniques are used to ensure that every member of the population has an equal chance of being included in the sample. A random sample is said to be representative of the entire population. The two methods of random sampling are lottery method and the use of the table of random sampling. Systematic sampling is a technique which selects every nth element of the population for the sample, with the starting point determined at random from the first n elements. Stratified random sampling is a technique of selecting simple random samples from mutually exclusive groupings or strata of the population. Graphical Representations of Data Histogram—a graphical picture of a frequency distribution consisting of a series of vertical columns or rectangles, each drawn with a base equal to the class interval and a height corresponding to the class frequency. The bars of a histogram are joined together, that is, there are no spaces between bars. Bar Chart—uses rectangles and bars to represent discrete classes of data. The length of each bar corresponds to the frequency or percentage of the given class or category. The categories are in turn placed in either horizontal or vertical. Frequency Polygon—a special type of line graph, where each class frequency is plotted directly above the midpoint or class mark of its class interval and lines are then drawn to connect the points. Pie Chart—an effective way of presenting categorized (qualitative) distributions, where a circle is divided into sectors—pie-shaped pieces—which are proportional in size to the corresponding frequencies or percentages. Pictogram—known as picture graph where picture symbols are used to represent values. MEASURES OF CENTRAL TENDENCY A measure of central tendency is a single, central value that summarizes a set of numerical data. The measures of central tendency are the mean, median and mode. MEASURES OF VARIABILITY A measure of variation or variability describes how large the differences between the individuals are on a trait. The common measures of variability are range and standard deviation. Measure s of Central Tendenc y Definition How to find Advantages Disadvantages Mean The sum of data divided by the number of data Ungrouped data: Grouped data: A single, unique value that is representative of all the scores. Stable from group to group. May be used in further computations. Not appropriate for skewed distribution as it is affected by extreme scores or outliers. The middles Ungrouped data: the More stable from Not necessarily 146 Median number of the set when the data are arranged in numerical order middle for the Grouped data: Mdn = L + group to group than the mode. Appropriate for skewed distribution. representative of all scores. Unstable from group to group. Cannot be used in further analyses Mode The number that occurs most Ungrouped data: The most frequent Easy to obtain. Not necessarily representative of frequently in the score all scores data Grouped data: Cannot be used The class mark of the in further class interval with the analyses highest frequency Measures of Variation Definition How to find Advantages Disadvantages Range The difference Ungrouped data: R = HS – LS Easy to compute. Unstable. Not between the Grouped Data: Gives a unique representative highest score R = Upper Limit of the value. of the set of and the Highest Class Interval – Easy to data. lowest score Lower Limit of the understand. Not used in Lowest Class Interval further computations. Standard deviation The square root of the variance of the set of data Ungrouped data: Grouped data: Most stable Gives unique value Most representative Used in further computations Affected by extreme scores More difficult to compute and understand