Math 2 The situtaion is well-defined. There is an explicit problem statement with all the necessary information (known and unkown). Drill The context of the problem is brand new (i.e., the student has never encountered this situation before). Problem There may be more than one valid approach. Problem Communication skills are not essential, as most of the solution involves math and sketches. Drill A method of instruction characterized by systematic repetition of concepts, examples and practice problems. Drill It involves memorization to gain proficiency in a skill. Drill The situation is ill defined. There is no problem statement and there is some ambiguity in the information given. Problem This often prescribe assumptions to be made, principles to be used and sometimes they even given hints. Drill Involves a process used to obtain to a best answer to an unknown subject to some constraints. Problem The process of constructing and applying mental representations of problems to finding solutions to those problems that are encountered in nearly every context. Problem Assumptions must be made regarding what is known and what needs to be found. Problem Using inductive reasoning, predict the next number: 1,5, 12, 22 35, 51 Math 2 1 Use inductive reasoning to decide whether the statement is true or false. Pick a counting number. Multiply the number by 6. Add 8 to the product. Divide the sum by 2. Subtract 4 from the quotient. The resulting number is twice the original number. False Use the inductive reasoning to decide whether the statement is true or false. The sum of any two even counting numbers is always an even accounting number. True During the past 10 years, a tree has produced plums every other year. Last year, the tree did not produce plums, so this year the trees will produce plums. This uses inductive reasoning. True Using inductive reasoning, predict the next number: 1,8,27,64? 125, 216 Using inductive reasoning, predict the next number: 5,11,17,23 29,35,41,47 Using the inductive reasoning, predict the next number: 1,2,5,10,17? 26, 37, 50, 65, 82, 101, 122, 145, 170 All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost P50,000. Thus, my home improvement will cost more than P50,000. This uses inductive reasoning False Using inductive reasoning, predict the next number: 100, 90, 81, 73,66? 60 There are 6 planets in the solar system. False, 8 Jaguars belong to the Felidae family. True A square of an integer is non-negative. True Two lines intersect at exactly 2 points. False, 1 The golden ratio is irrational. Math 2 2 True The capital city of Palawan is El Nido. False, Puerto Princesa Hogs are born without feathers. True Baby giraffes are called calves. True 15 is a prime number. False, composite The number 0 is on left of -2. False, right What is a problem? A situation that confronts the learner, that requires resolution, and for which the path to the answer is not immediately known What is a drill? A situation that requires resolution but the method is clear and the way to the answer is easily seen What is inductive reasoning? is a process of reaching a general conclusion by examining specific examples What is deductive reasoning? is the process of reaching a general conclusion by applying general assumptions, procedures, or principles Who outlined a strategy for solving problems from virtually any discipline? Goerge Polya (1887-1985) What are the steps in Polya’s Steps in Problem Solving? understand the problem, devise a plan, carry out the plan, review the solution What are routine problems? This involves using at least one of the four arithmetic operations to solve problems that are practical in nature Example: Cyril jogs four times around a rectangular plaza. If the length of the plaza is 60m and its width is 45m, how far does Cyril jogs every morning? Solution: Distance jogs = 4( perimeter of the plaza) Math 2 3 = 4( 2l + 2w) =4( 2 x 60 + 2 x 45)= 840 m What are non-routine problems? A non-traditional problem where one does not readily have an idea how to go about in solving the given problem Example: How are you going to put exactly 4 gallons of water on a scale. How would you do it if you only have a 5-gallon and a 3-gallon jug? ( the water source is unlimited) Solution: Fill the 5 gallon jug with water Pour water into the 3 gallon jug Empty the 3 gallon jug Pour the 2 gallon water into the 3 gallon jug Fill the 5 gallon jug with water Pour water into the 3 gallon jug with 2 gallons of water in it Which means exactly 1 gallon fills the 3 gallon jug, Thus remains exactly 4 gallons of water inside the 5 gallon jug What are logical connectives? ∧ ∨ conjunction and disjunction, which are denoted by and , respectively. The key word for conjunction is “and” and “or” is associated to disjunction. Statements joined by these connectives are called compound statements What is logical consequences? An implication or a conditional is a compound statement following the form “if p, then q” and is denoted by (Kwong, 2020), provided that p and q are propositions. A complicated form of implication is called biconditional written as (Baltazar et al., 2018). This is logically the conjuction of two implications It is a statement which either true or false but not both Proposition It is a statement corresponding to a statement with opposite truth value. The best way to describe the negation of a statement is to affix the word not Negation It is two or more simple proposistions combined by a logical operator commonly known as connective. The basic logical connectives are conjunction ( Math 2 ∧) ∨ and disjunction ( ) 4 Compound Proposition Suppose p and q are propositions. The proposition (read as “If p, then q”) is called an implication or a conditional statement. In this argument, p is called the premise and q is called the conclusion Implication Suppose p and q are propositions. The biconditional of p and q written (read as “p if and only if q”) is the conjunction of two conditional statements. In symbols, is equivalent to the statement Biconditional Statements Statement Symbol Description Implication p = q If p, then q Converse q = p If q, then p Inverse -p = -q If not p, then not q Contrapositive -q = -p If not q, then not p Example: p: a triangle is equilateral q: a triangle is equiangular (Implication): If a triangle is equilateral, then it is equiangular (p = q) Symbol Variation Statement q = p Converse If a triangle is equiangular, then it is equilateral -p = q Inverse If a triangle is notequilateral, then it is not equiangular -q = p Contrapositive If a triangle is not equiangular, then it is not equilateral What is a truth table? It shows the truth values of a compound statement for all possible truth values of its simple statements It is a compound proposition that is always true tautology It is a proposition that is always false self-contradiction / fallacy What is logical equivalence? Two compound statements are logically equivalent if they have the same truth values These are figures similar to Venn diagrams Math 2 5 Euler Circle Used to describe the variables in a statement Quantifiers Usually written in the English language as “for all” or “for every”. The symbol is ∀ Universal Quantifier Is expressed in words as “there exists” or “for some”. This quantifier is denoted by ∃ Existential quantifier type general form example universal affirmative All A is B all men are bipedal universal negative No A is B No dogs have feathers particular affirmative Some A is B Some dogs are hairy particular negative Some A is not B Some dogs are not hairy These arguments consist of two premises and a conclusion Syllogism Measurements or observations that are gathered for an event under study data Branch of mathematics that involves collecting, organizing, summarizing, and presenting data and drawing general conclusions from the data Statistics Consists of all subjects under study Population A representative subgroup or subset of a population sample Statistical techniques used to describe data. based on collecting, organizing, and reporting data without using the data to draw any wide-ranging conclusions descriptive statistics Statistical techniques used to make inferences. based on studying characteristics of a sample within a larger population and using them to draw conclusions about the entire population inferential statistics Each subject of the population must have an equal chance of being selected random sample Math 2 6 Taken by numbering each member of the population and then selecting every kth member, where kis a natural number systematic sample When a population is divided into groups where the members of each group have similar characteristics and members from each group are chosen at random stratified sample When an existing group of subjects that represent the population is used for a sample cluster sample The data collected for a statistical study raw data This is used when the data are categorical rather than numerical categorical frequency distribution Type of a frequency distribution that can be constructed uses numerical data grouped frequency distribution Each data value or number is separated into two parts. The very last digit is called the leaf, and what comes before is called the stem It is a circle that is divided into sections in proportion to the frequencies corresponding to the categories pie chart / circle graph This type of graph is used primarily to show trends, like prices rising or falling, for the time period time series When data are organized into grouped frequency distributions, two types of graphs are commonly used to represent them: histograms and frequency polygons Similar to a vertical bar graph in that the heights of the bars correspond to frequencies histogram Is the sum of the values in a data set divided by the number of values Mean, we use "X-bar" to stand for the mean Value in the middle if all values are arranged in order Median Midrange Math 2 7 lowest value + highest value / 2 The value that occurs most often in a data set Mode Data set is the difference between the highest and lowest values in the set Range, Highest value – Lowest value Data value indicates the percent of data values in a set that are below that particular value percentile Divides a data set into quarters quartile Is the same as the median, and divides a data set into an upper half and a lower half second quartile Is the median of the lower half, and the third quartile is the median of the upper half first quartile The distance between the first and third quartiles for a data set interquartile range, or IQR. That is, IQR = Q3 − Q1 Math 2 8
