Module 2 - 1 Language PDF

FOUR PICTURES, ONE WORD G E U N L G A A T ____ ____ ____ ____ ____ _____ _____ ____  ANSWER LANGUAGE  TOPIC 1: Characteristics and Conventions in the Mathematical Language Module 2 Mathematical Language and Symbols  S...

FOUR PICTURES, ONE WORD G E U N L G A A T ____ ____ ____ ____ ____ _____ _____ ____  ANSWER LANGUAGE  TOPIC 1: Characteristics and Conventions in the Mathematical Language Module 2 Mathematical Language and Symbols  Specific Objectives At the end of this lesson, the student should be able to: 1. Understand what mathematical language is. 2. Name different characteristics of mathematics. 3. Compare and differentiate natural language into a mathematical language and expressions into sentences. 4. Familiarize and name common symbols use in mathematical expressions and sentences. 5. Translate a sentence into a mathematical symbol. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  LANGUAG E Language is one of the most important things among people because it has an important role in communication. According to Cambridge English Dictionary, a language is a system of communication consisting of sounds, words and grammar, or the system of communication used by people in a particular country or type of work. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  A. Characteristics of Mathematical Language The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. It is: 1. PRECISE - able to make very fine distinction 2. CONCISE - able to say things briefly 3. POWERFUL - able to express complex thoughts with relative cases Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  B. Vocabulary vs. Sentences Every language has its vocabulary (the words), and its rules for combining these words into complete thoughts (the sentences). Mathematics is no exception. As a first step in discussing the mathematical language, we will make a very broad classification between the ‘nouns’ of mathematics (used to name mathematical objects of interest) and the ‘sentences’ of mathematics (which state complete mathematical thoughts)’ Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Importance of Mathematical Language Major contributor to overall comprehension Vital for the development of Mathematics proficiency Enables both the teacher and the students to communicate mathematical knowledge with precision Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  C. Comparison of Natural Language into Mathematical Language The table below is an illustration on the comparison of a natural language (expression or sentence) to a mathematical language. English Mathematics Expressions Noun such as person, place and things or this 2 is used to name things that we want to talk to while pronouns is another way in calling the 3–2 Name given to an object of interest. nouns 3x Example: a) Ernesto loves reading books. 3x + 2 b) Batangas City is part of the Philippines. c) He loves reading books. Note: Here, the variable x represents any quantity Sentence Group of words that express a statement, question or command. 3+2=5 It has a complete thought. Example: a+b=c a) Ernesto is a boy. b) He lives in Batangas City. ax + by + c = 0 c) Allan loves to read books. d) Run! e) Do you love me? (x + y)2 = x2 + 2xy + y2 Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  D. Expressions versus Sentences Ideas regarding sentences are explored. Just as English sentences have verbs, so do mathematical sentences. In the mathematical sentence; 3+4=7 the verb is =. If you read the sentence as ‘three plus four is equal to seven, then it’s easy to hear the verb. Indeed, the equal sign = is one of the most popular mathematical verb. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  D. Expressions versus Sentences Example: a) The capital of the Philippines is Manila. b) Rizal park is in Cebu. c) 5+3=8 d) 5+3=9 Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Connectives A question commonly encountered, when presenting the sentence example 1 + 2 = 3 is that if = is the verb, then what is + ? The answer is the symbol + is what we called a connective which is used to connect objects of a given type to get a ‘compound’ object of the same type. Here, the numbers 1 and 2 are connected to give the new number 1 + 2. In English, this is the connector “and”. Cat is a noun, dog is a noun, cat and dog is a ‘compound’ noun. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Mathematical Sentence Mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. It makes sense to ask about the TRUTH of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false? Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Mathematical Sentence Example: a) The capital of the Philippines is Manila. b) Rizal park is in Cebu. c) 5+3=8 d) 5+3=9 Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Truth of Sentences Sentences can be true or false. The notion of “truth” (i.e., the property of being true or false) is a fundamental importance in the mathematical language; this will become apparent as you read the book. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Conventions in Languages Languages have conventions. In English, for example, it is conventional to capitalize names (like Israel and Manila). This convention makes it easy for a reader to distinguish between a common noun (carol means Christmas song) and proper noun (Carol i.e. name of a person). Mathematics also has its convention, which helps readers distinguish between different types of mathematical expression. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Expression An expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. An expression does NOT state a complete thought; in particular, it does not make sense to ask if an expression is true or false. Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Conventions in mathematics, some commonly used symbols, its meaning and an example Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Sets and Logic SYMBOL NAME MEANING EXAMPLE ∪ Union Union of set A and A∪B set B ∩ Intersection Intersection of set A ∩B A and set B ∈ Element x is an element of A x∈A ∉ Not an element of x is not an element x ∉A of set A { } A set of.. A set of an element {a, b, c} Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Sets and Logic SYMBOL NAME MEANING EXAMPLE ⊂ Subset A is a subset of B A⊂ B ⊄ Not a subset of A is not a subset of B A⊄ B … Ellipses There are still other a, b, c, … items to follow/before …,-2,-1,0,1,2,… a + b + c + …. ∧ Conjunction A and B A∧ B ∨ Disjunction A or B A ∨B Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Sets and Logic SYMBOL NAME MEANING EXAMPLE ~ Negation Not A ~A → Implies (If-then If A, then B A→B statement) ↔ If and only if A if and only if A↔B B ∀ For all For all x ∀(x) ∃ There exist There exist an x ∃(x) Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language  Sets and Logic SYMBOL NAME MEANING EXAMPLE ∴ Therefore Therefore C ∴C | Such that or divides x such that y x|y x divides y x|y End of proof (QED) ≡ Congruence / equivalent A is equivalent to B A ≡B a is congruent to b modulo a ≡ b mod n n a, b, c, …, z Variables *First part of English Alphabet uses as fixed variable* (lower case) *Middle part of English alphabet use as subscript (axo)p (5x2)6 and superscript variable* *Last part of an English alphabet uses as unknown variable* Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language Basic Operations and Relational Symbols SYMBOL NAME MEANING EXAMPLE a plus b + Addition; Plus sign a added by b 3+2 a increased by b a subtracted by b - Subtraction; minus sign a minus b 3-2 a diminished by b · Multiplication sign a multiply by b 4·3 () *we do not use x as a symbol for a times b (4)(3) multiplication in our discussion since its use as a variable* ÷ or | Division sign; divides a÷b 10 ÷ 5 b|a 5 | 10 Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language Basic Operations and Relational Symbols SYMBOL NAME MEANING EXAMPLE ο Composition of function f of g of x f ο g(x) = Equal sign a=a 5=5 a+b=b+a 3+2=2+3 ≠ Not equal to a≠b 3≠4 > Greater than a>b 10 > 5 Module 2 | Topic 1:Characteristics and Conventions in the Mathematical Language Basic Operations and Relational Symbols SYMBOL NAME MEANING EXAMPLE < Less than b