GEMMW01X_2 Mathematical Language and Symbols PDF

Mathematics in the Modern World MATHEMATICS IN OUR WORLD Mathematical Language and Symbols The Four Basic Concepts of Mathematics Learning Outcomes: At the end of this lesson, learners should be able to: 1. Discuss the language, symbols, and conventions of mathematics. 2. Explain the na...

Mathematics in the Modern World MATHEMATICS IN OUR WORLD Mathematical Language and Symbols The Four Basic Concepts of Mathematics Learning Outcomes: At the end of this lesson, learners should be able to: 1. Discuss the language, symbols, and conventions of mathematics. 2. Explain the nature of mathematics as a language 3. Perform operations on mathematical expressions correctly. 4. Acknowledge that mathematics is a useful language. Language systematic means of communicating ideas or feelings by the use of conventional symbols, sounds, or marks having understood meaning Language of Mathematics system used to communicate mathematical ideas consists of a substrate of some natural language using technical terms and grammatical conventions that are peculiar to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas Language of Mathematics Characteristics of Mathematical Language ►Precise – able to make very fine distinctions Example: The use of mathematical symbol is only done based on its meaning and purpose. ►Concise – able to say things briefly Example: The long English sentence can be shortened using mathematical symbols. Eight plus two equals ten. 8 + 2 = 10 ►Powerful – able to express complex thoughts with relative ease Example: The application of critical thinking and problem-solving skill requires the comprehension, analysis, and reasoning to obtain the correct solution. Some examples of commonly used symbols Expression versus Sentences ►expression (mathematical expression) - finite combination of symbols that is well-formed according to rules that depend on the context - does not state a complete thought - does not make sense to ask if an expression is true or false ►examples of types of expression 1. numbers 2. sets 3. functions 4. ordered pair 5. matrices 6. vectors ►examples of expression the use of expressions ranges from the simple: 8𝑥 − 5 linear polynomial 7𝑥 ! + 4𝑥 − 10 quadratic polynomial "#! rational fraction " ! $%! to the complex: !' ∫& 𝑥𝑑𝑥  Expression versus Sentences ►sentence (mathematical sentence) - analogue of an English sentence - correct arrangement of mathematical symbols that states a complete thought - can be determined whether it’s true, false, sometimes true/false Comparison between the English language and Mathematical language Let’s Practice! Classify as an English noun (EN), Mathematical Expression (ME), English Sentence (ES), or Mathematical Sentence (MS). 1. Cat 9. x=1 2. 2 10. x–1=0 3. The word ‘cat’ begins with 11. t+3 the letter ‘k’. 12. t+3=3+t 4. 1+2=4 13. This sentence is false. 5. 5–3 14. x+0=x 6. 5–3=2 15. 1,x=x 7. The cat is black. 8. x Common Words and Phrases used Try it out! 1. a number decreased by ninety-two 2. fifty-five times a number 3. the product of two numbers 4. three less than twice a number is two 5. the square of the sum of five and a number 6. the product of three consecutive integers 7. Jose’s age in ten years Mathematical Convention ►a fact, name, notation, or usage which is generally agreed upon by mathematicians Examples: Symbols + − × ÷ / 𝜋 ∞ = ≈ ≠ < ≤ > ≥ √ ° ∴ Order of operations ► MDAS ► PEMDAS ► BODMAS Order of Operations Step 1: Do as much as you can to simplify everything inside the parenthesis first Step 2: Simplify every exponential number in the numerical expression Step 3: Multiply and divide whichever comes first, from left to right Step 4: Add and subtract whichever comes first, from left to right Examples: 1. Evaluate: 11 − 5 ×2 − 3 + 1 2. Evaluate: 10 ÷ 2 + 12 ÷ 2×3 3. Simplify: 4 − 3 4 − 2(6 − 3) ÷ 2 4. Simplify: 16 − 3(8 − 3)! ÷ 5  Examples: 1. Evaluate: 11 − 5 ×2 − 3 + 1 2. Evaluate: 10 ÷ 2 + 12 ÷ 2×3 6×2 − 3 + 1 5 + 6×3 12 − 3 + 1 5 + 18 9+1 𝟐𝟑 𝟏𝟎 3. Simplify: 4 − 3 4 − 2 6 − 3 ÷ 2 4. Simplify: 16 − 3(8 − 3)!÷ 5 4 − 3[4 − 2 3 ] ÷ 2 16 − 3 5 ! ÷ 5 4 − 3(4 − 6) ÷ 2 16 − 3 25 ÷ 5 4 − 3(−2) ÷ 2 16 − 75 ÷ 5 4+6÷2 16 − 15 4+3 𝟏 𝟕 Try it out! 1. Evaluate: 10 − 4 3 − (3)" +4 2. Evaluate: −1 3 − 4×7 ÷ 5 − 2×24 ÷ 6 3. Simplify: 8×2 − 3 ! + (5×2) 4. Simplify: 9 ÷ 3 + 2 9 + 10 − 8 + 4×3 The Four Basic Concepts of Mathematics Set Relation and Function Binary Operation 1. SET Definition and Representation of Sets 1. Descriptive Form ► state in words the elements of the set Examples: The set of colors on the Philippine flag. The set of all the natural numbers less than 10. The set of all even numbers. The set of all vowel letters 2. List Notation/Roster Method ► list names of elements of a set, separate them by commas and enclosed them in brackets Examples: 𝐴 = 1,12,45 𝐵 = 𝐶𝑜𝑟𝑎𝑧𝑜𝑛 𝐴𝑞𝑢𝑖𝑛𝑜, 𝑅𝑜𝑑𝑟𝑖𝑔𝑜 𝐷𝑢𝑡𝑒𝑟𝑡𝑒 𝐶 = 𝑎, 𝑏, 𝑑, 𝑚 𝐷 = {1,2, … , 100} 3. Set-Builder Notation/Rule Method ► states a property of its elements Examples: 𝐴 = {𝑥|𝑥 is a natural number and x