MMW Prelims 1

Fundamentals of Mathematical Language

• Mathematical Expression:

  • Definition: A finite combination of symbols that is well-formed according to specific mathematical rules.

  • Structural Property: It does NOT state a complete thought; it only represents a SUBJECT and contains no verb.

  • Examples:

    • $-5.4$

    • $5x + 2$

    • $2e$

    • $(x + 1)^2$

• Mathematical Sentence:

  • Definition: A correct arrangement of mathematical symbols that states a complete thought. It can be classified as either true or false.

  • Structural Property: Unlike an expression, a mathematical sentence includes both a subject and a verb.

  • Structural Example: In the sentence $5 + 2 = \sqrt{19}$, the symbol $+$ acts as the connective, and $=$ acts as the verb. The sentence represents a complete thought that evaluates as False.

  • Operational Example: $(w \div 2)(3)$

• Classification Examples:

  • "Roxas is the city of capiz.": Sentence (True) [Subject: Roxas, Verb: is].

  • "The province of cavite": Expression / Incomplete statement (Subject only, lacks a verb).

  • "The number 5 is an even number.": Sentence (False) [Subject: The number 5, Verb: is].

  • $(x + 1)^2$: Expression.

  • $\sqrt{x - 1} = 3$: Sentence (Contains the verb $=$).

  • $11x = 10$: Sentence (Contains the verb $=$).

Characteristics and Vocabulary of Mathematical Language

• Characteristics of Mathematical Language:

  • Precise: Able to make very fine distinctions or state concepts clearly (e.g., $1 + 1 = 2$).

  • Concise: Able to express long or complex ideas briefly through symbols rather than lengthy verbal phrases (e.g., $2x + 8 + 2 = 18$).

  • Powerful: Able to express complex ideas in simpler forms (e.g., expressing the Pythagorean theorem alongside a triangle figure as $c^2 = a^2 + b^2$).

  • Nontemporal: It has no tense (lacks past, present, or future tense), which makes it unique compared to natural human languages.

• Vocabulary and Parts of Speech Analogies:

  • Nouns / Subjects: Represented by variables, constants, and terms (e.g., $x$, $y$, numbers).

  • Verbs: Represented by relation symbols (e.g., $=$, $>$, $<$).

Translation Between English Phrases and Mathematical Symbols

• Keyword Operations Mapping:

  • Addition Keywords: sum of, added to, increased by, plus, more than, total of, added with.

  • Subtraction Keywords: difference of, diminished by, decreased by, minus, subtracted from, less than, less, exceeds by.

  • Multiplication Keywords: product of, multiplied by, times, twice, thrice, square of, of.

  • Division Keywords: quotient of, divided by, ratio, split into, all over, divided into.

• Sensitivity to Order in Addition and Subtraction:

  • Care must be taken regarding the order of terms in algebraic translation, especially for subtraction and specific addition phrases:

    • "5 more x" translates to $5 + x$

    • "5 more than x" translates to $x + 5$

    • "5 added to x" translates to $x + 5$

    • "5 greater than x" translates to $x + 5$

    • "5 less x" translates to $5 - x$

    • "5 less than x" translates to $x - 5$

    • "5 subtracted from x" translates to $x - 5$

• Key Phrase Translation Table:

  • "the sum of m and 8" (Key word: sum) $\rightarrow$ $m + 8$

  • "7 minus a" (Key word: minus) $\rightarrow$ $7 - a$

  • "x increased by 20" (Key word: increased by) $\rightarrow$ $x + 20$

  • "the product of 9 and n" (Key word: product) $\rightarrow$ $9n$

  • "y divided by 8" (Key word: divided by) $\rightarrow$ $\frac{y}{8}$

  • "twice a number" (Key word: twice) $\rightarrow$ $2x$

  • "the quotient of a number and 10" (Key word: quotient) $\rightarrow$ $\frac{x}{10}$

  • "10 less b" (Key word: less) $\rightarrow$ $10 - b$

  • "three fourths of a number" (Key word: of / fraction) $\rightarrow$ $\frac{3}{4}x$

Complex Algebraic Expressions and Naming Rules

• Complex Phrase Translations:

  1. "the sum of 2 times a number and 7" $\rightarrow$ $2x + 7$

  2. "the product of 12 and a number, divided by 8" $\rightarrow$ $\frac{12x}{8}$

  3. "20 more than 4 times a number" $\rightarrow$ $4x + 20$

  4. "the quotient of thrice a number and 15" $\rightarrow$ $\frac{3x}{15}$

  5. "8 times the square of a number, diminished by 7" $\rightarrow$ $8x^2 - 7$

  6. "Thrice the sum of a number and 5" $\rightarrow$ $3(x + 5)$

  7. "4 times b divided by their difference" $\rightarrow$ $\frac{4b}{a - b}$

  8. "the quotient of a and b divided by their sum" $\rightarrow$ $\frac{a \div b}{a + b}$

• Naming Principle for Expressions:

  • Mathematical expressions are generally named by the last operation to be performed when evaluating the expression:

    • $2x + 5$: Named a sum (addition is the final operation performed).

    • $x - 35$: Named a difference (subtraction is the operation performed).

    • $3(2x + 9)$: Named a product (multiplication is the final operation performed after calculating the sum in parentheses).

    • $\frac{9x^2 - 4b}{54}$: Named a quotient (division is the final operation performed after completing the numerator operations).

    • $9 + 5$ vs. $9 - 5$: Named a sum vs. a difference.

Problem-Solving Strategies and Patterns in Nature

• Polya's Four-Step Problem-Solving Strategy:

  1. Understand the problem: Identify the given values, what needs to be solved, and all conditions.

  2. Devise a plan: Formulate a method, write equations, find patterns, or choose an appropriate strategy.

  3. Carry out the plan: Execute the chosen strategy step-by-step to arrive at the solution.

  4. Look back / Check: Verify the solution to ensure it satisfies the original problem constraints.

• Patterns in Nature:

  • Symmetry: Characteristics of a figure being identical on both sides.

  • Repetition: Recurrent features or structural patterns.

  • Rotational Symmetry: Symmetry around a central rotational point.

  • Translational Symmetry: Repeating structural patterns shifted across space (e.g., honeycomb structures).

  • Fractals: Self-similar mathematical objects or figures that look similar at both small and large scales.

• The Fibonacci Sequence:

  • Recursive Formula: $F(n) = F(n - 1) + F(n - 2)$

  • Initial Base Values:

    • $F(0) = 0$

    • $F(1) = 1$

  • Step-by-Step Recursive Computation Example:

    • $F(2) = F(2 - 1) + F(2 - 2) = F(1) + F(0) = 1 + 0 = 1$