Mathematics for Science (MATH0014) Course Guide & Topics 1-2 Study Notes

Course Overview and Administration

Course Focus and Scope

• Unit Title: Mathematics for Science (MATH0014), Centre for Enabling Pathways, Curtin University.
• Subject Matter: Pre-calculus algebra, functions (graphs), and trigonometry designed as foundational preparation for degree studies in STEM (Science, Technology, Engineering, and Mathematics).
• Curriculum Range: Numeracy and measurement, fundamental algebra, linear equations, literal equations, inequalities, basic trigonometry, and real-world scientific applications.

Weekly Study Workflow

• Topic Structure: The unit comprises 12 weekly topics to be completed individually at a rate of one per week.
• Learning Materials: Each week's content is hosted on Blackboard under "Weekly Content" in a dedicated Word document split into sub-topics containing concept explanations, worked examples, and practice problems.
• Preparation: Work through each topic document prior to attending the scheduled tutorial session.
• Tutorial Structure:
  • Detailed explanation of key concepts and worked solution examples.
  • Group and class discussions alongside interactive activities.
  • Dedicated time for student Q&A.
  • Key updates regarding upcoming assessments.
• Weekly Quizzes:
  • Diagnostic Pre-Test: Completed before commencing a topic to evaluate baseline knowledge.
  • End-of-Topic Quiz: Completed after finishing the topic material to measure topic mastery.
  • Status: Quizzes are unassessed, carry unlimited attempts, provide immediate feedback, and do not contribute to the final grade.
• Extension Tests: Optional extension tests are provided in each topic folder for students seeking advanced problem-solving practice.

Approved Technology and Calculator Policy

• Recommended Model: CASIO fx-82AU PLUS II non-programmable scientific calculator.

• Approved Equipment: Any standard non-programmable scientific calculator is permitted.
• Prohibited Equipment: ClassPad, graphing calculators, and any programmable technology are strictly prohibited during formal examinations.

Assessment Structure and Weighting

• Assessment 1 (20%): eTest Quiz covering Topics 1, 2, and 3. Administered online via Blackboard using Respondus LockDown Browser. Time limit: 130 minutes. Available for a full 7-day window.
• Assessment 2 (40%): Algebra, Trigonometry, and Functions Investigation. Released immediately following the Assessment 1 eTest in Week 4. Part C requires Respondus LockDown Browser. Submitted via the Blackboard assessment portal.
• Assessment 3 (40%): Comprehensive Final Examination covering all 12 topics. Conducted using Respondus LockDown Browser.
• Submission Protocols and Technical Escalation: Verify that every submission generates an online confirmation receipt. In the event of technical failure on Blackboard, email a copy of the completed assessment directly to the tutor prior to the submission deadline to avoid late submission penalties.
• Extensions: Requests must be submitted via OASIS under My Studies > Study Essentials using the official Application for Assessment Extension form prior to the deadline. Students are required to maintain personal backup copies of all submitted work.

Academic Integrity and Mindset

• Academic Integrity: All submitted assessment items must represent original individual work. Avoid collusion when working collaboratively with peers.
• Study Strategy: Strive for broad mastery across all topics rather than specializing in select areas. Versatility and adaptability across the full syllabus are essential for examination performance. Target difficult or unfamiliar problem types until confidence is achieved.

Topic 1: Numbers, Real Number Systems, and Fundamental Trigonometry

Place Value, Rounding, and Arithmetic Foundations

• Place Value System: Standard decimal numbers use position to establish digit value. In the number $51,493$:
  • $5$ is in the Ten Thousands position ($50,000$).
  • $1$ is in the Thousands position ($1,000$).
  • $4$ is in the Hundreds position ($400$).
  • $9$ is in the Tens position ($90$).
  • $3$ is in the Units / Ones position ($3$).
• Rounding Procedures:
  • Rounding $392,546$:
  • To the nearest hundred: $392,500$
  • To the nearest thousand: $393,000$
  • To the nearest ten thousand: $390,000$
  • Rounding $4.098$:
  • To the nearest tenth: $4.1$
  • To the nearest hundredth: $4.10$
  • To the nearest whole number: $4$
• Lowest Common Multiple (LCM): The smallest positive integer divisible by both numbers.
  • To find the LCM of $8$ and $12$:
  • Multiples of $8$: $8, 16, 24, 32, 40, \dots$
  • Multiples of $12$: $12, 24, 36, 48, \dots$
  • $\text{LCM}(8, 12) = 24$

Language of Algebra, Expressions, and Operators

• Translating Verbal and Algebraic Statements:
  • Algebraic expression $16 + 11$ translates to English as "The sum of sixteen and eleven".
  • English phrase "The difference of 14 and 9" translates to the algebraic expression $14 - 9$
• Expressions versus Equations:
  • Expression: A mathematical mathematical phrase containing numbers, variables, and operators without an equals sign (e.g., $5 \times 4 + 3$).
  • Equation: A mathematical statement declaring the equality of two expressions, separated by an equals sign (e.g., $9 \times 6 = 54$).
• Relational Operators ($<, >, =$):
  • $9 > 4$
  • $-7 < 3$
  • $-10 = -2(5)$ because $-2 \times 5 = -10$
• Absolute Value: Represents the magnitude or distance of a real number from zero on a number line, denoted by $|x|$.
  • Simplify $|-32|$: $|-32| = 32$
• Signed Integer Operations:
  • $15 - (-12) = 15 + 12 = 27$
  • $19 + (-13) = 19 - 13 = 6$
  • $-4 \times 8 = -32$
  • $-52 \div (-4) = 13$

Fractions, Decimals, and Percentages

• Fractions:
  • Equivalent Fractions: Multiplying or dividing numerator and denominator by the same non-zero integer yields equal value. For example, two equivalent fractions for $\frac{3}{8}$ are $\frac{6}{16}$ and $\frac{9}{24}$.
  • Simplification:
  • $\frac{-40}{88} = -\frac{5}{11}$ (dividing numerator and denominator by $8$)
  • $\frac{4}{10} = \frac{2}{5}$
  • $\frac{4}{3}$ remains $\frac{4}{3}$ or $1\frac{1}{3}$
  • Multiplication:
  • $\frac{8}{21} \times \frac{12}{35} = \frac{8 \times 12}{21 \times 35} = \frac{8 \times 4}{7 \times 35} = \frac{32}{245}$
  • Arithmetic with Fractions:
  • Subtraction with unlike denominators: $\frac{7}{12} - \frac{5}{16}$
  • Least common denominator is $48$:       $\frac{7 \times 4}{12 \times 4} - \frac{5 \times 3}{16 \times 3} = \frac{28}{48} - \frac{15}{48} = \frac{13}{48}$
• Decimals:
  • Word conversion: "Twenty nine and eighty one hundredths" = $29.81$
  • Addition: $16.92 + 7.56 = 24.48$
  • Subtraction: $3.84 - 6.1 = -2.26$
  • Multiplication: $0.24 \times 0.6 = 0.144$
  • Division: $4.75 \div 25 = 0.19$
  • Decimal to Fraction: $0.1982 = \frac{1982}{10000} = \frac{991}{5000}$
  • Fraction to Decimal: $\frac{3}{8} = 0.375$
• Percentages:
  • Convert decimal to percentage: Multiply by $100\%$
  • $2.254 = 2.254 \times 100\% = 225.4\%$

Real Number Classification System

• Classification Categories:

• Classification Examples:
  • $2$: Real, Rational, Integer, Whole, Natural.
  • $0$: Real, Rational, Integer, Whole.
  • $-5$: Real, Rational, Integer.
  • $\frac{3}{8}$: Real, Rational.
  • $\sqrt{7}$: Real, Irrational.
  • $\sqrt{-4}$: Non-Real / Imaginary ($\sqrt{-4} = 2i$).
  • Radical Expression Conditions for $\sqrt{x}$:
  • When $x > 0$ and $x$ is a perfect square: Natural, Whole, Integer, Rational, Real.
  • When $x > 0$ and $x$ is not a perfect square: Irrational, Real.
  • When $x = 0$: Whole, Integer, Rational, Real ($\sqrt{0} = 0$).
  • When $x < 0$ (e.g., $-1, -4, -9$): Non-Real / Imaginary.

Geometric Trigonometry Foundations

• Interior Angle Sum Property: The interior angles of any planar triangle sum to $180^\circ$.
  • Given angles $30^\circ$ and $19^\circ$:     $\text{Missing Angle} = 180^\circ - (30^\circ + 19^\circ) = 180^\circ - 49^\circ = 131^\circ$
  • Right-angled triangle with acute angle $24^\circ$:     $\text{Missing Angle} = 180^\circ - (90^\circ + 24^\circ) = 180^\circ - 114^\circ = 66^\circ$
• Pythagorean Theorem: In any right-angled triangle with legs $a, b$ and hypotenuse $c$:   $a^2 + b^2 = c^2$
  • Finding hypotenuse with legs $13\,\text{m}$ and $15.5\,\text{m}$:     $c = \sqrt{13^2 + 15.5^2} = \sqrt{169 + 240.25} = \sqrt{409.25} \approx 20.23\,\text{m}$
  • Finding unknown leg with hypotenuse $29\,\text{cm}$ and known leg $7\,\text{cm}$:     $b = \sqrt{29^2 - 7^2} = \sqrt{841 - 49} = \sqrt{792} \approx 28.14\,\text{cm}$
• Trigonometric Ratios in Right Triangles:
  • For an acute angle $\theta$ in a right triangle:     $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
  • For a triangle with legs $q$ (opposite to $\theta$), $r$ (adjacent to $\theta$), and hypotenuse $t$:
  • For angle $\theta$: $\sin(\theta) = \frac{q}{t}$, $\cos(\theta) = \frac{r}{t}$, $\tan(\theta) = \frac{q}{r}$
  • For complementary angle $\alpha$: $\sin(\alpha) = \frac{r}{t}$, $\cos(\alpha) = \frac{q}{t}$, $\tan(\alpha) = \frac{r}{q}$

Topic 2: Order of Operations, Algebraic Expressions, Linear Equations, Inequalities, and Applied Trigonometry

Order of Operations (BEDMAS / PEDMAS)

• Order Hierarchy: Brackets/Parentheses, Exponents/Orders, Division/Multiplication (left-to-right), Addition/Subtraction (left-to-right).
• Worked Examples:
  • Compare $3 + 8 \times 5$ and $(3 + 8) \times 5$:     $3 + 8 \times 5 = 3 + 40 = 43$$(3 + 8) \times 5 = 11 \times 5 = 55$
  • Simplify $5[2 + 4(3 + 2)^2]$:     $5[2 + 4(5)^2] = 5[2 + 4(25)] = 5[2 + 100] = 5[102] = 510$
  • Simplify Fractional Expressions:     $\frac{2^2 + 3}{5} = \frac{4 + 3}{5} = \frac{7}{5} = 1.4$$\frac{8 \times 9 - 7 \times 6}{5 \times 6 - 9 \times 2} = \frac{72 - 42}{30 - 18} = \frac{30}{12} = \frac{5}{2} = 2.5$

Algebraic Evaluation and Simplification

• Variable Substitution:
  • Evaluate $7x + 8$ when $x = 2$:     $7(2) + 8 = 14 + 8 = 22$
  • Evaluate $2x^2 + 3x - 7$ when $x = 4$:     $2(4)^2 + 3(4) - 7 = 2(16) + 12 - 7 = 32 + 12 - 7 = 37$
• Identifying and Combining Like Terms:
  • Given terms: $3x, 8y, 14x, 8, 5y, 83x$
  • Like terms in $x$: $3x, 14x, 83x$
  • Like terms in $y$: $8y, 5y$
  • Constant terms: $8$
  • Simplify $10x + 7y + 5x - 2y + 7x - 4y$:     $(10x + 5x + 7x) + (7y - 2y - 4y) = 22x + y$
• Algebraic Subtraction:
  • Subtract $(2x^2 + x - 3)$ from $(8x^2 - 3x + 4)$:     $(8x^2 - 3x + 4) - (2x^2 + x - 3) = 8x^2 - 3x + 4 - 2x^2 - x + 3$$= (8 - 2)x^2 + (-3 - 1)x + (4 + 3) = 6x^2 - 4x + 7$

Solving Linear Equations

• Standard Single-Variable Equations:
  • $x + 24 = 35 \implies x = 35 - 24 = 11$
  • $x - 45 = 76 \implies x = 76 + 45 = 121$
  • $x + 31 - 10 = 46 \implies x + 21 = 46 \implies x = 25$
  • $6(x - 2) - 5 = 4(x + 3) - 4(x - 1)$$6x - 12 - 5 = 4x + 12 - 4x + 4$$6x - 17 = 16 \implies 6x = 33 \implies x = \frac{33}{6} = \frac{11}{2} = 5.5$
  • $8x = 56 \implies x = 7$
  • $-\frac{x}{4} = 35 \implies x = 35 \times (-4) = -140$
  • $100x - 16 = 4 - 10 - x$$100x - 16 = -6 - x \implies 101x = 10 \implies x = \frac{10}{101}$
  • $-10(-4x) - 57 = -93$$40x - 57 = -93 \implies 40x = -36 \implies x = -\frac{36}{40} = -\frac{9}{10} = -0.9$
  • $\frac{9x}{10} = 90 \implies 9x = 900 \implies x = 100$
  • $19.36x = -0.2 \implies x = \frac{-0.2}{19.36} = -\frac{5}{484} \approx -0.01033$
  • $19x = 18 - 7x \implies 26x = 18 \implies x = \frac{18}{26} = \frac{9}{13}$
  • $-11x - 8 = -7 \implies -11x = 1 \implies x = -\frac{1}{11}$
  • $4[5 - 8(4x - 3)] = 12(1 - 13x) - 8$$4[5 - 32x + 24] = 12 - 156x - 8$$4[29 - 32x] = 4 - 156x$$116 - 128x = 4 - 156x$$28x = -112 \implies x = -4$

Verbal Translations and Equation Classification

• Translating Word Problems into Equations:
  • "187 is the product of x and -17":     $-17x = 187 \implies x = \frac{187}{-17} = -11$
  • "The sum of two-fifths and x is 15":     $\frac{2}{5} + x = 15 \implies x = 15 - \frac{2}{5} = \frac{73}{5} = 14.6$
• Classification Categories of Equations:
  • Conditional Equation: An equation true for a restricted set of values (has a discrete solution set).
  • Identity: An equation true for all possible values of the variable (simplifies to $\text{LHS} = \text{RHS}$ or $0 = 0$).
  • Contradiction: An equation false for all values of the variable (simplifies to an impossible statement like $0 = \text{non-zero constant}$).
• Classification Problems:
  • $23x + 19 = 3(5x - 9) + 8x + 46$$\text{RHS} = 15x - 27 + 8x + 46 = 23x + 19$$\text{LHS} = \text{RHS} \implies 23x + 19 = 23x + 19$Classification: Identity.
  • $5(x + 4) + 8(2x - 1) = 9(3x - 5) - 6(x - 2)$$\text{LHS} = 5x + 20 + 16x - 8 = 21x + 12$$\text{RHS} = 27x - 45 - 6x + 12 = 21x - 33$$21x + 12 = 21x - 33 \implies 12 = -33$Classification: Contradiction.
  • $18(5x - 1) + 29 = 47$$90x - 18 + 29 = 47 \implies 90x + 11 = 47 \implies 90x = 36 \implies x = 0.4$Classification: Conditional.

Literal Equations and Formula Manipulation

• Temperature Conversion Formula: Solve for $F$ in $C = \frac{5}{9}(F - 32)$:   $\frac{9}{5}C = F - 32 \implies F = \frac{9}{5}C + 32$
• Triangle Area Formula: Rearrange $A = \frac{1}{2}bh$ to solve for base $b$:
  • General solution:     $2A = bh \implies b = \frac{2A}{h}$
  • Specific evaluation when $A = 126$ and $h = 8$:     $b = \frac{2(126)}{8} = \frac{252}{8} = 31.5$

Linear Inequalities

• Number Line Representation:
  • $x \ge 2$: Represented by a solid (closed) circle at $2$ with a shaded ray extending right toward $+\infty$
  • $-2 < x < 4$: Represented by open circles at both $-2$ and $4$ with a shaded segment connecting them.
• Inequality Reversal Rule: Reversing the inequality sign is mandatory whenever multiplying or dividing both sides by a negative number.
• Solving Inequalities:
  • $-11x < 33$$\text{Divide by } -11 \implies x > \frac{33}{-11} \implies x > -3$
  • $\frac{3}{5}x \le -45$$\text{Multiply by } \frac{5}{3} \implies x \le -45 \times \frac{5}{3} \implies x \le -75$
  • $12x + 3(4x - 1) \le 19(x - 2) + 5$$12x + 12x - 3 \le 19x - 38 + 5$$24x - 3 \le 19x - 33$$5x \le -30 \implies x \le -6$

Applied Trigonometric Analysis

• Solving Single Trig Equations:
  • Solve for $x$ in $\sin(20^\circ) = \frac{x}{15}$:     $x = 15 \times \sin(20^\circ) \approx 15 \times 0.34202 = 5.1303$
  • Solve for $x$ in $\cos(25^\circ) = \frac{10.25}{x}$:     $x = \frac{10.25}{\cos(25^\circ)} \approx \frac{10.25}{0.90631} \approx 11.3096$
• Right Triangle Side Calculation:
  • Right triangle with given angle $26^\circ$ and adjacent side $15\,\text{cm}$, solve for opposite side $d$:     $\tan(26^\circ) = \frac{d}{15} \implies d = 15 \times \tan(26^\circ) \approx 15 \times 0.48773 \approx 7.32\,\text{cm}$
  • Right triangle with given angle $32^\circ$ and hypotenuse $38\,\text{cm}$, solve for opposite side $x$:     $\sin(32^\circ) = \frac{x}{38} \implies x = 38 \times \sin(32^\circ) \approx 38 \times 0.52992 \approx 20.14\,\text{cm}$
• Applied Angle of Depression Word Problem:
  • Scenario: From the top of a fire tower $45\,\text{m}$ in height, a forest ranger observes his partner on the ground at an angle of depression of $40^\circ$. Find the distance from the partner to the base of the tower to the nearest tenth of a meter.
  • Mathematical Model: The angle of depression from the top equals the angle of elevation from the partner ($40^\circ$). The vertical height of the tower forms the opposite side ($45\,\text{m}$) and the horizontal ground distance forms the adjacent side ($d$).
  • Calculation:     $\tan(40^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{45}{d}$$d = \frac{45}{\tan(40^\circ)} \approx \frac{45}{0.83910} \approx 53.6289\,\text{m}$
  • Final Answer: Rounded to the nearest tenth of a meter, the partner is $53.6\,\text{m}$ from the base of the tower.