Overview
This lecture covers foundational secondary school math topics including number concepts, algebra, geometry, rates, data representation, and practical math applications, with concise explanations and problem-solving strategies.
Working with Numbers
- A negative sign flips the value (e.g., -(-5) = 5).
- To round to significant figures, cover all but the first digits, check the next, and round up if it's 5 or more.
- Leading zeros don’t count as significant figures.
Prime Numbers & Factorization
- Prime numbers are divisible only by 1 and themselves (e.g., 2, 3, 5, 7, 11).
- Prime factorization splits numbers into products of primes using methods like "ladder" division.
- The highest common factor (HCF) uses the lowest powers of shared primes; the lowest common multiple (LCM) uses the highest and all unique primes.
Algebraic Manipulation
- Simplify expressions by combining like terms.
- Expand by distributing terms across brackets.
- Factorize by taking out common factors.
- Solve equations by isolating unknowns; cross-multiply for equations with two fractions.
Rates, Ratios, and Proportions
- Calculate rates (e.g., speed, consumption) as quantity per time.
- Add rates when working together.
- Convert units by multiplying/dividing by conversion factors.
- Average speed = total distance ÷ total time (account for all intervals).
Percentages & Finance
- Percentage change = (final - original) ÷ original × 100%.
- For installment plans with interest, use I = PRT and convert all time to years.
- Cumulative gain/loss is not neutral if percentages are applied successively.
Sequences & Patterns
- Arithmetic sequence formula: general term = first term + (n-1) × common difference.
- For fractional sequences, analyze numerator and denominator patterns.
Graphs and Linear Equations
- Points on coordinate plane are written (x, y).
- Straight line: y = mx + c, where m is gradient (rise/run), c is y-intercept.
- Gradient between two points: change in y ÷ change in x.
- To find intercepts, set x or y to zero and solve.
Geometry: Angles, Triangles, Polygons
- Adjacent angles on a straight line add to 180°, angles at a point add to 360°.
- Vertically opposite angles are equal; alternate and corresponding angles only with parallel lines.
- Interior angle sum of polygon: (n-2) × 180°.
- Each regular polygon’s interior angle = sum ÷ n; exterior angle = 360 ÷ n.
Construction & Area
- Construct triangles using ruler, protractor, compass.
- Area formulas: triangle = ½ × base × height; parallelogram = base × height; trapezium = ½(a+b) × height.
- Volume of cylinder = πr²h; surface area = 2πrh + area of bases.
Data Representation
- Bar graphs for categories, line graphs for trends, pie charts for proportions.
- To create a pie chart, convert each category’s value into a sector angle using 360 × (category/total).
Key Terms & Definitions
- Prime Number — integer greater than 1 with no divisors except 1 and itself.
- Significant Figures — digits conveying meaningful precision in a number.
- Factorization — expressing a number or expression as a product of its factors.
- Gradient (Slope) — steepness of a line; change in y per unit change in x.
- Y-intercept — point where a line crosses the y-axis.
- Arithmetic Sequence — sequence with a constant difference between terms.
- Polygon — closed shape with three or more straight sides.
Action Items / Next Steps
- Practice rounding to significant figures and performing prime factorization.
- Complete algebra, rate, and geometry exercises as assigned.
- Review area and volume formula applications.
- Prepare a bar graph, line graph, and pie chart using provided data sets.