Lecture Notes: Mathematics Problem Solving

Rounding and Estimation

• Rounding to Significant Figures:
  • Example: 6.32 rounds to 6; 6.51 rounds to 7; 0.503 rounds to 0.5.
  • Multiplication and division using estimates: 6 x 7 = 42, 42 ÷ 0.5 = 84.
  • Dividing by 0.5 doubles the number.

Prime Factorization

• Prime Factors of 180:
  • Prime factor tree: 180 → 18 and 10; 18 → 2 and 9; 9 → 3 and 3.
  • Write in index form: 180 = 2² x 3² x 5.

Lowest Common Multiple (LCM)

• Example: Packs of pencils (15) and boxes of pens (40).
  • LCM of 15, 40 is 120 (8 packs of pencils, 3 boxes of pens).

Highest Common Factor (HCF)

• Example: HCF of 72, 90.
  • Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
  • HCF is 18.

Inequalities and Number Lines

• Drawing Inequalities:
  • Example: x between -2 and 4, x ≥ 1.

Percentage Problems

• VAT Calculation:
  • Calculate 20% of £640, add to find total with VAT.
  • Subtract deposit, divide remainder into installments.
• Reverse Percentage:
  • Sale price is 80% of original; calculate original using division.

Simple and Compound Interest

• Simple Interest:
  • Calculate percentage of principal, add each year.
• Compound Interest:
  • Multiply initial amount by percentage factor raised to the power of years.

Depreciation of Assets

• Yearly Depreciation:
  • Calculate percentage of current value.
  • Reduce value annually by calculated amount.

Fractions

• Operations with Fractions:
  • Multiplication: Multiply numerators and denominators.
  • Division: Invert and multiply.
  • Addition/Subtraction: Common denominator needed.
• Converting Mixed Numbers:
  • Convert to improper fractions before operations.

Standard Form

• Conversion to/from Standard Form:
  • Move decimal to create a number between 1 and 10.
  • Negative exponents for small numbers.

Powers and Indices

• Operations with Powers:
  • Add powers when multiplying like bases.
  • Subtract powers when dividing like bases.
• Negative and Fractional Powers:
  • Negative: Reciprocal of the base.
  • Fraction: Root determined by denominator.

Combinations and Permutations

• Counting Combinations:
  • Multiply choices for each stage.

Error Intervals

• Understanding Bounds:
  • Determine upper and lower bounds for rounded numbers.

Calculator Usage

• Important Functions:
  • Understanding location and use of square root, cube root, trigonometric functions.

Recurring Decimals as Fractions

• Algebraic Method:
  • Set up equation with x to eliminate repeating part.

Surds

• Expanding and Simplifying Surds:
  • Expand double brackets, collect like terms.

Advanced Topics

• Compound Interest with Variable Rates:
  • Calculate year-on-year using different percentages.

These notes cover various mathematical techniques and problem-solving strategies, including rounding, prime factorization, working with percentages, interest calculations, handling fractions, and more. They are useful for students learning to approach these types of problems systematically.