Calculation, Addition and Subtraction Chapter Notes | Year 5 Mathematics IGCSE (Cambridge) PDF Download

Addition and Subtraction

• Addition and subtraction involve combining or removing quantities, often requiring estimation to predict results.
• Estimation involves rounding numbers to make approximate calculations:
  • Example: For 3,250 - 1,115, estimate by rounding to 3,200 - 1,100 = 2,100.
  • Example: For 1,972 + 2,853 + 287, estimate by rounding to 2,000 + 2,900 + 300 = 5,200.
• The commutative law applies to addition, allowing addends to be reordered:
  • Example: 1,972 + 287 = 287 + 1,972.
• Regrouping simplifies calculations:
  • Example: For 4,350 + 654, regroup 654 as 600 + 50 + 4 to add to 4,350.
  • Example: For 654 - 554, regroup 654 as 600 + 50 + 4 to subtract 554.
• Addends are the numbers being added in an addition calculation (e.g., in 1,972 + 287, the addends are 1,972 and 287).
• Mental methods are used for simpler calculations, while column methods are used for more complex ones.

Adding and Subtracting Decimal Numbers

• Decimal numbers include a whole part and a fractional part (e.g., tenths, hundredths).
• Addition and subtraction of decimals use place value and known facts:
  • Example: 3.4 + 2.3:
    • Decompose: 3.4 = 3 + 0.4, 2.3 = 2 + 0.3.
    • Add: 3 + 2 = 5, 0.4 + 0.3 = 0.7.
    • Total: 5 + 0.7 = 5.7.
  • Example: 6.8 - 2.5:
    • Decompose: 6.8 = 6 + 0.8, 2.5 = 2 + 0.5.
    • Subtract: 6 - 2 = 4, 0.8 - 0.5 = 0.3.
    • Total: 4 + 0.3 = 4.3.
• Number line representation:
  • For 3.4 + 2.3, start at 3.4, add 2 to reach 5.4, then add 0.3 to reach 5.7.
• Estimation for decimals:
  • Example: For 4.8 + 3.1, round to 5 + 3 = 8.
  • Example: For 9.3 - 3.7, round to 9 - 4 = 5.

Multiplying by a 2-Digit Number

• Multiplying a 3-digit number by a 2-digit number requires breaking down the calculation into manageable parts.
• Example: 247 × 26:
  • Break into 247 × 20 + 247 × 6:
    • 247 × 20 = 4,940.
    • 247 × 6 = 1,482.
    • Total: 4,940 + 1,482 = 6,422.
  • Place value method:
    • 200 × 20 = 4,000, 200 × 6 = 1,200.
    • 40 × 20 = 800, 40 × 6 = 240.
    • 7 × 20 = 140, 7 × 6 = 42.
    • Total: 4,000 + 800 + 1,200 + 140 + 240 + 42 = 6,422.
• Column method for 247 × 26:
  • Multiply 247 × 6 = 1,482.
  • Multiply 247 × 20 = 4,940.
  • Add: 1,482 + 4,940 = 6,422.
• Estimation helps predict the product:
  • Example: For 192 × 33, estimate 200 × 30 = 6,000.

Division

• Division involves splitting a quantity into equal groups, often resulting in a quotient and a remainder.
• Example: 694 ÷ 3 = 231 remainder 1:
  • 231 × 3 = 693, plus remainder 1 gives 693 + 1 = 694.
• Example: 85 ÷ 4 = 21 remainder 1:
  • 21 × 4 = 84, plus remainder 1 gives 84 + 1 = 85.
  • Remainder as a fraction: 1 ÷ 4 = 1/4, so 85 ÷ 4 = 21 1/4.
• Remainders represent the amount left after forming complete groups of the divisor.
• Checking division with multiplication:
  • Example: For 85 ÷ 4 = 21 r 1, check 21 × 4 + 1 = 84 + 1 = 85.

Order of Operations

• The order of operations determines the sequence for performing calculations involving multiple operations.
• Rule: Multiplication and division have priority over addition and subtraction, unless brackets are used.
• Example: 5 + 6 × 2:
  • Multiply first: 6 × 2 = 12.
  • Add: 5 + 12 = 17.
• Example: 10 - 6 ÷ 2:
  • Divide first: 6 ÷ 2 = 3.
  • Subtract: 10 - 3 = 7 (not 10 - 6 = 4, then 4 ÷ 2 = 2).
• Addition and subtraction are inverse operations and can sometimes be reordered without changing the result:
  • Example: 10 + 5 - 3 = 10 - 3 + 5 = 12.
• Multiplication and division order matters:
  • Example: 10 × 6 ÷ 2 ≠ 10 ÷ 2 × 6 (60 ÷ 2 = 30, but 10 ÷ 2 × 6 = 5 × 6 = 30 only coincidentally).