Calculation, Positive and Negative Numbers Chapter Notes | Year 5 Mathematics IGCSE (Cambridge) - Class 5 PDF Download

Calculating with Positive and Negative Numbers

• Positive numbers are greater than zero, while negative numbers are less than zero.
• The difference between two temperatures can involve positive and negative numbers, depending on whether the temperature increases or decreases.
• A number line is used to perform additions and subtractions with negative numbers, especially when crossing zero.
• Bridging zero is necessary when the calculation moves from positive to negative numbers or vice versa.
• Example calculations using number facts for 9 to bridge zero:
  • 4 - 5 = 4 + 4 - 4 - 5 = 9 - 5 = -1.
  • 5 - 9 = 5 - 5 - 4 = 0 - 4 = -4.

Addition and Subtraction

• Estimation involves approximating the result of a calculation before performing it exactly.
• Example: Estimating the distance left to row or the total mass of a team requires rounding numbers to make quick approximations.
• Regrouping or reordering numbers can simplify addition:
  • Example: 346 + 1,750 + 250 = (346 + 250) + 1,750 = 596 + 1,750 = 2,346.
• Subtraction does not always need to follow the given order; regrouping can help simplify calculations.
• Example sequence of operations starting with 4,653:
  • Add 3,000: 4,653 + 3,000 = 7,653.
  • Subtract 499:7,653 - 499 = 7,154.
  • Subtract 2,486: 7,154 - 2,486 = 4,668.
  • Add 32: 4,668 + 32 = 4,700.
• Mental calculations are suitable for simpler operations, while written methods (e.g., column method) are used for more complex ones.

Missing Number Problems

• Inverse operations (e.g., addition as the inverse of subtraction) are used to find unknown quantities.
• Symbols represent unknown quantities in number sentences.
• Example: For shopping bag problems:
  • $20 - $2 = $18 (total for three trucks).
  • $18 ÷ 3 = $6 (cost of one truck).
  • $15 - $6 = $9 (cost of the model airplane).
• Related facts help solve problems with multiple unknowns by using known values to deduce others.

Simplifying Multiplications

• The distributive law regroups numbers to simplify multiplication:
  • Example: 19 × 5 = (20 - 1) × 5 = (20 × 5) - (1 × 5) = 100 - 5 = 95.
  • Example: 21 × 5 = (20 + 1) × 5 = (20 × 5) + (1 × 5) = 100 + 5 = 105.
• The commutative law allows reordering of factors:
  • Example: 46 × 2 × 5 = 46 × (2 × 5) = 46 × 10 = 460.
• The associative law groups factors to simplify calculations:
  • Example: 46 × 2 × 5 = (46 × 2) × 5 = 92 × 5 = 460.
• Using factors simplifies large multiplications:
  • Example: 90 × 40 = (9 × 10) × (4 × 10) = (9 × 4) × (10 × 10) = 36 × 100 = 3,600.

Multiplying Numbers up to 1,000

Estimation is used to approximate products before calculating exactly.

Place value representation for multiplication:
Example: 234 × 3can be broken into:

• 200 × 3 = 600.
• 30 × 3 = 90.
• 4 × 3 = 12.
• Total: 600 + 90 + 12 = 702.

Example: 34 × 13 can be represented as:

• 30 × 10 = 300.
• 30 × 3 = 90.
• 4 × 10 = 40.
• 4 × 3 = 12.
• Total: 300 + 90 + 40 + 12 = 442.

Patterns in multiplication:

• Example: 27 × 10 = 270, 27 × 12 = 27 × (10 + 2) = 270 + 54 = 324.

Estimating before multiplying helps predict whether the actual product will be larger or smaller than the estimate:

• Example: For 597 × 4, estimate 600 × 4 = 2,400. The actual answer is slightly less.

Relationship between products:

• (234 × 4) + (345 × 4) = (234 + 345) × 4 = 579 × 4.