Exploring Measures Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) - Class 6 PDF Download

Getting Started

• Measures in mathematics:
  • Measures are practical applications of mathematics used to solve problems, investigate, and manage everyday tasks.
  • Examples include measuring length, area, perimeter, time, and quantities for planning or purchasing.
• Applications of measurement:
  • Ruler: Measures lengths (e.g., sides of a shape to calculate perimeter or area).
  • Calendar: Tracks days, months, or schedules (e.g., planning events or deadlines).
  • Airport departure board: Displays time and schedules for flights, aiding in time management.
• Time adjustments:
  • Clocks showing incorrect time require adjustments based on whether they are fast or slow.
  • Digital clock notation (e.g., 18:30) is used to express corrected times, considering AM/PM or 24-hour formats.

Rectangles and Triangles

• Objectives:
  • Estimate the area of a triangle.
  • Calculate the area of triangles using rectangles.
• Area and perimeter calculations:
  • Rectangle area: Area = length × width.
  • Triangle area (when formed by cutting a rectangle diagonally): Area = (length × width) ÷ 2.
  • Perimeter: Sum of all side lengths of the shape.
  • Units: Area uses square units (e.g., mm², cm², m², km²); perimeter uses linear units (e.g., mm, cm, m, km).
• Real-world application:
  • Example: Calculating the number of 1 kg bags of grass seed needed to cover a field (1 kg covers 20 m²).
  • Process: Measure the field’s area (e.g., approximate as a rectangle or triangle), then divide by 20 m² to determine the number of bags.
• Area of composite shapes:
  • Rectangle formed by two squares:
    • If each square has side length s, the rectangle’s dimensions are 2s × s.
    • Rectangle area = 2s × s = 2s².
    • Each square’s area = s².
  • Square formed by two identical rectangles:
    • If the square’s side is s, each rectangle’s dimensions are s × (s/2).
    • Square area = s × s = s².
    • Each rectangle’s area = s × (s/2) = s²/2.
• Dividing shapes:
  • Paper cut in half:
    • Original area = length × width (e.g., 21 cm × 30 cm = 630 cm²).
    • If cut along the length, each piece’s area = (length × width) ÷ 2.
    • If cut along the width, each piece’s area = (length × width) ÷ 2, but dimensions differ.
• Estimating triangle area:
  • Count squares on a grid to estimate the area of a triangle.
  • Half-covered squares are approximated as 0.5 square units.
  • Observation: A triangle formed by cutting a rectangle diagonally has half the rectangle’s area.
• Worked example:
  • Rectangle (4 m × 8 m) cut diagonally into two triangles:
    • Rectangle area = 4 m × 8 m = 32 m².
    • Each triangle’s area = 32 m² ÷ 2 = 16 m².
• Practical applications:
  • Example: Triangular biscuits made by cutting 5 cm squares diagonally.
    • Square area = 5 cm × 5 cm = 25 cm².
    • Triangle area = 25 cm² ÷ 2 = 12.5 cm².
    • Icing coverage: If 400 g covers 340 cm², number of biscuits = 340 cm² ÷ 12.5 cm².

Time

• Objective:
  • Convert between time intervals expressed as decimals and mixed units (hours and minutes).
• Challenges with time:
  • Time units are not decimal-based (e.g., 60 seconds = 1 minute, 60 minutes = 1 hour, 24 hours = 1 day).
  • Key numbers related to time:
    • 60: Seconds per minute, minutes per hour.
    • 12: Hours on a clock face, months in a year.
    • 365: Days in a non-leap year.
    • 30: Average days in a month.
    • 100: Seconds or minutes in some contexts (e.g., 100 seconds).
    • 7: Days in a week.
    • 366: Days in a leap year.
• Time conversion:
  • Convert decimal hours to hours and minutes:
    • Example: 3.7 hours.
      • Whole hours = 3.
      • Decimal part = 0.7 hours.
      • 0.7 hours × 60 minutes/hour = 42 minutes.
      • Result: 3 hours and 42 minutes.
    • Alternative method: 0.1 hour = 6 minutes, so 0.7 hours = 7 × 6 = 42 minutes.
• Practical examples:
  • Dividing time:
    • 5 hours shared equally among 4 children:
      • Each gets 5 ÷ 4 = 1.25 hours.
      • 1.25 hours = 1 hour + 0.25 hours.
      • 0.25 hours × 60 = 15 minutes.
      • Result: 1 hour and 15 minutes per child.
  • Marathon times:
    • Convert minutes to hours, minutes, and seconds (e.g., 159.1 minutes):
      • Whole minutes = 159.
      • Decimal part = 0.1 minutes = 0.1 × 60 = 6 seconds.
      • 159 minutes = 2 hours (120 minutes) + 39 minutes.
      • Result: 2 hours, 39 minutes, 6 seconds.