Fractions and Percentages Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) PDF Download

Getting Started

• Shapes are matched with their names to understand their properties.
• Lines of symmetry in triangles:
  • Isosceles triangle: Has one line of symmetry, passing through the vertex opposite the equal sides and bisecting the base.
  • Scalene triangle: Has no lines of symmetry, as all sides and angles are different.
  • Equilateral triangle: Has three lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
• Composite shapes (e.g., a semicircle on a rectangle):
  • Right angles are formed where straight edges meet perpendicularly.
  • Curved edge comes from the semicircle.
  • Straight edges include the rectangle’s sides and the diameter of the semicircle.
  • Parallel sides exist in the rectangle portion.
• Symmetry in rectangles:
  • A rectangle has two lines of symmetry: one horizontal (through the midpoints of the vertical sides) and one vertical (through the midpoints of the horizontal sides).
• Shapes in everyday life:
  • Grocery store items (boxes, tins, packets) feature shapes like rectangles, cylinders, and prisms.
  • Graphic designers choose shapes for packaging to enhance visual appeal and functionality.

Quadrilaterals

• Objectives:
  • Identify quadrilaterals by their properties.
  • Describe quadrilaterals using geometric terms.
  • Classify quadrilaterals based on sides, angles, and symmetry.
  • Sketch quadrilaterals accurately.
• Quadrilaterals in real life:
  • Used as tiles in kitchens and bathrooms because they tessellate (fit together without gaps).
  • Understanding properties is essential for ordering tiles.
• Key terms:
  • Bisect: A line (e.g., diagonal) divides another line into two equal parts.
  • Diagonal: A line segment connecting non-adjacent vertices.
  • Decompose: Break a shape into simpler shapes (e.g., a square into triangles).
  • Justify: Provide reasoning to support a geometric claim.
  • Parallel: Lines that never meet and are equidistant.
  • Trapezia: Plural of trapezium, a quadrilateral with one pair of parallel sides.
• Properties of a rectangle:
  • A quadrilateral with four sides.
  • Two pairs of equal sides (opposite sides are equal).
  • Two pairs of parallel sides (opposite sides are parallel).
  • Sides meet at 90° (four right angles).
  • Diagonals bisect each other (cross at their midpoints).
  • Two lines of symmetry (horizontal and vertical).
• Properties of a square:
  • A quadrilateral with four sides.
  • Four equal sides.
  • Two pairs of parallel sides.
  • Sides meet at 90° (four right angles).
  • Diagonals bisect each other at 90°.
  • Four lines of symmetry (two along midpoints, two along diagonals).
• Properties of a parallelogram:
  • A quadrilateral with four sides.
  • Two pairs of equal sides (opposite sides are equal).
  • Two pairs of parallel sides (opposite sides are parallel).
  • Two pairs of equal angles (opposite angles are equal).
  • Diagonals bisect each other.
  • No lines of symmetry (unless it’s a rectangle or rhombus).
• Properties of trapezia:
  • Isosceles trapezium:
    • One pair of parallel sides.
    • Non-parallel sides (legs) are equal.
    • Base angles are equal.
    • One line of symmetry (through the midpoints of the parallel sides).
  • Non-isosceles trapezium:
    • One pair of parallel sides.
    • Non-parallel sides are unequal.
    • No lines of symmetry.
• Properties of special quadrilaterals (summary):
  • Square:
    • Four equal sides.
    • Two pairs of parallel sides.
    • All angles 90°.
    • Diagonals bisect each other at 90°.
    • Four lines of symmetry.
  • Rectangle:
    • Two pairs of equal sides.
    • Two pairs of parallel sides.
    • All angles 90°.
    • Diagonals bisect each other.
    • Two lines of symmetry.
  • Parallelogram:
    • Two pairs of equal sides.
    • Two pairs of parallel sides.
    • Two pairs of equal angles.
    • Diagonals bisect each other.
    • No lines of symmetry (generally).
  • Trapezium:
    • One pair of parallel sides.
    • No specific equal sides or angles unless isosceles.
  • Isosceles trapezium:
    • One pair of parallel sides.
    • One pair of equal sides (non-parallel).
    • One pair of equal angles.
    • Diagonals bisect each other.
    • One line of symmetry.
  • Rhombus:
    • Four equal sides.
    • Two pairs of parallel sides.
    • Two pairs of equal angles.
    • Diagonals bisect each other at 90°.
    • Two lines of symmetry (along diagonals).
  • Kite:
    • Two pairs of equal adjacent sides.
    • One pair of equal angles (between unequal sides).
    • Diagonals meet at 90°.
    • One line of symmetry (through the diagonal joining equal sides).
• Tessellation:
  • Quadrilaterals like squares, rectangles, and parallelograms tessellate because they can fit together without gaps.
  • Not all quadrilaterals tessellate (e.g., kites may leave gaps).
• Decomposition:
  • Quadrilaterals can be broken into simpler shapes:
    • Square: Two triangles, two trapezia, or a rectangle and two triangles.
    • Other quadrilaterals can be similarly decomposed for analysis.

Circles

• Objectives:
  • Learn the names of circle parts.
  • Draw circles accurately using compasses.
• Parts of a circle:
  • Centre: The fixed point equidistant from all points on the circle.
  • Radius: The distance from the centre to any point on the circle.
  • Circumference: The perimeter of the circle (total length around it).
  • Diameter: A line segment passing through the centre, joining two points on the circumference (diameter = 2 × radius).
• Drawing a circle:
  • Use compasses set to the desired radius (e.g., 4 cm).
  • Mark the centre with a dot.
  • Rotate compasses around the centre to draw the circumference.
  • Label the centre, radius, diameter, and circumference.
• Relationship between radius and diameter:
  • Equation: diameter = 2 × radius.
  • Example: If radius = 3 cm, diameter = 6 cm.
  • Conversion: 1 cm = 10 mm, so a diameter of 60 mm = 6 cm.
• Circle properties:
  • All points on the circumference are equidistant from the centre.
  • Touching circles: The distance between the centres of two touching circles equals the sum of their radii.

Rotational Symmetry

• Objectives:
  • Identify shapes and patterns with rotational symmetry.
  • Describe rotational symmetry.
• Definition:
  • A shape has rotational symmetry if it looks the same after being rotated about a point (the centre) by a certain angle.
  • Order of rotational symmetry: The number of times a shape looks identical during one full 360° rotation.
• Examples:
  • Rectangle: Order 2 (looks the same after 180° and 360° rotations).
  • Parallelogram: Order 2 (looks the same after 180° and 360° rotations).
  • Isosceles trapezium: Order 1 (only looks the same after 360°).
  • Button with four-fold symmetry: Order 4 (looks the same every 90°).
• Rotational symmetry in quadrilaterals:
  • Square: Order 4 (every 90°).
  • Rectangle: Order 2 (every 180°).
  • Parallelogram: Order 2 (every 180°).
  • Rhombus: Order 2 (every 180°).
  • Kite: Order 1 (only 360°).
  • Trapezium: Order 1 (only 360°).
  • Isosceles trapezium: Order 1 (only 360°).
• Rotational symmetry in other shapes:
  • Equilateral triangle: Order 3 (every 120°).
  • Scalene triangle: Order 1 (only 360°).
• Patterns and tiles:
  • Combining shapes can change the order of rotational symmetry.
  • Example: Joining two tiles may create a pattern with a different order than the individual tiles.
• Relationship with line symmetry:
  • Not all shapes have the same number of lines of symmetry and order of rotational symmetry (e.g., a square has 4 lines and order 4, but a parallelogram has 0 lines and order 2).
  • Shapes with no line symmetry may still have rotational symmetry (e.g., parallelogram).