Numbers and Sequences Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) - Class 6 PDF Download

Counting and sequences

Geometric Sequences

• The objective is to count on and back using fractions and decimals.
• Find and use the position-to-term rule of a sequence.
• Sequences are ordered lists of numbers following a specific rule, e.g., 1, 4, 7, 8, 6, 6 or patterns in nature like beehive structures.
• Sequences can represent geometric patterns, e.g., the number of shapes (rectangles and triangles) in a growing pattern can form a sequence with a position-to-term rule.

Term-to-term rule

• Describes how to get from one term to the next, e.g., add 4 to get 4, 8, 12, 16.
• Example: Start at 4, add 7 each time to get 4, 11, 18, 25, 32, 39.

Position-to-term rule

• Relates the position (n) of a term to its value directly, e.g., for 4, 8, 12, 16, the rule is term = n × 4, so the 25th term is 25 × 4 = 100.
• Example: For a sequence starting at 2, adding 2 each time (2, 4, 6, 8, 10, 12, 14, 16, 18, 20), the position-to-term rule is term = n × 2, so the 50th term is 50 × 2 = 100.

Counting with fractions and decimals

• Count in steps of fractions, e.g., counting up in quarters: 1/4, 1/2, 3/4, 1, 5/4, 3/2.
• Count in decimal steps, e.g., start at 15, count back by 0.4: 15, 14.6, 14.2, 13.8.
• Example: Counting from 20 in steps of 1.001: 20, 21.001, 22.002, 23.003, ….

Special numbers

• The objective is to work out the square number in any position, e.g., the ninth square number is 9 × 9 = 81.
• Use the notation ² to represent squared.
• Know the cube of numbers up to 5, e.g., 5³ = 5 × 5 × 5 = 125.
• Square and cube numbers can be organized in diagrams like Carroll diagrams to classify properties, e.g., odd/not odd, cube/not cube.

Square numbers

• Result of multiplying a number by itself, denoted as n², e.g., 1² = 1, 2² = 4, 3² = 9, 5² = 25, 7² = 49, 10² = 100.
• Ninth square number: 9² = 81.
• Example: Two square numbers summing to 45: 9 + 36 = 3² + 6² = 45.

Cube numbers

• Result of multiplying a number by itself twice, denoted as n³, e.g., 1³ = 1, 3³ = 27, 5³ = 125.
• Example: In a Rubik’s cube, the number of small cubes is a cube number, e.g., a 3 × 3 × 3 cube has 3³ = 27 small cubes.
• Example: Two cube numbers summing to 152: 8 + 144 = 2³ + 6³ = 152 (noting 6³ = 216, so this may refer to a different context).

Comparing powers

• Example: Compare 2³ and 3²: 2³ = 2 × 2 × 2 = 8, 3² = 3 × 3 = 9, so 3² > 2³.

Common multiples and factors

• The objective is to find common multiples.
• Find common factors.

Multiples

• Numbers obtained by multiplying a number by integers, e.g., multiples of 3: 3, 6, 9, 12, 15, 18, …; multiples of 5: 5, 10, 15, ….
• Common multiples: Numbers appearing in the multiple lists of two or more numbers, e.g., common multiple of 3 and 5 less than 20 is 15.
• Example: Common multiples of 3 and 8 less than 50: 24, 48.

Factors

• Numbers that divide another number without a remainder, e.g., factors of 18: 1, 2, 3, 6, 9, 18; factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
• Common factors: Factors shared by two or more numbers, e.g., common factors of 18 and 24: 1, 2, 3, 6.
• Example: Factors of 30 that are also factors of 20: 1, 2, 5, 10.
• Common multiples and factors can be visualized using Venn diagrams, with overlapping sections for numbers that are multiples or factors of multiple sets.