Notes on Chapter 5: Prime Time

5.1 Common Multiples and Common Factors

The chapter starts with a fun game called the Idli-Vada Game, where children say numbers in a sequence but replace multiples of 3 with "idli" and multiples of 5 with "vada". If a number is a multiple of both, they say "idli-vada". This game serves to illustrate the concepts of common multiples and common factors.

• Common Multiples: Numbers that are multiples of two or more numbers (e.g., for 3 and 5, the common multiples include 15, 30, 45, etc.).
• Common Factors: Numbers that divide two or more numbers without leaving a remainder.

Engagement Questions

1. Players need to determine how many times they say idli and vada while playing up to 90, finding patterns in common multiples.

Jump Jackpot Game

This game introduces the concept of factors through a character named Jumpy, who jumps on multiples of a chosen number. Jumpy collects treasures only if he can land on both numbers Grumpy has hidden treasures on. Through this scenario, players learn how factors of a number determine multiples that can land on that number.

5.2 Prime Numbers

Definition of Primes and Composites

• Prime Numbers: Natural numbers greater than 1 with only two factors: 1 and itself (e.g., 2, 3, 5, 7).
• Composite Numbers: Numbers with more than two factors (e.g., 4, 6, 8).

Finding Prime Numbers

The Sieve of Eratosthenes is introduced as a method to identify all prime numbers up to a certain number by systematically crossing out multiples of each prime starting from 2. Examples include finding all primes from 1 to 100.

5.3 Co-prime Numbers

Co-prime numbers have no common factors other than 1. The game layout illustrates pairs that are co-prime versus those that are not by checking common factors. Pair examples include:

• 16 and 27 (co-prime)
• 12 and 18 (not co-prime)

5.4 Prime Factorisation

Every integer greater than 1 can be expressed as a product of prime numbers. This is called its prime factorisation. Key points:

• A number's prime factorisation is unique, with the order of factors not affecting the product.
• The process involves breaking down composites until all remaining factors are prime numbers.

5.5 Divisibility Tests

The chapter concludes with various divisibility tests for numbers (by 2, 3, 4, 5, etc.), emphasizing patterns in their digits that can simplify checking for factors without performing repeated division.

5.6 Fun with Numbers

The chapter wraps up with engaging activities aimed at observing properties of different numbers, vital for recognizing special characteristics such as primes or specific numerical patterns.

1. Multiples: Common multiples are shared multiples of two or more numbers.
2. Factors: Common factors are integers that divide two or more numbers evenly.
3. Prime Numbers: Primes have two factors: 1 and the number itself.
4. Composite Numbers: Composites have more than two factors.
5. Co-prime: Numbers with no common factors other than 1.
6. Prime Factorisation: Unique representation of a number as a product of primes.
7. Divisibility Tests: Simple methods based on last digits can determine factors without long division.
8. Sieve of Eratosthenes: A method to identify all prime numbers up to a specified limit.