Year 8 2013 Mathematics with Miss Hudson

Whole Numbers, Integers, Counting Numbers! The first numbers used are counting numbers {1, 2, 3, 4…} Then these with zero make up whole numbers {0, 1, 2, 3, 4…}. The whole numbers together with their opposites, -1, -2, -3…. make up the set of integers {… -3, -2, -1, 0, 1, 2, 3, 4…} Whole Numbers, Integers, Counting Numbers!

Prime Numbers A prime number has exactly two factors, itself and 1. 2 is the smallest prime number, 1 is not a prime number. (After 2 , every other even number is not a prime!) Prime numbers are: {2, 3, 5, 7, 11, 13 ………..} By yourself list the next 6 Prime Numbers?

Composite Numbers A Composite number is a natural number (positive integer) which is greater than 1 Has more than 2 factors. Example 1: The number 9 is a composite number because it has the factors of 1, 3 and 9. Example 2: 12 is a composite number because its factors are 1, 2, 3, 4, 6 and 12.

It’s just like listing out your times tables Multiples The multiples of a number are obtained by multiplying it by the natural (counting) numbers. example 1: The multiples of 10 are: 10, 20, 30, 40……… example 2: The multiples of 6 are: It’s just like listing out your times tables 6, 12, 18, 24, ………

Note: The factors of a number always include 1 and itself. The factors of a number are all the numbers that divide into it evenly with no remainders. e.g: The factors of 15 are {1, 3, 5, 15} Note: The factors of a number always include 1 and itself. Do you understand the difference between factors and multiples?

Quick Quiz! Multiples of: 4 = 7 = 16 = 25 = 32 = Factors of: 4 = 7 = 16 = 25 = 32 =

Working left to right subtraction comes first Order of Operations BIMDAS Brackets Indices Multiplication Division When both x and ÷ occur in a problem work from left to right. Addition Subtraction When both + and - occur in a question work from left to right Working left to right subtraction comes first examples: 1) 8  6 + 2  4 2) 16  3 x 4  4 _ 16 12

Examples of BIMDAS cont’d 13 x 4 4) 52  (2 x 32) + 4 ÷ 2 = 9 25  18 + 2 Work out the numerator first, the denominator second, and then do the division. 5) 2

(subtraction was done before multiplication) Order of Operations! It is very important to understand that it does make a difference if the order is not performed correctly!!!!  70 - 2x(5+3) = 70 - 2x(8) = 68 x (8) = 544 incorrect        = 70 - 2x( 8) = 70 - 16 = 54 correct       (subtraction was done before  multiplication)

Factor trees & Product of Primes E.g. Write 84 as a product of prime factors Start with ANY two numbers that multiply to give 84 84 We circle the prime numbers at the end of each branch. 2 42 2 21 3 7 Don’t forget to write the answer! 84 = 2 x 2 x 3 x 7 = 22 x 3 x 7

Highest Common Factor (HCF) The largest factor of two or more numbers! example: Find the HCF of 24 and 30 Method 1: The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24 The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30 The HCF of 24 and 30 is 6 Method 2: 2 24 30 12 15 3 4 5 HCF = 2 x 3 = 6

Lowest Common Multiple(LCM) The LCM of two numbers is the first multiple of each number that is the same. example: Find the LCM of 12 and 30 Method 1: Multiples of 12 are: 12, 24, 36, 48, 60, 72........ Multiples of 30 are: 30, 60, 90, 120…….. The LCM of 12 and 30 is 60 Method 2: 2 12 30 3 6 15 2 5 LCM = 2 x 3 x 2 x 5 = 60

Index Notation/Integer Powers 4 x 4 x 4 x 4 x 4 = 45 2 x 2 x 5 x 5 x 5 = 22 x 53 4 5 is read as “4 to the power of 5” Two special names are squared, e.g. 4 squared = 4 x 4 = 4 2 cubed, e.g. 2 cubed = 2 x 2 x 2 = 2 3 index/power/exponent base

  Powers on a calculator Calculators generally have for x squared for x cubed for other powers To calculate 4 9 we would key in: and get the answer 262144 x 2 x 3   4 9 = Always enter negatives with brackets. (-3)2 = ( (-) 3 ) x 2 = 9

1st Index Law: Multiplication When multiplying terms with the same base we add the powers. Eg 1: y3 x y4 = y7 Eg 2: 23 x 25 = 28 Eg 3: 3 m2 x 2 m3 = 6 m5 Eg 4: c2d5 x 6c5d = 6 c7d6

2nd Index Law: Division 98 ÷ 95 = = 93 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9 So although it has a divide sign, when we are dividing indices we actually subtract them. However this can only be done when there is the same base!

When dividing terms with the same base we subtract the powers. eg 1: p 8 ÷ p 2 = p 6 5 5 x4 eg 2: 20x7 ÷ 4x3 1 2 m eg 3: or 3

Square Numbers & Square Roots If you multiply a number by itself you get a square number ie 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9; 4 x 4 = 16 etc So 1, 4, 9, 16, 25, 36……. are square numbers The square root of 25 is 5, because 5 x 5 = 25 The symbol for square root is so On the calculator: 2 5 = It’s cool to be square

Divisibility Tests A number is divisible by Rule 2 If it is even (ends in 0, 2, 4, 6 or 8) 3 If the sum of the digits is divisible by 3 4 If the number formed by the last 2 digits is divisible by 4 5 If it ends in 0 or 5 6 If it is divisible by 2 and 3 8 If the number formed by the last 3 digits is divisible by 8 9 If the sum of the digits is divisible by 9 10 If it ends in 0 12 If it is divisible by 3 and 4