Mental Math Starters 1 of 60 Boardworks Ltd

Mental Math Starters 1 of 60 © Boardworks Ltd 2004

Work it out! 4 + 3 × 0. 6 43 – 7 8 5 ===– 17 133 5. 8 28 19 2 of 60 © Boardworks Ltd 2004

Work it out! 7 × 0. 4 22 – 3 6 9 2 ====– 10. 5 31. 5 1. 4 21 77 3 of 60 © Boardworks Ltd 2004

Work it out! 0. 2 12 – 4 3 9 2 +6 ===6. 04 150 15 22 87 4 of 60 © Boardworks Ltd 2004

Work it out! 2( – 13 3. 6 18 69 7 + 8) ===23. 2 154 – 10 30 52 5 of 60 © Boardworks Ltd 2004

Key Words and Introduction to Expressions 6 of 60 © Boardworks Ltd 2004

Writing expressions Here are some examples of algebraic expressions: 7 of 60 n+7 a number n plus 7 5–n 5 minus a number n 2 n 2 lots of the number n or 2 × n 6 divided by a number n 4 n + 5 4 lots of a number n plus 5 n 3 a number n multiplied by itself twice or n×n×n 3 × (n + 4) or 3(n + 4) a number n plus 4 and then times 3. © Boardworks Ltd 2004

Writing expressions Miss Green is holding n number of cubes in her hand: Write an expression for the number of cubes in her hand if: She takes 3 cubes away. n– 3 She doubles the number of cubes she is holding. 2×n 2 n or 8 of 60 © Boardworks Ltd 2004

Equivalent expression match 9 of 60 © Boardworks Ltd 2004

Simplifying Algebraic Expressions 10 of 60 © Boardworks Ltd 2004

Like terms An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ). A term is made up of numbers and letter symbols but not operators. For example, 3 a + 4 b – a + 5 is an expression. 3 a, 4 b, a and 5 are terms in the expression. 3 a and a are called like terms because they both contain a number and the letter symbol a. 11 of 60 © Boardworks Ltd 2004

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 5+5+5+5=4× 5 In algebra, a + a + a = 4 a The a’s are like terms. We collect together like terms to simplify the expression. 12 of 60 © Boardworks Ltd 2004

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, (7 × 4) + (3 × 4) = 10 × 4 In algebra, 7 × b + 3 × b = 10 × b or 7 b + 3 b = 10 b 7 b, 3 b and 10 b are like terms. They all contain a number and the letter b. 13 of 60 © Boardworks Ltd 2004

Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In arithmetic, 2 + (6 × 2) – (3 × 2) = 4 × 2 In algebra, x + 6 x – 3 x = 4 x x, 6 x, 3 x and 4 x are like terms. They all contain a number and the letter x. 14 of 60 © Boardworks Ltd 2004

Collecting together like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting together like terms. An expression can contain different like terms. For example, 3 a + 2 b + 4 a + 6 b = 3 a + 4 a + 2 b + 6 b = 7 a + 8 b This expression cannot be simplified any further. 15 of 60 © Boardworks Ltd 2004

Collecting together like terms Simplify these expressions by collecting together like terms. 1) a + a + a = 5 a 2) 5 b – 4 b = b 3) 4 c + 3 d + 3 – 2 c + 6 – d = 4 c – 2 c + 3 d – d + 3 + 6 = 2 c + 2 d + 9 4) 4 n + n 2 – 3 n = 4 n – 3 n + n 2 = n + n 2 5) 4 r + 6 s – t 16 of 60 Cannot be simplified © Boardworks Ltd 2004

Algebraic perimeters Remember, to find the perimeter of a shape we add together the lengths of each of its sides. Write algebraic expressions for the perimeters of the following shapes: 2 a 3 b Perimeter = 2 a + 3 b + 2 a + 3 b = 4 a + 6 b 5 x 4 y x 5 x 17 of 60 Perimeter = 4 y + 5 x = 4 y + 11 x © Boardworks Ltd 2004

Algebraic pyramids 18 of 60 © Boardworks Ltd 2004

Algebraic magic square 19 of 60 © Boardworks Ltd 2004

Using Index Notation 20 of 60 © Boardworks Ltd 2004

Using index notation Simplify: x + x + x = 5 x Simplify: x × x × x = x 5 x to the power of 5 This is called index notation. Similarly, x × x = x 2 x × x = x 3 x × x × x = x 4 21 of 60 © Boardworks Ltd 2004

Using index notation We can use index notation to simplify expressions. For example, 3 p × 2 p = 3 × p × 2 × p = 6 p 2 q 2 × q 3 = q × q × q = q 5 3 r × r 2 = 3 × r × r = 3 r 3 2 t × 2 t = (2 t)2 22 of 60 or 4 t 2 © Boardworks Ltd 2004

Expressions of the form (xm)n When a term is raised to a power and the result raised to another power, the powers are multiplied. For example, (a 5)3 = a 5 × a 5 = a(5 + 5) = a 15 = a(3 × 5) In general, (xm)n = xmn 23 of 60 © Boardworks Ltd 2004

Expressions of the form (xm)n Rewrite the following without brackets. 1) (2 a 2)3 = 8 a 6 2) (m 3 n)4 = m 12 n 4 3) (t– 4)2 = t– 8 4) (3 g 5)3 = 27 g 15 5) (ab– 2)– 2 = a– 2 b 4 6) (p 2 q– 5)– 1 = p– 2 q 5 7) (h½)2 = h 8) (7 a 4 b– 3)0 = 1 24 of 60 © Boardworks Ltd 2004

The zero index Look at the following division: Any number or term divided by itself is equal to 1. y 4 ÷ y 4 = 1 But using the rule that xm ÷ xn = x(m – n) y 4 ÷ y 4 = y(4 – 4) = y 0 That means that y 0 = 1 In general, for all x 0, x 0 = 1 25 of 60 © Boardworks Ltd 2004

Fractional Indices 26 of 60 © Boardworks Ltd 2004

$Fractional indices Indices can also be fractional. x ×x = x 1 2 So,$

Fractional indices Indices can also be fractional. x ×x = x 1 2 So, 27 of 60 = x 1 = x The square root of x. 1 2 x = x So, But, 1 +2 x × x = x But, Similarly, 1 2 1 3 1 3 x ×x ×x = x 3 1 3 + 3 1 3 = x 1 = x 3 x × x = x 1 3 3 x = x The cube root of x. © Boardworks Ltd 2004

Fractional indices In general, n 1 n x = x 1 m Also, we can write x n as x n × m. Using the rule that (xm)n = xmn, we can write 1 n x m n We can also write x as xm × 1 n ×m = (x 1 n )m n = ( x)m . 1 x m× n = 1 (xm)n n = xm In general, m n x = 28 of 60 n m x or m n n x = ( x)m © Boardworks Ltd 2004

Index laws Here is a summary of the index laws. x– 1 = 1 x x–n = 1 n x xm × xn = x(m + n) xm ÷ xn = x(m – n) (xm)n = xmn x 1 x 0 29 of 60 x = x 1 2 =x = 1 (for x = 0) n x = x 1 n m n x = n xm n or ( x)m © Boardworks Ltd 2004

Multiplying Algebraic Terms 30 of 60 © Boardworks Ltd 2004

Multiplying terms together In algebra we usually leave out the multiplication sign ×. Any numbers must be written at the front and all letters should be written in alphabetical order. For example, 4 × a = 4 a 1×b=b We don’t need to write a 1 in front of the letter. b × 5 = 5 b We don’t write b 5. 3 × d × c = 3 cd We write letters in alphabetical order. 6 × e = 6 e 2 31 of 60 © Boardworks Ltd 2004

Brackets Look at this algebraic expression: 4(a + b) What do you think it means? Remember, in algebra we do not write the multiplication sign, ×. This expression actually means: 4 × (a + b) or (a + b) + (a + b) =a+b+a+b+a+b = 4 a + 4 b 32 of 60 © Boardworks Ltd 2004

Using the grid method to expand brackets 33 of 60 © Boardworks Ltd 2004

Expanding Algebraic Expressions 34 of 60 © Boardworks Ltd 2004

Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example, 2(5 – x) We need to multiply the bracket by 2. 10 – 2 x 35 of 60 © Boardworks Ltd 2004

Expanding brackets then simplifying Simplify 4 – (5 n – 3) We need to multiply the bracket by – 1 and collect together like terms. 4 – 5 n + 3 = 4 + 3 – 5 n = 7 – 5 n 36 of 60 © Boardworks Ltd 2004

Expanding brackets then simplifying Simplify 2(3 n – 4) + 3(3 n + 5) We need to multiply out both brackets and collect together like terms. 6 n – 8 + 9 n + 15 = 6 n + 9 n – 8 + 15 = 15 n + 7 37 of 60 © Boardworks Ltd 2004

Expanding brackets then simplifying Simplify 5(3 a + 2 b) – 2(2 a + 5 b) We need to multiply out both brackets and collect together like terms. 15 a + 10 b – 4 a – 10 b = 15 a – 4 a + 10 b – 10 b = 11 a 38 of 60 © Boardworks Ltd 2004