MATHS Week 12 Algebra

Starter! How much does 1 pineapple weigh? What about 1 apple?

What did we do last week?

Before we begin……

Have you done your directed study?

What are we going to do this week? Algebra Terminology Expand with two brackets Factorise Quadratics Laws of indices Solving, writing & rearranging

Algebraic Terminology

Can you identify which is which? You need to pick four different colours/highlighters. Can you colour code the grid to find which of the boxes contain expressions, formulae, identities and equations?

Answers

Expand 3(y + 10) Expand h(h – 5)

Expanding two brackets using the FOIL method

You may be asked to expand and simplify two brackets that look like this: (y + 6)(y + 3)

F O I L irst utside nside ast

Question – to do together (x + 3)(x + 6) F O I L

Question – to do together (4x – 3)(3x – 4) F O I L

Question – to do together (y – 7)2 F O I L

Your Turn! Remove the brackets and simplify: (x + 4)(x + 1)

ANSWERS x2 + 5x + 4 x2 – 5x + 6 x2 + 10x + 25 What do you notice about all these answers? x² + bx + c Do you know what these are called?

Factorising Factorise 5p – 25 Factorise mn + mt Factorise 9xy – 3y Factorise w2 + 3wz

Factorising Quadratics

Factorising Quadratics You can factorise quadratic expressions of the form x² + bx + c Find two numbers whose products is + c and whose sum is +b Use these two numbers, p and q, to write down the factorised form (x + p) (x + q)

Example – do together

Example – do together

Example – do together

Example – do together

Your turn x² + 3x – 18 b) x² - 6x + 5 c) x² + 9x + 20 d) x² + 3x -28 e) x² + 2x – 24 f) x² + 8x + 7

Answers a) ( x - 3) (x + 6) b) ( x – 5) ( x – 1) c) ( x + 4) (x + 5) d) ( x - 4) (x + 7) e) ( x - 4) (x + 6) f) ( x + 1) (x + 7)

Multiply out and simplify (w – 3)(w + 3) (x + 1)(x – 1) (y + 4)(y – 4) (z + 5)(z – 5) What do you notice?

Factorise x² - 36 b) x² - 49 c) y² - 144

Index Notation and Rules of Indices

2 x 2 x 2 x 2 x 2 25 is read 2 to the power of 5 can be written 25 5 is the power or index 2 x 2 x 2 x 2 x 2 25 is read 2 to the power of 5 2 is the base can be written 25

Example Calculate 34 34 = = 81 3 x 3 x 3 x 3

Write 7 x 7 x 7 x 7 x 7 x 7 in index form. Example Write 7 x 7 x 7 x 7 x 7 x 7 in index form. 76 7 x 7 x 7 x 7 x 7 x 7 =

Example Simplify 23 x 24 23 x 24 = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27 Provided the base numbers are the same you can add the powers when multiplying.

ax x ay = ax + y 43 x 45 = 4(3 + 5) = 48 Example Simplify 43 x 45 In general ax x ay = ax + y

Example Simplify 85 ÷ 83 85 ÷ 83 = 8 x 8 x 8 Provided the base numbers are the same you can subtract the powers when dividing. 8 x 8 x 8 x 8 x 8 = 82

ax ÷ ay = ax - y 57 ÷ 54 = 5(7 - 4) = 53 Simplify 57 ÷ 54 Example In general ax ÷ ay = ax - y Example Simplify 57 ÷ 54

34 ÷ 34 = 3(4 - 4) = 30 But 34 divided by itself is 1 a0 = 1 Example 30 = 1 In general anything to the power of zero is 1

Indices: Tick or Trash

Solving Equations

2x = 18 What is x?

Solving equations 2x – 1 = 7

Solving equations 3x + 2 = 23

Solving equations t + t + t = 4.5

Solving equations x + 5 = 7 3

Solving equations 2(x – 4) = 12

Solving equations 3x + 7 = x + 12

Solving inequalities – same method 2x - 4 < 5x - 16

Solving inequalities 3(3x + 1) ≥ 21

Algebra to solve problems

Algebra - writing expressions Ten quick questions

1 A: x + 2 b: 2x c: x - 2 d: 2 I have x biscuits on a plate. I eat 2. How many do I have left? A: x + 2 b: 2x c: x - 2 d: 2

2 Write an expression for the sum of two numbers v and w. a: v + w b: vw c: v - w d: v ÷ w

3 There are ‘t’ people in a queue. 7 people join. How many are in the queue now? a: 7 b: t + 7 c: 7 - t d: t - 7

4 a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y A milkshake costs x pence and a fizzy drink costs y pence. I buy 5 milkshakes and 6 fizzy drinks. Write an expression for the total cost. a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y

5 I am p years old. 3 years ago my brother was twice my age. How old was my brother? a: p - 3 b: 2(p – 3) c: 2p d: 6

6 There are n sweets in a packet. I buy 4 packets. How many sweets do I have? a: n + 4 b: 4n c: n4 d: 4n + 4

7 Cakes cost x pence and doughnuts cost y pence. I buy 5 cakes and 8 doughnuts. How much do I pay? a: 5x – 8y b: 8x + 5y c: 40xy d: 5x + 8y

8 My sister is y years old. In 4 years time I will be twice my sister’s age. How old will I be? a: y + 4 b: 2(y + 4) c: 2y + 4 d: 2y

9 There are d marbles in a bag. I give 3 to my friend. How many marbles do I have now? a: 3 b: d + 3 c: 3d d: d - 3

10 a: v + w b: v - w c: w - v d: -vw Write an expression for the difference between the numbers v and w. a: v + w b: v - w c: w - v d: -vw

Answers

1 A: x + 2 b: 2x c: x - 2 d: 2 I have x biscuits on a plate. I eat 2. How many do I have left? A: x + 2 b: 2x c: x - 2 d: 2

2 Write an expression for the sum of two numbers v and w. a: v + w b: vw c: v - w d: v ÷ w

3 There are ‘t’ people in a queue. 7 people join. How many are in the queue now? a: 7 b: t + 7 c: 7 - t d: t - 7

4 a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y A milkshake costs x pence and a fizzy drink costs y pence. I buy 5 milkshakes and 6 fizzy drinks. Write an expression for the total cost. a: x + y b: 30x + 30y c: 5x + 6y d: 6x + 5y

5 I am p years old. 3 years ago my brother was twice my age. How old was my brother? a: p - 3 b: 2(p – 3) c: 2p d: 6

6 There are n sweets in a packet. I buy 4 packets. How many sweets do I have? a: n + 4 b: 4n c: n4 d: 4n + 4

7 Cakes cost x pence and doughnuts cost y pence. I buy 5 cakes and 8 doughnuts. How much do I pay? a: 5x – 8y b: 8x + 5y c: 40xy d: 5x + 8y

8 My sister is y years old. In 4 years time I will be twice my sister’s age. How old will I be? a: y + 4 b: 2(y + 4) c: 2y + 4 d: 2y

9 There are d marbles in a bag. I give 3 to my friend. How many marbles do I have now? a: 3 b: d + 3 c: 3d d: d - 3

10 a: v + w b: v - w c: w - v d: -vw Write an expression for the difference between the numbers v and w. a: v + w b: v - w c: w - v d: -vw

Writing our own equations

Mary and John have £50 altogether. Mary has £12 more than John Mary and John have £50 altogether. Mary has £12 more than John. Write an equation for the money they have altogether and using the equation, work out how much money they have each.

Rearranging the Subject of a Formula

What does that mean? The subject of the formula is the letter that is by itself. To change the subject of a formula, we have to rearrange the other terms to get a certain letter by itself.

y = x + 5 If x = 6 what would y be? y = 6 + 5 y = 11

y = x + 5 You have just rearranged formula! If y = 15 what would x be? 15 = x + 5 We know that must be 10. But what is the calculation you have done? 15 – 5 = 10 This is the same as y – 5 = x You have just rearranged formula!

y = x + 5 became y – 5 = x What do you notice? The +5 has moved side and the + has turned to –

y = 5a This means y = 5 times ‘a’ If a = 8 what would y be? y = 5 x 8 y = 40

y = 5a You have just rearranged formula! If y = 20 what is a? 20 = 5 x a We know that must be 4. But what is the calculation you have done? 20 ÷ 5 = 4 This is the same as y ÷ 5 = a You have just rearranged formula!

y = 5a became y ÷ 5 = a What do you notice? The 5 has moved side and the multiply has turned to divide

When changing the subject of a formula the term that needs to be the subject has to be by itself. Everything else needs to be moved to the opposite side and do the inverse operation. + a would become – a x y would become ÷ y

y = a + b - c At the moment y is the subject but I want to make a the subject. I will leave it where it is Everything else needs to be moved to the other side and reversed I need to move c to the other side and + it instead of - y + c = a + b I need to move b to the opposite side and – it instead of + y + c – b = a

This means we want to rearrange the formula so it says Rearrange the formula to make a the subject b = 5a + 21 b – 21 = 5a b – 21 = a This means we want to rearrange the formula so it says a = -21 -21 ÷5 ÷5 5 Our answer should say ... a = b – 21 5

This means we want to rearrange the formula so it says Rearrange the formula to make t the subject h = 13 + 7t h – 13 = 7t h – 13 = t This means we want to rearrange the formula so it says t = -13 -13 ÷7 ÷7 7 Our answer should say ... t = h – 13 7

Hint: You need to expand the brackets first Exercise One Rearrange each formula to make s the subject u = 11s + 3 w = 8s + p q = 3s + 4t 7s + m + t = l a = 3(s + 4) 4 = g(s – 7) 22 + 5s = g Hint: You need to expand the brackets first

Rearrange each formula to make s the subject u = 11s + 3  s = u - 3 w = 8s + p  s = w – p q = 3s + 4t  s = q – 4t 7s + m + t = l  s = l – m – t a = 3(s + 4)  s = a - 12 4 = g(s – 7)  s = 4 + 7g 22 + 5s = g  s = g - 22 11 8 3 7 3 g 5

This means we want to rearrange the formula so it says Rearrange the formula to make v the subject e = 3v + t 5e = 3v + t 5e – t = 3v 5e – t = v This means we want to rearrange the formula so it says v = 5 x5 x5 - t - t ÷3 ÷3 3 Our answer should say ... v = 5e – t 3

Topic Test 13 minutes

Answers

12 Quick multiple choice questions And finally 12 Quick multiple choice questions

Recognise Words Used In Algebra Identity Equation Expression Formula Recognise Words Used In Algebra

6x + 12 A : Identity B : Equation C : Expression D : Formula

6x + 12 A : Identity B : Equation C : Expression D : Formula

4x + 2 = 18 A : Identity B : Equation C : Expression D : Formula

4x + 2 = 18 A : Identity B : Equation C : Expression D : Formula

6x + 12 = 3(2x + 4) A : Identity B : Equation C : Expression D : Formula

6x + 12 = 3(2x + 4) A : Identity B : Equation C : Expression D : Formula

v = u + at A : Identity B : Equation C : Expression D : Formula

v = u + at A : Identity B : Equation C : Expression D : Formula

a² + 2a = a ( a + 2 ) A : Identity B : Equation C : Expression D : Formula

a² + 2a = a ( a + 2 ) A : Identity B : Equation C : Expression D : Formula

A = π r² A : Identity B : Equation C : Expression D : Formula

A = π r² A : Identity B : Equation C : Expression D : Formula

9b - l0 A : Identity B : Equation C : Expression D : Formula

9b - l0 A : Identity B : Equation C : Expression D : Formula

2x² + 3x – 7 = 0 A : Identity B : Equation C : Expression D : Formula

2x² + 3x – 7 = 0 A : Identity B : Equation C : Expression D : Formula

An expression is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

An expression is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

A formula is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

A formula is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

An identity is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

An identity is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

An equation is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

An equation is … A : 8x + 2 = 12 B : 8x + 2 C : 8x + 2 = C D : 8x + 2 = 2(4x + 1)

Directed Study

Revise You have your next assessment next week so make sure you revise Moodle Folders Drop in sessions