Getting Used to Algebra Algebra is where you use letters to represent numbers Aims: To get to grips with the algebra idea To solve simple algebraic problems Monday, 24 April 2017

Example Alex has some sweets, we do not know how many sweets Alex has…. so we can say ‘Alex has x sweets’ If Alex is given 5 more sweets, how many sweets has he got? x + 5 Monday, 24 April 2017

Example Bob has some toys, we do not know how many toys he has…. so we can say ‘Bob has m toys’ If Bob buys 6 more toys, how many toys has he now got? m + 6 Monday, 24 April 2017

Another Example Bill catches y fish. Ben takes 3 away from him. How many fish does Bill now have? y - 3 Monday, 24 April 2017

A Third Example Fred has x DVDs Frank has y DVDs How many DVDs do they have altogether? x + y Monday, 24 April 2017

Questions Use algebra to write: 2 less than w 3 more than d 5 together with c f more than g p less than q m less than 7 w – 2 d + 3 c + 5 f + g q - p 7 - m Monday, 24 April 2017

Adding and Subtracting with Letters a + a + a = b + b + b + b + b = c + c – c + c = 3a 5b 2c Monday, 24 April 2017

Questions 1) a + a = 2) c + c + c + c = 3) p - p + p = 4) v + v + v + v - v + v = 5) b + b + b – b = 6) n - n + n + n + n - n + n = 7) h - h = 8) g + g + g - g 2a 4c p 4v 2b 3n 2g Monday, 24 April 2017

Adding Expressions & Terms 2a + 4a = 6a – 5a = 7a – 4a + 10a = 6a a 13a Monday, 24 April 2017

Questions 7w 1) 5c + 7c 2) 9d – 4d 3) 2s + 12s 4) 13a – 5a 5) 4e – e + 3e 6) 6e – 2e + 5e 7) 12e – 10e 8) 3h – 2h + h 9) 4r – r + 5r 12c 10) 3w + 9w – 5w 11) 2g + 5g – 3g 12) 7f – 4f + 9f 13) 5b + 50b 14) 75j – 43j 15) 34p + 12p – 5p 16) 3m – m + m + 5m 17) d – d + d - d 18) 4f + 10f - 13f 5d 4g 14s 12f 55b 8a 6e 32j 9e 41p 2e 8m 2h f 8r Monday, 24 April 2017

Going Negative 3a – 5a = 5p + 4p – 12p = -5a + 2a – 7a = -2a -3p -10a Monday, 24 April 2017

Working with Algebra Aims: To be able to collect like terms in order to simplify algebraic expressions To be able to multiply terms together and expand the brackets from an expression Monday, 24 April 2017

Example 1 3a + 4b + 2a + 6b = firstly collect like-terms… 3a + 2a + 4b + 6b = 5a + 10b Monday, 24 April 2017

-3a + 2b – 4a + 6b collect like-terms -3a – 4a + 2b + 6b -7a + 8b Example 2 -3a + 2b – 4a + 6b collect like-terms -3a – 4a + 2b + 6b -7a + 8b Monday, 24 April 2017

-4a + 9b + 3a - 12b collect like-terms -4a + 3a + 9b – 12b - a – 3b Example 3 -4a + 9b + 3a - 12b collect like-terms -4a + 3a + 9b – 12b - a – 3b Monday, 24 April 2017

Aims To be able to simplify expressions (including expressions with indices) To be able to expand brackets and simplify To be able to understand the 3 laws of indices Monday, 24 April 2017

Simplify each expression… 5a – 4y – 11a + 2y 2) -4f + 6g – 7f – 3g 3) 4m + 9n – 6m - 14n 4) -4t + 7y – 10t + 5y 5) 1 + 2r – 7 – 7r 6) 3d – 5e – 4d + 2e 7) p + 8r – 7p + 2r + 3p 8) 9x + 3x – 8 - 2x + 3 9) a – 6b + 2a + 3b - 3a 10) 3g + 2h – 3h + 2g -d – 3e -6a - 2y -11f + 3g -3p + 10r -2m – 5n 10x – 5 -14t + 12y -3b -6 – 5r 5g – h Monday, 24 April 2017

Harder Simplification… (remember, a, a2 and a3 are completely different terms) 4a – 5a2 + 2a + 3a2 = 2) 5h2 + 2h3 – 10h2 – 3h3 = 3) 3x2 – 4x – 5x2 + x = 4) -4t2 + 7t – 10t2 + 5t3 = 5) r2 + 2r – 7r2 – 7r2 – 2r = -5h2 –h3 – 3x-2x2 7t -14t2 + 5t3 -13r2 Monday, 24 April 2017

Multiplying Terms Together When two terms in algebra are being multiplied together, they are simply written next to each other. e.g. 3a means 3 x a and efg means e x f x g Monday, 24 April 2017

Multiplying Terms Together Simplify: 2a x 4b = 10c x 2de = 3w x 4y x 5z = 8ab 20cde 60wyz Monday, 24 April 2017

Questions – simple multiplication 1) 7ab x 8pq 2) 3f x 2h x 5a 3) 3abc x 4def 4) 5g x 25 5) 5mn x 2pq x 4 6) 5f x g x h x 2j 7) 3w x 4d x 2h x yz 8) 7pqr x 4abc x 2h 56abpq 30afh 12abcdef 125g 40mnpq 10fghj 24dhwyz 56abchpqr Monday, 24 April 2017

Questions – reverse multiplication 1) 7ab x ___ = 70abc 2) 3f x ___ x 5h = 60fgh 3) 3abc x ___ = 3abcd 4) 5g x ___ = 20g 5) 5mn x ___ x 4 = 20mnp 6) 5f x ___ x 2g = 40fgh 7) 3w x ___ x 2v x yz = 24vwyz 8) 4bc x ___ x 2ad = 80abcde 10c 4g d 4 p 4h 4 10e Monday, 24 April 2017

Dividing Algebraic Terms Simplify: 8a ÷ 4 = 10ab ÷ 2a = 20pq pq = 2a 5b 20 Monday, 24 April 2017

Powers Aims: To remember how to work out the HCF & LCM from any given pair of numbers To be able to understand how indices (powers) work in algebra To be able to manipulate the powers in an expression in order to simplify it Monday, 24 April 2017

Powers Rules a x a = a x a x a x a = But, what is a2 x a4? It’s (a x a) x (a x a x a x a) = What did you do with the powers? You added them! a2 Rule 1: When you multiply powers of the same letter or number you add the indices… a4 a6 Monday, 24 April 2017

a3 x a6 x a2 = c3 x c5 x c = 2y2 x 4y5 = 3m3 x 4m5 = Indices Questions a3 x a6 x a2 = c3 x c5 x c = 2y2 x 4y5 = 3m3 x 4m5 = a11 c9 1 8y7 12m8 Monday, 24 April 2017

Harder Indices Questions 2a3b6 x a6b2 = 3c5d4 x 5c2d4 = 2ab4 x 4a2b3 = 3mnp3 x 4mn2p5 = Q1. 2a9b8 Q2. 15c7d8 Q3. 8a3b7 Q4. 12m2n3p8 Monday, 24 April 2017

Powers a x a x a x a x a = a x a x a = But, what is a5 a3? It’s (a x a x a x a x a) (a x a x a) = What did you do with the powers? You subtracted them! a5 Rule 2: When you divide powers of the same letter or number you subtract the indices… a3 a2 Monday, 24 April 2017

Indices Questions a6  a2 = c3  c5 = 3y2 x 4y5 = 2y3 a4 c-2 6y4 Monday, 24 April 2017

Harder Indices Questions 4a6 x 8a2 = 2a9 30y2 x 4y5 = 6 x y4 Q1. 16a-1 20y3 Q2. Monday, 24 April 2017

Powers a x a x a = What do you think is… (a3)2 = a3 x a3 = a3 a6 Rule 3: When you raise a power by another power (separated by brackets) you multiply the indices… a6 what did you do with the powers here? multiplied them! Monday, 24 April 2017

(a2)4 x (a3)2 = 4(a2)3 = (4a2)3 = (5ab3)2 = Indices Questions (a2)4 x (a3)2 = 4(a2)3 = (4a2)3 = (5ab3)2 = Q1. a14 Q2. 4a6 Q3. 64a6 25a2b6 Q4. Monday, 24 April 2017

Have you really understood indices?? We’ll see… Simplify this… Monday, 24 April 2017

Expanding the Brackets Expand these expressions: 3(a + b) = 4(y + 6) = 2(2a + 3b – 4c) = 3a + 3b 4y + 24 4a + 6b – 8c Monday, 24 April 2017

Expand the brackets… 1) 4(2p – 12) = 2) 5(2a + 4b) = 3) 3(4c – 4b) = 4) 7(3e + 2f – 4g) = 5) 9(6w + 2y – 7z) = 6) 10(3b – 9m) = 7) 6(-6j – 7m) = 8) 5(4a – 3b) = 8p - 48 10a + 20b 12c – 12b 21e + 14f – 28g 54w + 18y – 63z 30b – 90m -36j – 42m 20a – 15b Monday, 24 April 2017

Factorising Aims: To remember how to expand brackets, including expressions with indices To learn what the process factorising is and be able to apply it to any expression Monday, 24 April 2017

Factorising Reminder of how to expand brackets: 4(2m – 5) = What is factorising then? Q1. 8m – 20 = 8m - 20 4 ( 2m - 5 ) Monday, 24 April 2017

Factorising Q2. 25a – 30b = Q3. 40a + 6a2 = 5 ( 2m - 5 ) 2a ( 20 + 3a Monday, 24 April 2017

Adding Bracketed Expressions Aims: To be able to multiply out the brackets from expressions… … and then collect like-terms and simplify Monday, 24 April 2017

Adding Bracketed Expressions 4(a + 5b) + 3(7a – b) = 4a + 20b + 21a – 3b = 25a + 17b Monday, 24 April 2017

Adding Bracketed Expressions 2(7a – 6b) + 6(3a + b) = 14a – 12b + 18a + 6b = 32a – 6b Monday, 24 April 2017

Questions 1) 7(2a – 4b) + 2(2a + b) 2) 2(3y – w) + 3(2y + 5w) 3) 4(p + q) + 5(2p – 3q) 4) 6(2n + m) + 2(n – m) 5) 2(4s + r) + 7(2s – 3r) 18a – 26b 12y + 13w 14p – 11q 14n + 4m 22s – 19r Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2a – 4b) – 2(a + 2b) = Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2a – 4b) – 2(a + 2b) = 6a – 12b – 2a – 4b = Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2a – 4b) – 2(a + 2b) = 6a – 12b – 2a – 4b = 4a – 16b Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2p + 3q) – 4(p – 2q) = Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2p + 3q) – 4(p – 2q) = 6p + 9q - 4p + 8q = Monday, 24 April 2017

Subtracting Bracketed Expressions 3(2p + 3q) – 4(p – 2q) = 6p + 9q - 4p + 8q = 2p + 17q Monday, 24 April 2017

Invisible 1 7(m – 2n) – (m – 3n) = Monday, 24 April 2017

7(m – 2n) – (m – 3n) = 7m – 14n – m + 3n = Invisible 1 7(m – 2n) – (m – 3n) = 7m – 14n – m + 3n = Monday, 24 April 2017

7(m – 2n) – (m – 3n) = 7m – 14n – m + 3n = 6m - 11n Invisible 1 7(m – 2n) – (m – 3n) = 7m – 14n – m + 3n = 6m - 11n Monday, 24 April 2017

Questions 1) 3(4n + 5m) – 5(2n – 4m) 2) 5(n – 2m) – (m + 2n) 3) 3(2n + 6m) – 6(n – 2m) 4) 2(3n – m) – 7(n – 5m) 5) 8(5n – m) – 2(2n + m) Monday, 24 April 2017

Questions 1) 3(4n + 5m) – 5(2n – 4m) 2) 5(n – 2m) – (m + 2n) 3) 3(2n + 6m) – 6(n – 2m) 4) 2(3n – m) – 7(n – 5m) 5) 8(5n – m) – 2(2n + m) Monday, 24 April 2017