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Algebra

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Published in: Mathematics
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This PPT introduces the basics of Algebra.

Sharon / Dubai

19 years of teaching experience

Qualification: Bsc mathematics and QTS (Qualified Teacher Status)

Teaches: English, Science, Maths, General Science, Phonics, Mathematics

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  1. 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 February 2025 Page 1
  2. 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? Monday, 24 February 2025 Page 2
  3. 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? Monday, 24 February 2025 Page 3
  4. Another Example Bill catches y fish. Ben takes 3 away from him. How many fish does Bill now have? Monday, 24 February 2025 Page 4
  5. A Third Example Fred has x DVDs Frank has y DVDs How many DVDs do they have altogether? Monday, 24 February 2025 Page 5
  6. Questions Use algebra to write: 1) 2 less than w — W—2 2) 3 more than d — 3) 5 together with c 4) f more than g 5) p less than q 6) m less than 7 Monday, 24 February 2025 Page 6
  7. Adding and Subtracting with Letters = 3a b+b+b+b+b Monday, 24 February 2025 Page 7
  8. Questions C b + b -b = 2b Monday, 24 February 2025 = 3n Page 8
  9. Adding Expressions & Terms = 6a 7a - 4a + IOa Monday, 24 February 2025 = 13a Page 9
  10. Questions 1) 5c + 7c 12c 2) 9d - 4d5d 12S 14S 4) 13a - 5a eg 6e 6) 6e - 2e + 5e 9e 7) 12e - IOeæ 2h + h2h Monday, 24 February 2025 10) 3w + 9w - 5w — 11) 2g + 5g - 3g _g 12) 7 f — + 12f 13) 5b + 50b 55b 14) 75j - 43j 321 15) 34p + 12p - 5p 41p 16) 3m - m + m + 5m 8m 17)d-d+d-d 18) 4f + IOf - 13f Page 10
  11. Going Negative — 12p = -3p 5p + 4p Monday, 24 February 2025 7a = -IOa Page 11
  12. 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 February 2025 Page 12
  13. Example 1 3a + 4b + 2a + 6b firstly collect like-terms... 3a + 2a + 4b + 6b = 5a + 10b Monday, 24 February 2025 Page 13
  14. Example 2 collect like-terms -3a - 4a + 2b + 6b -7a + 8b Monday, 24 February 2025 Page 14
  15. Example 3 b 3 -12b collect like-terms -4a + 3a + 9b - 12b - a - 3b Monday, 24 February 2025 Page 15
  16. 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 February 2025 Page 16
  17. Simplify each expression... 1) 5a-4y- 2y -d - 3e -6a - 2y 2) -4f + 69 - 7f - 39 7) p +8r-7p -lif + 39 -3p + IOr 3) 4m + 9n - 6m - 14n -2m - 5n lox - 5 4) -4t + 7y- 10t + 5y 9) a- 6b + 2a + 3b - 3a -14t + 12y -3b 5) 1+2r-7-7r 10) 39 + 2h - 3h + 29 -6 - 5r Monday, 24 February 2025 Page 17
  18. Harder Simplification... (remember, a, a2 and a3 are completely different terms) 1) 4a - 5a2 + 2a + 3a2: 6a - 2a2 2) 5h2 + 2h3 - IOh2 - 3h3: -5h2 -h3 3) 3x2 - 4x - + x 4) -4t2 + 7t - 10t2 + 5t3 = 7t -14t2 + 5t3 — —13r2 5) r2 + 2r - 7r2 - 7r2 - 2r - Monday, 24 February 2025 Page 18
  19. Multiplying Terms Together When two terms in algebra are being multiplied together, they are simply written next to each other. e.g. and 3a means 3 x a efg means e x f x g Monday, 24 February 2025 Page 19
  20. Multiplying Terms Together Simplify: 2a x 4b : 8ab 20cde IOC x 2de : 3w x 4y x 5z = 60wyz Monday, 24 February 2025 Page 20
  21. Questions - simple multiplication 1) 7ab x 8pq 56abpq 30afh 3) 3abc x 4def 12abcdef 4) 59 x 25 1250 5) 5mn x 2pq x 440mnpq 6) 5f x g x h x 2 j 10fghj 7) 3w x 4d x 2h x yz 24dhwyz 8) 7pqr x 4abc x 2h 56abchpqr Monday, 24 February 2025 Page 21
  22. Questions - reverse multiplication 1) 7ab x IOC 70abc 2) 3fx4g x5h 60fgh 3) 3abc x d = 3abcd 4) 5g x _ 209 4 5) 5mn x x 4 20mnp P 4h x 2g = 40fgh 7) 3w x x 2v x yz = 24vwyz 8) 4bc x 10e x 2ad = 80abcde Monday, 24 February 2025 Page 22
  23. Dividing Algebraic Terms Simplify: 8a 10ab + 2a = 20pq pq Monday, 24 February 2025 20 Page 23
  24. 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, 2 February 2025 Page 24
  25. Powers Rules ax a: a2 Rule 1: When you multiply powers of the same letter or number you ax ax ax a: a4 add the indices... But, what is a2 x a4? It's (a x a) x (a x ax a x a) = a6 What did you do with the powers? You added them! Monday, 24 February 2025 Page 25
  26. Indices Questions a3 x a6 x a2 - all C3 X C5 X cl- C : 8y7 2y2 x 4y5 = 12m8 3m3 x 4m5 Monday, 24 February 2025 Page 26
  27. Harder Indices Questions QI. Q2. Q3. 2a3b6 x a6b2 - - 2a9b8 15c7d8 3c5d4 x 5c2d4 = - 8a3b7 2ab4 x 4a2b3 _ Q4. 3mnp3 x 4mn2p5 : 12m2n3p8 Monday, 24 February 2025 Page 27
  28. Powers axaxaxaxa= a5 3 axaxa=a But, what is a5+ a3? Rule 2: When you divide powers of the same letter or number you subtract the indices. It's (a x ax ax a x a)+ (a x a x a) = a? What did you do with the powers? You subtracted them! Monday, 24 February 2025 Page 28
  29. Indices Questions c3 — c5 a 2 4 a 3y2 x 4y5 2y3 Monday, 24 February 2025 Page 29
  30. Harder Indices Questions 4a6 x 8a2 30y2 x 4y5 6 x y4 Monday, 24 February 2025 16a— 1 20y3 Page 30
  31. Powers 3 axaxa:a What do you think is... (a3)2 6 Rule 3: When you raise a power by another power (separated by brackets) you multiply the indices... what did you do with the powers here? multiplied them! Monday, 24 February 2025 Page 31
  32. Indices Questions QI. (a2)4 x (a3)2 Q3. Q4. Monday, 24 February 2025 a14 4a6 (4a2)3 - 64a6 : 25a2b6 (5ab3)2 Page 32
  33. Have you really understood indices?? We'll see... Simplify this... (2a4)5 x x d x al 9 x (2a9)3 Monday, 24 February 2025 Page 33
  34. Expanding the Brackets Expand these expressions: 3(a + b) = 3a + 3b + 6) 4y + 24 2(2a + 3b - 4c) + 6b - 8c Monday, 24 February 2025 Page 34
  35. Expand the brackets... 1) 4(2p - 12) - 48 IOa + 20b 2) 5(2a + 4b) 3) 3(4c - 4b) 12c - 12b 4) 7(3e + 2f - 49) + 14f - 5) 9(6w + 2y - 7z) :54w + 18y - 63z 30b - 90m 6) 10(3b - 9m) 7) 6(-6j - 7m) =-36j - 42m 8) 5(4a - 3b) - 15b Monday, 24 February 2025 Page 35
  36. 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 February 2025 Page 36
  37. Factorising Reminder of how to expand brackets: 4(2m - 5): 8m - 20 What is factorising then? QI. 8m-20: 4(2m-5) Monday, 24 February 2025 Page 37
  38. Factorising Q2. 25a - 30b 5(2m- 5) Q3. 40a + 6a2 - 2a(20+ 3a) Page 38 Monday, 24 February 2025
  39. Adding Bracketed Expressions Aims: To be able to multiply out the brackets from expressions... ... and then collect like-terms and simplify Monday, 24 February 2025 Page 39
  40. Adding Bracketed Expressions + 5b) + 3(7a - b) 4a + 20b + 21a - 3b 25a + 17b Monday, 24 February 2025 Page 40
  41. Adding Bracketed Expressions 2(7a - 14a — Monday, 24 February 2025 6b) + 6(3a + b) = 12b + 18a + 6b = 32a - 6b Page 41
  42. Questions 1) 7(2a - 4b) + 2(2a + b)18a - 26b 2) 2(3y - w) + 3(2y + 5w)12y + 13w 3) + q) + 5(2p - 3q)14p - llq 4) 6(2n + m) + 2(n - m) 14n + 4m 5) 2(4s + r) + 7(2s - 3r) 22s - 19r Monday, 24 February 2025 Page 42
  43. Subtracting Bracketed Expressions 3(2a - 4b) - + 2b) Monday, 24 February 2025 Page 43
  44. 3(2a - Monday, 24 February 2025 Subtracting Bracketed Expressions 6a - 12b - - 4b + 2b) Page 44
  45. 3(2a - Monday, 24 February 2025 Subtracting Bracketed Expressions 6a - 12b - - 4b 4a - 16b + 2b) Page 45
  46. Subtracting Bracketed Expressions 3(2p + 3q) - 4(p - 2q) = Monday, 24 February 2025 Page 46
  47. Subtracting Bracketed Expressions 3(2p + 3q) - 4(p - 2q) = 6p + 9q - 4p + 8q = Monday, 24 February 2025 Page 47
  48. Subtracting Bracketed Expressions 3(2p + 3q) - 4(p - 2q) = 6p + 9q - 4p + 8q = 2P + 17q Monday, 24 February 2025 Page 48
  49. Invisible 1 Monday, 24 February 2025 Page 49
  50. Invisible 1 - 2n) - — 14n — (m - 3n) = Monday, 24 February 2025 Page 50
  51. Invisible 1 6m - lln - 2n) - — 14n — (m - 3n) = Monday, 24 February 2025 Page 51
  52. Questions 1) 3(4n + 5m) - 5(2n - 4m) 3) 3(2n + 6m) - 6(n - 2m) 4) 2(3n - m) - - 5m) 5) 8(5n - m) - 2(2n + m) Monday, 24 February 2025 Page 52
  53. Questions 1) 3(4n + 5m) - 5(2n - 4m) 3) 3(2n + 6m) - 6(n - 2m) 4) 2(3n - m) - - 5m) 5) 8(5n - m) - 2(2n + m) Monday, 24 February 2025 Page 53

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