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Algebra Expressions

Algebra Expressions. Year 9. Note 1 : Expressions . We often use x to represent some number in an equation. We refer to the letter x as a variable . x + 5 means ‘a number with 5 added on’ x – 7 means ‘a number with 7 subtracted from it’. e.g. Note 1 : Expressions .

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Algebra Expressions

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  1. Algebra Expressions Year 9

  2. Note 1: Expressions We often use x to represent some number in an equation. We refer to the letter x as a variable. • x + 5 means‘a number with 5 added on’ • x – 7 means‘a number with 7 subtracted from it’ e.g.

  3. Note 1: Expressions • A few Rules: • A number should be written before a letter. • yx 2 = 2y e.g. • Terms should be written in alphabetical order. • xyz and 3b x 4a = 12ab e.g. • We don’t use the x or ÷ signs in algebra instead we write it like this: • 5 x y = 5yx ÷ 9 = e.g.

  4. Activity: Expressions • Match each algebraic expression with the phrase x – 8 Five times a number 3 + x Three plus a number A number multiplied by seven Half of a number 7x 3x + 1 A number plus six 5x A number divided by nine A number minus eight x + 6 Three times a number plus one

  5. Activity: Expressions • Write an equivalent algebraic expression for each phrase 12x Twelve times a number 1 + x One plus a number 3x A number multiplied by three A quarter of a number x + 100 A number plus one hundred A number divided by nineteen x – 4 A number minus four IWBEx 11.01 Pg275-276 8x – 1 Eight times a number minus one

  6. Note 2: Substitution • We replace the variable (letter) with a number and calculate the answer. • Algebra follows the same rules as BEDMAS! e.g. If a = 2, b = -3, c = 5then calculate: a + 5 a + b + c 3b = 2 + 5 = 3 x -3 = 2 + -3 + 5 = 7 = -9 = 2 – 3 + 5 = 4

  7. Note 2: Substitution BEDMAS Remember: Do multiplication and division before addition and subtraction Anything in brackets is worked out first A number in front of the bracket means multiply The fraction line means divide e.g. If d= 3, e= 7, f = -2then calculate: 4(d + e) = 5def – 2e = 4(3 + 7) = 5 x 3 x 7 x -2 – 2 x 7 = = 4 x 10 = -210 – 14 = 40 = -224 = 4

  8. Note 2: Substitution BEDMAS Remember: Do multiplication and division before addition and subtraction Anything in brackets is worked out first A number in front of the bracket means multiply The fraction line means divide e.g. If d= 3, e= 7, f = -2then calculate: 2(d - e)2 5f – 2d = = 2(3 - 7)2 = 5 x -2 – 2 x 3 = 2 x (-4)2 = -10 – 6 = = 2 x 16 = -16 = 2 = 32

  9. Note 2: Substitution e.g. 5x Number of tyres = the number of cars IWB Ex 11.02Pg278 -280 Ex 11.03 Pg282 Puzzle Pg 283 b Number of tyres = 5 x 60 c 5 x 40 = 200 = 300

  10. Starter - = 5 x 3 = 5 x -4 = 3 x -4 = 15 = -20 = -12 = 2 x -4 = 12 x 3 = 2 x -4 = -8 = 36 = -8 = 6 x 2 = 3 x 2 x -4 = 3 x 2 = 12 = 6 = -24

  11. Note 3: Formulas A formula is a mathematical rule that explains how to calculate some quantity. e.g. John baby sits for his neighbours. He charges a set fee of $10 plus $5 for every hour (h), that he baby sits. A formula to calculate this charge is given by: Charge = $10 + $5h Use the formula to calculate the amount John charges if he baby sits for: 5 hours 3 hours = 10 + 5 x 5 = 10 + 5 x 3 = $ 35 = $ 25

  12. Note 3: Formulas If John receives $65 how long did he baby sit for? e.g. Charge = $10 + $5h $65 = $10 + $5 x h 65 = 10 + 5h 65-10 = 10-10 + 5h 55 = 5h 5 5 11 = h IWB Ex 12.01 Pg299-302 Ex 12.02 Pg305-308 John baby sat for 11 hours

  13. Starter Multiply the base length by the height and divide by 2 = 20 cm2

  14. Note 4: Multiplying Algebraic Expressions Rules: Multiply the numbers in the expression (these are written first) Write letters in alphabetical order

  15. Note 5: Adding & Subtracting Algebraic Expressions Write in simplest form: y + y + y + y = 4y a + a + a + a + a = 5a a + a + a + a + a + b = 5a + b = 5a + 3b a + a + a + a + a + b + b + b x + y + x + y + x + y + x = 4x + 3y = 4y x + y + y + y + y - x

  16. Note 5: Adding & Subtracting Algebraic Expressions Write in simplest form: y + 3y = 4y 3x + 5x = 8x = 6x 5x – 2x + 3x 5x – 4x = x 10x + x + 19x = 30x IWB Ex 11.05 Pg285 Ex 11.06 Pg286 12p + 3r – 2p + 3r = 10p + 6r

  17. Note 6: Like and Unlike Terms Like terms are terms that contain the exact same variables (letters) or combinations of letters. e.g. Like Terms 2x, 5x, 25x, -81x, 13x, 0.5x… Remember: If we had written these terms properly (in alphabetical order), it would be more obvious that they are like terms. xy, 2xy, -4xy, ½xy, -100xy,… 2abc, 4bac, 6 cab, 9abc, …..

  18. Note 6: Like and Unlike Terms Like terms are terms that contain the exact same variables (letters) or combinations of letters. e.g. Unlike Terms 2, 2x, 3y 3a, 7ab, 8b 2p, 4r, 10s

  19. Note 6: Like and Unlike Terms Expressions can have a mixture of LIKE and UNLIKE terms. 2a – 7b + 5a + 8b = 7a + b e.g. 3x + 5y Like terms can be grouped together and simplified Unlike terms cannot be simplified 5a + 2b – 3a + 6c + 4b e.g. 2a + 6b + 6c

  20. Note 6: Like and Unlike Terms 2x + 10y 8x + 8y 4x + 1 6p + 9q x + 1 2x + 2 IWB Ex11.07Pg289 Ex11.08 Pg292-3 x 2x – 7

  21. Starter Find the perimeter of this shape in terms of x 2x + 1 2x + 3 x The perimeter is the sum of all side lengths 4x + 1 5x – 2 P = x + (2x + 1) + (2x + 3) + (4x + 1) + (5x – 2) = 14x + 3 If the perimeter is 31 cm. What is the value of x and which side is the longest? P = 14x + 3 28= 14x x= 2 14 14 31= 14x + 3

  22. Note 7: Powers (exponents) Recall: When a variable (letter) is multiplied by itself many times, we use powers e.g. Write the following in index form: p3 p x p x p = ________________ q x q x q x q x q = __________ s x s x s x t x t x t x t = _______ p x p x q x q x s x s = ________ s x t x t x t x s x s = __________ q5 s3t4 p2q2s2 s3t3

  23. Note 7: Powers (exponents) Substituting: Evaluate the following when a =2, b = 7 and c = -3 5a2 = 5 x 22 a2 = 22 = 5 x 4 = 4 = 20 (b + c)2 = (7 - 3)2 = (4)2 2a2c2 = 2 x 22 x (-3)2 = 16 = 2 x 4 x 9 (5a)2 = (5 x 2)2 = 72 = 102 = 100

  24. Note 7: Powers (exponents) Multiplying: Simplify:b3 x b4 = (b x b x b) x (b x b x b x b) = b7 When multiplying power expressions with the same base, we add the powers. an x am = am+n e.g.e2 x e6 5m3 x 4m3 g8 x g = e2+6 = g8+1 = 20m3+3 = e8 = g9 = 20m6

  25. Note 7: Powers (exponents) IWB Ex13.08Pg346 Ex13.09 Pg349 PUZZLE Pg 350 Dividing: Simplify: = = c3 When dividing power expressions with the same base, we subtract the powers. am = am-n an e.g. g7 ÷ g = g7-1 = 5q7-3 = f 6-3 = = g6 = 5q4 = f 3

  26. Starter How would you calculate 7 x 83 in your head? 7 x 80 + 7 x 3 560 + 21 = 581 What we have done in our head can also look like this: x x 7 (80 + 3) 7 x 80 + 7 x 3 = = 581

  27. Note 8: Expanding Brackets • To expand (remove) brackets: • Multiply the outside term by everything inside the brackets • Simplify where possible e.g. Expand: a.) 4(x + y) b.) −2(x – y) c.) 5(x – y + 2z) The Distributive Law = 4x + 4y = -2x + 2y = 5x - 5y + 10z

  28. Try These! e.g.Expand: a.) 8(x + y) b.) 4(x – y) c.) 2(x – y) d.) 3(-x + y) e.) 9(x + y + z) f.) -8(x + y) g.) 5(x – 3y) h.) -(x – 2y) i.) -7(-x + 7y) j.) -4(3x - y + 5z) = 8x + 8y = -8x – 8y = 4x - 4y = 5x – 15y = 2x - 2y = -x + 2y = -3x + 3y = 7x – 49y = 9x + 9y + 9z = -12x + 4y – 20z

  29. Lets do some more e.g.Expand: a.) 8(x + 4) b.) 4(x – 2y) c.) 2(x – 10) d.) 3(2x + 9) e.) -(x + 2y - z) f.) -8(x + 8) g.) 5(5x – 3) h.) -(x – 11) i.) -7(x + 12) j.) -2(x - y + 14) = 8x + 32 = -8x – 64 = 4x – 8y = 25x – 15 = 2x - 20 = -x + 11 = -7x – 84 = 6x + 27 = -x – 2y + z = -2x + 2y – 28

  30. Starter e.g.Expand: a.) 2(x + y) b.) 4(x – y) c.) 2(x – 3y) d.) 3(x + 30) e.) -(2x - 4y + z) f.) -8(x + 2) g.) -5(3x – 6) h.) -(x – 23) i.) -3(-2x + 3) j.) -5(x - 2y + 1) = 2x + 2y = -8x – 16 = 4x – 4y = -15x + 30 = 2x – 6y = -x + 23 = 6x – 9 = 3x + 90 = -2x + 4y – z = -5x + 10y – 5

  31. Note 8: Expanding Brackets • The terms on the inside can also be multiplied by a variable on the outside. e.g. Expand: a.) a(x + y) b.) 2a(a + b) c.) 5x2(x2 – x + 2) = ax + ay = 2a2 + 2ab = 5x4 – 5x3 + 10x2

  32. Your turn! e.g.Expand: a.) a(x + 4) b.) b(x – 5y) c.) x(x – 15) d.) y(2x + 2) e.) x(x2 + 2y – 8) f.) x5(x4+ x3y) g.) 5xy(3xy – 1) h.) -x(x – 11) = ax + 4a = x9 + x8y = bx – 5by = 15x2y2 – 5xy = x2 – 15x = -x2 + 11x = 2xy + 2y = x3 – 2xy – 8x

  33. Note 8: Expanding Brackets and Collecting Like Terms Expand the brackets first, then simplify! e.g.Expand & Simplify a.) 4(2x + y) + 3(x + 5y) = 8x + 4y + 3x + 15y * Collect like terms = 11x + 19y b.) 4(5x - y) – 3(x – 10) = 20x - 4y – 3x + 30 * Collect like terms = 17x – 4y + 30

  34. Your turn! = 5x + 5y + 2x + 2y = 7x + 7y IWB - odd only Ex15.02, 15.03Pg389 Ex15.04 Pg390 Ex15.05 Pg391 Ex15.06, 15.07Pg393 Ex 15.08 Pg 397 = 2x + 2y + 8x + 4y = 10x + 6y = 6x + 3y + 6x + 12y = 12x + 15y = 12x + 18y – 10x – 4y = 2x + 14y = 2x + 6 + 4x + 24 = 6x + 30

  35. Factorising Factorising is the reverse procedure of expanding. Expanding 3x + 6 3 (x + 2) Factorising

  36. Note 9: Factorising (put in brackets) Factorising is the reverse process of expanding. • We want to put brackets back into the algebraic expression • find the highest common factor and write it in front of the brackets e.g.Factorise 3x + 3y 4x – 4y 7x + 7y + 7z = 7( ) x + y + z = 3( ) x+y = 4( ) x – y

  37. Try These f.) 7x +7 g.) 7x + 14 h.) 24x + 36 e.g. Factorise: a.) 6a + 6b b.) 3p – 3q c.) 4x + 4y = 7( ) x+1 = 6( ) a+b = 7( ) x+2 = 3( ) p – q = 12( ) = 4( ) x+y 2x+3 = 6( ) x+2 d.) 6x + 12 You can check that your answer is correct by expanding = 24( ) x+y e.) 24x + 24y

  38. Try These f.) 7x + 49 g.) 9x + 63 h.) 45x + 81 e.g. Factorise: a.) 8a + 6b b.) 12p – 3q c.) 4x + 8 = 7( ) x+7 = 2( ) 4a+3b = 9( ) x+7 = 3( ) 4p – q = 9( ) = 4( ) x+2 5x+9 = 6( ) x+5 d.) 6x + 30 You can check that your answer is correct by expanding = 29( ) x+1 e.) 29x + 29

  39. Starter Factorise: a.) 4a + 8b b.) 3p – 6q + 3r c.) 4x + 8y + 12z = 4(a+2b) = 3(p – 2q +r) = 4(x + 2y + 3z) = 3(2x +7) d.) 6x + 21 IWB - odd only Ex15.11Pg400 Ex15.13 Pg401 Ex15.14 Pg402 Ex15.15 Pg403 Ex15.16 Pg405 e.) 24x - 32 = 8(3x – 4)

  40. Extension – More exponent rules How do we simplify an exponential term raised to another exponent? × × e.g. (2y3)2 e.g. (3a4)3 = (2y3) × (2y3) = (3a4) × (3a4) × (3a4) = 4y6 = 27a12 Notice that there is a shortcut to get the same result = 22y2×3 = 33a4×3 = 27a12 = 4y6

  41. Extension – More exponent rules 1.) Index the number. 2.) Multiply each variable index by the index outside the brackets. 3.) If the bracket can be simplified, do this first. × × e.g.Simplify (2x2)3 (-4h2g6)2 = 23 x23 = (-4)2h2×2g6×2 = (4x)2 = 8x6 = 16h4g12 = 16x2

  42. Extension – Expanding 2 Brackets QUADRATIC EXPANSION When we expand two brackets we use: F – first (multiply the first variable or number from each bracket) O – outside (multiply the outside variables together) I – inside (multiply the two inside variables together) L – last (multiply the last variable in each bracket together) Simplify, leaving your answer with the highest power first to the lowest power (or number) last. F O I L e.g. (x + 4) (x – 2) = x2 - 2x + 4x - 8 = x2 + 2x - 8

  43. Extension – Expanding 2 Brackets e.g. (x + 10) (x + 1) QUADRATIC EXPANSION e.g. (x + 3) (x – 5) = x2 + 10x + x + 10 = x2 + 3x - 5x - 15 = x2 + 11x + 10 = x2 - 2x - 15 e.g. (x - 4) (x + 4) e.g. (x - 3) (x – 8) = x2 - 4x + 4x - 16 = x2 - 16 = x2 - 3x - 8x + 24 = x2 - 11x + 24 • Notice the middle term cancels out • DIFFERENCE OF SQUARES

  44. Extension – Expanding 2 Brackets e.g. (x – 5) (x + 4) QUADRATIC EXPANSION e.g. (x + 7) (x – 9) = x2 - 5x + 4x - 20 = x2 + 7x - 9x - 63 = x2 - x - 20 = x2 - 2x - 63 e.g. (x - 9) (x + 9) e.g. (x - 2) (x – 6) = x2 - 9x + 9x - 81 = x2 - 2x - 6x + 12 = x2 - 81 = x2 - 8x + 12 • Notice the middle term cancels out • DIFFERENCE OF SQUARES

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