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 + 5means a number with 5 added on x 7means a number with 7 subtracted from it e.g. Note 1: Expressions A few Rules:
A number should be written before a letter. y x 2 = 2y e.g. Terms should be written in alphabetical order. xyz and b x 4a = 12ab e.g. We dont use the x or signs in algebra instead we write it like this: 5 x y = 5yx 9 = e.g. Activity: Expressions
Match each algebraic expression with the phrase Five times a number x 8 3 + x Three plus a number A number multiplied by seven Half of a number 7x 3x + 1 A number plus six A number divided by nine 5x A number minus eight Three times a number plus one x + 6 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 IWB Ex 11.01 Pg 8x 1 Eight times a number minus one Note 2:Substitution We replace the variable (letter) with a number and calculate the answer. Algebra follows the same rules as BEDMAS! e.g. Ifa = 2,b = -3,c = 5 then calculate: a + 5 3b a + b + c = 2 + 5 = 3 x -3 = = 7 = -9 = 2 3 + 5 = 4 BEDMAS Note 2: Substitution Remember: e.g.
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. Ifd = 3,e = 7,f = -2 then 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 BEDMAS Note 2: Substitution Remember: e.g.
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. Ifd = 3,e = 7,f = -2 then 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 = 32 = 2 Note 2: Substitution e.g. 5x Number of tyres = IWB
the number of cars IWB Ex Pg Ex Pg 282 Puzzle Pg 283 bNumber of tyres = 5 x 60 c5 x 40 = 200 = 300 Starter - = 5 x 3 = 5 x -4 = 3 x -4 = 15 = -20 = -12 = 2 x -4 = 12 x 3
= -8 = 36 = -8 = 6 x 2 = 3 x 2 = 3 x 2 x -4 = 12 = 6 = -24 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 = x 5 = x 3 = $ 35 = $ 25 Note 3:Formulas e.g. If John receives $65 how long did he baby sit for? Charge = $10 + $5h $65=$10 + $5 x h 65= h = h 55=5h 5 5 11 = h IWB Ex Pg Ex Pg John baby sat for 11 hours Starter Multiply the base length by the height and divide by 2 = 20 cm2 Note 4: Multiplying Algebraic Expressions
Rules: Multiply the numbers in the expression (these are written first) Write letters in alphabetical order 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 a + a + a + a + a + b + b + b = 5a + 3b x + y + x + y + x + y + x = 4x + 3y x + y + y + y + y - x = 4y Note 5: Adding & Subtracting Algebraic Expressions
Write in simplest form: y + 3y = 4y 3x + 5x = 8x 5x 2x + 3x = 6x 5x 4x = x 10x + x + 19x = 30x IWB Ex Pg 285 Ex Pg 286 12p + 3r 2p + 3r = 10p + 6r 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, xy, -100xy, 2abc, 4bac, 6 cab, 9abc, .. 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 Note 6: Like and Unlike Terms
Expressions can have a mixture of LIKE and UNLIKE terms. e.g. 3x + 5y 2a 7b + 5a + 8b = 7a + b Like terms can be grouped together and simplified Unlike terms cannot be simplified e.g. 5a + 2b 3a + 6c + 4b 2a + 6b + 6c Note 6: Like and Unlike Terms
2x + 10y 8x + 8y 4x + 1 6p + 9q x + 1 IWB Ex Pg 289 Ex Pg 292-3 Workbook Ex E page 117 2x + 2 x 2x 7 Starter Find the perimeter of this shape in terms of 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 31= 14x + 3 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 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 Note 7: Powers (exponents)
NOTE: b = b1 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 g8 x g 5m3 x 4m3 = e2+6 = g8+1 = 20m3+3 = e8 = g9 = 20m6 Note 7: Powers (exponents)
Dividing: IWB Ex Pg 346 Ex Pg 349 PUZZLE Pg 350 Simplify: = = c3 When dividing power expressions with the same base, we subtract the powers. am = am-n an e.g. g7 g = g7-1 = f 6-3 = 5q7-3 = = g6 = f 3 = 5q4 Starter How would you calculate 7 x 83 in your head? 7 x x 3 =581 What we have done in our head can also look like this: x x 7 (80 + 3) = 7 x 80 x 3 =581 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 Try These! e.g. Expand: = 8x + 8y = -8x 8y = 4x - 4y = 5x 15y
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 Lets do some more e.g. Expand: = 8x + 32 = -8x 64 = 4x 8y
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 = 6x + 27 = -7x 84 = -x 2y + z = -2x + 2y 28 Starter e.g. Expand: = 2x + 2y = -8x 16 = 4x 4y = -15x + 30
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 = 3x + 90 = 6x 9 = -2x + 4y z = -5x + 10y 5 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 Your turn! e.g. Expand: = ax + 4a = x9 + x8y = bx 5by = 15x2y2 5xy
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 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 Your turn! = 5x + 5y + 2x + 2y = 7x + 7y = 2x + 2y + 8x + 4y
IWB- odd only Ex 15.02, Pg 389 Ex Pg 390 Ex Pg 391 Ex 15.06, Pg 393 Ex Pg 397 = 2x + 2y + 8x + 4y = 10x + 6y = 6x + 3y + 6x + 12y = 12x + 15y = 12x + 18y 10x 4y = 2x + 14y = 2x x + 24 = 6x + 30 Factorising 3x + 6 3 (x + 2) Expanding Factorising
Factorising is the reverse procedure of expanding. Expanding 3x + 6 3 (x + 2) Factorising 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 = 3( ) x+y = 4( ) x y = 7( ) x + y + z Try These e.g. Factorise: = 6( ) a+b = 7( ) x+1 = 7( ) x+2 = 3( )
e.g. Factorise: a.) 6a + 6b b.) 3p 3q c.) 4x + 4y f.)7x +7 g.) 7x + 14 h.) 24x + 36 = 6( ) a+b = 7( ) x+1 = 7( ) x+2 = 3( ) p q = 4( ) x+y = 12( ) 2x+3 d.)6x + 12 = 6( ) x+2 You can check that your answer is correct by expanding = 24( ) x+y e.)24x + 24y Try These e.g. Factorise: = 2( ) 4a+3b = 7( ) x+7 = 9( ) x+7 = 3( )
e.g. Factorise: a.) 8a + 6b b.) 12p 3q c.) 4x + 8 f.)7x + 49 g.) 9x + 63 h.) 45x + 81 = 2( ) 4a+3b = 7( ) x+7 = 9( ) x+7 = 3( ) 4p q = 4( ) x+2 = 9( ) 5x+9 d.)6x + 30 = 6( ) x+5 You can check that your answer is correct by expanding = 29( ) x+1 e.)29x + 29 Starter Factorise: = 4(a+2b) = 3(p 2q +r) = 4(x + 2y + 3z)
Factorise: a.) 4a + 8b b.) 3p 6q + 3r c.) 4x + 8y + 12z = 4(a+2b) = 3(p 2q +r) = 4(x + 2y + 3z) d.)6x + 21 = 3(2x +7) IWB- odd only Ex Pg 400 Ex Pg 401 Ex Pg 402 Ex Pg 403 Ex Pg 405 e.)24x - 32 = 8(3x 4) 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 = 22y23 = 33a43 = 4y6 = 27a12 Extension More exponent rules
(am)n = amn 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 x23 = (-4)2h22g62 = (4x)2 = 8x6 = 16h4g12 = 16x2 Extension Expanding 2 Brackets
QUADRATIC EXPANSIONWhen 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. FOIL e.g.(x + 4) (x 2) = x x x-8 = x2 + 2x - 8 Extension Expanding 2 Brackets
QUADRATIC EXPANSION e.g.(x + 10) (x + 1) 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 - 3x- 8x+ 24 = x2 - 16 = x2 - 11x + 24 Notice the middle term cancels out DIFFERENCE OF SQUARES Extension Expanding 2 Brackets
QUADRATIC EXPANSION e.g.(x 5) (x + 4) 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 Extension Factorising Quadratics
e.g.Factorise simple quadratics: a.) x2 + 6x + 8 Find 2 numbers which multiply to 8 and add to 6 (x+2) (x+4) 4 and 2 Put these number into brackets The factors are (x+2) and (x+4) b.) x2 - 5x - 24 Find 2 numbers which multiply to -24 and add to -5 (x-8) (x+3) -8 and 3 Put these number into brackets The factors are (x-8) and (x+3) c.) x2 + 5x - 24 Find 2 numbers which multiply to -24 and add to 5 (x+8) (x-3) 8 and -3 Put these number into brackets