Maths Algebra

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Wed/Ma/3

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Aims and Objectives  To introduce and understand the

concept of algebra  To learn how to put words into

algebra  To understand basic algebra  To revise Basic Algebra

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Key Words  Consecutive numbers  Algebra  Coefficient  Variable  Number Pattern  Sequence  Term  nth term rule

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Quick-Fire

Have a go at these questions!!

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x+12=26

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2x+5=17

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3x+10=29. 5

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9x+57=75

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9x=207

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x+(2x)=?

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x(x+5)=?

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2(x+7)=?

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3x(2+1)=?

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(x+1)(x+2) =?

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Answers 1. 14 2. 6 3. 6 ½ 4. 2 5. 23

6. 7. 8. 9.

3x x2+5x 2x+14 either: 6x+3x or 9x 10. x2+3x+2

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Algebra  Algebra is the branch of maths that

uses letters, symbols, and/or characters to represent numbers and express mathematical relationships.  These symbols are called variables.

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9x+2y=18 9 Coefficients Variables

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Subject #1

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How to change from english to algebra… A number is doubled, then 3 is added to the total, and the result is 15. What was the Original Number?

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Long-Winded English Ink-Saving Algebra A number

x

Double it

2x

Then add Three

2x+3

The Result is Fifteen

2x+3=15

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Questions 1. A group of 3 workers were paid £50

per hour plus a tip of £30. They shared the takings and each got £110. How many hours did they work for? 2. I have a number. My number is doubled, then 11 is added, and the result is 25. what is my starting number? Ext. I have a number. My number is squared, then doubled. My answer

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Answers: Question 1 x3 -30 50

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Answers: Question 2 -11

2

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Answers: Question EXT 2

simplify

Reciprocal Word Problem  Suppose one painter can paint

the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

 If the first painter can do the entire job in twelve

hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job per hour, and the second guy can do 1/8 per hour. How much then can they do per hour if they work together? To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is:

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 hours to complete job:

  first painter: 12   second painter: 8   together: t  completed per hour:   first painter: 1/12   second painter: 1/8   together: 1/t

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Adding their labour:

1

/12 + 1/8 = 1/t 5 1 /24 =  /t 24 /5 = t

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Plenary Do you…  Understand what Algebra is?  Understand what is a coefficient?  Understand what is a variable? Can you…  Convert English into Algebra?

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Independent Work  Question 1:  1) 4 2)-1

3)-3 4)-2 5)6 6)-20 7)-9 8)-11 9)24 10)6

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Question 2 Part 1 2000: J = 11W  J + 12 = 3W + 36 or 3(W+12) J + 12 = 3W + 36  (11W) + 12 = 3W + 36  11W – 3W = 36 – 12  8W = 24  W = 3 William was 3 years old in 2000.

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Question 2 Part 2 m + g = 68  (m – 1) g + 3 = 6(m – 1) m = 68 – g g + 3 = 6m - 6 g + 3 = 6(68 – g) – 6 g + 3 = 408 – 6g - 6 g + 3 = 402 – 6g 7g = 399 g = 57 68 – g = m = 11

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Question 2 Part 3 Hint!

n

(n+1) 15

n + (n+1) = 15 2n + 1 = 15 2n = 14 n=7

Consecutive means “next to each other”. Consecutive numbers are numbers that are next to each other (e.g. 7, 8, 9, 10 etc.)

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Question 2 Part 4 First Guy : 1/6 of the job per hour New Guy: 1/8 of the job per hour

(Add their labour time) : 1/6 + 1/8 = 7/24 They work for 2 hours so: 2(7/24) = 7/12 1 - 7/12 = 5/12 of the job left to do

will take to do 5/12 of the job. Let "h" indicate the number of hours he nee ( 1/8 job / hour) × (h hours) = 5/12 job h /8 job = 5/12  job h /8 = 5/12 h = ( 5/12) × ( 8/1) = 10/3 = 3 1/3 The New Guy takes 3 hours and 20 minutes more to complete the job.

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Subject #2

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Terms Before you can do anything in algebra you have to really understand what a term is… A term is a collection of numbers, letters and brackets all mutiplied/divided together.

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Simplifying “Simplify”:

Collect like terms:

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Expanding Brackets

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Squared Brackets

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Factorising (putting BRACKETS in)

=

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Subject 2: Sequences and the nth term rule

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Number Pattern Types  Common Difference (e.g. +3)  Increasing Difference (e.g. +2, +3,    

+4) Decreasing Difference (e.g. -10, -9, -8) Multiplying Factor (e.g. x2) Dividing Factor (e.g. 5) Adding Previous Terms

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Finding the nth term: Common Difference

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Finding the nth term: Changing Difference

1, 4, 9, 16, 25, 36…

a=1 d=3 C=2

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Method (easy way out)  Recognise the sequence (if it is

increasing decreasing etc.)

1, 4, 9, 16, 25, 36… Increasing, Squared

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Find the Difference between the terms…

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Half the Number, then put the number in front of n2

•

2 2 2 2

2 Divided by 2 = 4 So the first part = 2 2 1n or n

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Compare the _n2 sequence to the original sequence 1 1 0

4 9 16 25 4 9 16 25 0 0 0 0 0

36 36

So that means the sequences are the SAME So:

T(n)=n

2

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Practice Find the nth term: 2.3, 7,11,15,19… 3.2,5,10,17,26… 4.97,89,81… 5.-1, 0, 3, 8… 6.11, 22, 33…

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Answers 1. 4n-1 2. N2+1 3. 105-8n • n2-2n 5. 11n

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Plenary Do you:  Understand what a term is?  Understand what a sequence is? Can you:  Work out an nth term rule?  Name the different sequence types?