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Simplifying
Why use letters?
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Words to know
Expression: A group of letters and/or numbers with
either ÷, +, – or x between them, there must not be
an = sign.
Sum: add
Difference: subtract
Product: multiply
Term: part of an expression
Like terms: terms with the same letters
Simplify: join terms together
BEDMAS: See next slide…
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BEDMAS Revision
1st
Last
(
(
Brackets
Exponents
Division
Multiplication
Addition
Subtraction
)
1st
)
Last
Note 1: This is not a strict order, hence blue brackets
Note 2: Fractions have „invisible‟ brackets
E.g. Using BEDMAS
1–3+2
Subtraction and addition have equal
importance so use Left to Right Rule
–2 + 2
0
E.g. Using BEDMAS
5x6÷2
Multiplication and Division have equal
importance so use Left to Right Rule
30 ‚ 2
15
E.g. Using BEDMAS
(
(
)
)
Insert ‘invisible’ brackets
Apply BEDMAS to numerator and denominator separately
EXPRESSIONS
Expressions/Terms
Which of the following is not an expression?
8x  6
tq
3x  3 y
4 x  6  26
How many terms
does this expression
have?
3
4 x  7t  15
SLO
To identify like terms
Examples
Find the three pairs of like terms from the
following list
5xy, 8y,
2
9z t,
10, 12xy,
2
14tz ,
5xy and 12xy
10 and 7
9z2t and 14tz2
(notice that 8y has no like term)
7
Your Turn: Find the like terms
1) Find all the terms which are like terms with 3X
2) Find all the terms which are like terms with 6T
3) Find all the terms which are like terms with 7P
X
5T
6
5X
8X
4T
9P
10P
3
1P
22T
7V
7X
Simplifying
SLO
Adding and subtracting terms
http://www.youtube.com/watch?v=Fb_tQnSAC4M (easyish)
http://www.youtube.com/watch?v=mc0pALxpTWU (hardish)
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Adding and subtracting terms
When we add or subtract like terms in an expression we
are simplifying.
Only like terms can be added or subtracted
For example,
1) 3a + 4a = 7a
2) 6x – 2x = 4x
3) 7y + 3y – 2y = 8y
The following two examples cannot be simplified.
4) 6a + 7b
5) 9x – 7
SLO
To add and subtract terms
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Add and subtract terms
E.g. simplify
15T + 5F – 6T – 8F
Step 1: circle like terms (include the + or –
before the terms)
Step 2: Join the like terms together
15T – 6T = 9T
5F – 8F = –3F
Step 3: Write these unlike terms together
9T – 3F
E.g. Simplify the following
+ 7x + 3y + 5y – 9x – 17y
= – 2x
= – 9y
Your Turn: Simplify the following
4 x  2 x  3a  5a  6 x  8a
Click for hint:
4 x  3 y  5x  7 y  9 x  4 y
Click for hint:
7d  2e  3e  4d  3d  5e
Click for hint:
4 x 2  5 x  x 2  2 x  5 x 2  3x
Click for hint:
Your Turn: Simplify 1 – 10 by matching them to one of a – j.
(The first one is done for you)
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
a+a+a+a
a + 2a + 3a + a
a+b+a+b
2b + a + b – b
6a – 4a
5a + 2b – a + b
5b + b -3b + a
a + b + b + 2b
3a + 3b – a
6b – 5b + a
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
2a + 2b
4a
2a
a + 3b
a+b
a + 2b
2a + 3b
4a +3b
7a
a + 4b
Your Turn: Simplify the following
1) 5x + 8y – 2x + 3y =
5x – 2x + 8y + 3y =
3x + 11y
2) 7x + 9y – 5x – 2y =
7x – 5x + 9y – 2y = 2x + 7y
3) 6x + 5y – x =
6x – x + 5y =
5x + 5y
4) 8x + 5y – 2x – y =
8x – 2x + 5y – y =
6x + 4y
E.g. Find the perimeter of the following shape.
5x + y
5x + y
6x – 2y
P = 5x + y + 5x + y + 6x – 2y = 16x
Algebraic pyramids
Algebraic magic square
In a magic square, all rows, columns and diagonals are equal.
a+b
b +2c
2a + b +c
MAGIC!!! b
2a + 2c + b a + b + c
b+c
2a + b
a + b + 2c
Is this magic? You try…
4a + b
2c
5a + 2b +c
MAGIC!!! 2a
4a +2b +2c 3a + b + c
a+c
6a + 2b
2c + 2a + b
Fill in the gaps on your grids. Make it magic…
p + 3q + r
p+q
p - q + 2r
Click squares
to reveal
p + q +2r
p + 2q + r
p + 3q
p + 2q
p + 3q +2r
p+q+r
Can you arrange the cards to make a magic square?
3a + 8b
7b
Click cards
for answers
7a + 6b
2a + b
a + 4b
6a + 9b
4a + 5b
8a + 3b
5a + 2b
Clue
Questions to do from the books
Achieve
Merit
P7 Ex1.03 Q1–11
Gamma
P10 Ex1.04 Q1–29
P7 Ex1.03 Q12
P10 Ex1.04 Q30
CAT 1.2
P5 Q29–36
P4 Q1–28
Excellence
P5 Q37–40
SLO
Simplifying products
http://www.youtube.com/watch?v=Mm4y_I8-hoU (2a5)3
http://www.youtube.com/watch?v=2OVBCPrbUvo (easy: mostly same base)
http://www.youtube.com/watch?NR=1&feature=endscreen&v=cV3xWPyZpQI (harder)
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Multiplication rules
1) In algebra we usually leave out the multiplication sign.
2) Numbers are written at the front.
3) Letters in alphabetical order.
4) A full stop can be used as a multiplication sign.
E.g.
5xa
axb
1xb
bx5
3xdxc
=
=
=
=
=
5a
ab
(this could be written as a.b)
1b = b (the 1 is not required)
5b
(numbers first)
3cd
(alphabetical order)
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Multiplying terms together
1) Numbers and letters are multiplied separately
E.g. 4R x 2T = 8RT
E.g. Simplify the following
E.g. 1) 4F x 6H
=4xFx6xH
=4x6xFxH
= 24 x FH
= 24FH
E.g. 2) 7T x 5Y = 35TY
Your Turn:
Simplify the following
1) 4F x 3T 12FT
2) 5H x 7G 35GH
3) H x 2P 2HP
4) 4P x 5Z x 2 40PZ
5) 4W x 2T x 3R 24RTW
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Exponents
exponent – the number of times a number is multiplied by itself.
base – the number that is being multiplied.
3
base
8
exponent
index form
This is read “8 to the 3rd power” or “8 cubed.” or “8 to the power of 3”
E.g. Exponents
5
1
2
3
4
5
2 =2x2x2x2x2
3
6 =6x6x6
3
A =AxAxA
3 2
G H =GxGxGxHxH
Video exponent to expanded form: http://www.virtualnerd.com/pre-algebra/factorsfractions-exponents/exponential-to-expanded-form-conversion.php?&sid=
Other Exponents
Any number, besides zero, to the zero power is 1.
E.g.
4 1
0
A negative power means it should be ‘flipped’
E.g.
1
x  3
x
3
37
Your Turn:
R
4
FxFxSxSxS
AxAxAxPxPxPxP
Your Turn
Write the following in index form
1) 4 x F x 3 x F 12F2
2) 5 x H x H x 6 30H2
Simplify (put in index form) the following
1) 6H x 2H 12H2
2) 4P x 5P x P 20P3
Simplifying Indices (Multiplication)
Can you spot a quick
way to do this?
Simplifying Indices
14
Can you spot a quick
way to do this?
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Products and exponents (I)
To multiply powers that have the same
base, add exponents i.e.
m
a
x
n
a
Note that the bases
are the same
=
(m+n)
a
For multiplication we
ADD the indices
http://www.youtube.com/watch?v=2OVBCPrbUvo (easy)
http://www.youtube.com/watch?NR=1&feature=endscreen&v=cV3xWPyZpQI (hard)
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E.g.
T4 x T8 = T12
6W5 x 3W6 = 18W11
7W3 x 4Y7 = 28W3Y7
(add exponents)
(multiply numbers, add exponents)
(multiply numbers, different letters
so do not add exponents)
Your Turn: Simplify
1) G4 x G3 = G7
2) 6G9 x 2G5 = 12G14
3) G4 x H3 = G4H3
4) 9H4 x 5G3 = 45G3H4
5) 6H3G4 x 5H8G2 = 30G6H11
6) 7H9G4 x G = 7G5H9
Your Turn: Simplify
1) (GH)2 = GH x GH = G2H2
2) (5G2H3)3 = 5G2H3 x 5G2H3 x 5G2H3 =
125G6H9
3) (3G4H7)3 = 27G12H21
Your Turn: Simplify
x 
2 3
 x 2 .x 2 .x 2  Expand
 x.x.x.x.x.x
 x6
2 y
4
 2 y.2 y.2 y.2 y
 16 y 4
 3x 
2 3
 3x 2 .3x 2 .3x 2  Expand
 27x 6
3 x

2 3
 3.x 2 .x 2 .x 2
 3x 6
Hexagon puzzle
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Products and exponents (II)
To simplify brackets and exponents
m
n
q
(a b )
=
mq
nq
a b
Each part is raised
to the new base.
E.g.
(5R4T8)2 = 52R2x4T2x8 = 25R8T16
http://www.youtube.com/watch?v=Mm4y_I8-hoU (2a5)3 explained
Your Turn: Simplify
 
1. 2 x
5 3
 8x
15
Note that the number is also raised
to the power outside the brackets


4 6 8
2. 3a b
 3a b
8
32 48
2) Simplify
2 2
2(4 xy )
 2.4 xy .4 xy
2
 32 x y
2) Simplify
2) Simplify
2(3 x 2 )3
3(2 xy 2 )3  3.2 xy 2 .2 xy 2 .2 xy 2
 2.3 x 2 .3 x 2 .3 x 2
2
 24 x y
3
6
 54 x 6
x 7  x 2 x.x.x.x.x.x.x.x.x

4
3
5
2
8
x x
x.x.x.x.x.x. x  x  x.x.x.x.x.xx.x.
x.x.x.x.x.x.x.x

9
8
x
x.x.x.x.x.x.x.x.x.  x
x
x.x.x.x.x.x.x.x
2
4
1
 2
x
1

x
3) Simplify
2) Simplify
4(2 x )  4.2 x .2 x .2 x
2 3
3 2
3(2 x )
 3.2 x .2 x
3
 12 x
 32 x 6
3
6
x 2  x3
2
x.x.x.x.x
4 x 2 y 2 x.x. y

2
6 xy
3.x. y. y
2
2
Questions to do from the books
Achieve
Merit
Excellence
P7 Q53–58
P10 Q76–79
Gamma P16 Ex2.05 Q9,10,19
CAT 1.2
P7 Q41–52
SLO
Fractions and exponents
Quotient rule video explanation:
http://www.youtube.com/watch?v=Mn4WuvIGUgI
http://www.youtube.com/watch?v=P7edpw6N_uc
(Quotient rule video explanation)
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Algebra and Division
a
b Means: a divided by b
E.g.
1 means 1 ‚ 2 = 0.5
2
5
means 5 ‚ a
a
W means W ‚ V
V
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Simplifying Indices (Division)
Cross off pairs of Ds from top and bottom
Write out what you have left
Simplifying Indices (Division)
Cross off pairs of Ds from the top and bottom
Write out in index form what you have left
Simplifying Indices (Division)
Simplifying Indices (Division)
14 ‚ 7 = 2
21 ‚ 7 = 3
Simplify the number part of the fraction as usual
Cross off pairs of Ds from top and bottom
Write out what you have left
Your Turn: Simplify
or T-2
2T
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Dividing terms with the same base
When we divide two terms with the same base the indices
are subtracted.
m
x
‚
n
x
Note that the bases
are the same
=
(m
–
n)
x
For division we
SUBTRACT the
indices
Your Turn: Simplify
1)
9
a
a3
= a6
3)
3
2)
y
4
y
=
y-1
or
1
𝑦
6p2
6p2
‚ 3p =
3p
2
6×p×p
=
3×p
= 2p
6 11
ab
4. 3 7
ab
5
 ab
3 4
8
20 x y
5.
6
35 xy
4
4x y

7
2
4) Simplify
7
6 Simplify
4)
2 3
12a b 12a.a.b.b.b

4
6ab
6a.b.b.b.b
2a

b
y1  y 2  y 3
 y  y. y  y. y. y 8
4) Simplify
y
6
12a 2b3 12a.a.b.b.b

3 2
16a b 16a.a.a.b.b
3b

4a
y 4  y 3  y1
 y. y. y. y  y. y. y  y
8

y
8a 4b3 8a.a.a.a.b.b.b

2 4
6a b
6a.a.b.b.b.b
2
4a

3b
y1  y 5  y 3
Your Turn: MERIT
1)
1
2cd 4ed

5e
5c

2cd 5c

5e 4ed
2

3)
2
c
2e 2
(3x3 ) 2 (2 x 2 )3

2
4x
6( x 4 ) 2
3x3 3x3 2 x 2 2 x 2 2 x 2

4 x2 6 x4 x4
2)
3
9
2
3
12
2
5
7
3
24 x y 18 x y

4
5
8
16 x y 36 x 2 y 9
3
3x 2
 2
4y
3x
 10
x
 3x 2
2
Questions to do from the books
Achieve
Merit
Excellence
Gamma P16 Ex2.05 Q3,14,23 P16 Ex 2.05 Q28
CAT 1.2
P9 Q59–67
P9 Q68–75
http://www.youtube.com/watch?v=koGVrCMtP8s&feature=related (help with
harder merit questions)
Algebraic areas
SLO
To write an expression from words
http://www.youtube.com/watch?v=6E1BUAldick
(you tube video: gets a little hard near the end: 3 minutes
Forming Expressions
Mathematicians convert many words into a few letters
E.g. Write the following as an expression
I think of a number and add 3 to it.
x+3
We can use any letter as the unknown
number, but x is used most commonly.
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Writing expressions
Some examples of algebraic expressions:
n+7
a number n plus 7
5–n
5 minus a number n
2n
2 lots of the number n or 2 × n
6
n
6 divided by a number n
4n + 5
4 lots of a number n plus 5
E.g. Write Algebraic Expressions
for These Word Phrases
• Ten more than n
• w decrease by 5
• 6 less than x
• n increased by 8
• The sum of n and 9
• 4 more than y
n + 10
w-5
x-6
n+8
n+9
y+4
Your Turn:
Write the following as expressions
1)
A number doubled then 3 is added.
2x + 3
2)
A number multiplied by 3 then 5 is subtracted.
3x – 5
3)
A number is divided by 5 and 2 is subtracted.
4)
A number divided by 6 and then 9 is added.
𝑥
−2
5
𝑥
6
+9
Your Turn:
Writing an expression
Suppose Jon has a packet of biscuits and he
doesn’t know how many biscuits it contains.
He calls the number of biscuits in the full
packet, b.
If he opens the packet and eats 4 biscuits, he can write an
expression for the number of biscuits remaining in the
packet as:
b–4
Your Turn: Writing an expression
Miss Green is holding n number
of cubes in her hand:
Write an expression for the number of cubes in her hand if:
1) She takes 3 cubes away.
n–3
2) She doubles the number of
cubes she is holding (at the start).
2 × n or
2n
Equivalent expression match