Download Algebra 1: Simplifying

Algebra 1:
Simplifying
SLO
To understand the words algebra, expression
7
and what 7x and mean in Mathematics
𝑥
Algebra
In the subject English we use letters
In Mathematics we use numbers
In Algebra we use letters and numbers
Why use letters?
Copy into
notes
Words that mean to Add
•
•
•
•
•
Plus
More than
Sum
Increased by
Altogether
Copy into
notes
Words that mean to Subtract
•
•
•
•
•
Minus
Difference
Subtracted from
Decreased by
Less than
Copy into
notes
Multiplying Phrases
• Times
• Product of
Dividing Phrases


Quotient
Divided by
Expressions
Copy into
notes
(take care: before you copy these notes can you spot which
of the below is not an expression).
Expressions
A group of letters and/or numbers with either ÷, +, –
or x between them, there must be no = sign e.g.
3x  3 y
4 x  6  26
8x  6
tq
4 x  7t  15
Copy into
notes
Algebra and Multiplication
In algebra we usually leave out the multiplication sign ×.
Any numbers must be written at the front and all letters should
be written in alphabetical order.
5 x a = 5 times a = 5a
a x b = a times b = ab
For example,
4 × a = 4a
1×b=b
We don’t need to write a 1 in front of the letter.
b × 5 = 5b
We don’t write b5.
3 × d × c = 3cd
We write letters in alphabetical order.
Your turn:
Write the following in another way
7c = 7 times c
or
7xc
8d = 8 times d
or
8xd
gh = g times h
or
gxh
Copy into
notes
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
Your turn:
Write the following in another way
10 means 10 ÷ 5 = 2
5
R =R÷3
3
x =x÷y
y
SLO
To write an expressions 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.
Copy into
notes
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)
I think of a number and add 6.
x+6
2)
I think of a number and subtract 8.
x–8
3)
I think of a number and multiply by 3.
3x
4)
I think of a number and divide by 7.
𝑥
7
Your Turn:
Write the following as expressions
1)
A number plus 5.
x+5
2)
A number multiplied by 3.
3x
3)
6 subtracted from a number.
x–6
4)
A number divided by 4.
𝑥
4
Your Turn:
Write the following as expressions (harder)
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 (hard)
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 (hard)
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
SLO
To use basic substitution (just one letter)
http://www.youtube.com/watch?v=_FqJIXTXxho (you tube video: starts very wordy
but some good examples explained of single variable: 5 minutes)
Copy into
notes
Substitution
In algebra, when we replace letters in an expression or
equation with numbers we call it substitution.
Copy into
notes
Substitution Example 1
If a = 3, find the value of 4 + a
swap a for 3 in the formula:
4 + 3a = 7
Substitution Example 2
If y = 6, find
3y
Remember 3y means 3 x y
3x6
y = 18
Substitution Example 3
If b = 5 calculate 3b + 4:
3 x b5 + 4
15 + 4 = 19
Substitution Example 4
Find the value of 2p – 5 when
P=6
P=4
p- 5 = 7
2 x6
2x p
4 –5=3
Your Turn:
If p = 4, find the answer (evaluate):
1)
2)
3)
4)
5)
6)
7)
3p
p+5
p–2
2p + 1
8+p
10 – p
20
p
3(4) = 12
4+5=9
4–2=2
2(4) + 1 = 9
8 + 4 = 12
10 - 4 = 6
20 ÷ 4 = 5
SLO
Using more than one letter with substitution
http://www.youtube.com/watch?v=q8iaBmPNJSk (You tube: easy example of
using 2 variables with substitution: 1 minutes)
http://www.youtube.com/watch?v=_GCWnYS3Gaw (you tube video: multiple variable
substitution, American so uses ‘dot’ for times and PEMDAS, a little hard: 5 minutes)
Copy into
notes
Substitution with more
than one letter (variable)
E.g. evaluate (find the answer) to 3T + TH if
T = 6 and H = 10
Substitute the values of T and H into the expression
(remember that TH means T times H)
3T + TH
3 x 6 + 6 x 10
Remember to multiply before you add
18 + 60 = 78
Your Turn
Evaluate each expression given that a = 6, b = 12, and c = 3.
1.
2.
3.
4.
5.
4ac
a÷c
a+b+c
ba
b–c
6. c ÷ b
= 72
=2
= 21
= 72
=9
1
=
4
Extension for Substitution
(you will need hard copy of the snakes and ladders table)
100
99
98
97
x - 90
81
82
83
79
95
84
85
86
63
59
58
76
75
74
38
Miss
A
Go
65
57
56
42
43
66
21
22
x2 20
19
1
2
68
90
72
71
69
x
54
70
x2
53
52
48
49
51
3(x-2)
44
45
46
38
Miss
A
Go
37
23
24
36
47
- 10
35
(x-2)2
2x
25
50
(x-1)2
34
33
32
31
x(x-1)
26
27
28
29
30
15
14
13
12
11
x - 10
18
17
16
3-x
4(x-2)
START
1-x
89
73
67
55
x2
39
91
3x
x2 -
x+15
40
92
x - 50
x+2
x-2
2x
41
88
10 -2x
77
2(x2)
60
87
x2
78
62
93
x-2
2-x
61
94
10 -3x
480
96
3
4x - 2
4
5
6
2x + 3
7
2(x-1)
8
9
10
The number on the dice is ‘x’.
4x - 2
4 x 3 – 2 = 10
So move forwards 10…
100
99
98
97
x - 90
81
82
83
79
95
84
85
86
63
59
58
76
75
74
38
Miss
A
Go
65
57
56
42
43
66
21
22
x2 20
19
1
2
68
90
72
71
69
x
54
70
x2
53
52
48
49
51
3(x-2)
44
45
46
38
Miss
A
Go
37
23
24
36
47
- 10
35
(x-2)2
2x
25
50
(x-1)2
34
33
32
31
x(x-1)
26
27
28
29
30
15
14
13
12
11
x - 10
18
17
16
3-x
4(x-2)
START
1-x
89
73
67
55
x2
39
91
3x
x2 -
X+15
40
92
x - 50
x+2
x-2
2x
41
88
10 -2x
77
2(x2)
60
87
x2
78
62
93
x-2
2-x
61
94
10 -3x
480
96
3
4x - 2
4
5
6
2x + 3
7
2(x-1)
8
9
10
If you threw a 6 and landed on this square,
what did you move?
(x –
2
2)
If you threw a 4 and landed on this square,
what did you move?
x+2
x-2
If you threw a 3 and landed on this square,
what did you move?
4–
2
x
Which squares are SNAKES? (Always negative)
Which squares are LADDERS? (Always positive)
Magic Square extension
for substitution
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
To finish off… a = 2, b = 3, c = 5
a+b
b +2c
2a + 2c + b a + b + c
b+c
2a + b
2a + b +c
b
a + b + 2c
On your
grids, write
down the
value in each
square
To finish off… a = 2, b = 3, c = 5
5
13
12
17
10
3
8
7
15
Is it magic?
Copy into
notes
Simplifying
Simplifying is joining letters together
SLO
To use simplifying and Multiplication
http://www.youtube.com/watch?v=zJxw8Ck493o (You tube video to explain
simplification with addition, subtraction and multiplication: last few questions
involving squaring gets a little hard 10 minutes)
Copy into
notes
Multiplying
3 rules for Multiplying
1) Any numbers are multiplied
E.g. 4 x 2T = 8T
2) Letters are written in alphabetical order
E.g. 3Q x 5T = 15QT
3) Numbers are placed in front of letters
E.g. Y x 4Q = 4QY
Example
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 40Z
5) 4W x 2T x 3R 24RTW
SLO
To know what like terms are
Copy into
notes
Like Terms
An algebraic expression is made up of terms
A term is made up of numbers and letters.
For example,
3a + 4b – a + 5 is an expression.
3a, 4b, a and 5 are terms in the expression.
3a and a are called like terms because they both contain a
number and the same letter(s), the letter a.
Examples
Find the three pairs of like terms from the
following list
5x, 8y, 9z, 10, 12x, 14z, 7
5x and 12x
10 and 7
9z and 14z
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
SLO
To simplify by adding
Copy into
notes
Collecting together like terms
One way to simplify an expression is to add or subtract like
terms.
An expression can contain different like terms.
For example,
3a + 2b + 4a + 6b
= 3a + 4a + 2b + 6b
= 7a + 8b
This expression cannot be simplified any further.
Copy into
notes
Collecting together like 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
Your Turn:
Simplify the following
1) T + T + T + T 4T
7F
2) 4F + 3F
3) 5H + 7H 12H
3HH
4) H + 2H
3P
5) 8P – 5P
6) 4W – W
3W
SLO
To add expressions that have unlike terms
Copy into
notes
To add expressions that have
unlike terms
E.g. simplify
5T + 7F + 6T + 2F
Step 1: Identify like terms (include the + before the
terms)
Step 2: Join the like terms together
5T + 6T = 11T
7F + 2F = 9F
Step 3: Write these unlike terms together
11T + 9F
E.g. Simplify the following
1) 4c + 3d + 2c + 2d =
2) 4r + 6s + t
4c + 2c + 3d + 2d
Cannot be simplified
3) 4r + 8s + 3r
4) 7r + 8r + 3r + s
5) r + r + 3r + s + 8s
= 7r + 8s
= 18r + s
= 5r + 9s
= 6c + 5d
Your Turn:
Simplify the following
1) 3T + 2Q + 5T 8T + 2Q
4F + 8T
2) 4F + 3T + 5T
3) H + 5H + 5R + 8R 6H + 13R
4) B + 2H + B + 5B 7B + 2H
5) 7P + 5Y + 5P + 4Y 12P + 9Y
6) 4W + 6A + A + W 5W + 7A
SLO
To add and subtract expressions that have
unlike terms
http://www.youtube.com/watch?v=Fb_tQnSAC4M (you tube video: starts a little
easy but goes on to good questions and explanation: 4 minutes)
Copy into
notes
To add and subtract expressions
that have unlike terms
E.g. simplify
15T + 5F – 6T – 2F
Step 1: Identify like terms (include the + or –
before the terms)
Step 2: Join the like terms together
15T – 6T = 9T
5F – 2F = 3F
Step 3: Write these unlike terms together
9T + 3F
E.g. 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
1) 6x + 5y – x =
6x – x + 5y =
5x + 5y
1) 8x + 5y – 2x – y =
8x – 2x + 5y – y =
6x + 4y
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
Algebraic perimeters
Remember, to find the perimeter of a shape we add
together the lengths of each of its sides.
Write algebraic expressions for the perimeters of the following
shapes:
2a
Perimeter = 2a + 3b + 2a + 3b
3b
= 4a + 6b
5x
4y
x
5x
Perimeter = 4y + 5x + x + 5x
= 4y + 11x
Find the perimeter of the following shape.
4x + 3y
3x + 2y
3x + 2y
2x
P = 3x + 2y + 2x + 3x + 2y + 4x + 3y = 10x + 7y
What is the perimeter if x = 5 and y = 2?
P = 10(5) – 7(2) = 64
Find the perimeter of the following shape when x = 2.
5x + y
5x + y
6x – 2y
To find the perimeter, add the sides together.
P = 5x + y + 5x + y + 6x – 2y = 16x = 32
Does the value of y matter in this problem? Obviously Not!
Algebraic pyramids
Algebraic magic square
Extension work
To add and subtract expressions that
have unlike terms (extension)
Simplify the following
+ 7x + 3y + 5y – 9x – 17y
= – 2x
= – 9y
Example with workings
Simplify 4x + 5x -2 - 2x + 7
• Collect all the terms together which are alike. Remember that
each term comes with an operation (+,-) which goes before it.
4x, 5x, and -2x
-2 and 7
• Simplify the variable terms.
4x+5x-2x = 9x-2x = 7x
• Simplify the constant (number) terms.
-2+7 = 5
• You have a simplified expression by writing all of the results
from simplifying.
7x + 5
Another example…
• 10x – 4y + 3x2 + 2x – 2y
3x2
10x, 2x
-4y – 2y
• 3x2 + 12x – 6y
Remember you cannot
combine terms with the
same variable but
different exponents.
Your Turn: Question 1
Simplify the following:
5x + 3y - 6x + 4y + 3z
5x, -6x
3y, 4y
3z
-x + 7y + 3z
Your Turn: Question 2
Simplify the following:
3b - 3a - 5c + 4b
3b, 4b
-3a
-5c
-3a + 7b – 5c
Your Turn: Question 3
Simplify the following:
4ab – 2a2b + 5 – ab + ab2 + 2a2b + 4
4ab, -ab
-2a2b, 2a2b
5, 4
ab2
3ab + ab2 + 9
Your Turn: Question 4
Simplify the following:
5xy – 2yx + 7y + 3x – 4xy + 2x
5xy, -2yx, -4xy
7y
3x, 2x
-xy + 7y + 5x
3
A farmer has two rectangular fields.
He wants to put a fence around both. In algebraic
terms, how much fence would he need?
3x – 6
2x + 5
Field 2
4
Field 1
P1 = 3 + 3 + 2x + 5 + 2x + 5
P1 = 4x + 16
Ptotal = 4x + 16 + 6x – 4
Ptotal = 10x + 12
P2 = 4 + 4 + 3x – 6 + 3x – 6
P2 = 6x – 4
How much fence would the farmer need if x =
5?
Ptotal = 10(5) + 12
Ptotal = 62
http://www.youtube.com/watch?v=UH0HuxtBhEM (you tube : harder substitution 4
minutes)