Transcript 21.use_of_symbols - GCSE Resources for teachers and students
Use of symbols
Objectives: F Grade E Grade D Grade C Grade Simplify expressions with one variable such as:
a
+ 2
a
+ 3
a
Simplify expressions with more than one variable such as: 2
a
+ 5
b
+
a
– 2
b
Multiply out expressions with brackets such as: 3(
x
+ 2) or 5(
x
- 2) Factorise expressions such as 6
a
+ 8 and
x
2 – 3
x
Expand (and simplify) harder expressions with brackets such as:
x
(
x
2 3(
x
- 5) and + 2) - 5(2
x
– 1)
Algebraic Definitions Expression
An expression is a mathematical "phrase" that stands for a single number; for example, 3
x
+ 1
Equation
An expression that equals something, that maybe another expression or a single value.
For example: 3
x
+ 1 = 7 or 3
x
+ 1 = 2
x
- 1
Variable
A variable is a letter used in an algebraic expression in order to represent any number.
Algebraic Definitions Term
A term is a number and / or variable(s) connected with x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation.
For example term 3
x
+ 4
y
term term 3
x
+ 4
y a b
term
Use of symbols
What combinations of letters and numbers mean
a
+
a
Can be read as 2 of those things called
a
So this can be written as 2 ×
a
because things get abbreviated in maths we write this as: 2
a a
+
a
+
a
3 of those things called
a
So this becomes: 3 ×
a
3
a
Also, because multiplication is commutable (the order of the Multipliers can be swapped and the answer remains the same)
a× b = b × a
or
ab = ba
Use of symbols
This is not to be confused with:
a
×
a
Can be read as
a
of those things called
a
So this can be written as
a
×
a
We know that 4 × 4 can be written as 4 2 because any number multiplied by itself like this Index number
a
×
a = a
2 3 2 So the following are also true: Base number
a
×
a
×
a
×
a
×
a
×
a = a
3
a = a
4 The index number tells us how many times the base number is multiplied by itself.
e.g. 3 4 means 3 x 3 x 3 x 3 = 81
Use of symbols
Collecting terms of add & subtract 2
a +
3
a a + a + a + a + a
So this can be simplified to 5 of those things called
a
5
a
7
a -
3
a a + a + a + a + a + a + a - a - a - a
So this can be simplified to 4 of those things called
a
5
a
7
b -
3
a b + b + b + b + b + b + v - a - a - a
So this cannot be simplified because
a
and
b
are different This remains as: 7
b -
3
a
Now do these: 1
. p + 2p + 3p
6p
4
. t + t + 4t
6t
7
. d + 4d − 2d
3d
Use of symbols
2
. p + 4p − 3p
2p
5
. f + 6f − 10f
–
3f
8
. h + h − 5h + 2h
–
1h
3
. 2ab + 3ab
5ab
6
. 5ad − 2da
5ad
9
. p 2 + p 2 + p 2
3p
2
Use of symbols
The index number of a letter or number only applies to the number or letter immediately preceding it.
a
3
x = a
×
a
×
a
×
x abc
3
= a
×
b
×
c
×
c
×
c ab
3
c = a
×
b
×
b
×
b
×
c
Mathematical convention is that where we have more than one letter in a term, they are written in alphabetical order.
Use of symbols
Further rules for the use of letters
a
+
a
= 2
a a
×
a = a
2 These are different types of terms and cannot be mixed If the index number in two terms is different they cannot be added If the index number is the same they can be added / subtracted Example: Simplify this expression 5
x
2
+
2
x
+ 2
x
2 – 5
x
The same power of
x
these can be collected The same power of
x
these can be collected 7
x
2 - 3
x
Use of symbols
Now do these: 1
. x +
3
x +
5
+ x
5x + 5
4
. y +
2
+ y +
4
2y + 6
7
. d +
5
+ 2d −
3
3d + 2
2
. g +
2
g + h
5
3g + h
. 7w +
6
−
2
w +
2
5w + 8
8
.
2
x +
3
y +
4
x +
2
y
6x + 5y
3
.
7
ab +
2
b +
4
a +
2
ab
9ab + 4a + 2b
6
. p
2
+p
2
+
2
p+p
2
+
4
p
9
.
4
3p
2
+ 6p
w +
2
+ y −
3
4w + y - 1
10
. a
2
+ a
3
+
2
a
2
3a 2 + 6a 3
13
. w
5
+
2
w
5
+ w
15
.
c
t
2
3w 5
+ w
+ 3t
2
+ t − t
2
2ct 2 + 2t 2
+ t
11
.
4
a
2
b +
5
a
2
b
12
.
3
cd
2
+
4
cd
2
−
2
dc
2
−
3
c
2
d
9a 2
b
7cd 2 - 5c 2
d
14
.
5
x
3
y +
2
xy
3
+
2
x
3
y
7x 3 y + 2xy 3
16. 7
abc
2
+
4
ab
2
c +
5
abc
2
+
3
ba
2
c
12abc 2 + 4ab 2 c + 3ba 2
c
Use of symbols
Summary
F Grade Simplify expressions with one variable such as:
a
+ 2
a
+ 3
a =
6
a
E Grade Simplify expressions with more than one variable such as: 2
a
+ 5
b
+
a
– 2
b =
3
a +
3
b
Algebraic meanings:
a
+
a
+
a =
3
a a
×
a× b = b × a a
×
a = a
3 or
ab = ba
• If the index number in two terms is different they cannot be added • If the index number is the same they can be added / subtracted • The index number of a letter or number only applies to the number or letter immediately preceding it.
• Mathematical convention is that where we have more than one letter in a term, they are written in alphabetical order.