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Algebraic Notation
Here we will learn about algebraic notation, including writing expressions and forming equations.
There are also worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
What is algebraic notation?
Algebraic notation is a system for writing mathematical expressions and equations using letters, symbols, and operations.
It can be used to solve problems posed in worded form or added to the annotation of a diagram in geometrical problems to make a solution easier to find. Writing expressions using algebraic notation is a skill vital for GCSE Mathematics.
To do this we must first understand how to convert a mathematical expression in word form into algebraic notation. We need to know how to apply algebraic notation to the operations of addition, subtraction, multiplication and division.
The phrase “ 2 more than m ” can be written as m+2.
“ 5 less than h ” can be written as h-5.
“ a less than 3 ” can be written as 3-a.
In algebra, when numbers and letters are written next to each other it indicates that they are multiplied together.
For example,
We write 4 \times a as 4a.
Multiplication is commutative so 4 \times a is the same as a \times 4, but that does not mean we write a4. When using algebraic notation for multiplication we always put the numerical coefficient before the letter.
When letters, or numbers and letters, are being divided, they are written in fraction form.
For example,
y \div 3 would be written in algebraic notation as \, \cfrac{y}{3}.
So if we had a problem that said the length of a rectangle is one more than 3 times the width. We could write l=3w+1, where l is the length and w is the width.
Algebraic notation is used across mathematics and science. In GCSE Mathematics and GCSE Science you will see many algebraic expressions and formulae.
Across both subjects you will use algebraic notation when plotting graphs, solving equations, inequalities, expanding brackets, factorising, simplifying expressions or algebraic fractions.
What is algebraic notation?
How to use algebraic notation
In order to use algebraic notation:
- Read the worded phrase to identify the mathematical operations.
- Use letters to represent any variables.
- Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
- Collect any like terms.
Explain how to use algebraic notation
Simplifying expressions worksheet (includes algebraic notation)
Get your free algebraic notation worksheet of 20+ simplifying expressions questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE Simplifying expressions worksheet (includes algebraic notation)
Get your free algebraic notation worksheet of 20+ simplifying expressions questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEAlgebraic notation examples
Example 1: writing a sum using algebraic notation
A triangle has side lengths of 3 \ cm, 6 \ cm and x \ cm.
Write an expression for the perimeter of the triangle.
- Read the worded phrase to identify the mathematical operations.
The perimeter of a shape is found by finding the sum of the sides. We need to add the side lengths together.
2Use letters to represent any variables.
We already have been told that the variable is called x.
3Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
There are no multiplications or divisions, just additions.
3+6+x4Collect any like terms.
We can collect the 3 and 6 to give the perimeter as
9+x.Example 2: writing a subtraction using algebraic notation
Sally is h \ cm tall. Her little brother Peter is shorter than Sally. Peter is 130 \ cm tall.
Write an expression for the difference in their heights.
Read the worded phrase to identify the mathematical operations.
The difference in their heights is found by subtracting Peter’s height from Sally’s height.
Use letters to represent any variables.
We already have been told that Sally’s height is the variable h.
Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
There are no multiplications or divisions, just a subtraction. Because Sally is taller than Peter, we need to subtract Peter’s height from Sally’s height.
h-130
Collect any like terms.
There are no like terms to collect.
The difference in their heights is h-130.
Example 3: writing a product using algebraic notation
A rectangle has a length of 5 \ cm and a width of y \ cm.
Write an expression for the area of the rectangle.
Read the worded phrase to identify the mathematical operations.
The area of a rectangle is found by multiplying the length by the width.
Use letters to represent any variables.
We already have been told that the variable is called y.
Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
We need to do 5\times y.
This is written with the number first without the multiplication symbol, 5y.
Collect any like terms.
There are no like terms to collect.
The area of the rectangle is 5y.
Example 4: writing a division using algebraic notation
Amrit has a bag of counters. He knows one quarter of the counters are blue counters in the bag but does not know how many counters there are in total.
Write an expression for the number of blue counters.
Read the worded phrase to identify the mathematical operations.
The number of blue counters will be the number of counters divided by 4.
Use letters to represent any variables.
We have not been given a letter to represent the total number of counters so let’s call this variable n.
Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
There is a division n \div 4, so we will write this as a fraction. n is being divided by 4.
So n will be the numerator and 4 will be the denominator.
\cfrac{n}{4}
Collect any like terms.
There are no like terms to collect.
The number of blue counters is \cfrac{n}{4}.
Example 5: forming an equation using algebraic notation
A triangle has three angles. The middle sized angle is 10^{\circ} more than the smallest angle. The largest angle is 30^{\circ} less than double the smallest angle.
Form an equation that could be used to find the size of the smallest angle.
Read the worded phrase to identify the mathematical operations.
The sum of the angles in a triangle is 180^{\circ}. Therefore, we have to add the three angles together. The largest angle is 30^{\circ} less than double the smallest angle, therefore there is a multiplication and a subtraction.
Use letters to represent any variables.
We have not been given a letter to represent the size of the smallest angle, let’s call this variable a.
Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
The smallest angle is a^{\circ}.
The middle angle is (a+10)^{\circ}.
The largest angle is (2a-30)^{\circ}.
We need the sum of these terms which is equal to 180^{\circ}.
a+a+10+2a-30=180
Collect any like terms.
We can collect like terms to give
4a-20=180.
This equation can be solved to give a=50^{\circ}.
The smallest angle of the triangle is 50^{\circ}.
The middle angle of the triangle is 60^{\circ}.
The largest angle of the triangle is 70^{\circ}.
Example 6: writing a formula using algebraic notation
To find the original price of an item in a sale, we can divide the sale price by the percentage remaining if written as a decimal or fraction.
Write a formula to find the original price O of an item in a p\% sale with a sale price S.
Read the worded phrase to identify the mathematical operations.
The formula will involve a division.
Use letters to represent any variables.
We have been given the variables O, p and S.
Write the expression with any multiplications without the multiplication symbol and divisions as fractions.
To get the percentage remaining as a decimal or fraction we need to first subtract p from 100 , and then divide by 100.
This will be written as \cfrac{100-p}{100}.
We then need to divide S by this fraction.
S\div \cfrac{100-p}{100} is the the same as S\times \cfrac{100}{100-p}.
This can be written as \cfrac{100S}{100-p}.
Collect any like terms.
There are no term to collect but we need to finish writing the formula with O as the subject.
O=\cfrac{100S}{100-p}
Common misconceptions
- Not putting numerical coefficients in front of the letter
It is important to write h\times 8 as 8h and not h8.
- Not using powers correctly
A common error is to write a\times a\times a as 3a, when it should be a^3.
- Writing a subtraction in the wrong order
6 less than x is x-6 and not 6-x.
Practice algebraic notation questions
1. Write “ 4 more than y ” using algebraic notation.
“ 4 more” involves adding 4, so we add 4 onto y to make the expression y+4.
2. Write “ 9 less than x ” using algebraic notation.
“ 9 less” involves subtracting 9, so we subtract 9 from x to make the expression x-9.
3. Write “ g less than 6 ” using algebraic notation.
“ g less” involves subtracting g, so we subtract g from 6 to make the expression 6-g.
4. Write “ 1 less than 2 times x ” using algebraic notation.
“ 2 times x ” is written as 2x. “ 1 less” means subtracting 1, so we subtract 1 from 2x to make the expression 2x-1.
5. A rectangle has a length which is 2 \ cm less than 4 times its width. Write an expression for the perimeter of the rectangle.
Let w be the width of the rectangle.
The length of the rectangle will be 4w-2.
The perimeter will be the sum of all 4 sides of the rectangle,
w+4w-2+w+4w-2.
Collecting the like terms will simplify this expression to
10w-4.
6. The price of 2 apples and 3 bananas is £1. The price of 4 apples and 7 bananas is £2.20. Write this information as a pair of simultaneous equations where a is the cost of one apple and b is the cost of one banana.
The first piece of information can be written as
2a+3b=1.
The second piece of information can be written as
4a+7b=2.2.
These can be solved as simultaneous equations to give a=0.2 and b=0.2.
This means that the cost of one apple is 20p and the cost of one banana is also 20p.
Algebraic notation GCSE questions
1. Each small square on a chessboard used in a chess game has a side length of x \ cm.
(a) Write an expression for the perimeter of the chessboard.
(b) Write an expression for the area of the chessboard.
(4 marks)
(a)
Side length = 8x
(1)
Perimeter = 32x
(1)
(b)
8x \times 8x
(1)
64x^2(1)
2. The width of a rectangle is 1 \ cm less than half its length.
(a) Write an expression for the perimeter of the rectangle if its length is l \ cm.
(b) Given that the perimeter of the rectangle is 22 \ cm, find its area.
(7 marks)
(a)
\cfrac{l}{2}-1
(1)
\cfrac{l}{2}-1+\cfrac{l}{2}-1+l+l(1)
3l-2(1)
(b)
3l-2=22
(1)
l=8(1)
8\times (\cfrac{8}{2}-1)(1)
24 \ cm^2(1)
3. To give the correct dose of a medicine (in mg ) to a child, the following steps must be taken.
Divide the child’s mass (in kg ) by 20, then add 15.
Write a formula for the dose, D, of a child of mass, m \ kg.
(3 marks)
(1)
\cfrac{m}{20}(1)
D=\cfrac{m}{20}+15(1)
Learning checklist
You have now learned how to:
- Form expressions and equations using algebraic notation
- Identify algebraic terms and state their variables and coefficients
The next lessons are
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