Addition and subtraction (1)
Using letters to represent numbers
In maths we often use a letter to stand for a number. Sometimes the letter has one value we can work out, and sometimes it is a variable that can take many different values.
Letters stand for numbers
A letter such as d, s or x can stand for a number. A value that never changes is a constant, but a letter that can take different numbers is a variable. For example, if d is a dice score then d could be 1, 2, 3, 4, 5 or 6.
Finding the value of an expression
To find the value of an expression, replace the letter with its number and then calculate. If d = 5, then d + 4 = 5 + 4 = 9.
Expressions that are equal
Adding can be done in any order, so d + 4 is always equal to 4 + d. But subtraction cannot be swapped: 5 - d is NOT the same as d - 5.
A letter can be a variable
An unknown is not always one fixed number. If a rectangle has a perimeter of 20 cm, then its two side lengths s and t must add to 10, and there are many possible pairs such as s = 3, t = 7 or s = 4, t = 6.
Worked examples
Finding the value of an expression
- d stands for a dice score and d = 5.
- Replace d with 5 in the expression d + 4.
- Work out 5 + 4.
Answer: When d = 5, d + 4 = 9.
Using a perimeter formula
- The perimeter of a square is p = 4 × s because it has 4 equal sides.
- When s = 5, work out p = 4 × 5.
- You can also go backwards: if p = 32, then s = 32 ÷ 4.
Answer: When s = 5, p = 20 cm. When p = 32, s = 8 cm.
Deciding if two expressions are equal
- Compare d + 4 with 4 + d: adding can be done in any order, so they are equal.
- Compare 5 - d with d - 5: try d = 2. Then 5 - 2 = 3 but 2 - 5 is different.
- So subtraction cannot be swapped around.
Answer: d + 4 = 4 + d, but 5 - d is not equal to d - 5.
A variable with many values
- A rectangle has a perimeter of 20 cm with sides s and t.
- s + t is half the distance round the rectangle, so s + t = 10.
- List pairs that add to 10, such as s = 1 and t = 9, or s = 4 and t = 6.
Answer: s + t = 10, so there are many possible pairs of values.
Finding two lengths from a total
- Two strips a and b are placed end to end with total length 15 cm, and b is 3 cm longer than a.
- So a + b = 15 and b = a + 3, which gives a + a + 3 = 15.
- That means 2 × a = 12, so a = 6, and then b = a + 3.
Answer: a = 6 cm and b = 9 cm.
Where you see this in real life
- Using d for a dice score to work out how many spaces to move in a board game.
- Writing a formula for the perimeter of a shape using a letter for the side length.
- Sharing puzzle pieces between two people so that x + y equals the total.
- Comparing two lengths of card when one is a few centimetres longer than the other.
Key words
- Variable
- A letter that can stand for different numbers.
- Constant
- A value that does not change.
- Expression
- A group of numbers and letters joined by operations, such as d + 4.
- Unknown
- A number we do not yet know, often shown by a letter.
- Formula
- A rule written with letters, such as p = 4 × s for the perimeter of a square.