# Algebra

## What is algebra?

Algebra is a part of mathematics that helps represent a problem as a mathematical expression. It gives us rules about how equations should be made, and how they can be changed.

## What is an equation?

An equation is a mathematical sentence that has two equal sides separated by an equal sign, e.g.

3 + 5 = 8             3 + 5 = 4 + 4             8 = 2 + 6

## What is algebra used for?

Algebra is used when we do not know the exact number in an equation. A variable (usually written as a letter, such as a, b, x) is used in place of an unknown number, e.g.

3 + 5 = a             3 + b = 8              x + 5 = 8

Algebra is used in many everyday life situations, such as working out the amount of each ingredient for a recipe, or daily budgeting with money. Many professions require the use of equations too, including air traffic controllers, architects, computer programmers and carpenters.

## Forming and solving equations:

Algebra can be used to solve problems involving maths. Here is a simple example:

Barney looks out of his window in the morning and counts 16 birds sitting in his apple tree. In the afternoon, he counts again - this time there are 27 birds in the tree. How many more birds are there in the afternoon than the morning?

First, we need to write the problem as an equation, using a variable for the missing number:

16 + x = 27

The rules of algebra allow the equation to be changed until we can see what number the variable is representing. One of the rules of algebra is that whatever you do to one side of the equation, you must do to the other. This is called the balance method. So, we can subtract 16 from both sides, leaving x on its own, because 16-16 is zero:

16 - 16 + x = 27 - 16

x = 11

There were 11 more birds in the apple tree in the afternoon than in the morning.

## Solving one-step equations:

Here are some one-step equations and their solutions, using the balance method:

(Note: if a number is placed directly next to a letter, this means that the number and letter need to be multiplied together.)

a + 13 = 22

a + 13 - 13 = 22 - 13

a = 9

b - 8 = 15

b - 8 + 8 = 15 + 8

b = 23

3c = 24

3c = 24

3     3

c = 8

d = 5

4

d x 4 = 5 x 4

4

d = 20

Each answer can be checked by substituting it back into the equation in place of the letter.

## Solving two-step equations:

Here are some two-step equations and their solutions, using the balance method:

2e + 8 = 14

2e + 8 - 8 = 14 - 8

2e = 6

2e = 6

2     2

e = 3

5f - 6 = 19

5f - 6 + 6 = 19 + 6

5f = 25

5f = 25

5     5

f = 5

h + 7 = 4

5

h + 7  x 5 = 4 x 5

5

h + 7 = 20

h + 7 - 7 = 20 - 7

h = 13

k - 4 = 8

3

k - 4 + 4 = 8 + 4

3

k = 12

3

k x 3 = 12 x 3

3

k = 36

## Equations with two variables:

If an equation has two variables, then it cannot have a single solution. For example:

a + b = 6

In this case, we know that the total of a and b cannot be higher than 6.

We also know that a and b must be different numbers.

We could begin by substituting a for 0. This means that b would be 6.

Next, we could substitute a for 1, meaning b would equal 5.

There are six possible solutions for this equation: a = 0, b = 6   a = 1, b = 5   a = 2, b = 4   a = 4, b = 2   a = 5, 1 = 6   a = 6, b = 0

## Algebra in KS2

Algebra is formally mentioned in the National Curriculum in Year 6. However, 'algebraic thinking' can be encouraged before this:

• Make sure children understand that the equals sign does not stand for a calcuating instruction, rather it means 'equivalent to'. This idea can be illustrated visually using a balance. Take a look at this FREE Balancing Equations worksheet.
• Use pictures or symbols instead of letters in missing number problems. This familiarises children with the idea that a number can be represented by something else. Take a look at these FREE Maths Picture Puzzle worksheets.

## Algebra Year 6

According to the National Curriculum, pupils in Year 6 should be taught to:

• use simple formulae
• generate and describe linear number sequences
• express missing number problems algebraically
• find pairs of missing numbers that satisfy an equation with 2 unknowns
• enumerate possibilities of combinations of 2 variables

The non-statutory notes and guidance suggest that pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as:

• missing numbers, lengths, coordinates and angles
• formulae in mathematics and science
• equivalent expressions (for example, a + b = b + a)
• generalisations of number patterns
• number puzzles (for example, what 2 numbers can add up to)

These objectives are covered in our two ready-to-teach Year 6 lesson packs; Algebra and More about Algebra.