Solving linear equations

You can adopt these techniques and tasks when planning your lesson sequences on solving linear equations.

All pupils should aim to achieve mental agility with the linear form. Their ability to solve algebraic equations at the higher National Curriculum levels can be greatly improved if their responses to these stages of a problem become automatic.

The expectations listed below refer exclusively to the mental facility that will support written solution at level 5 and above.

The progression in the listed examples below is described in two ways.

• The list show how the form of the equation is perceived to become more difficult to deal with because of the change in position of the unknown term.
• Within each list the solution involves values that are harder to deal with, such as non-integer or negative integer values.

Construct and solve linear equations, selecting an appropriate method provides contexts in which pupils could develop mental processes in algebraic solution.

One-step linear equations

The position of the unknown value is not a source of difficulty if pupils are practised in thinking flexibly about the form of an equation.

Encourage them to generalise their knowledge of arithmetical commutativity and inverse in ‘families of facts’ such as $$3\times 5=15$$, $$5\times 3=15$$, $$3=15÷5$$, $$5=15÷3$$.

Progression

One-step linear equations with the unknown in a ‘standard’ position

$$x+4=7$$	positive integer solutions
$$\frac{x}{4}=6$$
$$x-9=34$$
$$8x=56$$
$$3x=5$$	non-integer solutions
$$x+14=9$$	negative integer solutions

One-step linear equations with the unknown perceived to be in a ‘harder’ position:

$$13=8+x$$	positive integer solutions
$$\frac{20}{x}=10$$
$$\frac{20}{x}=3$$	non-integer solutions
$$13-8=x$$	negative integer solutions

Equations involving brackets

One way of solving equations with brackets would involve multiplying out the brackets.

Another, often simpler, way is to encourage pupils to see the expression in brackets as an object or term.

For example, $$3(x+4)=27$$ has the same structure as $$3p=27$$, so pupils can mentally move through this step and see that [mathmml:38852].

Progression

Equations involving brackets, with positive integer solution:

• $$3x+4=27$$positive integer solutions
• $$\frac{\left(x-5\right)}{3}=7$$

Inequalities

Pupils can better understand an inequality (or ‘inequation’) if they have the ability to think clearly about the meaning of the ‘equals’ sign in an equation. Reading ‘equals’ as ‘makes’ is a limited interpretation.

Use the language of ‘equals’ meaning ‘is the same as’ to develop understanding of equations. The consequent step to ‘is greater than’ can develop from this much more clearly.

Consider building understanding by using visual and mental images such as number lines and coordinate axes.

Ask pupils to express inequalities both ways round, to develop flexible and thoughtful use of language and symbols.

Progression

Inequalities:

• $$5x<10$$ one boundary to solution set, one-step solution
• $$-4<2x<10$$ two boundaries to solution set
• $$5x+3<10$$ two-step solution