Dr Duncan James > UPmaths > Algebra

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Algebra is the building block of so much mathematics! The UPmaths approach is to learn a few simple skills that will enable you to answer any question. No tricks here! Just simple, intuitive maths with an emphasis on doing and not memorising.


Simplifying is a skill on its own and is often done as a part of other questions.

Splitting Fractions Challenge (YouTube)


An equation is a way of writing down a pattern or relationship using maths language.

Same to Both Sides

Always do the same to both sides of an equation!

Same to Both Sides of an Equation Advanced (UP3) (YouTube)

Factorising and Expanding

Factorising and Expanding are the opposite of each other. They are used regularly in mathematics so they are an important basic skill.

When Can We Ignore Brackets? (YouTube)

Spot the Algebra Mistake with these Brackets (YouTube)

Linear and Quadratic Factorising a Combined Approach (UP3) (YouTube)

Universal Factorise Approach (UP4) (YouTube)

Changing the Subject of an Equation

Changing the subject of an equation uses the skill "do the same to both sides" and also simplifying and factorising/expanding skills.

Sometimes Factorising Helps With Rearranging Equations (UP3) (YouTube)

Solving Equations

Solving equations is a crucial skill! Professional mathematicians will often make an equation based on a real-life situation: solving the equation can then make a phone, bridge, car or other thing work better.

Combined Solve Approach for Linear and Quadratic Equations (UP3) (YouTube)

The Two Ways of Doing Simultaneous Equations (UP3) (YouTube)

Inequalities Universal Solve Approach (UP4) (YouTube)


Inequalities are like equations but refer to a range instead of a point on a graph. Yes an equation can refer to more than one point on a graph but I was trying to give the big picture!

Equations and Inequalities

It can be useful to test your knowledge of solving equations and inequalities at the same time. This helps you notice the similarities and differences between these two closely-related skills.


There are many applications of equations. For example, a consistent proportional relationship between two variables can generate a useful equation.

Wording of Proportionality Questions (UP3) (YouTube)

At higher levels of mathematics algebra gets more difficult. Why? I think it is because so many different skills are combined at the same time. I train my students in pattern recognition so they can better deal with this confusing amount of information.

Exponentials and Logarithms

This has many applications. For example, it is a way of dealing with rapidly changing variables or situations where that involve exponential growth/decay.


Often taught in terms of triangles these patterns also occur naturally in circular motion and waves.

Partial Fractions

In mathematics we often see the same pattern of problem appear over and over again. Partial fractions is just one example of where mathematicians have developed a standard approach for a particular pattern.

Partial Fractions Overview (UP4) (YouTube)

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