Algebra (Mathematics Curriculum)

What Is Algebra?

Algebra is a branch of mathematics that uses letters and other symbols to represent numbers.

Dictionary Definition

The Oxford English Dictionary defines algebra as "the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations."

Here is an example of an algebraic equation. In this example, the letter x represents a number. Our job is find the value of x.

Where Does the Word Algebra Come From?

Algebra comes from the Arabic word al-jebr, which means "reunion of broken parts".

Albert Einstein and Algebra

The famous scientist Albert Einstein learned algebra from a young age.

His Uncle Jakob gave him books on the subject and called algebra "a merry science".

He compared algebra to hunting a little animal. You didn't know the name of the animal, so you called it "x". When you finally caught the animal you gave it the correct name.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning.

A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Algebra Definitions

There are many different words used in algebra: such as equation, variable, coefficient, term. Understanding algebra is easier when you understand what each of these words mean.

In this mini-curriculum, you will learn all the definitions.

Equation

An equation has an equals sign (=) and tells us two values are equal. Here is an example of an equation:

Constant

A constant have a fixed value. In the equation below, the 2, 1, and 9 are constants.

Variable

A variable is a symbol that stands for a number. Its value is not fixed - it can take any value. In the equation below, the x is a variable.

Coefficient

A coefficient is a number that is placed in front of a letter. The coefficient is multiplying the variable. In the equation below, the 2 is a coefficient.

Exponent

An exponent tells you how many times a number (or symbol) is multiplied by itself. It is written to the right and above the number. In the equation below, the 2 is an exponent.

Operator

An operator defines an operation, such as +, , ×, and ÷. In the equation below, the + is an operator.

Term

A term is a collection of one or more numbers, letters and brackets written next to each other (they are multiplied together). Terms are separated by + or operators. In the equation below, the 2x2, 1 and 9 are terms.

Expression

An expression is one term or several terms that are added (+) or subtracted (−) together. In the equation below, the 2x + 1 and 5 are expressions. They are separated by the = sign.

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Arithmetic with Letters

Algebra uses letters and other symbols to represent numbers. Just like numbers, the letters and symbols can be added, subtracted, multiplied and divided.

In this mini-curriculum, you will learn arithmetic with letters.

Adding Letters

Adding letters works when the letter is added to numbers, other letters and the same letter.

Subtracting Letters

Subtracting letters works when the letter is subtracted by numbers, other letters and the same letter.

Multiplying Letters

Multiplying letters works when the letter is multiplied by numbers, other letters and the same letter.

Dividing Letters

Dividing letters works when the letter is divided by numbers, other letters and the same letter.

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Arithmetic with Terms

Terms are a collection of one or more numbers, letters and brackets written next to each other. Terms can be added, subtracted, multiplied and divided.

In this mini-curriculum, you will learn arithmetic with terms.

Adding Terms

Terms can be added.

Subtracting Terms

Terms can be subtracted.

Multiplying Terms

Terms can be multiplied. The coefficients are multiplied, and each letter must appear in the result as many times as they appear in the multiplication.

Dividing Terms

Terms can be divided.

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Laws of Exponents in Algebra

An exponent tells you how many times a number, letter or term is multiplied by itself.

In this mini-curriculum, you will learn how to use the law of exponents in algebra.

Laws of Exponents

The laws of exponents in algebra let us understand how a letter can be multiplied by itself.

Exponent of −1

A letter with an exponent of −1 is equal to 1 divided by the letter.

Multiplying Powers

When the same letters with different exponents are multiplied together, the exponents can be added.

Dividing Powers

When the same letters with different exponents are divided, the exponents can be subtracted.

Powers of Powers

A power of a power is where a letter with an exponent is itself raised to an exponent. When a letter with an exponent is raised to another exponent, the exponents can be multiplied.

Powers of Products

A power of a product is where a term containing letters multiplied together is raised to an exponent. Each letter in the term is raised to the exponent.

Negative Exponent

A letter with a negative exponent is equal to 1 divided by the letter with the exponent made positive.

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Like Terms

Like terms are terms which have the same combination of letters, each with the same exponent. The only difference allowed between terms that are like terms are the numbers and signs in front of them.

In this mini-curriculum, you will learn how to find like terms and how to collect like terms to simplify expressions.

Like Terms

Like terms have the same letters with the same exponents. x, 2x and −x are like terms, they all contain x; only the number or sign in front of them is different. a2b, 5a2b and 12a2b are like terms, they all contain a2b; only the number or sign in front of them is different.

Identifying Like Terms

Identify like terms in an expression by finding terms with the same letters and exponents.

Adding Like Terms

Like terms can be added. Add the coefficients together.

Subtracting Like Terms

Like terms can be subtracted. Subtract the coefficients from each other.

Collecting Like Terms

Collect like terms by adding and subtracting the like terms you have identified. This simplifies the expression.

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Expanding Brackets

Expanding brackets means multiplying terms inside of a pair of brackets in order to remove the brackets from an expression.

In this mini-curriculum, you will learn how to expand brackets.

Expanding Brackets

Expanding a single pair of brackets means multiplying the term outside of the brackets with each term inside the brackets. This removes the brackets.

Expanding Brackets with the Grid Method

Brackets can be expanded using the grid method. This is a useful method if you prefer using pictures to understand a method.

Expanding Double Brackets

Expanding double brackets means multiplying the term in two pairs of brackets that are written next to each other. This removes the brackets.

The FOIL Method

FOIL stands for Firsts, Outsides, Insides, Lasts. It helps you remember an order to multiply terms when expanding two pairs of brackets.

Multiplying Expressions

Multiplying expressions means writing each expression in a pair of brackets and expanding them.

Expanding Double Brackets with the Grid Method

Double brackets can be expanded using the grid method.

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Solving Equations with Brackets

Some equations in algebra contain brackets.

In this mini-curriculum, you will learn about how to solve equations with brackets.

Solve Equations with Brackets

To solve equations with brackets, expand the brackets and rearrange the equation to find what x equals.

Solve Equations with Two Sets of Brackets

To solve equations with two sets of brackets, expand both brackets and rearrange the equation to find what x equals.

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Factoring

Factoring is the opposite of expanding brackets. It is a way of simplifying an expression.

In this mini-curriculum, you will learn about factors in algebra, the greatest common factor and how to factor expressions.

Factors

A factor is one of the numbers, letters and brackets (or a product of them) that are multiplied together to make a term.

Finding Factors

You can find factors of terms by looking at the numbers, letters and brackets (with any exponents) that appear in the term. The factors of 2x are 1, 2, x and 2x.

Greatest Common Factor

The greatest common factor is the largest factor that is common to two or more terms. The greatest common factor of 2xy and 2xz is 2x.

Finding the Greatest Common Factor

Find the greatest common factor by looking at the numbers, letters and brackets that appear in two or more terms.

Factoring Expressions

Factoring simplifies an expression by writing it as a product of factors. It lets you write an expression where terms are added together to one where they are multiplied together to make a single term.

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Algebraic Fractions

Algebraic fractions are fractions which contain letters and other symbols.

In this mini-curriculum, you will learn about algebraic fractions, including arithmetic and simplifying.

Algebraic Fractions

An algebraic fraction is a fraction which contains letters and other symbols.

Adding Algebraic Fractions

Algebraic fractions can be added.

Subtracting Algebraic Fractions

Algebraic fractions can be subtracted.

Multiplying Algebraic Fractions

Algebraic fractions can be multiplied.

Dividing Algebraic Fractions

Algebraic fractions can be divided.

Simplifying Algebraic Fractions

Algebraic fractions can be simplified by cancelling out terms that appear on the top and bottom of the algebraic fraction.

Powers of an Algebraic Fraction

An algebraic fraction can be raised to an exponent. The top and bottom of the fraction are both raised to the same exponent.

Reciprocal of an Algebraic Fraction

The reciprocal of an algebraic fraction is found by turning the algebraic fraction upside down.

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