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Goal:

## Algebra

### Evaluating expressions

##### Evaluate algebraic expressions

How to **substitute values for variables in algebraic expressions** and how to **evaluate the expressions** to give a numerical answer.

### Expanding brackets

##### Expand brackets: (a+b)(c+d)

How to **expand algebraic expressions with brackets.**

### Factorising

##### Factorise expressions like: ab + ac

How to factorise expressions by **finding common factors.**

##### Factorise quadratic expressions

How to **factorise quadratic expressions** using several methods.

##### Splitting the middle term of a quadratic

How to split the middle term of a quadratic expression so that it can be factorised.

##### Other factorising techniques

How to use Algematics to do these operations:** difference of squares, sum and difference of cubes, split middle term of quadratic, group and factorise four terms.**

### Quadratic Equations

##### Finding roots by factorising

How to **calculate the roots of a quadratic equation algebraically** after the quadratic expression is factorised.

##### Using the quadratic formula

How to find roots of a quadratic equation by using the quadratic formula.

### Simplifying

##### Simplify algebraic expressions

How to **simplify algebraic expressions where no preparation is necessary.** Several examples are discussed.

##### Advanced simplifying techniques

How to simplify algebraic expressions **where some expansion of brackets or factorising is necessary** before the expression can be simplified.

### Simultaneous Equations

##### Solve simultaneous equations by substitution

This topic explains the concept of simultaneous equations, and explains how to solve simultaneous equations algebraically by the method of **substitution**.

##### Solve simultaneous equations by elimination

How to solve simultaneous equations algebraically by the method of **elimination**.

##### Solve simultaneous equations graphically

How to solve simultaneous equations by finding** intersection points of plotted graph lines.**

##### Solve systems of linear equations

How to use your computer to automatically solve simultaneous linear equations in **any number of variables.** (A powerful numerical method called ‘partial pivoting’ is used.)

### Solving Equations

##### Solve linear equations with algebra

How to solve several common types of linear equations using algebraic techniques. Ten worked examples explain the methods clearly.

##### Solve equations graphically

How to solve implicit functions in one variable using a graphical technique. The method is** very accurate, and works on very difficult equations.**

Changing the subject of a Formula

Uses two worked examples and the principle of inverse operations to explain how to change the subject of a formula.

## Analytical geometry

### Distance formula

How to use the distance formula to **find distances between (x,y) points.** The download file displays the working steps and a diagram for your distance formula problems.

### Midpoint formula

How to use the midpoint formula to **find the midpoint of a line between two (x,y) points.** The download file solves midpoint formula problems. Enter two points to display working steps and a diagram.

### Divide a line in a given ratio

How to **find the point ‘P’ that divides the line segment AB in a given ratio**. This topic covers both internal and external divisions. The download file solves problems of this type, displaying working steps and a diagram.

## Angles

### Angle conversions between radians and degrees

This topic explains the concept of radian measure, as well as how to convert between radians and degrees. The download file converts between radians and degrees, showing full working steps and a labeled diagram.

## Area and Surface Area

### Area unit conversions

How to convert area measurements between 10 common units. Includes tables of conversion factors and worked examples. The download file converts areas in any of the ten area units to any of the remaining 9 types.

### Area of triangles, ˝b×h

How to find areas of triangles using the formula: A = ˝base×height. The download document displays working steps and a labelled diagram for this problem type.

### Area of triangles, ˝ab×sinC

How to find areas of triangles using the formula: A = ˝ab×sinC. The download document displays working steps and a labelled diagram for this problem type.

### Area of triangles, Heron’s formula

How to find areas of triangles using Heron’s formula. The download document displays working steps and a labelled diagram for solving a triangle using Heron’s formula.

### Area of parallelograms

How to find areas of parallelograms using the formula: A = base×height. The formula is explained pictorially, and the download document allows you to experiment with the formula, while displaying working steps and a labelled diagram.

### Area of circles

How to **find areas** of circles using the formula: **A = pr ^{2}**, or to

**find the radius**if the area is given. The download document solves both kind of problem, and also displays a labelled diagram.

### Surface area and volume of prisms

How to find the surface area and volume of prisms. Several prism types are discussed. The download document calculates the volume and surface area of rectangular prisms, showing all working steps. The labelled diagram shows a **3d model** of the prism, as well as the 2d net of the prism.

### Surface area of a sphere

How to** find the surface area** of a sphere, or the **radius if the surface area is given**. The download document solves both problem types and displays both the working steps and a labelled diagram.

## Calculus

### Plot derivative graphs

This topic explains relationships between the graphs of a given function and its derivative, and shows you how to graph your own derivative functions.

### Calculate definite integrals

A detailed introduction to definite integrals. Includes helpful diagrams and examples. Shows you how to calculate definite integrals using a variety of numerical methods.

### Area under a curve

An introduction to finding areas under curves using definite integrals. Helpful diagrams and examples are included.

### Area between two curves

A brief introduction and diagram explain the concept, then the rest of the article shows you how to find areas between graphs using Maths Helper Plus. The download file is set up for the worked example from the article, but you can easily edit the example functions to calculate your own areas.

### Stationary points

This article discusses both turning points and points of horizontal inflection of functions. Helpful worked examples and diagrams are given. The download file is a starting point for your own problem solutions.

### Tangents and normals

Worked examples and diagrams explain how to find tangent and normal equations to functions at a given point. The steps for calculating and plotting tangent and normal equations in Maths Helper Plus are explained, and the download file is a starting point for your own problem solutions.

## Complex Numbers

### About complex numbers

This topic **introduces complex numbers** and demonstrates how complex number calculations can be done using complex calculator data sets in Maths Helper Plus.

### Complex number operations

This topic shows you how to add, subtract, multiply and divide complex numbers. It also shows how to display full working steps for complex number operations in Maths Helper Plus.

## Co-ordinates

### Plotting ordered pairs in four quadrants

This topic introduces **the concept of an** **ordered pair** and shows you **how to plot ordered pairs** on a graph. The download file helps to clarify the concepts involved in plotting ordered pairs.

### Plotting polar coordinates

This topic introduces the concept of polar coordinates and shows you** how to plot polar coordinates **on a graph. The download file displays a polar grid and marks in the angle and radius for a given plotted polar point.

### Converting between rectangular and polar coordinates

This topic uses diagrams and examples to explain how to convert between polar and rectangular coordinates. The download file displays full working steps and a labeled diagram to help you convert your own coordinates.

## Fractions

### Comparing fractions

Explains one way of **comparing different kinds of fractions with each other.** The download file displays a number line on which you can plot up to five fractions of any type. The decimal value of each fraction is calculated and marked on the number line for easy comparison.

### Adding and subtracting common fractions

This topic explains how to add and subtract common fractions, and also how to use Algematics to create a worked solution for problems of this type.

### Converting decimals to common fractions

How to convert decimal fractions to common fractions, and also how to use Algematics to create a worked solution for problems of this type.

### Multiplying and dividing common fractions

How to multiply or divide common fractions, and also how to use Algematics to create a worked solution for problems of this type.

## Functions

### Create a table of function values

This topic **introduces the concept of a mathematical function** and shows you how to create a table of function values.

### Find the tangent or normal to a function at a given point

Worked examples and diagrams explain how to find tangent and normal equations to functions at a given point.

### Linear functions: y = mx + c, gradient and y intercept

This topic** introduces the straight line function, y = mx + c, discusses ‘m’ and ‘c’,** and shows you how to draw graphs of some different forms of this function.

### Learn about polynomial functions

An **introduction to polynomial functions**, and how to draw graphs of polynomials, as well as find **zeros** and **turning points.**

### Learn about trigonometric functions

An **introduction to sine and cosine functions,** and how to draw their graphs. The download file graphs sine functions and does the calculations involving period and phase shift, showing the working steps.

## Gradient

### Gradient of line from the origin to a fixed point

How to find the gradient of a line through the origin and another fixed point. The download file displays complete working and a labeled diagram.

### Gradient of line segment between two fixed points

How to find the gradient of a line through two fixed points. The download file displays complete working and a labeled diagram.

## Money

### Discount

This topic shows you **how to solve problems related to marked price, selling price and discount.** The download file solves a wide range of discount problems with detailed working and helpful diagrams.

### Profit and loss

This topic shows you **how to solve profit and loss problems.** The download file solves a wide range of profit and loss problems with detailed working and helpful diagrams.

### Simple interest

This topic shows you** how to solve simple interest problems.** Several worked examples and a powerful download file will help you solve your own simple interest problems easily.

### Compound interest

This topic shows you** how to solve compound interest problems** based on the compound interest formula. The worked examples and download file help you solve your own compound interest problems easily.

## Trigonometry

### Pythagoras

##### About the rule of Pythagoras

This topic introduces you to the** rule of Pythagoras** for right triangles, as well as **Pythagorean triples**. The download file is a demonstration of the rule of Pythagoras by areas. It lets you experiment with a triangle to verify the rule.

##### Using the rule of Pythagoras to find unknown triangle sides

How to use the Rule of Pythagoras to find unknown sides of right triangles. Worked examples and the download file show you all of the steps and include labelled diagrams.

### Tangent Ratio (right triangles)

##### Introduction to tangent ratios

This topic introduces the ‘tangent ratio’ for right-angled triangles and defines the **term ‘opposite’ and ‘adjacent’ for right triangles**. The download file calculates tangent ratios for right triangles with working and diagram.

##### Using the tangent ratio to find unknown angles

This topic shows you how to find unknown angles in right triangles using tangent ratios. The download file calculates tangent ratios for a given angle, and also finds an angle when the tangent ratio is given.

##### Finding unknown opposite or adjacent sides with the tangent ratio

This topic shows you how to find unknown opposite or adjacent sides in right triangles using tangent ratios. Worked examples and diagrams explain the mathematics, then the download file displays the complete working and a labelled diagram for solving your own problems.

### Sine Ratio (right triangles)

##### Introduction to sine ratios

This topic introduces the ‘sine ratio’ for right-angled triangles. The download file calculates sine ratios for right triangles with working steps and diagrams included.

##### Finding an unknown opposite side or hypotenuse with the sine ratio

This topic shows you how to find an unknown opposite side or hypotenuse in a right triangle using sine ratios. The download file shows full working steps and a diagram to help you solve your own problems of this type.

### Cosine Ratio (right triangles)

##### Introduction to cosine ratios

This topic introduces the ‘cosine ratio’ for right-angled triangles. The download file calculates cosine ratios and displays a helpful diagram when you enter an adjacent side and a hypotenuse.

##### Finding an unknown adjacent side or hypotenuse with the cosine ratio

How to find an unknown adjacent side or hypotenuse in a right triangle using cosine ratios. The worked examples and diagrams show you the steps, while the download file solves your problems showing full working and a labelled diagram.

### General triangle solution

##### given three (x,y) points

How to find the unknown sides and angles of a triangle defined by three (x,y) points. The working steps are fully explained, and the download file creates full working steps and a labelled diagram for your own problems.

##### given three sides

How to find the unknown angles of a triangle when given the three side lengths. The working steps are fully explained, and the download file creates full working steps and a labelled diagram for your own problems.

##### given two angles and one side

How to find the unknown angle and sides of a triangle when given the two angles and one side. The working steps are fully explained, and the download file creates full working steps and a labelled diagram for your own problems.

##### given two sides and the included angle

How to find the unknown side and angles of a triangle when given two sides and the angle between them. The working steps are fully explained, and the download file creates full working steps and a labelled diagram for your own problems.

##### given two sides and a non-included angle

How to find the unknown side and angles of a triangle when given two sides and an angle not between them. In this case, there may be two possible solutions. The working steps are fully explained, and the download file creates full working steps and a labelled diagram for your own problems.

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