Manipulating Formulae

In a formula like that for speed we can say that speed is the subject of the equation because the formula tells us what speed is. We can rearrange or manipulate the formula so that either distance or time is the subject of the equation. See image below.

Instead of asking you to make distance the subject of the equation you may be asked to express distance in terms of speed and time. In the Khan Academy videos they say solve for distance. These all mean the same thing.

These videos and exercises are quite short and are worth working your way through them. 

Remember:

1. Anything you do to one side of an equation you must do to the other.
2. Be able to identify how many terms an equation has.
3. When multiplying or dividing an equation each term must be multiplied or divided.
4. If the letter that you are looking for is in more than one term then you need to bring these terms together and then factorise.
5. In the video Sal says "cancelling" - What he means is "divides into once".

There are some more examples done as a pencast below.

Subject of Equation 1
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Long Division in Algebra

This can be tricky, it is not just the method students find difficult - it is easy to make simple mistakes. Be especially careful when subtracting.

Don't spend too long on this topic, it is only a small section of your course.
Note: A polynomial is an algebraic expression with 2 or more parts with different powers of x (or an x term and a number).

Dividing polynomials 1: Dividing polynomials 1




The second example is about as hard as they get!



How does she write backwards so well!

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Patterns 2 M

The aim in this module is to see the connections between

• Word Descriptions
• Tables of Values
• Graphs
• Equations

These are all different ways of looking at the same relationship. Each has its own advantages.

In this activity you will use the maths program GeoGebra to investigate further, especially graphs and equations.

Click here to find the applets for this activity - use your worksheet to go through them in order, make sure to complete each step.

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Solving Inequalities

Inequalities - you need to know

• The inequality signs. 

• The number sets (N, Z and R) and 
• how to graph them on the number line. (see this post)
• How to solve double inequalities (3^rd year only).
• How to solve practical problems based on inequalities.

Inequalities are like equations but with one important difference - if you multiple or divide both sides by a negative number the sign changes direction. To see this try the examples here. The solutions and some simple examples can be found here.

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Factors in Algebra

It is important to know the four methods for finding factors that we use in algebra. These notes cover all four methods. they include guide number method for finding the factors of quadratic expressions, this method works for all quadratic expressions that have factors with whole numbers and does not require guessing or trial and error. Initially it may look complicated but that is because it includes explanations, once you get used to working this way it is quick and effective. It is a good idea to practice multiplying your factors together to check your answer and to help your understanding of factorising.

Tip: When factorising by grouping copy the first bracket and then work out what to put in front of it.

When factorising this

2cd – 2ca –3bd+ 3ab

factorise the first two terms

2c(d – a)

then repeat what is inside the bracket

2c(d – a)     (d– a)

then work out what should go in front of the bracket

2c(d – a) –3b(d– a)

finish by grouping the common factors

(2c–3b) (d –2a)

You are less likely to make mistakes with the signs when you do this.

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Factors - Third Year Revision

In third year the same types of factors are covered, but the questions might be a little harder.

Be able to recognise the four types of factors that you are likely to come across and  know how to factorise each type.

• Highest Common Factor - always check for a HCF first.
• Factorise by Grouping - you may need to regroup before you can factorise.
• Quadratic Expressions - the guide number method works well here especially if the coefficient of x squared is not one.
• The Difference of Two Squares.

Sometimes after you find the HCF you can break it down further - you might end up with 3 factors.

Notes and Examples here - also see second year post.

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Inequalities and Number Sets

It is important to know the sets N, Z and R and how to graph these on the number line.

On page 23 of the Maths tables the sets of numbers that you need to know are listed - however it is easier to learn them here. When written in this order each set includes the numbers in the previous set.

Natural Numbers - Symbol N = {1, 2, 3, 4, ... }, whole positive numbers.

Integer Numbers - Symbol Z = { ... – 3, – 2, – 1, 0,  1,  2,  3 ...}, whole positive and negative numbers (and zero)

Rational Numbers - Symbol Q = any number that can be written as a fraction using two integers, but you cannot have zero as the denominator because it is impossible to divide by zero. You will only be asked to show the sets N, Z and R on the number line. The set Q is used when you are asked to write your answer as a fraction.

Real Numbers - Symbol R = all the numbers on the number line.  This includes the rational numbers and the irrational numbers (numbers that cannot be written as fractions e.g. pi, √2 etc. 

An open or empty circle is placed around the 2 because 2 is not included. For less than or equal use a full or coloured in circle.


This video explains what rational and irrational numbers are.

Simultaneous Equations

These consist of two equations and two unknowns. They can be solved in two ways; you can be asked to use either method so learn both.

Method 1

Solve the equations algebraically there are notes here –  and  a pencast here.

(See p.124-126 in your book for examples and questions)
Worksheet 1 
Worksheet 2
Worksheet 1 answers
Worksheet 2 answers

Method 2

Draw a graph of each line on the same page and find the point at which they meet, this point is the solution. There is a pencast on plotting lines below, for simultaneous equations just do both lines on the same page. Note this method finds where the line crosses the x-axis and the y-axis. If it is convenient you can pick any number for x or y and substitute this in to the equation and find the value of the other variable.

Graph of a Line
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Quadratic Equations - Introduction

Before we start we will solve these simple equations.

Ex:1    8x = 24, what is x?

               x = 3 (÷ both sides by 8)

 Ex:2       4y = 10, what is y?

                   y  = 2½ (÷ both sides by 4)

Remember the rules for multiplying and dividing Integers*

Same signs, answer is positive

plus by plus is plus

minus by minus is plus

Different signs, answer is negative

 plus by minus is minus

minus by plus is minus

Find the value of the letter in each equation

1.     5a = 20

2.     4b = 32

3.     3c = 0

4.     6d = –18

5.     3e = 0

6.     –7f = 56

7.     3g = 10

8.     2h =  –4

9.     –6k = –12

10.  –5m = 15

11.  –8n = 0

12.  6p = –3

13.  Can you solve this?

xy = 0

Can you say anything about the possible solutions?

14.   Solve these without removing the brackets.

a)    4(x – 3) = 0

b)   5(2x – 6) = 0

c)    (x + 4)(x – 6) = 0 , there are two answers to this one.

* Integers are positive and negative whole numbers, the same rules apply for all positive and negative numbers

Revision notes and examples for quadratic Equations can be found here.

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Solving Equations - A New Approach

I really like this way of showing how to solve equations. It is neat and easy to follow yet the steps are all explained very clearly. To see this method visit this page; you can see the original long method first and then the new method near the end of the page. 

In Project Maths it is recommended that you put a line on each side to emphasize that you are doing the same to each side. These are like "stabilizers" when you are learning to ride a bike. As you learn this method you can take off the "stabilizers". The important thing is to understand why this method works. Take your time and write out the answers in full. Later we will shorten this method, but don't rush it! The aim is to understand why this works. 

On this website you drag blocks onto the balance so that it matches the equation and then click continue. Pick an operation - from add, subtract, multiply or divide - then pick the number or letter etc. Khan Academy uses a similar method in these lessons. They also have an animation here.

Harder equations with negative numbers are here.

Further discussion here.
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Solving Equations in Algebra

In Project Maths students study patterns and relations before starting algebra. The aim is to see equations as describing relationships between variables. It is hoped that this approach will aid understanding and also make it more relevant.




A balance

In first year algebra we introduce simple equations and  how to solve them. The aim is to simplify the equation so that we know what the missing letter (usually x ) is equal to. We do this by doing the same thing to both sides of the equation. Since both sides are equal they stay equal if we do the same to each side. Think of of simple weighing scales or a seesaw.

It is important to understand what you doing and not to just learn rules that might give the correct answer. When you use a rule or a shortcut you should be able to explain why it works. Problems  arise in learning maths from rules being using incorrectly or without proper understanding. The Inspectors in the Department of Education have repeatedly identified this as a major problem at all levels, up to and including leaving cert maths.


There is more information and links to pages on solving equations in this post.

Image 1, image 2 and image 3 used under Creative Commons License.

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Graphing Lines

2x + 3y = 4 and 3x = 4y are typical examples of linear equations. The solutions for these come in pairs - for every possible value of x there is one value of y that will work and vice versa. If the solutions of these equations are plotted is it found that all the points lie on a straight line.

The pencast below shows the method for finding the points where the line crosses the X-axis and the Y-axis. This method is useful when you want to draw a graph of the line.

You can use this method to solve simultaneous equations, just put both lines on the same graph.

Graph of a Line
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Co-ordinate Geometry Part 2

Make sure you are familiar with the formulas in your tables on p.18 - remember you only use the first five formulas for now. 

Revision notes for second year and third year on co-ordinate geometry. Graphing Linear Equations - try these two examples. Solving simultaneous equations without drawing a graph - examples and notes. 

The notes are not enough unless you work on them - try doing the questions in the book using the notes to help you.

   There are some more interactive pages on co-ordinate geometry here.

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Simultaneous Equations

This is a pencast showing how to solve simultaneous equations. Notes on Simultaneous Equations are here and more on Co-ordinate Geometry can be found here.

Click the orange circle to see the solution and then click Full Screen to see it in full size. You can use the controls to zoom in on the solution. You can also use the controls to show all the writing immediately. If you don't want to listen to the solution you can just jump to the end. The the solutions are on three pages, click on the right to jump to the other pages. If this does not work with Chrome try Firefox etc.

Simultaneous Equations
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