Transformation of Functions

This is a useful topic as it brings a lot of ideas abut patterns, equations, functions, tables and graphs together.

All the transformations can be studied here. 


Also, make sure that you know how to get the roots of a quadratic equation from the graph. this is often included in the same question.

_

Co-ordinate Geometry Part 1
Worksheet - remember to find the hypotenuse you will need to use the theorem of Pythagoras. Spend a bit of time on question 1-3 it will help you to understand some of the ideas in co-ordinate geometry. You will find the applet for Q.3 here. Although you have the formulas in the tables it is very important to know where to find them, which ones you need to use and how to use them. Some notes and questions on co-ordinate geometry are available here.

Earlier this year you did an exercise on the equation of a line. The last part of the revison worksheet is on the equation of a line - remember that there are two formulas for the equation of a line in the maths tables - make sure that you use the right one. some answers for Q1-3 are here and the answers to Q4 to 6 are available below as a pencast.

Equation of a line worksheet Q.4 to Q.6
brought to you by Livescribe

_

Circle Theorems

Play around with these so that you become familiar with the different shapes and patterns.

Interactive animations on circle theorems and corollaries.

Four things to know about angles and circles.

1. The angle at the centre is twice the angle on the circle that marks out the same arc - the diagram below shows different examples of this.

2. Angles  standing on the same arc are equal.

3. The angle in a semi-circle is a right angle (the angle on a diameter is a right angle).

4. The sum of the opposite angles in a cyclic quadrilateral is 180.

_

Sample Answer Using Chomp

2013 Paper 2 Sample Paper Click on each image to see the question, read it carefully then close it and watch the solution, don't forget to make the video larger so that you can see the writing.

Q.1 (a)

Video of Solution

(b) 

Video of Solution

(c)

Video of Solution

(d)

Video of Solution

(e)

Video of Solution

3rd Year Junior Cert

Probably the most important advice is to really read the question carefully - there can be a lot of writing and students often misinterpret the question. Make sure that you clearly identify what the question is asking you to do.

Read (and heed!) the advice on pages (iii) to (v) of your exam papers. Also see the instructions for students e.g. on p.2 of your exam papers.

Below is an outline of topics on Paper 1 and 2 and how they relate to the chapters in your book.

The Maths course is divided into 5 Sections or Strands and the content is divided between the two papers as shown in the table. However, a question may require knowledge or skills from more than one strand. See below table for examples.

  

Paper 1	Topic	Chapter in book
	Number	1 Sets 2 Number Systems 9 Indices 10 Applied Arithmetic 11 Speed and Time
	Algebra	4 Patterns 5 Expressions 6 Factorising 13 Algebraic Fractions 14 Changing Formulae 15 Linear Equations 16 Inequalities 17 Quadratic Equations
	Functions	22 Graphing functions See also Chapters 30, 31 and 32 in Book 1
Paper 2	Topic	Chapter in book
	Probability	3 Counting and Probability
	Statistics	8 Statistics
	Applied Measure	12 Area and volume
	Geometry	7 Constructions 18 Geometry 19 Formal Proofs 20 Co-ordinate Geometry
	Trigonometry	21 Trigonometry

Probability on Paper 2 may require you to draw a Venn diagram. Algebra may be used in lots of questions for example Area or Trigonometry on Paper 2. These are just some examples – prepare to be flexible in your approach.

_

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
brought to you by Livescribe

_

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!

_

Sets - 3 Set Problems

There are a number of questions and partial solutions in this document. It is important that you try these yourself, if stuck look at the solution and then finish it.

_

Pythagorean Theorem

Pythagorean Theorem - notes and examples. These animations show demonstrations of the Pythagorean theorem - they are visual proofs. 

There are more examples and questions to try from Project Maths here. Note how the answer is written out - this is how you should write your work (at least three or four lines).

with this online calculator you can enter values and check your answers.

Click on the picture to see this turn

Angry Surds - this is a great game. You can practice the Pythagorean Theorem, Arithmetic and Surds all at the same time.

The Pythagorean theorem is very important in maths - you will use it regularly all the way up to sixth year and possibly beyond! Make sure that you have a full understanding of it.

_

Sets - Activity Book Solutions

3rd Year Sets Activity Book 2 Solutions

Exercise 1.2

Exercise 1.3

Exercise 1.4

Exercise 1.5

_

Indices for 3rd Years

Make sure that you understand the work covered in second year. 

Summary for third year.

_

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.

_

Probability

These notes should help to get you started on revising probability. Use these buttons to help you see the notes more easily.

Don't forget the different ways of showing the Sample Space

• Listing
• Two Way Tables
•  Tree Diagrams 

Also the "Fundamental Theorem of Counting"

Remember n! is called factorial n, the easiest way to explain what it means is with some examples

3! = 1 x 2 x 3
6! = 1 x 2 x 3 x 4 x 5 x 6

On your calculator you should have a button to do this for you 
- it is usually x! but you might have to press "shift" first.

We also use Venn Diagrams in some probability questions - sometimes we use the numbers in each group and sometimes the probabilities - see the page on Sets. 

Thanks to "Mr Barton" and the TES (Times Educational Supplement) for the notes.
_

Venn diagrams - 3 Sets

 Below are some points about Venn diagrams. Click on each image to enlarge it.

_

Sets

These are links to the Project Maths activities on Sets

• Class Worksheet - Check your answers here 
• Quiz on the symbols used in Sets
• Venn Diagram Quiz - only ten minutes to finish, how quickly can you do it?
• Sets Quiz - each time you click on an answer it tells you if you are correct.
• Crossword - click exactly on each number to get the clue.

Below are some examples of the symbols used in sets. Click on the image to enlarge it. You should know what each of these means.

_

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.

_

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.

_

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
brought to you by Livescribe


_

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.

_