Module Two: Foundation Algebra Skills - Mathematics Pathways

Module Two: Foundation Algebra Skills Pre-test

Click on the link below to take the Pre-test for Module Two.   You will be given 5 random questions to test your knowledge of foundation maths skills.  

If you receive more 80% or more, you can move onto the next module.  If your score less than 80%, work through the module lessons and take the post-test at the end of the module.

Make sure you enter your full name and email address so your results can be emailed to you.  You will need to print out or save these results for your records.  You may need to show them to your university.

Module Two:  Pre-test

Application of Algebra in Business

Everyday contexts where algebra is used, include:

• Investing money (interest rates, profits)
• Estimating costs
• Understanding computers
• Designing rooms
• Analysing the stock market
• Analysing risk
• Calculating prices and discounts
• Working out wages
• Planning and monitoring budgets
• Calculating mortgage and loan repayments
• Calculating quantities of materials needed to produce products
• Calculating time needed to perform parts of a process
• Calculating the break-even point and making decisions on the optimal level of production.

Watch the following videos to see how mathematics concepts can be used, these and other interesting maths video can be found at 

8 Videos That Prove Math Is Awesome

Another interesting application of mathematics, from a TEDx Talk

Lesson 1 - Patterns and Expressions

About this lesson

Patterns

Discovering patterns helps us to understand underlying mathematical relationships.

Example

Square numbers can be made by adding up a series of odd numbers

1=1

4 = 1 + 3

9 = 1 + 3 +5

16 = 1 + 3 + 5 + 7

25 = 1 + 3 + 5 +7 + 9

and so on

Expression

An algebraic expression is a way of describing something mathematically.

An expression is sometimes part of a formula.

Example

A rectangle has a length of L and a width of W

the expression 2L +2W would describe its perimeter and the expression L x W would describe its area.

L and W are called variables

The formula for the perimeter is:

P = 2L + 2W

The formula for area is :

A = L W

(In Algebra we don't use x for multiply as it can get confused with the letter x for a variable. L W is the same as L x W) 

Test Yourself

This interactive tutorial from Centre for Innovation in Mathematics Teaching (CIMT) will help you to understand how algebraic expressions are a way of describing patterns.

CIMT Extending Number Sequences helps to build expressions. 

Learn more

Learn more with these resources

• Maths is Fun  Algebra - Definitions
• Khan Academy Series Number Patterns - Includes tutorials and a self test

Go back and try these again if you had any problems with them the first time.

• CIMT Extending Number Sequences helps to build expressions
• Khan Academy self test from the series above Relationships between patterns

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 2 - Substitution in and evaluation of expressions

About this Lesson

Using the example of the rectangle from Lesson 1, when we know the actual numerical values of a particular rectangle we can replace L and W with the numbers and calculate the perimeter and area in that particular case.

Replacing L and W with actual numbers is called substitution and calculating the value is called evaluating the expression.

If L = 20cm and W = 10cm

Perimeter
2L + 2 W = 2 x 20 + 2 x 10 (Substitution in the expression)
2 x 20 + 2 x 10 = 60
Perimeter = 60cm (Evaluating expression)

Area
L x W = 20 x 10 (Substitution in the expression)
20 x 10 = 200
Area = 200cm2 (Evaluating expression)

Test Yourself

CMIT - Substitution in formulae

Learn More

Maths is Fun - Substitution
Khan Academy Series - Variables and Expressions - Includes tutorials and self tests

Khan Academy Self Test - Evaluating expressions in one variable
Khan Academy Self Test - Evaluating expressions in two variables

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 3 - The Commutative, Associative and Distributive Laws

About This Lesson

In Module 1 Basic Arithmetic Lesson 2, we looked at these;

• Khan Academy - Commutative Law of Addition
• Khan Academy  - Commutative Law of Multiplication
• Khan Academy - Associative Law of Addition
• Khan Academy - Associative Law of Multiplication

If you are not confident with this spend some time revising.

In Algebra, it is important to have a sound understanding of these topics and also the Distributive Law which tells us how to work with brackets

Please Note: some people call these Properties rather than Laws

Test Yourself

Maths is Fun Tutorial and Test - Commutative Associative and Distributive Laws
(test is at the bottom of the page)

Learn More

Khan Academy Series - Distributive Property
Khan Academy Self Test - Distributive Property

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 1 - Index Notation

About This Lesson

Superscripted numbers are used to show when a number or variable is multiplied by itself one or more times.

Example

3² = 3 x 3 = 9
5³ = 5 x 5 x 5 = 125
a⁵ = a x a x a x a x a

In the expression a^b

a is called the base and b is called the power, exponent or index. All these terms can be used.

Test Yourself

This interactive tutorial and self test will help check your understanding of these ideas. You don’t need to complete it all if you are confident that you understand it. Move on to the next lesson.

CIMT - Index Notation

Learn More

Maths is Fun - Laws of Exponents is useful for all the lessons in this topic.
Watch the following Khan Academy Video - Introduction to Exponents - Work carefully through the tutorial

CIMT - Index Notation and make use of the built in help and self tests

This MathTutor series covers all the lessons in this topic
MathTutor - Powers or indices

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 2 - Laws of Indices

About This Lesson

The Laws of Indices define how we work with indices so that there is only one way to evaluate an expression. 

Image source: http://www.cimt.plymouth.ac.uk/projects/mepres/book9/bk9i3/bk9_3i2.html

Test Yourself

This interactive tutorial and self test will help check your understanding of this.

CIMT - Laws of Indices

Learn More

Maths is Fun - Laws of Exponents is useful for all the lessons in this topic.

If you need to get more practice work carefully through

• CIMT - Laws of Indices and check your results at the end of the page.
• MathTutor - Powers or indices This MathTutor series covers all the lessons in this topic

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 3 - Negative and Fractional Indices

About This Lesson

Negative indices are used to represent reciprocals as shown below 

Fractional indices are used to represent square roots, cube roots and other roots.

Two special rules to note carefully:

1.  Any base raised to the power one is equal to itself

  a¹ = a
10¹ = 10

2.  Any base raised to the power 0 is equal to 1

 a⁰ = 1
10⁰ = 1

Test Yourself

CIMT - Negative Indices includes explanations and a self test
CIMT - Fractional Indices includes explanations and a self test

Learn More

Maths is Fun - Laws of Exponents is useful for all the lessons in this topic.
CIMT - Negative Indices includes explanations and a self test
CIMT - Fractional Indices includes explanations and a self test
Khan Academy Series - Negative and Fractional Exponents has video explanations and a self test. Please watch the videos below before trying the self test;

This MathTutor series covers all the lessons in this topic

MathTutor - Powers or indices

All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 4 - Indicial Equations

About This Lesson

An indicial equation is one where the unknown quantity is a power of a number.

Example

A simple example is:

3^x = 9
So x must be 2 because 3² = 9

More difficult examples are where the unknown is not a whole number such as the question:

"How long will I need to leave $500 in the bank before it doubles at 5% compound interest?"

The required formula is

500 x (1.05)ⁿ = 1000 where n is the number of years.

It can be solved by trial and estimation

Try n = 7, n = 8, n = 9

But to get an accurate answer we need to use logarithms.

This is explained in the link below.

If you find this difficult, there will be more work in a later modules.

For now focus on the examples that don't require the use of logarithms.

Test Yourself

Mathcentre - Indicial equations

Learn More

Maths is Fun - Variables with Exponents - How to Multiply and Divide them

Lesson 1 - Simplifying Expressions

About This Lesson

When working with algebraic expressions, there are three major areas that you need to understand.
You need to be competent at these skills including the use of negative and fractional values.

Example:  Collecting like terms

2x + 3y + 4x + 6y
= 6x + 9y

Example:  Expanding Brackets

7(3x + 5)
= 21x + 35

Example: Expanding binomials

(3x + 2)(2x + 4)
= 6x² + 12x + 4x + 8
= 6x² + 16x + 8

Test Yourself

This interactive worksheet and self test covers all these areas and is very comprehensive. It may take you some time to work through. Use the built in help if you need reminders.

CIMT - Simplifying Expressions

If you are confident in these skills you can move on to the next lesson.

Learn More

• Maths is Fun - Algebra - Expanding

Please watch the following Series from Khan Academy;

• Khan Academy Series - Manipulating Expressions
• Khan Academy Series - Multiplying and Dividing Monomials
• Khan Academy Series - Multiplying Binomials
• CIMT - Simplifying Expressions
  

 All Khan Academy content is available for free at http://www.khanacademy.org/

Lesson 2 - Factorising Expressions

About This Lesson

Factorising expressions is the opposite of expanding them. Look for common factors in expressions that can be taken out to leave the rest of the expression in a bracket. It is an import skill in Algebra.

Example

3x(x + 5) when expanded becomes 3x² + 15X

3x² + 15X when factorised becomes 3x(x + 5)

Test Yourself

CIMT Tutorial - Factorising Expressions

If you are confident in these skills you can move on to the next lesson.

Learn More

• Maths is Fun - Factoring in Algebra Work through the tutorial in detail using the built in help
• CIMT Tutorial - Factorising Expressions

These Khan Academy series include tutorials and a self tests.

• Khan Academy - Factoring Simple Expressions
• Khan Academy - Factoring Quadratic Expressions
  

All Khan Academy content is available for free at http://www.khanacademy.org/

Reference Sites for Module 2

Maths is Fun - Algebra Index
A simple introduction to high school Algebra

Khan Academy - Algebra 1
Comprehensive set of tutorials and tests

Mathtutor - Algebra
Detailed video lessons of around 20 to 40 minutes, PDF text books, diagnostic tests and exercises

All Khan Academy content is available for free at http://www.khanacademy.org/

When you have completed the module, check your knowledge with this multiple choice post-test. 

• This test will confirm your level of understanding of the skills in this module.
• Please attempt all questions on the test.
• To achieve 'mastery' (the level at which you are truly competent with these skills) you will need to achieve at least 80%
• You will need to enter your name and email address in the test to have the results emailed to you.

• If you want to use these results to apply to study business at UTAS or UNDA, ensure that you save them to use as evidence that you have passed the module.

Module Two: Post-test