UTAS Home › › Mathematics Pathways › Pathways to Business › Module Two: Foundation Algebra Skills
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.
Everyday contexts where algebra is used, include:
Watch the following videos to see how mathematics concepts can be used, these and other interesting maths video can be found at
Another interesting application of mathematics, from a TEDx Talk
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
Go back and try these again if you had any problems with them the first time.
All Khan Academy content is available for free at http://www.khanacademy.org/
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/
About This Lesson
In Module 1 Basic Arithmetic Lesson 2, we looked at these;
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/
In this topic you will revise the following
Lesson 1 - Index Notation
Lesson 2 - Laws of Indices
Lesson 3 - Negative and Fractional Indices
Lesson 4 - Indicial equations
Please note: Indices are also referred to as exponents.
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 ab
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/
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
All Khan Academy content is available for free at http://www.khanacademy.org/
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/
About This Lesson
An indicial equation is one where the unknown quantity is a power of a number.
Example
A simple example is:
3x = 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
In this topic you will revise the following
Lesson 1 - Simplifying expressions
Lesson 2 - Factorising 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
Please watch the following Series from Khan Academy;
All Khan Academy content is available for free at http://www.khanacademy.org/
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
These Khan Academy series include tutorials and a self tests.
All Khan Academy content is available for free at http://www.khanacademy.org/
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.
PLEASE NOTE:
Authorised by the Director, Centre for University Pathways and Partnership
2 May, 2018
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