Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Algebraic Thinking


1 What is Algebra?
2 Algebra and Graphs
3 Algebraic Manipulations
4 Conclusion

Examples of Algebra

Many people find the thought of algebra, equations, and variables
intimidating. But it is just generalized arithmetic.

Algebraic Examples

Consider the following common uses of algebra.

•Formulas – (C = πd)
Relation between two or more variables
•Equations – (5x = 30)
Finding an unknown value
•Identities – (sin2x + cos2x = 1)
An expression true for any x
•Property – (a + b = b + a)
Expression of a general rule
•Function – (f (x) = 3x + 1)
An input (independent) and output (dependent) variable

Examples of Using Algebra

Let’s look at a few examples from this list.

The formula   relates temperatures in Farenheit to
temperatures in Celsius. Use this formula to convert 20 degrees
cesius to farenheit and 41 degrees farenheit to celsius.
Fill in the blank.

Use symbols to express the fact that every number has an additive

Algebra as a Study of Structure

There are many different ways to view algebra.

Study of Structure
Algebra can be seen as a study of structure. That is, what is the
structure of arithmetic? How does it work?
For all real numbers a, b and c the following laws hold:
a + b = b + a commutative law of addition
(a + b) + c = a + (b + c) associative law of addition
a + (−a) = 0 additive inverses
a + 0 = a additive identity

Algebra as a Study of Relationships

Another way to view algebra is a method to express relationships.

Relationships Between Quantities
Algebra can be seen as a study of the relationship between
quantities. The concept of a function is important here as there is
usually an “input” quantity and an “output” quantity.
A phone card has a connection fee of $0.25 plus a $0.05/minute
charge for the actual time of the call. Describe the price of the call
as a function of the number of minutes spent on the call.
Use the following methods to express the relationship above
•a table •a graph •a function rule

Using Graphs to Visualize Algebraic Relationships

As we saw in the previous example, graphs can be an important
way to visualize a relationship between quantities which can be
expressed algebraically.

A beautician charges $15 for haircuts. Each week she has fixed
expenses of $150. Express her profit as a function of teh number
of haircuts she gives. Use a graph to describe this function.
Are the table, graph, and formula we used to answer the previous
question really accurate? In particular, is it possible to give
haircuts? How is this problem seen in the graph?

Interpreting Graphs

Many times a graph is enough to describe a relationship.

Below are descriptions of three runners in a race. Match each
description to the correct graph and explain your choice.

•Alex started slowly, then ran a bit faster, and then ran even faster at the end of the race.
•Manuel started quickly but then tired and slowed down a bit and then slowed down even more at the end.
•Sara started quickly, stoped to tie her shoe, and then ran even faster than before.

Recalling Algebraic Rules

There are several symbolic rules to working with algebra which you
have learned in the past. Let’s review some of these rules.

Adding Polynomials
When adding polynomials group like terms together and use the
distributive property (a(b + c) = ab + ac) to combine like terms.
Add the following polynomials.

Multiplication and Factoring

Finally, we will briefly review multiplying and factoring.

Multiply the following polynomials.

Factor the following polynomials completely.

Important Concepts

Things to Remember from Section 2.2
1 Various uses of algebra
2 Solving equations
3 Representing relationships using fuctions and graphs
4 Adding, multiplying, and factorying polynomials