What Is Algebra?

Algebra is a combination of:

1. A notation system for representing quantitative relationships, and
2. A set of rules for manipulating notation without changing the underlying quantitative relationship that it represents.

Why is algebra needed? Because:

1. The notation provides a concise and commonly accepted way of accurately communicating quantitative relationships, and
2. Changing the appearance of the notation describing a relationship, often repeatedly, allows us to develop insights into the relationship and/or determine the answer to quantitative problems.

While algebra may be “only a set of rules”, applying the rules effectively and efficiently in complex problems can often seem to be as much art as procedure. Having multiple approaches that can work sometimes makes a task more complex.

When students are first introduced to algebra, it is often with the short term goal of Continue reading What Is Algebra?

A number of students seem to find the introduction of quadratic equations frustrating. After spending much time learning about linear equations, and finally just getting to the point where everything seems to be starting to make sense and be “easy” again, all of a sudden the teacher starts in on a totally different and seemingly unrelated topic…

Life Skills Learned In Math Class

One of the hardest questions for many math teachers to answer in a way that is relevant to students is: “why do I need to know this?”  “For the next course you take”, the easiest answer in many cases, does not answer the question that was usually being asked. My answers to this question obviously depend on the topic being studied at moment, and I don’t have “good” answers for all topics…  but here is my list of key life skills I learned directly or indirectly from math class, with Continue reading Life Skills Learned In Math Class

Analyzing Linear Equations: a summary

Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed.  The following is an attempt to summarize and link the key concepts you need to be comfortable with.

Lines

What is the least amount of information you need to Continue reading Analyzing Linear Equations: a summary

Equivalence Deserves More Attention

Most students taking courses in Algebra or higher seem quite comfortable with the idea of “equivalent fractions”: improper or unsimplified fractions all of which evaluate to the same decimal value. An example would be

$\dfrac{2}{3}=\dfrac{4}{6}=\dfrac{12}{18}=\dfrac{60}{90}=0.\overline{666}$

To create such fractions, multiply whatever fraction you wish to start with by 1 (the multiplicative identity) in the form of a fraction whose numerator and denominator are the same:

$\dfrac{2}{3}\cdot \dfrac{2}{2}=\dfrac{4}{6}=0.\overline{666}$

$\dfrac{2}{3}\cdot \dfrac{6}{6}=\dfrac{12}{18}=0.\overline{666}$

The key concepts here are that
a) an infinite number of equivalent fractions can easily be created, and
b) while all these equivalent fractions sure look different, they all represent the same decimal value or simplified fraction.

Turning to algebra, the very similar concept of “equivalent equations” is helpful in Continue reading Equivalence Deserves More Attention

On the origin of algebra problems

As you are solving a math problem, have you ever wondered how textbook authors come up with algebra problems that have simple answers?

Just as you use inverse functions to solve an algebra problem, problem authors reverse the solution process to create a problem: they start with the answer.  From there they use the same principles that are used to solve an algebra problem, except instead of simplifying the equation at each step, they seek to make it more complicated by substituting a series of equivalent expressions and/or performing a series of operations to both sides of the equation.

Suppose you want the answer to be

$x=3$

Pick something to do to both sides, say multiply by 2:

$2x=6$

Do something else to both sides, say add 4:

$2x+4=10$

I could add 4x to both sides… why 4x?  Because I felt like it:

$4x+2x+4=4x+10$

I could rewrite the 4 on the left side as 2+2:

$4x+2x+2+2=4x+10$

Now to add another step to the solution process, I’ll factor part of the left side:

$4x+2(x+1)+2=4x+10$

And factor part of the right side too:

$4x+2(x+1)+2=2(2x+5)$

And voila!  An algebra problem which is guaranteed to have x = 3 as a solution.  Try creating several problems this way yourself.  Now that you know how to create your own algebra problems, and once you get some practice creating them, you may find it easier to solve algebra problems.

Problems fall into four categories

Math and science problems fall into four categories: Easy, Medium, Ugly, and Hard.

Easy Problems are ones you can solve with no difficulty in a short time.  An example from Algebra I might be:

$3x+2=8$

The problems that come at the beginning of each group of problems in a textbook are usually Easy Problems. If you had Continue reading Problems fall into four categories

Operations are taught in pairs

Many High School students I have worked with have not spent much time pondering the sequence in which math topics were taught to them. So, it can be interesting to step into the “waaay-back” machine and investigate this question a bit:

What was the very first arithmetic operation you were taught (probably in first grade)? What operation were you taught next? Why did your teacher choose this sequence?

If you followed the usual path, the first operation you learned was addition, and the second subtraction. Addition is the operation that describes things being joined or collected together: if I have three cookies, then two more are given to me, I add the two numbers to determine how many cookies I have. Addition is probably the most frequently Continue reading Operations are taught in pairs