Combining or Collecting Like Terms

The phrases “combine like terms” or “collect like terms” are used a lot in algebra, and for good reason. The process they describe is used a lot in solving algebra problems. Two approaches, one intuitive and the other algebraic, can help in understanding why some terms are “like” terms, and others are not.

Quantities With Units

Suppose you are sitting in front of a table that holds three piles of fruit:
– five apples
– three oranges
– four apples
If someone asks you “What do you see on the table?”, how would you answer the question?

Chances are you answered “nine apples and three oranges”. Why did you combine the two piles of apples with one another, but not with the oranges? How did you know that you could do that?

The quantities of apples may be combined because addition or subtraction only work with  Continue reading Combining or Collecting Like Terms

Sigma Notation (Summation Notation) and Pi Notation

Sigma (Summation) Notation

$\sum$ is a capital letter from the Greek alphabet called “Sigma”… it corresponds to “S” in our alphabet (think of the starting sound of the word “sigma”). It is used in mathematics to describe “summation”, the addition or sum of a bunch of terms (think of the starting sound of the word “sum”: Sssigma = Sssum).

Sigma can be used all by itself to represent a generic sum… the general idea of a sum, of an unspecified number of unspecified terms:

$\displaystyle\sum a_i~\\*\\*=~a_1+a_2+a_3+...$

But this is not something that can be evaluated to produce a specific answer, as we have not been told how many terms to include in the sum, nor have we been told how to determine the value of each term.

A more typical use of Sigma notation will include an integer below the Sigma (the “starting term number”), and an integer above the Sigma (the “ending term number”). In the example below, the exact starting and ending numbers don’t matter much since we are being asked to add the same value, two, repeatedly. All that matters in this case is the difference between the starting and ending term numbers… that will determine how many twos we are being asked to add, one two for each term number.

$\displaystyle\sum_{1}^{5}2~\\*\\*=~2+2+2+2+2$

Sigma notation, or as it is also called, summation notation is not usually worth the extra ink to describe simple sums such as the one above… multiplication could do that more simply.

Sigma notation is most useful when Continue reading Sigma Notation (Summation Notation) and Pi Notation

Function Notation

Many students seem puzzled by function notation well after it has been introduced, and ask “why can’t we just write $y = 3x$ instead of $f(x) = 3x$ ?”. To motivate the use of function notation and improve understanding, I advocate using multi-variable functions instead of single-variable functions in introducing this notation. My introduction usually proceeds something like this:

A Problem

Suppose you are talking to a friend over the telephone, and they have a piece of paper in front of them on which they have drawn a circle using a ruler and a compass. You need the clearest, most concise instructions possible so that you can exactly duplicate the appearance of the paper your friend has… your circle must be exactly the same size as theirs, and in the exact same location on the piece of paper. What information do you need in order to be able to do this?

For the size of the circle, either the radius or the diameter will work nicely (most students come up with this answer quickly). If you are using a compass to draw the circle, the radius will probably be more convenient. However, the radius alone does not tell you where to locate the circle on the paper. What aspect of a circle will do the best job of Continue reading Function Notation

Notation & Concept

As a math tutor and teacher, I use notation a lot. It is familiar to me, comfortable even. So it is all too easy to forget that this same notation can be getting in the way for math students.

In teaching algebra, I have conceptual, procedural, notational, intuitive, and experiential goals for my students. I want them to develop work and thought habits that help them to both understand the material and solve problems efficiently. I am also often trying to cram much material into a short period of time.

The result is often more time spent on concept and procedures (the crucial parts) than notation (which they’ll pick up from the way I use it, right?). Yet, I suspect many “typical” algebra errors may result from Continue reading Notation & Concept

It’s True! 3x = (3)(x)

Many students I work with perceive

$3x$

as being something different than

$(3)(x)$

Yet, if I ask “what operation connects the “3” to the “x”, most students will think a second and respond “multiplication”. So, they can figure out what it stands for – but they do not perceive it that way initially.

This mis-perception contributes to a number of Continue reading It’s True! 3x = (3)(x)