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

Algebra Intro 6: Multiplication

We have all known our multiplication tables for years, and have successfully answered questions like “what is 6 times 7?”, but do we really understand what multiplication represents?

Repeated Addition

One interpretation of multiplication, which only works when multiplying by an integer, is “repeated addition”. From this perspective, “6 times 7” is a compact way to Continue reading Algebra Intro 6: Multiplication

Algebra Intro 7: Properties of Multiplication

Properties Of Multiplication

Do the patterns that applied to addition also apply to multiplication… do the following all produce the same result?

3 \cdot 5 \cdot 7 \\*3 \cdot (5 \cdot 7) \\*(3 \cdot 5) \cdot 7

After carefully following the order of operations, we see that they all result in a value of 105. Therefore, it is reasonable to conclude that multiplication is associative, similar to addition.

Continue reading Algebra Intro 7: Properties of Multiplication

Algebra Intro 10: Fractions and Multiplication

Many people seem a bit phobic about “fractions”. This anxiety likely has two sources: not really understanding what a fraction represents, and having memorized a bunch of rules way back in elementary school without understanding why they work.

Revisiting fractions using variables as well as constants, with Continue reading Algebra Intro 10: Fractions and Multiplication

Multiplication

In reading through the multiplication-related blog postings of others while pondering multiplication and division as inverse operations, I came across Keith Devlin’s articles (originalfollow-upmore,  most recent), which led me to wonder about my own concept (or lack thereof) of multiplication.

I have a vague recollection of learning multiplication tables from flash-cards at home. When I could not remember a particular product, I would figure it out via the repeated addition model. So, I think my primary concept of multiplication (even today) uses the Continue reading Multiplication

Multiplication Notation

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 Multiplication Notation