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 quantities that have the same units, such as apples in the example above. If you were asked to add three apples and two oranges, the only answer you could give is “three apples and two oranges”, since neither addition nor subtraction make sense with quantities that have differing units. The concepts of addition and subtraction are based on combining or separating quantities with the same units.

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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?

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.

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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 repeated addition model. This is probably due to two factors: being taught that way and the inherent appeal of, as Mr. Devlin put it, “reducing all of arithmetic to addition” (describing all arithmetic operations as being tied back to addition in a strong way).

Multiplication Notation

Many people perceive

$3x$

as being something different than

$(3)(x)$

Yet, when asked “what operation connects the “3” to the “x”, most 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 algebra errors. Most often, people do not seem to see that