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Algebraic Expressions and English Phrases

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Two plus five

The sum of three and a number

Four less than seven

Twice a number, plus two more

Four more than a number squared

The temperature, after it has increased by 12 degrees

Three times the sum of five and a number

The difference between the product of 4 and a number, and the sum of 8 and another number

Lesson Plan/Article

Algebraic Expressions and English Phrases

One of the early steps in developing a useful understanding of Algebra is to be able to take English language phrases and convert them into expressions. I say useful because applying Algebraic understanding to real-world situations usually requires taking English language phrases and converting them into expressions to be simplified algebraically. If you can't do this, your understanding of Algebra is entirely theoretical rather than practical. 

As a precursor to this lesson, I've already explored the topic of variables and operations and grouping symbols. Understanding these concepts is key to being able to convert language-based phrases to math-based expressions. Just as I compared variables to pronouns, I extend that metaphor to say that an expression is a "math phrase," and that English languages phrases involving mathematics can be rewritten as algebraic expressions. In other words, you can't do math without grammar. The students invariably groan.

Most of what I do with students from here is based on examples. Each of the examples I use is included in the slide show, so you can put them on a screen instead of writing them each. On the other hand, you may want to mark the phrases up (I often do), so you might want to write them out.

Example #1: Two plus five

This one is almost ridiculously easy, but it's a good one to start out with, because it can be used to highlight the process of converting to an expression. I tell them that the phrases they'll deal with are likely to contain number words, operation words, and grouping words (after all, that's what expressions contain, right?). In this case, it's obvious what the number words are - two and five. The operation word is "plus." I point out to students that the addition operation takes two numbers and puts them together, and conveniently, there's a number before, and a number after. This makes the process very easy: "two plus five" is the same as 2 + 5.

Example #2: The sum of three and a number

Again, we first identify our number words. There are two of them: "three" and "a number." I remind them they don't know what "a number" is equal to, so they have to use a math pronoun (variable). We'll use x here. We also expect to see an operation word/phrase, and "the sum of" is it. "Sum" means addition, so we now know our operation. I remind the students that addition requires two numbers. But unlike the previous example, there isn't a number before and a number after the operation. In this circumstance, we expect to see the word "and" (or another joining word) connecting the two numbers that will be added. Sure enough - that's the way it is structured; there's a number word before and a number word after the word "and." "The sum of three and a number" is the same as 3 + x.

Example #3: Four less than seven

This is one that confuses a lot of students. we quickly identify the two number words (four and seven) and the operation (less than means subtraction). The confusion comes when they put it together, because they want to put it together from left to right: 4 - 7. Here I point out to them that four less than seven is actually positive three, but their expression equals negative three. This helps them to realize that what they're really doing is subtracting four from seven, not the other way around. "Four less than seven" is the same as 7 - 4. Note: I deliberately avoid using a variable in this example, because it's easier for students to see the reversal of the two number words if they use two numbers.

Example #4: Twice a number, plus two more

Here, students will want to tell you that "twice" is a number word, and it represents the number 2. That is not 100% correct. The word twice serves two purposes - it is both a number word and an operation word. It means "two times". If the students see it that way, they will easily recognize that they have their standard number-operation-number setup; twice a number means 2n. If you haven't yet talked about the fact that we can skip writing the multiplication symbol when multiplying by a variable, do so now. After that, it's easy to finish off the problem: "Twice a number, plus two more" is the same as 2x + 2.

I also point out that commas can be helpful - they may indicate how to group things; without the comma, someone might interpret this as 2(x + 2). But the comma makes it clear that we're doing the multiplication first, then the addition. This leads into the next one, which doesn't have a comma.

Example #5: Four more than a number squared

Again, students may want to tell you that "squared" is a number word for two. After the previous example, you may be able to coach them by hints into recognizing that "squared", like "twice" is a combination of a number and an operation. Squared means "raised to the second power." So it is an exponentiation operation with the number two. So if we think of the word as "carat 2", we read that as x2. Then we add in the number four. "Four more than a number squared" is the same as x2 + 4. Students may protest that it should be 4 + x2. I tell them that it is analogous to "less than," and they should try to build the habit of structuring it the same way as they structured the "less than" expression. It will help to keep them from making careless mistakes (because even though it doesn't make a difference what order you write addition, it does make a difference in subtraction!).

Also, with this problem, you can discuss that this could be interpreted as (4 + n)2, if a comma had been placed after the word "number".

Example #6: The temperature, after it has increased by 12 degrees

Students will want to use the variable T. I'll say, "What does T represent?" and they'll say "The temperature!" My response will be that that's not precise enough; the problem talks about TWO temperatures. The tempurate before the increase, and the temperature after. They have to specify which temperature they're talking about. I use this as an opportunity to explain subscripted variables: Tb could mean "the temperature before". "The temperature, after it has increased by 12 degrees" is the same as Tb + 12.

Example #7: Three times the sum of five and a number

This will be the trickiest one yet, because it requires grouping symbols.  We begin by noticing that we have a multiplication operation (times) and a number before it, but not a number after it.  This is a brand new situation - we've got two operations back-to-back: times followed immediately by the sum of. What do we do? We remember that the result of an operation is a number, so if we deal with the "sum" part first, we'll have a number to put into the multiplication operation. So we look first at "sum of five and a number." By now, the students will be quick to do this: 5 + n. If you ask them to finish it for themselves, they'll likely come up with 3 · 5 + n.  Then you'll need to point out to them that because of order of operations, they're not multiplying 3 by the sum; they're multiplying it by just 5.  In order to make sure that 3 gets multiplied by the whole sum, we need to insert a grouping symbol.  "Three times the sum of five and a number" is the same as 3(5 + n). At this point, if you haven't talked about the fact that you can skip writing the multiplication symbol when it's in conjunction with parentheses, you should do so now.

Example #8: The difference between the product of 4 and a number, and the sum of 8 and another number

This example will likely make your students groan. Walk them through it carefully. You should point out the following things:

  • There are two unknowns, so we need two variables. We'll use x for the first, y for the second.
  • We have a comma, which helps to separate it into groups (warn them that the comma may not always be there - it's a matter of preference from one writer to the next!).
  • We also have three occurrences of the word and.
  • We have one instance of two operations back-to-back (difference and product).

Accordingly, we skip over "difference" for the moment, and recognize that the word product goes with 4 and x, so that is 4x. Then I move on to "sum" which is followed by 8 and y: 8 + y. Now we have the difference between two quantities: 4x, and 8 + y. Since these are both results of operations, being put into another operation, I use grouping symbols when I write them just as I did with the previous example. Strictly speaking, of course, putting parentheses around 4x is not necessary, but I'd rather err on the side of too-many-parentheses. Later we can talk about why those parentheses are not necessary. "The difference between the product of 4 and a number, and the sum of 8 and a number" is the same as (4x) - (8 + y)

Lesson by Mr. Twitchell

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