*This is part of a continuing series of key math topics in various grades. It will eventually be a book (Traditional Math: An Effective Technique that Teachers Feel Guilty Using), to be published by John Catt Educational.* (*Readers are encouraged to provide examples of mistakes that students will make for the particular topic being discussed. They will be incorporated into the ever-evolving text, so you can be a part of this next book!)*

**Introduction**

Students have had some exposure to solving equations in their earlier courses, having to solve problems such as 3 +n = 10, and 13 – n = 5. These are solved using the arithmetic properties of numbers using the relationships known as “number families” or “number bonds”. These concepts view a number addition equation in three ways. For example 8 + 4 = 12 lends itself to two other related equations, namely 12 – 4 = 8 and 12 -8 = 4.

Faced with the problem 3 + n = 10, the number family approach teaches students that n can be expressed as 10 – 3, and therefore n = 7. For a problem like 13 – n = 5, the student know the number family 5 + 8 = 13, and therefore 13 – 8 = 5. He can also approach the problem by exchanging the n and the 5 to obtain 13 – 5 = n.

In seventh grade, students are given an introduction to the basics of algebraic expressions and learn to solve simple equations using the tools of algebra. This represents a different approach than they are used to and is a much more powerful method that allows them to solve more complex equations and provides an approach for solving word problems.

I like to start the unit by giving them a problem, and prefacing it with the following statement: “I’m going to give you a problem that you will think you know the answer to and you will probably be wrong.” This serves as a challenge and a dare and also defuses the fear of making a mistake because they will want to prove me wrong. The problem is: “John and his sister have $110 between them. John has $100 more than his sister. How much do each of them have?”

Almost instantly hands are raised and students will call out confidently: “John has $100 and his sister has $10.”

I say that that is incorrect, because the problem says John has $100 more than his sister. If she has $10 how much will John have?”

They quickly figure out that he would have $110. “And what is the sum of $110 and $10?” Seeing their error, some of the students then resort to a “guess and check” procedure, trying various combinations. Someone will inevitably say “John has $90.” I respond that if so, then his sister has $100 less. What is $100 less than $90?” I cross my fingers that they remember how to work with negative numbers and one of the braver students will volunteer that it is -$10. Since you can’t possess a negative amount, we know that is wrong. Eventually after enough guessing and checking they come up with John has $105 and his sister has $5.

While “guess and check” is a strategy that can solve problems, I point out that it took some back and forth before they came up with the right answer. “There is a way to get that answer on the first try,” I announce. That way, of course, is algebra. I tell them that they will learn to solve this problem and others using algebra in this unit.

**Writing and Evaluating Algebraic Expressions**

**Warm-Ups. **Typical Warm-Ups for this lesson should incorporate past concepts.

- A 5 foot length of ribbon is cut into 2 ½ inch strips. How many strips are there? (Answer: 5 ft = 60 in, so 60 ÷ 2 ½ = 60 x 5/2 = 24 strips)
- -2 ½ x 3 2/5 (Answer: -5/2 x 17/5 =-17/2 = -8 ½
- (8/3)/(5/6) (Answer: 8/3 ÷ 5/6 = 8/3 x 6/5 = 16/5 = 3 1/5
- If you lose $2 every week for four weeks in a row, what is your loss after 4 weeks, and the fifth week you make $7, do you have a net gain or loss, and by how much? (Answer: -2 x 4 = -8; -8 + 7 = -1; a loss of $1.)
- What is 5 times n if n = -3? (Answer: 5 x -3 = -15).

Students have had experience using letters to represent numbers as discussed above. Now we take it further with the goal of being able to represent English in terms of algebraic expressions. To do this students need to know the general rules of variables, and what a variable is.

A letter used to represent a number is called a **variable**. I liken it to a fill in the blank. The sentence “I have __ apples” can have different meanings depending on the number that is used to fill the blank. “I have x apples” does the same thing. The variable x in this case represents a changing—a fill in the blank type—number. The formal definition that I give to students is:

*A variable**is a letter or symbol used to represent an unknown value. Any letter can be used as a variable.*

**Addition and Subtraction.** To be able to work with variables it is helpful for students to plug numbers in to various expressions that have variables, and to calculate the value of that expression. We start the process by looking at addition and subtraction. I ask students if I have some unknown amount of apples, I can represent that by a variable. I then ask students if I let n represent the unknown amount of apples and then want to represent 3 more than that amount, how would I write it?

I point to the definition of variable which states that the variable represents an unknown value. I will tell them: “In algebra, when we don’t know what the value is of a quantity, we represent it with a letter.”

Students are generally slow to respond to this but eventually someone will say the correct answer of n + 3.

I give more examples of this nature: “I have an unknown amount of apples and I give 5 away. How do we represent this? x -5.

I will have them look at the first example of n + 3, and ask how many apples does the expression represent if n equals 5. I continue with other numbers: 100, 2,538, etc. In the second example of x – 5, I might ask “What is the number of apples if x equals 10, 27, 5?” and so forth.

Now I might ask: “If I have an unknown amount of apples and call that amount m, and I get more apples of the same amount; how would I write this?” They may hesitate a bit, but I am after m + m.

I now want to take it out of variables representing objects and just letting the variables represent “a number”. I have an unknown number; I’ll ask how do I represent an unknown number? By now they’re in the rhythm of the questioning and will call out letters; usually x, but I want them to know x isn’t the only one they can use. I’ll now ask I want to represent 5 more than that unknown amount. Next, with the x + 5 written on the board, I ask what is the value if x equals 4; then 3, then 0, then -5, -10 and so forth.

Finally, I’ll ask what happens if I have an unknown number and I add another *different *unknown number to it. How would I represent that? I tell them to write it in their notebooks, or on a mini-whiteboard. I’m looking for two different variables added: a +b, x + y, and so on.

**Like Terms. **At this point I introduce some additional vocabulary: Term, like terms, and numerical coefficient.

When addition or subtraction signs separate an algebraic expression into parts, each part is a **term. **For example a + b consists of two terms, a and b. Suppose I had 2a + 3b. Then 2a and 3b are terms. ** **The numerical part of a term that contains a variable is called the **numerical coefficient** of the variable. I will ask what the numerical coefficient is of 5x, of -24y.

If two terms have the same variable, or combination of variables, they are called “like terms”. I provide examples such as 2x and 34x, 8ab and 5ab. Because the terms are the same, they can be added, just like we added x + x previously. This tends to be confusing at first, so I will liken it to “like objects”. For example, if I have 2 apples and then get 3 more, I am adding 2 apples + 3 apples for a total of 5 apples. The variables can be thought of similarly. So 3x – 2x + 5x, is the same as adding the numerical coefficients—that is 3 – 2 + 5—and adding the variable afterward: 10x. The summary statement I give them is:

**To combine like terms that have variables, add or subtract the coefficients.**

**Multiplication. **With the above as an introduction, I move on to multiplication. I ask students if I have 3 boxes and each box contains 5 apples, how would I calculate the total amount of apples? They immediately know it is 3 x 5, but I write it as 5 + 5 + 5, and then next to it I write 3 x 5. If each box has 7 apples, then similarly we have 7 + 7 + 7, which is written 3 x 7.

What if I don’t know the number of apples in each box, I’ll ask. How would I represent the total number of apples?

If x is the variable, then 3 boxes with an unknown amount of apples in each one can be represented by x + x + x. How can we write this as a multiplication statement like before? In algebra we represent it as 3x, meaning 3 times x.

After stating that, I ask students to tell me what 3x equals if x equals, 7, -3, 5/3, and so on, so they get the idea that values for the variable are substituted, and that 3x means multiplied by 3.

**Division. **So far in previous lessons and discussions, whenever we’ve talked about fractions, we have mentioned that fractions are division. So 5/2 is the same as 5 divided by 2; 2/3 is 2 divided by 3 and so on. I tell students that in algebra, we are going to represent division by a fractional form, rather than by using the symbols they have been using. The divide sign, ÷, is not used, particularly when using letters. The expression x ÷ y is written as x/y. Similarly, x ÷ 3 is written x/3; 3 ÷ y as 3/y.

**Examples for them to solve. **I will write on the board various expressions with the values of the variables given for each problem, and ask them to find the value of each. These include expressions such as 5/x where x = 2. 5ab where a = -1, b = 2; z + 2x where z = 5 and x = – 2, and so forth. I instruct them to leave improper fractions in that form; it is not necessary to convert them to mixed numbers.

**Ending the Lesson. **While the lesson may move fast, much of the information is new. It is good to focus on these basics so they are familiar with combining like terms and being able to write expressions such as “three times some number” and “the sum of two different numbers” as a + b. The next lesson will focus more on translating more complex English expressions into algebraic ones as well as the use of parentheses and order of operations.

Homework may include evaluation problems such as:

When y = 2, evaluate the expressions:

- y + 23 2. 6y 3. 8/y 4. 4 + y – 7

When m = 8 evaluate the expressions:

- 3m 6. m/m 7. m x m x m

I think perhaps one of the things we need to do better as math teachers is to give a lot more leeway to the idea that a variable is just a letter. I encounter two types of problems with this. The first is that students who take geometry after an algebra class struggle with the idea that AB (meaning the length of segment AB) is a single variable. They want it to be two variables because it is two letters, A times B. Along those same lines, when coding – or even in AP Statistics – we often name a variable with a word. So I am trying to get better about that.

The flip side is that they think a variable is just a value, when oftentimes it can be an expression. For example, when I am trying to teach factoring a sum or difference of cubes, sometimes the A in (A^3 + B^3) is 3x. I am not really sure when or how to introduce this at a reasonable time, but we do need to get them away from “a variable is a number” view. I suppose it works initially to introduce them to the concept, but when and where they have to shift that view is a tough one to call, at least for me.

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I introduce that concept in the algebra 1 course. We have to start somewhere, so we let the letters stand for a number. Technically though, even if x = y + 7, or 3y^2, x is still a number because y + 7 and 3y^2 are numbers. So introducing the variable as standing for a number isn’t wrong–the concept just needs to be expanded. Doing so in 7th grade is a bit early; they need time to get used to working with variables.

By the time I introduce multiplying binomials, like (x + 3)(x – 2), they don’t find it strange when I explain it as follows:

Let’s let (x -2) = A. Then we have (x + 3)A. We distribute the A, and we get xA + 3A. Now we substitute back (x-2) for A in that expression and we get x(x-2) + 3(x-2).

They’ve had enough experience with “chunking” (i.e., letting (x + y) act like a single variable) that they follow the explanation fairly well, based on what I’ve seen from teaching it.

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