The introductory lessons to eighth grade algebra are mostly a review of the algebra topics covered in seventh grade—only a bit deeper. I typically do not spend a lot of time on solving elementary equations, since students did a lot of that in seventh grade.
I do like to focus on three things: 1) basic vocabulary, 2) more detail on what an equation is, and 3) representation of English into algebraic symbols. With respect to the latter, I also go the opposite way—the correct way of expressing algebraic symbols in English, which can be considered a part of the vocabulary of algebra.
In the first lesson (which would be prior to the one discussed here) I go over the following key terms: Variable, Constant, Variable expression, Factor, Numerical coefficient, Exponent, Power, and Term. I will not go into detail on these here, except for the word “Term” since it is used often, and frequently confused.
“Term” is a mathematical expression using numbers or variables or both to indicate a product or a quotient. In general, mathematical expressions that are separated by addition or subtraction signs are terms. Examples of terms include:
These are all products or quotients. 3a-3b are two terms—they are separated by a minus sign. The expression 3(a-b) is one term, because it is a product of 3 and (a-b). Using the rule that plus and minus signs in an expression separate the terms in an expression, one can see that there are three terms are in the following expression:
I also start writing algebraic fractions on the first day. Students must get used to seeing division represented as fractions.
Warm-Ups.
1. If x=3, what number does the expression 5x2+4 equal? Answer: 5(9) +4 = 45 + 4 = 49
2. -2 + 6.5 = ? Answer:-4.5
3. Express in lowest terms:
Answer: 20/54 – 1/54 – 1/54 = 18/54 = 1/3
4. 12 × 6 +12 × 4 = ? Answer: 72 + 48 =120
5. Answer: 6(9) – 3 = 54-3 = 51
These are review problems that include order of operations, evaluating expressions and working with fractions.
Equations. I start out with a more formal approach to equations than was presented in earlier grades.
“You’re all familiar with equations. 3 + 2 = 5 is an equation. So is 4 + 2 = 6. What about 5 - 2 = 500? Is that an equation?”
I’ll hear both “yes” and “no” and will ask why they said “no”. The usual answer is that the statement is untrue. “What about those of you who said it is an equation. Why do you say that?” If I hear something on the order of “It doesn’t have to be true” that’s pretty much what I’m looking for.
“Yes, it is an equation; it is an equation that happens to be untrue. Equations are like sentences and are sometimes called ‘open sentences’.”
I’ll write the following statement on the board:
W is a city in Texas.
“Is this sentence true or false?”
I’ll hear many answers, but I’m looking for “You can’t tell.”
“Yes, as this is written, it is neither true nor false. If I replace W with “Dallas” it’s a true statement. What can I replace w with to make it a false statement?”
I’ll hear a variety of answers. Sometimes I’ll ask about “Paris” replacing W, and upon hearing that it’s wrong, will say “Actually there’s a Paris, Texas, so in that case, the sentence is true.”
“In this sentence, W is the variable. It can be replaced by many different cities. And number sentences with variables are called open sentences. 30- x = 4 is an open sentence. It serves as a pattern for various sentences. We can substitute the number 3 in there. Would it be true or false?”
Hearing “false” I’ll ask why, and I hope they will say that 27 is not equal to 4, or something resembling that. I write the following on the board and they copy it in their notebooks:
An algebraic sentence is a statement composed of algebraic expressions related by one of the following symbols:
“Any sentence using the symbol “=” is called an equation.”
I write the equation 3x + 1 = 16 on the board.
“Is the equation true for x = 4?” No.
“Is it true for x = 6?” No.
“We can find out the value for which it’s true for either by substituting values—guess and check—or solving. You know how to solve this equation and we’ll be getting into more equations and solving them in the next few lessons. So can anyone solve this one?”
Some people will remember how to do it, and after hearing a correct answer (x = 5) I have the student come to the board to show how he or she solved it.
Each number that makes an equation true is called a root of the equation.
Inequalities. Now we move into inequalities which we also covered in seventh grade. “Now we’ll take the same equation we had before: 3x + 1 = 15, and replace the equal sign with a ‘greater than’ sign.”
(Inequalities have been discussed in a previous post.)
“Anyone remember how to do this?” There usually are a few students, and it comes back fairly quickly. The answer is x >5.
“So this inequality has more than one answer; it has many answers, an infinity of answers. Any number greater than 5 will satisfy the inequality. We can see this easily when we graph it. Anyone remember how to graph it?” I’ll have a student graph it on the board, and then have the class do two more.
(Graphing inequalities has been discussed previously in Post 17 and others.)
Now I write the following on the board:
(Inequalities have been discussed in a previous post.)
“Anyone remember how to do this?” There usually are a few students, and it comes back fairly quickly. The answer is x >5.
“So this inequality has more than one answer; it has many answers, an infinity of answers. Any number greater than 5 will satisfy the inequality. We can see this easily when we graph it. Anyone remember how to graph it?” I’ll have a student graph it on the board, and then have the class do two more.
(Graphing inequalities has been discussed previously in Post 17 and others.)
Now I write the following on the board:
“This is something new, but it’s done the same way as before, except we have different end-points. Let’s first plot the two numbers, and think of this as two inequalities”:
“The same rules for hollow and solid dots apply, but the line will not go on forever as it does when we only have one number.”
I lead them through it as they work it in their notebooks. It will look like this:
We do some more so they feel comfortable with the procedure:
“These are called ‘compound inequalities’ which we’ll get into in a few weeks.”
No Solutions. I write the following on the board:
“If we substitute numbers in for y can we find any that makes this true? Give it a try.”
After a minute or so, students will see that they’re coming up empty. But since they know basic solving techniques, I direct them to subtract y from each side. “We can do that, can’t we? If we subtract the same amount from each side, that’s allowed, right?” General agreement ensues, but when they try it they get 3 = 2 which they are quick to tell me that’s what they got, was that supposed to happen?
In case that is asked, I answer in the same way my high school physics teacher would when an experiment would go awry and we would ask if that was supposed to happen. “Did it happen?” I’ll ask. The student will look puzzled and say “Uh, yeah.”
“Then it was supposed to happen.” I wait for the puzzled look to appear and then ask “Does 3 = 2 make sense?”
Of course it doesn’t and the student will say so.
“So this equation has no solution. No matter what number—the same number—that we add to both sides, it will never be true, because 3 does not equal 2.”
Inevitably a student will say “So the answer is ‘no solution’?” To which I answer “Yes.”
Many Solutions. “But now suppose I write it as an inequality:
“Substitute some numbers in for y. See what happens.” I give them some time to experiment and then ask their opinion. “Does it look like this inequality is true?”
General consensus has been (for me at least) that it does. “Like before, let’s subtract y from each side. What do we get?”
They will see that what remains is 3 > 2. “So since 3 > 2 is true, no matter what number we add to both sides, that inequality will hold. This inequality has many solutions—an infinite amount of solutions.”
Again, someone will ask: “So the answer is ‘many solutions’?”
And again I answer “Yes.”
Homework. I like to give them very simple equations, one and two step, as well as inequalities—some simple, and some compound. They solve the equations, and graph the inequalities.