AE 4 Traditional Math: Basic Axioms of Equality, Properties of 1 and 0, and Combining Like Terms
Eighth Grade
The first chapter of most algebra courses is mostly a review of the algebra that was introduced in seventh grade. It affords an opportunity to present basic axioms and principles in a more formal manner than what the students are used to. In this section we address the reflexive and symmetric properties of equality as well as the multiplicative properties of 1 and 0. Following that is a discussion of division by zero and why it is not allowed.
Associative, commutative and the distributive property have been discussed in the previous Posts 8 and 9 for seventh grade. I would teach those as a review in a separate lesson previous to this.
Combining like terms is also addressed in Post 7 for seventh grade. We include this topic here, and introduce more complex examples that require distribution.
Warm-Ups.
1. Is there a solution to this equation? x + 2 = x + 3. Answer: There is no solution; subtracting x from each side results in 2 = 3 which is untrue. No number for x will therefore be true.
2. Translate to an algebraic expression. One-third of the sum d and e. Answer: 1/3(d+3)
3. Write an equation for this and solve. The sum of three times a number and 17 is 98. Answer: 3x + 17 = 98; 3x = 81; x = 27
4. Multiply. -3(-2 + x) Answer: 6 – 3x
5. If a = 2 and b = 2, does a + 1 = b? Answer: Since a and b both equal 2, then a + 1 = 3, and 3 does not equal 2.
These problems are basically review of what has been discussed in previous lessons and in seventh grade. Problem 4 is application of the distributive property. Problem 5 is a substitution problem which should be fairly obvious but some students may find it confusing. A prompt might be for them to plug in the values as they have done when evaluating expressions.
Key Axioms. “Today will be mostly things you learned last year, but we’re going to go a bit deeper into it. We’ll start with what are called ‘axioms’. An axiom is a statement that is accepted as true without any proof. This is because they are so obvious.”
The Reflexive Property of Equality:
I write on the board:
Reflexive Property of Equality: Any number is equal to itself. a = a.
Sometimes a student will ask why something so obvious needs to be stated. I answer along these lines: “In math, we often have to prove that things are true. You’ll get work with proofs in high school, but when we prove things, sometimes it comes down to something obvious. You’ll see why in tomorrow’s lesson.” (The next lesson will include a proof of the additive and subtractive properties of equality which rely on this axiom and the property of substitution.)
The Symmetric Property of Equality:
I announce: “The second axiom is like unto the first.” (I taught at a Catholic school.)
The symmetric property of equality: For any numbers a and b, if a = b, then b = a.
The Transitive Property of Equality:
On the board:
Transitive Property of Equality: For any numbers a, b and c, if a = b and b = c, then a =c.
“Let’s see how the transitive property works. If I have 4 + 6 = 10 and 10= 5+5, how can I write this by using the transitive property?
I let them discuss it. Usually someone will come up with the answer. If not, I’ll write it out: 4 + 6 = 5 + 5
“Now rewrite x + 2 = y, y = 2x -3. Using the transitive property. Answer: x +2 = 2x -3
“Last one: 5x = t, 3x + 2 = t. Rewrite it using the transitive property.” Answer 5x = 3x + 2.
This last might be confusing at first because it’s written differently. So I restate the transitive property: “If two numbers are equal to the same number, they are equal to each other.” I have them write that in their notebooks.
Multiplicative Properties of 1 and 0. “What can you tell me about multiplying by 1?” I ask. Students are quick to respond that the product is the same number that is being multiplied by 1. “What about multiplying by zero?” They are also quick to tell me that anything multiplied by zero equals zero.
“What about if I divide zero by any number?” Again, zero.
“Now what about if I divide a number by zero?” Most are quick to respond that it’s zero, and now starts an important discussion.
“Let’s explore this. We know that 6 divided by 3 is 2, and that means 2 times 3 equals 6. So if I have five divided by zero, I have to find a number which when multiplied by zero, equals zero. Is there any such number?”
I recall when I first learned this I was amazed that I hadn’t noticed this before. Perhaps I had just assumed, like my students, that anything divided by zero is zero. Students will likely be going through a similar disorientation.
“There is no number that satisfies 5 divided by zero, or any number divided by zero. You cannot divide by zero. No number will work; it contradicts the multiplicative property of zero. So we call any number divided by zero ‘undefined’. That includes fractions like 4/0 and so forth.”
Inevitable question from a student. “So on a test if you give us a problem like 6 divided by 0, the answer is ‘undefined’?”
“Yes, it is undefined, or ‘no solution’.
I write on the board: No number is divisible by zero.
“What about zero divided by zero? Think about it for a minute.” Soon, students begin to see that any number will work. “There is not just one answer; there are an infinite number of answers. So when we have zero divided by zero we call it ‘indeterminate’.
Combining Like Terms. I start out by asking “Someone simplify this: 2x + 3x”
Students will generally know this: 5x
“You are familiar with combining like terms from last year. One way of looking at 2x + 3x is that we’re counting how many x’s we have. But another more formal way of seeing why this works is to look at it as a distribution. 2x + 3x can be written as x(2 + 3). We can see if we distribute the x, we get 2x+3x. But what’s the value in the parentheses?”
Students should see that it equals 5.
“So then we have x(5), which is 5x. When we combine terms we are adding or subtracting numbers, or variable terms where the variables are the same. 8x and 9x are like terms. Why?”
I’m looking for “Because x is the variable in both terms.”
“What about 3a and 4a? What are they when combined?”
They should get 7a.
“When adding or subtracting like terms, we add or subtract the numerical coefficients. What about 3a and 4ab? Like or unlike?”
I’m looking for unlike. I’ll ask for a reason and I want to hear that the variables are not exactly the same. “We can’t combine something like a with ab. They are not the same thing.
Examples.
Students are to say whether they are like or unlike terms and if like terms, combine them.
1. 8x + 3a. Answer: Unlike; can’t be combined.
2. 3xyz – 4xyz Answer: Like terms; -xyz (I usually have to remind them to combine the coefficients, and not to write the 1, or -1 in this case.)
3. 8x + 3ab – 2x Answer: I give a prompt with this. “Are there any like terms that can be combined? What are they? Can we combine them with 3ab? What is left after combining is 6x + 3ab. These cannot be combined.
Distribution and like terms. In an allusion to the painter Bob Ross and his TV series which has a cult following and many students know of him, I will announce “Let’s get crazy.” This generally serves as a signal that things are about to get more complicated, and as students come to know what it means will groan when I say it. Which is fine with me.
“To combine some terms we need to apply the distributive property. For example if I have 2(a+b) + 8(a + 2b) we have two terms, both of which need distribution. We do we get for the first one? The second?”
They will get 2a+2b+8a+16b, which can then be combined to 10a + 18b
Examples. I work through the first two with them.
1. 8v + 5(7-v) Answer: 8v + 35 – 5v = 3v + 35
I point out that we can’t combine the 35 with the variable terms; a common mistake is that students will try to do that, and will end up with 38v in this example. I fully expect the mistake to be made and make a point to talk about it.
2. 2(a+1) – 2 Answer: 2a+2 – 2 =2a
3. 6v + 7(3+m) Answer: 6v + 21 + 7m It cannot be combined any further though students will feel compelled to do so.
4. x –x + x –x Answer: 0
5. 3y + 12 + y + 8 Answer: 4y + 20
Homework. The homework should consist of simplifying expressions by combining like terms, including problems that entail distribution.