AF 3 Traditional Math: Adding and Subtracting Algebraic Fractions (Parts I, II and III)
Eighth Grade
Adding and subtracting algebraic fractions is a topic that algebra students find confusing. It is therefore worth taking the time to ensure that students master and fully understand the procedures.
I generally use three lessons to go through it. On the first day I present fractions with like denominators which students find quite easy. The next two days are devoted to fractions with unlike denominators. On the first of those two days, I focus on problems for which the denominators do not need to be factored; e.g., denominators such as a and 2a, or x and 2x²y. The third day focuses on more complex denominators such as x -1, and x² - 1. I have presented all three days in this particular section.
Addition and subtraction of algebraic fractions is an abstraction of the procedures students have learned in lower grades for numerical fractions. A sketchy background in arithmetic fractions will make the transition difficult if not impossible. Students are used to seeing 3 and 6, 3 and 9, 4 and 16 and so forth as multiples. Now they must extend the idea to letter representation, so that a and 2a, a and ab, and b, b² and generally bⁿ are also seen as multiples. Finally, students must learn to factor denominators and see that a² - 25 is a multiple of a + 5.
It is worth taking the time to go over the material carefully, and to work in such problems in subsequent homework assignments. Facility with addition and subtraction of fractions is a prerequisite for understanding the procedure for solving equations with algebraic fractions.
Warm-Ups.
Students will likely need guidance for Problem 1; a diagram helps. In this problem the time travelled by each person is given—4 hrs, i.e., from noon to 4 PM. Problem 2 requires students to see that whatever the multiplier is to change the denominator to must then be multiplied by the numerator of the first fraction. For Problem 3, students must see that the numerator is the difference of two squares, and then to factor it and cancel the terms. Problem 4 is given as a prompt for students to be able to solve Problem 5. A prompt for students who are confused might be “How did you solve Problem 4? Is Problem 5 similar?” These two problems are a segue to the day’s lesson.
Fractions with Like Denominators. After going through the last two Warm-Ups, I point out what should now seem obvious. “Adding and subtracting algebraic fractions is done the same way as with numeric fractions. And in fact, in either case, algebraic or numeric, we are really using the distributive rule. Look at Problem 5. I’m going to write it a bit different.”
“Remember, we can write x/2 as ½ x. So now our problem looks like Problem 4. And what can I factor out? Discuss it a bit.”
I wait and see what they come up with. Hopefully someone comes up with ½ x:
“You can see that we wind up with 4 in the parentheses.”
“All this is by way of saying that algebraic fractions follow the same rules as numeric.”
I write on the board for students to copy in their notebooks:
Examples: I work through them with the class.
Simplifying Sums and Differences. “Looking at the last example, let’s say we had something slightly different.”
I have them do this on mini-whiteboards.
“Can this be simplified?” If no response is forthcoming I’ll ask “Can anything be factored. Like, say, in the numerator? And can anything be cancelled?”
They will see that the 3 can be factored:
Homework. Problems are a mixture of fractions that can be simplified and those than cannot. For some problems, I’ll ask that excluded values be identified.
Day 2: Unlike Denominators
“In our last lesson we worked with fractions that have like denominators. So it will be no surprise to anyone that today we will be working with differential equations in multiple dimensions.” Students of course are expecting me to say “unlike denominators” and to my knowledge, there has never been any laughter when I make this little joke. I thought I would include it here for you to do with it as you wish.
“Yes, today we talk about unlike denominators which you’ve worked with before. It involves finding the least common multiple between two numbers. What is the least common denominator for 1/3 and 1/9?” They answer it easily. “What about 1/6 and 1/9?” A slight pause and they answer “18”. “What about 5 and 7?” 35.
For this last, I point out that sometimes the least common multiple is the product of the two numbers.
“Algebraic multiples are similar. What is the common multiple of a and a²?
The answer: a². What about 6a and 9ab? It is 18ab. How about 5a and 7a? It is 35a. “What about 5a and 7x?” Some hesitation and a few students’ hands are up. The answer is 35ax.
“Now let’s suppose we have these two fractions to add. We need to find the least common denominator.”
“And there’s our answer. Now you try with this one.”
I walk around to see what they’re doing. They generally get this one, though some who do not see b² as a multiple of b might write the common denominator as 12b rather than 12b².
I have a student write the steps and the final answer on the board.
Homework. I work several of the more complex homework problems with the students as they do the homework in class. The homework is limited to the more straightforward problems, and is mixed with problems with fractions that have like denominators. It is important to have like denominator problems in the mix, because I have had students forget how to do such problems in the face of a lot of new information.
Day 3: More Complex Denominators
I make sure to answer questions about homework problems that proved difficult and have students come to the board to solve them.
“Yesterday, you had problems where the denominators might be something like ab and 2a. What’s the common denominator?” They’ve seen this before and identify the common denominator as 2ab.
I then ask student to tell me how to add the following fractions:
This is similar to the problems we did in the previous lesson.
“Now suppose I have two denominators like this.”
“What’s the first thing you’re going to do?” Someone might suggest that we factor, but if no one does, I’m prepared to suggest it.
“Let’s factor each one and see what we get.”
“Suppose now that I let (x -5) = a and (x + 5) = b. Then what I have on the board can be represented by ab, and 2a which you’ve just seen. And unless you were lying to me, you said the common denominator is 2ab. So what would be the common denominator for what I have on the board?”
After allowing them to discuss it—and such discussion is usually short to non-existent—the verdict on the common denominator is:
I now write on the board:
I proceed to add these fractions by asking the students what to do as I write down the steps.
Multiplying by -1. “What if I had something like this?”, I ask and write the following on the board:
Someone might say the LCD is (x-1)(1-x) which is technically a way to do this. “We could do that,” I say, but there’s an easier way. At which point someone might remember that we can reverse the order of a binomial by factoring out -1.
“Let’s do that.”
“Now we have -1 in the denominator which is making the whole fraction negative. So we can rewrite this.”
“And what if we had had this?”
“After we factored out -1 in the denominator we would have this.”
“So now we’re subtracting a negative—what happens when we subtract a negative?”
By now, since I’ve asked this question before in other contexts, they know the answer is “It becomes positive.”
“Good. So how would I rewrite this?” Usually many students know at this point.
I then write the rule which they copy in their notebooks:
When factoring out -1 in the denominator of a fraction, reverse the order of the terms, and change the sign of the fraction.
Examples. For the first two I work with the class, and then have them work independently.
Homework: Homework should be a combination of fractions with unlike denominators with both simple and complex form as discussed in this lesson including those with reversed terms that require factoring out -1. Also, fractions with like denominators should be included. In particular, I like to have them work the following:
The answer simplifies to “1” if students see that the numerator is a perfect square trinomial: