In this unit, students work with algebraic fractions, or rational expressions. The arithmetic concepts of fractions that students learned in lower grades are now extended to algebraic forms. It is therefore important that students have mastery of numerical fractions in order to understand the rules for algebraic fractions.
An initial lesson in fractions occurs before the one presented here. In the initial lesson, students learn that a fraction is defined as the quotient of two algebraic expressions. Fractions are defined only when the denominator is not zero. A fraction like 3/x has meaning, therefore, only when x is not equal to zero, and this is often written as a restriction. Zero is called an “excluded value”; that is, a number excluded from the set of numbers that x can be. Finding excluded values proves to be difficult for students when the fractions are more complex; e.g.:
For the first fraction, it’s obvious that x cannot equal 2, but it can also be solved by writing an equation: x -2 = 0, or to ensure that students understand what x cannot equal, writing it as x - 2 ≠ 0. As mentioned previously, although students have solved such a simple equation before, in new contexts the familiar seems strange—even more so for the second fraction, which requires factoring:
To alleviate confusion I increase students’ familiarity by including these simple equations as part of the daily Warm-Ups.
Simplifying algebraic fractions is done by cancelling; e.g.:
More complex fractions require factoring which students have previously. For example, the following simplification requires knowledge of and facility with factoring:
Students have had word problems such as in Problem 1 in the unit on factoring. Rectangle area problems are based on length x width = area. Negative values are discarded as answers, since all lengths must be positive. Problems 2 and 3 require putting the equation in standard form first. For Problem 3, I am on the lookout for students who obtain only one answer; both x and x-17 are equal to zero, so x = 0 is a solution frequently overlooked. Students have had problems like Problem 4 in working with powers, and now as part of the unit on fractions. For Problem 5, the excluded value is that which would make the denominator equal to zero.
Simplifying Fractions by Factoring. I refer to Problem 4 of the Warm-Ups as an example of simplifying a fraction. “You’ve been doing these for a while. Here’s one that’s a little different. See if you can do it.” I write on the board:
It doesn’t take long for students to say it equals 1. “Correct. Any number divided by itself is equal to 1. I wanted you to see that we treat x + 5 as a single number that can be cancelled just as if we had x/x. What if I had something like this? Could we simplify it?”
There might be a response, but generally there will be the usual blank stares. Assuming that’s true, I am ready with a prompt. “Can we factor the numerator?”
In fact the numerator can be factored into 2(x + 2), and I then ask if we can cancel anything. They see what’s going on and proclaim that the answer is 2 because the (x +2) expressions can be cancelled.
“When you simplify a fraction, look to see if it can be factored; factor as much as you possible can and then cancel. Here’s one that’s slightly more complex.”
I check their notebooks as they work it and select a student who has it correct to write the steps and solution on the board:
“Can anyone tell me what the excluded value is?” I anticipate that some students will still have difficulty with this, but more are starting to get it: x cannot equal 3.
Factoring out -1. “Some time in the past when we were on the factoring unit, you may recall that for problems like we did something. What was it?”
This jogs some memories as they recall that factoring out -1 changes the signs inside the parentheses. “So what can we do with our problem to get the numerator to match one of the expressions in the denominator?” I have them work on mini-whiteboards to show me:
Illegal Cancelling. “There is one thing you cannot do, and I’m willing to bet $100 that at least one of you will do this—and hopefully not on a test or quiz.” I write out the offending operation:
“The only time you can cancel terms is if the terms are multiplied. If we have 3x/3 he 3’s can be cancelled by dividing the 3’s and getting . But not if the 3 and x are being added or subtracted.” I have had a poster to this effect on the wall, and have even embedded it on tests or quizzes, and had students do this illegal cancelling. Usually after getting problems marked wrong on a test because of such cancelling they get the idea.
“While I’m at it, I’m still seeing people writing (x + y)² as x² + y². That’s wrong! Use the rule for squaring binomials, or multiply the binomials if you have to.”
Homework. Homework should be problems simplifying fractions, which involve some factoring—including factoring out -1 to reverse terms. For at least ten problems I will require that they identify excluded values.