Have you ever wondered why so many fairy tales are dark and scary? They're cautionary tales; don't follow the pied piper, don't go wandering in the woods, don't eat houses made of candy. Well, I'd like to tell you a mathematical fairy tale...
Let a = b.
Then a + a = b + a.
That is, 2a = b + a.
Then 2a - 2b = b + a - 2b,
or 2a - 2b = a - b.
In other words, 2(a - b) = 1(a - b).
Therefore, 2 = 1.
Now we're like Hansel & Gretel captured by the witch. How are they going to avoid becoming her dinner? How are we going to show that 2 can't equal 1?
The fun of it is that the students know you've pulled a fast one somewhere, but they can almost never figure out where it is. We can seemingly justify each and every step, but what they're missing is the fact that division by zero is undefined. The final step - dividing both sides by (a - b) - is only valid as long as (a - b) is not equal to zero. But in this case it is zero because we let a = b in the first step!
The moral of the story: Division by zero makes a mess of mathematics. Keep this in mind as you manipulate rational equations.
Hand out Solving Rational Equations 2.
As students work individually to solve the three equations on the front, circulate and assist. Make sure that the students are identifying the domain before attempting to solve the equation. Prompt students to consider why you're directing their attention to this important step, but don't tell them at first that it has to do with extraneous solutions. We want them to notice the connection on their own, if possible. Check out this video for some thoughts on this.
Also, encourage students to work efficiently. Perhaps suggest they consider how to write the equation without "fractions" [i.e. clear the polynomial denominators by multiplying by a single polynomial that incorporates the factors of all of the denominators]. The point is examine the structure of the equation to find the smallest polynomial that will accomplish this. (MP 7)
A simple example may help.
For the remainder of the class, students should work individually or in small groups to complete the problems on the back of the worksheet.
They will have to pay attention to the domain of the functions in order to identify the points of discontinuity. Be sure to watch their graphs carefully - they might not recognize the asymptotes! Encourage them to evaluate the function at a number of points close to the excluded x-values in order to really see what the graph is doing. In part c, the directions do not make this explicit, but many students will find it helpful to graph the line y = 2x + 1 before solving the equation analytically. Encourage them to do so.
The final equations are intended to remind students to look for helpful elements in the structure of the equation. Common denominators, differences of squares, and polynomials that differ by a factor of -1 are all included. (MP 7)