Students will learn how to identify the domain of a rational function. They'll start with rational functions where the denominator is linear. Next, students will consider functions where the denominator is quadratic. After that, students will solve problems where the denominator is a radical expression. In any case, the problem can be solved by considering the denominator as a function, then finding its roots. Students will also learn how to find the range of a rational function by finding the domain of its inverse. Here's a video on that. This topic builds off not only the ability to solve linear and quadratic equations, but also, the understanding that division by zero is undefined.
Next, give your students these challenges:
- A Tricky Length by MAA
Tiles in the Garden by NRICH: There are two elegant solutions to this problem, both of which, are illustrated below. The first way, is to remove the area of the four external triangles from the area of the encompassing rectangle. The second way, is to sum the area of the four internal triangles, then add the area of the internal rectangle which was never counted, or subtract the area of the internal rectangle which was counted twice. The first and second columns, of images below, correspond to the first and second methods of solution, respectively. The bottom right image is an example of when the area of the internal rectangle must be subtracted, as it will be counted twice.
Conclude by leading this investigation:
Portuguese Man o’ War (addition, scientific method)