Sometimes unexpected answers arise when solving for a point of intersection, therefore multiple tools may be necessary such as visualizations, alternative algebraic strategies, and reformulating a problem.

1. Solve for the point of intersection of the following lines: 2. Solve for the point of intersection of the following lines: 3. Solve for the point of intersection of the following lines: 4. Questions 2 and 3 probably yielded unexpected answers. If you haven’t already, graph the lines and explain why the answers looked the way they did. Once you’ve done this, create additional lines with the same properties and see if you get similar solutions.
5. Head to Dan Meyer’s Ditch Diggers task then watch Act One, consider the information you need to solve the problem in Act Two, then come up with a solution and check it with Act Three. Finally, attempt the Sequel, but then go beyond by generalizing a way to solve for any two lines.
6. There is no requirement for paths to be lines. Find the point(s) of intersection for the following curves: 7. Find the point(s) of intersection for the following curves: 8. Find the point(s) of intersection for the following curves: 9. Assuming there was an answer to problem 8, does it make sense? Why or why not? If you haven’t already, take a look at the graph and explain the source of such strange answers.
10. Although the point(s) of intersection may not exist on a normal plane, what about the complex plane, where the x-axis is the real numbers and the y-axis is the complex numbers?
11. Check out the applications of complex numbers here. Although it is a Wikipedia page, see if you can wrap your mind around one of the applications or topics, especially the Steiner Inellipse.