Strategy: 3-D Viewer
This strategy would work well when students are solving a system of equations. There are two algebraic methods (elimination and substitution), the graphing method (find the point of intersection) and a numeric method (examining a table of values or guess and check). These methods will all give the same solution, but each has advantages in different situations. Students should be encouraged to use at least two methods (for example, solve by elimination, then check the table, or solve by graphing and check by substitution) so that they are sure their answer is correct, but using all the methods for every problem would be redundant. It is important for students to know there are multiple representations because then they have a variety of strategies in their toolkit. One does not need to use every tool to solve a problem, they should choose the most effective tool they have available - which means the more tools available the better!
Strategy: Inductive Learning
This strategy works well when students have a set of cards to sort. For this particular activity I would put equations, descriptions and graphs of conic sections on the cards. First, I ask students to sort by whatever characteristic they would like (most students will probably group by similar looking cards - all graphs in one pile, etc.) and then predict what they think we are going to do with the cards (ideally students will realize there’s an equation for a circle somewhere in the pile and they’re about to learn to match it with a graph and description). The cards will be useful beyond previewing the lesson, I can ask students to match equivalent cards (one of each type) and use the examples to learn the characteristics of each conic section.
Learning Activity: Solving Systems of Equations
Grade Level: 11
Topic: Graphing Conic Sections
Learning Objective: Students will translate between the geometric description and the equation for a conic section.
The goal of this lesson is to review the methods of solving systems of equations and apply them to non-linear equations. Students will have seen all four methods (elimination, substitution, graphing and tabular) applied to linear equations in previous courses. As students enter the room project a system of equations and the four methods along with instructions “Use each method to solve the system of equations below.” The goal is to emphasize that all four strategies will give the same solution and that using more than one strategy is a smart way to check your work. The rating scale from Compare/Contrast would apply here as students moved forward to solving systems using the method of their choice.
- Slides with a mixture of systems of equations (conducive to elimination, substitution and graphing)
During class I will circulate the room, checking in with each student to make sure that they are not using the same method for every problem and that they are solving the problems accurately. Students will have more mixed problems for homework, which I will check at the beginning of the next class. At the beginning of the following class students will complete a short (3 question) quiz with three different systems of equations (one solved for y, two solved for x, linear/circle combination). They will check their own answers at correcting stations around the room when they are done. This gives students immediate feedback and enables me to flip through the quizzes quickly to see how the class did and which students need remediation.
Introducing multiple representation before setting students loose on the problems allows students to consider the different methods of solving systems of equations. This comparison encourages them to consider what type of system they are solving before blindly following the steps of a procedure. Mixing different types of problems together results in a deeper analysis than when students complete an algorithm without understanding. When students are tuned in to the characteristics of the problems they are solving it results in deeper meaning, not to mention more efficient calculations.