Strategy: Collaborative Summarizing
Students are given cards with examples of equations for circles, ellipses, hyperbolas and parabolas. They work in groups of three to determine the characteristics of each conic section by looking at all the examples and finding the differences between each type. Then, given a general form equation for each type, students determine how to describe the graph (vertices, width, height, etc.) given an equation. Finally, the group will summarize all of this information in a chart or other graphic organizer. Giving students examples and asking them to determine the important characteristics results in two things: first, students have to carefully analyze the equations, looking for details; second, the group has to come to a consensus on which details are important, distinguishing characteristics. Creating their own graphic organizer forces students to organize the information, but also means that the tool is one they understand. Frequently students do not use tools we consider useful because they do not know how to navigate them. In this case they create a tool that they find useful. Different students think differently, so they should have the opportunity to lay out the information in a way that matches their thought process.
Strategy: Paired Learning
Students work in pairs to discover the properties of trigonometric ratios using an applet. Each pair gets one computer and two handouts. Their task is to complete tables of trig ratios for different triangles (the applet allows students to change the angle measure and base length of a right triangle). They start with sine of varying angles for a triangles with a constant base, then change the base and find the sine of the same angles. In pairs students discuss their observations and particularly focus their attention on comparing the two tables (ideally recognizing that the values are the same regardless of the length of the base). They repeat this exercise for tangent and cosine. The questions which accompany the tables are open ended, asking students for anything they notice. Since the more observations they have, the better, working in pairs encourages sharing and discussion rather than blindly copying down data without analysis.
Learning Activity: Intro to Trig
Grade Level: 10
Topic: Introduction to Trigonometry
Learning Objective: Students will learn the definition of trigonometric ratios and identify characteristics of these functions.
The goal of this lesson is to introduce sine, cosine and tangent. Students are familiar with similar triangles but have not seen these functions before. In pairs students work on the computer to complete trig tables. They start with sine of varying angles for a triangle with a constant base, then change the base and find the sine of the same angles. In pairs students discuss their observations and particularly focus their attention on comparing the two tables (ideally recognizing that the values are the same regardless of the length of the base). They repeat this exercise for tangent and cosine. Working in pairs encourages discussion rather than blindly copying down data without analysis. The pairs will then share their observations in a whole class discussion.
- Laptops with internet (http://www.geogebratube.org/material/show/id/9694)
- Handout with blank tables, definitions and prompts to make observations
During class I will circulate the room, checking in with each pair to make sure that they are working together and making observations, (for students whose values are not exactly the same I will point out that measurements are not perfect and inform them those values should be the same). Students will apply these newly defined functions to the 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 on solving right triangles and the properties of trigonometric functions. 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.
Pairing students encourages discussions, which requires students to use the vocabulary. Students will automatically begin using the terms opposite, adjacent and hypotenuse to specify which side they are talking about. Since there is one computer for every two people the students take on the roles of computer manipulator and data recorder. Midway through class I will announce that students should switch roles if only one person has been “in the drivers seat” on the computer. Open ended questions asking for students’ observations results in all contributions being valued; there is no one right answer, the best answer is the one that has the most observations, which requires teamwork.