We've just finished a lesson on Reflecting Figures Across a Line. As a segue from that lesson, and as a lead-in to this project, I demonstrate how we reflect figures across lines using technology applications like Geometer's Sketchpad.
My discussion as I'm demonstrating goes something like this:
How convenient it is to have an application that will simply take a figure that we've drawn and reflect it across a line. We saw yesterday how much work goes into reflecting just one point across a line. This program reflects multiple points in less than a second. It almost seems as if it's magic. But no, someone had to do a lot of work behind the scenes in order to make it that way. Hundreds of hours of work...pages and pages of computer programming code...mistakes and restarts...not to mention developing the computer that runs the program. Somehow we take all of that for granted. Think of all the products that we depend on...cell phones, computers, mp3 players, televisions, the internet....they are all very complicated devices that took lots of work to produce. For the most part we only think of these devices as consumers...in other words, we're only concerned about what they can and cannot do for us. But what if there were no producers? Or better yet, what if you had to be one of the producers?
In this project, you'll get a little taste of what it's like to be a software developer. It's a job that takes creativity, mathematical skill, and communication. You'll get to design your own software platform that can reflect a figure across a line. We'll start by making sure you have the necessary math knowledge to get the project done. Then we'll get to work on the actual product. So here we go...time to get into production mode. Get ready to produce.
For this project, I choose homogeneous groups of 3 to 4 students each. I begin by handing students the Cooperative Activity_Reflecting a Figure Across a Line resource. In this section, we address the first problem on the handout. I let students know what the deliverables are: each student must produce an organized, communicative work flow showing the process they used to determine the coordinates for the vertices of the reflected image.
In the previous lesson, I provided a worked example of a problem similar to problem 1 and students should have this worked example recorded to use as a reference. So this will be a time for them to refresh memory, make sense of the worked example, clear up confusion, and transfer what they have learned to this new problem. I am careful not to give any pointers on this new problem. I am happy to go back and explain aspects of the worked example for students who really need it, but the burden for solving the new problem rests squarely on each group's shoulders. I don't get involved with solving the new problem.
I do get involved with the groups as they solve the new problem, though. My role is to spark up conversations between students. I get them to bounce ideas off of one another, and I urge them to be critical of each other's arguments (MP3). Typically I get (from one student in the group who is bypassing the rest of the group) "Is this right?". I say (gesturing to invite the rest of the group to participate in the conversation that's about to take place), "Well, explain your thought process and show us what you did." After the student has laid out his/her basic argument, I ask the other group members, "Do you understand what (s)he did?...Do you agree with the reasoning or is there anything you'd like to question/challenge?" And so it continues in this fashion, but I don't give the final verdict on anything. I let the groups work it out.
Although some students may not finish by the deadline, I still emphasize and call attention to the deadline so that students are conscious of it and work to their full capacity trying to meet it. When time is up, though, we do move to the next phase of the lesson for groups that are ready. The other groups will remain working on this first part.
This part of the lesson begins in the classroom and is completed in the computer lab or on students' home computers.
I start by introducing the task. Then I do a brief refresher on how to create formulas in MS Excel. I model creating a spreadsheet to determine ordered pairs for the function y=2/3 x + 8.
Next I specify what the current deliverable is: Each student must show the preliminary math work that will inform the excel spreadsheet formulas. When they have done that, they will be ready to get onto the computer and create their actual spreadsheets.
So what is that preliminary math work? Problem 2 requires students to develop formulas for the x- and y-coordinates of an arbitrary point (x,y) after reflection across y=½x−4. Students have seen a worked example for a similar problem. The key is for them to recognize that this is what the problem is calling for.
So without giving it away, I ask questions like: "What is the essential difference between problem 1 and problem 2?" The answer, of course, is that problem 1 involves reflecting points with known coordinates and problem 2 deals with points whose coordinates are not yet known. The spreadsheet they are creating must be able to handle arbitrary points. This is a connection students need to make on their own. Lots of tongue biting here. Or I ask things like, "What points are we reflecting in problem 2?"
When the first wave of students are ready to go to the lab, I take the whole class (even better if you have a laptop you can hand to each group). Some groups will still be working on the preliminary math piece. Other groups will still be working on problem 1. I don't want to hold any group back, though, and I also want to take advantage of the opportunity for students to see their peers advancing through the project successfully. Hopefully it's motivating.