In this section, my goal is to front load some important concepts that students will need in order to complete the dilation tasks which are the focus of this lesson. I will be leading students through Perimeter and Area Ratios of Similar Figures. There is a heavy emphasis on structural analysis in this activity, so I am careful to model this approach for students.
See the following screencast for specific examples of how I model structural analysis.
For the rest of the answers to the activity, see Perimeter and Area Ratios TEACHER
Students will be working in pairs, so as we begin this activity I have them move next to their A-B partners. Once students have paired up, I will hand out Dilation Tasks. I explain to students that they will be using what they've learned in previous lessons (Experimenting With Dilations and Verifying Properties of Dilations) and today's lesson to solve problems and complete tasks. I explain that I will be walking around to check on their progress and understanding. I also advise students to take care to show their work precisely and neatly as they may be coming to the document camera to present their work later.
As students are working on Dilation Tasks, I walk around the room looking for students who appear to be stuck. Below I highlight some of the moments during the lesson when I expect students to struggle, and perhaps require my assistance.
On Problem 1 my students tend to have trouble getting started. I typically ask these students to describe as precisely as they can where D' would have to be. I wait for them to in some way communicate that D' must be on ray AD and twice as far from A as D is. Once we've established this, I ask direct them to think about how they might achieve it using a compass and straightedge. Then I try to walk away before I give any definitive indication or confirmation that their approach or solution is correct. I want the students to feel responsible for doing the problem and believing that their approach is correct.
When they reach Problem 2a, my students tend to struggle with the fact that the center of dilation is not one of the labeled points. They are expected to determine the location of this point and label it themselves. So when students ask, for example, "Is one of these points the center of dilation?" I typically respond, "There is only one center of dilation and one place it could be; Where is that?"
With respect to this problem, my students also struggle with what it means to define a dilation completely. For that, I ask them, "What two things do we need to specify to define a dilation (center and scale factor)". They should recall this from a previous lesson.
Problem 2b challenges students to figure out that the shaded area is the difference of the triangle areas. For these students, I point out that the area of triangle FGH is given then ask what additional information we would need in order to find the area of the shaded region. Other students are not sure how to find the area of the smaller triangle. For these students, I ask how they might use what they learned in the previous section of this lesson in order to find the area of the smaller triangle.
Although Problems 3 and 4 challenge my students, by the time that they reach this point in the activity my intent is for them to work relatively independently. For this reason, my plan is to allow students to grapple with these on their own (MP1) and keep my input to a minimum. If anything, I might say "In what ways are these tasks similar to those you already completed in #1 and #2?"
During the previous section, I would have been walking around reviewing student work and looking for exemplars: work samples that are correct, use precise labeling and terminology, and show work neatly. I also look for novel approaches to a problem. I will now call on several students to present.
today, however, even though I have pre-selected exemplars, I tell all of the students that they will have 5-7 minutes to review their work and mentally rehearse what they might say if they are randomly called up to present. I tell them also to make sure their work is legible and clear. This serves as an opportunity for students to reflect on their work and thinking and also as a quality control measure.
After 5-7 minutes I will begin to call up the pre-selected students to share their work. Here are some prompts and probing questions I might employ during the presentations:
At the end of each presentation, students can ask the presenter questions?
The purpose of this section is to make sure that all students have completed and understood the tasks, and that they have made a quality work product. For up to the first five minutes, as needed, I give students the opportunity to revise their work after the presentations. I also remind students that they should have quality explanations where required. I let them know at this time that others will see their work so that they will put extra care into making it good.
When students have had enough time to review and revise their own work, I ask them to exchange papers with another student. I ask each student to read over all four problems and for each problem to think of one thing they like about the students' work sample and one constructive criticism they could offer.
Then it's time to collect the papers. In terms of checking for understanding, there are problems on the Unit Assessment that are similar to the problems in this activity. I explain this to students when I pass the papers back with feedback. I advise them to study the problems in preparation for the test.