Ratios of Similarity and 3D Solids Generated by Revolving 2D Figures
Lesson 13 of 14
Objective: Students will be able to solve volume problems and identify 3-D solids generated by revolving 2-D figures.
At the start of today's lesson I divide the class into four Mastery Groups, which will be responsible for solving one of the four green practice problems. I ask students to show high-quality work on their Green Problem and to check in with me after they have solved.
After I check their work for correctness and quality, I hand each student in the Mastery Group a white piece of paper containing all four practice problems. Each student is then tasked with finding a partner from the other three mastery groups with whom they can engage in reciprocal teaching. In each pair students will take turns teaching their partner how to solve the problem from their Mastery Group.
As the students complete their reciprocal teaching partnerships, each should record his/her own work on their white paper. Once a partner understands a problem being taught, the "teacher" should initial their partner's paper to indicate they have taught the other how to solve this type of volume problem.
I give students time to review their Homework in their groups, circulating the room to clarify confusions or misunderstandings about the problem. Since Problems #1-3 are fairly straightforward, I display the answers and take questions from the class.
Problem #4, however, requires students to have a strong sense of how to visually represent the packaging problem, which is why I ask for a student volunteer to project his/her visual representation of the problem and volume work.
Lastly, I select two students to each present one method for solving Problem #6, which requires students to compare volumes of extremely large sizes and to use ratios of similarity to solve.
Several of the activities throughout this unit have required students to think about the relationship between dimensions of similar solids and their corresponding surface areas and volumes, but we have yet to formalize any “rules” to describe this concept. I ask students to work on this Ratios of Similarity Task in their groups, and to check in with me to explain how they have generalized the ratios of surface areas and volumes of similar solids (MP1, MP3). After groups can explain their rules, I have them move on to the back page, where they apply their rules to solve three practice problems.
To help bring today's lesson to a successful closure I display an image of real-world objects (post-it notes in the shape of an apple, pear, and pumpkin), for which students can determine the 2D objects generate 3D objects when revolved around an axis. I ask students to work in groups to consider:
(a) the 3D solid generated by revolving a 3”x5” index card
(b) to determine whether the 3”x5” index card revolved around the 3” side has the same volume as revolving around the 5” side.
I ask volunteers to report out about their groups’ findings and to defend their answer or explain how their thinking may have changed. We end with a whole-class discussion around identifying the 2D figure that can be revolved to generate a cone and a question:
Would a cone be generated regardless of which side of the figure the figure is revolved about?
Students come up to the whiteboard to justify their ideas with a sketch of possible 3D solids.