I can statements:
Level 1: I can use formulas to calculate area and volume. 100% Strongly Agree or Agree
Level 2: I can identify cross-sections of 3-D objects, and I can identify the 3-D objects formed by rotating 2-D objects. 90% Strongly Agree or Agree
Level 3: I can explain the formulas for area and volume. 100% Strongly Agree or Agree
Level 4: I can calculate the area and volume of geometric objects to solve problems. 100% Strongly Agree or Agree
Geometric Measurement and Dimension G-GMD
Explain volume formulas and use them to solve problems
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Which Standard of Mathematical Practice did you use most often in this unit? In which other Standard of Mathematical Practice did you engage often during the unit?
Students answered that they used look for and make use of structure the most. They used make sense of problems and persevere in solving them and reason abstractly and quantitatively next.
Think back through the lessons. Did you feel that any were repeats of material that you already knew? If so, which parts?
10A Length Area
10B Surface Area
10E Cross Sections
10F The Best Box
10G Performance Assessment
10H Mastering Arc Length and Sectors
- I already knew length and area. I had been taught the formulas for surface area, but I didn’t actually understand why those formulas worked until this unit. Cross sections were completely new to me.
- I felt like I already knew some of the information for calculating volume and area of 3D forms, but in this unit I learned why and how the formulas work.
- The length and area was a bit of a repeat of what we’ve learned in the past. It was nice to review and be able to bring to memory what we’d already learned as a basis for the new things we were going to learn this unit.
- This Volume lesson was a tiny bit of something I already knew, but I never got as much into it as we did. Before, I just learned the formula and never understood. I also only ever learned the volume of a rectangular prism.
- The very first lesson in which we used the basic formulas was a bit of a repeat from past years. However, in the past we had to memorize the formulas, whereas this year we were given the opportunity to understand the why the formula is what it is. I find that understanding it helps more than simply memorizing it.
Think back through the lessons. Was there a lesson or activity that was particularly helpful for you to meet the learning targets for this unit?
- Yes, Length and Area helped me with Surface Area and Volume by making sense of structure within the problem.
- When you got the orange peels and separated them into four circles to show how the surface area of a sphere worked, it really helped me understand how surface area works. After that I started thinking about how the formulas worked instead of just memorizing them.
- 10G Performance Assessment helped me out the most. It summed up most of the chapter and helped my apply what I learned to many different problems.
- I think the lesson of volume helped the most in understanding the formula. When we talked about the different figures fitting into one another it helped me understand why this certain formula was there. It helped show me that all formulas are taken from a common one.
What have you learned during this unit?
- In this unit I have learned how to use surface area, lateral area, volume, and cross sections. I now have a better understanding from where the formulas come from by looking at the figures and in a way putting together the puzzle pieces to come up with the final formulas.
- I learned exactly how area, surface area, and volume are all different.
- I have learned how to calculate lateral area, surface area, and volume of forms and I also learned why the formulas work and not just to plug in numbers to those formulas.
- I have learned how to identify cross sections and explain the formulas for surface area and volume.
- I’ve learned how to use formulas to calculate area and volume and why we use those formulas. I’ve learned how to make sure to convert problems to the same dimensions and same ratios, and I’ve learned what cross sections are and the different ones in different 3D figures.
- I have learned to understand the formulas and not just memorize, but really understand them. I can use these formulas to solve word problems and figure out tough questions. I’ve also learned about cross-sections and how they differ for each 3D shape. I have also understood the surface areas of 3D shapes and how they work with all shapes.
- During this unit, I have come to understand the geometric formulas, and I believe that understanding will help me later on in tests like the ACT if I were to forget the formula that I memorized in 7th grade, I would remember the concepts and understandings of why the formula is what it is.
And so the journey continues … still struggling to find a balance between teaching conceptual understanding of mathematics and using mathematics to solve problems … although I do think the students’ language when reflecting on their understanding of geometric measure and dimension is a good sign.