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Wednesday, November 30, 2011

Curriculum Writing for the Reluctant

I am really trying to beef up my Area, Surface Area and Volume unit for Geometry this year. It gets the job done regents-exam-wise, but it is so dissatisfying and I feel it could be so much better. Overall it basically boils down to plugging things into formula-sheet-provided formulas, and isolating variables in formula-sheet-provided formulas. There are some good things in there... we find composite areas and perimeters using aerial and other images, for example. Finding the areas of regular polygons is a good application of right triangle trig. There is an investigation of how areas change when dimensions change, which is serviceable but I suspect kids don't really see the big picture. We "do" volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. My students tend to do very well on questions from this unit on the Regents exam, and I don't want to mess that up, but in this case I don't believe that the exam is valid for measuring understanding.

These are the kinds of things I want them to understand and/or be able to do:
  • what physical property you are actually calculating when you calculate a volume or a surface area
  • why the formulas are what they are
  • how changing a 2D or 3D figure's dimensions affects its area or volume. for example, I think they understand that if you order a pizza that has twice the diameter, you get way more than double the amount of pizza. But I don't think that intuition has any ties to math class.
  • isolate a variable in a formula. for example, solve S = lw + wh + lh for w.
I have a bunch of great resources and problems and tasks that I have collected in my Evernote over the past few years that could potentially work very nicely here.

1. Design a new label for a given tennis ball canister, oatmeal canister, or soda can. (a) Create a prototype label so that it covers the entire lateral surface of the canister with little to no overlapping paper. (b) Congratulations! The company chose your design and wants to produce 100,000 labels. Calculate how much material (paper, aluminum, whatever) you will need to order.

2. This game at NLVM is quite nice for challenging your intuition about how volumes are related to dimensions.

3. This video features people with charming accents complaining about how the volume of their chocolate bar decreased even though it appears that the surface area stayed the same or possibly increased. I've shown this in the past and found that students are unable to articulate what these people are upset about using the word "volume" (much less intelligently discuss surface area.) The word "volume" from math class is not connected in their brains to "how much stuff inside."

4. Starting with a piece of copier paper, roll it into a cylinder both the long way and the short way. Will it contain the same amount either way? If not, which way holds more? Mathematically justify your response.

5. Starting with a sheet of copier paper, cut four congruent squares out of the corners and fold up the sides to make a box. Who can make the box that holds the most? Kristen Fouss did something like this but in pre-calculus. Geometry probably doesn't need to get into deriving and optimizing a polynomial equation.

6. Starting with a sheet of copier paper, design, cut out, and assemble a right pyramid with a square base. First pass: any pyramid will do. Second pass: make the area of the square exactly ___. Third pass: make the overall height of the pyramid a specified length. Present your best-looking pyramid, including the area of its base, its overall height, its lateral surface area, its total surface area, and its volume.

7. Investigate what happens to area when dimensions change. What happens to volume when dimensions change. (Somehow.)

8. The car talk fuel-tank problem.

9. Some version of the PCMI volume/surface area problems. (If you know the perimeter and area of a rectangle, can you determine its dimensions? Are there any rectangles whose perimeter = area? If you know the surface area and volume of a rectangular prism, can you determine its dimensions? Are there any rectangular prisms whose volume = surface area?)

10. Derive the formula for the volume of a sphere without calculus. From Exeter Book 3. Would pose quite a challenge for my students. They would not be able to do it on their own. In fact, as it is written, it would completely mystify them.

What I am struggling with and probably will be for the next week or so is, how do I take any of these things and fit them into a logical, coherent unit of study of surface area and volume? NY/my district/my school does not provide us with a curriculum. We have : a list of standards, a collection of previous exams, a pacing calendar, and a kind-of crappy textbook, which are all useful in their own limited ways, but none of them tells you what to do in class. I have lessons already written that get it done, so there is no incentive to bother, other than it bothers me when I feel I could be doing a better job. Part of the dilemma is, I feel that any of this would have to be added to what I already do, not replace it. I still need them to be able to, for example, identify that the bases of a prism are the parallel sides, even if they are not on the top and the bottom. And I'm already about two weeks behind in this course.

How do you take a compelling resource and turn it into an effective lesson?


  1. Kate, this doesn't seem to go quite along with some of your objectives, but last year I had a fantastic time doing surface area and volume of composite solids with my 9th-graders! They designed their own 3-D solids (of concave and convex parts and prisms and pyramids) and drew 2-D nets and made calculations of surface areas and volumes before they went ahead and constructed them:

    In the end, they were able to do quite complex calculations on the exams (including needing to calculate slant height and pyramid height and other things in order to find volume and SA). This is a lovely project to do if you have time!

  2. I have my students build robots out of construction paper, and figure out surface area/volume. They are required to use X of each type of solid.

    It's a time burner (so "already two weeks behind" probably disqualifies this) but lots of fun.

  3. In response to: "Investigate what happens to area when dimensions change. What happens to volume when dimensions change. (Somehow.)"

    With respect to volume, how about what happens to the dimensions when the volume changes by looking at 25%/33%/50% extra free cartons of orange juice, the same applied to Pringles cans and so on?

  4. @Mimi those look very cool. Do you have any written materials for the project you would share with me?

  5. I got these for my son for his birthday:

    I fell in love with them and got a grant for 5 sets for my classes. You can make some really wonderful prisms and pyramids with them and I love how you can see through the tiles and 'open' a door (since they are all magnets, they're easy to open) and really dissect what's happening with height and other features.

  6. Hey Kate. I know exactly how you feel. In Canadia we don't have a "geometry" course, but in grade eight we start surface area and volume of prisms and cylinders. Kids don't get it at all (no fault of their own I feel). I have one #wcydwt that I have prepared that I intend to use, but I would love to have a great teacher like yourself take a gander and try it out.

    Another take is from one of our teachers at the high school connected to my middle school. He has the kids design and addition to the school using google sketch up. He just lets them go for it and uses socratic dialogue to talk about why we need to know the volume and SA of the additions.

    I look forward to stealing whatever you end up posting!

  7. There are superb ideas here. I think they could be combined into lessons and a whole unit with the help of your teacher talk and whole class discussion with the students. What about if you tried to do some of the plugging-into-formulas into lesson starters or enders just to satisfy yourself that they have the exam skills. Then use the discussion times to ask (or guide) students to make connections between the tasks and the more traditional questions. Does this make sense?

  8. These sounds like great ideas. I think that you have enough here for a good unit of work. Collect and sequence these ideas into lessons and draw out core ideas in whole class discussion. What about reducing the amount of plugging-into-formulas and moving it to lesson starters and enders; just to convince yourself they have test skills? And then in the discussion times ask (or guide) the students to make connections between the tasks and the formulas. Does this make sense?

  9. Here is a handwavy way to get the volume of a sphere from knowing the surface area:

    Imagine cutting up the sphere into infinitely many "square" pyramids with the vertex at the center and the base on the surface of the sphere. Each of these pyramids has volume (1/3)*r*(area of the "square"). If you add them all up, the square areas add up to the surface area of the sphere. So the total volume is (1/3)*r(4*pi*2^2) = (4/3)pir^3.

    Obviously not rigorous, but it is intuitively and introduces calculus-like thinking. It helps to have a styrofoam sphere with a pyramid cut out.

  10. Also, I split volume formulas into three categories:

    "Non-pointy things" (cylinders, prisms): Volume = (Base)*height

    "Pointy things" (cones, pyramids): Volume = (1/3)*(Base)*height

    Sphere: Volume = (4/3)*pi*r^3

  11. You might also like this lesson that LTF (they do teacher training for Pre-AP math and are developing content for Common Core also) has posted on their Common Core page. It is specifically for Geometry classes. When I use it, the only thing I change is that I provide graph grids instead of just the scaled axes that are on the student pages. The student pages are at the end of the pdf. Enjoy!

  12. Kate,

    I think many of us are in similar situations... bits and pieces of excellent materials can be accumulated all over the place, but organizing good resources into coherent and effective units is a monumental task.

    I enjoy the area/volume unit and I can't really help you with an entire unit, but I had a similar need last year in this unit and made a little bit of progress toward some cohesion:

    1. I began the unit with the big question: "Why are things shaped the way they are?" The students were required to identify items with geometric properties and provide a rationale for WHY they thought it was shaped that way. It led to a pretty good discussion about maximization (of area, volume, perimeter, etc.) and minimization (of costs, resources, etc.)

    2. I didn't let them use any formulas that we didn't derive first. They've been using those formulas since the third grade and I think the formulas have developed somewhat of a mystical nature to them. I emphasized counting units, which is only easy to do when right angles are involved, so we did a lot of chopping things up.

    3. I ended the unit with a debate about the existence of giants and elves which I remembered from my own geometry student days. It's a fun way to address the volume/area of similar figures in a way that gets to understanding instead of memorization of formulas. I cannot say I did it well enough last year and will revise next time, but here's the source I started with:

    Thanks for sharing so much, Kate!

  13. Sure. Here are the instructions for the project: .

    The timeline was more or less:

    Day 1 - they come up with a 3-D design, with labeled dimensions, that fits the stated requirements, and we review basic volume and area formulas as a class. They get this checked off by me.

    Day 2 - they draw the 2-D nets and label all sides in the 2-D nets. We learn to apply Pyth Theorem to calculate missing slant heights in Pyramids or (as the groups needed) use proportions to find missing lengths in cones. They get this checked off by me and THEN start surface area calculations.

    Day 3 - they finish surface area calculations, get that checked off by me, and then they start volume calculations. (I had to teach the class first how to find pyramid heights if that was missing.) Some faster groups were able to finish all of their calculations in class; others were asked to come back on their own time to check in with me on their finished calculations.

    Day 4 - the groups did constructions out of cardboards in class; asked to finish the gluing and construction on their own time.

    Day 5 (about a week later) - they bring their glued projects and decorate/paint them in class; polishing their calculations as necessary before turning everything in by the end of class.

    It sounds like a lot of time, but that included all the teaching of the volume, SA skills as well! Like I said, afterwards we still had to do a day or two of concentrated paper practice of problems, but the kids were already very confident with the concepts and formulas by that point.

  14. Mimi! You are the cat's pajamas.


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