my usual judge of the difficulty of a problem is the time it takes to figure out. I spent about 5 minutes on the problem so that would mean my average student would need about 15 minutes to get a decent estimate.

For me this problem was nothing more than triangles and rectangles. When I imagined the water rising above the bottom rectangle I saw another triangle but didn’t see a simple way to measure the height. (mainly because I wasn’t letting it shift in my minds eye). I was thinking at first the exact solution would requite some messing around with tangents and not having time I dropped the matter.

But when I do let the triangle shift I see it becomes a triangle with a base of the rectangle and a height measured from the highest point straight down to the imaginary rectangle underneath. And we are back to a simple area problem.

In the end one of those not very difficult problems as long as you can get your head out of the box so to speak.

A class may well need some guidance and hints to solve the original problem. But, isn’t it a win if we sacrifice a little of the “discovery” approach in exchange for students working at a higher cognitive level — higher than seeing how many cones can be poured into a cylinder for example.

I do like the “filling up” exercises you mentioned for building intuition on volume. An extension to breaking area up into familiar pieces might be something like looking at the volume of an office building whose top is slanted like a cheese wedge. Or, estimating the cone volume by slicing it up into small pieces, each of which is nearly a cylinder.

No I think this is a great problem for first year algebra students. However, if the water level is greater than the area of the side rectangle then the exact area would be more difficult to prove making it beyond the patience of a normal 8th or 9th grade student. Most but not all.

When we find the volume of cones and such basically we build a cylinder around it and find the volume of that and see how many cones it takes to fill it up. It is an extension of the way we learned area, by breaking the area into smaller parts we know or drawing boxes around and subtracting the bits we know.

How to set this problem up do students “discover” it on their own is a bit more involved.

Also, why do you feel this is a better problem for pre-calc? The crux of the “area” problem is easily and clearly stated. A 6th grade class could approach it through guess and check — at the very least you provide a reason to practice otherwise boring computations. For an algebra class, this is a practical problem, where, for once, algebra actually is a good way to approach the problem. The vast majority of students are capable of thinking about complicated problems like this one — they have just been trained not to.

Finding the exact answer for that would rise above the normal 8th or 9th grade Algebra class student. It would probably be a nice Trig/PreCalc problem if you went that far. I’m not sure most students would want too unless you changed the problem to rocket fuel or something.

But if the liquid rose above the rectangle I would then have simply used the remaining area with the formula for a triangle.

Yeah, this is key, ain’t it? I’m realizing right now that a student could probably divide the area by the base of the rectangle, get a height that’s within the right range of the problem, and write it down confidently. That’s going to be a tricky conversation to manage.

Area: the first area is a trapezoid. I can never remember the formula so I break it into a rectangle and triangle, but you don’t have to do that. When I tilt I see that the bottom part is a rectangle so I take my original area and set it as the answer to the area of rectangle formula. I know the base and solve for the height.

In my case with my estimations (My eyes had trouble counting the tiny squares) that was that. But if the liquid rose above the rectangle I would then have simply used the remaining area with the formula for a triangle.

This is a great questions no only for area but also for estimation. While finding exact numbers would add some complexity for students who finish early.

Dan- Can you say more about this…”students can work on more concrete tasks using more concrete representations, then abstract tasks using more abstract representations.”

This will be the subject of a longer series this summer so I’ll keep my powder a little bit dry. In short, I find it difficult to ask students concrete questions about an abstract representation. ie. It’s hard to ask students to guess, predict, and debate a context when its already been distilled into a graph, a table, and an equation.

Likewise, it’s downright impossible to ask students to generalize for a certain variable of a context without them having abstracted the context first. ie. If I’m just staring at a concrete photo of a basketball, I can’t answer the question, “How much air will fill basketballs of different sizes?” without abstracting the context first, turning the whole thing into variables and expressions.

Paper is the problem. It forces the designer to choose a certain level of abstraction for a task whether or not that level matches the questions being asked. Multimedia lets me change the representation as we change the level of abstraction.