Here is what my students saw on my television screen as they walked into the room. With almost any task I do, I give them the needed information and they start in whatever way necessary without my direction. I had the pennies available on my desk. I didn't want them to have too many pennies...I wanted to make some sort of conversion and be able to explain what this accomplished. We haven't gotten into any ratio discussion at this point of the year...so I wanted to get that seed planted as well. I love tasks that teach the mathematical concepts without directly teaching the mathematical concepts.
Here are the Common Core Standards we focused on:
I ended up with many different variations of the answer upon completion. While different, each answer had plenty of mathematical reasoning to be plausible. For example, if a group measured seven pennies across the bottom instead of eight, the answer will be different because there will be 15 fewer stacks of pennies. If one group identified the height of the pennies as 10 per half inch and another group said one is close to 1/16 of an inch, the answers will be quite different. An argument can be made for any of these scenarios, so as the teacher/facilitator, you have to be very quick thinking so you can talk about each variation with the student groups.
The difference in volume reasoning was also exciting to me. I think I only had one student who initially rationalized that if you multiply the length, width, and height of the pennies together, you would end up with the total amount of pennies. Here are a few examples of coming up with this formula organically:
*Finding the amount of pennies on the bottom of the cube, then figuring how high each of those pennies on the bottom would stack up
*Finding the pennies in one row on the bottom, finding how high each of those pennies stack, adding the stacks of one row together, and then adding the amount per row to complete the cube
*Finding the height of a stack first, then figuring out how many pennies would fit on the bottom of the cube, and realizing each penny in the stack would represent a layer, so the cube would have that many layers of how ever many pennies fit on the bottom
...and so on. All of these variations are showing me that my students understand the concept of volume and will be successful later when we expand greatly on this concept.
The second question with the cubic foot was also very interesting. Many students simply did what they did on question one with an expanded digram which was very effective. I let these students continue this way because it was what they thought of...and then I introduced these groups to new ideas through discussion later. The other way to do this was to see how many of the original penny cubes fit into the cubic foot. Not surprisingly, middle school students normally jump to the first idea that pops into their heads...it would fit in twice...it's twice as big. Obviously that is not correct because all the sides are twice as big, but many middle school students don't see this right away. For those groups that did not catch on, I broke out the blocks, and had them draw diagrams with these blocks. This was a nice way for the students to see that if each block represents six inches on all sides, it would take four blocks on the bottom and would stack two blocks tall resulting in eight blocks total.
One word of advice before we get to the student work. If your students aren't the greatest spellers (and apparently mine are not), watch out for some of the variations of the word "pennies". I had some pretty interesting spellings if catch my drift...Enjoy!