by Dan Meyer
June 22, 2014
What is the cheapest way to package up these candies?
- ImageAll Four Packages
Rank these packages in order of which uses most cardboard to least.
Rank these packages in order of which uses most ribbon to least.
How many candies do you think we're trying to package? How many do you think you saw?
What do you think are the dimensions of each of those four packages are?
- Teacher noteAfter they've guessed, tell them that there are 24 candies. Rewind the original video and let them gather the dimensions of each package from the video.
- Teacher noteNow help them /calculate/ the volume and surface area of each package. The formula for surface area is straightforward: 2wl + 2wh + 2hl. The formula for ribbon length is trickier: 4h + 2l + 2w.
- Teacher noteOnce they determine the /actual/ ranking of the packages by cardboard and ribbon, ask them to make sure there isn't /another/ option that's even better.
- FileDandy Candy Packaging Options
- Teacher noteA question I have: is it always going to be a guarantee that one option will be the best for both ribbon /and/ cardboard? If we were packaging a different number of candies, might there be an option that was best for cardboard while another option was best for ribbon? I don't know the answer.
- Teacher noteHere's an unverified strategy:
1. Take the number of units you're packaging.
2. Take the cube root of that number.
3. Decrease that number until you have a factor of the number of units you're packaging.
4. Divide that number into the total.
5. Now take the square root of this new number.
6. Again decrease the number until you have a factor of the number of units.
7. Divide that number into the remainder. Now you have the third factor.
8. The smallest factor should be the height of the box.
Is there a better option than the four we originally selected? Can you prove it is the best?
Does the way we position the box of candies matter? Up, down, left, right, top, bottom?
Pick other numbers of candies. 80 candies, for instance! What's the best way to package them? Can you figure it out faster this time?