## Checked baggage

Last week, James Cleveland (@jacehan) shared this:

It is weird. You would think the size limit would be volume, not combined length, right?

The first question that came to my mind was “What are the dimensions of the bag with the greatest volume?”

A “cubey” bag with a length and width of 21 inches and height of 20 inches would have a volume of 8820 cubic inches, or 5.1 cubic feet. The airlines are banking on your bag looking more like the one pictured above. The dimensions are not shown, so let’s assume the golden ratio is at play here:

w + l + h = 62
w + w(1.618) + w(1.618Â²) = 62
w(1Â âˆ’ 1.618Â³)/(1Â â€“ 1.618) = 62
w = 11.84
w = 12 in, l = 19 in, h = 31 in

A “golden” bag would have a volume of 7068 cubic inches, or 4.1 cubic feet. If passengers were able to check a “cubey” bag, they’d be able to pack about 25% more. Of course, the airlines would still get ’em with the weight limit.

I guess it does make sense to express the size limit in inches rather than inches cubed. After all, a bag with a length and width of 1 inch and height of 7068 inches would also have a volume of 7068 cubic inches.

Math teachers have seen this type of problem before, but never like this. We’ve seen farmers with x feet of fencing faced with the challenge of enclosing the largest possible pig pen. In later grades, we insist that the farmer use the exterior of the barn as one side. Length is given and area is maximized. This can be reversed. That is, given the size of the pen, our farmer must use the least amount of fencing.

We’ve seen problems in which surface area is given and volume is maximized (like the popcorn box problemÂ or theÂ rolling paper into cylinders thing). Again, this can be reversed. Timon’s Piccini’s pop box design taskÂ is in this family.

The checked baggage problem, on the other hand, jumps a dimension. We’ve never seen problems in which length is given and volume is maximized. I wonder if this opens up some interesting possibilities.

## Building Capacity

This week, we spent one day with 15 math teams (almost 50 teachers and administrators) from 15 elementary schools in Surrey. (I’ll blog about this project soon.) Part of this day was Â devoted to having teachers work together to solve problems. These problems help set the stage for some of the important themes schools will be exploring by participating in this project over three years.Â These include:

• conceptual understanding
• concrete, pictorial, and symbolic representations
• use of manipulatives
• communication
• connections between mathematical ideas
• learning and teachingÂ through problem-solving
• multiple solutions
• reasoning
• attitudes and self-confidence

We gave the following problem, from Figure This!:

Take two identical sheets of paper (8Â½ inches by 11 inches). Roll one sheet into a short cylinder and the other into a tall cylinder. Does one hold more than the other?

A common misconception is that the two cylinders hold the same because the two pieces of paper are the same size. Teachers use a variety of strategies to explore the relationship between surface area and volume.

The first approach most teams take is to calculate the volume of each cylinder. They’ll ask for (or google!) formulas. Even after determining the volume of each cylinder, many will remain unconvinced. Prompted with “How might young children solve this?” teachers will fill each cylinder with manipulatives available at their tables and compare the results, similar to the third act of Dan Meyer’s Popcorn PickerÂ or John Scammell’s Surface Area vs. Volume.

One of the things that I enjoy most about posing these problems to teachers is that each time someone will come up with a solution that I haven’t seen before. For example, this week one team solved this problem by solving a simpler problem. That is, they compared rectangular prisms. (Is ‘squarular prism’ a thing? It should be.)

They argued that the conclusion would be the same but the calculations would be easier. They found a way around the formulas V = Ï€rÂ²h and C = 2Ï€r. True problem-solving!

Part of me geeks out at seeing innovative solutions. The other part of me kicks himself for not making this a bigger part of my own classroom. A lot of missed opportunitiesâ€“ maybe one day I’ll get a do-over.