## 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.

“I couldn’t help but admire your large triangular prism,” I wrote. Sadly, this is not the strangest way I have begun an email to a colleague.

“Are you talking about the giant Toblerone-shaped thing? You math guys are weird,” she replied.

Anywayâ€¦ my three-act math task:

act one

• About how many regular size Toblerone chocolate bars fit inside the giant Toblerone-shaped thing?
• Give an answer that’s too big.
• Give an answer that’s too small.

act two

• What information would be useful to know?

act three

63. Relax. The video is coming soon.

sequel

• If 72 regular size Toblerone chocolate bars fit inside a mega Toblerone-shaped thing, how large would it be?
• If 112 regular size Toblerone chocolate bars fit inside a mega Toblerone-shaped thing, how large would it be?

better stillâ€¦

• AÂ mega Toblerone-shaped thing is a little bigger than a giant Toblerone-shaped thing.Â What could its dimensions be?
• How many regular size Toblerone chocolate bars would fit inside?

I like the phrase “a little bigger.” Probably “borrowed” from Marian Small. The ambiguity here allows for multiple solutions. Students could increase the length of the prism or the size of the triangle base. Which has the greater effect?

Also, there’s something interesting happening here with the sum of consecutive odd numbers.

Oh yeahâ€¦ a shout-out goes to Andrew Stadel for his Couch CoinsÂ task.

## Don’t mean to burst your bubble

via Colossal, an art, design, and photography blog:

While waiting for a train, commuters can help themselves to square sheets of bubble wrap labelled with how long it would take to pop them.

I love this idea. The world is a better place because of it. I hestitate to bring this up, but â€¦

the math is wrong.

It looks like the approximate times are based on length.Â Above, the ratio of side lengthsÂ is 3 to 5 to 10, or 1 to 1.67 to 3.33. Let’s assume that the small sheet does, in fact, take 3 minutes to pop, one bubble at a time. The large sheet does not have 3.33 times more bubbles; it has 3.33 times as many rows and 3.33 times as many columns. Therefore, it has 3.33^2, or 11.11, times as many bubbles. A better approximation for the large sheet would be 30 minutes. If we base the approximate times on area, the ratio of sides lengths would be 3 toÂ âˆš(5/3) toÂ âˆš(10/3), or 1 to 1.29 to 1.83, as shown below.

I’m thinking about how I could use this image or idea in class. Some possibilities:

1. As-Is

Display the photos. Ask students, “Are the times accurate?” Have students apply their understanding of the relationship between scale factor and area. M’eh.

2. Hands-On

Display the photos. In pairs, have students record how long it takes to pop a small square sheet of bubble wrap. Pose the problem,Â “A square sheet takes twice as long. What are its dimensions?” Have students test their predictions. In this activity, students develop their understanding between scale factor and area. They poke holes in the common misconception that when dimensions are doubled, area is doubled, too.

3. Three-Act

Play a video of a small square sheet of bubble wrap being popped. Include a timer. Maybe a soundtrack, too. Play the beginning of a video showing a large square sheet of bubble wrap being popped.Â Have students guess how long it will take. Ask, “What information would be useful?” Show the dimensions of the squares.Â Play the answer video.

I see this task being similar to Dan’s Penny Circle. Dan filmed himself filling a circle with 663 pennies so that the rest of us wouldn’t have to. I have a roll of bubble wrap measuring 24″ by 30′. Before I take one for the team and spend a ridiculous amount of time enjoying bubble wrap, any suggestions?

## I see math, people

Last Friday night, while the rest of the world was lining up to see The Avengers, I took my daughters to see The Pirates! Band of Misfits. Our hero, The Pirate Captain, desires to win the Pirate of the Year Award. He explains to his rag-tag crew,Â “Every time I’ve entered I’ve failed to win, so I must have a really good chance this time.” The gambler’s fallacy! In a children’s movie, no less.Â [The gambler’s fallacy is the belief that previous failures indicate an increased probability of success on subsequent attempts. It’s why I renew my (share of) Canucks season tickets every year.]

Fast-forward to Tuesday, lunch. I’m in the line-up to pay for my fish taco when I spot thisÂ¹:

I immediately ask myself, “How many Rice Krispies is that?”Â OtherÂ (more interesting?)Â questions soon follow:

• “What size of Rice Krispie square could you make with these?”
• “How many big marshmallows would I need to make this giant Rice Krispie square?”
• “How many calories would that be?”
• “How many ‘Snap, Crackle, Pops’ could I expect from 22 lbs of Rice Krispies?”

These sequels/extentensions offer more than “How many Rice Krispies are there in 50 kg?” In addition to proportional reasoning, there are connections to volume and probability.

I’ll upload this to 101questions. I’m curious if other math teachers will find it perplexing but that’s not really what’s important to me. What is important is that I’m starting to see math everywhere.

I blame Dan Meyer.

Â¹ From ages 15 to 24 I worked as a grocery clerk at Safeway. Sometime during a new hiree’s first shift, we’d ask him to run and do a price check on some seemingly mythical item such as pork wings, ice mix, or a 20 kg bag of puffed wheat. Huh. Who knew?