Our little baby's all grows up!
A couple weeks ago, it was (unexpectedly) senior skip day at my school. I found myself with no more than a half dozen kids in class and was torn about whether to actually teach/review something or do something more “fun.” (It’s sad that I, too, separate the two too often.)
This was my opportunity to try out something I’ve seen all over blogs and have wanted to do, but never got around to it. It’s a classic story in math circles and that’s all I told on that day (not having more ready), but I thought I’d update it with more modern visuals. You will already recognize the story by having glanced at the pictures below, but many students have not heard this yet by the time I see them in their lives, so it will still be novel to them, but the questions may venture in different directions.
I think a “pure WCYDWT” would mean just presenting the pictures. Of course, this one comes with a cute and somewhat compelling story (especially if you include the “ending” that the emperor of India realized what was going on by about the 20th square and executed the man--these age kids love death), so you should gauge your own class as to whether you tell the story or not. I think the lesson will go similarly either way.
If you’re not tied to the story, though, doing this with pennies would be another interesting way to go (especially if you don’t mind getting a little political and talking about our national debt and other large money numbers). There is much less variability in the measurements and the visuals could be created more easily (and is more familiar to more students than rice). There are more extensions to other forms of currency (with pictures to be found on the internet fairly easily) and they all stack fairly nicely.
Here is the opening picture:
And some other angles:
Some other useful pictures that can be revealed if questions take such turns:
Somewhat political in nature, but good visualizations of rice in piles
I did it in my two standard precalculus classes and they each went in slightly different directions with it. In one class we got to the large scale by using the weight of a grain of rice (see the bottom of this page), then envisioning a 10lb bag of rice as a backpack. In the other class they didn’t know what a 10lb bag looked like (and google images is blocked at school), so we happened to find that the average length of a grain of rice was about 7mm and then just assumed it was a sphere (clearly too big, but we justified it by saying it puffs up when cooked and that might account for some of it and we truncated most of our decimal answers).
After we got to the large scale, the class who used the weight was more interested in the ability to produce the rice. The second class was more interested in how to consume the rice.
Our district is VERY strict about internet censorship, so my computer is the only one in the room that can access the internet (the kids could use their phones, but that’s also against school policy). Even my computer is blocked from such useful resources as google images and wikipedia, so we had to make do with what we had. So, I looked up what I could on my computer, but this would’ve been much more fun and interesting in smaller groups of students or if each student had his own computer to pursue his own questions.
By the nature of the problem I couldn’t get a picture of “the answer” so that’s perhaps somewhat unsatisfying. My students trusted the basics of the math, though, and I didn’t have any problems with this in class.
The questions I expected (and they asked)::
- How much rice is already there?
- How many are on the last square?
- How much rice would it take to finish the board?
- How long did it take you to make that?
- How much room would that take up?
- What’s the volume of a grain of rice?
- How big is this room? (counted 1’x1’ floor tiles and cinder blocks on walls)
- How big is the school? (some tennis athletes in the room said it was 0.7mi to run around, then we estimated school has a square floor plan)
- How big is Tennessee?
- Along the line of thinking, “Could we grow that much rice on the planet?”
- How does rice grow?
- How many grains does a single plant produce?
- How much room does a plant need?
- They grow in water? Can we look up a picture?
- How much water is in a “paddy?”
- Where can rice grow?
- How much rice does China already grow? How much in the US?
- How does rice grow?
- Along the line of thinking, “What would the guy do with it?”
- How much rice do people eat per year on the planet already?
- How do we already store rice?
- How big is this guy’s chess board?
- How long would it take them to count them all out?
- How much bigger is a cooked grain than a regular one? (my wife found this article)
- Along the line of thinking, “Let’s go on a tangent trip!”
- Do birds really explode if they eat uncooked rice?
- If I make some rice, can we have a party? Can we order Chinese food?!
Some of my own questions brought up by theirs:
- Does rice dumped in a pile always form a similar cone? If so, what are the relative dimensions? (thinking about calculus related rates problems here where that is often a given in the setup)
- I know there are experiments that do things like “rice avalanches” to simulate landslides and such. I should look into this for fun tangent ideas to this.
- If I wanted to get the exact volume of a rice “ellipsoid,” how could I derive that formula? (And would it be accessible to precalculus students)
- Obvious packing problems.
Other interesting things:
- When I asked for estimations on how much rice there would be in total, all students thought it would fit within our school and most thought it would fit in our classroom. So, the end result was both interesting and shocking.
- Many students were bothered by the “estimation” parts of this and wanted a single, correct answer as in their entire math career to this point.
- While looking up the length of a grain of rice one of the first hits was for a math “word problem” that said something like, “If the length of a grain of rice is 1mm....” I told the class, “Well, this is coming from a word problem, so I’m not sure if we can trust this. Let’s keep looking.”
- Most students wish we could do this every day. It was interesting and they learned! The sample size for my trial, though, was limited to those students who actually come to school on senior skip day, so this lesson may go differently in another classroom.
- At some point in the year I like to talk about what you can do with a math degree and what math research looks like. I could tie the Knight’s Tour in to this problem.
- My other math-chess related story is about how Alan Turing wrote a computer program to play.
- I did my undergraduate studies at Rice University, so this was particularly fun for me!