Yesterday’s opener question:
Count the circles.
Several students tallied the left half of the pyramid, doubled it, and then added the middle column. One student not only counted the circles one-by-freaking-one but kept a current tally inside each circle.
There are 324.
He was somewhere in the low hundreds when I drew his attention to the numbers at the end of each row: 1, 4, 9, 16 ….
“What do you notice? How can we use that to save ourselves time?”
The tedium of busy work can motivate student invention.
pcarduneFebruary 4, 2010 - 8:51 pm -
I remember being given a whole bunch of quadratic equations to factor. I programmed my TI-83 to figure out the factors for me. My teacher did not appreciate this much. Would you have appreciated that sort of solution? Busy work is only legitimate when “cheating” around the busy work is encouraged.
Dan MeyerFebruary 4, 2010 - 9:25 pm -
Definitely. Screw that guy.
GilbertFebruary 5, 2010 - 12:56 am -
This exercise reminds me of some figures from a book I have called ‘sacred mathematics’ written about edo period Japanese temple geometry. Apparently these sorts of “here is a pile of barrels. How many are?” problems show up quite often in older (pre-edo and Chinese) mathematics texts. One very cool variant they have is a trapezoidal stack of barrels, and pictures showing the barrels stacked say 3 deep. They also have cool 3d versions of the problem, with various pyramidal cannonball stacking versions of the problem. Perhaps also interesting from a pedagogical point of view is that these texts routinely shirk algebraic statements and questions in favor of specific quantities or pictures, often sufficiently large to make their techniques actually seem like an attractive alternative to brute forcing a solution.
I’m just saying, you’re in good company.
Jackie BallariniFebruary 5, 2010 - 3:45 am -
I think it is only “busy work” if the teacher doesn’t ask the “what do you notice” and “how can you use … ” questions.
Matt EFebruary 5, 2010 - 3:47 am -
The tedium of busy work can motivate student invention.
YES. (I say the same thing about wrenching their calculators from their hands.)
I trace my interest in math back to 5th grade, when our teacher presented us with a challenge: a 10×10 square grid, and the question, “How many squares are there?” After the initial realization that the answer was in the realm of “Whoa, like, a LOT”, I started counting, and realized I could break my counting up into smaller groups of squares, notice and use patterns, and get the right answer. What had seemed next to impossible mere minutes ago, I had suddenly made short work of. THAT kind of power will get you hooked.
Matt EFebruary 5, 2010 - 3:53 am -
I’m torn about the “Program my calculator to do it” solution, though. While, yes, it does require some solid understanding to be able to write such a program successfully, the program does not care about tedium. “Brute force” will always work, but it requires the least from of the programmer. If we want to talk about efficiency, though… well, I feel as though that is a) worthwhile, but b) a topic for a Comp Sci class.
CraigFebruary 5, 2010 - 3:59 am -
Why is it that students always have a hard time recognizing perfect squares? The problem comes to mind as well.
CraigFebruary 5, 2010 - 4:00 am -
I should proofread more often. I meant to say the Locker Problem comes to mind.
KevinFebruary 5, 2010 - 6:24 am -
On the subject of the programming a calculator, I probably agree that doing 30 questions when you know how to do them is tedious, but I would have asked for the student to explain the program and how could he take that further.
KevinFebruary 5, 2010 - 6:47 am -
My appologies, that is a very valid point, there I was considering the elegant way you could program something to actually factorise a quadratic when in actuallity just using the formula would get the same result.
JohnFebruary 5, 2010 - 2:16 pm -
I agree with the comment above, teaching students to see “the trick” to this is indeed the issue. Not this particular problem but all like it.
I have worked with people that will just start counting. You come back 3 days later and they tell you it will be 10 more days. A good engineer will either find the pattern, write a program or make a machine to solve the problem.
One last note though, there are others that will always “automate” the process even for super simple issues. This is taking it to the other extreme. Common sense is really hard to teach.
sam bFebruary 6, 2010 - 4:21 pm -
gonna give this to my 14/15 yr olds (in britain) on monday and see what they make of it. i really really like this because there are two methods to solve it (one which you haven’t mentioned) and i will be giving it to them on a print out…
1) algebraic – notice sums of rows total square numbers – see 18 rows and hence answer is 18^2
2) geometric – cut out one of the 17 high right angled triangles on the side (the ones your students tried to double) and stick it onto the other 18 triangle to make an 18×18 square
any other methods? i love the link between algebraic result and shape here and it’s going to be the perfect intro to the algebra work we’re about to start
phansonFebruary 6, 2010 - 9:09 pm -
My 5th graders would love something like this. Why? Because of the different ways it can be solved. If you aren’t confident mathematically, you count the circles. But you realize (quickly) that there needs to be a better way. They are savvy enough to notice the number pattern, and could solve it that way as well. Personally I solved it geometrically, seeing the 2 right triangles and simply multiplying LxW (or squaring 18).
It certainly illustrates how math problems can be solved in a myriad of ways… and that knowing your math facts can save you some time (I’m picturing some counting the circles, as that is where they’re at).
danielFebruary 7, 2010 - 3:44 pm -
Not directly related to this example, but shouldn’t some math operations become ‘automatic’ so that we can focus on the deeper issues? i.e. I don’t want to be thinking (or worrying) about adding, subtracting, multiplying, and dividing integers when I’m learning to solve systems of equations. In my experience, students who are not solid in the fundamentals quickly become overwhelmed when faced with more abstract concepts. It would seem that ‘busy’ work has its place.
TouzelFebruary 23, 2010 - 9:41 pm -
I counted the rows a little differently and came up with a pattern I’ve never used before in problems like this.
I expressed each row, not as the total number of circles on that row like:
1 + 2 + 3 + … + 17 + 18
But rather as a sum of two numbers, the first being the row number and the second being whatever is leftover:
(1 + 0) + (2 + 1) + (3 + 2) + … + (17 + 16) + (18 + 17)
This large sum can be re-written into two separate sums: the sum of the first number of each group and the sum of the second. The former is simply a summation of the row number, and the latter is almost identical, missing only number from the previous sum (e.g. the second sum would go up to 17, but not 18. Obviously, the sum of the two summations works out to be a perfect square.
This seems obvious to me now, but, for some reason, seemed unique five minutes ago.