Assuming you're one of those students who does what he/she is asked to do, how would you respond to this question if it were, say, a homework problem?
Would you respond any differently if I just gave you the picture (in the context of this being part of your math homework)?
Thanks Heather for the suggestion to put this out here front and center.
This is hard to describe but...
ReplyDeleteHow does the number of boxes affect the total distance I can travel around the perimeter & dividers without traveling twice over any lines?
1 box - 4 units
2 boxes - 7 units
3 boxes - 9 units, unless I'm missing something
etc.
Is it predictable? How?
I'm not sure how different the results would be with/without the instructions. What you do when they arrive in class could drastically change the results on future similar assignments, though. What *would* you do?
Are we assuming that the student has been given problems like this before? Are we assuming that she has had discussions about what a "mathematical question" is or what it means to "explore a mathematical question"? Has she been given examples of such things?
ReplyDeleteIt's very hard for me to see how a student without any background or context could approach that assignment in a productive way. After all, when you give a problem like that to someone like me or you, we've seen hundreds of examples of mathematical questions and their exploration; we're not working without context. Who could do such a thing without context?
@ Ms. V: Love it, and I found your problem plenty clear. I'll talk more later about what I envision the "in class" portion to look like.
ReplyDelete@ Max: "It's very hard for me to see how a student without any background or context could approach that assignment in a productive way."
Agreed, and there will be scaffolding put in place next year. Any suggestions for this scaffolding would be welcome. For now, though, I'm just interested in the question(s) YOU would devise.
O via buzz: I think I would be initially frustrated because the question is so open ended and there doesn't appear to be a right answer. I think it can be frustrating to do work which you are uncertain if it is in the right direction because you can feel the work is pointless. Then I think I would do the minimal work to create an "answer" that would satisfy the teacher. Perhaps something like, "What is an equation that would describe the number of squares." I think the word "explore" is too vague.
ReplyDeleteH via buzz: Here are some questions I have (being an English teacher) and having no idea if these count as mathematical or not:
ReplyDelete1. What is the relationship between the top square and the bottom row of squares?
2. Does this pattern repeat until infinity?
3. What is the length of the lines around the outside of each row and how do the lengths around the outsides relate to one another?
4. What's inside the boxes? (I know, that's more a figurative English question.)
5. If I wanted to put all of them in a straight line by flipping them on their side, or telling a computer program to do it for me, how could I figure out how much I would need to increase the distance between the rows by the right amount for each row?
O -- many students like right answers because it affirms their sense that they are "smart" or have done the right thing or pleased the teacher. If, as a teacher, you can starting getting around those motivations to what they actually find interesting or are curious about, they'll have a much more fulfilling and impactful learning experience.
Maybe it is because my head is in Alg 2 sequence mode right now, but I can myself giving the students the picture and a bunch of sequences and asking them how the sequences apply to the picture.
ReplyDeleteFor example:
1) 1, 2, 3, 4, 5,... (number of boxes)
2) 4, 7, 10, 13, 16,...(number of edges..I picture making these out of matcn sticks for some odd reason, so I guess, it is the number of match sticks needed)
3) 1, 3, 6, 10, 15, ... (Adding each successive row...also the Triangular Numbers of Pascale's triangle fame..)
4) 6, 10, 14, 18, 22, ... (Surface area)
And many, many, many more...
Carry it further, to ask whether the sequences are geometric, arithmetic, or neither...
Hi Avery,
ReplyDeleteI'm excited for your blog!
The questions that are coming to mind at the moment are about fitting the boxes inside of each other. At first this isn't terribly interesting, as each box can fit into the one below it in just two ways--shifted to the left and to the right. Generalizing, the number of ways that a box m wide can fit into a box n wide would be (n-m)+1, since there would be n-m empty slots to shift over into, plus the first one all the way to the left.
Then I started wondering about how many ways there would be to pack all of the smaller boxes into the next biggest box up. If they all have to fit inside of one another, there's a left-right choice at each stage (shoving a box into the next largest box), so if I'm putting the first m boxes into the (m+1)st, there would be 2^m ways of doing it. Still thinking about the first m boxes, but without having to put them all inside each other, I'd still be forced to anyway up until the 1-box, which could go, what, anywheres in the (m+1) slots. So that would be (m+1)*2^(m-1) ways of packing them in there.
Things get interesting when instead of taking all of the first m and putting them in the (m+1)st, you only take some of them. The same variation as above could be asked about whether each smaller box must be included in the next larger. Finally, now I'm wondering about stable packings--ones where none of the boxes can shift around when I'm shipping them to you cross-country. Like fitting a one and a two into a three. I guess that'd mean that at each stage there'd have to be a partition...partitions within partitions...and I'd have to decide if I'm allowed at most one box of each size.
Pretty complicated stuff in just a few steps. I can't wait to see what other prompts you're excited about! One that occurred to me after reading your post was a ring of say ten circles (or eight, or whatever) that are connected one to the next by line segments. Like a bracelet.
Heyvery. Of course, because I have toyed with the boxes-in-boxes problem as a course in itself, I'd go with the "how many of each kind of box in any of these" problem. I think Justin's narrative approach is nice and accessible (as well as ultimately more interesting) but I would also start talking about pallets of boxes and get into the second dimension, and questions of orientation.
ReplyDelete@moxieman: These are great. All sort of fall into the category of "How many ____ are in the nth shape?" If you mess with the shape, the counting problems are endless.
ReplyDelete@Justin: Thanks for the contribution. You're really good about creating a narrative in your "solutions." I've been thinking about this problem for a while, and hadn't thought of it as a packing problem.
@bnachimi: Nice. Your ideas inspired me to play around with this problem from a net perspective.
I had this wonderful comment that just got eaten without me copying it. So you'll have to settle for what I can remember:
ReplyDelete- Might it be interesting to compare what kinds of student responses different scaffolds elicit? Could the end of the school year be a good time for this?
- Could the prompt be slightly modified to come up with as many questions as possible? Might this be an interesting assessment (for a student, a group of students/classroom, a teacher, or a group of teachers)? If so, what would it assess? How would what it assesses be similar to or different from "traditional" assessment items?
- In what ways is it significant that the "boxes" are "center-aligned" rather than "left-aligned"?
- I like the counting problems others have proposed (and which were among my first thoughts); here's a different one (perhaps not really related to the picture, but it reminded me of it): What are the squares of 1, 11, 111, ...
Enjoy!
Sendhil
I was such a student (and I went on to get a PhD in pure maths) and I'd have plotted murder in my heart if faced with such a thing. If I'd had the gumption I have now I'd have written "What comes after 5? Answer: 6". Not sure what I'd have done then, but you'd have lost me for ever.
ReplyDeleteExplore....math class never seemed the place to explore, although I am certain it should be.
ReplyDelete