kitchen table math, the sequel: The chairs

Thursday, October 31, 2013

The chairs

From the "High School" section of
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary.

This video shows an excerpt of a conversation between two students comparing approaches to solving a problem and trying to understand why they got different answers and where one of them made an error.


Three halls contained 9,876 chairs altogether. One-fifth of the chairs were transferred from the first hall to the second hall. Then, one-third of the chairs were transferred from the second hall to the third hall and the number of chairs in the third hall doubled. In the end, the number of chairs in the three halls became the same. How many chairs were in the second hall at first.
I haven't watched the video, but I hope one of the approaches the two students consider involves asking the teacher what the problem means by "the chairs."

By happenstance, my students and I read the first half of G.K. Chesterton's essay on fairy tales in class today. When I say "read," I mean that my students and I read each sentence individually and out loud and then stopped so I could explain what the sentence meant and why after first asking the person who had just read to take a crack at it.

Chesterton's opening lines:
Some solemn and superficial people (for nearly all very superficial people are solemn) have declared that the fairy-tales are immoral; they base this upon some accidental circumstances or regrettable incidents in the war between giants and boys, some cases in which the latter indulged in unsympathetic deceptions or even in practical jokes. The objection, however, is not only false, but very much the reverse of the facts.
I asked my students what the words "the objection" meant. When nobody knew, I pointed out that the words "the objection" function as an anaphora: the definite determinative (the) tells you that you already know which (or what) objection because you've seen it before, in the text. It's the objection, not "an objection."

So, if you've already seen "the objection," and you've only read one other sentence, what is the objection? It's got to be inside that one other sentence.

At that point, my student who was educated outside the United States in his early elementary years (and who speaks very lightly accented English) figured it out.


My answer to the Illustrated Math problem, which I arrived at on my own and without a lot of conjecturing and solution-pathway-planning and special-case mongering and the like is: 4,938.


The Common Core era is going to be an unhappy one for mathematically gifted children.

I predict.


SteveH said...

"Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards."

The CC standards can be interpreted and implemented in so many ways that "faithful" is meaningless. The "faithful" implementation we have in our state is driven by the PARCC test and their highest "distinguished" level only means that students would probably pass a course in college algebra. They state that in black and white. It's not opinion.

Why on earth would anyone think that CC is anything but a low end minimal college standard - one that provides NO STEM preparation starting in the earliest grades. Only those kids who get help at home or with tutors will be ready for algebra in 8th grade and calculus in 12th grade. It didn't use to be that way. Even if your goal is not a STEM career, the best that CC can offer is the academic mountaintop of pseudo algebra II. As I have mentioned before, this is not a base camp for college math and for applying all of the wonderful "understanding" one supposedly has.

SteveH said...

"My answer to the Illustrated Math problem..."

Doesn't matter because it's a stupid-bad problem. Even if it was expressed accurately, it would still not be a high school problem. It sounds like a 5th grade Singapore Math problem. Haul out the bar diagrams.

"The Common Core era is going to be an unhappy one for mathematically gifted children."

There are at least three levels of needs in math. CC is for the lowest level only. Many schools will continue to offer a proper algebra class in 8th grade, and high schools will continue to offer a path to AP calculus. This should handle most "gifted" students assuming they get help at home to get past K-6 math. For those who are at the gifted and high gifted range, some schools offer math clubs and teams that meet their needs. Some schools allow students to skip grades in math (if tested out), but that preparation usually has to be done at home. And more schools help kids take college courses.

Nobody questioned the needs and the paths for the upper end with the old state tests. The question is whether CC will dumb down those paths.

Anonymous said...

Does "One-fifth of the chairs were transferred from the first hall to the second hall" mean one fifth of the chairs in the first hall, or one fifth of all the chairs? The English is ambiguous. Are we supposed to solve the problem both ways?

I love good word problems, but this one sucks.

Allan Folz said...

The english is slightly ambiguous, but like any high school math problem the english barely matters because it has to have only one solution.

Working backwards, the 1/3 has to refer to the chairs in hall "B", because if it referred to the total chairs, and we were also doubling the number of chairs in hall "C", we couldn't have an equal number of chairs in all three halls. Hall C would have 2/3rd of the total.

So, if 1/3 refers to the chairs in hall "B", then likewise 1/5 refers to the chairs in hall "A".

My answer to the Illustrated Math problem... 4938.

Actually, that is incorrect. That's the number of chairs in hall "B" after the 1/5 of hall "A" was added to it. The question asked how many in hall "B" _at first_. :)

Working backwards....

Hall A + Hall B + Hall C = Total
final 3292 + 3292 + 3292 = 9876
+ 1646 - 1646 <-- it doubled at this step
mid 3292 + 4938 + 1646 = 9876
+ x - x
first ? + ? + 1646 = 9876

(3292 + x) + (4938 - x) + 1646 = 9876

Or (extra credit?), this wasn't intuitively obvious to me at first, but upon reflection I think it makes sense:

3292 + x = 4938 - x

Cheers. :)

Allan Folz said...

Ugh, it chopped out my my white space. Let me try again. Hope it works.

Aagin, working backwards...

| Hall A + Hall B + Hall C = Total
| 3292 + 3292 + 3292 = 9876 <- finish
| 0000 + 1646 - 1646 <-- because C doubled on this step
| ----------------------
| 3292 + 4938 + 1646 = 9876 <- mid-point
| +__x - __x
| ___? + ___?___ + 1646 = 9876 <- initial condition

Solve for x :: (3292 + x) + (4938 - x) + 1646 = 9876

BTW, it takes a lot longer to type than to solve. I can empathize with the school kids having to explain their answer. :)

Auntie Ann said...

Yes, it is poorly worded, but in that terribly thought out language, they do tell you what they mean--but you really have to think about it.

If you are moving a third of all chairs into the third hall, and then doubling their number, that room would end up with 2/3rds of all the chairs. There would be no way for the other rooms to each also have 1/3rd of them.

So the wording must mean: a fraction of the chairs in one room are being transferred to another room.

I agree that SM kids could do this by the end of 5th grade. It's not that hard of a problem. Unless they actually mean it to focus on the clarity of language and how you need to be careful about ambiguity.

bky said...

Yes, kids doing Singapore 6 or maybe 5 could do this problem, assuming it was written more clearly. And so it shouldn't be considered a high school problem -- except that high school kids could not do this problem, by and large.

I recently have had some teaching contact with 10-11 graders in a private school, and I am completely sure that given this problem they would have no idea what to do, AND most would make no attempt to think it through. A few might try to muddle through but I cannot imagine success.

And in our local public schools, in the accelerated algebra and geometry courses, there are almost no word problems at all, and then only easy ones, so I don't think that the success rate in the public schools would be much greater.

So maybe Common Core sucks and all, but so does the status quo.

Our schools use Everyday Math in middle school, for what it's worth. No word problems there either, despite the overwhelming verbiage in so many of the problems.

Auntie Ann said...

I think you forgot to subtract off the 1/5th that Room 1 gave Room 2 to get back to the initial conditions.

At the start, I get:

R1 = 4115
R2 = 4115
R3 = 1646

At the end, all have 3292.


1/5 * R1 = 823
R1 - 823 = 3292

R3 * 2 = 3292
-- R2 must give R3 1646

(R2 + 1/5 * R1) / 3 = 1646
(R2 + 823) / 3 = 1646
R2 + 823 = 4938
R2 = 4115

Glen said...

Understanding the question is part of solving any problem, but it doesn't have to be the first part. When I encounter an ambiguous problem statement, I often solve more than one interpretation of the problem and figure out what the problem was along with its solution.

In this case, transferring a fraction of "the chairs" could mean a fraction of the hall's chairs or that fraction of total chairs. I do the former first, because it seems more common in math questions, but frequency of problem type is not a mathematical principle. It could be the latter question (or some other), but I start with the former.

"In the end the number of chairs is the same" is relatively unambiguous: each hall had 1/3 of the chairs. Okay, so work backwards from there.

Since the problem never mentions a specific number of chairs in a transfer, only a fraction, I don't convert 1/3 of chairs into a number of chairs. I leave it as a fraction (of total chairs).

Since the final transfer doubled the number of chairs in the third hall, resulting in 1/3 of total chairs, it started with 1/6 and received another 1/6 from second hall. Since we're now talking about sixths, I'll switch to those units: after the second hall gave the third hall 1/6 of total chairs, all three halls ended up with 2/6 of total chairs.

So after the first transfer, the first hall was in its final state with 2/6, the second hall had 3/6, and the third hall still had the 1/6 of total chairs it started with. Then the second gave 1/6 to the third, and they all ended up with 2/6.

The first transfer dropped the first hall down to 4/5 of what it started with or, in reverse, it started with 5/4 of what it ended with. It ended with 2/6 of total chairs so it started with (5/4)x(2/6)= 5/12 of total chairs. It then gave away 1/5 of what it had, reducing it from 5/12 to 4/12. Okay, so I'll switch to units of twelfths.

The first hall started with 5/12, the third started with 2/12 (the 1/6 we figured out before), so the second started with the remaining 5/12 (of total chairs).

So, start with 5/12, 5/12, and 2/12, and assume "the chairs" means "its chairs." Does it work? Transfer 1/5 of first hall's chairs (one of its twelfths) to second hall, leaving 4/12 and giving second hall 6/12. Second hall then gives 1/3 (of its six twelfths, meaning two of its six twelfths) to third hall, leaving 4/12 in second hall, and those 2/12 double third hall's 2/12 to 4/12. They all end up with 4/12. It works.

Repeating the process with the other reasonable interpretation, where "the chairs" means total chairs doesn't work out. There is no solution that simultaneously matches all the given information if what is given is interpreted this way.

Since the first interpretation has an answer and the second doesn't, it's reasonable to assume that we now know what the question was and that the answer (how many chairs did second hall start with) is our 5/12 of total chairs. Since total chairs was 9,876, the answer is (5/12) x 9876 = 4115.

Anonymous said...

^This is a nice explanation.

Also, suppose you thought that 1/5 referred to all of the chairs. When you come to the part where they tell you that 9876 was the total, a number that is clearly not divisible by 5...

I do think this is poorly worded and even properly worded, still pretty ugly. But if I were teaching it, I would raise the following questions with my students: How do you suppose that the author of this problem found the solution? Was it just luck that it came out to a nice whole number? I'd want my class to realize that these problems are constructed in reverse -- and then write their own. It's like all of those "the supplement of an angle is 10 less than three times the complement" type problems. The guy who writes them starts by picking an angle.


Anonymous said...

One more thing about this problem: it gives unnecessary info. I just looked at it again and solved it without using the fact that the last room doubles.

Here's what I did:
1. You know the total,9876 and you know that each room ends up with 1/3 of that which is 3292.

2. If the first room ended on that number after losing 1/5 of its chairs, then it now has 4/5 of what it started with. So it started with 5/5 of 3292 which is 4115. And it transferred 823 to the second room.

3. The second room gained those 823 chairs and then gave 1/3 of its new total to the third room and ended up with the 3292 chairs that they each finished with.

So 2/3(x+823) = 3292 gives you the starting value for the second room. It comes out to 4115 just as the first room did.

If we feel like, we can now check the initial and final total in the third room to confirm that it doubled, but that would just be to check our work. The problem is over-described. You'd have to ask the author if it was an intended feature or a bug.


SATVerbalTutor. said...

Catherine: Did your students think that the first sentence was a run-on? A lot of kids I've worked with have difficulty with the idea that a sentence can be really long and still be correct. To them, anything long, esp. stuff containing multiple clauses, is weird, awkward, and probably wrong.

Glen said...

Phil, yes, I noticed that over-description, too, when I noticed that there were more solution paths than I expected as I checked my work. I assumed you'd have to calculate chairs using the transfer information, but as I was doing that, I kept noticing that, knowing the number of chairs in two rooms, I could just assign the remaining chairs to the remaining room.

Except for the ambiguity in wording, I like this problem---I seem to be the only one who does---and I even like it WITH the ambiguity if you use it to teach how to deal with such (all too common) ambiguities. Students could crawl all over this problem, finding multiple paths to the same solution, ways of checking that their answer is right, ways of checking that the question means what they think it means, etc.

And I don't see why this problem is inappropriate for high schoolers. Even smart adults can slip up and get it wrong without careful reasoning, careful reading, careful checking....

Anonymous said...

I usually don't like when you have to do that kind of meta-thinking to figure out what was intended. The relentless barage of imprecise writing that students encounter througout their schooling -- from sorces that they don't think to question -- leaves them with no idea about what is wrong with their own writing.

I will say that this problem's ambiguiy is minor: if you interpret it incorrectly, you find out almost immediately. And I actually watched the video so I can tell you that the two students discussing the problem jumped directly to the intended meaning.


Glen said...

LOL, yes, when I say I like this problem, I don't mean for English class. I would want to point out explicitly that the problem was poorly worded, so they wouldn't unconsciously treat it as a model, then maybe even have them discuss how it should have been worded.

In that case, though, maybe rewriting a handful of ambiguously worded problems WOULD make a good lesson in English class.

lgm said...

I have noticed NY's high school Regent's Math Program, as done by my zoned school, totally ignores some of the concepts that my midwestern high school math teacher thought important: Proof, the meaning of conjecture, definition of an axiom, the stating of assumptions. Never here has any teacher stated "Let x be defined as ...." at the beginining of a solution. I chalk it up to full inclusion....

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