Home » Discussion Forums » Blog Post Discussion » Maths, Chaos, Darkness and Handbells
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| Re: Maths, Chaos, Darkness and Handbells [message #46923 is a reply to message #46922 ] |
Tue, 13 December 2011 22:03   |
 Aaron
Messages: 319
Registered: June 2009
Location: California |
Senior Member |
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††† And her eight-year-old, advanced-maths-placement son was sent home with some maths problems this week, one of which she couldn’t do. You have a series of squares, one each of which has sides [insert measurement system of choice, I think it was centimetres] of the following lengths: 1, 4, 7, 8, 9, 10, 14, 15, 18. You want to make a rectangle of these squares—and it has to be a proper rectangle, no leftover bits and no overlaps. What size is it? What is/are the equation or equations for this?
She and her son ended up cutting little bits of paper into squares and shoving them around till they made a rectangle. And they did make a rectangle, but even working backwards she wasn’t seeing how to solve it as a problem.
As a mathsless English major and writer of fantasy novels where lately she seems to be finding herself anthropomorphising chaos^ I still don’t see why you don’t add something up and then divide it by something else—I know areas are different from straight lines, but even so. Anyone out there feel like explaining it to a level I might attain?
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SERIOUS geek warning but you did ask:
Since the problem states that the rectangle will be exactly covered by the squares the area of the rectangle will be the same as the collective area of the squares. If you sum up the squares of 1,4,7,8,9,10,14,15,18 you get 1056. So we know the area of the rectangle but not (yet) its shape.
The rectangle has two sides and the length of its side multiplied together equal its area. An arbitrary rectangle of area 1056 could have one side of any length up to 1056 and the other side equal in length to 1056 divided by the length of the first side. This rectangle has sides that are made up of the sides of the squares so they can't have just any length they have to be sums of the lengths of the sides of some of the squares (the ones on the edges of the rectangles once we assemble them. We don't yet know which ones these are). At the very least this means that the sides will be whole numbers. The whole numbers that can be multiplied to make a known whole number are its prime factors. The prime factors of 1056 are 2,2,2,2,2,3,11 (that is 2 x 2 x 2 x 2 x 2 x 3 x 11 = 1056) and these are the only numbers for which this is true.
So far we have a rectangle of area 1056 and sides made up of some combination of 2,2,2,2,2,3, and 11. We know something else about the sides, since the 18 square has to fit in the rectangle the sides of the rectangle both have to be more than or equal to 18. With the available factors we can have 22 (11 x 2) by 48 (3 x 2 x 2 x 2 x 2) or 33 (11 x 3) by 32 (2 x 2 x 2 x 2 x 2) or 44 (11 x 2 x 2) by 24 (3 x 2 x 2 x 2) but the other possibilities have a side smaller than 18 and the 18 square won't fit. In other words the rectangles with this area that are longer are too narrow for the 18 square.
So our options are 22x48, 33x32, and 44x24.
[Updated on: Tue, 13 December 2011 22:13]
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| Re: Maths, Chaos, Darkness and Handbells [message #46924 is a reply to message #46922 ] |
Tue, 13 December 2011 22:09 |
emnwalker
Messages: 6
Registered: October 2008
Location: Utah |
Junior Member |
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Well, as a sometime algebra teacher, my first inclination about your squares problem is to add up all their areas and then try to write an equation that would be for a rectangle of that area. If I didn't make a typo, those squares' areas add up to 1056. So, if it really is possible to create that rectangle (you didn't say, but I am presuming they want the length and width to be whole numbers), L x W = 1056
Here I am at least temporarily stuck since we have two variables and only one equation (you may remember from algebra that you generally need as many equations as variables if you want to solve the problem.) Off the top of my head, I don't see that we know anything else about the length, width, or their relationship. If I take the square root of 1056, I find out that a square of that area would have sides roughly 32.5 long. From that, I can say (because a square is the most efficient way to enclose an area with a rectangular perimeter) that the length and width have to add up to more than 65, that one of them must be more than 32.5 and the other must be less than 32.5... but none of those give me an exact relationship.
I can factor 1056, and find out that it is 33 x (2 to the fifth power).... now that's interesting. The rectangle's dimensions must be 32 x 33. I solved the problem, but not by finding an equation.
I don't know whether this example of mathematical reasoning is useful... but I'd be very curious to hear whether someone else solves the problem with equations.
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| Re: Maths, Chaos, Darkness and Handbells [message #46927 is a reply to message #46925 ] |
Wed, 14 December 2011 01:52 |
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| Diane in MN wrote on Tue, 13 December 2011 21:16 |
| Aaron wrote on Tue, 13 December 2011 21:03 |
[ . . . ]we will leave that, as the saying is, as a exercise for the reader.
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This is a construction like "hence . . ." or "it is intuitively obvious that . . ." that's generally much more fun for the person saying it than for the person addressed. 
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To any curious souls...
My approach was very similar to Aaron's, and I did find the solution. It's easier to explain with pictures, however, and I can't post those here. PM me with an e-mail address if you'd like me to send you either a .doc or a .pdf file with the solution.
[Updated on: Wed, 14 December 2011 01:52]
FairyTales - http://xkcd.com/872/
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| Re: Maths, Chaos, Darkness and Handbells [message #46928 is a reply to message #46922 ] |
Wed, 14 December 2011 05:46 |
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AJLR
Messages: 2564
Registered: September 2008
Location: England, UK |
Senior Member
[Moderator] |
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Well . . . unfortunately I think you’re suffering from Making a Difference Syndrome.
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No, I don't think I am - but you're equally entitled to your view.
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I’m willing to believe that what’s wrong is different than it used to be—and I’ve no doubt there are resources out there that weren’t available when you and I were still on the wrong side of the desk
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And a lot more people are making use of those resources these days. I wouldn't argue with the fact that there are still too many things wrong, and still people being disadvantaged by the system as it currently stands, but I don't think the abuses are as severe as they used to be and I do think that the part of the problem that was teachers/systems being resistant to any questioning about Their Way Being The Only Way has diminished under the barrage of development encouraging alternative ways of helping people learn in their own way. And the regulation/inspections - I have a lot of sympathy with the good teachers who are driven (almost) to distraction by the inspection regime and who feel that the teaching profession is being pulled in too many directions by government initiatives. But inspection isn't all bad.
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I still know bright—or off the wall, or both—kids who dislike school and are failing to learn what they are absolutely capable of learning, because of bad or blinkered teaching.
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Yes, and it shouldn't be so. There are still schools (and colleges) and individual classes where students aren't encouraged - and supported - in finding the most effective individual way to learn. Part of that is the system and part of it is individual teachers; even as an optimist I doubt we'll ever achieve a 100% success rate. And each student who is put off learning is a real tragedy. All I'm saying is that the evidence, across the country, is of a slow improvement in many areas. And the schools and other organisations who are really good are now expected to share how they do things with others, so that good practice has a chance to be taken up elsewhere. That sort of sharing and peer review just didn't happen to any significant effect before.
Now if we could just change the culture in some parts of society (and among some parents, sadly) that was for ever rubbishing education and learning, then I'd feel more hopeful over all.
"Never let a computer know you're in a hurry."
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| Re: Maths, Chaos, Darkness and Handbells [message #46931 is a reply to message #46922 ] |
Wed, 14 December 2011 10:02 |
CateK
Messages: 9
Registered: August 2011
Location: upstate New York |
Junior Member |
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"Meanwhile in another part of the forest, one of the things that yanks my chain about all this out-there physics and maths I’m (still) reading about is how similar the frelling creative process is, whatever name the drooling monster you’re trying to subjugate is refusing to answer to."
Yes!!!! As a former physics major, former social worker, and former engineer (all pre-ME), it absolutely drove me bonkers when someone would inform me that scientists and engineers have no imagination. Or that they can't do math/science because they can't limit themselves to non-creative thinking. Arrgghhh!!!
Of course, it would also drive me bonkers when scientists and engineers would look down their noses at 'liberal arts types' because 'they' are just fuzzy-headed wool-gatherers.....
It is, however, true that the types of thinking involved in those 3 different fields are different (yes, scientists and engineers tend to be different beasts). I can't nail down the difference, but I can *feel* the difference in how I use my brain.
Cate
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| Re: Maths, Chaos, Darkness and Handbells [message #46933 is a reply to message #46927 ] |
Wed, 14 December 2011 10:59 |
Aaron
Messages: 319
Registered: June 2009
Location: California |
Senior Member |
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| rainycity1 wrote on Tue, 13 December 2011 22:52 |
| Diane in MN wrote on Tue, 13 December 2011 21:16 |
| Aaron wrote on Tue, 13 December 2011 21:03 |
[ . . . ]we will leave that, as the saying is, as a exercise for the reader.
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This is a construction like "hence . . ." or "it is intuitively obvious that . . ." that's generally much more fun for the person saying it than for the person addressed.
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To any curious souls...
My approach was very similar to Aaron's, and I did find the solution. It's easier to explain with pictures, however, and I can't post those here. PM me with an e-mail address if you'd like me to send you either a .doc or a .pdf file with the solution.
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I should mention that all of these ways of looking at the problem rely heavily on the statement that there is a solution. emnwalker's work identifies a possible solution, mine limits us to a single possible solution. I don't think that anything either of us did proves that such a solution actually exists. All we proved was that if there is a solution it has to have certain features.
[Updated on: Wed, 14 December 2011 11:05]
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| Re: Maths, Chaos, Darkness and Handbells [message #46935 is a reply to message #46925 ] |
Wed, 14 December 2011 12:06 |
Aaron
Messages: 319
Registered: June 2009
Location: California |
Senior Member |
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| Diane in MN wrote on Tue, 13 December 2011 21:16 |
| Aaron wrote on Tue, 13 December 2011 21:03 |
[ . . . ]we will leave that, as the saying is, as a exercise for the reader.
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This is a construction like "hence . . ." or "it is intuitively obvious that . . ." that's generally much more fun for the person saying it than for the person addressed.
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True. Which is why, as a long time reader, it was so much fun to right it. At least I was up front about not having the answer.
[Updated on: Wed, 14 December 2011 12:07]
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| Re: Maths, Chaos, Darkness and Handbells [message #46943 is a reply to message #46922 ] |
Wed, 14 December 2011 17:23 |
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Kathy_S
Messages: 313
Registered: October 2008
Location: Indiana |
Senior Member |
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The references to "bad teaching" are really starting to get to me. I'm afraid they come across as a variant of "This book sucks dead bears" rather than "This book doesn't work for me."
It is very true that individuals learn differently. However, it is not the case that being unable to match teaching style to every learner makes someone a bad teacher. This is particularly true in a classroom situation. Some students find stories and examples and diverse media helpful for understanding and reinforcing material; others consider any moment not filled with orderly, numbered lists – preferably with each nugget labeled as to how important it will be on the exam – nothing but disorganized wandering. Some rebel if a teacher doesn't follow a text as though it were a script; others think a teacher dependent on the text isn't doing the job. Some learn by doing; for others, doing without first understanding is a total waste of time. Some desperately need repetition; some find it boooorrrring and lose interest. Some do best in a group; others need to work things through on their own. Within a single class, some reviewers will rank a course as the best ever, some as the absolute pits, and some come back years later saying that what they hated then was actually the most valuable -- or vice versa.
Ideally, I would like to see a system in which teachers can teach to their strengths, learners are partnered with appropriate teachers, and all are valued. Unfortunately, too often my experience has been that teachers are ordered to replace true-method-A with true-method-B and then true-method-C, leaving different students to fall through the cracks and the teachers' best talents wasted. Or, we have the all-methods-at-once method, in which almost everyone's learning seems to be minimized.
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| Re: Maths, Chaos, Darkness and Handbells [message #46945 is a reply to message #46943 ] |
Wed, 14 December 2011 18:08 |
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AJLR
Messages: 2564
Registered: September 2008
Location: England, UK |
Senior Member
[Moderator] |
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| Kathy_S wrote on Wed, 14 December 2011 22:23 |
Ideally, I would like to see a system in which teachers can teach to their strengths, learners are partnered with appropriate teachers, and all are valued. Unfortunately, too often my experience has been that teachers are ordered to replace true-method-A with true-method-B and then true-method-C, leaving different students to fall through the cracks and the teachers' best talents wasted. Or, we have the all-methods-at-once method, in which almost everyone's learning seems to be minimized.
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Yes, and that's a horribly frustrating situation isn't it. One of the concerns in the wider education community (this doesn't include government education ministers!) here at the moment is a risk of the 'de-professionalisation' of teachers, with people told to teach in particular ways rather than being allowed to use their skills as professionals to find and use the most appropriate and up-to-date way of encouraging their students' learning. For the enormous numbers of committed teachers we have, that's extremely demotivating.
"Never let a computer know you're in a hurry."
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| Re: Maths, Chaos, Darkness and Handbells [message #46948 is a reply to message #46922 ] |
Wed, 14 December 2011 21:40 |
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equus_peduus
Messages: 437
Registered: September 2009
Location: France |
Senior Member |
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Commenting prior to taking the time to read about (after thinking about) the rectangle question, because my brain is a little fuzzy right now...
Can't speak for the UK, but obesity IS the number one health concern of pet dogs and cats in the US. Well over half my patients are obese. A surprising number (though really, not that many, ultimately) *say* they go for x-mile walks every day. Everybody says the dog or cat doesn't eat "that much." Many of these dogs have free run of the yard, or are outdoor cats (presumably, they get some exercise). If you can manage to winkle out of the owner how much the animal actually eats (as opposed to "there's always food in the bowl, I top it off every couple days, and she only eats a mouthful or two at a time"), it usually adds up to one heck of a lot of calories.
So presuming you've got dogs or cats who actually eat, and people who think that love must be expressed with food, and that they are not aware how calorically dense most dog and cat foods are, much less people foods, and how few calories a dog or cat actually needs, especially if it's not getting all that much exercise (yard access does not mean the dog is actually running around very much, yards are boring after the first week).
And listing out the consequences of obesity, even after some of them have happened to one or more of their pets, does not usually work to motivate them. Then again, *I* am overweight, and I know the consequences... :-/
OK, done with my rant. I will consider the mathematic problem in a little bit.
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| Re: Maths, Chaos, Darkness and Handbells [message #46998 is a reply to message #46982 ] |
Fri, 16 December 2011 00:03 |
Aaron
Messages: 319
Registered: June 2009
Location: California |
Senior Member |
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| rainycity1 wrote on Thu, 15 December 2011 18:41 |
(let me know if the link doesn't work for you.)
Robin is obviously guilty of nerd sniping:
http://www.xkcd.com/356/
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The link worked fine, thank you. Since I don't see any approaching trucks I will mention that I think the construction can be formalized. I outline this below.
The Rest of the Geek Material:
The 1 square cannot be in the corner because it would have to have something bigger than it on both sides and they would have to overlap. Nor can it be on a side of the rectangle because whatever larger squares were on either side would create a narrow area of width 1 into which nothing would fit.
The 4 cannot go into a corner because it would have a larger square on one side or the other that would create a place on the other side that would fit nothing else except the 1 square and the 1 square would not fill the space. It cannot go on a side of the rectangle because that would leave an area of width one and length at least 3 (if the 7 and 8 were used) and, again, the 1 is the only thing that would fit and that would leave a unfillable space.
In fact the 7 is similarly restricted since the 1 and 4 squares cannot be used to fill the 7 wide gap that would be left between the two larger squares on either side of the 7 (even if the gap is only one high none of the larger squares can contribute to filling it).
Once we know that the 1, 4, and 7 cannot touch the sides of the rectangle we know that the perimeter will be made up of sides of the 18, 15, 14, 10, 9, and 8 squares. Any given square may contribute one side (if it is in the middle of a side of the rectangle) or two (if it is in the corner). These sides will have to sum up to the total perimeter of the 33x32 rectangle which is 130 (33x2 + 32x2).
Of the six squares four must be in the four corners and two will touch only the sides of the rectangle. This means that of the twelve available sides of the six squares only ten will actually be touching the sides of the rectangles. The total length of the twelve side is 148 (18x2 + 15x2 + 14x2 + 10x2 + 9x2 + 8x2). This is 18 greater than the perimeter of the rectangle so the two sides that are not used (because those two squares are in the middle of a side of the rectangle) must have sides that sum to 18. The only two of the six "perimeter" squares that fit this requirement are the 10 and the 8. The four corners must be occupied by the 18, 15, 14, and 9 squares with the 10 and the 8 filling in the middles of two sides.
The 15 and 14 squares can not be in adjacent corners since that would leave a gap on the side of no more than 4 (the longer side of the rectangle is 33 and 4 = 33 - 29 where 29 is 15+14. While there are two squares that would fit in this space (the 1 and the 4) we showed above that they cannot be on the side of the rectangle since they would leave a space that we could not fill with the remaining squares.
Since the 15 and 14 cannot be next to eac
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