Justin Does Math

Figured out something else:

(note that the blue triangle is supposed to be within the purple triangle)

The big purple triangle has an area of 400. It’s composed of 1S + 4T + 4P. So 1S + 4T + 4P = 400. We knew that already.

The little blue triangle has area 100. It’s composed of 2T + 1P. So 2T + 1P = 100.

If we cut the blue triangle out of the purple triangle, we have a purple “band” where 1S + 2T + 3P = 300.

That’s a good thing to notice, but unfortunately, I think that one is not going to turn out mathematically convenient to get an answer from.

I think your main problem is some kind of suppression of brainstorming.

But if the question in the book was to find every square in my pic, of any size, and count them, I think you could do it. And then write down a formula for each different type of square.

Ah doh.

That sounds plausible.

I looked it up, and my assumptions about their intended solution were wrong, btw.

What they do is closer to what Elliot did than what I did.


Hmm this square seems important (writing more)

I think there’s something a bit weird about what you do & do not block. I’m not really sure what’s going on.

I think there are things you know that you didn’t list in brainstorming.

And you are doing things like not using pythagorean theorem, because you think that’s not part of the intended solution, but you are using equations in a way that looks like it could be leading to systems of equations. But I don’t think those would be part of the intended solution either (I think it’s a pre-algebra book).

One thing you could do is try to solve it any way you know how, including with pythagorean theorem, and then go back and try to find other ways to solve it too.

With your brainstorming, you could try to write down everything you can figure out about it, without actually trying to put it all together, and then go back through & look at that. I don’t think you ever did a step like that.

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I actually there may actually be an important point here.

I think something like this may be happening: you are not allowing yourself to use pythagorean theorem because you think the book didn’t intend pythagorean theorem to be used. So you want to solve it the way the book intended. You think that the students reading the book could solve it without using pythagorean, so you should be able to too.

Are you doing something like that?

(I have more to say, I just wanted to check if this is happening before I write more.)

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The square in the previous post has the formula 4T + 4P. If we relate it to one of the other squares with the black square, like the one in my earlier example, we could say 4T + 4P = 1S + 3T + 3P. That simplifies to 1T + 1P = 1S. We can connect this to the information from the original square (that 400 = 1S + 4T + 4P and get 5S = 400 and S = 80. So the shaded area is 80.

You’re still avoiding anything resembling brainstorming. How many squares are in my pic, and how many different types are there? (Or don’t answer but introspect about why you won’t do this or something.)

I started writing a reply to your earlier question about identifying the squares but then noticed the center square and wanted to follow my thinking about it - like i got excited about that thought or something. I’m going to answer ingracke’s question then come back to thinking about the squares.

That sounds about right, but I would also add the following: I think that maybe the book makers maybe thought using the pythagorean theorem would make the problem too easy, and it’s supposed to be a challenge problem. I have a personal issue with doing things “the easy way” where I don’t learn as much, so I’m trying to respect the challenge of the problem .

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Ok, that sounds like another problem, besides the one I identified, and could explain part of your blocking brainstorming.

The actual solution was much easier than you were making it. It is possible you were systematically blocking anything you thought was “too easy”, in order to try to make the problem “hard” enough. But the actual solution was simple. So you failed to brainstorm the things required for the actual solution.

And, btw, I solved the problem multiple different ways, and the Pythagorean theorem way was the hardest one, imo. And doing it first didn’t block me from discovering the other simpler & more elegant solutions.

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By didn’t block me from discovering, I mean it didn’t even give me “hints” towards the other solutions. Like, knowing the answer from pythagorean didn’t make it so that I couldn’t discover the other solutions, it didn’t just give me the answer in a way that spoiled the problem.

Counting only larger squares (so not counting each separate trapezoid + small triangle combination), and counting only actual squares (and not e.g. rectangles) there are 10 squares divided into 4 types:

1 square consisting of the entire picture, whose formula is 4S + 16P + 16T = 1600
4 instances of the original square, whose formula is 1S + 4P + 4T = 400
4 squares whose formula is 1S + 3T + 3P
1 center square whose formula is 4T + 4P.

There are other shapes that almost get to squares but they get “clipped off” at the edges of the picture.

Why? This is exactly the type of suppression I’m talking about.

BTW, besides excluding squares, this is correct. The answer you got above was also correct.

So, for this part, my thought was:

You were trying to put yourself into the same position as the students using the book, who didn’t already know pythagorean.

But you were not doing that.

You were systematically blocking knowledge you already had. That is very very very different from just not having that knowledge in the first place.

The thing that you were doing was hurting your ability to brainstorm and think about the problem at all. You can’t properly brainstorm and think of new, elegant solutions when you are trying to systematically block off a large part of your math knowledge.

The students who were using the book and didn’t already have that knowledge were not systematically blocking out knowledge. They just didn’t have it. So they are still able to freely brainstorm and think up solutions.

So you weren’t putting yourself in their shoes at all. You were doing something entirely different & much harder. The thing you were doing makes it impossible to think about things well. You didn’t even know what knowledge was required, what sorts of ideas were required, but you still blocked out a bunch of stuff. And knowledge is all connected, so in blocking out one thing, you don’t know what else you might also be blocking out.

And you weren’t just blocking out your knowledge of Pythagorean theorem.

So you were blocking out anything you thought would make the problem too easy.

So of course you had trouble finding the intended solution, which was actually elegant & simple.

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Hmm. I think I was making some guesses about what sort of squares would be interesting or useful in some way and filtering for those. I guess that does sound like the sort of suppression you were talking about :slightly_frowning_face:

To answer your question (hopefully!), in addition to the squares I mentioned above, there are 12 smaller squares in the picture - 8 squares consisting of a small triangle and trapezoid, and 4 black squares.

yes. so the answer is 22 squares of 6 types. and you still need to write the formulas for the 2 types of small squares.