

Math visualization: (x + 1)^2 - niyazpk
http://www.billthelizard.com/2009/12/math-visualization-x-1-2.html

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sethg
Cf. the binomial cube
([http://homepage.mac.com/montessoriworld/mwei/sensory/sbinoml...](http://homepage.mac.com/montessoriworld/mwei/sensory/sbinoml.html)),
which has been part of Montessori preschool classrooms since, umm, probably
since Maria Montessori herself was teaching them a hundred years ago. I think
the Montessori cube was based on a similar object designed by Freidrich
Fröbel, the man who invented kindergarten.

~~~
brlewis
When I saw something like this at my kids' Montessori school, I thought, they
could do that in a textbook. (Most Montessori materials involve manipulation
that can't be done on paper.) I wonder if modern textbooks have this
illustration.

[http://ourdoings.com/brlewis/photo.html?th=gw/cq/eviq.jpg...](http://ourdoings.com/brlewis/photo.html?th=gw/cq/eviq.jpg&d=2006-01-21#p)

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shaunxcode
Thanks to one of the comments on that post I found
<http://www.betterexplained.com>! This is a great resource that I plan on
sharing with anyone who asks the "but WHY?" question (whichis a good thing).

~~~
vitaminj
I agree it's a great site. Kalid is also a regular contributor on HN.

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shaunxcode
Awesome, I bought and printed a copy of his book this afternoon!

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kalid
Hey, just saw this thread in my referral logs -- thanks for the support! :)

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pkrumins
I have been collecting visual proofs for over a year now. I think I am gonna
start writing an article series about math visualizations.

~~~
cruise02
I keep meaning to write more of these too. My blog is (ostensibly) about
programming, but I find that I keep drifting further and further into
mathematical topics without tying them back into programming somehow. I should
probably commit to one or the other, I just can't decide.

~~~
eru
You don't need to. There are surprising connections. E.g. between partial
differentiation and algebraic datatypes:
[http://en.wikibooks.org/wiki/Haskell/Zippers#Differentiation...](http://en.wikibooks.org/wiki/Haskell/Zippers#Differentiation_of_data_types)

~~~
cruise02
I guess my third option is to concentrate more on finding those connections.
Thanks for the feedback.

~~~
eru
Zippers are surprisingly practical. E.g. for a file system:
[http://okmij.org/ftp/Computation/Continuations.html#zipper-f...](http://okmij.org/ftp/Computation/Continuations.html#zipper-
fs)

If you want to read about Zippers and don't mind using your brain, then the
presentation in "he Derivative of a Regular Type is its Type of One-Hole
Contexts" (<http://www.cs.nott.ac.uk/~ctm/diff.pdf>) is right for you. He
talks about the connection between math and computer science:

"Apart from the implementation of this technology, and the development of a
library of related generic utilities, this work opens up a host of fascinating
theoretical possibilities - one only has to open one’s old school textbooks
almost at random and ask ‘what does this mean for datatypes?’."

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vicehead
Here's all of permutations and combination (written in processing) in 3D
space.

<http://pastebin.com/f18f24d8b>

    
    
      Permutations and combinations.
      Taking 3 (visualized as dimensions) from 7 (length of side).
      License: Public domain. Attribution requested.
      
      KEY:
      Permutation: (order matters).
       Repeating (Sequence): All (little cubies). Eg: Passwords.
       Non-repeating (Arrangement): All except the reds. (ie transparent + blues). Eg: Number of ways of podium finishes in a race.
      Combination: Selection (order doesn't matter).
       Repeating: Cubies with white outline. Eg: Number of 3 scoop icecream serving from a set of flavours.
       Non-repeating (Subset combination): The blues. Eg: Number of ways to choose 3 member committies from a group.

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iamwil
I use to wonder why there was an outside and inside part when doing FOIL. It
was after that I saw the visual proof, and doing other types of multiplication
with multiple 'parts/terms', that it slowly dawned on me that there's cross-
components. It's the same sort of thing when you multiple matricies or
covariances. You need the contribution from every combination of terms.

~~~
philwelch
Yeah...I was in a program called Kumon and learned most of my math there, more
than I learned in school. There was a lot of multiplying together polynomials,
so I got used to the idea of handling each combination of terms.

There was also a fair amount of factoring. Let me rephrase that: an
unreasonable amount of factoring. Factoring down 4th and 5th and I think 6th
degree polynomials by hand, for instance, in the equivalent of 10th grade
algebra.

By the way, there are analogous patterns between even degreed polynomials, 4th
degree polynomials look kind of like quadratics and 6ths do as well, with a
kind of rippling pattern in the intermediate terms.

By the time anyone told me about FOIL, I just squinted and shrugged it off. I
was so used to taking these polynomials apart that enumerating through all
possible combinations was trivial to me.

Incidentally, I always considered it more artful to factor quadratic equations
to solve them rather than applying the quadratic formula.

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jgrahamc
That was pretty simple. See also this fun post of mine:
[http://www.jgc.org/blog/2008/02/sum-of-first-n-odd-
numbers-i...](http://www.jgc.org/blog/2008/02/sum-of-first-n-odd-numbers-is-
always.html)

~~~
cruise02
I had also included that one in an earlier post:
[http://www.billthelizard.com/2009/07/six-visual-
proofs_25.ht...](http://www.billthelizard.com/2009/07/six-visual-
proofs_25.html) (scroll down to the 5th "proof").

I think I like the image you used to visualize the relationship a little bit
better than the one I used.

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rick_2047
Some people might be surprised to hear this but such visualizations are part
of the curriculum for the CBSE of 10th grade (maybe 9th also) students in
India. They have to do such things for lots of equations and some geometrical
problems. This is called practical Maths, and comprises about 20% of our
marks. We have to solve puzzles and do other fun stuff with math
visualization.

~~~
DarkShikari
This is how the EPGY software (I think? It might have been Hopkins CTY, this
was over a decade ago) taught factorization in algebra: it was incredibly
effective and simply _made sense_ , and was IMO far more effective than what I
saw being taught in the classrooms.

