Math "Whys"

[shinyhappypeople]

Is there a site that explains WHY the traditional math algorithms work? Bonus points if it has video explanations.

 

[Farrar]

I feel like the Khan Academy videos I've seen explain it reasonably well.

 

If you understand the concepts, the traditional algorithms for most things are pretty clear why they work.  A few things, like long division, take a few extra steps, I guess, but mostly it seems to me that if you understand the concepts, the algorithms explain themselves.

[Blessings2all]

I'm not sure which algorithms you mean, but sometimes manipulatives are the best tools for demonstrating why something works. Also, drawing pictures for the problem being solved can help demonstrate algorithms.

[....]

Not sure if this helps you, but have you seen this website (for younger kids)?  www.educationunboxed.com  

 

This guy also has some great videos, but his website is hard (for me) to navigate:  http://www.crewtonramoneshouseofmath.com

 

I also second the Khan Academy videos and the pre algebra videos on the Art of Problem Solving website:   http://www.artofproblemsolving.com/Videos/index.php?type=prealgebra   We've watched a ton of those videos.  The guy is kinda goofy.

[CaffeineDiary]

Is there a site that explains WHY the traditional math algorithms work? Bonus points if it has video explanations.

 

Can you be more specific?

[shinyhappypeople]

Can you be more specific?

 

Sure.  For example:  357 x 3 = ??   We all know the traditional algorithm (multiply 3 by 7, carry the 2, multiply 3 by 5, ADD the 2 you carried, etc.) I'd like to find a way to explain WHY we do it that way, why we add the carried numbers, etc.  

 

There must be a reason....  I've just never really thought about it before.  

 

My 11 yo DD is like a little robot when she does her math.  She just plugs numbers into the right places, but doesn't really understand why she's doing what she's doing.  This causes a lot of careless mistakes (e.g. in the example above she might carry the 1 of 21 instead of the 2).

 

We're halfway through TT4 at the moment.  On some levels it works brilliantly for her, and we're sticking with it.  But conceptual understanding.... no.  Homefry needs more explicit instruction.  At this level TT is almost purely algorithm-based.  DD's not making the connections on her own, so I need to step in and provide an extra layer of instruction.

 

Make sense?

[wapiti]

Sure.  For example:  357 x 3 = ??   We all know the traditional algorithm (multiply 3 by 7, carry the 2, multiply 3 by 5, ADD the 2 you carried, etc.) I'd like to find a way to explain WHY we do it that way, why we add the carried numbers, etc.  

 

There must be a reason....  I've just never really thought about it before.  

 

My 11 yo DD is like a little robot when she does her math.  She just plugs numbers into the right places, but doesn't really understand why she's doing what she's doing.  This causes a lot of careless mistakes (e.g. in the example above she might carry the 1 of 21 instead of the 2).

 

We're halfway through TT4 at the moment.  On some levels it works brilliantly for her, and we're sticking with it.  But conceptual understanding.... no.  Homefry needs more explicit instruction.  At this level TT is almost purely algorithm-based.  DD's not making the connections on her own, so I need to step in and provide an extra layer of instruction.

 

Make sense?

 

That particular example goes back to place value. 3 hundreds (300) + 5 tens (50) + 7 ones (7), each multiplied by 3, and then added together.  

 

MM does an excellent job of explaining multiplying in parts, IMO, before leading into the actual algorithm (shortcut).  Here is one sample sheet from MM, though multiplying in parts is covered more extensively than this sample shows:

http://www.mathmammoth.com/preview/Multiplication_2_Multiply_in_Parts.pdf

[kitten18]

Sure.  For example:  357 x 3 = ??   We all know the traditional algorithm (multiply 3 by 7, carry the 2, multiply 3 by 5, ADD the 2 you carried, etc.) I'd like to find a way to explain WHY we do it that way, why we add the carried numbers, etc.  

 

There must be a reason....  I've just never really thought about it before.  

 

My 11 yo DD is like a little robot when she does her math.  She just plugs numbers into the right places, but doesn't really understand why she's doing what she's doing.  This causes a lot of careless mistakes (e.g. in the example above she might carry the 1 of 21 instead of the 2).

 

We're halfway through TT4 at the moment.  On some levels it works brilliantly for her, and we're sticking with it.  But conceptual understanding.... no.  Homefry needs more explicit instruction.  At this level TT is almost purely algorithm-based.  DD's not making the connections on her own, so I need to step in and provide an extra layer of instruction.

 

Make sense?

There are videos at Education Unboxed that show exactly what you are asking.

 

She does a great job of relating the concept to the algorithm. 

[stripe]

Is there a site that explains WHY the traditional math algorithms work? Bonus points if it has video explanations.

http://www.mathsisfun.com/  ??

[CaffeineDiary]

The "why" of any particular algorithm can be unhelpful, because the job of an algorithm is fundamentally to teach you how to get something done, not to deepen your understanding.  However, most algorithms involving whole numbers can be successfully approached by understanding two things: 

 

(1) The concept of a prime number, which is to say a positive integer greater than 1 that can only be divided by itself and the number 1, and

(2) the Fundamental Theorem of Arithmetic, which is the fact that any whole number greater than 1 is either prime or is the product of a unique list of primes.

 

So for example, one can express the composite number 35 as 7 x 5.  The composite number 80 can be expressed as 2 x 2 x 2 x 5 x 5.  And these factorizations are unique; there is no other combination of prime products that will yield "80" other than 2 x 2 x 2 x 5 x 5 (rearranging the order of the primes is meaningless, and so doesn't count.  This is also why "1" is correctly not defined as a prime number - if you treat it as a prime number, then your composite numbers no longer have unique factorizations).

 

Turning to your example, 357 x 3, there are many ways to solve it.  No one way is more right or wrong than another.

 

You could turn it into an addition problem:

 

  357

  357

+ 357

 

You could turn it into a simpler multiplication-and-addition problem (which is actually what you're doing via 'long multiplication', just in a sneakier way)

 

357 x 3 = (300 x 3) + (50 x 3) + (7 x 3) = 900 + 150 + 21 = 1071

 

Or you could solve this problem with logarithms and either precomputed tables of logarithms or with a slide rule, which - when the multiplication was complex enough - is what everyone in the entire world did from about 1615 until the latter part of the 20th century and the advent of calculators.

 

The practical reason we use the "long multiplication algorithm" stems from the fact that through experience and practice we have memorized many of the simple multiplication tables from 1 to 12.  So in the particular case of (357 x 3), doing long multiplication doesn't really help you, but if we were doing (357 x 178), the "long multiplication algorithm" converts the problem from adding 357 to itself 178 times into a much simpler series of calculations that we already know the answers to.  I have absolutely no idea what 357 times 178 is, but I know without doing any calculation that (in no particular order) 7 times 8 is 56, 7 times 5 is 35, 7 times 3 is 15, 8 times 5 is 40, and 8 times 3 is 24, and 3, 5, and 7 times 1 are themselves.  If I know those facts, then what "long multiplication" actually does for me is it converts the problem in to a simple addition problem, and all of the bookkeeping is just making sure that I add the right things to the right other things.

 

Video resources: the challenge you'll face is that what you're currently working on isn't technically math, but "just" computation.  But if you are interested and want to get into it a little more, you want to search youtube for "Number theory".  There's a channel called "Numberphile" that is quite excellent, although probably above your kid's current level.  But I bet they actually cover this particular topic somewhere.  In terms of accessible (for you) books on Number theory, I like David Berlinski's book "1, 2, 3."
 

[Blessings2all]

Base Ten Blocks might help. You can usually find printable versions on the web.