Date: Sep 24, 2012 3:07 AM
Author: Jonathan Crabtree
Subject: Re: Long Divison...Again
STYLE NOTE I have included lots of dots in an attempt to preserve places and line up columns correctly. This looks ok in the preview, yet may or may not work as a live post as I discovered when my axioms were garbled in this post http://mathforum.org/kb/thread.jspa?forumID=206&threadID=2390540&messageID=7849273#7849273
Maybe I'm being too pedantic and missing the meaning in your post, yet to me, division is also asking, "What is the missing factor?" The following pedagogy may also work well for long division of arithmetic and algebraic expressions.
So the long division of 3680 divided by 46 is also asking us to find the missing product AB, that when multiplied by 46, equals 3680.
solve for AB
If we solve for A first...
Four doesn't go into 3, so how many times does 4 go into 36? (Just a conversation starter for children, before we include place value logic.)
NOTE: You can also ask, "What number A tens, times 4 tens will be as close as possible yet won't exceed 36 hundreds?" This gives 8 tens as if we say 9 tens, when we also cross multiply 9T x 6U we get more hundreds and we already have the 36 hundreds from our 9!
So 4 tens go into 360 tens safely 80 times with 40 tens as a Remainder (aka R)
So we get A = 8
Now solve for B units given that A tens = 80
Solve for .8B
less......3200 from 80 x 40
solve for B
What must B units equal so that the cross products 8T x 6U plus 4T x BU = 48T (aka THE TOTAL REMAINDER)
As 8T x 6U = 48T already, B must equal zero units
AB = 80
Alternatively, using place value letters (Th, H, T, U) we can write it as...
..................AT + BU
times...........4T + 6U equals
3Th + 6H + 8T + 0U
SOLVE B FOR UNITS COLUMN
B must be zero or BU times 6U would not = 0U.
Given B = 0 then BU = 0
SOLVE A FOR TENS COLUMN
So what will A be so that AT times 6U plus 4T times BU (zero) creates 8T?
What digit multiplied by 6 results in a product ending with 8?
Knowing our times tables, only 8T multiplied by 6U will create a cross product ending with 8T and it will have 4H as well. Ie 480
So A must equal 8.
HUNDREDS AND THOUSANDS COLUMNS CHECK
8T times 4T = 32H plus the 4H (from 8T x 6U) equals 36H to go along with the 8T
Check Answer for A = 8 and B = 0 becomes 3680 so missing factor is proven to be 80.
Place Value Multiplication Can Be Understood Via Manipulative Arrays so...
U x U = T and/or U
T x U = H and/or T
H x U = Th and/or H
T x T = Th and/or H
H x H = HTTh and/or TTh
The multiplication and division of algebraic expressions also works the same way and may be intuitively simpler when set out as above.
Dividing a quadratic 8x^2 +16x + 6 by 2x + 3 can be done as follows
..............Ax + B
times..... 2x + 3 equals
8x^2 + 16x + 6
B must equal 2 in order that 6 is the initial product.
Hence once B is known to be 2, A is revealed to be 4 so the simple cross product create 16x.
Then 4x times 2x gives you the required 8x^2.
So Louis (and others) I probably haven't explained myself as simply as possible (if at all) so if you can better re-write my explanation of this way of doing long division in this post then please do.
I will as usual, benefit from any help!
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