Here is another horror which I found recently:

The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

The current school math explanation:

You take the a and distribute it to the b to get ab

and then you distribute the a to the c to get ac

and then you add them together to get ab + ac

I have come across this explanation in several places, and once again real damage is done to the language, and real confusion sown. “Distribute” means “to spread or share out” as in “The Arts Council distributed its funds unevenly, as usual. Opera got the lion’s share.” So it is NOT the a that is distributed. I tried to find a definition of the term in wordy form as it applies to algebra systems but failed. Heavy thinking produced the “answer”. What is being distributed is the second factor on the left.

Example:

Take 3 x 7. We know that the value of this is 21

Distribute, or spread out, the 7 as 2 + 5 . . . . . . . . the b + c

Then 3 x (2 + 5) has the value 21

But so does 3 x 2 + 3 x 5. To check, get out the blocks !

So 3 x (2 + 5) = 3 x 2 + 3 x 5 ……… The Law !

Regarding the “second” version of the distributive property, a(b – c) = ab – ac, this cannot just be stated, and you won’t find it in any abstract algebra texts. Since the students are looking at this before they have encountered the signed number system, a proof must not involve negative numbers, as a, b and c are all natural numbers. It can be done, and here it is:

set b – c equal to w (why not!)

then b = c + w

multiply both sides by a

ab = a(c + w)

expand the right hand side by the distributive law

ab = ac + aw

subtract ac from both sides

ab – ac = aw

replace w by b – c, and then

ab – ac = a(b – c)

done !

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OMUnbelievable! Hopefully teachers out there will pass that by!

I have a feeling if you were my math teacher the course of my time in college and my current job would be utterly different.

Not to say I dislike public/social services, but I always wanted to be good at math, but was never approved to take calculus because of the way our school determined our courses. My performance in pre-cal wasn’t good enough, so I was put into statistics with the idea I’d slow the class down.

At the time, math scared me, and I felt inadequate. But I think with an engaging teacher and clear language, I wouldn’t have been so afraid.

Reblogged this on Saving school math and commented:

I have to put this one up at least twice a year, so here it is again.

That’s a nice way of getting the subtractive form of the distributive law and without throwing additive inverses into the mix.

/*The distributive law of addition: a(b + c) = ab + ac (OK, it’s a definition)

distributive property really */

bad language indeed (and worse graphics,

considering, for example, that the color serves only as

a distraction).

*”it” (what?) is *not* a definition.

(proof by cases)

(1) the distributive law is an axiom.

(2) the whole mishmosh

“The distributive … ab + ac” defines, not

the “distributive law of addition”

but (with a little cleaning up) defines instead

“the distributive law”

or

“the distributive law (of multiplication over division)”.

\endofproof

footnote. i’d say that multiplication-by-a

is what’s being distributed, not “a” itself.

(call it “left-hand factor” if you prefer; i don’t mind.

is “*second* factor on the left” a typo or am i

missing something?)

The second factor on the left hand side of the “equation” is what I meant, namely the 7, which is distributed as 5+2

“distributed” means “shared out” in normal language, and so it does here.

“He distributed the loaves and fishes among the five thousand”

i get it; thanks. this is indeed a good way to put it.

thIs “breaking up” of a sum into summands is

indeed the heart of the matter. the standard

rectangle-model graphics would be useful here.

meanwhile, my mind’s-eye somehow “sees”

the “distribution” of the “stretching”-by-a-factor-of-“a”

every bit as vividly as ever it did that fella in the gospel

feeding the multitudes… and my mind’s-ear recognizes

“operation A is distributive over operation B”

as a very useful verbal “formula” that i’ll never

give up unless for something even better.

cool post.

All done with graphics – check out my post:

https://howardat58.wordpress.com/2015/02/03/commutative-distributive-illustrative-ly/

mea culpa. having the “a” in a different color

from everything else is not, by itself, a bad idea.

more than two colors here is damage, though.

i’ll stand by that.