Invert and Multiply

Date: 11/08/97 at 14:24:45
From: Anonymous
Subject: Fractions!

Hi Doc,

I am a student teacher, currently taking a methods course in 
elementary mathematics. I am struggling with how to explain to a class 
"why" we invert and multiply when dividing fractions. I read your FAQ 
on this, but still don't understand how I would explain it using 
concrete materials in a classroom.  Please help.

Thanks ever so much!
Laura 

Date: 11/09/97 at 09:56:11
From: Doctor Chita
Subject: Re: Fractions!

Hi there:

Explaining "why" using concrete materials may be difficult. However, 
this is an opportunity, perhaps, to show students how mathematics 
relies on proof, rather than pictures or models.

Take an "easy" problem such as, 1/2 divided by 3/4.

Write this as a compound fraction: (1/2)/(3/4). (Use horizontal 
fraction bars, not diagonal bars as here. The computer has its 
handicaps!)

Everyone knows that 1 is the multiplicative identity element. 
Therefore, 

   1 * (1/2)/(3/4) = (1/2)/(3/4)

Substitute 4/4 for 1. Write:

   4/4 * (1/2)/(3/4)

Think of 4/4 as the compound fraction: (4/1)/(4/1). Then multiply the 
numerator and deminator of the compound fraction by 4/1:

   [4/1 * (1/2)]/4/1* (3/4)]

The new numerator is 2: the new denominator is 3. The answer is 2/3.

Therefore, the rule "invert and multiply" works because what you are 
really doing is multiplying the numerator and denominator of a 
compound fraction by the LCD of the fractions in the compound 
fraction.

Try 3/8 divided by 9/5. Write as (3/8)/(9/5) using horizontal bars.

Multiply the fraction by 1 = 40/40, in the form of (40/1)/(40/1) where 
40 is the LCD of 8 and 5:

   (40/1)/(40/1) * (3/8)/(9/5)

   [40/1 * 3/8]/ [40/1 * 9/5]

   15/72 = 5/24

After a while, you can drop the 1s in the fraction that represents the 
LCD.

Got it? Put this information together with what's in the archives and 
write back if you still need more help. Good luck.

-Doctor Chita,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/