Using Doubling to Multiply

Date: 02/20/2002 at 09:28:37
From: Ron Javenes
Subject: Using doubling to multiply

My 4th grade grandson comes home with a math worksheet. One of the 
problems he has to solve is, "Can you think of a way to use doubling 
to multiply 6 x 7? Explain your answer." Please explain to me the 
term "doubling to multiply" and how it would work with 6 x 7. 

Thank you.

Date: 02/20/2002 at 12:55:27
From: Doctor Peterson
Subject: Re: Using doubling to multiply

Hi, Ron.

I think this is just an opportunity for him to invent a way to use 
doubling in order to multiply two numbers without having to know the 
multiplication table. Such methods have been used for a long time in 
various cultures; the so-called "Russian peasant multiplication" and 
"Egyptian multiplication" are examples:

   Russian Peasant Multiplication
   http://mathforum.org/dr.math/faq/faq.peasant.html   

   Russian Peasant Method of Multiplication
   http://mathforum.org/dr.math/problems/brown10.7.98.html   

   Egyptian Method of Multiplication
   http://mathforum.org/dr.math/problems/shelley.6.26.96.html   

But you don't want to do research and dig too deep. Let's just look at 
what he might think of himself, and how you can guide him.

You want to multiply 7 by 6. If you double 7, you have multiplied it 
by 2:

    2*7 = 14

If you double it again, you've multiplied 7 by 4:

    2*2*7 = 28

If you double it yet again, you've gone too far and multiplied it by 
8:

    2*2*2*7 = 56

How can we use these numbers to find 6 times 7? Well, 6 is 2+4, so 6*7 
is (2+4)*7, which we can find as 2*7 + 4*7. (Technically, this is 
called the distributive property.)

Put more simply, 2 sevens plus 4 sevens is 6 sevens, which is what you 
want. So you can just add 14 (2 sevens) and 28 (4 sevens) to get 42, 
which is 6*7.

You'll want to give hints to gete him moving in this direction. He 
might think of ways that are just as good on his own; for instance, he 
might want to multiply 6 by 7, and do it as 8 sixes minus 1 six. 
That's fine! The idea is to get him thinking and inventing if 
possible.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/22/2002 at 12:25:11
From: Ron Javenes
Subject: Using doubling to multiply

Thank you for the answer to my question...I understand why they want 
young minds to explore different ways to solve a problem. But old 
minds sometimes have a problem understanding why something so simple 
can become so complex.

Have a nice day,
Ron