Demonstrating Equivalent Fractions

Date: 10/25/2000 at 10:24:54
From: Karen Harris
Subject: Equivalent fractions

My daughter is a 5th grader in a gifted class. She has been given an 
assignment to teach other children in her class about equivalent 
fractions. The demonstration must last at least 10 minutes, and she 
must use visuals. She does not understand the concept well enough, she 
feels, to explain and demonstrate to others. 

I am trying to help her with a lesson plan that would be unique and 
simple so that she can learn as well as teach others her age. I have 
gone to various sites and am having a difficult time finding something 
I can explain to her and that she can use.

Date: 10/25/2000 at 13:09:55
From: Doctor Peterson
Subject: Re: Equivalent fractions

Hi, Karen.

One thing to do first is to go to our search page and enter the phrase 
"equivalent fractions" (don't use the quotes, and check the button for 
'that exact phrase'). There your daughter will be able to look through 
examples of how we have explained the concept to others, which may 
give a number of different approaches to try - one of which may be 
just what she needs in order to feel more confident about it.

Here's my favorite way. Take a sheet of paper and divide it into some 
number of columns, and shade some of them in:

     +---+---+---+
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     +---+---+---+

Now make several copies of this, by hand or with a copier. Label the 
shaded part on the original "2/3" (in my example).

Now take one of the copies and draw a line across the middle

     +---+---+---+
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     +---+---+---+
     |XXX|XXX|   |
     |XXX|XXX|   |
     |XXX|XXX|   |
     +---+---+---+

Suddenly the 2/3 has changed to 4/6! Write "4/6" on the shaded part. 
You could have fun with this and act like it's a magic trick, making 
the change behind a scarf. If someone puts the act down, saying 
nothing really changed, well, that's the whole point. Fractions aren't 
magic, they're just common sense. 

Now ask someone in the audience if (s)he can see how to change the 
thirds (on another sheet) into ninths, or twelfths. After repeating 
this a few times, you will have a set of equivalent fractions taped to 
the board: 2/3, 4/6, 6/9, 8/12, and so on. These are all different 
ways to name the same fraction.

Now here comes the real math. Write on the board:

      2     2     4
     --- x --- = ---
      3     2     6

This explains what the first equivalent fraction means: we have twice 
as many pieces in all (the denominator - which means "namer," since it 
says that each piece is a "third"); and also twice as many pieces in 
our fraction (the numerator - which means "numberer," since it tells 
how many of those pieces we have.) In multiplying both by the same 
number, we do not change the meaning of the fraction.

If there is time, your daughter might like to think up a way to do the 
same sort of thing, but starting with a fraction that is not in lowest 
terms, such as 4/6. One way is to color in 4 of 6 columns (probably 
best done with the paper in landscape position), then cut them all 
apart and rearrange them from 1 by 6 into a 2 by 3 formation, with the 
shaded pieces looking the way they do in my picture for 4/6. Proper 
use of tape can change the fraction to 2/3. The fact that both 
numerator and denominator can be evenly divided by 2 allows this to be 
done; you couldn't make such a rearrangement for 5/6, though you could 
for 3/6. 

There's a lot to learn by playing like this!

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