Q&A #105

Math 'to the power of'

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From: Cindy Wilkins (for Teacher2Teacher Service)
Date: Jun 04, 1998 at 16:56:13
Subject: Re: Montessori Math "to the power of" - exponents

As a Montessori teacher and a public school teacher, I was intrigued and 
dismayed by your question. The entire foundation of Montessori education is to 
prepare the classroom environment in such a way as to stimulate a child's 
natural curiosity about the world. Also, lessons are to be taught from the 
concrete to the abstract. It seems as if your son's teacher skipped the 
concrete and went to the semi-abstract. It is only through the concrete 
activities that interest and excitement of learning have a chance to develop.

Although circumstances prevent me from teaching Montessori now, I insert 
Montessori activities and teaching structure into my public school classes at 
every opportunity. One activity that I've found helpful and enjoyable comes 
from "Mathematics Teacher."

Tape two pieces of construction paper together so that the resulting rectangle 
is as long and narrow as possible. Find or make several sheets of grid paper - 
I like 2 cm square paper the best. You are going to make a model of the powers 
of 2.

First, cut out a 2 x 1 rectangle. This represents 2^1. Glue it onto the paper, 
midway between top and bottom of the sheet (once you've read the entire 
activity, you'll be better able to visualize where the best position is for 
this first placement). Above this rectangle, label it 2.  Below this 
label it 2^1. Below this label, label it 2.

Now, if you were to double the size of your rectangle, you would have 4 
instead of 2. Cut out a 2 x 2 rectangle (for this activity you always want a 
shape that is as close to a square as possible, so don't cut out a 4 x 1 
rectangle.)  Glue this shape next to your first one. Above, label it 4. Below, 
label it 2^2, and below that label 2 x 2 (for length x width).  

Double the size of your new rectangle - getting a 2 x 4. Cut it out, glue it 
down, and label accordingly. Usually, I can go up to 4 x 4 without the shapes 
getting too large.

Here's the part of this exercise I like best. Look at the patterns you've 
created. Each rectangle doubles in area as you go up the chart. Exponents go 
from 1 to 2 to 3 to  4. Extend the pattern downward.

If you multiply by 2 as you move up the chart, what would you do to move down 
the chart? You would divide by 2. Look at your 2 x 1 rectangle. What would you 
have if you divided it by 2?  a 1 x 1 shape. Cut out this shape and glue it to 
the paper.

Label above the shape 1, and label below the shape 2^0 with no second label 
below that.

What would you have if you cut that shape in half?  1/2 x 1. Cut it out, glue 
it down, and label:  1/2 above the shape, 2^-1 below the shape, and 1/2 below 

The next shape would be 1/2 x 1/2, and its labels would be 2^-2 and 1/(2 x 2).  
Continue until you have shapes made and labeled up to 2^-4.

This activity really helps my students visualize negative exponents, exponent 
of zero (do the same activity with powers of 3 - every level is tripled 
of doubled, or divided by 3 instead of 2, and 3^0 still = 1) and the rapid 
growth of exponential patterns.

Hope this helps!

Cindy Wilkins

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