Geometers Sketchpad modeling of division by fractions

[Steve New]

I have some ideas on this topic, but want to have input from those who have already tried and found success.

I want to model division by fractions in a constructivist way. For example. 1 divided by 1/3 = 3. Or 1/2 divided by 1/4 = 2.

I am wondering if it is better to use 2 dimensional figures like squares/rectangles. Or if it is better to use segment lengths (one dimensional).

Is it better to use the built-in calculator when starting to show the results of this algebra concept or is it better to not rely on that feature.

Any insights you can give would be helpful.

Steve New
Math & Science Teacher.

[N. Silver]

Steve New wrote:

>I have some ideas on this topic, but want to have input from
> those who have already tried and found success.

> I want to model division by fractions in a constructivist way.

Using Geometer's Sketchpad is guilding the lily.
It's the wrong tool.

For example. 1 divided by 1/3 = 3.

Simply denote unity by a sum of a number of X's, say 3.

Let 1 = X+X+X.

Then 1/3 = X

How many 1/3rds are there in 1? 3

Or 1/2 divided by 1/4 = 2.

Let 1 = X+X+X+X.

Then 1/2 = X+X.

And 1/4 = X

How many 1/4ths are there in 1/2? 2

> I am wondering if it is better to use 2 dimensional figures
> like squares/rectangles. Or if it is better to use segment
> lengths (one dimensional).

> Is it better to use the built-in calculator when starting to show
> the results of this algebra concept or is it better to not rely on that
> feature.

> Any insights you can give would be helpful.

Of course, everything depends on the level of the students.
Using arrays of X's with no plus signs works well also.

XXXXX
XXXXX
XXXXX
XXXXX
XXXXX
XXXXX

Begin as follows:

1) If the above array represents 1, represent:

A) 4/5 B) 5/6 C) 2/3

2) If the above array represents 5/6, represent 1.

3) If the above array represents 6/5, represent 1.

4) If the above array represents 2/3, represent 8/9.

5) If the above array represents 1, model 2/3 of 4/5.

Answer: First, cross out one column to get 4/5.
Then cross out 2 rows to get 2/3 of 4/5.
How many Xs's remain? 16/30

5) If the above array represents 1, model 2/3 divided by 1/5.
2/3 = 20 X's.
1/5 = 6 's
How many 6's are there in 20? 20/6

In general, it's better to be brief when modeling
operations with fractions. After the students get
the hang of it, move on to something new, New.