Fraction Multiplication and Division Derivations

The following are from Alan Kay's derivations of Fraction Multiplication and Division (w/o Algebra)

Below is an excerpt from an email from Alan about the project:

Through the magic of how the Squeak VM and the Squeak image are done, this image from Disney times (before 2000) can be run today on a completely different Mac CPU and OS.

About 80% of the elementary school teachers I asked back then what turned them away from math said "invert and multiply". Schools have almost always introduced this as an article of faith in 5th or 6th grade. So it is quite incomprehensible and completely non-math.

If they waited for a few years, this can easily be derived by algebra. But they don't wait.

So I decided to try to do a completely iconic derivation (that is also a proof) that would be more in the 10 and 11 year olds' mental wheelhouse. 

I wanted to avoid algebra, and I also wanted to avoid having to use the "multiplication of fractions" rule. No one worries about this one because "it looks reasonable", but in fact it is the harder of the two relations. It was one of the triumphs of Greek mathematics.

So the division of fractions project directly derives the "invert and multiply" result without needing the multiplication rule.

The white arrow in the yellow explanation oval will go to the next step. I thought back then about being able to also run this backwards but didn't do it. Being able to freely go in either direction really helps I think, so this should be added.

There is a project link to a similar sequence that shows an iconic derivation for the more difficult and subtle multiplication of fractions.

I still think that fractional arithmetic and its understanding should be delayed -- but if schools don't have the wisdom and courage to do this, they should at least teach it -- and all math -- only via understanding, not via faith in a rule.

Cheers,

Alan

Fraction A Day - Fraction Multiplication

Today I revisited Fraction Multiplication adding controls to the Fraction Multiplication visualization to simplify modifying the fractions.  So that when you mouse over the fraction controls appear to allow you to modify the numerator and denominator.  I also played around with larger buttons to make it easier on touch screens, but that still needs some work.

I would also like to make it easier to facilitate comparisons, so kids can compare 2/4 x 2/4 with 1/2 x 1/2 etc.  The goal being to make the comparisons easy to make so they can focus on the fractions rather than the artifacts of making copies of fraction images.

Etoys/Google Apps Integration takes 2nd place at Google Hackfest

Steve Thomas and Brett Schlueter 

A lot of fun at Google Apps Hackfest in NYC (although I thought it was a Google App Engine Hackfest and had to do some quick thinking to come up with a new project idea).  There's a lot to be said for having the API authors there with you to get you quickly over technical hurdles (special thanks to Vic Fryzel for his help).

We were able to use exported images from an Etoys book to create a Picasa Web Album and a Google Doc with images from the Etoys project, that could be commented on by the teacher.  We were also able to integrate this into Google App Engine to play a slideshow of the project pages and have a link to the google doc, so a teacher could comment on it, or the student could use it, to further describe the project.

Fraction A Day - Day 36+ - Tangrams

Fun with Tangrams in Etoys.  Create shapes and stories with Tangrams.  You can also explore area and fractions.  Ed Emberly also has a wonderful activity using shapes here (click on the animation icon, below "Cut and Paste, then try it yourself).

The Noonday Project - Kids measuring the circumference of the Earth

Measuring the Circumference of the Earth

The Goal of the Noon Day Project is to have students measure the circumference of the earth using a method that was first used by Eratosthenes over 2000 years ago.

Students at various sites around the world will measure shadows cast by a meter stick and compare their results.

From this data students will be able to calculate the circumference of the earth. Click here to get to their site and register.

          

The Eratosthenes Experiment

Watch as Carl Sagan describes some of the background surrounding Eratosthenes' experiment, which is the basis for this project.

Fraction A Day - Day 35+ - Multiply Me

Kids love pictures and especially pictures of themselves (don't we all). So this Etoy allows them to use their own pictures to play with Multiplication (and fractions).  By clicking on an image they can create multiple copies of themselves. Which can be used to ask how many of you are there in row 1, row 2, etc or what is the size of an image in row X if the image in row 1 is our unit size of 1?



Below is a copy of the script:

If you click "show code textually" (from the menu at the top left of the Script editor) you can see the smalltalk/squeak version as well.

Fraction A Day - Day 34 Is it Half a Star?

Using a gridded playfield allows kids to see the area for different shapes. Using the Fractionator (and a playfield with gridding turned on) I was able to get the images below, which are 1/3, but are they 1/3 of the Star? Let the kids ponder this for a while, then ask "What's your unit/whole/one?"