# The "water triangle" proportional reasoning task

(I previously asked this at The Mathematics Teaching Community, but I'm hoping it would attract further answers here.)

The Wikipedia page on proportional reasoning mentions a "water triangle" task. It seems that the task was created in the 1970s, possibly by Robert Karplus. However, I was not able to find any reference to it. Does anyone know of any publication that mentions its source?

Consider a container of colored liquid inside a right triangle where the triangle can be tilted and the water levels on the left and right side can be measured on a built-in scale. This is called a "water triangle":

The water triangle is rotated until it shows a measurement of 4 units on the left side and 6 units on the right side. Suppose the triangle is tilted even more until the water level on the right side is at 8 units. Predict what the water level in units will be on the left side.

• I didn't find the reference but while looking, got lost in a whole array of fun things, including the Karplus Workshop here: digitalcommons.unl.edu/karplusworkshop Thanks for sending me on this fun hunt! Commented Mar 25, 2014 at 22:19
• @ChrisCunningham, I've been many times to the link you provided. The documents there have been scanned as graphics, and so the only way to search them for certain terms is to download and read them. So far I haven't found the info I'm looking for. But I do agree that the works of Karplus and the others are very important.
– JRN
Commented Mar 25, 2014 at 23:54
• Does the link to the Mathematics Teaching Community still work? The page doesn't load for me, but it could be a problem on my end...
– user7990
Commented Aug 29, 2020 at 8:08
• @Brahadeesh, the MTC website has been down for quite some time now; I don't know why.
– JRN
Commented Aug 29, 2020 at 9:35

The instrument pictured was created by Barry Kurtz. He writes by email:

I completed my PhD under Bob Karplus at UC Berkeley. I was his last PhD student. My dissertation dealt with teaching for proportional reasoning. I invented the idea of a "water triangle" to teach inverse proportions. These were all made by the workshop at the Lawrence Hall of Science; they were not a commercial item. I doubt any exist today; I certainly don't have any.

The thesis can be found through DigitalDissertations; the citation is:

Kurtz, Barry Lloyd. A Study of Teaching for Proportional Reasoning. University of California, Berkeley, ProQuest, UMI Dissertations Publishing, 1976.

An image of the water triangle appears on page 34:

Here is an additional picture of an actual water triangle:

• Can you share how you found it? I am in awe. Commented Mar 27, 2014 at 21:05
• The trick is to dig through wikipedia until you find the photograph. (This is also at the OP's earlier link.) The photographer, Kurtz, uploaded it as original content; so I googled his name with Karplus to see if they had worked together - they did. Searching DigitalDissertations for his thesis and looking at the table of contents leads to the reference above; emailing Kurtz was the final piece. Commented Mar 27, 2014 at 21:27
• (For what it's worth, I think Professor Kurtz was probably pretty surprised to receive my email. His response ends: "Thanks for your interest. You did a good job tracking me down!") Commented Mar 28, 2014 at 2:34
• Thank you very much for this!
– JRN
Commented Mar 28, 2014 at 13:05
• @BenjaminDickman, I used the reference you provided in a poster I made.
– JRN
Commented Jun 12, 2015 at 14:05