Introduction >>

The big holiday party is today. You are supposed to take four gallons of punch. All you have is two unmarked jugs, one holding 5 gallons of liquid and one holding 3 gallons, and no time to go shopping. How can you get exactly four gallons using only the two containers you have?

Sound like combinatorics and number theory to you?

“Fill 'n' Pour” is an activity in which students use two containers with relatively prime capacities to discover ways to make specified quantities of liquid. It is suitable for both middle and secondary school students in any mathematics course. The activity gives an example of a problem-solving task. Students may initially try to solve it by means of trial and error. Through trial and error, they may develop a strategy that works—or one that produces other quantities of liquid.

This article suggests two different strategies for the group activities, one with and one without the availability of liquid in the classroom. Also included is a link to an applet featuring the activity. Using the applet allows students to explore multiple number combinations in a short amount of time and, therefore, be able to concentrate on underlying concepts. The applet also replicates the pouring of liquids from one vial to another, a benefit for classrooms without water supplies. In classrooms with a water supply students can physically explore the mathematics. In addition to the exploration of the mathematics, there is intrinsic value to doing the activity both ways. If students do this activity using an applet, they will learn more about using applets and will benefit from the ability to start over any time they want. If they do this activity in groups, they will learn from working with one another and may be more likely to solve the problem than if they were working individually.

Note: This material is partially based upon work at the Park City Mathematics Institute, supported by the National Science Foundation under Grant No: 0314808 to the Institute for Advanced Study, Princeton, New Jersey. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect those of the National Science Foundation.

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