I went away for a 2 week vacation in December. When I arrived home I realized I had left the faucet dripping by accident. About how much water did I waste in the 2 weeks? Your answer should be in a form that will give me a clear idea or picture of how much water I wasted. Please use the chart of equivalent weights and measures. Since I did not tell you how fast my faucet was dripping, be sure to include the information about your dripping faucet that will help me to compare it to my dripping faucet.
I teach sixth grade math and science and I wanted a problem that would link the two subjects. We just finished a unit on density where we did a fair amount of measuring volume of regular and irregular shaped objects. We were working with cubic measure and I wanted to assess how many students could use their understanding of volume to give me a solution that really gave me a feel for how much water I wasted. This problem has students experimenting to collect data to use in their solution. This relates to the NCTM Curriculum and Evaluation Standard 1: Mathematics as Problem Solving - students can use problem-solving approaches to investigate and understand mathematical content; and develop and apply a variety of strategies to solve problems, with emphasis on multi-step and non-routine problems. The students will also need to communicate their solution in writing which covers Standard 2 - Mathematics as Communication - discuss mathematical ideas and make conjectures and convincing arguments.
The problem also has students using Standard 5: Number and Number Relationships - students can understand, represent and use numbers in a variety of equivalent forms in real-world situations. Hopefully it has students thinking more about volume and different ways of expressing their solution, which is in Standard 12: Geometry - develop an appreciation of geometry as a means of describing the physical world. Finally, the problem also has students thinking of how to use a chart of equivalent measures to change from one unit of measure to another. This skill is described in Standard 13: Measurement - students can make and use measurements to describe and compare phenomena and understand the structure and use of systems of measurement. All the students went home and collected their water. Some did not record enough information about their drip rate to allow me to compare it to my drip. Almost all students could work the computation (amount of water in one minute, to one hour, to one day, to two weeks) using calculators. The most difficult part was getting them to express their solution in a meaningful way. Some found the equivalent measures chart confusing and difficult to work with.
The students were given the problem at the end of a class period and some discussion was necessary. They brought in their individual water dripping information the next day and worked the whole period. Some kids still needed more time because at the end of the period we talked more about making their solution meaningful to themselves and to me.
This problem could be used in science or an environmental education unit. I made sure everyone understood the problem. We talked about fast drips and slow drips and that I would be able to compare their results to my drip if they could describe how fast their drip was. I told them to collect their information by experimenting with a drip at home (we talked about how to get your faucet to drip). I told them if they wanted to, they could bring in their water and use some of the measuring tools we had in school, or they could use any measuring tools they had at home. We looked over the equivalent tables and discussed some of the equivalents so they would be familiar with the chart. I found it necessary to repeat many times and give specific examples as I tried to get my students to understand what "a meaningful solution" was. I asked them if a student said that I wasted 143,859 drips in the two weeks - would that have any meaning to them? Did that solution (although maybe correct) give them a clear and meaningful idea or picture of how much water I wasted? I would then refer them to the chart to see if they could find a better equivalent.
- Chart of equivalents (see page 5)
- Graduated cylinders
- Measuring cups
- Teaspoons
- Tablespoons
- Calculators
Answers vary, but the best solutions were students who related the amount of water to the sink at home or at school, gallon milk cartons or some kind of cube or rectangular solid that they drew.
The student attempts to find the number of cubic centimeters wasted in two days not two weeks. The student multiplies 48 (hours) by 14,400 (cc in 24 hours) and thinks s/he is finding the number of cc in two days, so the student is applying inappropriate concepts. Their strategy is not helping to solve the problem.
This solution is not complete. The student tells me how many drips s/he got in two weeks, but failed to work with any standard measure in order for me to get a feel for how much water I wasted. Since this was part of the problem, the student's strategy was only partially useful, leading toward a partial solution. There was some evidence of mathematical reasoning when the student successfully found the number of drips for two weeks, but there is also some confusion. The student records that 1 drip = 1 cc + 1 ml. Since a cubic centimeter equals a milliliter, does that equation mean that a drip is the same as 2 cc? Or is the student confused. The student never uses that information, so it is hard to tell. The solution is not clearly presented.
This student shows a clear understanding of the problem. S/he works the problem successfully although not very directly. S/he gives the solution in three different ways; first the number of gallons that was wasted; then s/he uses the chart to change that to cubic feet; and then goes back to the gallons and uses milk cartoons to give me a better picture of how much water was wasted. S/he has a clear explanation (especially by his/her use of labels), detailing how the problem was solved. All the steps are included.
This student has a deep understanding of the problem and also uses his/her knowledge of volume and proportions to express his/her solution in a more meaningful way. His/her explanation is very clearly shown and explained by labeling all the computational steps. S/he uses a very efficient strategy. S/he also adds at the end, the cost of the wasted water by finding the cost/gallon of town water.
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