Jed bought a generator that will run for 2 hours on a liter of gas. The gas tank on the generator is a rectangular prism with dimensions 20cm by 15cm by 10 cm as shown below

If Jed fills the tank with gas, how long will the generator run? Explain how you arrived at your answer.
 Circumstances of performance This sample of student work was produced under the following conditions: - alone in a group - in class as homework with teacher feedback with peer feedback - timed opportunity for revision with manipulatives with calculator

This student had a conversion table which contained many conversions including the two needed to solve this problem.

 This work sample illustrates a standard-setting performance for the following parts of the standards: c Number and Operation Concepts: Have facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications. g Number and Operation Concepts: Carry out proportional reasoning. k Geometry and Measurement Concepts: Work with geometric measures of length. b Problem Solving and Mathematical Reasoning: Implementation. c Mathematical Communication: Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

The key to solving this problem lies in these three relationships:

1. (time in hours the generator will run) = (liters of gas in the tank) • (0.5);

2. (liters of gas in the tank) = (volume of tank in cc’s) • (1,000);

3. (volume of tank in cc’s) = (length in cm) • (width in cm) • (height in cm).

Students can derive (1) using basic proportional reasoning from the statement that says the generator will run for 2 hours on 1 liter of gas.

Relationship (2) is the conversion between cubic centimeters and liters.

Relationship (3) is the basic volume formula for rectangular solids.

Putting the three together provides a solution to the task:

4. (time in hours the generator will run) = (0.5) • (1,000) •(length in cm) • (width in cm) • (height in cm).

(Note the units of the constants: 0.5 has the units “hours per liter” and 1,000 has the units “liters per cubic centimeter.” A dimensional analysis of the formula (4) shows that combining all the units on the right hand side of the formula yields just “hours,” as it should.)

What the work shows
 c Number and Operation Concepts: The student has facility with the mechanics of operations as well as understanding of their typical meaning and uses in application. The student correctly showed the calculations involved in finding the volume. g Number and Operation Concepts: The student carries out proportional reasoning in cases involving part-whole relationships. The student successfully converted cubic centimeters to milliliters and milliliters to liters. k Geometry and Measurement Concepts: The student works with geometric measures of length and volume. The student computed the volume of the gas tank using the length, width, and height of the gas tank. b Problem Solving and Mathematical Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution. The student employed the formula to find the volume of the gas tank, converted the volume from cubic centimeters to milliliters and milliliters to liters. Finally, the student concluded that with 3 liters of gas the generator will run for 6 hours. c Mathematical Communication: The student organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly. The student provided a step-by-step explanation of the procedure used to arrive at a solution. Although it is clear what the student intended, the use of the equals sign here is incorrect. The minor error (the inclusion of “the” in the fifth line from the bottom) does not detract from the overall quality of the work.