How Large is a Ton of Rock? -- Thinking about Rock Density

Quantitative Goals: Students will work with the concept of weighted average in a variety of ways. They will also work with ratios, in particular density, as they go from weight (mass) to volume. They will gain more experience with unit conversions. Finally, they will visualize the contrasting sizes of cubes and spheres as they work with the formulas for the volume of cubes and spheres and their edge lengths and diameters, respectively. In the process of solving the problem and building the spreadsheets, they will increase their know-how of weighted average, volume formulas, and school algebra (rearranging equations).

Spreadsheet Goals: Students will build spreadsheets to work through the step-by-step calculations. Some of the cell equations get complicated because of the embedded unit conversions. In the process of building their spreadsheets, students will learn to organize their thinking about a calculation. They will learn to pay attention and be careful when they work with equations (watching for parentheses, for example). They will learn that their first attempts are not necessarily correct (when their cell equations do not result in the same numbers as shown in the module), but that Excel will indeed produce the same answer as the module when they get the mistakes out of the cell equations -- that Excel does exactly what they tell it to do, which is not necessarily what they think they are telling it to do. They will come to appreciate the built-in SUMPRODUCT function.

Geology (Content) Goals: Students will work with the concepts that:

• a rock consists of one or more minerals,
• the density of a rock depends on the kind and relative amounts of the constituent minerals,
• the volume of a given weight (or mass) of rock depends on the rock density, and
• porosity, too, affects rock density.

I use this module in my Computational Geology course, GLY 4866 (Acrobat (PDF) 39kB Sep25 06). The course is aimed at geology majors who have completed or are about to complete the mathematics courses required for the geology major. The class consists of students who anticipate graduating in three or fewer semesters.

This module, the first of the course, comes in the second week of the semester. It accompanies the first problem-solving session. The preceding session is a computer-lab session which introduces Excel. The Ton of Rocks problem-solving session happens in a lecture room equipped with computer and projector. I start the session by posing the question: "How large is a ton of rocks?" I introduce the class to the fact that the class sessions will consist of such questions, and that they will be dividing up into groups to consider how to solve those questions. We discuss strategies of working in groups, and they go after the ton of rocks question.

The students soon decide that they need to know what kind of rock they are thinking about. What kind of minerals are in it? And what do I mean by how large anyway? The calmer tables think I mean volume. The more demonstrative tables motion the dimensions (length, width, height) of volumes. So we discuss the first step of Polya's heuristic: understanding the problem and drawing a figure. With agreement as to what the problem is (and a decision to start with a monomineralic rock), they discuss amongst themselves strategies to come up with a calculated length. They successfully design a plan in time for me to show them the first spreadsheet (Slide 4) and discuss it before the end of the 1-1/2 hr session. They leave the session to complete the module (polymineralic rocks), which has "gone live" on Blackboard during the class session, and to send the first set of end-of-module assignments to the TA within a week.

The class following the problem-solving session is in the computer lab. The objective of the computer-lab session is to tutor the students in Excel to the extent that they leave with working spreadsheets for the first set of end-of-module assignments (through Slide 6). The students leave to complete the second set of end-of-module assignments and send it to the TA within a week.

The module is a PowerPoint presentation with embedded spreadsheets. The PowerPoint includes links to information about the setting of the Duluth Gabbro, Stone Mountain Georgia granite, and Fountain Arkose.

If the embedded spreadsheets are not visible, save the PowerPoint file to disk and open it from there.

This PowerPoint file is the student version of the module. An instructor version is available by request. The instructor version includes the completed spreadsheet. Send your request to Len Vacher (vacher@usf.edu) by filling out and submitting the Instructor Module Request Form.

The module is a computer-based activity. It is helpful to remind the students that it is still useful to have a pad of paper and a pencil at the ready. They will find that such equipment, although primitive, helps their thinking, especially when they try to invert the formula for the volume of a sphere.

Remind the students of the first rule of building spreadsheets: save your work. Early and often.

End-of-module assignments can be used as a post-test.

The instructor version includes a pre-test.