In order to construct a mathematical model, students need to become familiar with the context of the problem. They need some time to think about the various quantities involved and to describe their relationship informally. Then they need to be guided toward a reasonable choice for a mathematical model of that relationship. All of this is the aim of the opening conversation.
When harvesting timber from a forest, there are many factors to consider. On the one hand, the lumber company must be able to cut enough trees to make a reasonable profit. At the same time, however, the health of the forest must be preserved (or enhanced) by the harvesting, which means the company typically cannot cut down all of the trees, or simply cut down all the biggest trees. (Alternatively, you might open the class with a video like this one.)
To help make decisions about harvesting, the lumber company needs to be able to predict how much lumber can be produced from trees of different sizes. In general, taller trees produce more wood. But tree height isn't easy to measure until after the tree is cut down. On the other hand, the diameter of the tree is easy to measure, and the diameter is directly proportional (in general) to the height of the tree.
A mathematical model can help! The diameter of a tree is generally an indicator of the amount of lumber that the tree will produce. In mathematical terms, the amount of lumber is a function of the diameter, L(d).
What kind of function? Well, diameter is a linear measure (feet), but the amount of lumber is a cubic measure (board-feet, or 1ft. x 1ft. x 1in.), so L would reasonably be modeled with as a cubic function of d.
Ok, so in general L(d) = aL^3 + bL^2 + cL + d.
In order to turn this into a usable model, you'll need to find appropriate values for the coefficients a, b, c, and d, and to do that you'll need some data on the actual relationship between diameter and lumber.
Now, hand out the Lumber Problem and let students begin!
Now students will work in small groups (three students, max.) to use a system of equations to find the particular solution to this problem. Their goal is to write an equation to model this situation, and then to produce a data table and graph of the function.
Please see this video for details and suggestions.
Now that the students all have a working model - an equation and its associated graph - it's time to solve the problem. This is going to push them to think about the context of the problem and move freely back and forth between the mathematical model and the actual quantities it represents.
Please see this video for the details and for some tips on what to expect.