(using technology to explore a local enterprise)

By: Richard Francisco and Katie Gilbert


We based our instructional unit on the premise that many people have, that one does not need mathematics if the they do not plan on atttending college or pursuiting careers known for involving mathematics. A huge misconception many students have is that mathematics has no relevance in the blue-collar industry. The goals of this lesson are to dispell this myth, to demonstrate, explore and discuss mathematics through a field operation, and through investigative activities to extend the concepts from the farm to general mathmematics with the use of technology. We will use a field trip to a local Dairy Farm, Gilbert and Sons Dairy Farm, to achieve these goals.

Here is a brief description of the mathematical rationale behind the trip to the dairy farm:

The Gilbert family, who own Gilbert and Sons Dairy Farm, were kind enough to allow our class to visit their farm and examine mathematics in their workplace. Russ and Katie Gilbert, whose family farm is located at 2281 Appalachee Rd. in Madison, GA 30650, were the primary contacts and facilitators for the field experience. They proved quite the instructional duo, as Russ is an intelligent, thoughtful second generation farmer, while his wife Katie is well-versed in mathematics, having received a bachelor’s degree in mathematics from Emory University and is currently pursuing a master’s degree in mathematics education at the University of Georgia.

Dairy farming is a class of an animal husbandry enterprise. The goal of the Gilberts’ farm is long-term production of milk. In general, milk produced by dairy farms may either be processed on-site or transported to a dairy factory for processing and eventual retail sale. The Gilberts usually sell their unprocessed product to the Publix grocery chain’s dairy division for pasteurization, but in certain months they may opt to pasteurize the milk themselves in order to increase profits. In addition to milk production, there is a secondary byproduct of their farm, namely the male calves born by their cows, which are usually sold as studs to nearby farms or for (unfortunately as admitted by the family, who have strong ethical beliefs regarding the humane treatment of animals) veal production. Finally, the farm also grows and produces their own feed, usually alfalfa and hay, but these products are usually used solely for consumption by the cattle, and rarely sold on the market.

Quantitative literacy in mathematics is vital (and perhaps surprisingly so to urban populations) to successful dairy farming. In addition, a dairy farmer is an entrepreneur whose trade requires that they be well versed in zoology, botany, economics, and other academic subjects. Moreover, the dairy farmer must be skilled in management across each of these academic domains. We assert that mathematics is the core subject which links the content specific knowledge in these respective domains into a coherent production schedule that dictates the daily activities of farm management. We have divided the mathematics of dairy farming into (but not limited to) four main content areas: mathematics of maintenance/construction, mathematics of field management, mathematics of animal management, and mathematics of economic/general farm management.

1. Mathematics of Maintenance/Construction:

The dairy farmer who wishes to incur few extra expenses on his business must be proficient at general construction and maintenance. The Gilberts recently constructed and installed a new roof to their barn. They were required to first formulate a theoretical model of the new roof and its elevation (using arctangents I am told!), and conduct accurate measurements that determined the required raw materials. The ability to conduct accurate measurements in woodwork and assemble raw materials is a valued skill. For this particular project, not having to hire outside contractors saved the Gilberts several thousands of dollars.

In general, constructing facilities without having to hire outside contractors will greatly increase the farmer's profits. This observation will hopefully provided motivation to students that mathematical knowledge would facilitate monetary conservation. In particular, they need to be able to have an understanding of geometry (and possibly trigonometry) in order to construct new barns or other storage facilities. We are considering possible avenues of exploration in these subjects.

Furthermore, the task of creating fencing contains some interesting mathematics as well, as barbed wire must be wrapped around fence posts. We were also explicitly told that it is imperative to obtain perfection in construction of cow shoots for waste disposal. The shoots should be the most economical and practical - for obvious reasons! Mathematical modeling becomes important for both of these situations. It is impossible to predetermine the amounts of needed materials without first constructing an algebraic (or analogous) model. My preliminary ideas for problems include setting physical geographic constraints and asking students to create optimal (or simply exact) calculations of required materials using mathematical language to simplify the task.

The farmer must also maintain stable field growth to sustain the livestock. On the Gilberts' farm, the food for the dairy cows is a mixture of hay, feed, and brew. It is imperative that there are sufficient quantities of each crop for the mixture. Accordingly, there is a necessary margin of security required, as you don't want weather fluctuations rendering your fields inadequate. Analyzing past data on weather and climate helps the Gilberts make informed decisions on appropriate quantities to plant. Possible activities for high school students would be to analyze some previous data and interpret confidence intervals to determine appropriate field allocation.

2. Mathematics of Field Instruction:

There are some rather obvious examples of mathematics in field management that will (hopefully) be clear to high school students at first glance. Namely, calculating the size of the field, the necessity to divide up the field to grow different crops, etc. will be concepts with which even pre-algebra students will (hopefully) have had previous exposure. We can hopefully extend these notions with problems involving the time it will take to cut a field with a tractor taking into consideration the size of the mowing blade.

We also have some particular classic, more challenging problems in mind here, which pertain more to general problem solving, including a field sustenance problem and a 'tethered goat' problem.

3.Mathematics of Animal Management:

Supporting the cattle is also a delicate matter in which interesting mathematics occurs. On the Gilberts' farm, milk is weighed each week, and with aid of computer they come up with average milk weight per cow. We will have to probe a little further to analyze the software they use to perform these calculations. They use the data of cow production to calculate production cycles of the cows to determine impregnation cycles. After all, pregnant cattle produce the milk! There could be some interesting problems determining possible milk productivity after a certain number of years, based on the probability of having a male/female cow.

Furthermore, the physical process of milking contains interesting mathematics. Some methods and technological inventions of milking make the process more manageable.We briefly brainstormed some different 'milking structures', i.e. linear or circular configurations and the amount of time required to complete the daily milking using each method, respectively.

4. Mathematics of Economic/General Farm Management:

The Gilberts must manage their farm in a manner similar to all other American enterprises. They must manage the aforementioned concerns, and examine the farm holistically to determine possible improvements to productivity. We have in mind some general troubleshooting problems for students in this regard. We may present them certain scenarios, and ask them to select the optimal solution to maximize profit. Linear programming has uses in almost all economic endeavors.

Creation of the Lesson Plan:

The lessons were (hopefully) intentionally created to allow for tiered instruction in a heterogeneous classroom to accommodate the diverse aptitudes and readiness levels of individual students. The content of the activities requires students to have a prerequisite working knowledge of the concept of a variable, and some experience at representing physical situations algebraically. It is also recommended, though not essential, that students demonstrate prior proficiency with graphing lines and computing areas of basic shapes. Moreover, perhaps the most essential prerequisite knowledge required is specific process training with Microsoft Excel and Geometer’s Sketchpad software. Successful delivery of the activities mandates that students be both comfortable and proficient at explorations with these computer software packages. These prerequisites make the activities ideal (depending upon student aptitude and readiness) for incorporation into the later portion of an Algebra I course, a Geometry course, or an Algebra II course. It should be explicitly mentioned that the mathematics involved in dairy farm management is goal-directed, in that the farmers’ desire to optimize production and maximize profits is at the core of their farm management.

This post-field trip lesson was created with certain goals in mind. The first goal was to provide students with the opportunity to engage in the practice of representing real world problems algebraically. Translating properties of the real world into mathematical dimensions is a key type of reasoning that should be developed during a high school mathematics curriculum. The second major goal of the activity is to engage students in the process of investigative optimization, particular of the areas of polygonal shapes. In particular, this goal depends on the arithmetic/geometric mean and calculation of local maxima of parabolas. The third primary goal is to have students investigate the areas of lattice polygons. The investigations will hopefully allow them to hypothesize a formula to calculate the area of lattice polygons based on the number of boundary and interior points. The final goal of the activity is to provide students with some challenging problem solving activities, which will most likely (depending on time) be assigned as homework and then discussed the next day. These problems will be connected to the assignment, but will be significantly harder than the activities develop during the in-class portion of the lesson. These activities will help achieve the goal of providing differentiated curriculum for identified gifted students in the class, as well as providing all students the opportunity to engage in higher order thinking tasks according to Bloom’s taxonomy.