Consortium for Mathematics and its Applications

Product ID: 6903
High School

Tools of the Trade (Teacher)

Author: Project Co-Directors: Dave Engelhard, Michael Kallaher & Jack Robertson

Table of Contents

Front Matter
Unit 1 : Unit Conversions
Unit 2 : Agricultural Aviation
Unit 3 : Washington Apples
Unit 4 : Erosion Problems
Unit 5 : Negotiating the Maze
Unit 6 : Growing Degree Days
Unit 7 : Growing Degree Days Projects
Unit 8 : More on Growing Degree Days
Unit 9 : Effect of Burning Stubble on Wheat Yields
Unit 10 : Growth of Wheat
Unit 11 : Estimating Wheat Yield
Unit 12 : Wheat Yield Table
Unit 13 : Washington Farm Sales
Unit 14 : Farm Product Sales
Unit 15 : Present and Future Value of Money
Unit 16 : Buy or Lease
Unit 17 : Pests, Population Growth, and Chaos
Unit 18 : Interpreting Slopes and Intercepts
Unit 19 : Using Algebraic Expressions
Unit 20 : Managing Residue Cover
Unit 21 : Why is Crop Residue Important?
Unit 22 : Converting Residue Cover Units
Unit 23 : Estimating Residue Cover
Unit 24 : Finishing the Field Work
Unit 25 : Volumes
Unit 26 : Tanks and Fields—How Big?
Unit 27 : Cutting Logs into Lumber
Unit 28 : Calibrating Tanks—An Overview
Unit 29 : Calibrating Tanks—Estimations
Unit 30 : Calibrating Tanks—Putting Them Together
Unit 31 : Calibrating Tanks—Using Calculus
Unit 32 : Rain and Rain Gauges
Unit 33 : Center-Pivot Irrigation Systems
Unit 34 : Reservoir Storage
Unit 35 : The Farmer and the Elevator—Geometric Constructions in the Field
Unit 36 : The Farmer and the Elevator—Hyperbolas in the Field
Unit 37 : The Shadow Knows: Rectangles in the Sun
Unit 38 : The Shadow Knows: Three-Dimensional Views
Unit 39 : Making the Most of Things

1.What is the source and scope of the material in the book?

The book contains 39 units, designed to supplement any existing secondary course, and all are based on mathematics applications in agriculture. The units, which are appropriate for any student, rural or urban, were generated by onsite visits with a variety of agriculture professionals. Therefore, many units start with an actual real world problem arising in an agricultural setting. The sources are quite varied and come from the farm, agriculture literature, the agriculture businessman, the agriculture economist, the researcher, or other agriculture professionals. The units that do not have a specific application as their primary focus cover important mathematical skills areas used widely in agricultural (and other) settings. Also we have, in some cases, followed the lead of the application into a more extensive examination of the mathematics used to solve the specific problem.

The mathematics content is quite varied. Many problems for younger students requiring no more than routine arithmetic skills are given in the applied context. Geometry problems occur in center-pivot irrigation systems, rain gauge and tank calibrations, reservoir storage, and deciding where to market a crop. Applications of exponential functions come from deciding whether to buy or lease a combine and computing residue covers in fields. The hyperbola appears as a border for regions in which the farmer will haul wheat to a specific elevator and also in the analysis of what can be said about farm product sales based on a newspaper article. Calculus material appears in certain volume problems as well as in applications of growing degree days.

A first draft of the materials was prepared by a writing team that included secondary and college mathematics teachers, a farmer, a computer professional, and an agricultural engineer. Three authors wrote the final version. The materials, which have served as the content for a course taught to four different classes of secondary teachers, have been field tested and revised on the basis of classroom performance.

2. What is the nature of the materials?
The materials are in a pick and choose format. The units are essentially independent and by making appropriate choices can be used to supplement any secondary mathematics class.
Five brief videotapes introduce a number of the units. The student sees the problem in the field that is the basis of the mathematics in the unit. Thus, we see how various types of farm cultivators retain or cover up residue. We see the wheat plant developing and we go into the field for harvest. The large variety of tanks and other storage facilities can be seen before their geometry is studied. We see center pivot irrigation systems in operation. Also there is one umbrella videotape that contains information about mathematics related careers in agriculture, and students hear practicing professionals advise them on the importance of learning mathematics and other necessary skills.

The type of units include standard student worksheets, units based on table and graphical interpretation of information, and mini-research projects in which simple mathematical models are constructed and tested. All of the units can be used in a small student group setting even though many are appropriate for individual work.

The calculator gets a good workout as many computations are required. The graphing calculator is a suggested aid in the residue units and computer projects are suggested in some units. Software* has been developed for the unit “The Farmer and the Elevator” utilizing Geometric Sketchpad.

Besides the abundance of computational problems, there are many problems that are conceptual in nature and require little or no computation.

3. Where do the materials fit in existing courses?
The matrix that follows the preface is designed to help answer this question. Some units are clearly appropriate for the prealgebra, algebra, geometry, pre-calculus or calculus course. However, because the units are based on a real application, it is inevitable and proper that some units will cut across traditional course boundaries. In some units there are pages designed for different classes. Exercises are written in graded format so that the unit can be tailored to meet class preparedness.

* This software can be obtained by sending a 3 1/2” floppy disk to Michael J. Kallaher, Dept. of Mathematics, Washington State University, Pullman WA 99164-3113. The software runs on the MAC version of Geometer Sketchpad.

Most units contain material that we hope would interest any secondary student. Thus the calculus student can profit from a unit that uses no calculus because much of the material will have a new and fresh perspective and application. The self contained units can be used as worthwhile over-and-above work for the gifted student who is ahead of classmates.

The ultimate placement of the materials into existing courses is in the hands of the able classroom teacher, which is the purpose and strength of the pick-and-choose format. And every teacher in every course should have no problem finding material that will help answer that recurrent and legitimate student question, “what is it good for?”

4. What aids are available for the teacher?
Probably the most important aid is the complete solution of every exercise in the book. Also, there are teacher aids for every unit stating purpose, required prerequisite skills, the nature of the unit, and references to related units. Videotape introductions are referenced. Where experimental work is required, a full report on others’ experiences is given with advice based on experience. No activity is suggested that is untested. All of the above is found under the title of “Just Between Us Teachers”.

Two other features are intended to aid the teacher by heightening student interest. One is “Just Between You and Your Parent” where students are encouraged to talk over certain issues with their parents who may have valuable experience to share. The other is “Just Between You and Me.” This is an informal bit of extra information given to the student about the subject at hand. Usually this is what you might say to a student as an extra aside on the material you are covering.

5. Who was involved in producing the book?
As stated above, many have contributed in various ways to the development of the book. Many deserve special thanks.

Agricultural professionals representing a broad spectrum of the industry who helped direct the project in an advisory capacity were Tom Fischer, Robin Herrman, Dana Herron, Glen Jacklin, John Massey, Larry Meyer, and Jon Ochs.

Some of these also helped by providing material for specific units as did others including Gayland Campbell, Ken Casavant, Jim Cook, Ray Huffaker, Don McCool and Gayle Willitt from the College of Agriculture and Home Economics at Washington State University, and Frank Moslemi and Tom Gorman from the College of Forestry at the University of Idaho. Thanks also go to Mike and Gary Boone who allowed us to interrupt their cultivation and harvest work.

Many teachers were involved in the development on these materials. Beth Hendrickson was a resource for both materials and ideas. Part of her contribution can be seen in the format of the book. Our first line pilot teachers Beverly Bunch, Dave Engelhard, Marge Gary, Evelyn McCoard, and Larry Yetter made many valuable suggestions for material revision. Other
secondary teachers, too numerous to list, joined them in summer courses based on the materials and also provided other needed “reality checks” for which we are most grateful. We also thank Dana Herron, Larry James, Troy Joshua, Jon Ochs, and Steve Williams who made contributions to the project through instructional duty in those summer courses.

We are also grateful to the National Science Foundation for the support that made the project possible. The specific help of Jim Sandefur at NSF even reached the pages of the wheat marketing unit. Larry James was a co-director of the project and provided valuable guidance of the effort.

Drex Rhoades from the Information Department of the College of Agriculture and Home Economics is responsible for the creative work done with the videotapes. He was notably patient as he explained many times to us “I am making these tapes for students, not you” and the product attests to the wisdom of the strategy. And as always, there are the indispensable support staff who keep budgets straight, type manuscripts, copy materials, and put up with all that involves.

We especially thank Bonnie Collins, Jaimie Dahl, and Judy Hobart who simply couldn’t have been more gracious or capable in their help.

Project Co-Directors
Dave Engelhard
Michael Kallaher
Jack Robertson

© 2000, COMAP, Inc.
Tools of the Trade
510 Pages

Mathematics Topics:

  • Pre-Algebra ,
  • Algebra ,
  • Geometry ,
  • Precalculus & Trigonometry ,
  • Calculus

Application Areas:

  • Environment & Sustainability ,
  • Ancillary Lesson Materials, Agriculture, Farming, Crops, Climate, Produce, Irrigation


Elementary Math

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COMAP develops curriculum resources, professional development programs, and contest opportunities that are multidisciplinary, academically rigorous, and fun for educators and students. COMAP's educational philosophy is centered around mathematical modeling: using mathematical tools to explore real-world problems.

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