Math for Measurement - following a theme through the materials.

Because the course "spirals" it is not so easy to see what depends on what as you skim the book. The purpose of this page is to provide some pointers. This is not a complete summary of what is in each topic, but is enough to help you see the dependence you'd need if you want to cover some of the later topics.

Modeling

The crucial background for modeling is the introduction to spreadsheets in Topic C. In this, students learn the basics of graphing a formula in a spreadsheet and also learn to graph a parameterized formula, so that they can change the values of the parameters and see how the graph changes. Most students need to practice on this for awhile to become comfortable with it.

In Topic H, we cover word problems with linear equations, so this is the basis of linear modeling. Here the students do practice with the algebra of linear equations, but they also deal with many aspects of mapping the real world to mathematical models: input and output variables, adjusting the "zero" by using "years since 1900", interpreting the results, and making predictions.

Since, in modeling, we want to use data, which is noisy, the next important topic is Topic J on data, where we introduce graphing data and understanding and estimating the standard deviation of a set of data points.

Then start in the topics specifically labeled "Modeling." First, in Topics K and P, we introduce linear, quadratic, and exponential models and we have provided template spreadsheets, so students just have to deal with copying the data into the template spreadsheet, making the graph, and estimating the parameter values to get a reasonable fit.

In Topic S, we start automating the fitting and so really discuss the deviations from the model, why we square the deviations to get an overall measure, what the pattern of positive and negative deviations means, and other ideas. At the end of this lesson, we ask the students to model a new formula - one for which they do not already have a template, so here is where they start from a blank spreadsheet.

Topics U, W, and Y are not central to the ideas of modeling - they are good extensions. Topic U introduces semi-log and log-log graphs which can be used to distinguish between data fitting an exponential model and data fitting a power model. Topic W introduces four additional functions: normal, logistic, sinusoidal, and logarithmic. Topic Y addresses combining formulas.

Applied Trigonometry

Topic F provides a critical background. It is designed to teach the students to draw careful diagrams of word problems - meaning diagrams that you can use to obtain good numerical approximations to the answers instead of using algebra, geometry, and trig to find the exact answers. As they practice making careful diagrams, they review facts about angles and diagrams, and learn how to read and use both types of reporting bearing.

The first few topics in the review portion of the course are relevant: solving proportion equations, graphing by point-plotting, rounding numbers, using a calculator, in Topics A, B, D, E.

Our introduction of trig builds on the idea of similar triangles and, in Topic L, immediately uses the tangent function to find the angle of a right triangle when given the lengths of the two sides. Students don't really understand the inverse tangent function in the way we expect trig/precalculus students to understand that function, but they do learn to use it and feel that it is a useful tool. From here we go to full solutions of right triangles and a few related ideas in Topic M. Topic N is really about propagation of error in rounded numbers, but uses right-triangle trig for the examples. There is also a fuller discussion of significant digits as a second part of this topic. I consider that optional, since most students don't need to learn that much detail about using significant digits and it is confusing to them. This completes the right-triangle portion.

In Topic Q, we define trig functions of angles larger than 90 degrees, both from a unit circle approach and by using graphs of trig functions (where the domain is expressed in degrees.) Students learn to solve equations like sin(A) = -0.32 and cos(B) = 0.76 on any given interval by using a calculator to find one solution and a graph and the symmetry to find the others. We do not deal with the tangent function here because it isn't needed to do general triangles and the asymptote at x = 90 degrees is confusing. To deal with spreadsheet graphing of functions across vertical asymptotes would require more materials that we have not found necessary and have not developed for this course.

Then, in Topic R, we introduce the Law of Sines and the Law of Cosines, and students learn to solve general triangles with these, doing the computations by hand with a calculator. The "ambiguous case" is handled in a separate supplement.

In about the 4th week from the end of the course, the students are divided into pairs and each pair works on one case of the trig problems and writes a worksheet to automate the solution of that type of problem. The instructor writes one case as an example and the SSA (ambiguous case) and the SAA case. These are compiled into a trig workbook and the students use that to solve some applied trig problems at the end of the course. This has been one of the highlights of the course. Students have made and leared to use a tool that they see as very useful.

Error Propagation

This begins with having students fully understand how to take a rounded number and determine what interval of actual values would be rounded to that in Topic D. Students think this is trivial, but it needs emphasis because many students are not able to reliably answer these questions - they lapse back into the math course idea that 12 means the same thing as 12.0, etc. In this topic we also underline significant digits - defining those as digits that you really mean that indicates the measured value. We find that doing the rounding and underlining significant digits together is a better introduction than the typical "counting significant digits" that students do in trig courses.

Topic I on error propagation of rounded numbers is next, and then Topic N uses these same ideas on right-triangle trig problems and gives a more classical discussion of significant digits, and then relates the two ideas.

Topic J on data is where we discuss the difference between rounded numbers (errors represented by a box - uniform distribution) and noisy numbers (errors represented by a bell-shaped curve - normal distribution.) We describe how it is easier to see the ideas with the sharp cut-offs of the rounded numbers, but the ideas of error propagation apply to both.

Topics O and T address input sensitivity, graphically and numerically. This is a important concept, and the graphical presentation is generally very easy for the students. Often I don't do the numerical discussion because it seems to distract them from the main point.

In thinking about error propagation of noisy numbers, a major question to think about is whether to assume independence. Standard texts have a major emphasis on the results for independent measurements, but the formulas are often confusing to the students and the concept of independence is deep. We do expect students to learn about reducing variation by averaging independent measurements in Topic X, and the sigma divided by the square root of n result. However, after that, we rely on an empirical method, which is appropriate without assuming independence. - Topic Z. The materials include one last topic, Topic ZA, giving the usual formulas, which I don't cover in my classes.


Last updated January 8, 2009 . Mary Parker