Mathematics For Measurement: “Math for practical arts”
Materials at www.austincc.edu/mparker/talks/math_mar07/
By Mary Parker and Hunter Ellinger
OUTLINE
What: An accessible, stand-alone mathematics course using a theme of measurement to provide
tools for thinking and communicating mathematically
By whom: A math/statistics teacher interested in technical education and an application scientist
experienced in industrial measurement interested in math/science education
For whom: College students who need only one math course for their degree.
What is taught: practical trigonometry, modeling, error analysis (bias, noise, sensitivity, propagation)
Why: Learn to analyze and solve measurement problems and to talk with colleagues about them.
Learn to understand discussion about these topics by more highly trained people in their field.
How (tools): Algebra (linear and proportional equations), formulas (including non-linear),
spreadsheets, graphing (with spreadsheet), diagram construction (simplified technical drawing)
How (tactics):
• Connection between algebraic, numerical, and graphical representations
• Spiraling of topics
• Use of alternate representations to check answers
• Cross-connections between the major strands of the course.
How (strategy): Sophisticated application of simple techniques
Example: techniques: spreadsheets & graphing → application: graphing and exploring a
parameterized curve
Example: technique: graphing (by hand and with spreadsheets) → application: solving equations by graphing.
Example: techniques: spreadsheets & graphing → application: fitting formulas to data and
assessing fit using residual plots
Example: technique: diagram construction → application: solving geometric / trig problems
A hunter is lost in a forest, but can communicate by portable radio to two ranger stations. The rangers at each station
figure out how far away he is from them by how long it takes the sound of a shot to reach them through the air after
hearing it through the radio. Station A is 5.1 miles due west of Station B, and the hunter is 4.6 miles from Station A
and 3.7 miles from Station B. Construct a diagram indicating the possible locations of the hunter.
Example: techniques: spreadsheets & trig → application: students construct an Excel workbook
to solve all types of triangle problems.
Example: technique: Understanding slope
→ application: understanding the sensitivity of formulas (mostly non-linear formulas)
to changes in input as the slope of a curve at a point. Using that to do “forward”
calculations and “backward” calculations similar to those with differentials in calculus.
Example: technique: Understanding slope and intercept of straight lines → applications:
→ understanding slope and intercept as parameters to be varied to find best formula for data
→ fitting exponential formulas and quadratic formulas in vertex form in the same way and
understanding the effects of changing the parameters
How (strategy): Spiraling topic sequences
•
Approximate numbers → rounding → intervals and identification of significant digits
→ error propagation
•
Data → fitting a line by minimizing the sum of squared deviations → residuals
→ standard deviation
•
Approximate numbers → noise → standard deviation → error propagation
•
Data → fitting linear formulas → fitting exponential and quadratic formulas
→ fitting more types of formulas, including some composite formulas
•
Formula → graph → solutions → numerical zoom
•
Linear equations → graphs → slope → input sensitivity
→ fitting → calibration / bias: offset (intercept) & scale (slope)
•
Word problems → diagram construction → right-angle trigonometry → general trigonometry
•
Noise → std dev → Noise in averaged measurements → other measurement combinations
•
Solving equations → word problems (linear equations) → word problems (trigonometry)
Surprises:
Student enthusiasm about using spreadsheets
Impact of diagram construction on word-problem capability
Classroom organization: Daily handouts with outline of day’s class.
Reviews – list of topics.
Very specific guidelines for students organizing their notebooks.
Calendar for Spring 2007
(Each part is four weeks, and followed by a test.)
Part 1. Review, Basic Tools, and Introduction to Error Propagation for Rounded Numbers
Topics A – I. except F
Part 2: Linear Modeling and Right-Triangle Trigonometry
Topics F and J, K, L
Part 3:
• Extension of Approximate Number Discussion: From Rounding to Measurement Errors and
Sensitivity of Formulas to Changes in Input
• Extension of Modeling to Non-Linear Modeling
• Extension of Trig to General-Triangle Trigonometry
Topics N, O, P, Q, R, S
Part 4:
• Calibration
• Reduction Measurement Noise by Averaging
• Modeling – Summary
Topics V, W, X and others as time permits
Table of Contents
Topic A. Solving Equations and Evaluating Expressions
Topic B. Formulas - Computing and Graphing
Topic C. Using a Spreadsheet
|Topic D. Approximate Numbers, Part I. Communicating Precision by Rounding
Topic E. Using a Calculator
Topic F. Angles and Construction of Diagrams
Topic G . Linear Equations – Algebra and Spreadsheets
Topic H. Linear Formulas - Word Problems
Topic I. Approximate Numbers, Part II. Propagation of Errors in Computing with Rounded
Numbers
Topic J. Modeling, Part I. Separating Data into a Linear Model and Residual Noise
Topic K. Trigonometry, Part I. Tangent Ratio in Right Triangles
Topic L. Trigonometry, Part II. Ratios and Relationships in Right Triangles
Topic M. Approximate Numbers, Part III. Communicating the Result of Computations
(Supplement on Significant Digits)
Topic N. Modeling, Part II. Nonlinear Modeling
Topic O. Approximate Numbers, Part IV. Summarizing Noise in Measurements using Standard
Deviation
Topic P. Approximate Numbers, Part V. Sensitivity of a Formula to Errors in Input Values
Topic Q. Approximate Numbers, Part VI. Computing with One Input Value - Empirical Method
Topic R. Trigonometry, Part III. Sine and Cosine Formulas on Larger Intervals
Topic S. Trigonometry, Part IV. Solving General Triangles
Topic T. Trigonometry, Part V. Solving General Triangles: The Ambiguous Case
Topic U. Trigonometry, Part VI. More Applications
Topic V. Modeling, Part III. Correcting Systematic Errors (Bias) in Measurements
Topic W. Approximate Numbers, Part VII. Reducing Noise by Averaging Multiple Measurements:
A Mathematical Formula
Topic X. Modeling, Part IV. Additional Topics
Topic Y. Approximate Numbers, Part VIII. Computing with Multiple Input Values - Empirical
Method
Topic Z. Approximate Numbers, Part IX. Computing with Multiple Input Values - Mathematical
Formulas
Example 1: Hourly Temperature
[a] For this graph, at what
time would being 10 minutes late
in measuring the temperature
result in the largest overestimate?
[b] At what times would taking
the measurement 10 minutes late
make almost no difference in the
measured temperature?
Temperature for different times of day
90
85
Air temperature (F)
The graph to the right shows
how the average air temperature
varies, based on measurements
taken each hour throughout the
day.
80
75
70
65
60
55
50
0
6
12
18
24
Hours since midnight
Answers to Example 1:
[a] Being late will cause an overestimate where the temperature is increasing, so the answer must be
during the period from about 8 am to 4 pm, where the graph is noticeably rising. The biggest mistakes will
happen where the slope of graph is steepest at about 12 hours after midnight, which is noon.
[b] A 10-minute delay will cause the smallest error in measurement when the temperature is changing
most slowly. This will be where the graph of the temperature is nearly level, which in this case at about 7 am
(when the temperature has stopped falling and is about to start rising) and 5 pm.(when the temperature has
stopped rising and is about to start falling).
----------------------------Classroom discussion:
8. The following formula describes the growth of the number of bacteria in a culture in a lab
over a period of 12 days.
A=
1900
0.1 + 3(0.5)t
9. Graph the formula for values of t between 0 and 12.
10. Look at the graph and estimate where a change of about 1/24 of a day in measuring the time
would make the smallest difference in the computed number of bacteria. (Smallest input
sensitivity)
11. Look at the graph and estimate where a change of about 1/24 of a day in measuring the time
would make the largest difference in the computed number of bacteria. (Largest input
sensitivity)
12. Think about t = 1 days and t = 5 days. Which of those has the largest input sensitivity?
13. Notice that you’re looking at the slope of the graph to decide this. This gives us the
motivation for the definition of input sensitivity.
i. Definition: The Input Sensitivity of a formula measures how sensitive the
formula is to changes in the input values. It is a quotient. Its units are the units of
the numerator divided by the units of the denominator.
Input sensitivity =
erroroutput
errorinput
14. Compute the input sensitivity at t = 1 days and give the units of the input sensitivity.
15. Compute the input sensitivity at t = 5 days and give the units of the input sensitivity.
16. Suppose we are at day 5 and we want to find the number of bacteria correct to the nearest 50.
How precisely must we measure the time to do that? (Use one of the two methods of
Example 6, page 4-5, of Topic P.)