Finding the best water line: the least squares method in action
This material was originally developed by the the National Numeracy Network
as part of its collaboration with the SERC Pedagogic Service.
as part of its collaboration with the SERC Pedagogic Service.
Summary
Unreviewed Activity submitted in preparation for the NNN Writing with Numbers Workshop
In this project, students are introduced to several different ways of measuring the goodness of fit of a line representing a perforated pipe or straight hose used to water several bushes. They use the least squares method to compare different possible "water lines", and are guided in the use of the derivative to make the sum of squared errors as small as possible.
CollapseLearning Goals
Consider various ways of measuring the total error of a line used to fit data points on a scatterplot. Master the use of the least squares method to determine which of several proposed lines fits the data best. Estimate the best line and find the resulting sum of squared errors. For students with calculus background, use calculus to optimize the choice of line.
Context for Use
I use this in Calculus I as a project done over 2 to 3 weeks, after they have learned how to take the derivative of a polynomial. Many of the activities in it could also be used in an introductory statistics class or an introductory quantitative reasoning class, with no modification.
Description and Teaching Materials
Two- or three-week project using written project description and a calculator.
Project description (Microsoft Word 2007 (.docx) 89kB May16 08)
Teaching Notes and Tips
Question 1d): students usually need to be reassured that all that is sought here is a verbal description of an alternative method for measuring error, and that no calculations are required.
Question 5: most students should have an "aha" when they find that the slope m, previously encountered only as a constant, can be the variable in a function giving total error.
Question 8: this may be the first time students construct a function of two variables, so they often need guidance here.
Question 9: For Calculus I students, this question provides a bridge to the idea of partial derivatives. That is, if we imagine that we have found the ideal m and b that give a minimum total squared error, then it is natural to think that holding either one of these constant should reveal the value of the other that makes the sum of squared errors as small as possible.
Question 5: most students should have an "aha" when they find that the slope m, previously encountered only as a constant, can be the variable in a function giving total error.
Question 8: this may be the first time students construct a function of two variables, so they often need guidance here.
Question 9: For Calculus I students, this question provides a bridge to the idea of partial derivatives. That is, if we imagine that we have found the ideal m and b that give a minimum total squared error, then it is natural to think that holding either one of these constant should reveal the value of the other that makes the sum of squared errors as small as possible.
Assessment
The syllabus for the course stresses that all project reports must be self-contained, including all given information and identifying problems to be solved, and typewritten in correct English. Those criteria typically account for 30% or 40% of the grade. Correctness and clarity of operations, including clarity of algebraic steps accounts for another 30% or 40%. Insights obtained and the clarity with which they are conveyed account for the remainder. I check the obvious fill-in-the-blank calculations for correctness, but am even more interested in how the student deals with the open-ended questions, such as #1d) and #9.