Illustrative Examples of Units of
Deltalyzers - An intuitive approach to finite differences
This short activity uses
Apple Works 5.0 spreadsheets to provide a concrete, visual, goal driven exploration
that bridges the gap between number patterns and the technique of finite differences.
Learning Levels: Grades 10-12
Author(s): Terry Thorsen
Students are lead to develop their own algorithm for finding
the equation for a table of values. Their explorations provide motivation and
develop understandings for finite differences.
A series of 10 spreadsheets (Deltalyzers) require students
to guess the equation for a table of values. They accomplish this by comparing
the tables and graphs of the given data with those of their guessed equation.
Eventually, students will develop their own algorithm. This can lead to a good
discussion prior to the introduction of #5 below.
Students will develop skills and ideas concerned with:
- Using spreadsheets.
- Guess and check.
- Successive approximation.
- Visualization of lines, quadratics and cubics.
- Connections between tables, equations and graphs.
- The leading coefficient determines the shape.
- The constant term is the y intercept.
- An algorithmic approach to finding the equation is possible.
The method of finite differences is developed in the following
- Number patterns arise naturally in many areas.
- Calculating successive differences (delta ys) is
a valuable technique for analyzing these patterns.
- Polynomial equations of degree n have constant nth differences.
- The usual technique of finite differences involves building
a table of values with algebraic expressions as y values, and then comparing
the original table with the algebraic table. This sets up a system of n equations
in the coefficients a, b, c ... that is easily solved.
Students will have considered realistic problems involving
number patterns as well as items 1 through 3.
As an introductory experience, Deltalyzers shouldnt
take more than half a period. If one gets involved with writing up their discoveries,
extended discussion, formal evaluation, mastery learning or one of the extensions
it takes two or three periods.
4 Deltalyzers: Have the students try each spreadsheet in
turn. Students should focus not only on getting the equation but on describing
their method. If they develop a purely visual technique (using only the graphs),
then shrink the window and have them focus on the tables. Conclude by having
students share their algorithms.
This activity can be extended in a number of ways:
- Students bring in their own data for others to guess
- Students formalize their ideas as spreadsheet formulas.
- Students can guess equations by acting out their algorithm
using pencil and paper. This can be computationally intensive so keep the
- Students build or customize the deltalyzer template.
- This technique can be extended to data tables with errors
in the y values (just use the average of the nth differences to calculate
the leading coefficient and subtract to produce a data table of degree n-1
and repeat). Doing this will provide an understandable alternative to polynomial
regression using least squares.
The critical issue is
that if students develop a purely visual technique (using only the graphs), then
respond by shrinking the window and have them focus on the tables. The teacher
will have to watch for this.
This activity is self checking and provides the student
with a fair bit of feedback. Remembering that this activity is more of an introductory
exploration, the teacher can evaluate student understandings by:
- Watching them work with the spreadsheets.
- Having them pass in a description of their algorithm.
- Acting out their algorithm using pencil and paper.
- Building a spreadsheet that acts out their algorithm.
The spreadsheets are imbedded in Apple Works 5.0 drawing
Click here to download
The files are currently locked so that student changes are not saved. You can
unlock them and change them however you want.
If you create your own data tables or translate the spreadsheets onto another
program or version of Apple Works, submit your own unit of practice. To help
others to find it, include Deltalyzer as a keyword.