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.

Subject: Math

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 way:

- Number patterns arise naturally in many areas.
- Calculating successive differences (delta y’s) is a valuable technique for analyzing these patterns.
- Polynomial equations of degree n have constant nth differences.
- Deltalyzers.
- 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 shouldn’t 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 the equation.
- 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 tables simple.
- 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 documents.

Click here to download Deltalyzers.sit

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.