The research of the past two decades is clear: algebra in the secondary school must be firmly based upon understanding of numbers and their relationships, and so it makes sense to begin the study of algebra using the table of values in a “guess my rule” scenario. Mathematics is, after all, a search for patterns and relationships. Using a teacher ViewScreen™, simply enter your function with the Overhead Projector turned OFF, go to the table of values, and switch it on. Students quickly learn to look for the “magic number” in the linear pattern. The rule follows easily.
Number patterns of the form ax + b provide the basis for students’ early understanding of the function concept and, indeed, the notion of variable, even though they may or may not need to be introduced to these terms at the outset. Once they can confidently predict the rule for a linear pattern and appreciate that different values of x produce differing outputs, they can begin to consider the symbolic aspects. A very effective introduction to the symbols of algebra can be achieved using areas of shapes, beginning with a unit square (called a “one”) and a rectangle, 1 unit wide but of unknown length (call its length – and hence its area – “x”). StudyCards™ offers a nice way to present these ideas. Once students are comfortable with symbolic simplifying, then they may benefit from access to simple algebra software to support their growing understanding of the processes of simplification, expansion and, later, equation solving.
The nice people at Detached Solutions (http://www.detachedsolutions.com) have written some wonderful Apps for the 83 Plus, including one called Symbolic. As the name implies, this adds basic symbolic algebra capabilities to the 83Plus: in particular, a SIMP command to perform simplification and expansion of algebraic expressions, and a symbolic differentiation command, D. These functions operate on string variables: i.e. they require inverted commas around the input. This makes them a little awkward to use, and the output tends to look a little messy.
Fortunately, when combined with another App, called PrettyPrint (also available from DetachedSolutions but written by SoftTheiss at http://www.softheiss.com) this problem disappears. PrettyPrint does what it promises and presents algebraic expressions in correct formatting, making it easy for students at all levels to interpret their algebraic output. These two apps can work seamlessly together, but to make the process even easier, I wrote a couple of TI-BASIC programs which offer a simple front-end for teachers and students who would like to add some simple algebra capabilities to their 83Plus calculators.
ALG83 takes an expression (or allows the user to pick up whatever is in Y1), presents it correctly and then offers a choice of options: simplify (or expand), substitute a value, differentiate or view the function using table of values or graph. It also links directly with a second program for equation solving.
EQNSOLV takes the left-hand and right-hand sides of an equation (or uses Y1 and Y2 if desired) and offers the choice of a “fast solve” (using the in-built SOLVE command, or a step-by-step linear equation solver.
In the example shown, the first step would be to SIMPLIFY, expanding the left-hand side. From the menu shown, choosing 2 produces a simpler form.
Next step involves, perhaps, taking 3x from both sides, so the user enters –3X.
This is automatically simplified to give the next step of the process. In this way, students may work through the equation solving process, supported by the software until they are capable of working unassisted.
So what algebra did you want to be able to do with your high school students: simplify, expand, substitute, solve equations, even some calculus? It is all here. Have fun!
