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Dynamic Algebra Using TI-Nspire CAS

TI-Nspire technology, especially in its CAS (Computer Algebra System) form, offers great possibilities for enhancing and transforming the teaching and learning of algebra across the secondary school years. It may even be in the formative years when algebra is first being introduced, that the potential may be greatest.

Introduced without a great deal of forethought and caution, however, CAS stands to do at least as much damage as good in the early years of high school - a key problem with traditional algebra instruction lies in its focus on the syntax (the rules and manipulations with apparently meaningless symbols) at the expense of the semantics (the meaning and concepts behind the variables and functions being used). In their basic form, CAS tools are purely syntactic - they are manipulative tools which blindly carry out the processes of algebra without recourse to the underlying significance.

A tool such as TI-Nspire offers much in this regard. It can be used to embody the manipulations, the variables and the functions within meaningful context - often underpinned by dynamic geometric models which can serve, not just to provide context, but also to provide measured data which can be graphed; this graphical model then forms the basis for verifying algebraic models constructed by the students.

Within the geometric context, too, concrete models can be constructed which can strongly underpin student understanding and correct manipulation of the symbols of algebra.

There lies within the Lists & Spreadsheet application, too, the potential for what might be termed dynamic algebra in which algebraic processes may be developed and verified in an interactive way. Some early explorations are presented here: linear equation solving, simultaneous equation solving, and completing the square.

### Equation Solving with CAS

 A key challenge of using computer algebra with students in the early years of secondary school lies in not allowing the tool to do all the work! In the example shown here (CAS_Eqns.tns), a student enters the equation to be solved in cell B2, and then enters each step of the solution in the cells below. As each step is entered, the "check" column indicates whether this is correct or incorrect. The final result may be checked against the graph on the next page. Further scaffolding is available through the possible use of the equation labels, eqn1, eqn2, etc. The student can refer to the required step in terms of the action upon the preceding equation: for example, in the sample shown, eqn3 could be defined as eqn2 + 3. Notice that in this example, at least, two or three wrongs DO make a right!

These ideas are explored in more detail in the TI-Nspire file, Equation_Solver.tns.

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### Solving Simultaneous Equations

 Three possible models present themselves: two substitution methods and one elimination. The first and last allow equations to be entered in any form. The second requires these to be entered in "y =" form - but they can be entered from either spreadsheet or graph.

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### Completing the Square

 As with linear equation solving, two methods present themselves: students specify the steps of the process and the CAS follows these instructions ("student as spectator"), OR students actually enter each line (either literally or in terms of some action upon the previous line (e.g. eq2 + (b/2a)^2) and the CAS engine verifies that this line agrees algebraically with the previous line (high student involvement).
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