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TI-Nspire Concrete Algebra Sampler

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.

Used to complement concrete models, however, computer algebra tools can strongly underpin student understanding and correct manipulation of the symbols of algebra.

## Visualising Algebra - Concretely

(with thanks to Steve Thornton, Charles Darwin University, Northern Territory)

Algebra (at least at school level) is generalised arithmetic, and students must be comfortable with the rules for numbers and their operations before they can generalise these. Pattern building is always a great place to start, but wherever possible, offer context, meaning and visualisation.

It is easy to build squares using matchsticks, and chains of squares can lead to some interesting patterns. In this TI-Nspire document, the same pattern is physically built in four different ways - and each corresponds to a different algebraic form. Students are challenged to describe each pattern, first in words and then using mathematical notation. Very quickly, they will learn two important things:

- That it is often easier to describe such rules using algebra than it is using real language, and
- That an important aspect of algebra is learning to say the same thing in different ways!
These ideas are explored in more detail in the TI-Nspire file, Visualising_Algebra.tns.

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## Building Algebraic Expressions - Concretely

Area models which serve as concrete referents for the symbols of algebra are well-established through research and classroom practise as highly effective tools for introducing the symbolic forms of early algebra. After working with these models (using paper or cardboard cut-outs, or their virtual counterparts) even for a short time, students become confident in their understanding of the symbols and the ways in which they can be manipulated. Never afterwards do students confuse expressions such as

x + 2and2x + 1orxand^{2}2x- they are different shapes!There are two key problems identified by research with the use of actual concrete materials (such as cardboard shapes) here: there is no clear link between the concrete shape and the algebraic form, but, more importantly, these cardboard shapes, once cut out, are

staticand cannot be changed - hence they can build misleading understanding of the true nature of algebraic symbols as representingvariables. Both these problems are clearly overcome using the virtual manipulatives shown here: the algebraic form is dynamically built as the shapes are dragged onto the "stage", and the value of the variable can be changed at any time - and the corresponding shapes change to match!These ideas are explored in more detail in the TI-Nspire file, expressions_v2.tns.

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## Solving Equations - Concretely

The idea of using a scale or balance to represent the equality of the two sides of an equation is brought to life using this virtual concrete representation. Students build their equation by dragging variables and units onto each side of the scale, observing the changes to the scale lines as the values on each side change. They then use the "variable controller" to adjust the scale until the two sides are clearly equal - they have physically established this equality. They may now begin removing the same things from each side until they are left with a single variable on one side, and its value on the other.

These ideas are explored in more detail in the TI-Nspire file, equations_v2.tns.

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