They came up with drawings such as a butterfly when they discussed plotting points on a

Cartesian plane.

Given a set of coordinate data on a

Cartesian plane, Excel and other spreadsheets offer a simple yet powerful means of mapping the data, checking for data errors, and performing geometric calculations to find perimeters and areas.

To illustrate many of the features discussed above for cubic polynomials with real coefficients plotted in the Cartesian plane, consider the curve y = [x.

The complex conjugate roots do not correspond to the locations of either turning point or the PoI shown in Figure 1 and have often been misplaced by well-meaning but ingenuous authors like Stroud (1986, see Programme 2, Theory of Equations, where he [incorrectly] places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation).

Note also that although each of the points of intersection is found by considering points of intersection of equations (1) and (2) in the Cartesian plane, these points correspond directly to the roots in the complex plane.

The quadratic equation with complex coefficients can readily be solved by considering the intersection of two hyperbolas in the Cartesian plane.

Meanwhile, in the Cartesian plane, a closely related topic deals with the solution of polynomials (ACARA, n.

This also leads to the n-th roots of unity, although the location of these n-roots relative to the Cartesian plane is commonly misunderstood (see Stroud (1986, Programme 2, Theory of Equations); he (incorrectly) places the complex conjugate roots in the Cartesian plane at one of the turning points of a cubic equation.

This approach is much simpler than the comprehensive analysis presented by Bardell (2012, 2014), but it does not make the full visual connection between the

Cartesian plane and the Argand plane that Bardell's three dimensional surfaces illustrated so well.

Australian Curriculum ACMNA296), and subsequently explore its various features using techniques such as two-dimensional plots in the

Cartesian plane, factorisation, completing the square, and the general root formula shown in Equation (3).

The game is played by two participants on a grid that mimics the x- and y-coordinates of a

Cartesian plane and can be played in any classroom.