Blue Meanies- Where's the Math?

Blue Meanies! - Notes for Teachers (and parents, and interested students)

Where's the Math?

One of the motivations for developing this game and related tools is the belief that multimodal presentation (including the kinesthetic experiences provided by a gestural interface) may help to ensure a more robust understanding of concepts.

The concepts addressed in this game have to do with the graphs of equations, and the specific skills involved are the accurate recognition and visualization of graph shapes corresponding to equations of various types and an understanding of how the graphs depend on the numerical parameters occuring in those equations.

Level 1 - Linear Graphs

The first level activity starts by encouraging the student to experiment with various parameter values in the equation y=mx+b. In the course of this 'trial and error' part of the activity, the students are expected to notice (or reinforce their memory of) the facts that m and b are the slope and y-intercept of a straight line. It will be useful, both in applications and in future math classes, for the student to develop a sense of what different slopes look like. In particular, being able to quickly estimate slopes 'by eye' when x and y scales are equal will be very helpful in various aspects of calculus (such as exam questions asking for a sketch of the graph of f ' given that of f). In fact, a survey of BC postsecondary educators has identified a robust understanding the concept of slope of a line as the single most important prerequisite skill for both calculus and statistics.

When working with sliders, the first strategy suggested (of first rotating and then sliding up) should foreshadow (or recall to mind) the algebraic process of first finding m=rise/run and then plugging in to solve for b.

On the other hand, the second strategy (of visualizing a line through two points and extrapolating to its intercept) exercises basic geometric imagination skills in which many students are surprisingly lacking even when they enter calculus courses. Such skills, while not often explicitly identified as testable learning outcomes in elementary courses are nonetheless very important for success in more advanced courses where they are often taken for granted.

The intent of the second version of this linear equation activity (without sliders) is mainly for the student to reinforce this geometric visualization skill by trying to find quick approximate answers rather than to explicitly "solve for b" by computing the slope as rise/run and then plugging in one of the points. But it is certainly worth seeing how that works in one or two cases. So students should be encouraged to try the process of reading off the exact coordinates (x₁,y₁) and (x₂,y₂) of the centres of the blobs and computing m as rise/run=(y₂-y₁)/(x₂-x₁) and b as (eg) y₁-mx₁(and also to think of how this relates to the mental process of visualizing the line and tracing along it to the y-axis)

Level 2 - Quadratic Graphs

This activity starts by encouraging the student to experiment with various parameter values in the quadratic equations y=a(x-h)^2+k and y=ax^2+bx+c. In the course of this trial and error part of the activity, the students are expected to notice (or reinforce their memory of) the facts that h and k are the coordinates of the vertex and that a is related to the curvature (and equal to the change in y as x moves one unit right or left of the vertex), and that c and b are the y-value and slope at x=0.

With regard to the h and k values, having a strong sense of these facts is helpful for future work on shifting and scaling of graphs which in turn is an important component of the practical skill of mathematical modelling (ie finding equations suggested by sets of experimental data). The b and c interpretations are less immediately important but having experience of them will be helpful when the student later studies derivatives and Taylor series.

The actual fact that a quadratic function can be found through any three non-collinear points with different x-values may not be all that important for its own sake (though it is an example of a general quasi-principle about constraints and parameters which has its rigorous version in Linear Algebra, and also it is applied specifically in the derivation of Simpson's rule). But the act of visualizing a parabola through three points and inferring its properties can help develop various aspects of geometric imagination which are useful in many contexts. These include data analysis, modelling, drafting, and the creation of useful diagrams to aid in analysis of physical problems (either in the real world, or on exams in "word problems").

It should be emphasized that the intent of the second stage (without sliders) is mainly for the student to reinforce this geometric visualization skill and knowledge of graph properties by trying to find quick approximate answers, rather than to repeatedly engage in the tedious solution of systems of three equations in three unknowns. This is not to say that the algebraic approach should not be used from time to time to confirm that it gives the same answer - and to show that the quick visualization method can be used to check the answers if one is ever required to actually compute more exact parameters.

Level N - General Graphs

Here there are many options.

Lower level students might just use it as several linear or quadratic problems presented at the same time and look for lucky line-ups to reduce the number of lines or parabolas required.

Others might be encouraged to try cubic and higher degree polynomials (eg by imagining a smooth curve and using its intercepts to generate the factors), to use conic sections other than vertically oriented parabolas, or to try other function types such as trig functions.

At present the applet just presents 4 random points, but in future it might be worthwhile to increase the number and/or to generate patterns that are amenable to specific kinds of solutions.