graphing software Archives - Mathematics for Teaching

GeoGebra is a great software for teaching and learning mathematics. It offers geometry, algebra and calculus tools in one environment, a great support indeed for linking mathematical concepts. On top of that it is free and an open-code software. Click here to download the latest version of the software.

Is it easy to use? Yes and No. Yes, for math teachers because they know the mathematics and can therefore easily understand the ideas and logic behind the tools. Yes, for students who have been instructed on how to use the tools and understand the mathematics and logic behind it. They can use it in solving problems and for investigating mathematical relationships. But, for the majority of students, especially those who have not learned the basic of graphing, equations, and geometric relationships, the use of GeoGebra is limited to manipulating ready-made GeoGebra applets. (Click here for my posts on solving problems about quadrilaterals or here for introducing function using Geogebra applets.) Well, yes, GeoGebra applets are easy to use but most of the time if you do not know the mathematics behind the construction or can’t construct it yourself, then the learning of the mathematics may be superficial.

Construction of math models using the software is not accessible for many younger students just starting to learn basic Algebra and Geometry. In order for them to construct a model, they will have to follow a set procedure (constructed by the teacher) without really understanding why they do what they do. So I thought why not teach GeoGebra tools and mathematics at the same time? This is a challenge I set for myself and I have no idea if it will work or not. I am thinking of doing a research of it later in the year. GeoGebra is free and faithful to mathematics so for countries like us that can’t afford to buy licensed softwares, we get the same quality teaching tool with Geogebra. I think all students need to know how to learn mathematics with it.

TIMSS (Trends in international Math and Science Study) classifies test items in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g. Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x) b. g(x) = f^-1(x) c. g(x) = f^-1(-x) d. g(x) = -f(x) e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x². Which of the following function is represented by the other graph on the right (the blue one)?

a. $g(x)=\frac {4}{x^2}$ b. $g(x)=4+\frac {4}{x^2}$ c. $g(x)=\frac {4}{(x-2)^2}$ d. $g(x)=\frac {4}{(x-4)^2}$ e. $g(x)=\frac {4}{(x+4)^2}$