CLG 1



CLG 1.1.4 – Describing Non-linear Functions

Terri Sprout

Susana Davidenko, February 1997, “Building the Concept of Function from Students Everyday Activities,” Mathematics Teacher, Pages 206-212.

This article shows how often functions are used in students’ everyday activities and also demonstrates ways in which these examples of functions can be incorporated into the classroom. Davidenko believes that students should not just be given the definition of a function, rather than being shown how functions are relationships between sets of objects. This article gives examples that can be used to make students aware of how functions are present in many activities that are normally not thought of as a function. These examples can be used to teach students about the domain and range of functions. For example if the function is the temperature in different cities, the domain is the cities and the range is the temperature. Lastly, this article showed the advantages in using spreadsheets when learning about functions. This use of spreadsheets can be beneficial is helping students understand the difference of functions and also the composition of functions. They are also very helpful in showing students the correspondence between the tables and graphs of functions.

One strength of this article is the way that it encourages teachers to make sure that students have an understanding of what a function is. It is helpful in the lesson when students can connect the ideas they learn in class to what they see on a day to day basis. This article can be used in the classroom by showing teachers everyday examples of functions which they can give their students so that students can have a basic understanding of what a function is.

One weakness of this article is that it may over simplify a function and it does not do much to describe functions, other than to give students examples of what the domain and range of a function is.

Sharon E. Taylor and Kathleen Cage Mittag, May 2001, “Seven Wonders of the Ancient and Modern Quadratic World,” Mathematics Teacher, Pages 349-350, and 361.

This article describes seven ways that a quadratic function can be solved. Taylor and Mittag assert that although factoring, completing the square, and the quadratic formula is what is typically written about in textbooks they are often not put together so students rarely understand that all three will accomplish the same thing. Most of the time these methods had no graphical or numerical reasoning to explain them in textbooks. This article also explained four newer methods that can be used on a graphing calculator to find the zeros of a quadratic equation. These include graphing the equation and then using the trace program, the table, a quadratic-formula program, and also the solver that can be found in most graphing calculators. Taylor and Mittag believe that students should be given as many of these methods as possible so that they can connect them to each other. By showing all of the methods to students at one time they are able to solve the formulas by hand but also see how the calculator produces the same answer. Also by using the graph on the calculator they are also more able to visualize what they are trying to find.

A strength of this article is the way that it shows how many methods of solving a quadratic formula are related. This could even be used in a current classroom setting when each method is taught separately, by reviewing the old method before the new method is taught.

A weakness of this article is its failure to mention how calculators can also be used to visualize many other concepts of functions. These include the minima, and maxima, and the domain and range of a function.

Discussion Questions:

1. Taylor and Mittag describe a classroom where calculators are used to visualize functions, and in the process students are also taught ways to solve for the zeros of the function using the calculator. Are students becoming calculator dependent or does the calculator add to the understanding of what they are actually solving for?

2. Should all of the ways of solving an equation by hand be shown before those on the calculator, or should the methods be mixed (teach a hand method then a calculator method then another hand etc.)

3. Do non-numerical function examples help students understand the concept of functions better or make things more complex?

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