Function Machine
http://nlvm.usu.edu/en/nav/frames_asid_191_g_3_t_2.html?from=category_g_3_t_2.html

Description
This technology plays to the common metaphor that functions are like “machines”- you put a number in (input), you get a number out (output). This applet allows students to put a given set of numbers into the function “machine” and have the machine generate an output. Students must also try and decipher the function “rule” in order to determine missing outputs for inputs.

Evaluation
What mathematics is (potentially) being learned?
NCTM Standard(s) addressed?
-Algebra, 6-8: Students will be able to understand patterns, relations, and functions through analysis of data and input-output tables.

What is the nature of the mathematics?
The main concept being shown is functions. Students are going to recognize functions as “machines” that process inputs and outputs. Students will also become familiar with reading and using an input-output table and analyzing an input-output table for patterns.

How does learning take place?
Learning takes place as students begin to analyze the outputs of the function machine for patterns. Using the input-output table, students have the opportunity to analyze the values the function machine has produced. By doing this, a pattern will begin to emerge (for example, for every input, the output is the input plus three). Using this pattern will help determine the nature of the function and allow the students to figure out the three missing outputs that are omitted from the input-output table. By analyzing the function’s data, examining relationships between the inputs and outputs, and recognizing patterns, student learning of functions can occur.

What role does technology play?
This technology allows students to physically move inputs into the function machine. This plays to the metaphor of functions being a “machine” as it brings the metaphor to life. The technology also generates the outputs for the student, and places both the inputs and outputs into an input-output table that is presented. The applet also allows for students to enter the values for three missing outputs in the table, which the students must determine from the pattern of the function.

This technology affords students the opportunity to represent knowledge and thinking based upon the student’s ability to show the recognition of a function’s pattern by naming the three missing outputs to the applet’s function. The technology also automates and simplifies the task, as students do not have to do any work in figuring out the first four outputs of the input-output table (the function machine does that for them) and do not have to fill out must of the input-output table (the applet fills out the first four input-output pairs for the students).

How does it fit within existing school curriculum?
This technology works well as an introduction to functions. Students in secondary schools will see this technology as beneficial as they are familiar with functions being termed “machines”. This applet would crystallize this image for them, as well as provide an engaging and interactive activity for students to practice recognizing patterns, filling out input-output tables, and learning about functions. This technology is to be used as a supplement to algebra lessons dealing with patterns and functions.

How does the technology fit or interact with the social context of learning?
There are some functions that involve non-numeric outputs. For example, in one particular function an input of ‘one’ will result in an output of ‘F’. This abstract thinking tests students’ abilities to think outside the box as well as think critically about patterns. Because some of the patterns aren’t as easily recognizable as others, for beginning classes first dealing with functions, it is recommended that students pair together and work with this function machine together.

What do teachers and learners need to know?
An understanding of functions is a must for students. This technology is not to be used as an introduction to a unit or the concept of functions, but rather as an assessment or skill builder. It is important for students to understand the terminology associated with functions to give this technology more meaning and purpose.