Dueling Calculators
http://nlvm.usu.edu/en/nav/frames_asid_312_g_4_t_1.html?from=topic_t_1.html

Description
“Dueling Calculators” an applet allows students to visually and numerically see the differences calculator truncating and rounding can play in mathematics. When a calculator truncates (due to the size of its viewing window), answers are not usually effected greatly. Over time, however, repeated truncation can drastically change the solutions and results we get. This applet provides a resource for students to see that change.

Evaluation
What mathematics is (potentially) being learned?
NCTM Standard(s) addressed?
-Numbers and Operations, 6-8: Students will be able to understand large numbers (numbers with many digits) as they are written in calculator notation.
-Data Analysis, 6-8: Students will be able to interpret data in a table as well as in a graph and understand the connection between the two.
-Communication, 6-8: Students will able to clearly communicate their ideas using mathematical language about rounding and truncating.

What is the nature of the mathematics?
The concepts being displayed are calculator truncating, rounding, and the limitations of calculators. These concepts are displayed in the technology using two calculators, a table of created data, and a graph of the data. The students must interpret the data through discussion and collaboration with peers to gain understanding of the concepts displayed.

How does learning take place?
Learning takes place through meaningful discussion with peers. While the technology provides great data for analysis and visuals for interpretation, what the technology does not do is offer answers or conclusions for the data presented. Thus it is the students’ responsibility, through careful thinking and fruitful discussion, to use what is provided in the technology to reach conclusions about calculator truncation and rounding in general, and their effects on a data set.

What role does technology play?
The technology shows two calculators that are set side by side, one that displays nine-digits and the other that displays eight-digits. The technology allows students to set an input that is the same for each calculator, and the calculators put the input into a given function (provided by the technology), and display an output, which is truncated to the last spot in each calculator’s display. The calculators then take their truncated output and put them back into the original function as inputs. Because of truncation, the outputs for each calculator begin to change, and students have the opportunity to see the change both in numerical data displayed, as well as in a visual graph.

This technology affords peers the opportunity for communicating and collaborating with one another. The technology acts as a starting point for a discussion on the effects of rounding and calculator truncating. By interpreting the data presented in a table and on a graph provided by the technology through discussion, students are able to reach conclusions about calculator truncation and the effects rounding have on numbers.

How does it fit within existing school curriculum?
This technology would be a great beginning of the year activity. Some students nowadays are becoming too reliant on technologies, calculators especially, and are becoming weak in computational skills. By discussing the pitfalls of relying on calculators entirely and examining their limitations, students will begin to understand that complete reliance on calculators can be a detriment to their learning. It is also a fantastic introduction to some math vocabulary, like “truncate”, and provides a great visual of what it means for a calculator to truncate a number.

How does the technology fit or interact with the social context of learning?
This program can be a great tool for leading a whole-class discussion on the cautions of technologies. What is problematic about relying on technology completely? What do users have to know when using a technology? What limitations do calculators have? Teachers can even move to a discussion on rounding. How does rounding affect a set of data? This technology is designed to stimulate discussion and communication amongst students and the classroom.

What do teachers and learners need to know?
While students do not need have a particular knowledge of the functions the program provides (some may be a little abstract for early-secondary students) or how to graph the functions, students and teachers must be able to interpret the graph the technology generates and compare the two sets of data generated by the two calculators. A basic understanding of how functions work (there is an output for every input) is essential.