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Teaching decimal – binary conversionthrough an interactive exhibitApostolos Papanikolaoua & Aris Mavromatisa

a National Science Center of Athens, Athens, GreecePublished online: 28 Feb 2013.

To cite this article: Apostolos Papanikolaou & Aris Mavromatis (2013) Teaching decimal – binaryconversion through an interactive exhibit, International Journal of Mathematical Education inScience and Technology, 44:5, 718-721, DOI: 10.1080/0020739X.2012.756544

To link to this article: http://dx.doi.org/10.1080/0020739X.2012.756544

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International Journal of Mathematical Education in Science and Technology, 2013

Vol. 44, No. 5, 718–777

CLASSROOM NOTES

Teaching decimal – binary conversion through an interactive exhibit

Apostolos Papanikolaou∗ and Aris Mavromatis

National Science Center of Athens, Athens, Greece

(Received 11 April 2011)

This note is a presentation of a didactic proposition about how to comprehend theconversion from the decimal into the binary system and vice versa with the aid of aninteractive exhibit.

Keywords: interactive exhibits; mathematical challenge; decimal and binary systems

The present article is a presentation of a didactic proposition about how to comprehendthe conversion from the decimal into the binary system and vice versa with the aid of aninteractive exhibit.

A great number of these interactive exhibits are found in many scientific centres world-wide [http://www.experiencingmaths.org, http://exploratorium.edu/math_explore, http://www.technorama.ch].

The positive role of exhibits of scientific museums in mathematical education has occu-pied the 16th ICMI Study.[1]

This interactive exhibit is a variant of ‘Seven Segment display’ [http://en.wikipedia.org/wiki/Seven-segment_display] and comes from an earlier collection of Cite des Sciences etde Industrie [http://www.cite-sciences.fr/fr/cite-des-sciences]. It is located in the NationalScience Center of Athens [http://www.eee-athens.edu.gr] and exhibits similar to this arefound in the Knowledge Center of Halkida [http://www.estiagnosis.gr] and in the ScienceCenter of Patras [http://www.eduscience.gr].

The exhibit is a device that consists of a keyboard with the 10 digits – numbers in thedecimal system, similar to that of a phone and an array of eight lamps. Each time a studentpunches in a number on the number pad, some of the eight lamps come on and some donot. By pressing the letter keys A and B, all the previous orders are cancelled and studentscan restart (see Figure 1).

The instructive approach described later was applied to 15-member groups of students,aged 16, at the National Science Center of Athens. Each group was allocated about onehour with the exhibit.

The same activity has also been implemented to pupils, aged 12–18, with a diversifica-tion in the development of the mathematical level.

What students are required to do is check the operation of the exhibit, that is whatnumbers should be typed in order to light all or some of the specific lamps.

The electronic circuit, which supports the exhibit, relies on the conversion from thedecimal system of the pad into the binary system of the lamps but this is not known to thestudents.

∗Corresponding author. Email: apapanik@sch.grhttp://dx.doi.org/10.1080/0020739X.2012.756544

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Figure 1. The interactive device.

Figure 2. The correspondence between the two states of the lamps and the empty–full circles.

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720 Classroom Notes

Figure 3. Numerical representation of lamp situation.

Figure 4. The states of the lamps and the powers of number 2.

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Figure 5. The sum of powers of the number 2 with coefficients 0 or 1.

As a rule, the first reaction of the students is to press numbers at random, hoping thatthey will find the one that lights all the lamps by chance. Of course, this is statisticallyimpossible, so the students attempt to deal with the exhibit in a more systematic recordingof the trial–experimentation data.

The two states (on–off) are symbolized by the students, with the pair empty–full circle(see Figure 2).

When students are prompted to replace the symbols representing the two situationswith numerical symbols, the students normally propose 0, 1. So each eight-lamp situationis represented by an arranged octet of 0s and 1s (see Figure 3).

Students are expected to observe that the numbers which light up just one lamp arepowers of the number 2 and any other condition of the lamps is a sum of powers of thenumber 2 (see Figure 4).

By following this process, they normally find out that the number 255 lights all thelamps since it is the sum of 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128, and they are thusenabled to predict which of the eight lamps will light up every time they key in any naturalnumber from 0 to 255. This will be made possible by having students to write it as the sumof powers of the number 2 with coefficients 0 or 1 (see Figure 5).

At this point, students are normally able to generalize this process for numbers higherthan 255 by converting them from decimal to binary system and vice versa.

Reference[1] Barbeau E, Taylor P. Challenging Mathematics in and beyond the classroom. Springer; 2009.

p. 70.

A geometric approach to calculus

A.V. Grebea,b∗

aDepartment of Mathematics, Pine View School, Osprey, Florida 34229, USA; bDepartment ofMathematics, Washington University in St. Louis, St. Louis, Missouri 63130, USA

(Received 4 December 2011)

This investigation explores whether it would be possible to derive the calculus from ageometric basis. The article proves several rules of calculus for polynomials and explainshow to differentiate and integrate arbitrary polynomial functions. Differentiation isperformed by calculating the slope of a tangent line drawn to a function. Definiteintegration calculates the area under the graph of a function by comparing this area to ahypersolid’s content using Cavalieri’s principle.

Keywords: calculus; geometry; integration; differentiation; hypersolid; Cavalieri’sprinciple

∗Email: agrebe@wustl.eduhttp://dx.doi.org/10.1080/0020739X.2013.790499

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