Graphing calculator

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A typical graphing calculator by Texas Instruments

A graphing calculator (also graphics / graphic calculator) typically refers to a class of handheld scientific calculators that are capable of plotting graphs, solving simultaneous equations, and performing numerous other tasks with variables. Most popular graphing calculators are also programmable, allowing the user to create customized programs, typically for scientific/engineering and education applications. Due to their large displays intended for graphing, they can also accommodate several lines of text and calculations at a time.

Some of the more recent graphing calculators are capable of color output, and also feature animated and interactive drawing of math plots (2D and 3D), other figures such as animated Algebra theorems, preparation of documents which can include these plots and drawings, etc. This is giving the new graphing calculators a presence even in high school courses where they were formerly disallowed. Some calculator manufacturers also offer computer software for emulating and working with handheld graphing calculators.

Many graphing calculators can be attached to devices like electronic thermometers, pH gauges, weather instruments, decibel and light meters, accelerometers, and other sensors and therefore function as data loggers, as well as WiFi or other communication modules for monitoring, polling and interaction with the teacher. Student laboratory exercises with data from such devices enhances learning of math, especially statistics and mechanics.

Since graphing calculators are usually readily user-programmable, such calculators are also widely used for gaming purposes, with a sizable body of user-created game software on most popular platforms. Even though handheld gaming devices fall in a similar price range, graphing calculators offer superior math programming capability for math based games. However, for developers and advanced users like researchers, analysts and gamers, 3rd party software development involving firmware mods, whether for powerful gaming or exploiting capabilities beyond the published data sheet and programming language, is a contentious issue with manufacturers and education authorities as it might incite unfair calculator use during standardized high school and college tests where these devices are targeted. Nowadays graduate (Masters) students and researchers have turned to advanced Computer Aided Math software for learning as well as experimenting.

Types[edit]

Graphing calculators can be sub-divided into two categories (both need the graphical display):

1. Numerical (or just graphing) Calculators - non-CAS and producing numerical results, at most represented as a fraction. In some countries, graphing calculators are not permitted in high school science tests or certain basic math tests. See examples below.

2. CAS (or symbolic) Calculators - most advanced calculators capable of producing a symbolic result (in expression or equation form), usually utilizing a Computer Algebra System (CAS). Symbolic/CAS calculators are posing a challenge to high school and undergraduate educators. They can make math easier to learn for high schoolers, provided school curriculum evolves towards this advantage.[1][2] Testing based on tedious hand calculation is also being forced to evolve towards more creative testing.[3] Such tests are often more challenging and expensive to design and can't be recycled as much, but encourage a genuine deeper appreciation of the art of mathematics and critiqueing a fallacy. CAS calculators are therefore usually permitted only in select advanced math or calculus tests, thereby being more of a classroom learning tool for many users who then switch to a permitted and speedy numerical non-CAS graphing or scientific calculator for tests and exams.

History[edit]

Casio fx-7000G; The world's first graphing calculator

Casio produced the world's first graphics calculator, the fx-7000G, in 1985. After Casio, Hewlett Packard followed shortly in the form of the HP-28C. This was followed by the HP-28S (1988), HP-48SX (1990), HP-48S (1991), and many other models. Recent models like the current HP 50g (2006), feature a computer algebra system (CAS) capable of manipulating symbolic expressions and analytic solving. An unusual and powerful CAS "calculator" is the now obsolete year 2001 Casio Cassiopeia A10 and A11 (flip top) stylus operated PDAs which actually ran the Maple V symbolic engine. The HP-28 and -48 range were primarily meant for the professional science/engineering markets; the HP-38/39/40 were sold in the high school/college educational market; while the HP-49 series cater to both educational and professional customers of all levels. The HP series of graphing calculators is best known for its Reverse Polish notation interface, although the HP-49 introduced a standard expression entry interface as well.

Texas Instruments has produced graphing calculators since 1990, the oldest of which was the TI-81. Some of the newer calculators are similar, with the addition of more memory, faster processors, and USB connection such as the TI-82, TI-83 series, and TI-84 series. Other models, designed to be appropriate for students 10–14 years of age, are the TI-80 and TI-73. Other TI graphing calculators have been designed to be appropriate for calculus, namely the TI-85, TI-86, TI-89 series, and TI-92 series (TI-92, TI-92 Plus, and Voyage 200). TI offers a CAS on the TI-89, TI-Nspire CAS and TI-92 series models with the TI-92 series featuring a QWERTY keypad. TI calculators are targeted specifically to the educational market, but are also widely available to the general public.

Graphing calculators are also manufactured by Sharp but they do not have the online communities, user-websites and collections of programs like the other brands.

Graphing calculators in schools[edit]

TI-89 Titanium, capable of doing Symbolic Manipulation, Computer Algebra System (CAS)
  • North America – high school mathematics teachers allow and even encourage their students to use graphing calculators in class. In some cases (especially in calculus courses) they are required. Some of them are disallowed in certain classes such as chemistry or physics due to their capacity to contain full periodic tables.
  • College Board of the United States – permits the use of most graphing or CAS calculators that do not have a QWERTY-style keyboard for parts of its AP and SAT exams, but IB schools do not permit the use of calculators with computer algebra systems on its exams.
  • United Kingdom – a graphics calculator is allowed for A-level maths courses (excluding the C1 module where no calculator of any kind is permitted), however they are not required and the exams are designed to be broadly 'calculator neutral'. Similarly, at GCSE, all current courses include one paper where no calculator of any kind can be used, but students are permitted to use graphical calculators for other papers. The use of graphical calculators at GCSE is not widespread with cost being a likely factor. The use of CAS is not allowed for either A-level or GCSE [4] The Scottish SQA allows the use of graphic calculators in maths exams (excluding paper 1, which is exclusively non-calculator), however these should either be checked before exams by invigilators or handed out by the exam centre, as certain functions / information is not allowed to be stored on a calculator in the exam. SQA exams do not favour a graphic calculator, and since working must be shown for full marks, do not give a significant advantage over candidates who do not use them.
  • Finland and Slovenia – and certain other countries, it is forbidden to use calculators with symbolic calculation (CAS) or 3D graphics features in the matriculation exam. This will change in the case of Finland, however, as symbolic calculators will be allowed from spring 2012 onwards.
  • Norway – calculators with wireless communication capabilities, such as IR links, have been banned at some technical universities.
  • Australia – policies vary from state to state.
    • Victoria – the VCE specifies approved calculators as applicable for its mathematics exams. For Further Mathematics an approved graphics calculator (for example TI-83/84, Casio 9860, HP-39G) or CAS (for example TI-89, Classpad 300, HP-40G) can be used. Mathematical Methods and Mathematical Methods CAS have a common technology free examination consisting of short answer and some extended answer questions. They also each have a technology assumed access examination consisting of extended response and multiple choice questions: a graphics calculator is the assumed technology for Mathematical Methods and a CAS for Mathematical Methods CAS. These two exams have substantial material in common but also some distinctive questions. Specialist Mathematics has a technology free examination and a technology assumed access examination where either an approved graphics calculator or CAS may be used. Calculator memories are not required to be cleared. In subjects like Physics and Chemistry, students are only allowed a standard scientific calculator.
    • Western Australia – all tertiary entrance examinations in Mathematics involve a calculator section which assume the student has a graphics calculator; CAS enabled calculators are also permitted. In subjects such as Physics, Chemistry and Accounting only non programmable calculators are permitted.[5]
    • New South Wales – graphics calculators are allowed for the General Mathematics Higher School Certificate exam, but disallowed in the higher level Mathematics courses.
  • China - calculators in general are banned in primary and secondary education.[citation needed]
  • India - Calculators are banned in primary and secondary education. University degree and diploma courses have their own rules on use and permitted models of calculators in exams.
  • New Zealand – calculators identified as having high-level algebraic manipulation capability are prohibited in NCEA examinations unless specifically allowed by a standard or subject prescription. This includes calculators such as the TI-89 series [1].
  • Turkey – any type of calculator whatsoever is prohibited in all primary and high schools except the IB and American schools.[citation needed]
  • Singapore – graphing calculators are used in junior colleges; it is required in the Mathematics paper of the GCE 'A' Levels, and most schools use the TI-84 Plus or TI-84 Plus Silver Edition.
  • Netherlands – high school students are obliged to use graphing calculators during tests and exams in their final three years. Most students use the TI-83 Plus or TI-84 Plus, but other graphing calculators are allowed, including the Casio CFX-9860G and HP-39G. Graphing calculators are almost always allowed to be used during tests instead of normal calculators, which sometimes results in cheat sheets being made on forehand and exchanged before the test starts using link cables.
  • Israel – Graphing calculators are forbidden to use in the Bagrut (equvalent to the American SAT) math exam, in addition to programmable calculators.[6] University degree and diploma courses have their own rules on use and permitted models of calculators in exams.

Programming[edit]

The game Tetris is being played on a TI-83 Plus.
Graphing calculators are sometimes used for gaming.

Most graphing calculators, as well as some non-graphing scientific and programmer's calculators can be programmed to automate complex and frequently used series of calculations and those inaccessible from the keyboard.

The actual programming can often be done on a computer then later uploaded to the calculators. The most common tools for this include the PC link cable and software for the given calculator, configurable text editors or hex editors, and specialized programming tools such as the below-mentioned implementation of various languages on the computer side.

Earlier calculators stored programs on magnetic cards and the like; increased memory capacity has made storage on the calculator the most common implementation. Some of the newer machines can also use memory cards.

Many calculators, such as earlier TI graphing and scientific calculators will tokenize the code for a program or function, using ISO 8859 type character codes for the statements and other programming elements. The TI-92 Plus and many HP calculators read the code much like computers do and they have functions such as Chr$, Chr, Char, Asc, and the like in Basic (sometimes renamed) in addition to using somewhat modified or unmodified versions of 7-bit, 8-bit or 9-bit ISO 8859-derived character sets and other character sets running from values of 0 to 127 (07F hex), 255 (0FF hex), or 511 (1FF hex) -- and many of them have a tool similar to the Character Map on Windows.

Three websites with extensive programming information are www.ticalc.org, www.hpcalc.org, and www.casiocalc.org. The official sites of the manufacturers and of other people like professors & teachers, students, statisticians, scientists, and organizations like university business and computer science departments, SourceForge, and the 27. November Spreadsheet Macro Programming Club are also useful. A broad array of third-party software including 3D function graphing tools, web browsers, chat, email and NNTP clients, telnet/SSH, spreadsheets, word processors, sound & graphics tools, network tools, and programming tools can be located on the internet.

A cable and/or IrDA transceiver connecting the calculator to a computer make the process easier and expands other possibilities such as on-board spreadsheet, database, graphics, and word processing programs. The second option is being able to code the programs on board the calculator itself. This option is facilitated by the inclusion of full-screen text editors and other programming tools in the default feature set of the calculator or as optional items. Some calculators have QWERTY keyboards and others can be attached to an external keyboard which can be close to the size of a regular 102-key computer keyboard. Programming is a major use for the software and cables used to connect calculators to computers, other calculators &c.

The most common programming languages used for calculators are similar to keystroke-macro languages and variants of BASIC. The latter can have a large feature set—approaching that of Basic as found in computers—including character and string manipulation, advanced conditional and branching statements, sound, graphics, and more including, of course, the huge spectrum of mathematical, string, bit-manipulation, number base, I/O, and graphics functions built into the machine.

Languages for programming calculators fall into all of the main groups, i.e. machine code, low-level, mid-level, high-level languages for systems and application programming, scripting, macro, and glue languages, procedural, functional, imperative &. Object-Oriented Programming can be achieved in some cases.

Most calculators capable to being connected to a computer can be programmed in assembly language and machine code, although on some calculators this is only possible through using exploits. The most common assembly and machine languages are for TMS9900, SH3, Zilog Z-80, and various Motorola chips (e.g. a modified 68000) which serve as the main processors of the machines although many (not all) are modified to some extent from their use elsewhere. Some manufacturers do not document and even mildly discourage the assembly language programming of their machines because they must programmed in this way by putting together the program on the PC and then forcing it into the calculator by various improvised methods.

Other on-board programming languages include purpose-made languages, variants of Eiffel, Forth, and Lisp, and Command Script facilities which are similar in function to batch/shell programming and other glue languages on computers but generally not as full featured. Ports of other languages like BBC Basic and development of on-board interpreters for Fortran, Rexx, Awk, Perl, Unix shells (e.g., bash, zsh), other shells (DOS/Win95, OS/2, and WinNT/2000 shells as well as the related 4Dos, 4NT and 4OS2 as well as DCL), COBOL, C, Python, Tcl, Pascal, Delphi, ALGOL, and other languages are at various levels of development.

Some calculators, especially those with other PDA-like functions have actual operating systems including the TI proprietary OS for its more recent machines, MS-DOS, Windows CE, and rarely Windows NT 4.0 Embedded et seq, and Linux. Experiments with the TI-89, 92, 92+ and Voyage 200 machines show the possibility of installing some variants of other systems such as a chopped-down variant of CP/M, an operating system which has been used for portable devices in the past.

Tools which allow for programming the calculators in C/C++ and possibly Fortran and assembly language are used on the computer side, such as HPgcc, TIgcc and others. Flash memory is another means of conveyance of information to and from the calculator.

The on-board Basic variants in TI graphing calculators and the languages available on HP 48 type calculators can be used for rapid prototyping by developers, professors, and students, often when a computer is not close at hand.

Most graphing calculators have on-board spreadsheets which usually integrate with Microsoft Excel on the computer side. At this time, spreadsheets with macro and other automation facilities on the calculator side are not on the market. In some cases, the list, matrix, and data grid facilities can be combined with the native programming language of the calculator to have the effect of a macro and scripting enabled spreadsheet.

See also[edit]

References[edit]

  1. ^ Wildstrom, Stephen. "A Smarter Math Machine". Business Week. Archived from the original on 9 Oct 2006. Retrieved 5 July 2011. 
  2. ^ Mathematics Educators Exploring Computer Algebra Systems. "Let no child be left behind". MEECAS. Retrieved 5 July 2011. 
  3. ^ Pointon, A; Sangwin (20 March 2003). "An analysis of undergraduate core material in the light of hand held computer algebra systems.". International Journal of Mathematical Education in Science and Technology 34 (5): 671–686. 
  4. ^ http://jcq.org.uk/attachments/published/898/17.%20ICE%2010-11.pdf General, Vocational and Diploma Qualifications Instructions for conducting examinations.
  5. ^ http://www.curriculum.wa.edu.au/internet/_Documents/Policy/Calculators_use_in_the_WACE_examinations_2010_pdf.pdf Calculator use in the 2011 WACE examinations
  6. ^ http://meyda.education.gov.il/sheeloney_bagrut/2012/1/HEB/35807.PDF The instructions for conducting the 5 point exam Bagrut in math, 2012 in Hebrew.

Further reading[edit]

  • Dick, Thomas P. (1996). Much More than a Toy. Graphing Calculators in Secondary school Calculus. In P. Gómez and B. Waits (Eds.), Roles of Calculators in the Classroom pp 31–46). Una Empresa Docente.
  • Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education. 34(5), 433-463.
  • Heller, J. L., Curtis, D. A., Jaffe, R., & Verboncoeur, C. J. (2005). Impact of handheld graphing calculator use on student achievement in algebra 1: Heller Research Associates.
  • Khoju, M., Jaciw, A., & Miller, G. I. (2005). Effectiveness of graphing calculators in K-12 mathematics achievement: A systematic review. Palo Alto, CA: Empirical Education, Inc.
  • National Center for Education Statistics. (2001). The nation's report card: Mathematics 2000. (No. NCES 2001-571). Washington DC: U.S. Department of Education.