Exploring Manipulatives on the Web

Deubel, P. (2007). Math manipulatives [Online]. Retrieved [2007] from Computing Technology for Math Excellence at http://www.ct4me.net/math_manipulatives.htm

Calculators, their Tutorials, and Software enhancements:

• Basic Calculator:  Enter your basic problem followed by the equal sign, as in 6 * 2 =   Press "c" to clear the calculator for the next problem. The Javascript code for this calculator was provided free by http://javascript.internet.com/
   

• Calculator City from 1728 Software Systems has online converters and calculators for topics studied in basic math, algebra, geometry, trig, calculus, and more.  Several include additional graphics and brief discussions of concepts associated with the calculator.

• Calculators On-Line Center features over 19,000 calculators for mathematics, statistics, science, and engineering.  Calculators for mathematics range from those suitable for basic mathematics through calculus and higher level mathematics.  Don't miss this vast collection of specialized calculators by topic from J. Martindale.

• Graphing Calculator Help from Prentice Hall for TI-82, TI-83, TI-85, TI-86, TI-89, TI-92, HP48G, CFX-9850, Sharp EL9600C, and Casio FX2. 

• SimCalc MathWorlds software for TI-graphing calculators, computer, and TI-Navigator developed by the University of Massachusetts-Dartmouth's James J. Kaput Center for Research and Innovation in Mathematics Education.  Animations, real life examples, narrative stories, and more are used to explain math concepts.

• Tutorials from Texas Instruments.  TI provides a number of interactive tutorials for basic and graphing calculators.

• Tutorials from Atomic Learning on Texas Instruments calculators: the TI-30XS, TI-84 and TI-Nspire handhelds.

• WebGraphing.com provides online graphing (1D, 2D, & Interactive 3D) of functions, equations, systems of equations, inequalities in one and two variables, and piecewise functions, with tutorial analyses appropriate for students of algebra, precalculus, and calculus. There is also a forum for the different math levels.  What sets the function graphing calculators apart from other graphing calculators is the automatic display of asymptotes and discontinuities in standard mathematical notation, and the automatic determination of an optimal graphing window--one that includes all mathematical features of interest.  Dr. Barry Cherkas of the Mathematics and Statistics Department at Hunter College (CUNY) is President of WebGraphing.com.

Tired of using 3, 4, 5 right triangles?  Get a quick list of Pythagorean Triples using Nicholas Exner's (MSTE division of University of Illinois-Urbana-Champaign) simulation or learn more about Pythagorean Triples using several online calculators from Dr. Ron Knott of the University of Surrey in the UK.

Using positive integers s and t with s< t, calculate legs x and y and hypotenuse z as follows:

K-12 Algebra Applets--Arlen Strader of the Texas A & M University has two algebra tiles applet pages to help middle and high school students explore factoring of quadratic equations by arranging given tiles into a rectangle: Algebra Tiles1 (easy version-coefficients positive) and Algebra Tiles2 (hard version-coefficients positive and negative).  There are also applets for Cartesian coordinates, slope, parallel and perpendicular lines, and Pythagorean Theorem.

Ambleweb Numeracy Hour of Ambleside C.E. Primary School (UK) contains excellent interactive online Tools for Teachers to use with elementary students for demonstrating concepts and helping students to master basic skills.  There is a large buttoned online four function calculator for demonstrating how to use a calculator, an online resource for teaching about angles and how to use a protractor, a virtual tangram, counter squares of various sizes and purposes on which you can scribble using the left button on your mouse, applets for studying number sequences and patterns, a simple bar graph to introduce graphing data, teaching time on an analog clock, and interactive geometry.  There is also a section for interactive skill building at three levels of difficulty, puzzles, and games.  

Arcytech's Educational Java Programs are primarily for elementary and middle school students.  Award-winning applets help students learn to count money, tell time, use patterns to explore fractions and geometric shapes; use base-10 blocks to explore whole and decimal numbers, place value and algebra with blocks; use integer bars to explore basic math facts, fractions, and more; use fraction bars to explore equivalent fractions, ordering fractions, adding fractions, and more.  Additional applets help students explore the Pythagorean theorem and the value of Pi, and fractals.  Lesson plans are included.  Site developed by J. Bulaevsky.

Argyll Centre of Edmonton Public Schools in Canada hosts Jim Reed's Math and Science Resources.  His resources address elementary and secondary topics in numbers/operations, patterns and relations, shape and space, probability and statistics, algebra, geometry, and trigonometry. Students can engage with interactive learning activities, see movies, and manipulate excellent Java math applets to study concepts.  The site is also an excellent resource for the home-schooled student.

Collaborative Whiteboard from GE allows individuals or groups to draw anything freehand or type text.  You can display a  graphing grid, select and then draw geometric shapes, save the work and share it with others via email.

Erathothenes' Sieve--watch the primes up to 400 being generated with this Java display.

Euclid's Elements Dr. Dave Joyce of Clark University in Massachusetts brings all13 books of the Elements alive using a Java geometry applet to illustrate definitions and postulates.

ExploreLearning.com is a subscription based set of over 300 virtual manipulatives (simulations of math and science concepts), which the authors call "Gizmos."  The math series addresses the five NCTM strands for grades 6-12 and college developmental math, college algebra, and pre-calculus.  ExploreLearning Gizmos, with associated Exploration Guides, help students develop critical thinking skills and conceptual understanding.  The series is also correlated with national and state standards and major textbooks, and includes assessment and real time reporting.  A 30-day free trial is available for the teacher or home.  Sample Gizmo at the site can be immediately activated.  ExploreLearning.com contains all Gizmos and lesson plans from the award-winning ExploreMath.com and ExploreScience.com sites, in addition to hundreds of new Gizmos.

Geometry Gallery from Saltire Software contains some excellent Java applets, which are easy to use and suitable for illustrating several theorems presented in a high school or post-secondary geometry course.  These relate to angles in parallel lines, circular configurations, Pythagorean triples and incircles. Students can drag points to explore concepts.  Four triangle calculators are presented.  By entering SSS, SAS, or ASA, students can see the unique triangle that is formed, but if SSA is entered the resulting triangle is not unique (two are possible) or no triangle is possibly formed with given information.  Some advanced geometrical configurations are explored.  There is a geometry atlas of algebraic formulas with associated visuals. Saltire's work has been assisted with a number of National Science Foundation grants.

Harcourtschool.com E-Lab: Extensive collection of interactive applets for grades 3-8.  Activities to accompany each are provided.  You can download an E-Lab activity and use it without being connected to the internet.  While working on internet, Microsoft Java VM is recommended.  Later versions of Java will be supported soon for online activity.

Hispanic Math, available in English and Spanish, is an interactive multimedia tutorial program for grades 4-6 that combines complex action script with flash animation to deliver developmentally appropriate instruction on the relationship between perimeter and area. "To complete each lesson , students must help the animated characters "Kiri" and "Karlo" through a variety of adventures that require the use of measurement devices both on and off the computer screen. Each bilingual lesson features diagnostic testing, multiple learning paths, and flash animations in which working adults model various occupations such as cartography, cattle ranching, and banking." This is a project of Arizona State University and Technology Based Learning and Research.  Excellent and engaging!

HKCEE Mathematics contains several Java applets for investigating trigonometry ratios and the geometry of circles, which high school students can manipulate.

Illuminations Activities from the National Council of Teachers of Mathematics (NCTM) contain Java applets identified by grade band to investigate various concepts within NCTM identified strands.

Interactive Mathematics promotes learning math while you play with it!  Primarily for upper secondary and college level students; basic math addresses algebra, complex numbers, statistics, logarithms, trigonometry, plane analytic geometry, and graphing.  Calculus includes differentiation, integration, and higher topics in calculus.  Lectures and full examples are provided.  Site provides links to LiveMath plug-in, Scientific Notebook, and Flash.

Interactive Resources from the UK has four math packs, each with a wide range of excellent interactives for classroom use with preK-6 students.  Explore numerous concepts related to NCTM strands.  Some include whiteboard (drawing on the figures) capabilities.  There are rulers, protractors, geoboards, Venn diagrams, number lines, and so on. Don't miss these teaching tools.

Key Curriculum Press JavaSketchpad has several applets for use with secondary math students and above.  See the interactive manipulative on the Pythagorean Theorem and the JavaSketchpad DR4 Gallery.  In the Gallery you will find demonstrations from geometry (e.g., centroid), trigonometry and algebraic geometry (e.g., sine waver and conic sections), calculus (e.g., box volume and falling ladder), mechanics, and an applet on least squares.

The Learning Equation (TLE) is brought to you by R. Blond of Edmonton, Alberta, Canada.  TLE features JAVA applets that help students to interactively explore complex mathematics concepts typically studied in depth from grades 9 through post-secondary.  Among topics are triangle relations, slopes, circle geometry, conic sections, exponential and trig functions; translations, reflections, stretching; and more. 

Learning Objects from Brock University (Canada) Department of Mathematics are tools to help students explore mathematical concepts interactively.

• The Great Balancing Act: explore the mean and median

• Parabola games: explore parabola properties, transformations and their use

• Leap Frog: explore number patterns

• Fire, Fire!: explore trigonometric ratios

• Transformations: an introduction to transformational symmetries

• Markov chains: What happens after years of movement in cities and suburbs, an introduction to Markov chains

Manipula Math with JAVA by International Education Software contains a collection of over 280 applets suitable for teaching math concepts to middle school, high school, and college students.  Applets address middle school geometry and the Pythagorean Theorem, trigonometry, calculus, vectors, complex numbers, conics, and miscellaneous topics.

Math Playground: Animated characters created with Flash add to the appeal of this learning site for grades 1-6.   Among resources are some virtual math manipulatives, such as a protractor, fraction bars, equivalent fractions, percents, a spinner for probability, function machine, pattern blocks, and a geoboard.

Mathematics Open Reference is a free plane geometry textbook for high school learners.  Its focus is on using interactive materials and animations to develop concepts.

MathTools from the Math Forum contains a number of interactive tools for understanding concepts related to K-12 mathematics and calculus.  Help kindergarten students understand concepts.  Introduce topics with pattern blocks, number lines, fraction bars and more.  Algebra concepts include number systems, integers and integer operations, properties (commutative, associative, distributive), multiplying/dividing, using variables, equivalent equations, inverse operations, graphing, linear equations and systems, quadratic equations, factoring, and more.  Geometry interactives for plane and solid topics are extensive.  Trigonometry includes the Law of Sines and Law of Cosines.  There is a range of calculus tools for differentiation and integration topics.

Mathsnet.net is a highly interactive site for doing mathematics, featuring concepts in number, geometry, algebra, graphs, data handling.  JAVA, Javascript, or plug-ins may be needed.  The geometry is of particular interest.  There are curriculum, articles, books, and a section for download, including free software to explore topics interactively.  Don't miss this award winning site brought to you by B. Dye in England.

MegaMaths, brought to you by the BBC, is a School Radio series for elementary students aged 7-9.  Each 15 minute episode promotes a whole-class approach to mental maths activities through a series of lively and stimulating quizzes. 

National Library of Virtual Manipulatives for Interactive Mathematics contains manipulatives and associated online activities for each of the five strands identified in standards set forth by the NCTM.  This is an ongoing project of Utah State University and a must see site for grades preK-12.  Select a manipulative from the grade level you teach and an activity from one of the strands.

Number Line Applet: Introduce addition and subtraction of integers with this number line.  Students see the concept after clicking onto the integers  (-35 to 35) for the problem.  From MSTE at University of Illinois-Urbana_Champaign.

PBS Teacherline Interactives --explore with cubes, floor tiles, frog in a well (number patterns and relations), geoboard, histograms, making a rule, mixing orange juice, number cruncher, the race, real-life data, space blocks, surface area, and tesselate.

Principles and Standards for School Mathematics Electronic Examples--interactive applets for investigating concepts from NCTM.  Examples are divided into grade bands.

• preK-2: Investigate patterns, triangles and properties of polygons, geometry and measurement; develop spatial skills by using tanagrams, learn estimation strategies

• 3-5: Communicate about math using games, use  simulation software to investigate distance, rate, and time; investigate data

• 6-8: Visualize the concept of multiplication, investigate rate of change, length, perimeter, area, volume, congruence, similarity, and symmetry, and the Pythagorean Theorem

• 9-12: Explore vectors; use graphs, equations, and tables to investigate elimination of medicine from the body; explore inscribed figures, linear regression, and linear functions.    

Project Interactivate, which is funded, in part, by the Office of Dependent Education of the Department of Defense Education Activity (DoDEA), includes a set of JAVA-based tools to support courseware for grades 6-8, although some activities are appropriate for grades 3-5.  Activities that use the tools support NCTM standards and concepts in number and operation, geometry and measurement, algebra and functions, probability and data analysis.  Lessons supported by discussion of concepts are included.  The project is listed by The Shodor Education Foundation, Inc.

PDA Handheld Computing Resources on the Web is brought to you from the Shiawassee Regional Education Service District in Michigan.  The collection includes links to ebooks, journals, peripherals, tutorial videos, hardware/software, assessment programs, math programs, and more to integrate the PDA into your instruction.

Seeing Math by the Concord Consortium has eight free secondary math interactives written in Java to assist algebra learners in the study of functions.  "Each interactive provides a real-time connection between representations of the mathematics (symbolic, graphical, etc.), so that changes in one representation instantly cause changes in the other."  Interactives include a qualitative grapher, piecewise linear grapher, linear transformer, quadratic transformer, function analyzer, system solver, plop it and proportioner.  A user's guide, warm up exercise, frequently asked questions, and sample activity are provided. 

Statistics: Are you teaching your students about statistics and survey research?  Read Statistics Every Writer Should Know by Robert Niles.  Learn about mean, median, percent, per capita, standard deviation, margin of error, data analysis, and more.  Link to sites for data sources and interactive help to select the right statistical test.

• Descriptive Statistics (introduction to Mode, Median, and Mean) by Jay Hill at the University of Illinois.

• Chi-Square by Amar Patel. The material in this lesson, which is correlated to NCTM standards for grades 9-12,  will help students to understand statistical terms such as fairness, expectation, significance, and chi-square.  Excel spreadsheet is needed.  Content could be used with grade 8 and above, however.  Cartoon characters add humor to the discussion. 

• Tools:  Generate random numbers, sample sizes, charts and graphs, and perform statistical calculations using these tools:

• Research Randomizer

• Random.org

• The Survey System Sample Size Calculator 

• Create a Graph, brought to you by the National Center for Education Statistics, is an interactive site that helps students understand how to create area, line, and bar graphs, and pie charts.  You might point out to students the 10 Common Mistakes Found on Graphs, noted by CanTeach in Canada.  

• If you are looking for data to use when creating graphs, see the work of Robert Niles on finding data on the Internet.

• If you have developed a bar graph using Microsoft Excel, show students how to Create a Pictograph from that data. 

• Web Pages that Perform Statistical Calculations

References:

Clements, D. H. (1999). Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60. [Update online]. Available: http://www.gse.buffalo.edu/org/buildingblocks/Newsletters/Concrete_Yelland.htm

Hirsch, E. D., Jr. (1999). The schools we need and why we don't have them. New York: Doubleday. ISBN: 0-385-49524-2.

Klein, D. (2005, January). The state of state math standards 2005. Washington, DC: Thomas B. Fordham Foundation.  http://www.edexcellence.net/foundation/publication/publication.cfm?id=338&pubsubid=1117

Matawa, C. (1998). Uses of Java applets in mathematics education.  Paper presented at Asian Technology Conference in Mathematics, 1998.  Available: http://www.atcminc.com/mPublications/EP/EPATCM98/ATCMP016/paper.pdf

Moyer, P. S., Bolyard, J.J., & Spikell, M.A. (2002). What are virtual manipulatives? [Online]. Teaching Children Mathematics, 8(6), 372-377.  Available: http://my.nctm.org/eresources/article_summary.asp?URI=TCM2002-02-372a&from=B 

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Available: http://standards.nctm.org/ 

Reimer, K., & Moyer, P. S. (2005). Third graders learn about fractions using virtual manipulatives: A classroom study. Journal of Computers in Mathematics and Science Teaching, 24(1), 5-25.

Additional reading on math manipulatives:

For a current review of the literature, see David Young's (2006, April) Virtual Manipulatives in Mathematics Education. Available: http://plaza.ufl.edu/youngdj/talks/vms_paper.doc

Selected references relating to the use of math manipulatives, a list of resources by Dr. Garry Taylor of Northern Arizona University.  Note that CT4ME is one of the Web resources. 

Durmus, S., & Karakirik, E. (2006, Jan). Virtual manipulatives in mathematics education: A theoretical framework. The Turkish Online Journal of Educational Technology, 5(1), article 12. Available: http://www.tojet.net/articles/5112.htm  [Note: CT4ME is cited in this article.]