﻿ Math Manipulatives: Essay and Virtual Manipulatives on the Web

Are you confused by terms that educators use? The ASCD Lexicon of Learning might be what you need.

# Math Manipulatives

## Page 1: Essay and Virtual Manipulatives on the Web

Math Manipulatives contains two pages of resources:

Math Manipulatives (Page 1 of 2):

• A short essay about virtual manipulatives:

Math Manipulatives (Page 2 of 2): Calculators and PDA resources, including calculator tutorials, activities, software enhancements, and calculator apps for mobile devices

#### What is a virtual manipulative?

In What are Virtual Manipulatives?, Patricia Moyer, Johnna Bolyard, and Mark Spikell (2002) defined a virtual manipulative as "an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (p. 373).  Static and dynamic virtual models can be found on the Web, but static models are not true virtual manipulatives.  Static models look like physical concrete manipulatives that have traditionally been used in classrooms, but they are essentially pictures and learners cannot actually manipulate them.  "...[U]ser engagement distinguishes virtual manipulative sites from those sites where the act of pointing and clicking results in the computer's providing an answer in visual or symbolic form" (p. 373).  The key is for students to be able to construct meaning on their own by using the mouse to control physical actions of objects by sliding, flipping, turning, and rotating them.

Virtual manipulatives have a range of characteristics, such as pictorial images only, combined pictorial and numeric images, simulations, and concept tutorials, which include pictorial and numeric images with directions and feedback (Moyer-Packenham, Salkind, & Bolyard, 2008).  Currently, virtual manipulatives are modeled after concrete manipulatives such as base ten blocks, coins, pattern blocks, tangrams, spinners, rulers, fraction bars, algebra tiles, geoboards, and geometric plane and solid figures, and are usually in the form of Java or Flash applets.  Patricia Deubel of CT4ME developed the figure above to illustrate virtual manipulatives found on the Web, which are useful for mastery of basic skills and conceptual understanding of K-12 mathematics and calculus.

#### What role can virtual manipulatives play in learning mathematics?

Virtual manipulatives can be used to address standards, such as those in  Principles and Standards for School Mathematics (NCTM, 2000), which calls for study of both traditional basics, such as multiplication facts, and new basics, such as reasoning and problem solving.  Using manipulatives in the classroom assists with those goals and is in keeping with the progressive movement of discovery and inquiry-based learning. For example, in their investigation of 113 K-8 teachers' use of virtual manipulatives in the classroom, Moyer-Packenham, Salkind, and  Bolyard (2008) found that content in a majority of the 95 lessons examined focused on two NCTM standards:  Number & Operations and Geometry. "Virtual geoboards, pattern blocks, base-10 blocks, and tangrams were the applets used most often by teachers. The ways teachers used the virtual manipulatives most frequently focused on investigation and skill solidification. It was common for teachers to use the virtual manipulatives alone or to use physical manipulatives first, followed by virtual manipulatives" (p. 202).

Virtual manipulatives provide that additional tool for helping students at all levels of ability "to develop their relational thinking and to generalize mathematical ideas" (Moyer-Packenham, Salkind, & Bolyard, 2008, p. 204). All students learn in different ways. For some, mathematics is just too abstract. Most learn best when teachers use multiple instructional strategies that combine "see-hear-do" activities.  Most benefit from a combination of visual (i.e., pictures and 2D/3D moveable objects) and verbal representations (i.e., numbers, letters, words) of concepts, which is possible with virtual manipulatives and is in keeping with Paivio and Clark's Dual Coding Theory .  The ability to combine multiple representations in a virtual environment allows students to manipulate and change the representations, thus increasing exploration possibilities to develop concepts and test hypotheses.  Using tools, such as calculators, allows students to focus on strategies for problem solving, rather than the calculation itself.

According to Douglas H. Clements (1999) in "Concrete" Manipulatives, Concrete Ideas there is pedagogical value of using computer manipulatives. He said, "Good manipulatives are those that are meaningful to the learner, provide control and flexibility to the learner, have characteristics that mirror, or are consistent with, cognitive and mathematics structures, and assist the learner in making connections between various pieces and types of knowledge—in a word, serving as a catalyst for the growth of integrated-concrete knowledge. Computer manipulatives can serve that function" (Section: The Nature of "Concrete" Manipulatives and the Issue of Computer Manipulatives, para. 2).

Christopher Matawa (1998, p. 1) suggested many Uses of Java Applets in Mathematics Education:

• Applets to generate examples. Instead of a single image with a picture that gives an example of the concept being taught an applet allows us to have very many examples without the need for a lot of space.

• Applets that give students simple exercises to make sure that they have understood a definition or concept.

• Applets that generate data. The students can then analyze the data and try to make reasonable conjectures based on the data.

• Applets that guide a student through a sequence of steps that the student performs while the applet is running.

• Applets that present ''picture proofs''. With animation it is possible to present picture proofs that one could not do without a computer.

• An applet can also be in the form of a mathematical puzzle. Students are then challenged to explain how the applet works and extract the mathematics from the puzzle. This also helps with developing problem solving skills.

• An applet can set a theme for a whole course. Different versions of an applet can appear at different stages of a course to illustrate aspects of the problem being studied.

While the research is scarce on mathematics achievement resulting from using virtual manipulatives, Moyer-Packenham, Salkind, and Bolyard (2008) found, overall, results from classroom studies and dissertations "have indicated that students using virtual manipulatives, either alone or in combination with physical manipulatives, demonstrate gains in mathematics achievement and understanding" (p. 205).  Generalizability might be a concern, however, as found in Kelly Reimer's and Patricia Moyer's action research study (2005), Third-Graders Learn About Fractions Using Virtual Manipulatives: A Classroom Study.  The study provides a look into the potential benefits of using these tools for learning.  Interviews with learners revealed that virtual manipulatives were helping them to learn about fractions, students liked the immediate feedback they received from the applets, the virtual manipulatives were easier and faster to use than paper-and-pencil, and they provided enjoyment for learning mathematics.  Their use enabled all students, from those with lesser ability to those of greatest ability, to remain engaged with the content, thus providing for differentiated instruction.  But did the manipulatives lead to achievement gains?  The authors do admit to a problem with generalizability of results because the study was conducted with only one classroom, took place only during a two-week unit, and there was bias going into the study.  However, results from their pretest/posttest design indicated a statistically significant improvement in students' posttest scores on a test of conceptual knowledge, and a significant relationship between students' scores on the posttests of conceptual knowledge and procedural knowledge.  Applets were selected from the National Library of Virtual Manipulatives.

Boston Public Schools has a professional development initiative to provide teachers and students access to virtual manipulatives and technology equipment that directly support the district's math and technology curricula.  Partially funded by a NCLB state grant, SELECT Math contains alignments for Grades 6-8, Algebra I and II, and Geometry with a Scope and Sequence calendar describing which book or chapter is being used in math classes during each month of the school year.  Click on the individual book/chapter to see the related SELECT Math alignments, worksheets, and links to supporting virtual manipulatives. The project began in 2002 as a collaboration between the Boston Public Schools' Secondary Math and Instructional Technology departments, in conjunction with their partner, the Education Development Center, Inc.  CT4ME believes this initiative to be valuable for middle and high school math educators throughout the country.  Visit Teacher2Teacher for more on the role of manipulatives.

#### Using Virtual Manipulatives and Calculators and Overusing Manipulatives in General

In order to effectively use virtual manipulatives in the classroom, "teachers must have an understanding of how to use representations for mathematics instruction as well as an understanding of how to structure a mathematics lesson where students use technology...Teachers must also be comfortable with technology and be prepared for situations where computers may not be available or Internet connections are not working properly" (Reimer & Moyer, 2005, p. 7).  This author's own experience confirms that virtual manipulatives may take a while to download, and in some cases, the wait time might be frustrating.  Imagine the frustrations for a learner anxious to begin.  Plus, even when successfully downloaded, they might not work fast enough for learners who are accustomed to playing high speed, interactive video games.  In some cases, the footprint on the screen might be too small for learners with poor mousing skills or for those with limited dexterity to click on relevant icons or to perform the spins, rotations, flips and turns required.

Teachers should be aware of problems that might arise from overusing both concrete and virtual manipulatives.  In The State of State Math Standards 2005, David Klein (2005) discussed nine problem areas in which state standards come up short.  Among those was concern for an overuse of calculators and manipulatives in that students might come to depend on them and focus on the manipulatives more than on the math.  "[M]any state standards recommend and even require the use of a dizzying array of manipulatives in counterproductive ways" (p. 11).  In this author's view such a reliance might have its roots in the quality of instruction, in part, and failure of the math educator to explicitly state and reinforce the link between the use of the manipulative, and development of concepts for understanding and properties of mathematics to be learned.  Such might be the case, for example when using algebra tiles for multiplying and factoring polynomials, if the educator failed to explicitly link the knowledge of the distributive property to that action.

I have an interesting personal story to relate on the use of calculators.  One day our newspaper person, who was a middle school student at the time, knocked on our door to collect our monthly payment for the newspapers.  He took out his calculator to multiply the weekly payment by four, which he should have been able to do mentally.  I asked him what he would do to figure out my bill, if his calculator no longer worked.  He said, "I'd go buy new batteries!" Klein (2005) stated that manipulatives are useful for introducing new concepts to elementary students, but, "In the higher grades, manipulatives can undermine important educational goals" (p. 11).  Among those are for students to develop skill fluency, conceptual understanding, and mathematical reasoning.  Many states' standards documents overemphasize calculator use, for example.

I agree with Klein (2005) in that educators should not overly rely on calculator use at the expense of having students master basic skills and memorize basic facts, which are essential for higher order learning in mathematics.  In this sense drill and practice still have a role in teaching and learning mathematics.  According to E. D. Hirsch (1999), drill and practice may have a disparaging connotation as a pedagogical tool to teach skills and runs contrary to the progressive movement, but the method should not be slighted as low level. It is just as essential to complex intellectual performance as drill and practice are to the virtuoso violinist or the athlete on the playing field.

Bottom line:  According to the National Mathematics Advisory Panel (2008) in its Foundations for Success:

Despite the widespread use of mathematical manipulatives such as geoboards and dynamic software, evidence regarding their usefulness in helping children learn geometry is tenuous at best. Students must eventually transition from concrete (hands-on) or visual representations to internalized abstract representations. The crucial steps in making such transitions are not clearly understood at present and need to be a focus of learning and curriculum research. (p. 29)

With this being said, CT4ME has a number of virtual manipulatives that can serve you well in the classroom.  As one educator recently told me at one of my own conference presentations on this topic, "I don't have to worry about students flicking rubber bands at each other any more!"  She was using virtual geoboards.

The Amazon widget below shows books using the search phrase: math manipulatives.  You can also use the widget to search with other key words or to find books with activities for use with calculators.  Suggestions include:

 TI-Nspire calculator graphing calculator algebra and calculator calculus and calculator trigonometry and calculator statistics and calculator, or any math subject and calculator. hands-on math math and tangrams math and geoboards visual math

### Manipulatives on the Web

#### Technology Requirements:

JAVA: You might need to download Java, a free plug-in to interact with virtual manipulatives.

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.

Analyze Math by A. Dendane (United Arab Emirates University) has an extensive collection of applets in the section called "Math Software."  Learners can explore and gain deep understanding of topics in mathematics including calculus, precalculus, geometry, trigonometry and statistics.

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.

British and Metric Conversions from Annenberg Media's Learner.org. is an interactive Web site to learn how to convert measurements between the metric system and the English (sometimes called British) system, which is commonly used in the United States.  Learn the symbols, more about length, mass, temperature, volume, and then test your knowledge.

C.a.R. (Compass and Ruler) is a dynamic geometry program simulating compass and ruler constructions on a computer.  It's by by Rene Grothmann, professor of mathematics at the University of Eichstätt (Germany).  For school geometry in the plane, there are lots of geometric objects like points, lines, rays, line segments, circles, arcs, angles, polygons and conic sections. Ruler and compass constructions can be changed by dragging one of the basic construction points. Arithmetic computations, numerical solutions, curves and functions go beyond classical constructions. It is even possible to construct in 3D using advanced macros.  Other geometries, hyperbolic or elliptic, can be explored. The software is free open-source and comes with documentation, tutorials, and demos.  Java needs to be installed.

CalcPlot3D is a Java applet from the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources, which provides an exploration environment for multivariable calculus.

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.

Complex Numbers--An Interactive Introduction to Complex Numbers includes two Java applets, explanatory text, and exercises for exploring basic principles of complex numbers (e.g., basic arithmetic, conjugation, powers, roots).  Both applets allow users to "input complex numbers in either cartesian or exponential form and display them in vector form."  Content is posted within the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources.

Core Math Tools from NCTM is a set of downloadable "interactive software tools for algebra and functions, geometry and trigonometry, and statistics and probability. The tools are appropriate for use with any high school mathematics curriculum and compatible with the Common Core State Standards for Mathematics in terms of content and mathematical practices. Java required" (Core Math Tools Home section).  You'll find general purpose tools, custom apps, sample lessons that use the tools, data sets, and more.

Crickweb.co.uk has interactive resources for use in primary grades.  Multiple core subjects, including math, are available.  These can be used with interactive whiteboards, or online with your computers.  Highly recommended.

Didax Math Manipulative Information Center provides general information on manipulatives and instructions for the use of specific manipulatives.  While the focus is on concrete manipulatives with downloadable activities, where possible associated internet and virtual manipulative activities are provided.

DreamBox Learning has over 20 free virtual manipulatives for K-5 math available on its website that can also be used with any whiteboard.  They help learners engage with number sense, addition, subtraction, multiplication, and division concepts.

Erathothenes' Sieve--watch the primes up to 400 being generated with this Java display.  Learn more about prime numbers and the largest known prime at Mersenne Prime Digits.  As of 2013, the largest prime is 257885161-1.  It has 17,425,170 digits!

Euclid's Elements Dr. Dave Joyce of Clark University in Massachusetts brings all 13 books of the Elements alive using a Java geometry applet to illustrate definitions and postulates.

ExploreLearning.com is a subscription based set of over 400 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. Gizmos for elementary math for grades 3-5 are also available.  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.

FlashandMath.com has an entire section devoted to math applets appropriate for use in precalculus, calculus I and II, multivariable calculus, discrete mathematics and more.  These are great for classroom demonstrations or for student exploration and self-paced study.  For example, use the derivative plotter to enter a function, see its graph, then test your understanding of its derivative by sketching it on the mathlet.  When done see the graph of the real derivative of the function.

Floorplanner "an online CAD (Computer Aided Design) system specifically developed for bringing good-looking, complex floor plans within reach of everyone" (section: About us).  Educators might consider using it with learners in upper elementary grades through high school for creating 2D and 3D plans--great for geometry studies.  Create an entire house with interactive floor plans.  A free version is available for personal use.

GeoGebra is free, open-source, dynamic mathematics software that joins geometry, algebra and calculus. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.  You can enter equations and coordinates directly. GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.  It received several international awards, including the European and German educational software awards. In Dynamic Mathematics with GeoGebra, authors Markus Hohenwarter, principle developer of the software, and Judith Preiner (2007) said, "You can use GeoGebra both as a teaching tool and to create interactive web pages for students from middle school up to college level. Specifically designed for educational purposes, GeoGebra can help you to foster experimental, problem-oriented and discovery learning of mathematics" (Abstract section).  They illustrate the basic ideas of the software and some of its versatile possibilities by discussing several interactive examples.  Additional resources for using GeoGebra include:

• GeoGebraWiki--Educators can post their creations and view the work of others.  For example, see the main page in English for free teaching materials created with GeoGebra.
• Exploring Geometric Transformations in a Dynamic Environment, posted within the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources.  You'll find eight activities and virtual manipulatives related to teaching reflections (also including reflections with parallel lines, intersecting lines, and concurrent lines), rotations, translations, and glide-reflections to middle and high school learners.

Geometry 3-D Shapes is one of the Interactives from Annenberg Media's Learner.org.  Students can interactively learn about three-dimensional shapes, calculate surface area and volume, and discover some of the mathematical properties of shapes. It comes with tutorials and an interactive test.

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.

Geometry Playground is a free Java applet from the Mathematical Association of America Mathematical Sciences Digital Library Loci Resources, which enables "ruler and compass" constructions in "Euclidean, Spherical, Projective, Hyperbolic, Manhattan and Conical geometries" (Description section).  Potential classroom activities are included.

Glencoe/McGraw-Hill virtual manipulatives include a series of common assets for preK-8, which are associated with their e-books.  Choose the grade level, background, and virtual manipulative.  For example, you will find game boards, story boards, various grids, graph formats, number lines, base ten blocks, cubes, clock, algebra tiles, currency, and so on.  Tools include a ruler, protractor, timer, stopwatch, straight line tool, pen, color options, text boxes, and more.

Harcourtschool.com E-Lab: Extensive collection of interactive applets for grades 3-6.  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 is recommended.

Harvey's Homepage: Interactive SMART Board Lessons for Mathematics.  This is a real find for educators who have a SMART board in their classroom--so engaging and colorful.  Download the notebooks of lessons, which were developed by Harvey Almarode (James Madison University). Topics include number sense, computation/estimation, patterns/algebra/functions, geometry, measurement, probability/statistics, teacher utilities, problems of the day, and more.

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.

Interactivate, which is a project of The Shodor Education Foundation, Inc., includes a set of interactive JAVA-based tools to support courseware in science and math.  Activities are divided by grade-band (3-5, 6-8, 9-12, and undergraduate).  Some activities allow students to keep score to monitor their progress, and are listed under assessments.  Activities that use the tools support NCTM standards.  Browse by topic, audience, type (activity, lesson, discussion), or subject (number and operations, geometry, algebra, probability, statistics, modeling, discrete, and other).  A math dictionary of terms used within the site is included.  Interactivate has also aligned its activities to several textbook series (e.g., Connected Math, Math Thematics, Mathematics in Context, and MathScape).

Interactive Mathematics promotes learning math while you play with it!  Primarily for upper secondary and college level students; basic math includes 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.

Larry Green's Applet Page has numerous applets within basic math, beginning algebra, intermediate and college algebra, statistics, and calculus.  This is well worth an investigation.

Learning and Teaching Scotland has a section on numeracy, which includes interactives for developing core skills to cope with the demands of everyday life.  These help learners to learn to be comfortable with numbers, graphs, symbols, diagrams, and calculators.  There are several levels of content.  Highly recommended.

The Learning Equation (TLE) is brought to you by Ron 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

Learning Today, which produces math and reading software, has free teacher resources with math applets, interactive math games, and K-5 interactive whiteboard lessons.  Applets include tesselations, pan balance, Venn diagram shape sorter, a fractal tool, ten frame, a concentration game, angle sums, congruence theorems, and a turtle pond for investigating distance and angle.  Games deal with number and operations, measurement and geometry.  These are definitely of value and worth investigating.

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 Doodles is by Daren Carstens, who is the developer of the award winning software Math Arena.  He offers several interactive puzzles and games for elementary learners to have fun.  But there is learning, too, as students practice addition, subtraction, money skills, and pattern matching (shape, number, color), for example.  Among current games are Connect Sums, Number Jump, Polyomino Shift,  Double Traits, Sum Stacker, Time Shuffle, and more.  Older learners also benefit, as students really need to think as they do these.

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.

MathinSite Applets address pure and applied mathematics.  Several pure mathematics applets are suitable for middle and high school learners.  Some applets come with work sheets and theory sheets.  You'll find: Straight Line, Parabola, Cubic, Exponential Function, Circle, Ellipses and Circles,Trig Functions, Adding Sinusoids; Piecewise, Odd/Even & Periodic Functions; Complex Numbers, Maclaurin Series; Arithmetic, Geometric & Linear Sequences; and Differentiation.  The site is from Bournemouth University (UK).

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.  There are numerous interactive applets with at least one accompanying nearly every concept.  This is highly recommended for visual learners and reinforcement for all.

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.  Math Tools also contains resources (e.g., tools, activities, lesson plans, and other support materials) for mobile devices such as iPad, Android phones, iPhones, and so on.

Math Warehouse is "dedicated to dynamic Math lessons, demonstrations, and interactive activities." There are topics in algebra, geometry, trigonometry.  Worksheets are available for download, too.

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.

Oswego City School District (NY) interactive games for math will help elementary school learners develop basic skills.  In some, levels of difficulty can be set.  Teachers can also design their own games (e.g., matching, or drag and drop).

PBS LearningMedia has over 600 resources for teaching mathematics using videos and interactives for grades preK-12.  These can also be used on interactive whiteboards.

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.

PhET Interactive Simulations from the University of Colorado at Boulder has " an extensive suite of simulations to improve the way that physics, chemistry, biology, earth science and math are taught and learned.   The simulations are interactive tools that enable students to make connections between real life phenomena and the underlying science which explains such phenomena" (About PhET section). The simulations are free, and can be easily used in the classroom. Flash and Java must be installed.

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.

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:

Tabula for grades 5-12 has dynamic geometry (DG) features similar to those in other DG programs.  However, developer David Hartmann indicated it can be distinguished from them as follows: "1. It has a set of virtual manipulation tools for folding, cutting, and taping. This set is designed to help teachers guide and extend classroom activities involving paper, scissors, etc.  Many teachers are familiar with Michael Serra’s Patty Paper Geometry book in which properties of shapes are discovered by folding and working with paper in other ways.  Tabula can be used to model the steps in most of the activities.  2. It has a different approach to transformations that incorporates pushpins and animation.  This is particularly designed to help students struggling with visual/spatial skills.  3. It has a set of virtual instruments that model their real counterparts.  This includes a traditional compass, a safety compass, a ruler, and a protractor.  Unlike models commonly used on electronic whiteboards, these instruments can be used directly in constructions and proofs.  4. It’s easy to use.  The research suggests that DG software can be beneficial, but that in order to realize the benefits, a serious commitment to preparation and classroom time must be made.  Tabula requires less of a time commitment and many of the same benefits." (personal communication, November 4, 2011)

Tangrams from Channel4.com has three levels, then the challenge level.

TeacherLED, produced by Spencer Riley, a teacher since 2002 in Birmingham, United Kingdom, "is designed to provide teachers with access to free high quality teaching resources for use on Interactive Whiteboards."  Math resources are categorized by algebra, data handling, number, shape/space/measure, general maths, and investigations.

Thinking Blocks teaches children how to visualize and solve math word problems. Using interactive blocks and cubes, children create models that illustrate the underlying math concepts within the problems. Thinking Blocks combines guided instruction, video explanations, and independent practice to help young students in grades 3-5 develop strong problem solving skills.

Using Virtual Manipulatives on the Web to Develop Number Sense by Margo Mankus and hosted by the Graduate School of Education at George Mason University contains three interactive applets and a host of activities to develop concepts of number, operations, and geometry.  Applets, developed by J. Bulaevsky, include pattern blocks, base ten blocks, and integer rods.  These are great tools to demonstrate concepts in the classroom and to give students practice in class or at home.

Utah Education Network K-12 Student Center features student interactives categorized by math topics for gradebands K-2, 3-6, and 7-12.

Virtual Polyhedra, by G. Hart, is a collection of thousands of polyhedra that you not only can look at, but move and spin.  Hart also provides extensive research on the topic and information on how to construct paper models.

Visual Calculus from the University of Tennessee Math Department at Knoxville is an excellent resource that includes tutorials, drills and programs for pre-calculus and calculus.  Animations, interactive pages, step-by-step solutions and illustrations are included for topics ranging from functions, graphing with technology, solving equations, and conic sections to limits and continuity, derivatives and applications, integration and applications, sequences and series.  As a veteran calculus teacher, P. Deubel highly recommends this site.

Visual Fractions, by Richard Rand, is an online tutorial with instruction and interactive practice in identifying, renaming, comparing, and operating on fractions, including mixed numbers (add, subtract, multiply, and divide). All examples are modeled with number lines or circles.  This site helps students, primarily in grades 3-8, to picture fractions and operations on them.

Visual Math: Functions contains JAVA applets for exploring linear and quadratic functions.  Linear explorations include representations of linear functions, functions defined on intervals, equivalent expressions for functions, addition/subtraction, transformations, rate of change.  Quadratic explorations equivalent quadratic expressions, comparisons and operations, solving equations, graph transformations (vertex form), add functions (polynomial form), products of linear functions, quadratic growth, graphic design, motion at changing speed, and economic decisions.

Visual Math Learning is a free educational web site developed by W. Bateman, who indicates that the site features "an interactive on-line tutorial for teaching elementary mathematics and basic arithmetic for grades K-12 at the pre-algebra level. It is an instructional aid for parents, teachers, and educators of primary, elementary, and middle school students, as well as a resource for lesson plans, homework help, and home schooling math lessons. The tutorial includes games, puzzles, interactive diagrams, and computer animated virtual manipulatives that emphasize active learning concepts by visualization."

Visualization of Elementary Math, by E. Kluk, M. Frank, and M. Mangru of Dickinson State University in North Dakota, contains several applets to help students with understanding meanings of elementary math operations.  Applets include a visual multiplication table, visual addition (single and double digit), and visual fractions (making and comparing, adding and subtracting).  Instructional examples are included.

Waldo's Interactive Maths Pages contains over 50 Java applets for teaching math concepts.  These are grouped within age ranges (11-14, 14-16, and 16-19) and then by topics usually studied within those ranges.  Topics are extensive from angles and intersecting and parallel lines, exploring triangles, solving equations, sequences, graphing linear and quadratic equations, trigonometry, calculus, and much more.  Waldo is a teacher in England who developed these.  Don't miss this work, which links theory to the visual.

Walter Fendt's Java Applets on Mathematics address arithmetic, elementary algebra, geometry (e.g., transformations, triangles, Pythagorean theorem, area and circumference of circles, and platonic solids), trigonometry (visual demonstration of graphs of sine, cosine and tangent of an angle), vector analysis, analysis (e.g., first and second derivatives), and complex numbers (arithmetic operations).  These are very engaging and illustrative of concepts.

WisWeb is the website of the Freudenthal Institute for middle and high school education (students of 12 to 18 years old). The main focus of the site is applets.  Subjects include number sense, number and estimation, measurement, algebra and calculus, geometry, discrete math, statistics and probability.  The Freudenthal Institute is a Research Group on Mathematics Education located in the Netherlands.

Wolfram Demonstrations Project from Wolfram Research, Inc. is a collection of interactive illustrations "to illuminate concepts in science, technology, mathematics, art, finance, and a remarkable range of other fields" (sec: About the Wolfram Demonstrations Project). You can preview Demonstrations directly in your browser, but to fully interact with them, download the live versions and run them in the free Mathematica Player, if you don't already have Mathematica (Version 6 or higher) or Mathematica Player Pro. Educators might also wish to download the "Experience Mathematica in Education" brochure to learn how to instantly make lessons interactive and use these Demonstrations to help you visualize classroom concepts.  Math demonstrations are provided for grades K-2, 3-5, middle school, and high school by subject, including calculus.  Highly recommended.

Cinderella, version 1.4, is free software for creating geometric constructions.  Its developers say, "Use it to experience geometry on your desktop or on the web.  Starting from simple triangle relations, continuing with trigonometric theorems up to fractals and transformation groups Cinderella lets you create and manipulate visualizations in an intuitive, yet powerful way. Using Java® technologies, constructions are seamlessly exported to the WWW."

FlashandMath.com has an entire section devoted to developing your own mathlets: ActionScript 3 Tutorials for Developers.

#### References

Clements, D. H. (1999). Concrete' manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45-60. [Update online]. Retrieved from 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, NY: Doubleday. ISBN: 0-385-49524-2.

Klein, D. (2005, January). The state of state math standards 2005. Washington, DC: Thomas B. Fordham Foundation.  Retrieved from http://www.edexcellence.net/publications/sosmath05.html

Matawa, C. (1998, August). Uses of Java applets in mathematics education.  Paper presented at Asian Technology Conference in Mathematics, Tsukuba, Japan.  Retrieved from 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.  Retrieved from http://www.nctm.org/eresources/article_summary.asp?URI=TCM2002-02-372a&from=B

Moyer-Packenham, P. S., Salkind, G., & Bolyard, J. J. (2008). Virtual manipulatives used by K-8 teachers for mathematics instruction: Considering mathematical, cognitive, and pedagogical fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), 202-218. Retrieved from http://www.editlib.org/index.cfm?fuseaction=Reader.ViewFullText&paper_id=26057

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

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.