# Teaching Mathematics with Virtual Manipulatives

*from Educators' eZine*

### What is a virtual manipulative?

The Center for Implementing Technology in Education (CITE) defines virtual manipulatives as digital objects often modeled on their physical or concrete counterparts such as Base 10 Blocks and Cuisenaire Rods. In the article "Virtual Manipulatives for K-12 Classrooms," a virtual manipulative is described as "an interactive, Web-based visual representation of a dynamic object that presents opportunities for constructing mathematical knowledge" (Moyer, 2002). Virtual manipulatives are distinguished from computer models or pictures by their interactive attributes and manipulability.

### Why use a virtual manipulative?

There are several reasons to use manipulatives. One aspect is the time, cost and space savings over use of concrete manipulatives. Virtual manipulatives are free, available to the entire class, and working with them does not require tables or other large spaces. Virtual manipulatives do require computers, ideally with Internet access, but many can be downloaded and used offline. Students enjoy using virtual manipulatives and find them easy to use (CITE).

Research has shown that concrete manipulatives help students better understand abstract concepts in math, and comparative studies show virtual manipulatives to be as effective as concrete ones. Some research shows Virtual manipulatives may lead to more complex, richer understandings of concepts than through use of concrete manipulatives (CITE).

### How to use Virtual Manipulatives Effectively

The virtual manipulative sites described below are accompanied by lesson plans for teachers describing how to use them effectively in teaching mathematics. Virtual manipulatives are often accompanied by usage instructions and hints to help students use them successfully, making independent use possible. However, researchers and authors caution "Although virtual manipulatives provide some support for individual student use, as with physical manipulatives, students benefit from teacher guidance to help them use the manipulative correctly and connect to the underlying math" (CITE).

The Center for Implementing Technology in Education lists several factors to consider when selecting virtual manipulatives:

- Can the level of difficulty be adjusted for different students?
- What type of feedback do they provide?
- Will teachers need to provide feedback and support?
- How clear are the instructions for use? (CITE)

### Examples and Resources for finding Virtual Manipulatives: Virtual Manipulative Websites

**Math Playground**

Teachers who are passionate about math and technology created this site for students Grades 1-6, and their expertise and experience is evident in this excellent site. Students can use the fraction bars to show equivalent fractions, examine area and perimeter, work with angles, and use a virtual geoboard. The Percentage Equation allows one to enter any two of the three quantities, Part, Whole, or Percent, and this manipulative will calculate the remaining quantity. The percentage relationship is displayed visually two different ways, and an equation shows how to compute the unknown quantity and the result is then stated in words.

The percentage pie chart is excellent; it shows the part of the whole represented by a number, and it is easy to change the numbers and observe the effect on the chart. Function machine helps students learn the basics of functions and how to write linear functions, has a variety of levels, and is challenging. There are many more manipulatives at this award winning site. The advertisements are a small drawback.

**Mathematics Lessons that are Fun**

A site created by Cynthia Lanius with 22 interactive activities. Each lesson gives a description of the purpose, topics covered, grade level, materials (the option to print the activity is often given), suggested teaching strategies and lesson procedures.

http://math.rice.edu/~lanius/Patterns/

In the lesson on Fractions online pattern blocks from the arcytech website are used to answer questions such as the following:

Accompanying material for Fractions lesson:

Description: These activities are designed to cause students to think; they are not algorithmic. They do not say, *To add fractions, do step one, step two, step three*. Students will explore geometric models of fractions and discover relations among them.

Appropriate Grades: 3rd - 6th, maybe. But precocious kindergarteners could do some of it, and middle schoolers needing another look at fractions could appreciate it as well. "Drawing Fun Fractions" would be good for most middle school students.

Lesson Procedures: Students will reinforce their knowledge of fractions by using online pattern block activities. The lessons are designed for students to work independently or with guidance from the teacher. Lessons should be printed so students can draw and color the appropriate shapes.

Mathematics Topics: Identifying fractional values, equivalent fractions, adding and subtracting fractions, ordering fractions, and identifying geometric figures and their properties.

**Arcytech's Educational Java Programs**

Ten areas of math such as Base 10, fractals, the Pythagorean theorem, and the meaning of pi are available, with interactive applets and lessons included for each. The manipulatives and lessons are primarily for elementary and middle school students. This is one of the earliest manipulative sites created, and a recipient of several awards. The manipulatives help students learn to count money and tell time, and mirror concrete manipulatives such as Cuisenaire Rods. Lesson plans are provided, as well as detailed instructions. The feedback from other teachers gives ideas for use, also.

The money program is the next best thing to working with real money.

http://arcytech.org/java/money/

In the section called "Money Lesson: Description for Teachers", the author, Jacobo Bulaevsky, includes a summary of the lesson, suggested grade levels, objectives, procedures and materials.

**National Council Teacher of Mathematics**

At the NCTM website 17 E-examples are linked to NCTM standards for grades PK-8. Each activity includes an explanation of the mathematics standard, instructions for using the manipulative or interactive object, a problem based lesson and reflective activities to extend the lesson.

E-Examples—interactive applets for investigating concepts from NCTM. Examples are divided into grade clusters.

- 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.

The NCTM site also contains the Illuminations page, including 102 manipulatives, lessons based on NCTM standards, and links to related online activities. The purpose of Illuminations is to "provide Standards-based resources that improve the teaching and learning of mathematics for all students" and "provide materials that illuminate the vision for school mathematics set forth in *Principles and Standards for School Mathematics*".

Fraction Model—grades 3-8: explore several models for representing fractions with adjustable numerators and denominators.

Version 1: The range of the numerator is limited to values from 0 to 20 and the denominator is limited to benchmark values 1, 2, 4, 5, 8, 10, and 20.

Version 2: The range of the numerator and denominator is limited to values from 0 to 20.

Version 3: The range of the numerator and denominator is limited to values from 0 to 100.

http://my.nctm.org/eresources/

**National Library of Virtual Manipulatives**

The NLVM combines the organization of sites such as NEIRTEC and the Select website (see below) with true, high quality manipulatives. The manipulatives are sorted by grades and linked to the standards set forth by the NCTM. To use, select a manipulative from the grade level you teach and an activity from one of the strands. Strands include the areas of number, algebra, geometry, measurement, and data & probability.

Geoboard: A strength of this site are the lessons available, and the suggested activities, as in this Geoboard example for PK-2.

Use geoboards to illustrate area, perimeter, and rational number concepts.

Measurement Grades 3-5

The Instructions given for this manipulative are much more detailed, with many more activities described.

**Shodor Educational Foundation**

This site holds Project Interactivate, described as "a set of free, online courseware for exploration in science and mathematics", including activities, lessons, and discussions. The site holds tons of instructor resources, including links to related activities, links to standards, and correlations to many commonly used math textbooks. It does not focus solely on virtual manipulatives, as included are many types of interactive activities, such as simulations and problem solving activities. It is designed for grades 6-8, but some activities are great for grades 3-5. Activities and lessons are based on NCTM standards and include concepts in number and operation, geometry and measurement, algebra and functions, and probability and data analysis.

Basic Navigation

The site is structured around collections of activities, lessons, and discussions. You can reach listings of each type of resource either by following the prominent links (2)(3)(4) on the front page, or by using the "Jump To:" pulldown menu (1) located at the top of every page.

Venn Diagram Shape Sorter—sort colored shapes into a Venn Diagram. This is an example of a lesson suitable for younger elementary students.

**PBS Teacherline Interactives**

The PBS site features 12 interactive math lessons that use geoboards, histograms, floor tiles, space blocks and more to solve mathematical problems. Objectives, suggestions for use, and often printables are included. To teach linear functions, two manipulatives (Make a Rule and Real-life Data) demonstrate the relationship between a real-life situation, the values it presents and the mathematical equation that generates those values. They can be used separately or instructions are provided for using them together.

Linear Functions: Make a Rule and Real-life Data

http://www.pbs.org/teacherline/resources/interactives.cfm

**NEIRTEC (Northeast and the Islands Regional Technology in Education Consortium)**

There are a few manipulatives on this site and they are called Interactive Math Applets. The best part of this site is in itsâ€™ organization of links to other virtual manipulative sites. The links are organized into these 12 categories of algebra readiness indicators:

1. Represent Integers, Fractions, Decimals, Percent

2. Compute Fractions, Decimals, Percent

3. GCF, LCM, Prime Factors

4. Ratios, Rates, Proportions

5. Geometric Figures and Properties

6. Perimeter, Area, Volume

7. Pythagorean Theorem

8. Data

9. Probability

10. Algebraic Equations

11. Patterns

12. Functions

The Integer Applet timeline is a great manipulative for exporting addition or subtraction of negative and positive numbers.

http://www.neirtec.org/activities/applets/numberline.htm

**SELECT Math website**

This is another site which organizes links to virtual manipulatives found on other sites. The site was developed by the Boston Public School District to link virtual manipulatives to the math and technology curricula, grades 6 through high school. Activities are aligned to MA state standards and to the *Connected Math* textbook series by Prentice Hall. The site contains links to many virtual manipulatives at sites such as Shodor and the National Library of Virtual Manipulatives, organized as shown in the chart below.

http://boston.k12.ma.us/teach/technology/select/

**References**

Center for Implementing Technology in Education. (CITE)

Moyer, P. Bolyard, J. & Spikell, M. (2002)."Virtual Manipulatives in the K-12 Classroom".

Oregon Technology in Education Council (OTEC). Virtual Manipulatives.

*Dr. Judy Donovan is an Assistant Professor of Instructional Technology at the HH 345 School of Education in Indiana.*