Though there are dozens of different manipulatives that can be used to educate students, the pedagogical basis for using one is the same: firsthand interaction with manipulatives helps students understand mathematics. Manipulatives provide concrete ways for students to bring meaning to abstract mathematical ideas. They help students learn new concepts and relate new concepts to what they have already learned. They assist students with solving problems.
When students explore with manipulatives, they have the opportunity to see mathematical relationships. They have tactile and visual models that help develop their understanding. Without these concrete references, students are too often lost in a morass of abstract symbols for which they have no concrete connection or comprehension.
Teachers need to learn how to make use of concrete manipulatives so that students learn the how and why of mathematics concepts. Students’ thinking and reasoning must be the top priorities when they are engaged in learning with manipulatives. The concrete manipulatives and the activities for which they are used are only as valuable as the students’ reflection on the mathematical concepts.
Algebra Tiles involve students in learning algebraic concepts, including adding and subtracting polynomials, factoring trinomials, the Zero Principle, and solving first and second degree equations. Each tile represents the quantities x, x2, and 1 along with their additive inverses.
AngLegs enable students to study polygons, perimeter, area, angle measurement, side lengths, and more. The set includes 72 snap-together AngLegs pieces (12 each of six different lengths) and two snap-on View Thru® protractors.
The Attribute Blocks set includes five basic shapes (triangle, square, rectangle, circle, and hexagon) displaying different attributes. The basic shapes come in three different colors, two different sizes, and two different thicknesses. Attribute Blocks can be used to teach sorting, patterns, and identifying attributes.
Base Ten Blocks
Base Ten Blocks are constructed in powers of ten, representing ones, tens, hundreds, and thousands. The materials include 1-centimeter unit cubes to represent ones, 10-centimeter rods to represent tens, and 10-centimeter square blocks to represent hundreds. They can be used to teach number and place value concepts, such as the use of regrouping in addition and subtraction. They can also be used to teach measurement concepts, such as area and volume. Place Value Mats serve as organizers.
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The Bucket Balance features removable ½-liter buckets. The buckets are clear to help students see what they are measuring. Measures 16"L × 5.75"W × 5"H. The balance helps students explore the measurement of mass with accuracy to 1 gram.
The set of Bug Counters contains counters in six different shapes (grasshopper, bumblebee, beetle, spider, dragonfly, and caterpillar) and six colors. Bugs can be used for sorting and counting activities.
These plastic Centimeter Cubes are 1 cm on a side and come in 10 colors. They can be used to teach counting, patterning, and spatial reasoning. They are suitable for measuring area and volume and may also be used to generate data for the study of probability.
Color Cubes are available in manipulate® and wood, and six different colors in a set: red, orange, yellow, green, blue, and purple. They help children through hands-on exploration of basic mathematics and geometric relationships as they stack, count, sort, and work with patterns.
Color Tiles are a collection of square tiles, one inch on a side, in four colors–red, blue, yellow, and green. The tiles have applications in all areas of the math curriculum. They are useful for counting, estimating, measuring, building understanding of place value, investigating multiplication patterns, solving problems with fractions, exploring geometric shapes, carrying out probability experiments, and more. A supply of these tiles provides versatile assistance to math instruction at all grade levels.
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The CounTEN Sorting Tray is an egg carton-shaped ten-frame carton used for building basic numeracy as well as for sorting activities.
Cuisenaire Rods include Rods of 10 different colors, each corresponding to a specific length. White Rods, the shortest, are 1 cm long. Orange rods, the longest, are 10 cm long. Rods allow students to explore all fundamental math concepts, including addition and patterning, multiplication, division, fractions and decimals, and data analysis.
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The vertical ten-frame tiles provide an intuitive and visual representation of patterns for numbers up to 10. They can be used to learn shortcuts, such as counting the spaces remaining instead of counting the number of dots. They emphasize the importance of 10 in place value.
Deluxe Rainbow Fraction® Circles
The set consists of nine color-coded, 3 ½ inch plastic circles representing a whole, halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. The circles enable students to explore fractions, fractional equivalences, the fractional components of circle graphs, and more.
Deluxe Rainbow Fraction® Squares
The set consists of nine color-coded, 10-cm plastic squares representing a whole, halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. The squares enable students to explore fractions, fractional equivalences, and more.
Fraction Circle Rings
These five plastic rings are used with the Deluxe Rainbow Fraction® Circles to make measurements related to circles and fractions of circles. The set consists of a Degree Measurement Ring, a Fraction Measurement Ring, a Decimal Measurement Ring, a Percent Measurement Ring, and a Time Measurement Ring.
Basic Fraction Circles have six circles that show halves, thirds, fourths, sixths, eighths, and one whole. Each circle is a different color, with plastic pieces that can be put together and taken apart to show different fractions. Circles are ideal for introducing students to basic fraction concepts.
Fraction Tower® Equivalency Cubes
Faction Tower Equivalency Cubes snap together to demonstrate fractions, decimals, and percentages. Each tower is divided into stacking cubes that represent a whole, halves, thirds, fourths, fifths, sixths, eighths, tenths, and twelfths. Each cube is labeled with the part of a whole that it represents. One side shows the fraction, another shows the decimal, and a third shows the percentage. The fourth side is blank. Students can turn the cubes or towers to see each of the representations of the same value. Towers, or portions of towers, can be compared with each other.
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These Geared Clocks are made of plastic and have hidden gears that reflect accurate hour and minute relationships. The hour and minute hands are color-coded to match hour and minute markings on the clock face. Clocks allow children to explore telling time on analog clocks and calculating elapsed time.
The double-sided Geoboard is 7.5 inch square and made of plastic. One side has a 5 &time 5 peg grid. The other has a circle with a 12-peg radius. Students stretch rubber bands from peg to peg to form geometric shapes. Geoboards can be used to study symmetry, congruency, area, and perimeter.
This mirror is made of colored, transparent plastic so that the mirror image of an object placed in front of the mirror appears superimposed on the background behind the mirror. The mirror can be used to help students understand transformations, symmetry, and congruence.
Graphing Mats are double sided and have square grids or a Venn diagram for graphing. Both sides are ideal for activities that use manipulatives or other real objects. The mat can be used to introduce graphing data. It can also be used for activities such as sorting and classifying geometric shapes.
Plastic Inchworms are 1 inch long. Pieces come in six different colors and can be snapped together to make a chain. Worms are ideal for children who are just starting to learn measurement with standard units, because the worms provide a transition to using a ruler. They can be used to measure length, width, and height.
The Inchworms Ruler is made of plastic. Each inch of the ruler is marked with an Inchworm to help children see the units of measurement clearly. The ruler can be used with compatible Inchworms products to explore using standard units to measure length, width, and height.
Link ‘N’ Learn® Links
Multicolored Link ‘N’ Learn Links are large and easy for children to interlock to make chains. The chains can be used to explore concepts such as number sense and operations. Use links to teach counting, addition, and subtraction. Links can also be used to explore measuring with nonstandard units.
Pattern Blocks are a collection of six shapes in six colors—green triangles, orange squares, blue parallelograms, tan rhombuses, red trapezoids, and yellow hexagons. The shapes are designed so that the sides are all the same length except for the trapezoid, which has one side that is twice as long. This feature makes it possible for the shapes to nest together and provides for a wide range of explorations.
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These dice come in 4-, 6-, 8-, 10-, 12-, and 20-sided varieties and are most typically used for probability activities. They may be used to generate data for number and operations activities and for data analysis.
Reflect-It™ Hinged Mirror
This mirror with hinge and clear protractor base allows you to see the multiple reflections created by controlling the angle size of the mirror. Create angles up to 180° using the base. Use the mirror without the base to hypothesize the properties of special angles; then draw conclusions and discover symmetry.
A Rekenrek is an arithmetic frame designed to help children visualize addition and subtraction strategies. The 20-bead Rekenrek features two rows of 10 beads. Each of these sets of ten are broken into two sets of 5 beads using contrasting colors–red and white–to help children see numbers, as well as to visualize how numbers can be composed and decomposed. The Rekenrek combines features of the number line, individual counters, and base-ten models such as Base Ten Blocks. This model allows for children to think in groups of those benchmark numbers, 5 and 10.
Relational GeoSolids are 14 three-dimensional shapes that can be used to teach about prisms, pyramids, spheres, cylinders, cones, and hemispheres. GeoSolids facilitate classroom demonstrations and experimentation. The shapes can be filled with water, sand, rice, or other materials to give students a concrete framework for the study of volume.
Each side of a Snap Cube can be connected to another cube. Cubes can be used to teach a variety of different math concepts. Use cubes to explore number sense and operations with activities involving counting, place value, addition, and subtraction. Or use cubes to show measuring using nonstandard units. Cubes can also be used to demonstrate patterning and basic geometry.
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These collapsible Sorting Circles can be used to teach beginning algebraic thinking by having children sort objects into sets. They can also be used for classifying geometric shapes by attribute.
Spinners enable students to study probability and to generate numbers and data lists for number operations and data analysis.
Tangrams are ancient Chinese puzzles made of seven three- and four-sided shapes. Each set of tangrams contains four tangram puzzles in four different colors. Each puzzle consists of five triangles (two small, one medium, and two large), a square, and a parallelogram. Tangrams can be used to solve puzzles in which all seven pieces must be put together to create a specified shape. Tangram puzzles teach many geometric concepts, including symmetry, congruency, transformations, and problem solving.
Three Bear Family® Counters
Three Bear Family Counters come in three different sizes and weights—Baby Bear™ (4 grams), Mama Bear™ (8 grams), and Papa Bear™ (12 grams). Bear Counters™ can be used to teach abstract concepts involving number sense and operations by allowing children to act them out. Use Bears to explore sorting and comparing sets, counting, estimating, addition and subtraction, and sequencing. Bears can be used to experiment with measuring mass, or to teach patterning concepts and early algebra.
These versatile Two-Color Counters are thicker than most other counters and easy for students to manipulate. They can be used to teach number and operations concepts, such as patterning, addition and subtraction, and multiplication and division. Counters can also be used to introduce students to basic ideas of probability.
These 4.5-inch-square clocks are laminated so that students can write the digital time below the movable hands of the clock face. Clocks can be reused over and over again to give students plenty of hands-on practice measuring time. Clocks also help students practice addition, subtraction, and problem solving.
XY Coordinate Pegboards
XY Coordinate Pegboards can be used to graph coordinates in one, two, or four quadrants; show translations of geometric figures; display data in various forms; and demonstrate numerous algebraic concepts and relationships.