Activities With Manipulatives for Middle Grade Math

An inexpensive manipulative, dice can be used to teach counting, logical reasoning and probability.

In her paper titled "Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete," Judith Bellonio likens mathematics manipulatives to the Chinese proverb: "I hear, and I forget. I see, and I remember. I do, and I understand." Simply, manipulatives are objects -- concrete or virtual -- that can be touched, moved and rearranged to enhance student learning. In mathematics, manipulatives help students transition from the concrete to more abstract levels. While research is lacking on the use of manipulatives at the middle school level, studies summarized by Dianne Golsby at Texas A&M University indicate mathematics achievement increased at the elementary level with the use of manipulatives.

Fractions

Pattern blocks are used to build patterns, explore geometric shapes and enhance critical thinking.

In a curriculum unit designed for grades five and six, Judith Bellonio of the Yale National Initiative describes teaching fractions using pattern blocks, a popular math manipulative. Each student is given three red trapezoids, three blue rhombuses and 12 green triangles. Ask the students how many triangles fit in one trapezoid, then ask them to build a number of triangles using different ratios of patten blocks, for example: 4/4 green meaning four green triangles make one big triangle, 2/4 (1/2) green and 2/4 (1/2) blue or 3/4 red and 1/4 green. Students can move on to build parallelograms and trapezoids using the pattern blocks: 6/8 blue and 2/8 green for the parallelogram and 12/12 green and 6/12 blue and 6/12 green for the trapezoid. The lesson reinforces geometric shapes while teaching fractions.

Geometry

Invented by the mathematician Caleb Gattegno in the 1950s, the geoboard grid tool -- a square lattice of pegs -- can be employed to illustrate area, perimeter and rational number concepts. Using rubber bands, students create geometric shapes. Cartesian coordinates can be introduced as a means of communicating the location of the geoboard pegs. In their paper titled "A New Algebra: Tools, Themes, Concepts," Henri Picciotto and Anita Wah provide sample exercises such as the following: Find the area of various geoboard triangles with one horizontal or one vertical side. Students develop techniques for finding shapes of the same size using simple arithmetic operations like addition and subtraction, dispelling the notion that area is only a multiplication problem.

Probability

"Rabbits and Wolves," a virtual manipulative, allows students to alter components of an ecosystem and evaluate impacts.

North Carolina-based Shodor disseminates interactive, Java-based mathematics manipulatives through the Interactivate Project. In the "Rabbits and Wolves" module, students learn about probability and chaos by manipulating a simple ecosystem composed of grass, rabbits and wolves. The activity allows students to simulate how plants and animals achieve balance in nature. Simple rules govern the students' experiments in the ecosystem. For example, two rabbits cannot occupy the same section of grass; wolves will not eat a rabbit if doing so causes it to surpass their maximum food capacity; and wolves and rabbits cannot reproduce until they have reached a certain age. Students are posed questions such as: "What would happen if there were lots more bunnies than there were wolves? Would the bunnies take over and live forever?" "How do the rabbits and wolves live in balance in this game?"

Algebra

Simple linear equations can be represented and solved using a manipulative based on a balance beam representation. Utah State University's National Library of Virtual Manipulatives provides an online algebra balance scale in which students drag x-boxes (the unknown) and unit blocks to either side of the beam until the beam is balanced, representing the linear equation. Arithmetic operations can be performed on either side of the fulcrum, with the goal of getting a single x-box on one side and a finite number of unit blocks on the other, all while keeping the beam balanced. Alternatively, one can create a concrete manipulative balance beam using a ruler and a block as the fulcrum. Use a large block of unknown size on one side and then add unit blocks or blocks of various sizes to the other side until the beam is balanced, thus yielding the size of the large block.

References

• National Middle School Association: Manipulatives in Middle Grade Mathematics; D. Goldsby; February 2009
• Yale National Initiative: Multi-Sensory Manipulatives in Mathematics: Linking the Abstract to the Concrete; J. L. Bellonio; 2001
• Journal of Mathematical Behavior: A New Algebra: Tools, Themes, Concepts; H. Picciotto and A. Wah; 1993
• Shodor - Interactivate: Rabbits and Wolves;
• Utah State University: National Manipulatives Laboratory: Algebra Balance Scales - Negatives

Resources

• Mathematics Lessons That Are Fun, Fun, Fun!; Pattern Block Fractions; Cynthia Lanius; 1997
• Math Playground: Geometry Board
• Shodor - Interactivate: Activities
• Utah State University - National Library of Virtual Manipulatives: Algebra