# Manipulatives Make Abstract Math Concepts Concrete

Using math lessons that include manipulatives can help cement learning.

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If a picture is worth a thousand words, a hand-held object is worth a million. Using manipulatives in mathematics can help bring conceptual ideas alive for students. Yet, because mathematics is traditionally taught more abstractly, relying on textbooks, paper, and pencil, teachers in middle school and high school are often reluctant to use manipulatives. However, manipulatives can help relieve student boredom, spark imaginations, and promote cooperative learning at any age level.

Things always get exciting when the props come out in a classroom, and just about anything can be used. Pennies, beads, sea-shells, M&Ms, calculators, rulers, dominos, dice, spinners, playing cards, buttons, and even pebbles are just some objects that allow students to make mathematics more concrete. Such manipulatives can easily be used when focusing on estimation techniques, fractions, measurement, ratios, probability, and properties of whole numbers, for example.

There are also many companies that make more specialized manipulatives. Tangrams, pattern blocks, geoboards, algebra tiles, Cuisenaire Rods, geometric solids, mini cubes, square tiles, and base ten sets are just some of the manipulatives available for purchase. Many of these manipulatives are familiar to elementary math teachers, but they can also be utilized by middle school and high school teachers by creating more complex challenges and experiments for students.

Manipulatives may also encourage practical applications of mathematical concepts, and increase the retention of information. Rather than assigning problem sets every day for homework, teachers could instead ask students to design a probability simulation using a manipulative. Additionally, manipulatives often allow for multicultural and historical connections to mathematics. For instance, tangrams have Chinese roots. Students might also study the evolution of the abacus through various cultures and number systems.

While it is true using manipulatives requires teacher to spend more time planning, setting up, and cleaning up, their use typically proves to be worth it because it provides a engaging, kinesthetic change for students.

## Math Lesson Plans Using Manipulatives:

Students strategize how to vary different numbers of tiles to form shapes, and then find minimum and maximum perimeters. Students examine area with either square tiles or geoboards.

Students analyze patterns of exponential models by collecting data using paper folding and M&Ms. They use graphing calculators to create scatter plots that lead to equations for the exponential models. They then apply this model to predict the future population changes in African rhinos.

Students listen as the teacher identifies and defines four math properties. They use manipulatives (small and large rocks) to experience non-traditional number systems. They use the "rocks and pebbles" to demonstrate the properties.

Students use paper cups and colored chips to observe properties of operations with real numbers. As a class, students brainstorm using manipulatives to demonstrate the associative, commutative, distributive identity and inverse properties.

Geoboard Exploration of the Pythagorean Relationship

Students create a right triangle on a geoboard or on dot paper. They construct a square on each side of the triangle, and complete a table and determine a pattern. Students use the Pythagorean Theorem to derive the distance formula.