Using Manipulatives: Introduction

Philosophy:

Students who use manipulatives in their math classes usually outperform those who do not. This benefit holds across grade level, ability level, and topic, given that using the manipulatives makes sense for the topic. Also, attitudes toward mathematics are improved when students are instructed with concrete materials. The student will move from using manipulatives to expressing problems in symbolic form as he or she advances from the concrete operational stage to the formal operational stage.

Role of the teacher:

The teacher is no longer the giver of knowledge. Instead the teacher is the coach or facilitator of the learner’s discovery of knowledge. When students learn from their own investigation, it is more meaningful to the student and the knowledge is retained longer.

Role of the student:

The student is no longer a spectator. Now the student is the active participant. Learners should use the hands-on materials to discover patterns, make conjectures, and test their conjectures before moving on to the more abstract stage of learning.

What about time? We DO have SOLs to cover?

This is true. It is time-consuming to introduce and use manipulatives in the classroom. However, the time you spend using these materials will pay off on future lessons when the students are comfortable with using them. Also, not only will you be able to cover the same amount of material, but students’ retention will improve as well.

Please keep in mind that all Algebra topics do not have to be taught in isolation. Several topics in Algebra are interrelated and more than one SOL may apply in one lesson and correlate with other activities. As time passes, students will enter the class with experience in using manipulatives. This will make your task easier and will save valuable class time.

Suggestions for using manipulatives:

·  Allow students to work in pairs. This will allow them to answer each other’s questions before your attention is needed. It also halves the class size in function. (The teacher only has to address 14 “students” instead of 28.)

·  Use your overhead set during the introduction and as the end as a summary. Your overhead set should be the same as theirs to avoid confusion. (See notes below about making a set.)

·  Not all students will approach a problem the same way. Ask the student to explain their method of solution and encourage appropriate strategies.

·  Guide students in their actions, even if they are making a mistake—MAKE THEM THINK!! Encourage them to reflect on their actions to improve their thinking and problem solving skills.

·  As students become comfortable with the manipulatives, they may want to move on to a “shorter method.” Encourage them to use symbols in place of the manipulatives. If they have difficulty, allow them to continue with the manipulatives as needed.

·  Students should be allowed to use manipulatives on the test as you would allow them to use a calculator.

·  At the end of the lesson, have students discuss their strategies and/or write about their learning. This will help them become focused on the learning that occurred.

Obtaining Manipulatives:

Several companies manufacture sets of manipulatives: Algebra Tiles, Algebra Mods, Algeblocks, etc. If your school does not have these, you can make a set for your own class. Making your own is a less expensive option that will allow each student to have a set of his or her own to take home and use. Also, the replacement cost of damaged or lost pieces is minimal.

The following page contains a blackline master that you can use to copy for each student. The colors you use do not have to match the commercial ones. There is so much variation from one company to another that is can get confusing (some companies use green as positive while others use green as negative). Choose one color for positive tiles and another color for negative tiles. Red is a good color for negative because students tend to associate negative with “being in the red.” However, the colored paper that is available at your school. To decrease the confusion, students may mark their tiles as either “+” or “—.”

Teacher Materials: If the teacher has an overhead set of tiles, use the same colors for the students packets that correspond to the overhead set. If you do not have an overhead set, make transparencies of the tile sheet and color it (or make colored ones on the computer).

Materials:

Each student should have 2 tile sheets—one of each color.

Each sheet contains 4 large square tiles (x2)

12 rectangles (x)

20 small square tiles (1’s)

SOLs:

Each activity will have a stated objective along with the appropriate SOL.

Sources:

The following activities are of my own creation unless otherwise stated. Most of them have been adapted from a compilation of many sources. For example, the section on solving equations is a mixture of Henry Borenson’s “Hands-On Equations” and the idea of a balance scale. Dr. Borenson’s activities also focus on equations being a balance scale, but his materials (pawns and dice) may be difficult to use for some. A criticism I have of “Hands-On Equations” is the use of a single pawn as the variable. A variable represents an unknown quantity and I believe a cup or container of some kind best represents this quantity. Also, several textbooks have their own version of balancing scale activity. After years of doing this, I have compiled a little of each one into these activities. Therefore, it is difficult to give credit to one single source.