An Algebra Strategy to Consider: Symbolic Manipulation An Excerpt from the Considerations Packet – Algebra Strategies for Middle School

by Donni Davis-Perry, M.Ed.

The Training and Technical Assistance Center (T/TAC) at the College of William and Mary (W&M) offers educators and parents user-friendly information on a variety of educational topics through Considerations Packets. These concise packets of information suggest strategies to consider using when working with students. Considerations Packets are available for free, either by requesting a copy to be delivered in the mail or for immediate download at the T/TAC W&M website. This article contains an excerpt from the Algebra Strategies for Middle School Considerations Packet written by Elizabeth M. O'Brian. The complete packet is available at the T/TAC website link above.

Algebra Strategies for Middle School Students

This Considerations Packet addresses strategies that middle school teachers can implement when teaching algebraic thinking. Algebra is a strand found in the Virginia Standards of Learning at each of the middle school grade levels. It is also one of the areas addressed by the National Council of Teachers of Mathematics (NCTM) in Principles and Standards for School Mathematics.

The algebra strategies included in this packet address the following topics: Exploring patterns, graphs, symbolic manipulation, technology as an aid for understanding algebraic concepts, discourse in the algebra classroom, and writing about algebraic thinking. Additional references and resources are provided at the end of the packet. This article explains symbolic manipulation, one of the many strategies featured in the packet.

Manipulatives for Solving Equations (Thompson & Mayfield-Ingram, 1998)
When solving the one-step equation, x + 4 = -5, represent the equation as follows:


Using a small plastic cup or an empty film canister to represent the unknown quantity and beans painted red on one side to represent known quantities enables students to represent the equation. Explain that students are looking for the unknown quantity of beans that will fill the cup to make both sides equal.

Concepts to be taught and reinforced using this method include:

  1. Using different colors to represent positive and negative numbers.

  2. How to represent a variable.

  3. Forming a zero by having equal quantities of positive and negative beans, emphasizing the idea that they will "cancel" each other out.

To continue this example, place 4 red beans on the left side of the equation to "cancel out" the 4 white beans. Then place 4 red beans on the right-hand side of the equation to keep the equation balanced.


Have students spend considerable time in class working with concrete manipulatives to problem solve. After students are comfortable with the process of representing variables and creating a zero, they can move to drawing pictures to represent the equations. Finally, after much practice, students can move to the abstract manipulation of numbers and symbols.

As students move to the symbolic stage, many will benefit from using a different-colored pencil to show the step where a zero is created and both sides of the equation are kept balanced. (The example below shows the use of bold numbers to indicate where color could be used.)

x + 4 = -5
    -4     -4
x      = -9


O'Brian, M. E. (2004). Algebra strategies for middle math students. (T/TAC Consideration Packets)

Thompson, V., & Mayfield-Ingram, K. (1998). Family math: The middle school years: Algebraic reasoning and number sense. Berkeley: Regents of the University of California.

Date: May/June 2009