Wicked Problem: Using Algebra Balance Scales Virtual Manipulative

Educational Need

Solving algebraic equations represents the focus of the educational need addressed with this project. More specifically, this project targeted the following two algebra standards:

• Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution (A.FO.06.12)
• Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solution (A.FO.06.13)

Proposed Technological Solution

After time spent both thinking and searching for strategies to teach solving equations, the decision was made to use the National Library of Virtual Manipulatives Algebra Balance Scales applet to address the educational need identified. The free Algebra Balance Scales applet provides students with a linked visual and symbolic representation of the equation which change concurrently once students modeled the equation and begin to solve to determine the value of the variable. As students selected an operation to perform to both sides, the changes are evidenced in both the equation as well as the blocks removed from the respective sides of the scale.

Technological Pedagogical Knowledge 

The selected technology, Algebra Balance Scales virtual manipulative, supports the teaching methods and strategies intended for the intervention. The applet scaffolds solving equations, models steps to solve an equation sequentially both symbolically and visually, and provided immediate feedback both verbally and visually to student responses. The applet offered students the opportunity to compare the created model and the given equation before proceeding to solving. Additionally, the virtual manipulative allowed for connections and observations to be made regarding how changes effect all representations of the equation, an advantage over using a physical manipulative to investigate solving equations.

Technological Content Knowledge

“Meaning does not reside in tools; it is constructed by students as they use tools.” Herbert and Colleagues (1997) quoted by Suh in Third Graders' Mathematics Achievement and Representation Preference Using Virtual and Physical Manipulatives for Adding Fractions and Balancing Equations. The Algebra Balance Scales applet helps make the content accessible by providing linked, multiple representations. The visual representation of the scale and blocks helps address student misconceptions regarding coefficients. The picture links to the symbolic representation by clarifying what coefficients actually mean. “One,” is a common response from students when given an equation such as 3x+7=13 and asked how many “x's” are on the left side. The manipulative helps address this misconception by providing verbal and visual feedback during the modeling phase. Students have the opportunity to recall prior information, or experience for the first time, that multiplication is repeated addition. Simultaneous manipulation of the scale and symbolic representation contribute greatly to the applet's support of the content and increased accessibility to students. The manipulative prompts students to work between both the visual and symbolic representation. As one representation is changed, students evidence how the change effected the other representation supports the learning objectives. Students watch as blocks are added, taken away, multiplied, or divided and the scale remains balanced. The differences in the two images demonstrate the linked representations previously described. The applet provides students an image of the intended learning goals.

Pedagogical Content Knowledge

The instructional strategies used for this intervention activity support the content much like the selected technology supports both content and pedagogy. Students ability to solve equations depends largely on a developed understanding of the symbolic representation. The misconception regarding coefficients discussed previously relates to the essential understanding of symbolic representations. Scaffolding supports the content by ensuring students have appropriately modeled the given equation. The scaffolding continues after the modeling phase also. This links to the importance and understanding of order of operations. Immediate feedback then continues the support of the content. Students proceed through the scaffold to solve the equation with appropriate mathematical moves; however, the feedback redirects students with a little hint to reconsider and manipulate the equation differently. Both symbolic and visual representations of equations further support the content's accessibility to students. As noted previously, verbal and visual feedback helps connect and develop an understanding of what manipulating an equation actually does to the equation. Again, the visual feedback contributes and supports the learning objectives by showing a balanced scale emphasizing the equality of both sides of the equation. Scaffolding, providing feedback and multiple representations, along with student manipulation of the applet all assist in making the content more accessible to students. 

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