An Algebra Balance is a balance scale for interactive and visual learning of various algebra concepts. The manipulative consists of a beam and pan that tilt depending on which side holds a greater value. Students place and manipulate objects to model equations and inequalities with constants and variables.
Using a physical Algebra Balance, positive integers are represented with weights or chips. Variables are represented using containers that conceal an amount of weights or chips (the value of the variable). Two-pan Algebra Balances are restricted to positive values only and negative values must be modeled by removing objects. Four-pan Algebra Balances use clever mechanics so that a negative pan on each side will tilts the side up when weights or chips are placed on it.
Virtual manipulatives, such as Brainingcamp's Algebra Balance Manipulative, use blocks to represent positive values and balloons to represent negative values. Virtual versions include a number of other conveniences that do not exist with physical Algebra Balance manipulatives. For example, Brainingcamp's Algebra Balance Manipulative performs an animation when positive values are dropped on negative values so that students can visualize the concept of a zero pair.
Create the expression 2 + (-2) on the left pan by placing 2 unit blocks and 2 unit balloons. The left pan balances with the empty right pan, so both sides have a value of 0. The additive inverse of any number is –a. The sum of a and –a is always 0.
Commutative Property of Multiplication
On the left pan, create the expression 2 × 3 by placing 2 stacks of 3 unit blocks. On the right pan, create the expression 3 × 2 by creating 3 stacks of 2 unit blocks. The scale balances because both sides have a value of 6. The commutative property of multiplication states that the way factors are ordered does not change the product. For any a and b, a × b = b × a.
Model an equation on the scales. To find the value of the variable that makes the equation true, isolate the variable by performing the same operations to both sides of the Algebra Balance. The solution to the equation is the value of the pan when the opposite pan has just a single variable. To solve using division, evenly distribute the pan contents into a number of groups equal to the divisor, and then delete the contents of all but one group.
One Solution, No Solutions, Infinite Solutions
An equation that simplifies to the form x = a has exactly one solution. An equation that simplifies to the form a = a has an infinite number of solutions because any value of x will balance the scales and make the equation true. An equation that simplifies to the form a = b (where a and b are different constants) has no solutions because there is no value of x that will balance the scale and make the equation true.