Mathematics/Physics

The Cuisenaire Four-Pan Algebra Balance: Limitations and Suggestions

 Bill Colson Morgan Park High School 425 W. Surf #817 1744 W. Pryor Ave. CHICAGO IL 60657 CHICAGO IL 60643 (773) 871-4890 (773) 535-2550

Objective(s):

Many teachers use a physical or metaphorical balance in presenting the performance of equivalent operations on both sides of an equation.  The design of the Cuisenaire balance (see illustration) facilitates the teaching of these and other algebraic properties using both positive and negative numbers.  The purpose of this lesson was: (1) to expose middle- and high-school teachers to its use as described in the manufacturer’s instructions,  (2) to explore with them any problems or limitations, and (3) to brainstorm, discuss, and evaluate ways to use the balance in the teaching of other concepts or properties.

Materials:

·       1 or more Cuisenaire Four-Pan Algebra Balance(s)

·       Chips and canisters (included in kit)

·       Instruction booklet (included in kit)

·       Sample problems

Strategy:

The lesson began with an interactive discussion regarding the intuitive connection between equations and balance.  Participants shared ways they had incorporated this concept in their own teaching.  Some had used an actual scale, and they were asked to model this using a two-pan balance.  They were then given a simple problem involving addition or subtraction of a negative term, and asked to model it using the same balance.  It quickly became obvious that this was not possible without rearrangement of terms and use of the “subtraction = adding the opposite” rule (or its converse), contradicting the introductory nature of the lesson.

Participants then watched as the Four-Pan Balance was put together, and commented on the differences from a traditional balance.  In particular, the upper beam is fixed; from each end, a short arm is attached through its middle, with a pan hanging from each end of the arm.  This would create two individual balances, except that the inner pan of each is further connected to a long arm passing through the perpendicular shaft containing the balance indicator.  The effect is to make all four pans interdependent at all times.

Next, it was explained that the inner pans would represent positive values and the outer pans, negative.  The terms of an equation are represented by putting the appropriate number of chips in each pan. Variables are represented by canisters, which are filled ahead of time with the number of chips corresponding to the solution, less one (to account for the weight of the canister).  Using examples from the instruction manual, operations such as adding opposites, adding and subtraction of integers, and solving linear equations were modeled.

Finally, participants were encouraged to suggest problems and ways to use the balance in their solution.  They were also asked to experiment with properties or problem-types not demonstrated in the manual, and to note any difficulties, either practical or conceptual.

Performance Assessment:

Participants were given random assignments from a list of problems and properties to demonstrate on the balance, some of which were expected to be impractical, if not impossible.  They were asked to explain their strategy, and comment on any advantages or limitations they saw to using the balance as opposed to another method.

Conclusions:

Participants were excited about being able to extend the concept of balance to equations containing both positive and negative numbers.  The Four-Pan Balance seemed especially appropriate for exhibiting the Zero Property (a + -a = 0), addition or subtraction of two numbers, and solution of simple linear equations (those not requiring combination of like terms).  Some pointed out possible confusion once the canisters are introduced: having to allow for their weight takes away (albeit slightly) from the conceptual simplicity.  Also, space on the pans and numbers of chips and canisters limited the demonstrations to problems containing small integers.  This was not a problem as long as the object was to demonstrate basic properties, rather than to solve problems.  Also, it was felt that merely watching the instructor demonstrate was not nearly as effective as actually doing it themselves.  This raised the question of how many balances it would be necessary to purchase per class for it to be a worthwhile investment.

The general conclusion of the participants was that they would be enthusiastic about using the Balance to introduce algebraic concepts, if they could afford a class set.  However, they would be impractical to use in routine problem solving after the introductory period.  This was not seen as a major drawback, since the objective of a lesson should be to master the concept, not the use of the teaching tool.  The Balances can be brought back out when introducing the Multiplication and Division Properties of Equality, and even combined when solving systems of equations, but they should not be used on a daily basis in the way we use calculators, for example.

References:

Kung, George and Vicchiollo, Ken, Four-Pan Algebra Balance (1997)

Instruction manual included with the kit.  May be ordered from:

Cuisenaire Company of America, Inc.

PO Box 5026

White Plains, NY 10602-5026

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