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Pan Balance Lessons to Help Students Make Sense of Symbols & Reason Abstractly and Quantitatively

Illuminations has three Pan Balance applets to help you and your students learn how to make sense of symbols and reason abstractly and quantitatively:


1) (3-8) Pan Balance – Shapes

  Exploring Relationships Among the Weights of Various Objects by Placing Them on Either Side of a Balance

2) (3-8) Pan Balance – Numbers

  Finding Equivalent Numerical Expressions Using a Balance Scale

3) (3-12) Pan Balance – Expressions

  Investigating the Concept of Equivalence by "Weighing" Numeric and Algebraic Expressions







 

In order to better help you use them, we’ve compiled a list of lesson plans that will maximize the potential of these applets.


Pan Balance - Shapes


1. Finding the Balance (Algebra; Pre-K–2)

This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form.  

2. Balancing Discoveries (Number & Operations; Pre-K–2)

This lesson encourages students to explore another model of addition, the balance model. The exploration also involves recording the modeled addition facts in equation form. Students begin to memorize the addition facts by playing the “seven-up game.”


3. Balancing (Number & Operations; Pre-K–2)

This lesson encourages students to explore another model of subtraction, the balance. This model leads naturally to recording with equations. Students use actual and virtual pan balances in their explorations and record the modeled subtraction facts and the related addition facts in equation form.


4. Balancing Act (Algebra; Pre-K–2)

Problems such as those in this activity help develop what students already know in preparation for writing equations and learning ways to solve for variables. Students use mathematical models to explore quantitative relationships. When presented with pictures of pan balances with one or more objects in each pan, they communicate relationships between the weights of the objects by comparing the balanced and unbalanced pans.

 

5. Balancing Shapes (Algebra; 6-8)

Students will balance shapes on the pan balance applet to study equality, essential to understanding algebra. Equivalent relationships will be recognized when the pans balance, demonstrating the properties of equality.  


Pan Balance - Numbers


1. Extending to Symbols (Algebra; 6-8)

In this investigation, students learn about the notion of equivalence in concrete and numerical settings. As students begin to use symbolic representations, they use variables as place holders or unknowns. This investigation illustrates the continued transition from the concrete balance view of equivalence to a more abstract view.  


2. Stability in Numbers (Algebra; 3-5)

In this lesson, students apply their knowledge of addition equations to create number sentences using an electronic balance tool.  


3. Extending to Symbols 2 (Algebra; 6-8)

In Parts I and II of this investigation, students learn about the notion of equivalence in concrete and numerical settings. As students begin to use symbolic representations they use variables as place holders or unknowns. This part of the i-Math investigation illustrates the continued transition from the concrete balance view of equivalence to a more abstract view.

 

4. Balance Beam Discoveries (Numbers & Operations 3-5)

This lesson encourages students to explore another model of multiplication, the balance beam, and another relationship, the inverse of multiplication. This exploration leads naturally into representing multiplication facts in equation form. In addition to extending their understandings of the concept of multiplication, students begin to practice the multiplication facts by playing the Product Game.


5. Balancing Algebraic Understanding (Algebra 6-8)

Using a balance in the classroom is a first step to algebraic understanding. Use this pan balance (numbers) applet to practice the order of operations in simplifying numerical expressions and to demonstrate the conventions of using algebraic logic in simplifying expressions.  


Pan Balance - Expressions


1. Exploring Equations Further (Algebra 9-12)

Starting from the concrete notion of weights and balance and moving to symbolic expressions and representations of functions, this i-Math investigation has focused on some of the issues that arise along the way. In this part, this connection is extended to functions. More sophisticated tools allow for a greater diversity of investigations.  


2. Algebra in Balance (Algebra 6-8)

Students use the Balance Pans - Expressions Tool to explore algebraic expressions. They determine if algebraic expressions are equal. They balance pans to solve a system of equations and use graphing to find the solutions to a system of equations.  


3. Exploring Equations (Algebra 9-12)

In this lesson, students use their knowledge of weights and balance, symbolic expressions, and representations of functions to link all three concepts.  


What now?

If you’ve used these lesson plans or activities successfully, please share it with the math education community! E-mail us at illuminations@nctm.org with a sentence or two about your experiences, and we would love to host it as part of a Success Story!


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