The Pan Balance: Hands-On Approach to Equations

In an effort to meet our goals of building conceptual understanding, we at Happy Numbers look for ways to provide students with multiple representations of mathematical ideas. One often-overlooked representation of equations is the pan balance. Like other tools we’ve written about in our blog – real objects, base-10 blocks, number line – the pan balance is a great concrete visual that helps primary students “see behind the veil” of complex math problems.

 

For example, instead of developing a new procedure for each problem such as:

 

5 + 2 = ?     ? = 3 + 2        4 + ? = 6         8 = 1 + 5 + ?    and      7 = ? + 3

the pan balance gives students an efficient system for understanding variables in an equation.

Wondering how? Then keep reading!

 

Exercises shown below represent a variety of lessons from Happy Numbers Grade 1. You can access these interactive exercises through your Happy Numbers account or our demo page.

 

First of all, don’t worry if your students are unfamiliar with using a pan balance! Because getting familiar with it is exactly where the learning happens! What is important is the proper scaffolding to ensure smooth and enjoyable learning for all students (and that’s where our virtual pan balance does much of the work).

 

Students start by identifying weights (addends) to balance (equal) an object (sum). This simple exercise familiarizes them with how the pan balance responds to changes. Students select from a range of labeled weights, attempting to balance the pans (one of which is holding a labeled box):

Once the pans balance, the box is opened to reveal a toy:

 

It’s very important to note that there are no “wrong” answers for this exercises. The balance responds to changes just as it would in real life. Students simply experiment (and thus learn!) with the weights until the green light indicates the pans are balanced.

As students progress, we offer problems that require a bit more experimentation than others:

 
Because our pan balance tool works exactly like a real one, students may solve a problem using a non-traditional method. The example below shows 7 + 1 = 5 + 3 rather than the more obvious 7 = 5 + 2. (Of course, students are not yet exposed to the written equations for these problems.)

 
We also prompt students to balance the pans by removing, rather than adding, weights:

 

This practice will prepare them for future exercises using subtraction.

Once students have mastered the use of the pan balance, we ask them to balance the pans when the weight of the toy is unknown (which models the variable “x”):

and then calculate the unknown weight, thus requiring some simple addition:

 

 

We continue to scaffold concepts with an exercise that requires only simple addition based on a given scenario. Here, students no longer manipulate weights but simply perform calculations to find the total.

They now work with problems that use two or three addends:

 

They are also presented with numbers on the right as well as on the left pan:

 

 

We then increase the complexity by presenting balanced pans with numbered weights on both the left and right, along with a toy. Now, students must subtract or calculate the missing addend to find the weight of the toy.

 

 
This challenging exercise really prepares primary students for more difficult numeracy problems in the future. It is a very visual, tangible way to develop strategies for solving problems like:

 

9 = 2 + x or x + 5 = 10

It also helps them to begin thinking of equations as two balanced expressions rather than one calculation that results in an answer.

We think this important lesson using the pan balance is a great way to get started with equations in your class. Our grade 1 and grade 2 curriculum include several other lessons that focus on more advanced learning of equations.

So stay tuned!

Evgeny & Happy Numbers Team

P.S.: Pan balance problems are easily adapted for offline use (try our free supplemental worksheets). And if you have a pan balance in your classroom, students will be able to use it at a center or in a small group with new understanding after using the Happy Numbers version!