By Patricia Helmuth
Many researchers have concluded that one of the reasons why students struggle with algebra is because of a common misconception: students perceive the equal sign as operational rather than relational. Thus, researchers recommend activities that focus on the meaning of the equal sign long before students are formally introduced to algebra. One such activity can be found at NCTM Illuminations – Pan Balance – Shapes, an interactive visual model that fosters algebraic thinking by getting to the heart of what equality means.
If you have access to a SMARTboard or a projector, you can explore the activity in your classroom as a whole group. You’ll need a computer with internet access and a Google Chrome browser. The following activity has worked well in my classroom:
- Access NCTM Illuminations – Pan Balance – Shapes.
- Click on “Set 3”.
- Start by dragging one yellow to one side of the balance scale and one red to the other side until the scale is balanced. Ask students for observations and write their responses on the whiteboard. Be sure to ask students to explain their thinking, drawing out and focusing on the concept of equality. Ask students if they agree, disagree with, or would edit any of the responses. “Reset balance.”
- Now, drag one blue shape to one side of the pan balance and one yellow to the other side until the scale is balanced. Ask: “What can you infer about the relationship between the red and blue?” What you want your students to pick up on here is that red weighs more than blue. Be sure to ask students to explain their reasoning. “Reset balance.”
- Test our their theory(ies). Drag one red to one side and one blue to the other side until balanced. Discuss as before, then “reset balance.”
- Final Set: Drag one pink to one side and one yellow to the other side until balanced. Again, ask for observations and discuss as before. (Your pan balance should look as pictured above.)
- Divide students into four groups and explain that each group will be assigned a value for the yellow shape and will need to figure out the values of the other shapes, based on the information that they see in the table on the right side of the pan balance. These activities have been ordered according to level of difficulty so that you can differentiate instruction. (Alternatively, you can assign values that are appropriate for your students.)
- Level 1: If the value of yellow is 5, what is the value of the other colors?
- Level 2: If the value of yellow is $1.00, what is the value of other colors?
- Level 3: If the value of yellow is 1 hour, what is the value of other colors?
- Level 4: If the value of yellow is ½, what is the value of other colors?
Each group should be given an easel- size piece of paper and asked to use drawings, number sentences that include the equal sign, and word sentences to explain how they figured out the value of the other colors. Explain that they will be presenting their findings to the entire class.
Alternatively, students can explore the pan balance activity in pairs (in a computer lab or sharing smartphones) using a print out of Internet Scavenger Hunt – Pan Balance. I recently organized a Student Numeracy Adventures Day and the Internet Scavenger Hunt was a big confidence booster, as is evidenced from the pictures below.
Be sure to follow up this activity by exploring the other sets in the same manner, noting the similarities and differences.
Finally, you can bridge to algebra problems such as the one on the right:
If Y = 5, find the values of B, P, and R.
5R = 3Y
Note that these variables represent the shapes as pictured in the pan balance above, exactly what you’ve already explored with your students!
For some other great ideas on how to help students to gain conceptual understanding of the equal sign, check out Integrating Arithmetic and Algebra, by Donna Curry.
Additional Research: Does Understanding the Equal Sign Matter?
Patricia Helmuth is an Adult Numeracy Consultant and Educator, who teaches for the Adult Program and SC BOCES in New York. She also works with the Hudson Valley RAEN Regional Staff Developers Network, as a Teacher Leader-Trainer, to provide support for Adult Education Instructors in CCSS mathematics instructional strategies. She recently became Co-Editor of the Math Practitioner, a newsletter published by ANN, The Adult Numeracy Network.