How Visual Models Can Help Solve Equations with Variables on Both Sides

Solving equations that have variables on both sides can be tough for students. It’s not just about getting both sides to balance. There are lots of variables and numbers to think about at the same time. Plus, understanding what it means for two expressions to be equal can be confusing. This can lead to mistakes and frustration.

Understanding the Challenge

1. Managing Variables:
   Take an equation like $3x + 5 = 2x + 8$. Students need to get the variable $x$ all by itself on one side. But separating $x$ from the other numbers can be tricky. Instead of really understanding the math, they often end up just managing the steps, which can be frustrating.

2. Making Arithmetic Mistakes:
   The more steps there are, the easier it is to make mistakes with math. For example, when someone subtracts $2x$ from both sides, they might accidentally write $3x - 2x + 5 = 8$ instead of the correct $x + 5 = 8$. These little errors can lead to wrong answers and more frustration.

3. Understanding Equality:
   Many students find it hard to see that both sides of the equation are just two ways of showing the same amount. This is very important for solving equations the right way. If they can’t picture the equation in their minds, it’s hard to know what steps they need to take to keep both sides equal.

How Visual Models Can Help

Visual models can be really helpful for getting past these challenges. But they’re not a magic fix. If students misunderstand these models, it can just make things more confusing.

1. Algebra Tiles:
   Algebra tiles show positive and negative values visually. They allow students to physically work with the equation. For the equation $3x + 5 = 2x + 8$, students can:

   • Use three tiles for $3x$ on one side.
   • Add five tiles to represent the +5.
   • On the other side, use two tiles for $2x$ and eight tiles for +8.

   This hands-on method can help explain how to isolate $x$. But not all students connect moving tiles to the abstract math they need to do, which can limit how effective these tools are.

2. Number Lines:
   A number line shows how the values relate to one another. By placing $3x + 5$ and $2x + 8$ on a number line, students can visually see where the two points meet. This meeting point represents the solution for $x$. However, this method might not click with students who don’t easily think about numbers on a number line.

3. Graphing:
   For more advanced students, graphing the two sides of the equation can be useful. In our example, students could plot $y = 3x + 5$ and $y = 2x + 8$ on a graph. The point where the two lines cross shows the solution for $x$. But not every student is comfortable with graphing, and if they can’t visualize it properly, it can be frustrating.

Conclusion

In conclusion, visual models can really help with understanding and solving equations that have variables on both sides. However, there are still many challenges that can make it hard for students to succeed. Teachers need to be aware of these challenges and help students navigate them. By using visual aids wisely, teachers can help students connect the dots without creating more confusion.