Understanding Equations In Mathematics
An equation introduces students to the idea of balance in math, showing that both sides of the equation can have the same value even if they look different. This idea of “different but equal” helps build a strong foundation for algebraic reasoning. By exploring equations, teachers can help students understand comparison, balance, and how numbers relate to one another.
Students should have the opportunity to work with different types of equations, such as a = a, a + b = c, c = a + b, or a + b = c + d, to see how numbers can be arranged in various ways while still maintaining balance. Furthermore, they should practice determining whether given equations are true or false, helping them develop a deeper understanding of the relationship between the parts of an equation and how they work together to form a balanced statement.
Teaching Strategies to Understand Equations
Begin With Hands-On Activities
Students learn best when they can see and work with examples that show how equations work. Using tools like balance scales, Unifix cubes, or counters, students can explore equations in a hands-on way. Start by letting students compare groups of objects and set up simple equations, showing how both sides can be balanced even if the numbers look different. This hands-on exploration helps students understand that an equation is about balance and equality, not just matching shapes or numbers. For example:
Unifix Cubes on a Balance Scale – Place 3 red Unifix cubes plus 4 blue Unifix cubes on one side of a balance scale and 5 green Unifix cubes plus 2 yellow Unifix cubes on the other side. Let students observe that, even though the cubes are arranged differently, the balance scale stays level because both sides represent the same total. Students can test equations like 3 + 4 = 5 + 2 and verify that both sides of the equation are balanced.
Counters on Ten Frames – Place 2 yellow counters and 3 red counters on one ten frame, and 4 yellow counters and 1 red counter on another ten frame. Even though the counters are arranged differently, both ten frames are filled to the same level, showing that 2 + 3 = 4 + 1. Students can use these ten-frame equations to compare and solve different combinations of numbers while reinforcing the idea of balance in equations.
Connect to the Symbolic Representation
To deepen understanding, these activities can be extended by having students move from physical manipulatives to drawings and, eventually, to symbolic equations. This progression allows students to first explore equality through hands-on materials, then represent their understanding visually, and finally connect it to abstract mathematical concepts. By gradually shifting from concrete to abstract, students build a stronger foundation for solving and testing equations. For example:
Balance Scale with Unifix Cubes to Test Equations – Students can test the equation 4 + 3 = 2 + 5 and observe that if the scale stays balanced, this confirms that the equation is true. Place 4 red Unifix cubes and 3 blue Unifix cubes on one side of the balance scale, and 2 green Unifix cubes and 5 yellow Unifix cubes on the other side. Students can also experiment with changing the numbers and checking whether the new equation balances.
Counters on Ten Frames to Solve Simple Equations – Students can solve the equation 3 + 4 = __ + 5 by placing 3 red counters and 4 yellow counters on a ten frame, and then placing 5 yellow counters on another ten frame. Next, they add red counters to the second ten frame until it is filled to the same level as the first ten frame. The number of red counters added is the number that belongs in the blank spot to make the equation true.
Eventually, students can start drawing a simple balance scale with an equal symbol at the fulcrum. Using a ten-frame template, they can also draw the counters instead of using the physical ones. These visuals act as a bridge, helping students transition from concrete manipulatives to abstract representations of equations.
Equations in Multiplication and Division
As students move into multiplication and division, their understanding of equations continues to grow. They begin to see equations not just as balanced addition statements, but as a way to show relationships between quantities, especially when solving problems or exploring fact families.
Students benefit from exploring multiplication and division equations visually and conceptually before working purely symbolically. Area models, arrays, and equal group drawings can help students connect the structure of an equation to the math it represents. For example:
Using Arrays to Represent Equations
Give students a set of counters to build an array with 3 rows of 4. Have them write a multiplication equation to match: 3 × 4 = 12. Then, ask them to turn the array and see that 4 × 3 = 12 is also true. This helps students see that equations can show the same total in different ways.
Relating Multiplication and Division Equations
Provide students with a group of 12 counters and ask them to divide them into 3 equal groups. They can record this as 12 ÷ 3 = 4, and then connect it to the related multiplication fact 3 × 4 = 12. Seeing how the two operations are connected through balanced equations helps reinforce fact families and inverse operations.
Equation Sorting with Multiplication and Division
Give students a mix of true and false multiplication and division equations, such as 5 × 3 = 15, 8 ÷ 2 = 5, or 4 × 6 = 24. Ask them to sort the equations and explain their reasoning using drawings, skip counting, or manipulatives. This encourages them to think critically about whether both sides have the same value.
Just like in the primary grades, students should continue testing equations for truth and balance as they work with multiplication and division. The goal is to help them understand that an equation represents a relationship between two equal values, no matter which operation is used.
Building Math Vocabulary With Equations
The language we use and encourage students to use around the word equation plays an important role in their understanding. It’s crucial to avoid framing an equation as simply “a problem to solve,” as this is a common misconception among young learners. Instead, we should emphasize that an equation shows a relationship where two sides have the same value, even if they look different. This understanding helps students grasp the idea of balance and equality in math.
Encourage students to describe an equation as “a statement that shows two sides have the same value” instead of just calling it “a math problem.” This helps them see equations as tools for representing relationships, not just something to solve.
