Understanding Evaluate in Mathematics
To evaluate expressions means to find their value. Students evaluate expressions whenever they simplify problems like 3 + 4 or calculate 23 + 15. As they practice, they develop a stronger sense of how operations work and build computational skills they’ll use in algebra.
As students progress through elementary school, evaluating expressions becomes more complex. In upper elementary grades, students encounter expressions with more than two numbers or operations, such as 6 + 3 × 2. They begin learning to apply the correct order of operations, an important skill needed to accurately evaluate expressions in multi-step problems.
Teaching Students to Evaluate Expressions
Start with Hands-On Activities
Hands-on models give students a tangible way to explore what it means to evaluate expressions. Using manipulatives such as base-ten blocks, counters, or number lines helps students see how numbers combine to produce a value (or result). When introducing this concept, start with simple, single-operation expressions that students can physically model. Here are effective approaches:
Base- Ten Blocks: Represent the problem 23 + 15 using base-ten blocks. Use 2 tens and 3 ones for 23, and 1 ten and 5 ones for 15. Have students combine the blocks and count the total to find the value of 38.
Connecting Cubes: Use interlocking cubes to model expressions. For example, if the problem is 3 + 4, have students build one tower of 3 cubes and another with 4 cubes, then combine them to find the total value. For subtraction, they can “remove” blocks to represent the operation.
Incorporate Visual Models
An example of incorporating visual models is an order of operations flow chart. To evaluate expressions with multiple steps, provide students with a flowchart where they fill in each step as they work. This helps them visualize the sequence of operations and reinforces the importance of following the correct order.
Evaluating More Complex Expressions
In upper elementary, students begin to evaluate expressions that involve multiple steps and operations. It’s very important for them to understand the importance of following the correct order of operations when evaluating these expressions. Explain to students that mathematicians established a standardized order so everyone arrives at the same value for the same expression, ensuring consistency and clarity.
The acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) is often used to teach the order of operations. However, it can sometimes cause confusion, as students may mistakenly believe that multiplication must always come before division, or that addition must always precede subtraction. In reality, multiplication and division should be solved in the order they appear from left to right, and the same rule applies to addition and subtraction.
For example, consider the expression 12 ÷ 3 × 2. Following PEMDAS strictly might lead students to believe they must do multiplication (3 × 2 = 6) before division, resulting in 12 ÷ 6 = 2. However, the correct approach is to work from left to right: 12 ÷ 3 = 4, then 4 × 2 = 8.
Using Precise Mathematical Language
When working with students, it’s important to use accurate terminology to help them distinguish between different mathematical tasks. Use “evaluate” when finding the value of an expression like 7 + 3, and reserve “solve” for equations like ☐ + 3 = 10. This consistent language use helps students understand that evaluating expressions and solving equations are fundamentally different processes.
Bringing It All Together
Teaching students to evaluate expressions effectively requires a combination of approaches: concrete manipulatives for building foundational understanding, visual models for seeing mathematical relationships, careful attention to the order of operations as complexity increases, and precise mathematical language throughout.
When students develop fluency in evaluating expressions through these varied approaches, they build the computational skills and mathematical reasoning needed for success.
