How to Teach Evaluating Expressions

Hazel Learning
Sep 19
5 min read

Evaluating expressions is one of the first major skills students encounter as they transition from arithmetic to algebra. At its core, it means taking a mathematical expression that includes variables, substituting specific values for those variables, and performing the operations in the correct order. On the surface, this looks straightforward. But for many students, it’s a challenging leap because it requires both procedural fluency and conceptual understanding.

As teachers, we know that if students develop shaky habits here, those misunderstandings will resurface in more advanced topics like solving equations, simplifying expressions, or even calculus later on. That’s why a strong foundation in evaluating expressions matters so much. This article provides a detailed roadmap for teachers who want to help their students build confidence and accuracy with evaluating expressions. It’s divided into five major teaching steps, with classroom strategies, sample problems, and tips for addressing common pitfalls.

Step 1: Start with What Students Already Know

Whenever we teach a new algebraic skill, it’s helpful to anchor it in familiar territory. Evaluating expressions builds directly on arithmetic skills students have practiced for years. They already know how to apply the order of operations, how to add and subtract, and how to multiply and divide.

One effective entry point is to start with purely numerical expressions and walk through the evaluation process step by step. For example:

3 + 2 × 5
(20 ÷ 5) × 2
2 × (4 + 6)

These are expressions without variables, but they mirror the structure of algebraic expressions.

Classroom idea: Put two problems side by side on the board. One with only numbers and one with a variable. For example:

Numerical: 3 + 2 × 5
Algebraic: 3 + 2x, where x = 5

Solve the numerical one first, then guide students to see that evaluating the algebraic expression is almost the same - the only difference is that we substitute the value for x before performing the operations. This builds comfort and helps students see algebra as an extension of what they already know, rather than a brand-new set of rules.

Step 2: Emphasize the Order of Operations

One of the biggest sources of error when evaluating expressions is misusing the order of operations. Students often memorize PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction), but many misapply it. They think multiplication must always come before division, and addition must always come before subtraction, when in fact multiplication and division share the same level of priority, as do addition and subtraction.

As a teacher, it’s critical to explicitly model this nuance. For example:

20 ÷ 5 × 2 should be read left to right: (20 ÷ 5) × 2 = 4 × 2 = 8.
If students incorrectly do division last, they’ll write 20 ÷ (5 × 2) = 20 ÷ 10 = 2, which is incorrect.

To prevent this misconception, avoid presenting PEMDAS as a strict top-to-bottom sequence. Instead, present it as layers: parentheses first, then exponents, then multiplication/division together from left to right, and finally addition/subtraction together from left to right.

Classroom strategy: Use number talks or short bell-ringers where students evaluate two similar expressions and explain why the results differ. For example:

(20 ÷ 5) × 2 = 8
20 ÷ (5 × 2) = 2

This builds conceptual awareness and makes students more thoughtful about the process, not just the acronym.

Step 3: Teach Substitution Carefully

Substitution is at the heart of evaluating expressions. The idea is simple: replace the variable with a given number. But students often stumble here, especially when coefficients and negative values are involved.

Example of a common error: If x = 3, evaluate 2x + 5.

A student might mistakenly write 2 + 3 + 5 = 10.

Correct approach: Replace x with 3 and multiply first: 2 × (3) + 5 = 6 + 5 = 11.

This mistake usually comes from students misreading 2x as “2 plus x” instead of “2 times x.” Since they’re used to always seeing multiplication explicitly written 2 × (3), the algebraic shorthand takes getting used to.

Teaching tip: Always use parentheses during substitution in the early stages.

For example: 2x + 5, with x = 3, becomes 2 × (3) + 5. This reinforces the multiplication idea and prevents errors when variables are negative. For example, if x = -4, writing 2(-4) makes it clear the result is -8. As students grow more confident, they can drop the parentheses, but requiring them early on prevents a lot of mistakes.

Step 4: Highlight the Role of Negative Signs

Negative numbers are notoriously tricky. Many students think of the minus sign as permanently “stuck” to a number, rather than part of the operation. This leads to major errors when substituting.

Example: If x = -5, evaluate -x. Many students write -5. They assume x already includes the negative, so they just repeat it.

Correct approach: -x means the opposite of x. So -(-5) = +5.

Another frequent issue arises when students square a negative number.

For example: If x = -3, then x² = (-3)² = 9. But many students write -3² = -9, forgetting the order of operations.

Teaching tip: Drill the use of parentheses. Encourage students to always substitute with parentheses:

For x = -5 in -x, write -( -5 ).
For x = -3 in x², write (-3)².

This simple practice prevents the majority of negative sign mistakes.

Classroom idea: Make a quick “negative number substitution chart” where x = -1, -2, -3, and evaluate expressions like -x, x², -x², and -(x²). Having students compare the different results helps them see how placement of the negative sign changes everything.

Step 5: Reinforce with Varied Practice and Real-World Examples

Once students understand substitution, order of operations, and handling negatives, the final step is practice - but not just routine drill. Students need varied practice that mixes straightforward problems, multi-step problems, and real-world contexts.

Example problems for variety:

Evaluate 3x + 4 when x = 7.
Evaluate 2x² – 5 when x = -3.
Evaluate (y + 2)(y – 2) when y = 4.

Example real-world context: If a taxi company charges 3 dollars per mile plus a flat 5-dollar fee, write an expression for the total cost of a ride. Then evaluate the cost when the ride is 10 miles.

This not only checks procedural skills but also deepens understanding of why evaluating expressions matters. Students begin to see algebra as a tool for solving real questions, not just a set of abstract exercises.

Pulling the Steps Together ...

By following these five steps, teachers can give students a clear, structured path toward mastering evaluating expressions:

Connect to prior knowledge (numerical expressions).
Emphasize order of operations (especially left-to-right rules).
Teach substitution with care (use parentheses early).
Clarify negative sign use (stress that minus is an operation).
Reinforce with variety (mix problem types and real-world scenarios).

When taught this way, evaluating expressions shifts from being a memorized procedure to a meaningful process. Students learn not just “how” but “why” - and that understanding builds a foundation for success in algebra and beyond.

As math educators, we know evaluating expressions is not just about getting the right answer. It’s about cultivating precision, logical reasoning, and flexibility. Students who master this skill will be much better prepared for later algebraic concepts like solving equations, factoring, and graphing functions. But just as importantly, addressing misconceptions early prevents frustration later. Each time a student realizes, “Oh, that’s why we can’t just add inside the parentheses,” or “That’s why -(-5) becomes positive,” they take ownership of their learning.

Your role as a teacher is to guide them through these moments - with patience, clarity, and plenty of opportunities to practice. Evaluating expressions isn’t glamorous, but it’s one of the most important stepping stones in a student’s mathematical journey.