Writing and Evaluating Expressions for 5th Grade: Variables and Rules

Expressions with variables represent quantities that can change. In Grade 5, students write expressions using variables (like n, x, or m) to stand for unknown or varying amounts, and they evaluate those expressions by substituting a specific value for the variable. This skill prepares students for algebra and helps them model real-world situations—cost of n tickets, money left after buying m items, distance traveled in h hours—where one quantity depends on another.

When we write “the cost of n tickets at $12 each,” we use the expression \(12n\) or \(12 \times n\). The variable n can be any number; when we replace n with a specific value (e.g., 5), we evaluate: \(12 \times 5 = 60\). Writing expressions requires identifying what varies (the variable) and how it relates to the other quantities (the operation). Evaluating requires careful substitution and following the order of operations.

DETAILED EXPLANATION

Writing expressions:

• Identify the variable (what varies or is unknown).

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• Translate the situation into math: “cost of n items at $d each” → \(d \times n\) or \(dn\).

• “x more than 10” → \(10 + x\). “5 less than y” → \(y – 5\).

• “twice a number” → \(2n\). “half of a number” → \(n \div 2\) or \(\frac{n}{2}\).

Evaluating expressions:

• Replace the variable with the given number.

• Simplify using the order of operations.

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Example: Evaluate \(5x + 3\) when \(x = 4\). Substitute: \(5(4) + 3 = 20 + 3 = 23\).

WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

A ticket costs $12. Write an expression for the cost of n tickets.

Solutions:

Step 1: The cost depends on the number of tickets. Let n represent the number of tickets.

Step 2: Each ticket costs $12. Total cost = price per ticket × number of tickets = \(12 \times n\).

Step 3: We can write this as \(12n\) (multiplication is implied when a number is next to a variable).

Answer: \(12n\) dollars

Example 2

Evaluate \(5x + 3\) when \(x = 4\).

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Solutions:

Step 1: Replace x with 4 in the expression: \(5(4) + 3\).

Step 2: Multiply first (order of operations): \(5 \times 4 = 20\).

Step 3: Add: \(20 + 3 = 23\).

Answer: 23

Example 3

Sarah has $50. She buys m markers at $2 each. Write an expression for the money left.

Solutions:

Step 1: Cost of m markers at $2 each = \(2 \times m = 2m\).

Step 2: Money left = money she had − money spent = \(50 – 2m\).

Step 3: The expression \(50 – 2m\) represents the dollars Sarah has left after buying m markers.

Answer: \(50 – 2m\) dollars

Example 4

Evaluate \(3a – 7\) when \(a = 5\).

Solutions:

Step 1: Substitute \(a = 5\): \(3(5) – 7\).

Step 2: Multiply: \(3 \times 5 = 15\).

Step 3: Subtract: \(15 – 7 = 8\).

Answer: 8