K-12 Math (US): 6-8 Algebra K-12 Math Expressions Evaluate expressions

Grade 6–8 Algebra Study Guide: Evaluating Expressions

1. The Driving Question

You’re at a video game arcade, and each game costs g tokens. You start with 50 tokens, play g games, then your friend gives you 10 more tokens. How do you write a single math sentence that tells you exactly how many tokens you have left—no matter how many games you play? And why can’t you just add and subtract the numbers right away?

2. The Core Idea — Built, Not Listed

Imagine you’re programming a robot to calculate your arcade tokens. The robot doesn’t know how many games you’ll play—it only knows the rule: start with 50, subtract g for each game, then add 10. That rule is an expression: 50 – g + 10. But to find out how many tokens you actually have, you need to evaluate it—plug in a number for g and simplify.

Here’s the key: expressions are like recipes. They tell you what to do, but they don’t give an answer until you pick the ingredients (the numbers). If you play 3 games (g = 3), the expression becomes 50 – 3 + 10. Now you can simplify: 47 + 10 = 57. But if you play 7 games (g = 7), it’s 50 – 7 + 10 = 53. The expression is the same; the answer changes based on g.

Key Vocabulary:
- Expression: A math phrase with numbers, variables, and operations (no equals sign).
Example: The cost of b books at $12 each plus a $5 bookmark is 12b + 5.
- Evaluate: To substitute a number for a variable and simplify.
Example: Evaluate 4x – 1 when x = 2 → 4(2) – 1 = 7.
- Variable: A letter standing in for an unknown number.
Example: In a pizza party, p could be the number of pizzas ordered.
- Order of Operations (PEMDAS): The rules for simplifying expressions (Parentheses, Exponents, Multiply/Divide, Add/Subtract).
Example: 3 + 2 × 4 is 11, not 20, because multiplication comes first.

(Grade 9–12 note: In algebra, expressions become functions—rules that take inputs and produce outputs. Later, you’ll study how expressions can be factored, graphed, or combined into equations.)

3. Assessment Translation

How This Appears on State Tests (Grade 6–8):
- Multiple Choice: Questions like "Evaluate 3m + 7 when m = 5" with distractors that ignore order of operations (e.g., 3 + 5 × 7 = 56).
- Short Answer: "The expression 2x + 10 gives the cost in dollars of renting x movies. What is the cost for 4 movies? Show your work." - Evidence-Based Writing (rare): "Explain why the expression 5(n + 2) is not the same as 5n + 2. Use an example."

What a Proficient Response Looks Like:
- Multiple Choice: Correctly substitutes and simplifies, e.g., 3(5) + 7 = 15 + 7 = 22.
- Short Answer: Shows substitution and simplification: "For 4 movies: 2(4) + 10 = 8 + 10 = $18." - Teacher Look-Fors: - Did the student substitute the number for the variable? - Did they follow order of operations? - Did they write the final answer with units (if applicable)?

Model Proficient Response (Short Answer):
Prompt: Evaluate 4k – 6 when k = 3.
Response: 1. Substitute: 4(3) – 6 2. Multiply: 12 – 6 3. Subtract: 6 Answer: 6

(Grade 9–12 note: On the SAT, expressions appear in "Heart of Algebra" questions, often combined with word problems or graphs. AP Calculus uses expressions to define functions for limits and derivatives.)

4. Mistake Taxonomy

Mistake 1: Ignoring Order of Operations
Prompt: Evaluate 5 + 2 × 3.
Common Wrong Answer: 21 (student adds first: 5 + 2 = 7, then 7 × 3 = 21).
Why It Loses Credit: Order of operations requires multiplication before addition.
Correct Approach: 1. Multiply first: 2 × 3 = 6.
2. Then add: 5 + 6 = 11.

Mistake 2: Substituting the Wrong Value
Prompt: The expression 10 – 2h gives the number of hours left in a school day after h classes. How many hours are left after 3 classes? Common Wrong Answer: 10 – 2(2) = 6 (student uses h = 2 instead of h = 3).
Why It Loses Credit: The question specifies h = 3, but the student misreads or guesses.
Correct Approach: 1. Substitute h = 3: 10 – 2(3).
2. Multiply: 10 – 6.
3. Subtract: 4 hours left.

Mistake 3: Forgetting Parentheses in Substitution
Prompt: Evaluate 3(x + 4) when x = 2.
Common Wrong Answer: 3x + 4 = 3(2) + 4 = 10 (student distributes incorrectly or forgets to multiply x + 4 by 3).
Why It Loses Credit: The expression is 3 × (x + 4), not 3x + 4.
Correct Approach: 1. Substitute: 3(2 + 4).
2. Parentheses first: 3(6).
3. Multiply: 18.

5. Connection Layer

1. Within Math: Expressions → Equations
   Why it matters: An equation (e.g., 2x + 5 = 11) is two expressions set equal. Evaluating expressions helps you solve equations by testing possible values for x.

2. Across Subjects: Expressions → Computer Programming (Coding)
   Why it matters: In coding, expressions like score = 10 * level + bonus work the same way—they calculate a value based on variables (e.g., level or bonus).

3. Outside School: Expressions → Discounts and Coupons
   Why it matters: A store’s "30% off" sale uses the expression original price × 0.7. Evaluating it tells you the final price (e.g., a $50 shirt costs 50 × 0.7 = $35).

6. The Stretch Question

If an expression has two variables (like 3a + 2b), can you evaluate it with only one number? Why or why not?

Pointer Toward the Answer: An expression with two variables needs two numbers to evaluate fully (e.g., a = 1 and b = 4 → 3(1) + 2(4) = 11). But you can simplify it partially—like 3a + 8 if b = 4—because some terms become constants. This is how algebra builds flexibility: variables let you "hold a place" until you’re ready to plug in numbers.