Progress auto-saved

Mission 13 · Unit 6

Expressions

6.EE.A.2 · Unit 6
Today's objective: Write, read, and evaluate algebraic expressions.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The school carnival is this Saturday! Your team is in charge of the game booth. Each game costs $2 to play, plus a $25 booth rental fee for the day. You need to figure out how much money the booth will collect depending on how many students play. Then the principal asks: “What if we add a $0.50 prize cost per player? Write an expression for the profit.” Your team must write, evaluate, and interpret expressions to plan the carnival booth budget.

CARNIVAL GAME BOOTH $2 PER GAME Revenue 2n + 25 Prizes 0.50n n = ? Expression Breakdown Revenue = 2n + 25 $2 per player (n) + $25 booth fee Profit Expression Profit = 2n + 25 − 0.50n Simplified: 1.50n + 25

Language Vocabulary Support

Variable
A letter (like n) that stands for a number we do not know yet.
n = unknown
Expression
A math phrase with numbers, variables, and operations (no = sign). Example: 2n + 25
Coefficient
The number multiplied by a variable. In 2n, the coefficient is 2.
2 n coefficient
Constant
A number that does not change. In 2n + 25, the constant is 25.
Evaluate
Replace the variable with a number and calculate the answer.
Term
Each part of an expression separated by + or −. In 2n + 25, there are 2 terms.
Equivalent Expressions
Two expressions that always equal the same value no matter what number replaces the variable.
Simplify
Combine like terms to make an expression shorter. 2n − 0.50n = 1.50n

Sentence Frames:

“The variable _____ represents _____.”

“When I evaluate the expression for n = _____, I get _____.”

“The coefficient _____ tells me _____.”

“These expressions are equivalent because _____.”

Investigation

The Problem: (A) Write an expression for the total money collected if n students play at $2 each, plus the $25 booth fee. (B) Evaluate the expression for n = 10, 20, 50, and 100. (C) Now write an expression for profit, knowing each player costs $0.50 in prizes. (D) Simplify the profit expression by combining like terms. (E) Are the original and simplified profit expressions equivalent? Prove it.

Visual Model: From Words to Expressions

Step 1: Identify Quantities $2 per player = 2n $25 booth fee = 25 $0.50 prize/player = 0.50n Revenue Expression: 2n + 25 coefficient = 2 variable = n constant = 25 terms: 2n and 25 → Step 2: Build Profit Revenue − Costs = Profit Revenue − Prize Cost (2n + 25) − 0.50n Combine like terms: 2n − 0.50n = 1.50n + 25 stays: + 25 Simplified Profit: 1.50n + 25 → Step 3: Evaluate n = 10: 1.50(10) + 25 = $40 n = 20: 1.50(20) + 25 = $55 n = 50: 1.50(50) + 25 = $100 n = 100: 1.50(100) + 25 = $175 Pattern: Every 10 more players = $15 more

Step-by-Step Investigation Guide

  1. Identify what changes and what stays the same. The number of players (n) changes. The price ($2) and booth fee ($25) stay the same. Which quantities are variable and which are constant?
  2. Write the revenue expression. Revenue = (price per player) × (number of players) + booth fee = 2n + 25. What does each part of 2n + 25 represent in the real situation?
  3. Evaluate for specific values. Substitute n = 10 into 2n + 25: 2(10) + 25 = 20 + 25 = $45 total revenue. Does $45 make sense for 10 players at $2 each plus $25?
  4. Write the profit expression. Profit = Revenue − Costs = (2n + 25) − 0.50n. What does 0.50n represent? Why do we subtract it?
  5. Simplify by combining like terms. 2n − 0.50n = 1.50n. The constant 25 stays. Profit = 1.50n + 25. Which terms are “like terms”? Why can we combine them?
  6. Prove equivalence. Test both expressions with n = 20: Original: (2(20) + 25) − 0.50(20) = 65 − 10 = 55. Simplified: 1.50(20) + 25 = 30 + 25 = 55. Same answer! Is testing one value enough to prove equivalence? What else could you try?

Interactive Expression Tools

Expression Builder

Build an expression by choosing a coefficient, variable, and constant.

×
2n + 25
2(10) + 25 = 45

Evaluation Practice Table

Evaluate the expression 1.50n + 25 for each value of n. Type your answer and press Enter to check.

n (players) 1.50n + 25 Profit ($)
10 25
20 25
50 25
100 25
200 25

Like Terms Sorter

Click each term to sort it into the correct group.

Variable Terms (with n)
Constants (numbers only)

Equivalence Tester

Test if two expressions give the same result. Enter a value for n and compare.

Multiple Representations

Words

“The profit is one dollar and fifty cents per player, plus a twenty-five dollar booth fee.”

Best for: explaining to someone who does not know algebra

Expression

Profit = 1.50n + 25
Coefficient: 1.50 (rate per player)
Variable: n (number of players)
Constant: 25 (fixed booth fee)

Best for: quick calculations and identifying parts

Table of Values

n = 0 → $25 | n = 10 → $40 | n = 20 → $55
n = 50 → $100 | n = 100 → $175

Best for: seeing the pattern and checking work

Visual / Bar Model

Draw a bar split into two parts: a green section that grows (1.50 per player) and a fixed purple block ($25). As n increases, only the green section stretches.

Best for: understanding what changes vs. what stays the same

Equivalent Expressions

Original: (2n + 25) − 0.50n
Simplified: 1.50n + 25
Proof: Both give $55 when n = 20, $100 when n = 50, $175 when n = 100.

Best for: connecting simplification to real meaning

Parts of an Expression

1.50 n + 25 coefficient variable constant term 1 term 2

Best for: identifying and labeling each part

Team Roles

Facilitator Keeps the group on track, watches the timer, makes sure everyone speaks. Mission 13 task: Make sure the team writes both the revenue AND profit expressions, evaluates for all values in the table, AND proves equivalence.
Expression Writer Translates the word problem into algebraic expressions. Labels coefficient, variable, and constant. Mission 13 task: Write Revenue = 2n + 25 and Profit = (2n + 25) − 0.50n. Label every part. Show the simplified form 1.50n + 25.
Precision Checker Evaluates the expression for each value of n. Checks the arithmetic. Verifies that both profit expressions give the same answers. Mission 13 task: Fill in the evaluation table for n = 10, 20, 50, 100. Verify the original and simplified forms match for every value.
Reporter Prepares the defense: interprets what the expression means in context, explains each part, and presents findings. Mission 13 task: Explain what 1.50 means per player, what 25 means for the booth, and why the simplified expression is easier to use. Present the equivalence proof.

Timed Lab Phases

Ready
Click a phase, then press Start.
03:00

Read the scenario out loud. Assign roles. Underline the key numbers: $2, $25, $0.50.

  • What is the variable in this problem? What does it represent?
  • What stays the same no matter how many students play?
  • Estimate: if 30 students play, will the booth collect more or less than $100?
Checkpoint: Every teammate can identify the variable, coefficient, and constant.

Write both expressions. Label every part.

  • Revenue = 2n + 25. Label: 2 is the coefficient, n is the variable, 25 is the constant.
  • Write the profit expression: (2n + 25) − 0.50n.
  • Simplify: combine 2n and −0.50n to get 1.50n. Keep + 25.
  • Start evaluating for n = 10 and n = 20.
Checkpoint: Both expressions are written. Simplified form = 1.50n + 25. Two evaluations started.

Complete the evaluation table and prove equivalence.

  • Evaluate 1.50n + 25 for n = 50 and n = 100.
  • Evaluate the original (2n + 25) − 0.50n for the same values.
  • Do both forms give the same results? That proves equivalence!
  • Look for a pattern: how does the profit change each time n goes up by 10?
Checkpoint: Full table complete. Both expressions match for all values. Pattern identified ($15 per 10 players).

Prepare your presentation. Interpret the expression in context.

  • State what each part of 1.50n + 25 means for the carnival.
  • Show the evaluation table as evidence.
  • Explain why the simplified expression is equivalent to the original.
  • Share one mistake your team caught and fixed.
Checkpoint: Reporter can present the expression, table, and equivalence proof in under 60 seconds.

Challenge Zone

Extension 1: The principal says: “If more than 80 students play, we will lower the price to $1.50 per game.” Write a new revenue expression for when n > 80. How does the profit expression change?
Extension 2: Write an expression using exponents. If the carnival runs for d days and the number of players doubles each day (starting at 20), write an expression for the number of players on day d. (Hint: 20 × 2d−1.) Evaluate for d = 1, 2, 3, and 4.
What If? What if there were no booth fee ($25 = 0)? How would the expression change? What if prizes cost $1 per player instead of $0.50? Can the booth still make a profit? What is the “break-even” point?
Real-Life Connection: Businesses use expressions every day! Revenue = price × quantity. Profit = revenue − costs. The variable is the number of items sold. When a company simplifies expressions, they find the “break-even point” — the value of n where profit = 0.

Defense Preparation

Questions Your Team Must Answer

  1. What expression represents the total revenue? Label each part (coefficient, variable, constant, terms).
  2. What expression represents the profit? How did you build it from the revenue and cost?
  3. Show that the original and simplified profit expressions are equivalent using at least 3 test values.
  4. Interpret: What does the coefficient 1.50 mean in the context of the carnival?
  5. What pattern do you see in the evaluation table? What happens to the profit as n increases by 10?

Sentence Starters for Your Defense

“The expression _____ represents _____ because _____.”

“The coefficient _____ means _____ in this problem.”

“When n = _____, the profit is $_____ because _____.”

“The expressions are equivalent because when we substitute n = _____ into both, we get _____.”

“We simplified by combining like terms: 2n − 0.50n = 1.50n because both terms have the variable _____.”

Writing Expressions (4 pts) Revenue and profit expressions are correct. Parts are labeled (coefficient, variable, constant, terms).
Evaluating (4 pts) Table is complete and correct for all values of n. Work is shown for substitution.
Simplifying & Equivalence (4 pts) Like terms are correctly identified and combined. Equivalence is proven with multiple values.
Interpreting (4 pts) Team explains what each part means in the carnival context. Pattern in table is identified.

Exit Product

Deliver: Carnival Booth Budget Report

Your team submits a one-page report that includes:

  • Revenue expression (2n + 25) with all parts labeled
  • Profit expression in original form: (2n + 25) − 0.50n
  • Simplified profit expression: 1.50n + 25
  • Evaluation table for n = 10, 20, 50, 100
  • Equivalence proof (both forms tested with same values)
  • Pattern observation (how profit changes as n increases)
  • A 3-sentence interpretation: what the expression means, what the coefficient tells you, and what happens at n = 0

Self-Assessment

  • I can write an algebraic expression from a word problem.
  • I can identify the coefficient, variable, and constant in an expression.
  • I can evaluate an expression by substituting a value for the variable.
  • I can simplify an expression by combining like terms.
  • I can prove two expressions are equivalent by testing values.
  • I can interpret what an expression means in a real-world context.

Student Work Space

Revenue Expression (label all parts):

Profit Expression (original and simplified):

Evaluation Table:

Equivalence Proof:

Interpretation (what does the expression mean?):

Defense (Claim, Evidence, Reasoning):