Mission 16 · Unit 7

Equations

6.EE.B.7 · Unit 7
Today's objective: Solve one-step equations of the form x + p = q and px = q.
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 store sells notebooks for $3 each. A student paid a total of $21 for a stack of notebooks. The cashier forgot to write down how many notebooks were sold. Your team must write and solve a one-step equation to find the missing quantity, then verify your answer using a balance scale model and substitution.

n n n 3 groups of n $21 3n = 21 What is n? Divide both sides by 3

The Investigation

The Problem: Notebooks cost $3 each. A student paid $21 total. Let n = the number of notebooks. Write an equation, solve it using inverse operations, and prove your answer is correct. Then solve two more equations: n + 8 = 15 and n ÷ 4 = 6. For each, explain which inverse operation you used and why.

Visual Model: Balance Scale

Solving 3n = 21 with a Balance Scale Step 1: Set Up 3n = 21 Both sides are equal → Step 2: Divide by 3 3n ÷ 3 = 21 ÷ 3 Same operation, both sides → Step 3: Solution n = 7 7 notebooks! Verify: Substitute n = 7 back into the original equation 3(7) = 21   ✔   21 = 21   True!

Step-by-Step Investigation Guide

  1. Read and identify. What are the known values? What is unknown? Define the variable. Guiding question: What operation connects the number of notebooks to the total cost?
  2. Write the equation. Cost per notebook times number of notebooks = total cost. Write: 3n = 21. Guiding question: Why multiplication? What does 3n mean in words?
  3. Choose the inverse operation. Since n is multiplied by 3, the inverse is division. Divide both sides by 3. Guiding question: Why must you do the same thing to both sides?
  4. Solve. 3n ÷ 3 = 21 ÷ 3, so n = 7. Write the answer in a complete sentence. Guiding question: Does 7 notebooks at $3 each really equal $21?
  5. Verify by substitution. Replace n with 7 in the original equation: 3(7) = 21. True! Guiding question: What would happen if we got n = 8? How would we know it was wrong?
  6. Solve the bonus equations. For n + 8 = 15, subtract 8. For n ÷ 4 = 6, multiply by 4. Show all steps and verify. Guiding question: How do you decide which inverse operation to use?

Language Support: Key Vocabulary

Equation
A math sentence with an = sign. Both sides have the same value.
Inverse operation
The opposite operation. Addition undoes subtraction. Multiplication undoes division.
Solve
Find the value of the variable that makes the equation true.
Variable
A letter that represents an unknown number.
Substitute
Replace the variable with a number to check if the equation is true.
Balance
Both sides of the equation must stay equal, like a scale.
Sentence Frames:

"The equation is _____ because _____ times _____ equals _____."

"I used _____ (division/subtraction/etc.) because the inverse of _____ is _____."

"I verified my answer by substituting n = _____ and getting _____."

Multiple Representations

Approach 1: Balance Scale

Place 3n on the left and 21 on the right. To isolate n, divide both sides by 3. The scale stays balanced. n = 7.

Approach 2: Inverse Operations (Algebraic)

3n = 21
3n ÷ 3 = 21 ÷ 3
n = 7
The inverse of multiplying by 3 is dividing by 3.

Approach 3: Bar Model

Draw a bar split into 3 equal parts, total length 21. Each part = 21 ÷ 3 = 7. So one notebook = $7? No! n = number of notebooks = 7 at $3 each.

Team Roles

Facilitator Read the store scenario aloud. Ensure everyone identifies the unknown and writes the equation. Manage phase transitions.
Model Builder Draw a balance scale diagram. Show 3n on one side and 21 on the other. Then show the division step on both sides.
Precision Checker Verify each solution by substituting back into the original equation. Check that the inverse operation was correctly identified.
Reporter Prepare the defense: state each equation, the inverse operation used, the solution, and the verification. Use complete sentences.

Timed Lab Phases

Launch Phase
03:00

Read the notebook store scenario. Assign roles. Identify the unknown and the known values.

  • What is the cost per notebook?
  • What is the total amount paid?
  • What are we trying to find?
Checkpoint: Everyone can state the equation 3n = 21 and explain what n means.

Model Builder draws the balance scale. Team writes the equation and identifies the inverse operation.

  • What is on each side of the balance?
  • What operation will isolate n?
  • Draw the "divide both sides by 3" step on the scale.
Checkpoint: Balance scale drawn with equation. Inverse operation identified.

Solve all three equations. Verify each answer by substitution.

  • 3n = 21 → n = ?
  • n + 8 = 15 → n = ?
  • n ÷ 4 = 6 → n = ?
  • Substitute each answer back to verify.
Checkpoint: All three equations solved and verified.

Reporter prepares the defense. Explain the pattern: each equation used a different inverse operation.

  • "For 3n = 21, we divided because..."
  • "For n + 8 = 15, we subtracted because..."
  • "For n ÷ 4 = 6, we multiplied because..."
Checkpoint: Defense explains the inverse operation pattern across all three equations.

Challenge Extensions

Extension Problem: The store has a sale: buy 5 notebooks, get $4 off the total. A student pays $11 during the sale. Write and solve an equation to find the price per notebook. (Hint: 5p - 4 = 11 is a two-step equation!)

What If?

  • What if notebooks cost $4 each instead of $3? How does the equation change? What about the answer?
  • Write your own word problem that leads to the equation 8m = 56. Trade with another team to solve.
  • Can an equation have a solution of 0? Write one and check.

Real-Life Connections

Equations help in cooking (recipe scaling), shopping (finding unit prices), sports (calculating scores needed), and medicine (determining doses).

Defense Preparation

  1. What equation did you write for the notebook problem? What does each part mean?
  2. Why did you divide both sides by 3? What is the inverse of multiplication?
  3. How did you verify that n = 7 is correct?
  4. Explain the pattern: how do you choose the right inverse operation?
Sentence Starters:
  • "We wrote the equation _____ because _____."
  • "The inverse operation is _____ because it undoes _____."
  • "We verified by substituting _____ and the equation is true because _____."

Rubric

Criteria Excellent (4) Proficient (3) Developing (2)
Equation setup Correct equation with context explanation Correct equation Equation attempted
Inverse operations All 3 equations with correct inverse identified 2 of 3 correct 1 correct
Balance model Clear diagram with both sides shown Diagram with minor gaps Diagram attempted
Verification All 3 verified by substitution 2 verified 1 verified

Exit Product

Your team submits: An Equation Solution Sheet with:
  • The notebook equation (3n = 21) with a balance scale diagram
  • Step-by-step solution using inverse operations
  • Verification by substitution for all three equations
  • A summary sentence: "To solve a one-step equation, I use _____ because _____."
  • A table showing each equation, its inverse operation, and the solution

Self-Assessment Checklist