Mission 14 · Unit 6

Equivalent Expressions

6.EE.A.3 · Unit 6
Today's objective: Generate equivalent expressions using properties of operations.
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 garden club is building 3 identical raised beds. Each bed needs soil along the length (x feet) plus 4 extra feet of border stone on every side. Two club members wrote different expressions for the total border stone needed. Your team must prove whether the two expressions are equivalent or not, and recommend which form is easier for the club to use when ordering materials.

x feet +4 +4 x feet +4 +4 x feet +4 +4 Expression A: 3(x + 4) Expression B: 3x + 12

The Investigation

The Problem: Club member Maya wrote the total border stone as 3(x + 4) feet. Club member Carlos wrote it as 3x + 12 feet. Are these expressions equivalent? Prove it using at least two different methods. Then determine: if each bed is 6 feet long, how much total border stone is needed?

Visual Model: Area Model for the Distributive Property

Area Model: 3(x + 4) 3x x 12 4 3 3(x + 4) = 3x + 12 When x = 6: 3(6 + 4) = 3(10) = 30     3(6) + 12 = 18 + 12 = 30 ✔

Step-by-Step Investigation Guide

  1. Identify the expressions. Write both expressions on your paper. What operation connects the 3 to (x + 4) in Maya's expression? Guiding question: What does the 3 represent in the garden context?
  2. Apply the Distributive Property. Expand 3(x + 4) by multiplying 3 by each term inside the parentheses. Write each step. Guiding question: What is 3 times x? What is 3 times 4?
  3. Compare the results. After expanding, does Maya's expression simplify to match Carlos's expression? Write your conclusion. Guiding question: If both expressions simplify to the same form, what does that tell us?
  4. Test with a value. Substitute x = 6 into both expressions. Do they give the same number? Show all work. Guiding question: Why is one test value not enough to be completely sure?
  5. Test with a second value. Substitute x = 10 into both expressions. Record results. Guiding question: Could two different expressions give the same answer for one value but not another?
  6. Write your recommendation. Which expression form is more useful for the garden club when ordering stone? Explain your reasoning. Guiding question: When might the factored form be better? When might the expanded form be better?

Language Support: Key Vocabulary

Equivalent
Equal in value. Two expressions that always give the same answer.
Expression
A math phrase with numbers, variables, and operations (no = sign).
Distributive Property
a(b + c) = ab + ac. Multiply the outside number by each term inside.
Variable
A letter (like x) that stands for a number we do not know yet.
Substitute
Replace a variable with a number to test or evaluate.
Factored form
An expression written with parentheses, like 3(x + 4).
Sentence Frames:

"The expression 3(x + 4) means _____ because _____."

"I know the expressions are equivalent because when I substitute x = _____, both give _____."

"The distributive property lets me rewrite _____ as _____."

Multiple Representations

Approach 1: Distributive Property (Algebraic)

Expand 3(x + 4) step by step:
3 · x + 3 · 4 = 3x + 12
Since 3x + 12 = 3x + 12, the expressions are equivalent.

Approach 2: Substitution (Testing Values)

Try x = 6: 3(6+4) = 30 and 3(6)+12 = 30.
Try x = 0: 3(0+4) = 12 and 3(0)+12 = 12.
Same result each time supports equivalence.

Approach 3: Visual (Area Model)

Draw a rectangle with height 3 and width (x + 4). Split the width into x and 4. The total area is 3x + 12, which matches the expanded form.

Team Roles

Facilitator Read the scenario aloud. Make sure everyone writes both expressions. Keep the team moving through each step.
Model Builder Draw the area model on the team paper. Label every part: height = 3, width sections = x and 4, area sections = 3x and 12.
Precision Checker Check every substitution calculation. Verify that both expressions give the same result for each test value. Flag any errors.
Reporter Prepare the defense statement. Write the claim ("These expressions are/are not equivalent because...") with evidence from at least two methods.

Timed Lab Phases

Launch Phase
03:00

Read the garden scenario aloud. Assign roles. Each member writes both expressions on their paper.

  • What does x stand for in this problem?
  • What does the 3 represent?
  • What does +4 mean in the garden context?
Checkpoint: Everyone can explain what the expressions mean in words.

Model Builder draws the area model. Team applies the distributive property step by step.

  • Can you label every part of the area model?
  • Does the total area match 3x + 12?
  • Write the distributive property steps clearly.
Checkpoint: Area model is drawn and labeled. Distributive property is written out.

Substitute x = 6 and x = 10 into both expressions. Calculate and compare results.

  • Do both expressions give 30 when x = 6?
  • Do both expressions give 42 when x = 10?
  • Write your conclusion: are they equivalent?
Checkpoint: Two substitution tests completed with matching results.

Reporter prepares the defense. Team reviews the claim, evidence, and recommendation.

  • State your claim: "These expressions are equivalent because..."
  • List your evidence: distributive property AND substitution results.
  • Which form should the garden club use to order stone? Why?
Checkpoint: Defense includes claim + two types of evidence + recommendation.

Challenge Extensions

Extension Problem: A fourth raised bed is added, but it needs 5 extra feet of border stone instead of 4. Write two equivalent expressions for the total border stone of all 4 beds if the new bed is also x feet long but with 5 extra feet. Prove they are equivalent.

What If?

  • What if the club builds 5 beds instead of 3? How do both expressions change?
  • What if the border stone for each bed was (x + 4) on two sides and just x on the other two? Write a new expression.
  • Can you write a third equivalent expression that looks different from both Maya's and Carlos's?

Real-Life Connections

Equivalent expressions appear in shopping (coupons applied before vs. after tax), construction (different ways to calculate lumber), and recipes (scaling ingredients).

Defense Preparation

Be ready to answer:

  1. How did you prove the expressions are equivalent? Name at least two methods.
  2. What does the distributive property do to 3(x + 4)?
  3. If x = 20, what is the total border stone? Show your work.
  4. Which expression form is more useful for quick mental math? Why?
Sentence Starters:
  • "We proved equivalence by..."
  • "The distributive property shows that..."
  • "For ordering materials, we recommend _____ form because..."

Rubric

Criteria Excellent (4) Proficient (3) Developing (2)
Proof of equivalence Two+ methods with clear steps One complete method Attempted but incomplete
Visual model Area model fully labeled Area model with minor gaps Model attempted
Substitution Two+ values tested correctly One value tested correctly Attempted with errors
Recommendation Clear choice with reasoning Choice stated No recommendation

Exit Product

Your team submits: A one-page Equivalence Proof Sheet that includes:
  • Both expressions written clearly at the top
  • An area model with all parts labeled
  • The distributive property shown step-by-step
  • Two substitution tests with calculations
  • A three-sentence defense: claim, evidence, recommendation

Self-Assessment Checklist