6.GR.2 Lesson 5-5-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

5.5 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

Visual Models · Volume with Whole-Number Edges · Volume · Procedural Fluency

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do we find the volume of a box?
1To find the volume of a box, multiply all three edges: V = length × width × height — and you can say why it is true, and where it would stop being true.
  • You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.
2Worked Example — Watch me
  1. My time capsule box is 6 long, 4 wide, and 3 tall. I write the formula: V = length × width × height.
  2. I put in the numbers: V = 6 × 4 × 3.
  3. First I multiply two of them: 6 × 4 = 24.
  4. Then I multiply by the last one: 24 × 3 = 72.
  5. So the volume is 72 cubic units. I use cubic units because volume fills space in 3 directions.
3Second Model — Try it together — then prove it
  1. Now a smaller box: 5 long, 2 wide, 2 tall. What formula do we use?
  2. Let's put in the numbers: V = 5 × 2 × 2. What is 5 × 2?
  3. We got 10. Now multiply 10 × 2. What is the volume?
  4. Yes, V = 20 cubic units. Don't forget the cubic units!
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Volume with Whole-Number Edges (Volumen con aristas de números enteros) — The number of unit cubes inside a rectangular prism. Formula: V = l × w × h.
  • Volume (Volumen) — How much space is inside a solid shape.
  • Rectangular prism (Prisma rectangular) — A solid box shape with six flat rectangle sides.
  • Cubic units (Unidades cúbicas) — The units used to measure space inside, like cubic inches.
  • Length, width, height (Largo, ancho, altura) — How long, how wide, and how tall a box is.
  • Net (Plantilla (desarrollo plano)) — A flat shape that folds up into a solid.
5Watch out
  • A common mistake in Volume with Whole Number Edges is multiplying only two of the three edges and stopping there — for example, seeing a box that is 6 in long, 4 in wide, and 3 in tall, multiplying just 6 × 4 = 24, and calling that the volume, when 24 is only the area of the bottom face (in square inches), not the volume of the whole box. The height must also be multiplied in: V = 6 × 4 × 3 = 72 cubic inches. A second common mistake is adding the three edges instead of multiplying them (6 + 4 + 3 = 13 instead of 6 × 4 × 3 = 72 cubic inches). Volume always MULTIPLIES all three edges, and the answer is always labeled in cubic units (in³), never square units (in²) or plain units (in).
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Complete the table. Find the missing dimension or volume for each time capsule.

    CapsuleLengthWidthHeightVolume
    Capsule X
    Capsule Y
    Capsule Z
    ✏️ Scratchpad / Reasoning
  2. 2 BAR MODEL

    Each section holds how many cubic inches?

    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A shipping crate is 12 ft long, 3 ft wide, and 2 ft tall. How many 1-foot cubes exactly fill it?

    1. A36 cubes
    2. B144 cubes
    3. C72 cubes
    4. D17 cubes
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Two prisms have the same volume of 48 cm³. Prism A is 4 × 4 × 3. Prism B is 8 × 3 × h. What is h?

    1. A6 cm
    2. B2 cm
    3. C4 cm
    4. D3 cm
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 5 ERROR ANALYSIS

    Find the Volume Error

    1. 1Dimensions:l = 9 ft, w = 4 ft, h = 6 ft
    2. 2Formula:V = l × w × h
    3. 3Calculate:V = 9 × 4 × 6 = 9 × 10 = 90 ft³
    4. 4Answer:Volume = 90 ft³

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 6 OPEN RESPONSE

    A box needs to hold exactly 60 cubic inches. Give three different sets of whole-number dimensions that work. Which set would make the box closest to a cube shape? Why might that matter?

    ✏️ Mathematical Justification & Response
🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."
👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."
✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The mathematical model proves the solution because each step preserves quantitative equivalence.
BUT An estimate provides a quick benchmark, but an exact proof is required for precision.
SO The quantities follow the standard rule, so the final result is verified with certainty.
📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim
State your direct mathematical answer/claim with units.
E Evidence
Cite exact numbers, calculations, dimensions, or graph data.
R Reasoning
Explain the mathematical theorem, property, or definition connecting evidence to claim.
✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

✏️ Author Workspace & Complete Solution Key
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It