5.MD.C.5 Lesson 1-4-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

1.4 Small Group · Group 2

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

argument · conjecture · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — When we do math, we develop arguments to defend our thinking
1We can defend a recommendation using equations, drawings, or words — and volume in cubic units gives the equations something exact to say — 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 argue with equations
  1. Suppose each large box is 8 inches long, 3 inches wide, and 10 inches tall. Its volume is 8 × 3 × 10 = 240 cubic inches, so three large boxes hold 240 × 3 = 720 cubic inches.
  2. Suppose each jumbo box is 10 inches long, 4 inches wide, and 10 inches tall. Its volume is 10 × 4 × 10 = 400 cubic inches, so two jumbo boxes hold 400 × 2 = 800 cubic inches.
  3. I recommend buying two jumbo boxes because 800 cubic inches is greater than 720 cubic inches — the total volume is greater, so there is more cereal for the same price.
3Second Model — Try it together — then prove it
  1. Not every good reason is an equation. Finish this argument: “I recommend buying two jumbo boxes because …”
  2. The book offers two word-based reasons: there is less packaging with two boxes, and two boxes are easier to store than three boxes.
  3. Now listen to the other side. One student recommends the three large boxes because the cereal will stay fresher — each box is open for less time — and because greater volume does not always mean more cereal inside.
  4. How convincing is each argument? That judgment is math too.
  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
  • argument (argumento) — A chain of reasons — equations, drawings, or words — used to defend your thinking.
  • conjecture (conjetura) — A mathematical statement you believe is true but have not yet proven.
  • counterexample (contraejemplo) — A single example that shows a statement is not always true.
  • volume (volumen) — The amount of space a solid figure takes up, measured in cubic units.
  • cubic unit (unidad cúbica) — A cube with edges 1 unit long — the unit used to measure volume.
5Watch out
  • Reporting volume without cubic units — saying the garden needs '48 feet' of soil, which is a length, or '48 square feet', which is an area.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Compute the volume of each cereal box.

    Box (l × w × h, inches)Volume (cubic inches)
    8 × 3 × 10
    5 × 8 × 10
    6 × 4 × 5
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each prism to a DIFFERENT prism with the SAME volume.

    1. 8 × 3 × 10
    2. 4 × 3 × 2
    3. 5 × 8 × 10
    4. 6 × 2 × 4
    • A6 × 4 × 10
    • B6 × 2 × 2
    • C20 × 10 × 2
    • D8 × 3 × 2
  3. 3 MULTIPLE CHOICE

    Which statement is a CONJECTURE?

    1. A“I think the two jumbo boxes hold more cereal than the three large boxes.”
    2. B“8 × 3 × 10 = 240.”
    3. C“Volume is measured in cubic units.”
    4. D“Yuzuki bought cereal.”
    ✏️ Workspace & Solution Steps
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 4 MULTIPLE CHOICE

    A student claims: “Greater box volume ALWAYS means more cereal inside.” Which is the best counterexample?

    1. AA jumbo box with more volume that is sold half-filled with air
    2. BA large box that costs more than a jumbo box
    3. CTwo jumbo boxes that are easier to store than three large boxes
    4. DA box of a different cereal brand
    ✏️ Workspace & Solution Steps
  2. 5 MULTIPLE CHOICE

    The garden needs 48 cubic feet of soil. One decomposition used two prisms of 24 cubic feet each. Which of these is ANOTHER decomposition of the same garden volume?

    1. AA 4 × 4 × 2 prism and a 4 × 2 × 2 prism
    2. BA 4 × 4 × 2 prism and a 4 × 4 × 2 prism
    3. CA 3 × 3 × 3 prism and a 3 × 3 × 2 prism
    4. DA 6 × 4 × 2 prism and a 2 × 2 × 2 prism
    ✏️ Workspace & Solution Steps
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 OPEN RESPONSE

    Use precise vocabulary to explain how you would find the soil needed for a bed 5 ft long, 2 ft wide, and 2 ft deep. Include the answer with its unit.

    ✏️ Mathematical Justification & Response
✍️ 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