1.4 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

argument · conjecture · Procedural Fluency · Real-World Applications

Name: Date: Period: Learning Goal:

■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — 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.

Getting an answer is only half the work. The other half is explaining it so clearly — with conjectures, examples, and counterexamples — that someone else can decide whether your argument is convincing. We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.

2Worked Example — Watch me argue with equations

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.
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.
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.

3Mathematical 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.

4Watch 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]

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 OPEN RESPONSE

Yuzuki can buy three large boxes or two jumbo boxes for the same price. Give ONE reason in words (no equations) for either choice.
💬 Sentence Starter: I recommend ___ because ___ .<br>A reason that is not about volume is ___ .
✏️ 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

Starter: My mathematical claim is that the solution is...

E Evidence

Starter: The evidence from the model/table demonstrates that...

R Reasoning

Starter: This proves my answer because the standard mathematical definition of...

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It