5.6 Small Group · Group 2

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

Visual Models · Surface Area Using Nets · Surface area · Procedural Fluency

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — How do we find the surface area of a box?

1A box has 3 pairs of matching faces, so add up all 6 faces to get the surface area — 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

My box is 8 in. long, 5 in. wide, and 3 in. tall. I unfold it into a net and see 3 pairs of faces.
Top and bottom: each is 8 × 5 = 40, and there are 2 of them, so 2 × 40 = 80.
Front and back: each is 8 × 3 = 24, so 2 × 24 = 48.
Left and right: each is 5 × 3 = 15, so 2 × 15 = 30.
I add all the pairs: 80 + 48 + 30 = 158. The surface area is 158 in².

3Second Model — Try it together — then prove it

Now a smaller box: 4 long, 3 wide, 2 tall. How many pairs of faces does it have?
Top and bottom: 4 × 3 = 12, times 2 = 24. Front and back: 4 × 2 = 8, times 2 = 16.
Left and right: 3 × 2 = 6, times 2 = 12. Now add 24 + 16 + 12. What is the surface area?
Yes, SA = 52 in². Remember to use square units.
Now prove it: say why that move had to work at all — not just that it did.
Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."

4Mathematical Word Bank

Surface Area Using Nets (Área de superficie usando redes) — The total area of all the faces of a solid. A net unfolds the solid flat so you can see every face and add their areas.
Surface area (Área de superficie) — The total area of all the flat sides of a solid.
Net (Plantilla (desarrollo plano)) — A flat shape that folds up into a solid.
Face (Cara) — One flat side of a solid shape.
Two-dimensional (Bidimensional) — A flat shape with length and width, but no thickness.
Edge (Arista) — The line where two flat sides of a solid meet.

5Watch out

A common mistake in Surface Area Using Nets is finding the area of only one face from each of the 3 matching pairs and forgetting to double it, instead of counting all 6 faces. For a box that is 9 in × 4 in × 5 in, a student might add just one face from each pair: (9×4) + (9×5) + (4×5) = 36 + 45 + 20 = 101 in², but the net actually has two of each face, so the correct total is 2(36) + 2(45) + 2(20) = 72 + 90 + 40 = 202 in² — exactly double. Before you submit, ask: 'Did I count both matching faces in each pair, or only one from each pair?'

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 FILL TABLE

Find the surface area of each time capsule box using its net.

Box Length Width Height Surface Area
Box A
Box B
Box C

✏️ Scratchpad / Reasoning
2 BAR MODEL

What is the combined area of the left and right faces?
✏️ Workspace & Solution Steps

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

3 MULTIPLE CHOICE
You unfold a time capsule that is a cube. Its net shows 6 equal squares, and each square has an area of 4 cm². What is the cube's total surface area?
1. A24 cm²
2. B10 cm²
3. C16 cm²
4. D12 cm²
Formula

Put the numbers in

Work it out

Answer with its unit
4 MULTIPLE CHOICE
Your team unfolds a rectangular time capsule box into a flat net. How many faces (flat sides) will the net show in all?
1. A6
2. B4
3. C5
4. D8
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
A time capsule box measures 2 in × 3 in × 4 in. Using its net, you add up all six faces. What is the total surface area?
1. A52 in²
2. B24 in²
3. C26 in²
4. D48 in²
Formula

Put the numbers in

Work it out

Answer with its unit

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

6 ERROR ANALYSIS
Level 2 Extension — The Missing Back Face
1. 1Box from the net:A time capsule box: l = 8 in, w = 3 in, h = 2 in
2. 2Label the faces:Top/Bottom: 8×3 = 24, Front/Back: 8×2 = 16, Left/Right: 3×2 = 6
3. 3Add the faces:SA = 24 + 24 + 16 + 6 + 6 = 76 in² (only one front face counted)
4. 4Answer:Surface Area = 76 in²
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation

🗣️ 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

Box	Length	Width	Height	Surface Area
Box A
Box B
Box C