6.GR.1 Lesson 5-9-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

5.9 Small Group · Group 2

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

Area of Regular Polygons · Regular Polygon · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — How do we find the area of a regular polygon?
1A regular polygon splits into the same number of triangles as it has sides, so total area = (one triangle's area) × (number of sides) — 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 hexagon has 6 equal sides. Each side is 6 feet, and each center triangle has a height of 5.2 feet.
  2. First I find one triangle: A = ½ × base × height = ½ × 6 × 5.2.
  3. I multiply: 6 × 5.2 = 31.2, then ½ × 31.2 = 15.6 square feet.
  4. A hexagon has 6 sides, so it has 6 equal triangles.
  5. I multiply: 6 × 15.6 = 93.6. So the total area is 93.6 square feet.
3Second Model — Try it together — then prove it
  1. Now a smaller regular pentagon: each of its 5 triangles has base 4 cm and height 3 cm.
  2. First find one triangle: ½ × 4 × 3. What is 4 × 3? Yes, 12, and half of 12 is 6 sq cm.
  3. A pentagon has 5 sides, so how many triangles? Yes, 5.
  4. Multiply: 5 × 6 = 30, so the total area is 30 square centimeters.
  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
  • Area of Regular Polygons (Área de polígonos regulares) — The space a regular polygon covers. Split it from the center into identical triangles, find one triangle's area, then multiply by how many there are.
  • Regular Polygon (Polígono regular) — A closed flat shape made of straight sides, where every side is the same length and every angle is the same size.
  • Decompose (Descomponer) — To break a shape into smaller, simpler shapes.
  • Triangle (Triángulo) — A shape with three sides.
  • Composite (Compuesto) — Made by putting two or more simple shapes together.
  • Formula (Fórmula) — A math rule written with symbols.
5Watch out
  • A common mistake in Area of Regular Polygons is stopping after finding the area of ONE triangle and reporting that as the total area — or adding the number of triangles instead of multiplying. Remember: total area = (one triangle's area) × (number of sides), not (one triangle's area) + (number of sides).
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    Solve the mathematical problem. Show all of your work and reasoning.

    PolygonNumber of TrianglesTriangle BaseTriangle HeightArea of One TriangleTotal Area
    Hexagon
    Pentagon
    Square
    Octagon
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each polygon to its total area.

    1. Hexagon: 6 triangles, each ½×6×5.2
    2. Pentagon: 5 triangles, each ½×8×5.5
    3. Octagon: 8 triangles, each ½×4×4.8
    4. Square: 4 triangles, each ½×5×4
    5. Decagon: 10 triangles, each ½×3×4
    6. Equilateral triangle: 3 triangles, each ½×4×3
    • A93.6 sq ft
    • B110 sq ft
    • C76.8 sq ft
    • D40 sq ft
    • E60 sq ft
    • F18 sq ft
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    A regular pentagonal tile splits into 5 congruent triangles. Each triangle has a base of 4 in and a height of 6 in. What is the total area of the tile?

    1. A60 in²
    2. B12 in²
    3. C24 in²
    4. D120 in²
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  2. 4 MULTIPLE CHOICE

    An octagon tile splits into equal triangles, each with base 6 ft and height 5 ft. How many triangles, and what is the total area?

    1. A120 sq ft
    2. B15 sq ft
    3. C90 sq ft
    4. D105 sq ft
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  3. 5 MULTIPLE CHOICE

    A regular hexagon is divided into 6 equal triangles from the center. Each triangle has a base of 6 ft and a height of 5.2 ft. What is the area of one triangle?

    1. A15.6 sq ft
    2. B31.2 sq ft
    3. C11.2 sq ft
    4. D15.6 ft
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 6 ERROR ANALYSIS

    Level 2 Extension — Find the Tile Area Error

    1. 1Decompose the tile:A regular hexagonal tile = 6 congruent triangles
    2. 2Identify one triangle:base = 8 in, height = 3.5 in
    3. 3Find the area of one triangle:A = ½ × 8 × 3.5 = 14 sq in
    4. 4State the total tile area:Total area = 14 sq in

    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