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

5.4 Small Group · Group 2

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

Area of Composite Figures · Composite Figure · 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 composite figure?
1Break the figure into simple shapes. ADD the areas when shapes are joined; SUBTRACT when a piece is cut out — 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 L-shaped room splits into two rectangles: one is 12 ft by 8 ft, the other is 6 ft by 5 ft.
  2. First rectangle: A = length × width = 12 × 8 = 96 square feet.
  3. Second rectangle: A = 6 × 5 = 30 square feet.
  4. The pieces are joined, so I add: 96 + 30 = 126.
  5. So the total area is 126 square feet.
3Second Model — Try it together — then prove it
  1. Now an L-shape made of a 5 ft by 4 ft rectangle and a 3 ft by 2 ft rectangle.
  2. First rectangle: 5 × 4 = 20 square feet.
  3. Second rectangle: 3 × 2 = 6 square feet.
  4. Are they joined or cut out? Joined, so we add: 20 + 6 = 26 square feet.
  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 Composite Figures (Área de figuras compuestas) — The total space inside a figure made of basic shapes. Formula: A = A₁ + A₂ — add the parts (or subtract a missing piece).
  • Composite Figure (Figura compuesta) — A shape made by putting two or more simple shapes together.
  • Decompose (Descomponer) — To break a shape into smaller, simpler shapes.
  • Add (Sumar) — Add up the areas of the smaller shapes to get the total.
  • Subtract (Restar) — Take away the area of a missing piece from a bigger shape.
  • Formula (Fórmula) — A math rule written with symbols.
5Watch out
  • A common mistake in Area of Composite Figures is adding every piece even when a piece has been cut out of the figure. For a 10 ft by 8 ft rectangle with a 3 ft by 2 ft rectangular notch cut from one corner, students compute the full rectangle 10 × 8 = 80 sq ft and stop, or they add the notch instead of removing it (80 + 6 = 86 sq ft) instead of subtracting it: 80 − (3 × 2) = 80 − 6 = 74 sq ft. Before submitting, check whether each piece is part of the figure (add it) or missing from the figure (subtract it).
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

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

    FigureShape 1 AreaShape 2 AreaOperationTotal Area
    L-shaped deck
    Floor with alcove
    Wall with window cutout
    Courtyard with fountain cutout
    ✏️ Scratchpad / Reasoning
  2. 2 MATCHING GAME

    Match each composite figure to its total area.

    1. L-shape: 10×6 + 4×3
    2. T-shape: 14×4 + 6×10
    3. Wall 15×10 minus window 5×4
    4. Pool 14×10 minus cutout 6×5
    5. L-shape: 8×5 + 3×6
    6. Floor 12×8 minus vent 2×3
    • A72 sq ft
    • B116 sq ft
    • C130 sq ft
    • D110 sq ft
    • E58 sq ft
    • F90 sq ft
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Plan 2 — L-shaped lobby: decompose the L into two rectangles, one 9 ft by 6 ft and one 7 ft by 3 ft, then ADD. What is the total floor area?

    1. A75 sq ft
    2. B63 sq ft
    3. C54 sq ft
    4. D21 sq ft
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  2. 4 MULTIPLE CHOICE

    Plan 4 — Design studio: the floor is a 16 ft by 12 ft rectangle, but a 5 ft by 4 ft storage closet is cut out of one corner. Find the usable floor area.

    1. A172 sq ft
    2. B192 sq ft
    3. C212 sq ft
    4. D20 sq ft
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  3. 5 MULTIPLE CHOICE

    An L-shaped room is made of two rectangles: 10 ft × 6 ft and 4 ft × 3 ft. What is the total area?

    1. A72 sq ft
    2. B60 sq ft
    3. C12 sq ft
    4. D72 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 — Catch the Architect's Area Mistake

    1. 1Identify the shapes:A clubhouse wall is a 18 ft by 11 ft rectangle with a triangular gable (base 18 ft, height 6 ft) joined on top.
    2. 2Find the rectangle area:A = 18 x 11 = 198 sq ft
    3. 3Find the triangle area:A = 1/2 x 18 x 6 = 54 sq ft
    4. 4Add the areas for the total:Total = 198 + 198 + 54 = 450 sq ft

    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