Mission 10 · Unit 5

Area of Polygons

6.G.A.1 · Unit 5
Today's objective: Find the area of triangles and polygons by composing and decomposing.
Need a hint?
Re-read the problem and underline the numbers and the question. Pick one representation (model, table, or equation), show your steps, and check that your answer makes sense for the situation.

The school is building a new outdoor reading garden. The garden plot is an irregular polygon shaped like an L. The groundskeeper needs your team to calculate the exact area so she can order the right amount of sod (grass). The L-shaped plot has these measurements: the full length is 18 meters, the full width is 12 meters, and the cutout rectangle in the upper-right corner is 8 meters long and 5 meters wide. She also wants a triangular flower bed with a base of 6 meters and a height of 4 meters placed inside the garden.

Cutout 8 m x 5 m 18 m 12 m 8 m 5 m Flower b=6m, h=4m Rectangle A 10 m x 12 m Rectangle B 8 m x 7 m Method: Decompose!

Investigation

The Problem: Find the area of the L-shaped garden plot (full rectangle is 18 m x 12 m with an 8 m x 5 m cutout). Then find the area of the triangular flower bed (base = 6 m, height = 4 m). Finally, calculate how much sod is needed: garden area minus the flower bed area.

Visual Model: Decomposing the L-Shape

Method 1: Full Rectangle Minus Cutout 18 x 12 = 216 m² − 8 x 5 = 40 m² 216 − 40 = 176 m² Then subtract the flower bed: ½ x 6 x 4 = 12 m² Sod = 176 − 12 = 164 m² Method 2: Add Two Rectangles A: 10 x 12 = 120 m² + B: 8 x 7 = 56 m² 120 + 56 = 176 m² Then subtract the flower bed: ½ x 6 x 4 = 12 m² Sod = 176 − 12 = 164 m²

Step-by-Step Investigation Guide

  1. Sketch the L-shape. Draw the full 18 m x 12 m rectangle on your paper. Then mark the cutout: 8 m wide and 5 m tall in the upper-right corner. Label all sides. How many sides does the L-shape have? Can you label every side length?
  2. Choose a decomposition method. You can either (a) find the area of the full rectangle and subtract the cutout, or (b) split the L into two smaller rectangles and add their areas. Which method feels easier for your team? Do both methods give the same answer?
  3. Calculate the L-shape area. Method A: 18 x 12 = 216 m², then 216 - (8 x 5) = 216 - 40 = 176 m². Method B: find the dimensions of each sub-rectangle, then add. If you use Method B, how do you figure out the missing side lengths of each rectangle?
  4. Find the triangle area. Use the formula: Area = ½ x base x height = ½ x 6 x 4 = 12 m². The flower bed takes up 12 square meters inside the garden. Why do we multiply by ½? What shape would it be without the ½?
  5. Calculate the sod needed. Subtract the flower bed from the L-shape: 176 - 12 = 164 m². This is the area that needs grass. Why do we subtract the flower bed area instead of adding it?
  6. Verify with both methods. If you used Method A, try Method B as a check. Both should give 176 m² for the L-shape. Why do compose and decompose methods always give the same total area?

Language Support: Key Vocabulary

area — the amount of flat space inside a shape, measured in square units (m²)
compose — to put smaller shapes together to make a bigger shape
decompose — to break a big shape into smaller shapes that are easier to measure
base (b) — the bottom side of a shape that you use in the area formula
height (h) — the straight-up distance from the base to the top, measured at a right angle
polygon — a closed flat shape with straight sides (triangle, rectangle, trapezoid)
square meter (m²) — a unit for measuring area; a square that is 1 meter on each side
composite shape — a shape made of two or more simpler shapes put together

Sentence Frames

"I decomposed the L-shape into _____ and _____, so the total area is _____ + _____ = _____ m²."

"The area of the triangle is ½ x _____ x _____ = _____ m² because a triangle is half of a rectangle."

"The garden needs _____ m² of sod because _____ minus _____ equals _____."

Multiple Representations

Labeled Diagram

Draw the L-shape on grid paper with 1 cm = 1 m. Label every side. Shade the cutout differently. Count squares as a check: you should count close to 176 shaded squares.

Best for: visualizing which part is garden and which is cutout

Equation Chain

Full rectangle: 18 x 12 = 216 m²
Cutout: 8 x 5 = 40 m²
L-shape: 216 - 40 = 176 m²
Flower bed: ½ x 6 x 4 = 12 m²
Sod: 176 - 12 = 164 m²

Best for: showing the calculation order clearly

Area Table

Make a table: Shape | Formula | Dimensions | Area. List the full rectangle, cutout, L-shape (full minus cutout), triangle, and final sod area in rows.

Best for: organizing all the parts in one place

Team Roles

Facilitator Keeps the group on track, watches the timer, makes sure everyone speaks. Mission 10 task: Make sure the team finds the L-shape area, triangle area, AND the final sod amount.
Model Builder Draws the shapes, labels dimensions, and creates diagrams showing decomposition. Mission 10 task: Draw the L-shape with labeled sides, show the decomposition lines, and shade the flower bed triangle.
Precision Checker Verifies formulas are correct, checks arithmetic, and confirms units (m²). Mission 10 task: Use BOTH methods (subtract cutout and add sub-rectangles) to verify the L-shape area is 176 m². Check that ½ x 6 x 4 = 12.
Reporter Prepares the defense: claim, evidence, and one mistake the team caught. Mission 10 task: Explain WHY the team decomposed the shape the way they did and how they know the answer is correct.

Timed Lab Phases

Ready
Click a phase, then press Start.
03:00

Read the scenario out loud. Assign roles. Underline all dimensions: 18 m, 12 m, 8 m, 5 m, 6 m, 4 m.

  • Sketch a rough L-shape in the air or on paper. Where is the cutout?
  • Which area formulas will you need? (rectangle, triangle)
  • What does "decompose" mean in your own words?
Checkpoint: Every teammate can point to the cutout and name the formula for rectangles and triangles.

Draw the L-shape with labels. Choose a decomposition method and calculate the L-shape area.

  • Label every side of the L-shape with its measurement.
  • Draw lines showing how you split or subtract.
  • Calculate each sub-area and add/subtract to get the total.
  • Write the area formula for the triangle flower bed.
Checkpoint: The team has a labeled diagram and the L-shape area (176 m²).

Calculate the flower bed area and final sod needed. Then verify using the second method.

  • Triangle area: ½ x 6 x 4 = 12 m²
  • Sod needed: 176 - 12 = 164 m²
  • Verify: use the OTHER decomposition method. Do you still get 176 m²?
  • Does the answer make sense? Is 164 m² a reasonable amount of grass?
Checkpoint: Both methods agree. The final answer is 164 m² of sod.

Prepare your presentation. Show both decomposition methods, the triangle calculation, and the final answer.

  • State your claim: "The garden needs 164 m² of sod."
  • Point to your diagram as evidence.
  • Explain why compose and decompose both work.
  • Share one mistake your team caught (wrong dimension, forgot ½, etc.).
Checkpoint: The Reporter can walk through the full solution in under 60 seconds.

Challenge Zone

Extension: The groundskeeper also wants to add a parallelogram-shaped herb garden with a base of 5 m and a height of 3 m. Find the area of the herb garden. Now how much sod is needed? (Hint: subtract the herb garden area too.)
What If? What if the cutout were 10 m x 6 m instead of 8 m x 5 m? How would the sod amount change? Would the L-shape still look like an L, or would it become a different shape?
Real-Life Connection: Think about the floor plan of your home or school. Can you find an L-shaped room or hallway? How would you measure the floor area to buy carpet or tile? Every composite shape in real life uses the same decompose-and-add strategy.

Defense Preparation

Questions Your Team Must Answer

  1. How did you decompose the L-shape? What sub-shapes did you use?
  2. What is the area of the full L-shaped garden? Show the calculation.
  3. Why is the triangle formula ½ x base x height? What would happen if you forgot the ½?
  4. How much sod does the groundskeeper need? What did you subtract and why?
  5. How do you know both decomposition methods give the same answer?

Sentence Starters for Your Defense

"We decomposed the L-shape by _____ because _____."

"The total area is _____ m², and we verified it using a second method that also gave _____."

"The flower bed takes up _____ m², so the sod area is _____ minus _____ equals _____ m²."

"One mistake we caught was _____. We fixed it by _____."

Accuracy (4 pts) All area calculations are correct with proper units (m²). Triangle uses ½ x b x h.
Diagram (4 pts) L-shape is drawn with all dimensions labeled. Decomposition lines are clearly shown.
Reasoning (4 pts) Team explains why decomposing works and why the triangle is subtracted.
Communication (4 pts) Reporter uses vocabulary (area, compose, decompose, base, height) and all teammates can explain.

Exit Product

Deliver: Garden Sod Order Report

Your team submits a one-page report that includes:

  • A labeled diagram of the L-shaped garden with all dimensions
  • The decomposition shown with lines on the diagram
  • Area calculation using Method 1 (subtract cutout)
  • Area calculation using Method 2 (add sub-rectangles) as a check
  • Triangle flower bed area calculation with formula shown
  • Final sod amount with units: _____ m²
  • A 3-sentence defense: claim, evidence, and reasonableness check

Self-Assessment

  • I can decompose a composite shape into rectangles and triangles.
  • I used the correct area formula for each shape.
  • I verified my answer using two different methods.
  • I included square units (m²) in my answer.

Student Work Space

L-Shape Diagram (label all sides):

Method 1 Calculation (Subtract Cutout):

Method 2 Calculation (Add Sub-Rectangles):

Triangle Flower Bed Area:

Final Sod Calculation and Defense: