A three-lesson unit guiding sixth-grade mathematicians to decompose composite polygons into rectangles and triangles to calculate total area. Students learn to spot component shapes, draw strategic slice lines, deduce missing dimensions, and justify multiple decomposition pathways.
Efficiency: "Why is decomposing into 1 rectangle and 1 triangle easier than decomposing into 3 triangles?"
Cut Line Quest Worksheet: Worked Solutions
Part 1: Hidden Shapes & Part 2: Dual Slicing
Figure 1 (L-Silhouette): Two rectangles (or one rectangle with a cutout).
Figure 2 (House): One triangle (roof) and one rectangle (base).
Strategy 1 (Vertical cut): Piece A: \(2 \times 5\) units; Piece B: \(4 \times 3\) units. Both are rectangles.
Strategy 2 (Horizontal cut): Piece 1: \(2 \times 2\) units; Piece 2: \(6 \times 3\) units. Both are rectangles.
Part 3: Slanted Edges & Complex Polygons
Figure A (Trapezoid): Vertical drop cut creates 1 rectangle (width 4, height 5) and 1 right triangle (base 3, height 5).
Figure B (House): Horizontal cut across eaves creates 1 triangle (base 6, height 2) and 1 rectangle (width 6, height 3).
Figure C (Step Polygon): 2 cuts yield 3 rectangles (e.g., \(2 \times 4\), \(2 \times 2\), and \(3 \times 1\)).
Figure D (Arrow): Vertical cut separating shaft from head creates 1 rectangle (shaft: \(4 \times 2\)) and 1 triangle (head: base 4.5, height 4).
Part 4: Mathematical Reasoning Sample Response
"Taylor's strategy (3 rectangles) is better than Jordan's (5 squares) because calculating 3 areas is faster than calculating 5, and each extra step increases the chance of a math mistake."
Differentiation Strategies
Support (Scaffolding):
Provide students with physical cardstock cutouts of the figures to fold or cut before drawing lines on paper.
Extension (Challenge):
Challenge students to design their own 8-sided polygon that can be decomposed into exactly 1 square, 1 rectangle, and 2 right triangles.
Polygon Slicers • Lesson 1 Teacher Guide Next Lesson: Lesson 2 Area by Parts Page 2 of 2
Garden Ramp Challenge
Find the total area of this trapezoid garden bed.
Hints to guide you:
• Decompose into 1 rectangle + 1 right triangle.
• Triangle base = 12 - 7 = 5 m.
• Remember to divide triangle base × height by 2!
7 m 6 m Total Base = 12 m Worksheet Ready! Let's solve!
Unit: Polygon Slicers Next: Area By Parts Worksheet
Teaching Note: Point out to students that the additive method is much simpler here because the subtractive method requires calculating non-labeled decimal widths (\(4.5\text{ ft}\)). This reinforces choosing the most efficient strategy!
Problem 6: Community Swimming Pool
Total Area = 120 m²
Main Lap Zone (Rectangle): \(\text{base} \times \text{height} = 12\text{ m} \times 8\text{ m} = \mathbf{96\text{ m}^2}\)
Wading Zone (Triangle): Base = \(8\text{ m}\) (shared edge with rectangle). Height = \(6\text{ m}\). Area = \(\frac{1}{2} \times 8\text{ m} \times 6\text{ m} = \frac{48}{2} = \mathbf{24\text{ m}^2}\)
Total Combined Pool Area: \(96\text{ m}^2 + 24\text{ m}^2 = \mathbf{120\text{ m}^2}\)
Observation Check: Confirm that students identified the triangle's base as the shared \(8\text{ m}\) vertical edge and its perpendicular height as the horizontal \(6\text{ m}\) distance.
Part 4: Sample Student Reflection & Evaluation Criteria
"I chose the additive method for the T-shape because it only took 1 cut across to make 2 rectangles (\(14 \times 4\) and \(5 \times 6\)). The subtractive box method would have made me calculate the width of two empty corners (\(4.5\text{ ft}\) each), which takes more time and has decimals."
Full Credit Criteria: Clearly names method used, cites specific dimensions, and provides a logical reason (e.g., fewer steps, avoiding decimals).
Partial Credit: Names method and calculates correct answer but offers vague justification (e.g., "because it was easier").
Polygon Slicers • Lesson 2 Teacher Key Ready for Lesson 3: Strategy Showdown Page 2 of 2
"To calculate the floor area of the Oakridge Youth Center, I placed two vertical cut lines along the natural room transitions. The first cut separates the Gym Hall (\(40 \times 30 = 1,200\text{ sq ft}\)) from the Maker Studio. The second cut separates the Maker Studio (\(20 \times 20 = 400\text{ sq ft}\)) from the triangular Reading Nook (\(\frac{1}{2} \times 20 \times 12 = 120\text{ sq ft}\))."
"An alternative method would be using a large bounding rectangle (\(72 \times 30 = 2,160\text{ sq ft}\)) and subtracting the empty space above the studio plus the two triangular cutouts outside the nook. However, that requires 4 separate subtractions and calculating multiple unstated dimensions."
"My 3-zone additive method is more efficient because all three zones use labeled dimensions with clean whole numbers, requiring only 3 multiplications and 1 sum. This minimizes arithmetic errors and accurately yields \(1,720\text{ sq ft}\) with a total flooring cost of \(\$7,740\)."
Sequence Wrap-Up Note: This capstone demonstrates students' mastery of CCSS 6.G.A.1 by marrying geometric decomposition with practical economic decision-making and formal mathematical communication.
Polygon Slicers • Complete Unit Key Sequence Ready for Classroom Delivery Page 2 of 2
