Grade 6 Math WebQuest · Unit 5 · 6.G.A.1

The Mosaic Commission

The city is reopening Riverbend Plaza, and the parks department has hired your design studio to lay a giant stone mosaic in the center. The catch: tiles are sold by the square foot, the budget is tight, and your mosaic uses every shape in the geometry toolbox. Find the area of each piece, total the cost, and submit a winning bid.

⏱ 2–3 class periods 📐 Area of polygons 🧮 Composite figures 💵 Real budget math

Your Studio's Big Break

A mosaic is a picture made of many flat shapes fit together with no gaps — exactly how mathematicians break a hard area problem into easy ones. In this WebQuest you become a mosaic architect. Every shape in the Riverbend design has a real area, a real tile cost, and a real place in your bid.

The Driving Question

How can decomposing a complex shape into familiar polygons help you measure and budget the real world accurately?

Level 1 · Support Area means "how much flat space is covered," measured in square units (like square feet). Picture filling the shape with 1-ft × 1-ft tiles and counting them. En español: El área es el espacio plano que cubre una figura, medido en unidades cuadradas (pies cuadrados). Imagina llenar la figura con baldosas de 1 pie por 1 pie y contarlas.
Level 2 · Enrichment Designers rarely measure a weird shape directly. They decompose it into rectangles, triangles, and trapezoids, find each area, then add — or they enclose it in a big rectangle and subtract the empty corners. Keep both strategies ready; pick the faster one for each piece.

The Task

By the end of this WebQuest, your studio will submit a Mosaic Bid Sheet for Riverbend Plaza. A winning bid must:

  • Find the area of six mosaic pieces: a parallelogram, a triangle, a trapezoid, a regular hexagon, and two composite figures.
  • Show each area in square feet with the formula you used.
  • Multiply total area by the tile price ($6.50 / ft²) to get the project cost.
  • Stay under the $4,000 budget — or explain exactly which piece to resize.
  • Pass the Check Your Understanding self-check at the bottom.
Blueprint of the six mosaic pieces in Riverbend Plaza P T Z H C1 C2 P parallelogram · T triangle · Z trapezoid · H hexagon · C1 / C2 composites

The Process

Work through the steps in order. Each step builds one row of your Bid Sheet.

Step 1 — Stock your formula toolbox

Before you measure anything, review the five area formulas you will use. Keep this reference open.

Parallelogram
A = b × h
Triangle
A = ½ × b × h
Trapezoid
A = ½(b₁+b₂) × h
Regular polygon
decompose into triangles
Composite
add or subtract parts

Step 2 — Measure the simple pieces (P, T, Z)

Use these blueprint measurements (in feet) and the formulas above. Record each area on your Bid Sheet.

Worked example — Piece Z (trapezoid):

A = ½ × (b₁ + b₂) × h = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 32 ft²

Level 1 · Support Stuck on the triangle? A triangle is exactly half of a parallelogram with the same base and height. Find b × h first, then take half. En español: Un triángulo es la mitad de un paralelogramo con la misma base y altura. Calcula b × h y luego toma la mitad.

Step 3 — Tackle the regular hexagon (H)

Piece H is a regular hexagon park bench. A regular hexagon splits into 6 identical triangles. Each triangle has base 4 ft and height 3.5 ft.

Find one triangle's area, then multiply by 6 for the whole hexagon.

Level 2 · Enrichment The height of each inner triangle is the hexagon's apothem. Notice your method (½ × perimeter × apothem) is the same idea professional surveyors use for any regular polygon. Predict: would a regular octagon split into how many triangles?

Step 4 — Decompose the composite pieces (C1, C2)

Composite pieces are made of simpler shapes joined together. You can add the parts or subtract from a big rectangle.

Worked example — Piece C1 (add the parts):

Rectangle 1 = 10 × 4 = 40 ft² · Rectangle 2 = 5 × 4 = 20 ft² · Total = 40 + 20 = 60 ft²

Step 5 — Total the area and build the bid

Add all six areas to get the total mosaic area. Multiply by the tile price $6.50 / ft² to get the project cost. Compare against the $4,000 budget. If you are over, decide which single piece to shrink and by how much.

Level 1 · Support Cost = total area × $6.50. To check if you are under budget, ask: is your cost less than $4,000? If yes, you win the bid. En español: Costo = área total × $6.50. ¿Tu costo es menor que $4,000? Si es sí, ganas la licitación.

Resources

Use these Neft Teacher tools as you work. They open in the same window — use your back button to return.

Unit 5 · 3D Area BuilderSnap polygons together and watch areas update live. Area Architect ActivityPractice decomposing composite figures step by step. Reveal Evidence StudioWrite your Claim–Evidence–Reasoning for the bid. Math Writing Help · EN/ESSentence frames and bilingual vocabulary support.

Key vocabulary · Vocabulario clave

Evaluation

Your Mosaic Bid Sheet will be scored on this rubric.

Criteria 4 · Architect 3 · Builder 2 · Apprentice 1 · Getting started
Area accuracy All 6 areas correct with correct formulas. 5 of 6 areas correct. 3–4 areas correct. 0–2 areas correct.
Decomposition Both composites clearly split and justified. One composite clearly split. Attempts splitting with errors. No clear strategy shown.
Budget & cost Cost correct; budget decision justified. Cost correct; decision unclear. Cost has a small error. Cost missing or incorrect.
Units & labels Every answer labeled ft² and $. Most answers labeled. Some labels missing. No units shown.
Reasoning Clear claim–evidence–reasoning explanation. Explains most choices. Brief or partial reasoning. No explanation.
Teacher Notes & Answer Key (not printed)

Mosaic Commission · Bid Sheet — pairs with the Evaluation rubric above.

Sample Answers — Mosaic Bid Sheet

  • Parallelogram P: A = b × h = 12 × 5 = 60 ft².
  • Triangle T: A = ½bh = ½ × 9 × 6 = 27 ft².
  • Trapezoid Z: A = ½(b₁+b₂)h = ½(10+6)×4 = 32 ft².
  • Regular hexagon H: 6 triangles, each ½ × 4 × 3.5 = 7 ft² → 6 × 7 = 42 ft².
  • Composite C1 (L-shape): 10×4 + 5×4 = 40 + 20 = 60 ft².
  • Composite C2 (house): rectangle 6×5 = 30 + triangle ½×6×4 = 12 → 42 ft².
  • Total cost: sum the six areas, then × $6.50/ft² for the bid total ($1,644.50 with the full set).

Facilitation

  • Require the formula beside each area; decompose composites into known shapes.
  • Only the final total area is multiplied by the tile price.

Standard

CCSS 6.G.A.1.

Check Your Understanding

Answer all six. Click Check My Answers when you are done. These match the pieces in your bid.

1.Piece P parallelogram: base = 12 ft, height = 5 ft. What is the area?
b = 12 ft h=5
Hint / Pista: A = b × h.
2.Piece T triangle: base = 9 ft, height = 6 ft. What is the area?
b = 9 ft
Hint / Pista: A = ½ × b × h.
3.Piece Z trapezoid: base₁ = 10 ft, base₂ = 6 ft, height = 4 ft. What is the area?
b₂ = 6 b₁ = 10 ft
Hint / Pista: A = ½(b₁ + b₂) × h.
4.Piece H regular hexagon: 6 identical triangles, each base = 4 ft, height = 3.5 ft. What is the total hexagon area?
Hint / Pista: one triangle = ½ × 4 × 3.5, then × 6.
5.Piece C2 is a rectangle (6 ft × 5 ft) with a triangle roof (base 6 ft, height 4 ft). Which plan finds its area correctly?
6.The six pieces total 253 ft² of tile. At $6.50 / ft², what is the project cost? (Round to the nearest cent.)
Hint / Pista: Cost = total area × price. Is it under $4,000?

Conclusion

You did what real designers, builders, and surveyors do every day: you broke a complicated space into shapes you understand, measured each one, and turned area into money. The total cost of $1,644.50 comes in well under the $4,000 budget — your studio wins the Riverbend Plaza commission!

Take it further · Level 2

Redesign one piece to be 25% larger and recalculate the bid. Stay under budget while making the mosaic bolder.

Reflect · Reflexiona

Which piece was hardest to measure, and what decomposition made it easier? Explain in 2–3 sentences. ¿Cuál figura fue la más difícil y qué descomposición la hizo más fácil?

Real careers · Carreras reales

Landscape architects, tile setters, and game level designers all bid jobs using polygon area — exactly the skill you just used.