📋 Mission Brief
6.G.A.1
Find the area of triangles, parallelograms, rhombuses, and polygons by composing or decomposing shapes.
🌐 Field Guide / Guía de Campo — Math Word Bank
area / área
base / base ✓
height / altura
multiply / multiplicar
diagonal / diagonal ✓
formula / fórmula ✓
perpendicular ✓
parallelogram / paralelogramo ✓
rhombus / rombo
triangle / triángulo ✓
square units / unidades cuadradas
"The area is…"
"El área es…"
"El área es…"
"I multiply base × height"
"Multiplico base por altura"
"Multiplico base por altura"
"I divide by 2 because…"
"Divido entre dos porque…"
"Divido entre dos porque…"
"The formula for ___ is…"
"La fórmula para ___ es…"
"La fórmula para ___ es…"
🗺️ Terrain Survey
Identify the Shapes — Click to Reveal Field Notes
Parallelogram
Opposite sides parallel
Rhombus
All sides equal
Triangle
Three sides
📚 Field Glossary
| Term / Término | Meaning / Significado | Example / Ejemplo | 🇪🇸 Español |
|---|---|---|---|
| Area | Number of square units inside a shape | How many 1×1 tiles fit inside? | Área ✓ cognate |
| Base (b) | The bottom side — the side the height is measured from | Any side can be the base | Base ✓ cognate |
| Height (h) | Perpendicular distance from base to opposite vertex — always 90° | Straight up from base | Altura |
| Perpendicular | Lines that meet at exactly 90° | Height is ALWAYS perpendicular to base | Perpendicular ✓ |
| Diagonal (d) | Line connecting two opposite corners | Rhombus has two diagonals crossing at 90° | Diagonal ✓ cognate |
| Decompose | Break a shape into simpler shapes to find area | Parallelogram = rearranged rectangle | Descomponer |
Parallelogram
A = b × h
base × perpendicular height
Rhombus — two ways
A = b × h
base × perpendicular height
— or —
A = (d₁ × d₂) ÷ 2
half the product of the diagonals
Triangle
A = (b × h) ÷ 2
half of base × height
🟦 Parallelogram Scroll
Formula: A = b × h
A
Area
Total space inside, in square units
b
Base
The bottom side of the shape
h
Height
Straight up at 90° — NOT the slant side
⚠️
The Slant Side Trap
A parallelogram has a slant side and a height. They are different! The height is the dashed gold line — it drops straight down at 90°. The slant side is the red line — it leans. Only the height goes in the formula.
Area = b × h
40
square units
📓 Field Log (TWR)
In A = b × h —
• b stands for
• h stands for (not the slant side!)
• To find area:
📐
Worked Example — Parallelogram
I
Identify the formula
For a parallelogram: A = b × h. Use the perpendicular height, not the slant side.
II
Label dimensions
Base = 12 cm, Height = 7 cm (slant side = 9 cm — ignore it).
b = 12 cm h = 7 cm
III
Calculate
A = 12 × 7 = 84 cm²
IV
Check units
Area is always in square units — cm², m², ft². Never forget the ²!
💎 Rhombus Scroll
Two Formulas — Use Either One
A = b × h
b
Base
Bottom side
h
Height
Straight up at 90°
Use when you know the base and height
A = (d₁ × d₂) ÷ 2
d₁, d₂
Diagonals
Corner to corner, cross at 90°
÷ 2
Halve it
Rhombus = half the diagonal rectangle
Use when you know the two diagonals
Try Formula 1 — b × h
A = b × h
40
square units
Try Formula 2 — Diagonals
A = (d₁ × d₂) ÷ 2
30
square units
📓 Formula Check — Fill in the Blanks
Formula 1 — A = b × h:
• b stands for
• h stands for (must be perpendicular to b)
Formula 2 — A = (d₁ × d₂) ÷ 2:
• d₁ and d₂ are the
• I divide by 2 because
• I choose Formula 1 when I know
• I choose Formula 2 when I know
📐
Worked Examples — Both Rhombus Formulas
1a
Formula 1: A = b × h
A rhombus floor tile has base = 9 cm and perpendicular height = 7 cm.
A = 9 × 7 = 63 cm²
1b
Formula 2: A = (d₁ × d₂) ÷ 2
A baseball diamond (rhombus) has diagonals d₁ = 127 ft and d₂ = 127 ft.
A = (127 × 127) ÷ 2 = 16,129 ÷ 2 = 8,064.5 ft²
⚠️
Which formula to use?
If the problem gives you base and height → use A = b × h.
If the problem gives you the two diagonals → use A = (d₁ × d₂) ÷ 2.
Both give the same answer when applied correctly.
If the problem gives you the two diagonals → use A = (d₁ × d₂) ÷ 2.
Both give the same answer when applied correctly.
🔺 Triangle Scroll
Formula: A = (b × h) ÷ 2
A
Area
Total space inside, in square units
b, h
Base & Height
Base = bottom side. Height = straight up at 90° to the base.
÷ 2
Divide by 2
A triangle is exactly half a parallelogram with the same base and height
Area = (b × h) ÷ 2
30
square units
📓 Field Log (TWR)
In A = (b × h) ÷ 2 —
• b stands for
• h stands for
• I divide by 2 because
📐
Worked Example — Obtuse Triangle (Tricky!)
I
Warning
For an obtuse triangle, the height falls outside the shape. The formula still works identically: A = (b × h) ÷ 2.
II
Label
b = 14 m h = 9 m (outside the triangle)
III
Calculate
A = (14 × 9) ÷ 2 = 126 ÷ 2 = 63 m²
IV
⚠️ Trap to avoid
Never use the slant side as height. Height always forms 90° with the base.
🔗 The Grand Codex — How All Formulas Connect
Every formula connects to the rectangle. A parallelogram is a rearranged rectangle. A triangle is half a parallelogram. A rhombus uses diagonals to find the same half-rectangle relationship.
0
Solved
12
Missions
0%
Score
🌐 ESOL Mission Support
L1–2
Point to the shape. Circle the formula. Use word bank above.L3–4
Use sentence frames below each mission. Write each step in your Field Notes.L5
Explain your reasoning. Compare shapes. Use math vocabulary.⛺ Sector Alpha — Single-Step Missions
Missions 1–4 · 5 XP each
1
Parallelogram
+5 XP
📍 Sector Alpha, Site 1: Explorers discover a stone panel shaped like a parallelogram at the edge of camp.
🌐 Field Support
Parallelogram / Paralelogramo / figura con 4 lados, 2 pares paralelos
🗣️ "The formula for a parallelogram is A = b × h. The area is ___ cm²."
"La fórmula es A = base × altura. El área es ___ cm²."
Parallelogram / Paralelogramo / figura con 4 lados, 2 pares paralelos
🗣️ "The formula for a parallelogram is A = b × h. The area is ___ cm²."
"La fórmula es A = base × altura. El área es ___ cm²."
🗣️ "The formula for a parallelogram is A = b × h. The area = ___ × ___ = ___ cm²."
La fórmula es A = base × altura. El área = ___ × ___ = ___ cm².
La fórmula es A = base × altura. El área = ___ × ___ = ___ cm².
Find the area of the stone panel. Base = 14 cm, Height = 9 cm.
cm²
📜 Field Notes — Show Your Work
Use A = b × h. Multiply the base (14) by the height (9). The slant side is a trap — ignore it!
2
Rhombus
+5 XP
📍 Sector Alpha, Site 2: A kite-shaped rhombus marker is found embedded in the canyon wall.
🌐 Field Support
Rhombus / Rombo / 4 lados iguales, usa diagonales
🗣️ "The rhombus formula uses diagonals. A = (d₁ × d₂) ÷ 2."
"La fórmula del rombo usa diagonales. A = (d₁ × d₂) ÷ 2."
Rhombus / Rombo / 4 lados iguales, usa diagonales
🗣️ "The rhombus formula uses diagonals. A = (d₁ × d₂) ÷ 2."
"La fórmula del rombo usa diagonales. A = (d₁ × d₂) ÷ 2."
🗣️ "The rhombus formula uses diagonals. A = (___ × ___) ÷ 2 = ___ m²."
A = (d₁ × d₂) ÷ 2. Multiplico ___ × ___ = ___, luego divido entre 2.
A = (d₁ × d₂) ÷ 2. Multiplico ___ × ___ = ___, luego divido entre 2.
A rhombus kite has diagonals of 16 m and 10 m. What is its area?
m²
📜 Field Notes — Show Your Work
Rhombus formula: A = (d₁ × d₂) ÷ 2. Multiply 16 × 10 = 160, then divide by 2.
3
Triangle
+5 XP
📍 Sector Alpha, Site 3: A triangular garden plaza is mapped at the expedition's base coordinates.
🌐 Field Support
Triangle / Triángulo / 3 lados — área = MITAD de base × altura
🗣️ "A triangle is half of a parallelogram, so I divide by 2."
"Un triángulo es la mitad de un paralelogramo, entonces divido entre 2."
Triangle / Triángulo / 3 lados — área = MITAD de base × altura
🗣️ "A triangle is half of a parallelogram, so I divide by 2."
"Un triángulo es la mitad de un paralelogramo, entonces divido entre 2."
🗣️ "A triangle is HALF of a parallelogram. A = (b × h) ÷ 2 = (___ × ___) ÷ 2 = ___."
Un triángulo es la mitad de un paralelogramo. Multiplico y divido entre 2.
Un triángulo es la mitad de un paralelogramo. Multiplico y divido entre 2.
The triangular plaza has base = 18 ft and height = 11 ft. Find its area.
ft²
📜 Field Notes — Show Your Work
Triangle: A = (b × h) ÷ 2. Multiply 18 × 11 = 198, then divide by 2.
4
Parallelogram — ⚠️ Trap!
+5 XP
📍 Sector Alpha, Site 4: A decoy marker has three measurements — only two are needed. Can you identify the trap?
🌐 Field Support
⚠️ Trap! / ¡Trampa! / El lado inclinado NO es la altura
🗣️ "The height makes a right angle (90°) with the base. Do NOT use the slant side."
"La altura forma un ángulo recto. NO uses el lado inclinado."
⚠️ Trap! / ¡Trampa! / El lado inclinado NO es la altura
🗣️ "The height makes a right angle (90°) with the base. Do NOT use the slant side."
"La altura forma un ángulo recto. NO uses el lado inclinado."
🗣️ "The slant side is a TRAP. I only use the base and the perpendicular height."
El lado inclinado es una trampa. Solo uso la base y la altura perpendicular.
El lado inclinado es una trampa. Solo uso la base y la altura perpendicular.
Base = 13 cm, slant side = 10 cm, height = 7.5 cm. What is the area?
cm²
📜 Field Notes — Show Your Work
Don't use the slant side (10 cm)! Only base × perpendicular height. A = 13 × 7.5
⚔️ Sector Beta — Multi-Step Missions
Missions 5–8 · 8 XP each
5
Missing Dimension
+8 XP
📍 Sector Beta, Site 5: The height measurement on this triangular marker was eroded away. Work backwards to find it.
A triangle has area = 48 in² and base = 12 in. Find the missing height.
in
📜 Field Notes — Show Your Work
Work backwards: 48 = (12 × h) ÷ 2 → multiply both sides by 2: 96 = 12 × h → h = 96 ÷ 12.
📓 Field Log (TWR)
I can work backwards from the formula: if A = 48 and b = 12, then h = because .
6
Real World · Cost
+8 XP
📍 Sector Beta, Site 6: The expedition needs to replace a diamond-shaped window in the field station.
A rhombus window: horizontal diagonal = 2.4 ft, vertical diagonal = 3.6 ft. Replacement costs $18/ft². Total cost?
dollars
📜 Field Notes — Show Your Work
Use A = b × h (base × height) because you're given base and height, not diagonals. Tile area = 11 × 8 = 88 m². Tiles needed = 440 ÷ 88.
7
Missing Dimension
+8 XP
📍 Sector Beta, Site 7: The base stone is missing from this parallelogram marker. Calculate it from the remaining data.
A parallelogram: area = 91 m², height = 7 m. What is the base?
m
📜 Field Notes — Show Your Work
Work backwards: A = b × h → 91 = b × 7 → b = 91 ÷ 7.
8
Composite · Fractions
+8 XP
📍 Sector Beta, Site 8: A triangular rooftop at the dig site needs solar panels installed on part of it.
Triangle roof: base = 24 m, height = 6.5 m. Solar panels cover ⅔ of the roof. Panel area?
m²
📜 Field Notes — Show Your Work
Step 1: Roof area = (24 × 6.5) ÷ 2 = 78 m². Step 2: ⅔ of 78 = 78 ÷ 3 × 2 = 52 m².
🏛️ Sector Gamma — Transfer Missions
Missions 9–12 · 12 XP each · DOK 3
9
Compare & Contrast
+12 XP
📍 Sector Gamma, Site 9: Two shapes share the same base and height — but their areas tell a different story.
Parallelogram: base = 10 cm, height = 8 cm. A triangle has the same base and height. How many triangles are needed to equal the parallelogram's area?
triangles
📜 Field Notes — Show Your Work
Parallelogram: 10 × 8 = 80 cm². Triangle: (10 × 8) ÷ 2 = 40 cm². 80 ÷ 40 = ?
📓 Field Log (TWR)
The triangle formula includes ÷ 2 because it equals exactly the parallelogram's area. So triangles always equal one parallelogram.
10
Error Analysis
+12 XP
📍 Sector Gamma, Site 10: A field assistant made a calculation error on this ruin marker. Find the mistake.
Marcus says: "Triangle with base 8 and height 5 has area 40 sq units — I just did 8 × 5." Find Marcus's error and give the correct area.
sq units
📜 Field Notes — Show Your Work
Marcus used the parallelogram formula (b × h), not the triangle formula. A = (b × h) ÷ 2 = (8 × 5) ÷ 2.
📓 Field Log (TWR)
Marcus's mistake was . The correct formula is . The answer is 20 because .
11
Transfer
+12 XP
📍 Sector Gamma, Site 11: Two ruin chambers have equal floor area — one rectangular, one rhombus-shaped.
A rhombus and rectangle share the same area. Rectangle: 9 cm × 8 cm. Rhombus: diagonal d₁ = 12 cm. Find diagonal d₂.
cm
📜 Field Notes — Show Your Work
Rectangle area = 72 cm². Set rhombus = 72: (12 × d₂) ÷ 2 = 72 → 12 × d₂ = 144 → d₂ = 12.
12
Open-Ended · Cost
+12 XP
📍 Sector Gamma, Site 12: Final mission — calculate the cost to tile the expedition's triangular command center floor.
Triangle floor: base = 15 ft, height = 8 ft. Tiles = 1 ft² squares at $3.50 each. Minimum total cost? (You can't cut tiles.)
dollars
📜 Field Notes — Show Your Work
Step 1: Area = (15 × 8) ÷ 2 = 60 ft². Step 2: Need 60 tiles (can't cut). Step 3: 60 × $3.50 = $210.
⚒️ Design Lab — Parallelogram (A = 60 ft²)
There are infinite valid answers. Find any base and height that multiply to 60.
—
⚒️ Design Lab — Triangle (A = 60 ft²)
For a triangle, (b × h) ÷ 2 = 60, so b × h = 120.
—
🏛️ Composite Ruin Challenge
This ancient building cross-section combines a parallelogram (base=140, h=80) and a triangle (base=140, h=60), all in feet. Find the total cross-section area.
ft²
✍️ Expedition Journal — Prove Your Thinking
📓 Trial I — Explain the Connection
The triangle formula is half of the parallelogram formula because . I can prove this by .
📓 Trial II — Apply Decomposition
When a complex shape is decomposed into simpler shapes, the total area equals because .
📓 Trial III — Compare Formulas
A rhombus has two formulas: A = b × h and A = (d₁ × d₂) ÷ 2. I use A = b × h when I know , and A = (d₁ × d₂) ÷ 2 when I know . Both give the same result because .
🗺️ Expedition Waypoints
Base Camp
Vault
Missions
Gauntlet
Summit
⚖️ Explorer Self-Assessment
How well can you find areas of these shapes?
1
Still learning. The formulas are new territory.
2
Can do it with hints or by referencing the scrolls.
3
Can solve most missions independently.
4
Can teach this and apply it to new expeditions.
📓 Reflection Log
My mastery level is _____ because . One concept I still want to explore is .
⭐ Gate I — Identify
A parallelogram has base = 11 cm and height = 6 cm. Which formula applies, and what is the area?
⭐⭐ Gate II — Explain
A triangle and a parallelogram share the same base and height. Explain why the triangle's area is exactly half the parallelogram's area. Use a TWR sentence.
TWR Frame
The triangle formula includes ÷ 2 because , which is similar to .
⭐⭐⭐ Gate III — Transfer
A rhombus has an area of 84 in². One diagonal measures 14 in. Find the other diagonal. Then explain: could a triangle with the same diagonal lengths have the same area? Why or why not?
Saves your answers, field notes, and XP to PDF
EduWonderLab — Area Quest ESOL Edition
Grade 6 · CCSS 6.G.A.1 · Interactive Digital Activity · Neft.Alba · For classroom use only. © EduWonderLab