The Ramp Works
A skate crew builds a ramp and a deck: the ramp is a triangular prism needing concrete (volume) and paint (surface area); the deck is an L-shape needing boards (composite area). Students decompose every figure into pieces they already own.
Common Core standards covered
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Teacher-facing overview
Big idea: Solve real-world problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Student target: “I can find areas of composite figures by decomposing, volumes of right prisms as base area × length, and surface areas as the sum of a complete face list.”
Success criteria: I can split a composite figure and total the pieces (or subtract a notch). · I can compute a prism's volume as (base area) × length, for any base I can area. · I can list every face of a solid and sum their areas. · I can tie each computation to what it buys: boards, concrete, paint.
Essential question
No formula exists for an L-shaped deck or a skate ramp — and none is needed. How far can DECOMPOSITION carry you?
Learning target: I can find areas of composite figures by decomposing, volumes of right prisms as base area × length, and surface areas as the sum of a complete face list.
Success criteria
- I can split a composite figure and total the pieces (or subtract a notch).
- I can compute a prism's volume as (base area) × length, for any base I can area.
- I can list every face of a solid and sum their areas.
- I can tie each computation to what it buys: boards, concrete, paint.
Why this matters
7.G.B.6 is the capstone of Grade 7 measurement: no new formulas, only the strategy that composite things are sums of simple things. Base-area-times-length is the master idea that makes EVERY prism computable (and sets up Grade 8's cylinders, where the base is the Circle Works' πr²). The face list is the honest path to surface area — and the volume-vs-surface distinction (concrete vs paint) is the applied literacy the standard's 'real-world' phrasing demands.
Students should already be able to
- Areas of rectangles and triangles.
- Cross-sections of prisms — The Slice Lab.
- Circle area (for the challenge) — The Circle Works.
Vocabulary to define explicitly
- Decomposition
Cutting a figure into pieces with known formulas — the L becomes two rectangles.
Example: 8×6 deck minus a 3×2 notch: 48 − 6 = 42 m².
Not an example: Multiplying the L's two longest sides (8 × 6 = 48) — that areas the notch too.
🔒 3 more terms with examples and non-examples — Right prism, Volume = base area × length, Surface area (the face list) — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- Warm-up dual routes.
- Build-sheet decompositions during floor work.
- The budget audit lines.
Summative — the graded evidence: The exit ticket (composite, volume, SA), scored with the four-criterion rubric.
The task students work on
The crew's build sheet: THE DECK: an L-shaped platform — an 8 m × 6 m rectangle with a 3 m × 2 m corner notch removed — decked in boards (area). THE RAMP: a right triangular prism — base triangle legs 3 m and 4 m (slope 5 m), prism length 2 m — poured in concrete (volume) and painted on ALL faces (surface area). THE PAINT BOX: supplies come in a 4 × 3 × 2 m storage box; the crew wants its volume and outside surface too. BUDGET DESK: boards $12/m², concrete $80/m³, paint covers 9 m² per can.
Build sheet — decomposition shown for every line.
| Item | Decomposition | Computation | Result |
|---|---|---|---|
| Deck area | |||
| Ramp volume | |||
| Ramp surface (paint) | |||
| Box volume | |||
| Box surface |
Anticipated student strategies
Two agreeing area routes build trust in decomposition; base-times-length generalizes it to 3D; the face list keeps surface area complete.
- Notch subtraction. The student areas the full 8×6 and subtracts the 3×2 notch.
Elegant when the notch is clean. Pair it with…
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Route 1: 48 − 6 = 42. Route 2: split into two rectangles and add to 42. The agreement is the lesson: composite figures fall to decomposition, and two routes are a free proof.
Listen for: Clean notches favor subtraction; sprawling shapes favor splits. Strategy follows shape.
The near-miss to name: 3·4·2 = 24 is the box around the ramp — the prism is half of it, and the base-times-length view says so instantly.
- Volume: area of the base, stacked along the length — ANY base you can area.
- Surface: list every face first; the slanted 5×2 is the one that escapes.
Listen for: The Slice Lab: a prism's cross-section never changes, so the solid is base area × how far it slides — the base's shape is irrelevant.
🔒 The remaining 4 steps — fully scripted with teacher-says lines, discourse questions, look-fors, and interventions — are in the full lesson. See plans →
Misconceptions and how to respond
- “The L's area is 8 × 6.” That charges the crew for the notch they cut away. Subtract it (48 − 6) or split the L — both land 42.
- “Ramp volume is 3 × 4 × 2.” — the teacher response is in the full lesson.
- “Surface area has one formula per solid.” — the teacher response is in the full lesson.
- “Volume and surface area rise and fall together.” — the teacher response is in the full lesson.
🔒 Every misconception ships with a tested classroom response. See plans →
Scaffolded practice
The workbook follows gradual release rather than a page of blank prompts: 1 fully worked, 1 partially completed, then 5 independent.
- I doThe deck: 8 m × 6 m with a 3 m × 2 m corner notch removed. Board area, two routes.
🔒 6 more prompts across all three release tiers are in the full lesson. See plans →
Worked solutions and scaffolds for every prompt are in the downloadable kit, included with membership. See plans and pricing.
Exit ticket
- An L-shaped patio: 10 × 4 plus a 3 × 2 tab. Area?
🔒 2 more exit-ticket questions are in the full lesson. See plans →
Worked answers, the completed answer tables, and the scoring rubric are in the downloadable kit, included with membership. See plans and pricing.
Printable handouts in the kit
All handouts are in Printable_Handouts.pdf inside the kit download.
Materials and classroom resources
Student experience
Students enter through a low-floor prompt, build or sketch a model, test a strategy, revise their thinking, collaborate, and present a final solution. The task remains accessible while allowing for sophisticated reasoning.
Detailed facilitation notes
- Step 1: Decompose the L-deck and total its area. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Compute the ramp's concrete: base area × length. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Paint the ramp: every face, listed then summed. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Audit the build against the budget. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Face lists keep surface area honest; extend with the paint-one-coat-skip-the-bottom variant and the cost audit.
Language support: “I started by…”, “My model shows…”, and “I know my answer is reasonable because…”.
Extension: Require a second solution, alter a condition, or have students write a challenge for another group.
Accessibility: Manipulatives, enlarged visuals, oral response choices, partner reading, and reduced-copy student pages.
Assessment plan
Formative checks: Strategy choice, accuracy, representation, collaboration, and explanation.
End evidence: One composite area, one prism volume, and one surface area, each with the decomposition shown.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.