- Read the scenario. Identify the 3 questions.
- Assign roles.
- Sketch the outdoor classroom with dimensions.
- Checkpoint: Can everyone identify which shape is the platform and which is the sail?
Mission 25
Unit 10
6.G.A.1 · 6.G.A.2 · 6.G.A.4
Geometry Models
6.G.A.3 · Unit 5
Today's objective: Use coordinates and nets to model real-world geometry problems.
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 wants to build a new outdoor classroom. It will have a rectangular platform that is 12 ft long, 8 ft wide, and 1.5 ft tall (filled with gravel). On top, there will be a triangular shade sail with a base of 10 ft and a height of 8 ft. Your team must calculate: (1) the volume of gravel needed for the platform, (2) the surface area of the platform that needs waterproof paint, and (3) the area of the shade sail. Use geometric models of real objects to solve each part.
The Problem
How much material do you need to build the outdoor
classroom?
- Gravel: How many cubic feet of gravel fill the platform? (Volume)
- Waterproof paint: How many square feet of the platform need paint? Consider: do you paint the bottom? (Surface Area)
- Shade fabric: How many square feet of fabric for the triangular sail? (Area of triangle)
Visual Model: Net and 3D Shape
Step-by-Step Investigation Guide
- Identify the geometric shapes. The platform is a rectangular prism (box shape). The shade sail is a triangle. Sketch both with dimensions. What real-life shapes do you see? How do you know the platform is a prism?
- Calculate the volume of gravel. V = length x width x height = 12 x 8 x 1.5 = 144 cubic feet. What does "cubic feet" mean? Why do we use volume for gravel?
- Decide which faces to paint. The platform sits on the ground, so you probably do NOT paint the bottom. That leaves 5 faces: top, front, back, left, right. Why is deciding which faces to paint a real-world decision, not just a math one?
- Calculate surface area to paint. Top: 12 x 8 = 96. Front + Back: 2(12 x 1.5) = 36. Left + Right: 2(8 x 1.5) = 24. Total = 96 + 36 + 24 = 156 sq ft. How many cans of paint do you need if each can covers 50 sq ft?
- Calculate the shade sail area. A = 1/2 x base x height = 1/2 x 10 x 8 = 40 sq ft. Why is the triangle formula half of the rectangle formula?
- Calculate total cost. Gravel: 144 x $2.50 = $360. Paint: 4 cans x $18 = $72 (since 156/50 = 3.12, round up to 4). Fabric: 40 x $8 = $320. Total = $752. Why did you round up the number of paint cans instead of down?
Language Support: Key Vocabulary
Volume
The amount of space inside a 3D shape. Measured in cubic
units (ft³).
Surface Area
The total area of all outside faces of a 3D shape. Measured
in square units (ft²).
Net
A flat pattern that folds up into a 3D shape. Helps you see
all faces at once.
Rectangular Prism
A 3D shape with 6 rectangular faces, like a box. Has length,
width, and height.
Area of a Triangle
A = 1/2 x base x height. Half the area of a rectangle with
the same base and height.
Cubic Foot (ft³)
A unit for measuring volume. A cube that is 1 ft on each
side.
Sentence Frames:
"The volume of the platform is ___ ft³ because I multiplied ___ x ___ x ___."
"We need to paint ___ faces because the bottom ___."
"The total cost is $___ because gravel costs $___, paint costs $___, and fabric costs $___."
Multiple Representations
The 3D model in the hero image shows the rectangular prism (platform) and the triangular shade sail. Think of the platform as layers: each layer is 12 x 8 = 96 sq ft. There are 1.5 layers, so 96 x 1.5 = 144 ft³ of gravel.
The net diagram above unfolds the prism so you can see every face. Count the faces: 1 top, 1 bottom, 2 long sides, 2 short sides = 6 faces total. Since the bottom touches the ground, subtract it from the painted area.
| Item | Calculation | Amount | Cost |
|---|---|---|---|
| Gravel | 12 x 8 x 1.5 | 144 ft³ | 144 x $2.50 = $360 |
| Paint | 96+36+24 = 156 ft² | 4 cans | 4 x $18 = $72 |
| Fabric | 1/2 x 10 x 8 | 40 ft² | 40 x $8 = $320 |
| TOTAL | $752 | ||
Team Roles
Facilitator
Read the 3 questions. Guide the team to tackle them in order:
volume first, then surface area, then triangle area. Keep
everyone on task.
Model Builder
Draw the 3D platform with labels. Draw the net showing all
faces. Sketch the triangular sail. Use color to show which
faces get painted.
Precision Checker
Check every multiplication. Verify units: volume in ft³, area
in ft². Make sure the paint calculation rounds UP. Confirm
cost arithmetic.
Reporter
Organize the final answer into a cost table. Prepare to
explain the difference between area, surface area, and volume.
State the total cost clearly.
Timed Lab Phases
- Calculate volume: V = 12 x 8 x 1.5.
- Draw the net. Decide which faces to paint.
- Calculate surface area of painted faces.
- Calculate triangle area: A = 1/2 x 10 x 8.
- Checkpoint: Do you have all three measurements with correct units?
- Build the cost table: gravel, paint cans, fabric.
- Calculate how many paint cans (round up!).
- Find the total cost.
- Checkpoint: Does the total cost seem reasonable for building an outdoor classroom?
- Reporter: explain each calculation and its real-world meaning.
- Practice answering: "Why did you round up for paint cans?"
- Explain the difference between area, surface area, and volume.
- Checkpoint: Can the team explain why they chose to skip the bottom face?
Challenge
Extension: The school wants to add a second
platform next to the first one (same dimensions) to make an
L-shape. How does the total volume change? How does the surface
area change? (Hint: two touching faces no longer need paint.)
What If...?
- What if the platform were 2 ft tall instead of 1.5 ft? How much more gravel and paint?
- What if the shade sail were a rectangle instead of a triangle? How much more fabric?
Real-Life Connection: Architects and builders calculate volume for materials (concrete, gravel, fill) and surface area for coverings (paint, tile, wrap). Getting these wrong can mean wasted money or a project that does not work.
Defense Preparation
- What is the difference between area, surface area, and volume?"Area measures ___, surface area measures ___, and volume measures ___."
- Why did you choose not to paint the bottom face?"We did not paint the bottom because in real life ___."
- Why did you round up the number of paint cans?"We need 3.12 cans, but you cannot buy part of a can, so we rounded ___ to ___."
- How did you use the net to find surface area?"The net shows all ___ faces laid flat. We calculated the area of each face and ___."
Rubric Quick-Check:
- Volume calculated correctly with cubic units
- Surface area calculated with correct faces selected
- Triangle area formula used correctly
- Cost table complete with all three materials
- Drawings include labeled dimensions and units
Exit Product
Submit a one-page Lab Report that includes:
- Labeled sketch of the outdoor classroom (3D view)
- Net of the platform with face areas calculated
- Volume calculation with correct units (ft³)
- Surface area calculation showing which faces are painted
- Triangle area for the shade sail
- Cost breakdown table with total
Self-Assessment:
- I can model a real object as a geometric shape
- I can calculate volume of a rectangular prism
- I can use a net to find surface area
- I can calculate the area of a triangle
- I can use measurements to solve a real cost problem
Work Space
Sketch:
Calculations:
Cost Table: