Ice Cream Container Challenge
An ice cream company needs a new container. Students compute the volume of cones, cylinders, and spheres to recommend the shape that holds the most for the least material.
Common Core standards covered
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Teacher-facing overview
Big idea: Know and apply the volume formulas for cones, cylinders, and spheres to solve real-world problems.
Student target: “I can use the volume formulas for cylinders, cones and spheres to solve real problems, give answers exactly in terms of π and as decimal estimates, and compare value by price per volume.”
Success criteria: I can identify radius and height on each solid — including halving a diameter. · I can compute cylinder, cone and sphere volumes, keeping π exact until the end. · I can explain the 1/3 in the cone formula with the same-r-same-h comparison. · I can rank containers by price per unit of volume, not by sticker price.
Essential question
Three containers, three shapes, three prices. The cheapest sticker is on the smallest container — how do you catch that without pouring anything?
Learning target: I can use the volume formulas for cylinders, cones and spheres to solve real problems, give answers exactly in terms of π and as decimal estimates, and compare value by price per volume.
Success criteria
- I can identify radius and height on each solid — including halving a diameter.
- I can compute cylinder, cone and sphere volumes, keeping π exact until the end.
- I can explain the 1/3 in the cone formula with the same-r-same-h comparison.
- I can rank containers by price per unit of volume, not by sticker price.
Why this matters
Volume formulas are where geometry starts costing money: every package, tank, silo and scoop is priced by what it holds. The 1/3 and 4/3 in the formulas reward respect — forget the third and every cone in the shop is triple-billed. And the exact-in-π discipline is last unit's irrational-number lesson cashing in: π has no exact decimal, so honest work keeps it symbolic until the final estimate.
Students should already be able to
- Area of a circle, πr², and radius vs diameter (7.G.B.4).
- π is irrational; decimals for it are approximations — Hunting for Irrational Numbers.
- Cubes and cube roots — Square Root Architects.
Vocabulary to define explicitly
- Cylinder volume
V = πr²h — the circular base area, stacked h high.
Example: r = 3, h = 10: V = π · 9 · 10 = 90π ≈ 283 cm³.
Not an example: 2πrh is the LATERAL SURFACE (the label's area), not the volume.
🔒 3 more terms with examples and non-examples — Cone volume, Sphere volume, Exact vs approximate volume — are defined in the full lesson. See plans →
Where this shows up outside the classroom
Assessment plan
Formative — for feedback during the lesson
- The cone bet: prior intuition made public.
- Lab logs during team time: the diameter conversion and the π-form column.
- Value verdicts: rate per 100 cm³ present and cited.
Summative — the graded evidence: The exit ticket (three solids plus the 1/3 explanation), scored with the four-criterion rubric.
The task students work on
The Frost & Cone ice-cream company is redesigning its containers and your team is the packaging lab. Three candidates: the classic CUP (a cylinder, r = 3 cm, h = 10 cm, priced $2.75), the WAFFLE CONE (r = 4 cm, h = 12 cm, $2.00), and the SNOWBALL (a sphere, r = 3 cm, $1.25). Compute every volume exactly in π and in cm³, then issue the lab's verdict: which container is the best deal per 100 cm³ — and which sticker price is lying to customers?
Packaging lab log — exact form first, then the estimate, then the value math.
| Container | Formula with values | Exact (π form) | ≈ cm³ | Price per 100 cm³ |
|---|---|---|---|---|
| Cup (cylinder, r=3, h=10) | $2.75 → | |||
| Waffle cone (r=4, h=12) | $2.00 → | |||
| Snowball (sphere, r=3) | $1.25 → |
Anticipated student strategies
Exact computation, then the decimal detour's cost, then structure. Keeping π symbolic is both honest (π has no exact decimal) and efficient (coefficients compare instantly) — the lesson should let students feel both advantages.
- Formula, substitute, simplify in π. The student writes the formula, substitutes r and h, and simplifies the coefficient — 90π, 64π, 36π — converting to decimals only at the end.
The lab's official method. The π-coefficients also make the RANKING readable before any decimal appears: 90 > 64 > 36.
🔒 2 more anticipated strategies with select-and-sequence guidance are in the full lesson. See plans →
Lesson script
Collect bets (2 is the crowd favorite). If a demo model is available, pour; otherwise reveal: three exactly. The cone formula's 1/3 is not decoration — it is this experiment, written down.
Listen for: The cone narrows the whole way down — it loses volume at every level compared with the straight-walled cylinder.
Repeat for the cone — (1/3)π · 16 · 12 = 64π ≈ 201 — narrating the 1/3 as the warm-up's pour. Then the sphere: (4/3)π · 3³ = 36π ≈ 113, flagging the CUBE on r. Close with the diameter trap: a 'd = 6' snowball has r = 3, not 6.
- Exact in π until the last line; the coefficient does the comparing.
- r³ for spheres — three dimensions, three factors of r.
Listen for: It is exact — π has no exact decimal — and comparisons like 90π vs 64π need no arithmetic at all.
🔒 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 cone formula doesn't really need the 1/3.” Three cone-fills top off the matching cylinder — the pour is the proof. Dropping the third triple-charges every cone.
- “The card says 6, so r = 6.” — the teacher response is in the full lesson.
- “Sphere volume uses r².” — the teacher response is in the full lesson.
- “282.6 cm³ is the exact volume.” — the teacher response is in the full lesson.
- “The cheapest container is the best deal.” — the teacher response is in the full lesson.
- “Doubling the radius doubles a sphere's volume.” — 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 4 independent.
- I doFind the volume of the Cup: a cylinder with r = 3 cm and h = 10 cm.
🔒 5 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
- A cylindrical juice can has r = 2 cm and h = 6 cm. Volume, exact and approximate?
🔒 3 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: Introduce the three volume formulas and identify radius and height on each solid. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 2: Teams calculate the volume of three candidate containers. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 3: Test predictions by filling physical models and comparing to calculations. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
- Step 4: Write a recommendation citing volume and estimated material cost. Ask: “What do you notice?”, “What could you try next?”, and “How can you prove that?”
Differentiation and inclusion
Support: Provide formula reference cards and whole-number dimensions for support; extend by holding volume fixed and minimizing surface area.
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: Students calculate one cone and one sphere volume and justify their container recommendation with numbers.
Mastery: The student completes the task accurately and explains the strategy with limited prompting.