Math practice · shared unit · both grades · Synthetic materials

The Materials Tournament

Engineers don't pick materials by vibes — they compute ratios. Strength per gram wins drones; cost per liter wins bottles. Then: the plastic curve, and what the recycling numbers actually say.

skills: ratios fold-change percent scaling ≈ 35 min
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Name Period Date
PART A

Strength per gram

You're building a drone frame: maximum strength, minimum weight. Strength-to-weight ratio = strength ÷ density. Higher wins. (Simplified values — real datasheets have more decimal places and more lawyers.)

Tournament bracket (simplified data)
MaterialStrength (MPa)Density (g/cm³)Strength ÷ density
Steel5007.8
Aluminum3002.7
Polymer plastic800.9

  1. Compute all three ratios (round to whole numbers). Steel is the strongest material — does it win the drone frame? What did the ratio catch that raw strength missed?

PART B

The plastic curve

Global plastic production, real figures (rounded), in million tonnes per year: 1950: 2 · 1976: 50 · 2000: 213 · 2019: 460.

  1. Fold-change 1950 → 2019: 460 ÷ 2 = ___×

  2. Fold-change 2000 → 2019 (just 19 years): ___×

  3. Someone says "plastic production grows about the same amount every decade." Test the claim with the data — do the decade-to-decade gaps stay the same, or do they grow? What kind of curve stays flat in gaps vs. grows in gaps?

PART C

Recycling reality

Of the roughly 400 million tonnes of plastic produced in a year, about 36 million tonnes gets recycled.

  1. What percent is recycled? What percent isn't?

  2. Write the recycled fraction in lowest terms (36/400). Now say the sentence a headline wouldn't: "For every 100 bottles produced, ___ are recycled and ___ go… where?"

  3. PET bottles (code 1) do better: about 29% recycled in the US. If a school throws away 1,000 PET bottles, how many likely get recycled? Is "better" the same as "good"? Defend with your two percents.

PART D

Bottle math: the scaling chain

Average American: about 50 plastic water bottles per year. One bottle: 12 g of plastic.

  1. Class of 30 students: bottles per year? Total plastic mass?

  2. School of 600: bottles per year? Mass in kg?

  3. City of 1,000,000: bottles per year? Mass in tonnes?

  4. Which step of the chain surprised you most, and why? (Estimations like this are how cities decide whether a bottle deposit is worth it.)

Answer key — teachers

Part A

Steel: ≈64 · Aluminum: ≈111 · Plastic: ≈89.

  1. Aluminum wins the drone frame despite being weaker — the ratio punishes steel's density. Per gram carried, aluminum gives more strength.

Part B

  1. 230×
  2. ≈2.2×
  3. Gaps GROW (48 per ~26 yrs, then 163, then 247 per similar span) — growth that grows its own gaps is exponential-ish, not linear.

Part C

  1. 9% recycled, 91% not.
  2. 36/400 = 9/100: "9 of every 100 recycled; the other 91 go to landfill, incineration, or the environment."
  3. ≈290 of 1,000. Better ≠ good: 29% still means ~7 in 10 escape the loop.

Part D

  1. 1,500 bottles → 18,000 g = 18 kg
  2. 30,000 bottles → 360 kg
  3. 50,000,000 bottles → 600,000 kg = 600 tonnes
  4. Accept any honest surprise; the point is scaling reveals what per-person numbers hide.