Math practice · Unit #05 · 8th grade · Matter

Thirty Cuts

Cut a strip in half thirty times and you're holding something a few atoms thick. The math that gets you there is just ÷2 — repeated until it's exponential. Then down the powers-of-ten ladder we go.

skills: exponents powers of ten density ratios ≈ 35 min
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Name Period Date
PART A

The halving table

width after n cuts = 10 cm ÷ 2ⁿ
Start with a strip 10 cm wide. Cut it in half, then in half again… Fill the table and watch the exponent do the work.

Repeated halving of a 10 cm strip
Cuts (n)2ⁿWidth (cm)
0110
125
24
38
416
532
101,024

  1. After 10 cuts the strip is under a hundredth of a centimeter — too thin to see, and you've only used one-third of the cuts. 2²⁰ = 1,048,576. Width after 20 cuts (in cm)?

  2. 2³⁰ ≈ 1.07 × 10⁹. An atom is about 10⁻⁸ cm wide. Compute the width after 30 cuts — does it land near atom-size?

  3. Why can't you actually DO 30 cuts with scissors? Where does the real world stop the math? (This is why the atom stayed hidden for so long.)

PART B

The powers-of-ten ladder

Scientists write tiny sizes as powers of ten. Match each object to its size — then write the missing sizes as powers.

  1. Write 1,000 as a power of ten: 10^___

  2. Write 0.001 as a power of ten: 10^___

  3. An atom: 0.00000001 cm = 10^___ cm

  4. Order these smallest → largest, by exponent only: grain of sand (10⁻¹ cm) · water molecule (10⁻⁸ cm) · flea (10⁰ cm… wait, a flea is 0.1 cm — check me!) · your thumbnail (10⁰ cm) · red blood cell (10⁻³ cm). Which one did I get wrong, and what's its correct exponent?

PART C

Mystery metal by density

density = mass ÷ volume g/cm³
Four mystery blocks, four computed densities, one published table. Identify each.

Mystery blocks
BlockMass (g)Volume (cm³)Density (g/cm³)Identity
A5420
B7910
C4550
D19310
Published densities (g/cm³)
AluminumIronIceGold
2.77.90.919.3

  1. Block C floats in water. Density of water is 1.0 g/cm³. State the float rule: "An object floats when its density is ___ than water's."

PART D

Counting atoms

Chemical formulas are recipes written in ratios. Subscripts count atoms.

  1. H₂O — how many atoms total? What's the H:O ratio?

  2. CO₂ and CH₄ — total atoms in each?

  3. Glucose, C₆H₁₂O₆ — atoms of each element? Total atoms?

  4. Three water molecules (3 H₂O) contain how many hydrogen atoms? How many oxygen?

Answer key — teachers

Part A

2.5 · 1.25 · 0.625 · 0.3125 · ≈0.0098 cm (10 ÷ 1,024).

  1. 10 ÷ 1,048,576 ≈ 0.0000095 cm (≈ 10⁻⁵ cm)
  2. 10 ÷ 1.07 × 10⁹ ≈ 9.3 × 10⁻⁹ cm — yes, right at atom scale (~10⁻⁸ cm).
  3. Scissors/eyes/hands stop around cut 5–8; tools can't divide forever — atoms stayed hidden until indirect evidence arrived.

Part B

  1. 10³
  2. 10⁻³
  3. 10⁻⁸
  4. Order: molecule (10⁻⁸) → red blood cell (10⁻³) → sand (10⁻¹) → flea & thumbnail (~10⁰). The flea: 0.1 cm is 10⁻¹, not 10⁰ — a flea is about a tenth of a centimeter wide? No — real fleas are ~0.1–0.3 cm long… accept either 10⁻¹ reasoning; the skill is checking exponents against reality.

Part C

A: 2.7 → aluminum · B: 7.9 → iron · C: 0.9 → ice · D: 19.3 → gold.

  1. "…less than water's."

Part D

  1. 3 atoms; H:O = 2:1
  2. CO₂ = 3; CH₄ = 5
  3. 6 C, 12 H, 6 O → 24 atoms
  4. 6 hydrogen, 3 oxygen (coefficient multiplies everything).