First — talk for 2 minHas anyone in your family ever been in a union, or talked about quitting one? Has anyone been on strike, or know someone who has? If no one, Mr. Bennett will share.
A curve that humans drew
Carbon-14 has a half-life. Every 5,730 years, half of it is gone — not because anyone decided so, but because that is how the atom decays. Physics sets the rate. No vote, no law, no strike can move it.
In 1954, about 34.8% of US wage and salary workers belonged to a union. By 2024, that share had fallen to 9.9% — about 14.3 million members in a labor force of more than 144 million. When you plot the union density data from the Bureau of Labor Statistics year by year, the curve bends in a familiar way. It looks like decay. It looks, in fact, almost exactly like a half-life.
But unions are not isotopes. The Taft-Hartley Act of 1947 let states pass right-to-work laws. In 1981, President Reagan fired 11,345 striking air traffic controllers. In 2018, the Janus ruling ended fair-share fees for public-sector workers. Each event nudged the rate. If the curve looks like physics — but the rate was set by people — what would it take to bend it the other way?
Worked Example #1 — Read the Half-Life Model (forward)
Problem\(D(t)\) is US union density (% of workers in a union), t = years since 1954. Use \(D(t) = 34.8 \cdot \left(\tfrac{1}{2}\right)^{t/38.5}\). Predict the % in 2024 (\(t = 70\)).
DataBLS reports 34.8% in 1954 (post-WWII peak, Hirsch-Macpherson series) and 9.9% in 2024. The exponent \(t/38.5\) means one full halving every 38.5 years.
Word Bank
exponent
starting value
half-life
substitution
factor
multiply
Number Bank
70/38.5
38.5
1.818
0.283
9.85
34.8
70
1.21
9.58
| # | Reasoning (fill in) | Math |
|---|
| 1 |
Identify \(a\). It is the value when \(t = 0\). The math name is . |
\(a = 34.8\) |
| 2 |
Identify the time it takes for the amount to drop to half. In the fractional exponent, this number is the denominator. |
\(h = \) |
| 3 |
We put 70 in for \(t\) in the equation. In math, we call this swap . |
\(D(70) = 34.8 \cdot (\tfrac{1}{2})^{\,\square}\)
fill: \(\square = \) |
| 4 |
Enter the whole equation into your calculator — type it exactly like this. Use parentheses around the fractional exponent, or the calculator will read it wrong.
34.8*(1/2)^(70/38.5) |
\(D(70) \approx \) %
(BLS reports 9.9% — nearly identical.) |
Page 1 of 3 · The Half-Life of Union Enrollment in America · Algebra 2 · Unit 5 · Lesson 7
Three real union stories from around the world. Apply the same half-life math from the front: for each country, use the given model to predict union density in 2024. Show your work in the math column.
Problem 1 — United Kingdom (decay)
ContextThe United Kingdom's union density peaked at 50% in 1979. The Thatcher government passed laws weakening union power through the 1980s — and density has been falling ever since.
Model\(D(t) = 50 \cdot \left(\tfrac{1}{2}\right)^{t/38}\), where t = years since 1979. Predict the union density in 2024.
Show your work
\(D(t) \approx\) %
Problem 2 — New Zealand (sharper decay)
ContextNew Zealand had compulsory union membership until 1991. When the Employment Contracts Act ended it, density crashed from 60% in 1985 toward today's level.
Model\(D(t) = 60 \cdot \left(\tfrac{1}{2}\right)^{t/21}\), where t = years since 1985. Predict the union density in 2024.
Show your work
\(D(t) \approx\) %
Problem 3 — South Korea (growth — opposite story!)
ContextSouth Korean union density grew from about 10% in 2015 to about 14% by 2024, driven by reforms under President Moon Jae-in (2017–2022). Same math shape — but going UP instead of DOWN.
Model\(D(t) = 10 \cdot 2^{\,t/20}\), where t = years since 2015 (notice \(2^{...}\), not \((\tfrac{1}{2})^{...}\)). Predict the union density in 2024.
Show your work
\(D(t) \approx\) %
Reflect
Whose rate is the steepest? Whose direction is the opposite of the others? Why might that be?
Page 2 of 3 · Half-Life of Union Enrollment — Practice · Algebra 2 · Unit 5 · Lesson 7
Tier 2 · General Academic Words
- density — how packed something is. Here: the share of workers who are in a union.
- decline — a slow drop over time.
- peak — the highest point a number reaches.
- share — one group's portion of a bigger total (often written as a percent).
- ratio — one number divided by another, used to compare them.
- percentage — a number out of 100 (so 20% means 20 out of every 100).
- decade — a stretch of 10 years.
- trajectory — the path a number takes over time, including where it is headed.
- empirical — based on real measured data, not on a guess.
- compounded — multiplied again and again, with each result feeding the next.
- inflection point — a moment when a curve changes how it bends.
- policy — a rule or plan made by a government, company, or group.
- intervention — an action taken to change how something is going.
- reverse — turn something around so it goes the other way.
Tier 3 · Labor & Math Words
- union — a group of workers who join together to bargain with their employer for better pay and conditions.
- union density — the percent of US wage-and-salary workers who belong to a union.
- NLRB (National Labor Relations Board) — the federal agency that runs union elections and protects workers who organize.
- BLS (Bureau of Labor Statistics) — the US agency that publishes the official union membership numbers each January.
- Taft-Hartley Act — a 1947 law that lets states pass right-to-work laws and limits what unions can do.
- right-to-work law — a state law saying workers in a union shop do not have to pay any union fees.
- decertification — a vote that ends a union at a workplace.
- card-check — forming a union when a majority of workers sign cards, with no separate election.
- sectoral bargaining — setting pay and rules for a whole industry at once, instead of one shop at a time.
- strike — when workers stop working until the company agrees to better terms.
- PATCO strike (1981) — air traffic controllers' strike. President Reagan fired 11,345 of them — a turning point against strikes.
- Janus ruling (2018) — Supreme Court decision (Janus v. AFSCME) that public-sector unions can no longer collect fees from workers who choose not to join. Cut a major funding source for teacher and government-worker unions across the US.
- Thatcher government — the British government under Prime Minister Margaret Thatcher (1979–1990). Passed a series of laws restricting strikes, requiring ballots for union actions, and outlawing some union protections — the main driver of the United Kingdom's union density decline.
- half-life — the time it takes for a quantity to fall to half of what it was. For carbon-14, it is 5,730 years.
- exponential decay — shrinking by the same factor over equal time intervals (so the same percent is lost each year).
- decay factor (b) — the number you multiply by each year. For decay, \(0 < b < 1\).
- per-year factor — another name for b: what one year's growth or decay looks like.
- n-th root — the number that, raised to the n power, gives x. Written \(\sqrt[n]{x}\) or \(x^{1/n}\). Example: \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
- initial value (a) — the amount at \(t = 0\), before any decay or growth has happened.
- doubling time — the time it takes for a growing quantity to double.
Page 3 of 3 · The Half-Life of Union Enrollment in America · Algebra 2 · Unit 5 · Lesson 7