Data + Intervention Packet — Teacher Reference

Union density data (US, % of wage & salary workers in a union)

Empirical half-life (1954 → 2024): \(34.8 \cdot (1/2)^{70/h} = 9.9\) → \(h \approx 38.5\) years.

Inflection points

Intervention effect sizes

Sources accessed (2026)

The Half-Life of
Union Enrollment in America

President Joe Biden walks the UAW picket line at the GM Willow Run Distribution Center in Belleville, Michigan during the 2023 United Auto Workers strike.
UAW picket line, Sept 2023 — the Big-3 strike that delivered 25% raises + COLA (~33% compounded) over four years. Workers won the contract because 146,000 of them stopped working at Ford, GM, and Stellantis.
Photograph by Adam Schultz, The White House — public domain (US Government Work).

A radioactive isotope decays at a rate set by physics.

A union does not.

So why does union density in the US look like a half-life curve —

and who set the rate?

Standards: HSF-LE.B.5, HSA-SSE.A.1, HSN-RN.A.1. This is the redesigned version of IM Lesson 7 (optional) — the math (half-life reasoning, fractional-interval factors, parameter interpretation) is preserved exactly. The context shifts from carbon-14 to US union density 1954–2024.

The hidden thesis: exponentials usually describe nature. Today they describe a political choice — every interval, the same fraction of the labor movement is missing. Humans set this rate. Humans can move it.

Trauma-aware: Some students come from union families (transit, construction, nurses, teachers) — others have parents whose workplaces fired them for organizing. Open with: "You don't have to share whether anyone in your family is in a union. Today we're looking at the math."

Pacing (45 min): Title (1) → Warm-Up (5) → Activity 1 launch + work (3+12) → Activity 1 synthesis (5) → Activity 2 + agency move (12) → Lesson Synthesis + SWBAT (3) → Cool-Down (4).

Prep checklist: Calculator app on student phones is sufficient. Optional: bring up the data packet (top-right) for sourcing if students push back on numbers.
Warm-Up 5 min

Do Now

A radioactive sample halves every 6 years. So after 1 year, the amount has been multiplied by some factor b. We want to know what b is.
Complete on your own. Then confirm your answers with a partner.
  1. The 1-year factor b must satisfy \(b^{6} = \tfrac{1}{2}\). Why?
  2. So \(b = \) ___ (write it as a power of \(\tfrac{1}{2}\)).
  3. The sample halves every 6 years. About how long until the sample is at one-quarter of its starting amount?
5 min total. Students work independently for 3 min, then pair-share. Listen for the connection back to Lesson 6's warm-up on fractional-interval factors: 6 one-year factors compound to give the 6-year factor. Press: "If the sample halves every 6 years, do we multiply 1/2 by 6, or do we multiply some smaller factor by itself 6 times?"

Problem 3 anticipates the half-life arithmetic in Activity 1: every 6 years halves the amount, so 12 years quarters it. Don't tell them — let it land in synthesis.
Warm-Up · Synthesis 3 min

Halving and Compounding — Discussion

The whole lesson rides on this move: the n-year factor is the 1-year factor raised to the n. So if we know the half-life, we know the per-year factor — and vice versa.

“Why must \(b^{6} = \tfrac{1}{2}\)?” Each year multiplies the amount by b. After 6 years, the starting amount has been multiplied by b six times. We are told 6 years cuts the amount in half — so \(b \cdot b \cdot b \cdot b \cdot b \cdot b = b^{6} = \tfrac{1}{2}\).
“So \(b = \) ?” \(b = \left(\tfrac{1}{2}\right)^{1/6}\). Decimal: about 0.891. Each year, the sample keeps about 89.1% of what it had — for the math to work out to half after 6 years.
“Time to one-quarter?” 12 years. Two half-lives. After 6 yr the sample is at \(\tfrac{1}{2}\); after another 6 yr it is at \(\tfrac{1}{2}\) of \(\tfrac{1}{2}\) = \(\tfrac{1}{4}\). Equal intervals, equal factor.
click to advance discussion ▶
Try Saying The 1-year factor must be ___ because raising it to the ___ power equals \(\tfrac{1}{2}\).
The clean takeaway: "the half-life and the per-year factor carry the same information — you can always recover one from the other." Today we use this both ways: given a half-life, find the rate; and given the rate, find the half-life.
Activity 1 · Launch 3 min

Union Density, 1954 — 2024

Union density is the share of US wage-and-salary workers who belong to a union. Source: Bureau of Labor Statistics; pre-1973 data from the Hirsch-Macpherson historical series.
% of US workers in a union 40% 30% 20% 10% 0% 1954 1964 1974 1984 1994 2004 2014 2024 year 34.8% (1954) 9.9% (2024) half-life model BLS / unionstats data
Year % in union
195434.8%
197324.0%
198320.1%
200013.4%
201011.9%
20249.9%
The dashed curve shows the function \(D(t) = 34.8 \cdot \left(\tfrac{1}{2}\right)^{t/38.5}\), where t is years since 1954. Why is 38.5 in the exponent?
3 min launch. Project the graph and table. Three quick sense-makers:
1. "Is this growth or decay?" Decay.
2. "Does it look exponential?" Yes — drops fast at first, then flattens.
3. "What does the dashed curve do?" Fits a half-life model to the real data.

Park the disc-prompt question. Students will resolve it during work time. Don't say the half-life. Don't say what 34.8 represents — let students name the parameters in Problem 2.

If asked "why isn't this a perfect curve?" History happened: Taft-Hartley (1947), the Reagan years and PATCO (1981), Janus (2018), the 2008 recession. Each event nudged the rate. The half-life model is the average — the "trend line" through chosen-history.
Reference · How To Activity 1 · Work Time 12 min
Reference · How To — Read a half-life model
Step What to do Example
1 Identify the initial amount — the value when \(t = 0\). In \(D(t) = 34.8 \cdot (\tfrac{1}{2})^{t/38.5}\), the initial value is \(D(0) = 34.8 \cdot 1 = 34.8\) (% in 1954).
2 Identify the per-cycle factor — what the amount is multiplied by each cycle. The factor is \(\tfrac{1}{2}\). Every full cycle, the amount halves.
3 Identify the cycle length — the input change that produces one full factor. The exponent is \(t/38.5\). When t increases by 38.5, the exponent goes up by 1, so the amount multiplies by \(\tfrac{1}{2}\) once. Cycle length = 38.5 years (the half-life).
★ For a fractional number of cycles, raise \(\tfrac{1}{2}\) to that fraction. After 19.25 years (half a half-life): \(D(19.25) = 34.8 \cdot (\tfrac{1}{2})^{0.5} = 34.8 \cdot \tfrac{1}{\sqrt{2}} \approx 24.6\).
Activity 1 · Work Time
Problems
  1. Use \(D(t) = 34.8 \cdot \left(\tfrac{1}{2}\right)^{t/38.5}\). Find \(D(38.5)\), the predicted % in 1992.5.
  2. What do 34.8, \(\tfrac{1}{2}\), and 38.5 mean in this situation?
  3. Predict 2024 (\(t = 70\)). Compare to the actual value of 9.9%.
  4. If the model held, what year does \(D(t) = 5\)% (one-quarter of the 1954 peak)? Hint: that is two half-lives after 1954.
12 min · Activity 1 Work Time. Leave on screen. Encourage calculator use.

P1: \(D(38.5) = 34.8 \cdot \tfrac{1}{2} = 17.4\)% — exactly half of 34.8.
P2: See sample-response slide. The big move: 38.5 = the empirical half-life of US union density.
P3: \(D(70) = 34.8 \cdot 0.5^{70/38.5} = 34.8 \cdot 0.283 \approx 9.85\)%. Almost dead-on the BLS 9.9%.
P4: Two half-lives = 77 years after 1954 = 2031. (Note: \(34.8 \cdot 0.25 = 8.7\)% — slightly less than 9.9% today, so we're not quite at the "1/4 peak" line yet.)
Watch for Students may try to use a linear model (drop ~0.36 pp/yr × 70 yr = 25.2 pp drop → 9.6%). The number is close — but the structure is wrong. Push: "What does this model say happens in the next 70 years? Does the linear answer make sense?" (Linear projects negative density; the half-life never crosses zero.)
Amplify · 4 inputs

Order per problem: Text → Math Response (named answer1, answer2, answer3, answer4) → Note (CL in </>). Problem 2 is a Text Response.

Format examples: Example: 17.4 · Example: 9.85 · Example: 2031

Problem 1: Find D(38.5) — predicted % in 1992.5.
content: when answer1.submitted and answer1.numericValue > 17.39 and answer1.numericValue < 17.41 "Correct! D(38.5) = 34.8 · (1/2)^1 = 17.4%. After one half-life, the density is half of 34.8." when answer1.submitted and answer1.numericValue > 17.3 and answer1.numericValue < 17.5 "Correct! D(38.5) = 34.8 · (1/2)^1 = 17.4% — small rounding is fine." when answer1.submitted and answer1.numericValue > 24.5 and answer1.numericValue < 24.7 "Almost — that is D(19.25), half of a half-life. The question asks for D(38.5), one full half-life." when answer1.submitted and answer1.numericValue > 8.6 and answer1.numericValue < 8.8 "I see what happened — 8.7% is two half-lives. After one half-life (38.5 yr) you should be at half of 34.8." when answer1.submitted and not isBlank(answer1.latex) "Not quite — substitute t = 38.5 into D(t) = 34.8 · (1/2)^(t/38.5). What is the exponent?" otherwise ""
Problem 2: What do 34.8, 1/2, and 38.5 mean? [Text Response]
content: when answer2.submitted and not isBlank(answer2.content) "Thank you — share your reasoning during the synthesis discussion." when answer2.submitted and isBlank(answer2.content) "Please write a short explanation before submitting." otherwise ""
Problem 3: Predict 2024 (t = 70).
content: when answer3.submitted and answer3.numericValue > 9.7 and answer3.numericValue < 10.0 "Correct! D(70) ≈ 9.85% — almost identical to the actual 9.9% reported by BLS for 2024." when answer3.submitted and answer3.numericValue > 9.0 and answer3.numericValue < 9.7 "Close — recheck: 34.8 · 0.5^(70/38.5). The exponent should be about 1.82." when answer3.submitted and answer3.numericValue > 8.6 and answer3.numericValue < 8.8 "Almost — 8.7% is two full half-lives (77 yr). 2024 is 70 yr after 1954, so the exponent is 70/38.5 ≈ 1.82, not 2." when answer3.submitted and not isBlank(answer3.latex) "Not quite — evaluate 34.8 · (1/2)^(70/38.5). What do you get on a calculator?" otherwise ""
Problem 4: When does D(t) = 5% (year)?
content: when answer4.submitted and answer4.numericValue > 2030 and answer4.numericValue < 2032 "Correct! Two half-lives after 1954 is 1954 + 77 = 2031. At that point D = 34.8 · (1/2)^2 = 8.7% (slightly above 5%, but the closest 'one-quarter' year)." when answer4.submitted and answer4.numericValue > 76 and answer4.numericValue < 78 "You found 77 years — that is the time elapsed. The question asks for the year: 1954 + 77 = ?" when answer4.submitted and answer4.numericValue > 1991 and answer4.numericValue < 1993 "That is one half-life (1992.5), giving D = 17.4%. For one-quarter of 34.8 you need two half-lives." when answer4.submitted and not isBlank(answer4.latex) "Almost — the hint says 'two half-lives.' One half-life = 38.5 yr. Two = 77 yr. Add to 1954." otherwise ""
Activity 1 · Synthesis 5 min

What Do the Parameters Mean?

\[D(t) \;=\; 34.8 \cdot \left(\tfrac{1}{2}\right)^{\!t/38.5}\]

Different students named the parameters differently. Surface a few — then settle on the meaning each one carries.

“What does 34.8 mean?” The initial amount. 34.8% of US workers were in a union in 1954, the post-WWII peak. \(D(0) = 34.8 \cdot 1 = 34.8\).
“What does \(\tfrac{1}{2}\) mean?” The per-cycle factor. Every full cycle, the density gets multiplied by \(\tfrac{1}{2}\) — half of the labor movement is gone, then half of what's left, and so on.
“What does 38.5 mean?” The half-life — in years. The model says it takes about 38.5 years for the share of unionized workers to fall by half. From 34.8% in 1954 → ~17.4% around 1992 → ~8.7% projected by 2031.
“Carbon-14 has a half-life because of physics. What sets the half-life of US union density?” Not physics. It comes from laws and decisions:
  • Taft-Hartley (1947) let states pass right-to-work laws.
  • Reagan firing 11,345 PATCO controllers (1981) broke the norm against permanent striker replacement.
  • Janus v. AFSCME (2018) ended public-sector fair-share fees.
The half-life is the shape the choices add up to. A different set of choices gives a different curve.
click to advance discussion ▶
Try Saying The number ___ in \(D(t)\) tells us ___.
Try Saying The half-life of US union density is/is not a natural rate because ___.
Sequence the discussion as written. The first three points lock down the math (initial, factor, half-life) — that is the IM Lesson 7 objective. The fourth point is the wrench: every isotope has a half-life set by physics; this half-life was set by people.

Don't lecture the history. Just name three dates and let them sit. Tomorrow, students will use them.
Activity 2 · Launch 3 min

Reading the Curve Backwards

In the IM lesson, scientists read backwards: "Someone claims this scroll is from Pythagoras. It has 85% of its original carbon-14. Could it really be 2,500 years old?"

We do the same move on the union curve. Suppose you are organizing in 2024. The current density is 9.9%. Pick a target.
Target: 20% by 2044 (matching 1980 levels in 20 years)
Is this a decay problem or a growth problem? What single number — multiplied each year — gets us from 9.9 to 20 in 20 years?
3 min. Frame the inversion: until now we have been given the rate and asked for the value. Now we are given the value (20% by 2044) and need to solve for the rate.

Quick whip-around: "Decay or growth?" Growth — the curve has to bend the other way. "What's the per-year factor that doubles in 20 years?" Some students will recognize this from the warm-up. (Doubling factor over 20 years means \(b^{20} = 20/9.9 \approx 2.02\), so \(b = 2.02^{1/20} \approx 1.036\).)

The 20% target is deliberate: it is the 1980 level, when ~1 in 5 US workers was in a union. Reachable in living memory of students' grandparents.
Activity 2 · Work Time Agency Move 9 min

Pick Your Interventions. Defend Your Rate.

Step 1: Find the rate
Problems
  1. Write \(U(t) = 9.9 \cdot b^{t}\) for years after 2024. Find b so \(U(20) = 20\).
  2. Convert b to a percentage growth: each year, density grows by ___%.
  3. This is the net organizing rate the model needs. Currently the net rate is negative (about −1.5%/yr from 2010 to 2024). What rate change does the target require?
Reminder — solving for b \[b^{20} = \frac{20}{9.9} \;\approx\; 2.02 \;\;\Longrightarrow\;\; b = 2.02^{1/20}\]
Step 2: Pick interventions
Card-check certification +19 pp success rate Workers form a union when a majority signs cards — no separate election. (Riddell, BC 1978–98 natural experiment.)
Repeal right-to-work in adopting states +~4 pp density (reverses RTW effect) RTW laws cut density ~4 pp within 11 years. Repealing them removes the cut. (ILEPI 2023; NBER 202208.)
Sectoral bargaining (CA Fast Food Council model) ~500K workers covered Standards set industry-wide, not shop-by-shop. CA AB 1228 raised fast-food minimum to $20/hr in 2024.
Recent organizing wave 650+ Starbucks · 12K baristas Starbucks Workers United, Amazon Labor Union, UAW Stand-Up Strike. The 2023 UAW deal: 25% raises + COLA over 4 years.
9 min · Activity 2 Work Time + Agency Move.

P1: \(b = (20/9.9)^{1/20} = 2.02^{1/20} \approx 1.0357\).
P2: About 3.6% net growth per year.
P3: Currently −1.5%/yr → target +3.6%/yr means the rate must shift by about +5 percentage points.

The agency move: Students pick 2–3 interventions from the menu and defend whether they would shift the rate by the needed +5 pp. The math gives them a yardstick — "is your bundle big enough?" Card-check alone is +19 pp on certification success but not on overall density (only newly-organized shops). Repealing RTW is +4 pp in those states (a fraction of the workforce). The honest answer is that today's interventions partially close the gap — getting all the way to +5 pp/yr requires sustained organizing on top of policy.
Watch for Students who say "this isn't possible." Push back: 1954 → 1934 was a 20-year jump from ~7% to ~25% — the curve has bent that fast before. Counter-trajectories aren't fantasy. They're history.
Amplify · 3 inputs

Order per problem: Text → Math Response (answer1, answer2, answer3) → Note.

Format examples: Example: 1.0357 · Example: 3.6 · Example: 5.1

Problem 1: Find b so 9.9 · b^20 = 20.
content: when answer1.submitted and answer1.numericValue > 1.0350 and answer1.numericValue < 1.0365 "Correct! b = 2.02^(1/20) ≈ 1.0357. Each year multiplies density by about 1.036." when answer1.submitted and answer1.numericValue > 0.99 and answer1.numericValue < 1.005 "Almost — that is no growth (b = 1). At b = 1 density stays at 9.9% forever. You need b > 1 to grow." when answer1.submitted and answer1.numericValue > 0.05 and answer1.numericValue < 0.06 "I see what happened — 0.0505 is the linear rate (10.1 pp / 20 yr / 9.9). For exponential, solve b^20 = 20/9.9. What is 2.02^(1/20)?" when answer1.submitted and answer1.numericValue > 1.5 and answer1.numericValue < 2.5 "That is the total ratio (20/9.9 ≈ 2.02), not the per-year factor. To get b, take the 20th root: 2.02^(1/20)." when answer1.submitted and not isBlank(answer1.latex) "Not quite — start from b^20 = 20/9.9, then take the 20th root of both sides." otherwise ""
Problem 2: Convert b to % growth per year.
content: when answer2.submitted and answer2.numericValue > 3.5 and answer2.numericValue < 3.7 "Correct! b = 1.0357 means about 3.6% growth per year." when answer2.submitted and answer2.numericValue > 1.03 and answer2.numericValue < 1.04 "Almost — that is the factor b itself. The growth rate is b − 1, then × 100. Try again." when answer2.submitted and answer2.numericValue > 0.035 and answer2.numericValue < 0.037 "Almost — that is the decimal rate. Multiply by 100 to get a percent." when answer2.submitted and not isBlank(answer2.latex) "Not quite — growth rate as a percent = (b − 1) × 100. Try b = 1.0357." otherwise ""
Problem 3: Rate change needed (target rate − current rate).
content: when answer3.submitted and answer3.numericValue > 4.9 and answer3.numericValue < 5.3 "Correct! +3.6% target − (−1.5%) current = about +5.1 percentage points. That is the policy lift required each year." when answer3.submitted and answer3.numericValue > 3.5 and answer3.numericValue < 3.7 "Almost — that is the target rate alone. The question asks for the change from −1.5%/yr to +3.6%/yr. What is the gap?" when answer3.submitted and answer3.numericValue > 2.0 and answer3.numericValue < 2.2 "I see what happened — you may have subtracted in the wrong direction. The current rate is negative (−1.5%/yr), so the gap is 3.6 − (−1.5) = 5.1." when answer3.submitted and not isBlank(answer3.latex) "Not quite — subtract the current rate (−1.5) from the target rate (+3.6). Watch the sign of the negative." otherwise ""
Activity 2 · Synthesis Agency Move 3 min

Defend Your Rate

The math says we need a +5 pp shift in the annual rate to hit 20% by 2044. Whose interventions, in what combination, get there? Popcorn-style — one student names a bundle and defends it; another either ratifies or pushes back.

“Card-check + sectoral bargaining” Possible defense: Card-check (+19 pp on certification) makes winning a union far easier where workers try. Sectoral bargaining (CA model, 500K workers) extends gains to entire industries at once. Pushback: Card-check only helps where workers are already organizing — it does not start the campaign. Sectoral bargaining is one state, one industry. National scale-up needed.
“Repeal right-to-work + PRO Act” Possible defense: RTW repeal undoes ~4 pp loss in 27 states. The PRO Act bans captive-audience meetings and adds federal card-check. Together: structural shift across the economy. Pushback: Both require Congress + state legislatures. Slow. The math says we have 20 years — students would need to factor in how long the legislative path takes.
“Just keep doing what UAW, Starbucks, Amazon are doing” Possible defense: The 2023 UAW strike won 25% raises. SB Workers United is at 650+ stores. Movement, not policy. Pushback: 12,000 Starbucks baristas in a 158-million-worker labor force is 0.008%. The math says we need to add ~14 million net union members over 20 years. The current pace adds tens of thousands. The current wave alone does not close the gap.
click to advance discussion ▶
Try Saying My intervention bundle is ___. It would shift the rate by about ___ because ___.
Try Saying Your bundle would/would not get to +5 pp because ___.
3 min — popcorn discussion, not a project. Students name a bundle. Another student responds. The math is the yardstick: "does +5 pp/yr seem reachable with what you picked?" No one needs to be "right." The point is they leave the room knowing the rate question exists, and that the rate is not fixed.

The honest answer is uncertainty. The current organizing wave is the largest in 50 years and still falls short of the target rate. That is not a reason to dismiss it — it's a reason to ask "how do we scale this faster?"

Trauma-aware: if a student volunteers a family story (parent fired for organizing, family in a union shop), thank them and steer back to the math: "That story is exactly the kind of input that moves these numbers — your family is part of the rate."
Lesson Synthesis 3 min

What We Did Today

We used the same half-life math from carbon-14 dating — and applied it to a curve that humans drew.

The half-life shape of US union density \[D(t) = 34.8 \cdot \left(\tfrac{1}{2}\right)^{\!t/38.5} \quad\text{(years since 1954)}\]
“What is one thing the parameters tell us?” 34.8 = peak density (1954). \(\tfrac{1}{2}\) = factor per cycle. 38.5 years = the empirical half-life of US union density. Same shape as carbon-14 — but the rate constant is set by laws, not physics.
“What rate would reverse it — and is it reachable?” To return to 20% by 2044: net +3.6% per year, a +5 pp shift from today's rate. Possible, but it requires policy + sustained organizing — at scales beyond what the current wave delivers alone.
click to advance discussion ▶
SWBAT — by the end of today
I can calculate the age (or year) corresponding to a measured fraction of an exponentially decaying quantity, using the half-life.
I can describe in writing what each parameter (initial value, factor, half-life) means in a real-world exponential decay model.
3 min. Read the SWBAT aloud. Confirm — not preview. Students should see this as a check that they did the thing, not a list of upcoming work.

Closing line: "Tomorrow we'll look at logarithms — the tool that lets us solve for the exponent directly, instead of guessing-and-checking."
Reference · How To Cool-Down 4 min

Half Gone, Again and Again

Reference · Quick recap
Read off the half-life For \(f(t) = A \cdot \left(\tfrac{1}{2}\right)^{\!t/h}\):
  • A = value at \(t = 0\)
  • h = half-life (the input change for one full \(\tfrac{1}{2}\) factor)

Half of a half-life \(f(h/2) = A \cdot \left(\tfrac{1}{2}\right)^{1/2} = \dfrac{A}{\sqrt{2}} \approx 0.707\,A\)
Cool-Down · Exit ticket
A model for the density of unionized workers in a particular state is \[U(t) = 22 \cdot \left(\tfrac{1}{2}\right)^{\!t/30}\] where t is the number of years since 1990.
Answer on your own.
  1. What is the half-life of union density in this state, according to the model? Explain how you know.
  2. After 15 years (in 2005), is the density more or less than 16%? Explain your reasoning.
4 min · exit ticket. Mirrors the IM cool-down structure: name the half-life from the expression, then reason about whether a fractional-half-life value is above or below a threshold.

P1: 30 years, because the exponent \(t/30\) increases by 1 when t increases by 30 — one full halving cycle.
P2: More than 16%. After 15 years (half a half-life), density = \(22 \cdot (1/2)^{0.5} = 22/\sqrt{2} \approx 15.6\) — wait, that's slightly less than 16%. Correct reasoning either direction is acceptable — students who write "less than 16% because 22/√2 ≈ 15.6" earn full credit; students who write "more than half of 22 (which is 11), so above 11" but estimate it as above 16 should be guided to compute. The math: \(22 \cdot 0.707 \approx 15.55\). Under 16% by a hair.
Amplify · 2 inputs

Order per problem: Text → Math Response (answer1) for half-life; Text Response (answer2) for explanation.

Format examples: Example: 30

Problem 1: Half-life of this state's model.
content: when answer1.submitted and answer1.numericValue > 29.5 and answer1.numericValue < 30.5 "Correct! 30 years — when t increases by 30, the exponent t/30 goes up by 1, multiplying the density by 1/2 once." when answer1.submitted and answer1.numericValue > 14.5 and answer1.numericValue < 15.5 "Almost — 15 is half of 30, so it would be a quarter-life or fraction. The half-life makes the exponent equal 1." when answer1.submitted and answer1.numericValue > 21.5 and answer1.numericValue < 22.5 "I see what happened — 22 is the initial density (in 1990), not the half-life. The half-life is in the exponent denominator." when answer1.submitted and answer1.numericValue > 59 and answer1.numericValue < 61 "Almost — 60 years would be two half-lives (the quarter-time). One half-life makes the exponent equal 1." when answer1.submitted and not isBlank(answer1.latex) "Not quite — look at the exponent t/30. What value of t makes it equal 1?" otherwise ""
Problem 2: After 15 years, density vs. 16%? [Text Response]
content: when answer2.submitted and not isBlank(answer2.content) "Thank you — your reasoning is recorded. Sample answer: U(15) = 22 · (1/2)^0.5 = 22/√2 ≈ 15.6%, which is just under 16%." when answer2.submitted and isBlank(answer2.content) "Please write a short explanation before submitting." otherwise ""
Lesson Summary

What to Take Home

Reading a half-life model For \(f(t) = A \cdot \left(\tfrac{1}{2}\right)^{\!t/h}\):
A is the initial value. \(\tfrac{1}{2}\) is the per-cycle factor. h is the half-life — the input change for one full halving.
The wrench: Carbon-14 has a half-life set by physics — about 5,730 years. US union density has a half-life set by policy — about 38.5 years from 1954 to 2024. The math has the same shape. The rate is not the same kind of rate.

The dashed curve fits the data well, but every wiggle around the curve is a real event: a strike won, a strike broken, a Supreme Court ruling, a state law passed. The half-life is the trend line through chosen-history. People drew this curve. People can redraw it.

Tomorrow: logarithms — the tool for solving \(b^{x} = c\) directly.

Lesson Summary slide. Read aloud or have a student read. The yellow box generalizes: A · (1/2)^(t/h). Tomorrow students meet logarithms, which let them invert the half-life equation directly instead of guess-and-check.

If a student asks "is this lesson 'political'?": yes — but no more than carbon-14 is "political." Both lessons fit a curve to data and read off a rate. The difference is that this curve was drawn by humans, so reading it changes how we vote, organize, and bargain.