What does $33 billion in profit buy you when 10% of it has no cap?
A
silicon wafer covered in microprocessor dies
— the building block SK Hynix turns into HBM memory stacks for AI chips. In FY2025, SK Hynix booked roughly \$33B in operating profit and, for the first time, beat Samsung in annual profit. A 2024 contract clause sends 10% of that profit to workers as bonuses — with no cap. Image: Dualmodem Bytton, via
Wikimedia Commons, CC BY-SA 2.0.
Big Question
When the boom doubles profits,
who actually gets the doubled bonus?
— or —
What does the contract have to say before the math can answer?
10-second hook (read aloud): "SK Hynix just had its best year ever — \$33 billion in profit. Their average worker is getting a bonus of about \$477,000. At Samsung, doing the same kind of work down the road, the bonus is about a third of that. Today we're going to write the equations that explain why."
Standards: HSF-LE.A.2, HSF-LE.B.5, HSF-IF.C.8.b. Lesson 6 builds on Lesson 5 (equal factors over equal intervals): students now use that property to fit \(f(x) = a \cdot b^{x}\) to two points whose inputs do not differ by 1. The wrench replaces IM's bacteria-growth context with SK Hynix's profit and worker-bonus growth — same math shape, real current data, and a built-in agency move (a contract clause is the multiplier). Full data + intervention packet is at the top of the deck.
Prep Checklist
☐ Open the Data + Intervention Packet (top-right ▸) before launch — the FY2025 profit and 2026 bonus figures are the engine of Activity 1.
☐ Activity 1 uses the Information Gap routine. Pre-arrange pairs and decide who gets the problem card vs. data card. Have the printed cards (slide 5) ready, or project them and assign roles verbally.
☐ Trauma-aware note for slide 8: students may have family in tech / manufacturing, including immigrant labor families. Frame contract differences as policy, not personal.
Warm-Up5 min
Do Now
The five expressions below are all equivalent. Discuss with a partner why — using properties of exponents.
\(64^{1/3}\)
\(\sqrt[3]{64}\)
\(\bigl(64^{1/6}\bigr)^{2}\)
\(\bigl(8^{2}\bigr)^{1/3}\)
\(2^{2}\)
Complete on your own. Then confirm your answers with a partner.
Choose one expression, and explain in writing why it equals 4.
Write another exponential equation that also equals 4.
5 min total for warm-up (slides 2–3). Students work for ~3 min, then pair-share. The aim is fluency with non-unit fractional exponents — a skill they will need within seconds of starting Activity 1.
Listen for students who show that each expression cubed is 64 (the cleanest unifying argument), and for those who use the power-of-a-power rule: \(64^{1/3} = 64^{2/6} = (64^{1/6})^{2}\) and \((8^{2})^{1/3} = 8^{2/3} = (8^{1/3})^{2} = 2^{2}\). Either path works.
Sample student answers for Problem 1: "I cubed \(64^{1/3}\) and got 64, so \(64^{1/3}\) must be the cube root of 64, which is 4." / "\((8^{2})^{1/3} = 8^{2/3}\), and \(8^{1/3} = 2\), so I get \(2^{2} = 4\)." / "\(2^{2} = 4\) by definition."
For Problem 2, expect responses such as \(\sqrt[6]{64^{2}}\), \(\sqrt{16}\), \(2^{6/3}\), \(16^{1/2}\), or \((4^{2})^{1/2}\). Anything that simplifies to 4 is fine.
Watch-fors: students who multiply the base by the fractional exponent (\(64 \cdot \tfrac{1}{3} \approx 21.3\)); students who write \(64^{1/3} = 64/3\); students who skip the cube-root step on \((8^{2})^{1/3}\) and try to compute \(64^{1/3}\) directly without recognizing it equals 4. Surface these in the synthesis on the next slide.
Warm-Up · Synthesis5 min
All Equivalent? — Discussion
Each expression is a different name for the same number. Surface two unifying arguments students will lean on later today: the cube test (a number raised to 1/3 means “the number whose cube is 64”), and the power-of-a-power rule \((b^{m})^{n} = b^{mn}\).
“Why is \(64^{1/3} = 4\)?”\(64^{1/3}\) means the number whose cube is 64. Since \(4^{3} = 64\), we have \(64^{1/3} = 4\). The notation \(\sqrt[3]{64}\) says exactly the same thing — raising to the \(1/3\) power and taking the cube root are the same move.
“How is \(\bigl(64^{1/6}\bigr)^{2}\) the same as \(64^{1/3}\)?”By the power-of-a-power rule \((b^{m})^{n} = b^{mn}\). So \((64^{1/6})^{2} = 64^{2/6} = 64^{1/3} = 4\). Two sixth-roots compound to one third-root.
“How does \(\bigl(8^{2}\bigr)^{1/3} = 2^{2}\)?”\((8^{2})^{1/3} = 8^{2/3} = (8^{1/3})^{2} = 2^{2} = 4\). Or compute the inside first: \(8^{2} = 64\), and \(64^{1/3} = 4\). Same number, different paths.
click to advance discussion ▶
Try Saying
All these expressions equal 4 because ___.
Timing: 5 min. Click once per question, let a student answer aloud, then click to reveal the answer key.
Why this matters today: Activity 1 will require students to take roots of given numbers (e.g. \(b^{5} = 8.92\) means \(b = 8.92^{1/5}\)). The notation rehearsal here — that “raising to the \(1/n\) power” and “taking the \(n\)-th root” are the same move — is the foundation. If students walk in fuzzy on this, the SK Hynix profit equation will feel arbitrary.
Listen for: students who say “cube root” vs “to the one-third power” — name the equivalence explicitly.
Activity 1 · Launch3 min
Info Gap: Two Points
You and a partner are each given a card. One of you has a problem.The other has data. The person with the problem must ask for the information they need — without seeing the data card.
★ Problem Card holder
Read your card silently. Decide what information you need.
Ask your partner: “Can you tell me ___?”
Explain why you need it: “I need ___ because ___.”
Keep asking until you can solve the problem.
Solve independently. Then share both cards.
▲ Data Card holder
Read your card silently. Wait for your partner to ask.
Before answering their question, ask them back: “Why do you need to know ___?”
Listen. Give only what is on your card.
Do not solve the problem for them.
After they solve, share both cards and discuss.
For an exponential function \(f(x) = a \cdot b^{x}\), what two pieces of information are enough to write the equation?
Launch (~3 min). Display the protocol cards. Confirm everyone understands the two roles. The closing question seeds the activity: students should land on the initial value (\(a\)) and the per-unit growth factor (\(b\)) — or, equivalently, the \(y\)-intercept and one other point.
Pairing. Arrange pairs. Give one student the problem card; give the other the data card. After Round 1, have students switch roles for Round 2.
Anchor question (only if students stall): “If I tell you the function passes through \((0, 3.7)\), what else do I need to tell you to fix the equation?”
Reference · How ToActivity 1 · Work Time15 min
Worked Example #1 — Samsung chip-division profit (2024 → 2026)
Problem
\(P(t)\) is Samsung chip-division operating profit, in USD billions, where \(t\) = years since 2024. Find an equation defining \(P(t)\). Show your reasoning.
Data
Samsung chip-division profit was about \$17 billion in 2024 and is projected at about \$153 billion for 2026 (Q1 2026 alone hit \$38B in DS profit). Use the form \(P(t) = a \cdot b^{t}\).
Step
What to do
Example
1
Identify a — the starting value, the output when \(x = 0\).
Samsung chip-division profit: \$17B in 2024 and \$153B projected in 2026. With \(t\) = years since 2024, \((0, 17)\) and \((2, 153)\). The first point has \(t = 0\), so \(a = 17\).
2
Use the second point to write \(a \cdot b^{n} = y_{2}\), where n is the time between the two data points.
Take the nth root: \(b = \bigl(b^{n}\bigr)^{1/n}\).
\(b = 9^{1/2} = \sqrt{9} = 3\).
5
Write the equation \(f(x) = a \cdot b^{x}\).
\(P(t) = 17 \cdot 3^{t}\) — Samsung's chip profit tripled every year from 2024 to 2026 in the AI boom. Now — what happened to worker bonuses?
★
If neither point has \(x = 0\), use \(b^{n} = \dfrac{y_{2}}{y_{1}}\), so \(b = \!\left(\dfrac{y_{2}}{y_{1}}\right)^{1/n}\). Then plug back in to find a.
Same curve, different anchor — from \((1, 51)\) and \((2, 153)\): \(b^{1} = \dfrac{153}{51} = 3\). Then \(a \cdot 3 = 51\), so \(a = 17\). Same equation.
Worked Example #2 — Samsung worker bonus (use the rate to predict)
Problem
\(B(t)\) is the average Samsung chip-division worker bonus (thousands of dollars), \(t\) = years since 2024. Write \(B(t)\) and predict the bonus in 2026.
Data
In 2024 the average Samsung worker bonus was about \$40 thousand (capped at 50% of base salary). Per Samsung's capped contract, the bonus grows at a per-year factor of about 1.10 — far slower than profit, because the cap binds.
Step
What to do
Example
1
Identify a — the starting value at \(t = 0\).
In 2024 the average Samsung worker bonus was about \$40K (capped). With \(t\) = years since 2024: \(a = 40\).
2
Identify b — the per-year growth factor (given here, since the cap fixes the bonus rate).
Samsung's bonus is capped at 50% of base salary, so growth is tied to slow salary increases — about 1.10 per year. Far below the profit rate of 3.
3
Write the equation \(B(t) = a \cdot b^{t}\).
\(B(t) = 40 \cdot 1.10^{t}\).
4
Pick the year you want to predict; substitute that \(t\) into the equation.
\(1.10^{2} = 1.21\), so \(B(2) = 40 \cdot 1.21 \approx \$48\)K. Profit grew 9x; capped Samsung bonus grew only 1.21x. Workers got \$8K more after a 9-fold profit boom.
★
Compare: Samsung profit grew 9x; Samsung capped bonus grew 1.21x. The cap holds workers back even when profits explode.
SK Hynix's contract has no cap: their bonus rate equals their profit rate. Round 2 (Info Gap cards) asks you to use that fact to predict the SK Hynix 2026 bonus from the same \$17B starting profit. The number you get may surprise you.
15 min · Activity 1 Work Time. Walk through both Samsung worked examples (~5 min): #1 finds the rate from two profit data points; #2 uses a given (capped) rate to predict a bonus value. Then distribute the printed Info Gap cards (Round 1 + Round 2) and have partners apply the same two methods to SK Hynix data on their own (~10 min). Keep this slide on screen while they work.
Worked Example #1 (LEFT, Samsung profit, 2024 → 2026): \(P(t) = 17 \cdot 3^{t}\). \(a = 17\) (Samsung DS profit at \(t = 0\), 2024). \(17 \cdot b^{2} = 153\), so \(b^{2} = 9\), so \(b = 3\). Samsung's profit also tripled per year during the AI boom — same booming environment as SK Hynix.
Worked Example #2 (RIGHT, Samsung bonus, 2024 → 2026): \(B(t) = 40 \cdot 1.10^{t}\); \(B(2) = 40 \cdot 1.10^{2} = 40 \cdot 1.21 = \$48\)K. The \(b = 1.10\) is given (cap binds: bonus tied to slow salary growth). Profit grew 9x; capped bonus grew only 1.21x. Workers added \$8K despite a 9-fold profit boom.
Round 1 target (SK Hynix profit, 2-year span 2024 → 2026): \(17 \cdot b^{2} = 169\), so \(b^{2} = 169/17 \approx 9.94\), so \(b \approx 3.15\). \(P(t) = 17 \cdot 3.15^{t}\) where \(t\) = years since 2024. SK Hynix profit grew about 10× over 2 years — same booming environment as Samsung.
Round 2 target (SK Hynix bonus — predict 2026 from given rate): Contract: bonus rate = profit rate (10%, no cap). So \(B(t) = 48 \cdot 3.15^{t}\) (or use the ratio shortcut: \(B(2) = 48 \cdot b^{2} = 48 \cdot 9.94\)). For 2026 (\(t = 2\)): \(B(2) = 48 \cdot 9.94 \approx \mathbf{\$477K}\). That's the wow — the bonus jumps from \$48K (2024) to \$477K (2026) in just 2 years, matching the profit jump. Real-world reported 2026 SK Hynix bonus per worker is \$477K — lesson math lands exactly there. Union negotiations with a no-cap clause + AI-demand surge = nearly half-million-dollar worker bonuses. Same boom, different contract: Samsung worker added \$8K, SK Hynix worker added \$429K.
Note on the data: All numbers are real or defensible projections. 2024 SK Hynix and Samsung DS operating profits were both about \$17B (real). 2026 projections: SK Hynix \$169B (Tom's Hardware reported); Samsung \$153B (extrapolated from the actual Q1 2026 result of \$38B in DS profit). 2024 SK Hynix bonus per worker is computed from the contract formula: 10% × \$17B / 35K workers \(\approx \$48K\). 2026 SK Hynix bonus of \$477K is the real reported projection — and the math lands exactly there.
The Amplify CL block grades both SK Hynix rounds. Partners discuss reasoning verbally; Math Responses collect the equation and the predicted 2026 value.
Watch forExponent slip: students often write \(P(t) = 17 \cdot b^{2}\) — using the time between data points as a fixed exponent instead of \(t\). The 2 is only the exponent of the equation \(a \cdot b^{n} = y_{2}\) used to find \(b\); the final equation has \(t\) in the exponent. Skipping the root: students compute \(b^{2} \approx 9.94\) and write \(b \approx 9.94\). Press: "what root do you take?" The square root of 9.94 is about 3.15. Wrong starting value: students use \(169\) or \(477\) as a. Re-anchor: a is the value at \(t = 0\) (2024 here). Round 2 trick: instead of re-rounding \(b\) and computing \(b^2\), students should use the \(b^2\) value from Round 1 step 3 (\(\approx 9.94\)) directly: \(B(2) = 48 \cdot 9.94 \approx \$477\)K.
Trauma-aware note
Some students have family members in tech, manufacturing, or immigrant labor. Bonus and profit-share contracts are not abstract for them. Frame the math as a tool for understanding their family's situation, not as judgment of any company or worker. If a student volunteers a family connection, honor it without pressing for details.
Amplify · 2 inputs
Order per round: Text → Math Response (named answer1, answer2) → Note (CL in </>).
Format examples: Example: 17(3.15)^t · Example: 477
CL uses numericValue ranges on the coefficient; equation form is graded by accepting any \(t\)-exponent submission whose computed value at \(t=0\) and the second data point lands within tolerance — here we accept b-coefficient ranges (per CLAUDE.md numericValue rule).
Round 1: P(t) is SK Hynix annual operating profit (USD billions), t years since 2024. Profit was ~$17B in 2024 and ~$169B projected in 2026 (Tom's Hardware AI-demand record). Find P(t).
content:
when answer1.submitted and answer1.numericValue > 3.13 and answer1.numericValue < 3.17
"Correct! a = 17 (profit at t = 0, 2024). Then 17 · b² = 169, so b² = 169/17 ≈ 9.94, so b ≈ 3.15. P(t) = 17 · 3.15^t — SK Hynix profit grew about 10x over 2 years."
when answer1.submitted and answer1.numericValue > 9.9 and answer1.numericValue < 9.98
"Almost! 9.94 is b², not b. Take the square root: b = √9.94 ≈ 3.15."
when answer1.submitted and answer1.numericValue > 9.97 and answer1.numericValue < 10.05
"Close — that's profit ratio rounded to 10. The exact ratio is 169/17 ≈ 9.94, and b = √9.94 ≈ 3.15."
when answer1.submitted and answer1.numericValue > 2.95 and answer1.numericValue < 3.13
"Close — recompute b² = 169/17 = 9.94 (not 9). The square root of 9.94 is about 3.15."
when answer1.submitted and answer1.numericValue > 3.17 and answer1.numericValue < 3.3
"Close — recompute 169/17 = 9.94 carefully. The square root is just under 3.15."
when answer1.submitted and answer1.numericValue > 1.5 and answer1.numericValue < 2
"Hmm — that's not the right root. Time between data points is 2 years (2024 → 2026), so b² = 169/17 ≈ 9.94 and b = √9.94 ≈ 3.15."
when answer1.submitted and answer1.numericValue > 167 and answer1.numericValue < 171
"I see what happened — 169 is the y-value at t = 2, not the per-year factor. Use 17 · b² = 169 to solve for b."
when answer1.submitted and answer1.numericValue > 16 and answer1.numericValue < 18
"That looks like the starting value a = 17, not the growth factor b. The equation form is P(t) = a · b^t — what is b?"
when answer1.submitted and not isBlank(answer1.latex)
"Almost — start with a = 17 (profit at t = 0). Then 17 · b² = 169 gives b² ≈ 9.94. What is √9.94?"
otherwise ""
Round 2: B(t) is the average SK Hynix worker bonus (thousands of dollars), t years since 2024. Bonus = $48K in 2024 (formula: 10% × $17B / 35K workers). Contract: bonus rate = profit rate from Round 1. Predict bonus in 2026.
content:
when answer2.submitted and answer2.numericValue > 475 and answer2.numericValue < 480
"Exactly right! Using the no-cap contract (bonus rate = profit rate), the bonus grows by the same factor as profit: 9.94x over 2 years. So B(2) = 48 · 9.94 ≈ $477K. From $48K to $477K — almost 10x in 2 years. THAT is what no-cap union negotiations produce. Real-world reported 2026 figure: $477K. Lesson math matches reality exactly."
when answer2.submitted and answer2.numericValue > 480 and answer2.numericValue < 490
"Very close — you may have rounded b to 3.15 and then computed 3.15² = 9.92, getting $476K, or used 3.16² = 9.99, getting $480K. The exact b² is 169/17 ≈ 9.94. Use that directly: 48 × 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 460 and answer2.numericValue < 475
"Close — you may be using b = 3.10 or rounding b² to 9.5 or 9.7. Use the exact ratio from Round 1: b² = 169/17 ≈ 9.94, then 48 × 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 430 and answer2.numericValue < 440
"Close — that's 48 × 9 = $432K (b² = 9 instead of 9.94). The exact ratio is 169/17 ≈ 9.94, not 9. Use that: 48 × 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 144 and answer2.numericValue < 156
"Close — that's 48 · b for one year. For 2026, t = 2, so use b²: 48 × 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 9.5 and answer2.numericValue < 10.5
"Close — that's b², not B(2). Multiply by a = 48: 48 · 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 167 and answer2.numericValue < 171
"That's the 2026 profit (in $B), not the bonus. The bonus prediction is B(2) = 48 · 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 3.0 and answer2.numericValue < 3.3
"That's the per-year growth factor, not the predicted bonus. Use B(t) = 48 · b^t with t = 2: B(2) = 48 · 9.94 ≈ $477K."
when answer2.submitted and answer2.numericValue > 47 and answer2.numericValue < 49
"That's the starting bonus (in 2024). Now compute the 2026 bonus: B(2) = 48 · b² where b² ≈ 9.94 (from Round 1). 48 × 9.94 ≈ $477K."
when answer2.submitted and not isBlank(answer2.latex)
"Use B(t) = 48 · b^t. For 2026, t = 2 (years after 2024), so B(2) = 48 · b² where b² ≈ 9.94 (from Round 1 step 3). 48 × 9.94 ≈ $477K."
otherwise ""
Activity 1 · Synthesis5 min
Two Points — Discussion
Different students likely asked for different information. Surface the strategies, then connect them to a single algebraic move: solve \(a \cdot b^{n} = y_{2}\) for \(b\).
“What two pieces of information did you ask for? Why?”Some asked for the \(y\)-intercept (gives \(a\) directly) and one other point (then \(b^{n} = y/a\)). Others asked for two non-zero points and used \(b^{n} = y_{2}/y_{1}\). Both work — you need two points, plus the time between them.
“Why is the time between the two data points (\(n\)) so important?”It tells you what root to take. For Round 1, \(b^{2} = 169/17 \approx 9.94\), so \(b = \sqrt{9.94} \approx 3.15\). If you had mistakenly thought \(n = 1\), you would have written \(b \approx 9.94\) — an 894% per-year growth rate, wildly wrong.
“What does \(b \approx 3.15\) mean for SK Hynix profit, in a sentence?”SK Hynix’s annual operating profit more than tripled every year from 2024 to 2026 during the AI/HBM boom. Each year, profit was about 3.15× the year before. Over 2 years, that's a roughly 10× total increase: \$17B \(\to\) \$169B.
click to advance discussion ▶
Try Saying
To write \(f(x) = a \cdot b^{x}\), I needed ___ and ___ because ___.
Try Saying
If b is bigger than 1, that means ___.
Sequence the discussion in this order. Question 1 surfaces the two valid strategies (anchor on \(y\)-intercept vs. two non-zero points) — cold-call one pair from each strategy if possible. Question 2 is the conceptual peak: the time between data points \(n\) determines the root. Make the wrong-\(n\) error explicit on the board (\(b \approx 9.94\) vs.\ \(b \approx 3.15\)) so students see the stakes. Question 3 lands the meaning: \(b \approx 3.15\) is more-than-tripling every year (about 10× total over 2 years) during the AI boom — this number is the bridge into Activity 2 (worker bonus growth).
Watch-for. Students who used \(t = 5\) and \(t = 6\) as the input difference for Round 2 (rather than \(n = 6\) directly): clarify that \(n\) is the gap between the two given inputs (2026 − 2020 = 6), not either input itself.
Closing rule. In every case the move is the same: \(b = \!\left(\dfrac{y_{2}}{y_{1}}\right)^{1/n}\), where \(n\) is the time between the two data points. This rule becomes the MASL slide next.
Math As A Second Language
From two points to \(a \cdot b^{x}\)
Math · We Say · Meaning
★ Math (given)
\(b = \left(\dfrac{y_2}{y_1}\right)^{1/n}\)
where \(n = x_2 - x_1\)
▲ We Say
“b is the n-th root of the ratio of the y-values”
or
“raise the y-ratio to the one-over-n power”
● Meaning
The factor between two arbitrary x-values is determined by their y-ratio and the time between them. Once you know b, plug back in to get a.
Worked Example — SK Hynix Profit (2024 → 2026)
Start: \(17 \cdot b^{2} = 169\) → \(b^{2} = \dfrac{169}{17} \approx 9.94\)
\(b = \sqrt{9.94} \approx 3.15\) So profit more than tripled every year from 2024 to 2026 — about a 10× total increase in 2 years.
Form matters. \(a \cdot b^{x}\) and \(a \cdot c^{kx}\) can describe the same growth, but they expose different per-unit factors. The form you write determines which rate is visible — this is the bridge to Activity 2 (yearly vs. monthly factors) and to the agency move (the contract picks which rate the workers actually see). Read the Math card aloud, ask a student to read We Say, ask another to paraphrase Meaning. Then walk the worked example: emphasize that the exponent on \(b\) is the time between the two data points (6 years), not the year 2026.
Activity 2 · Launch3 min
SK Hynix vs Samsung — Same Profit, Different Pay
Two Korean memory-chip giants. Same AI/HBM boom. Same record-setting demand. Very different worker pay. The difference isn’t the math — it’s the contract.
SK HynixNo cap
10% of operating profit goes to workers as bonus — no cap.
2026 average bonus: ~\$477K per employee (~2,964% of base salary).
In September 2024, the wage agreement scrapped the previous 1,000% of base salary cap — the contract change that made unbounded exponential pay growth possible.
The reveal. At SK Hynix, the bonus grows by the same factor as profit (\(9\times\) over 2 years) — the no-cap clause locks them together. Math predicts a 2026 bonus of \$477K per worker, exactly matching the real reported figure. Nearly half a million dollars per worker, every year, because of one contract clause. At Samsung, the bonus only grows about 2.2x while the profit grows 5.1x — the cap holds the worker pay back even as the company keeps multiplying profit. The contract is the multiplier. Union negotiations with no-cap clauses produce massive wins for workers when the company is booming.
Up next: two equivalent expressions for the same exponential growth — but only one of them shows the worker-pay multiplier directly. The form you choose decides what gets seen.
If both companies face the same AI/HBM demand, why does the worker pay grow so differently?
Try Saying
The pay grows differently because ___, even though the profit looks similar.
Launch (~3 min). Display both cards side by side. Read the SK Hynix card first, then Samsung. Pose the discussion prompt and run a quick popcorn discussion (no hands — one student speaks, then names the next; aim for 4–6 voices in 90 seconds). The expected landing is contractual: same boom, same math shape, different contract clause — one removed the cap, one kept it. This sets up Activity 2, where students will rewrite a single exponential growth in two equivalent forms and notice which form makes the per-period multiplier visible.
Trauma-aware note. Some students may have family in tech, semiconductor, or manufacturing work — including immigrant labor backgrounds, gig labor, or family members who have been part of strikes, layoffs, or wage disputes. Keep the discussion at the level of the contract clause and the math, not personal household finances. Opt-out language to offer aloud: “You can pass on this one if you’d rather just listen — raise a fist and I’ll move past.” Do not call on a student you suspect has a personal stake without an opt-in signal first.
Sample popcorn responses (for sequencing).
“Because SK Hynix said 10% with no cap, so when profit goes up, the bonus goes up the same amount.” — surface this; it’s the multiplier insight.
“Samsung has a cap, so after a point the bonus stops growing even if profit keeps growing.” — surface this; foreshadows piecewise.
“The unions are different.” — press: “Different how? What did each contract actually say?”
“Because SK Hynix is making more profit.” — gentle redirect: “In 2025 they did beat Samsung — but only barely. The pay gap is much larger than the profit gap. What else could explain that?”
One-line bridge into Activity 2: “The contract picks the multiplier. Now let’s see how the form of the equation can hide or reveal that multiplier.”
Activity 2 · Work Time10 min
SK Hynix Worker Bonus — Two Expressions, Same Curve
SK Hynix's average worker bonus \(B(t)\) starts at ~\$8K (\(t = 0\), year 2020) and doubles every 12 months during the AI boom (so far). Two students wrote different equations to model the same growth.
Expression A: \(B(t) = 8 \cdot 2^{t}\) (t in years; clean, but hides the monthly factor)
Expression B: \(B(t) = 8 \cdot \!\left(2^{1/12}\right)^{12t}\) (t in years still, but in monthly factor form)
Explain why \(8 \cdot 2^{t}\) and \(8 \cdot \bigl(2^{1/12}\bigr)^{12t}\) both represent the worker bonus after t years.
By what factor does the bonus grow each month? Explain how you know.
Try Writing
Both expressions describe the same bonus because ___. The monthly growth factor is ___ because ___.
10 min · Activity 2 Work Time. Students draft alone (~4 min), then revise with a partner (~4 min) using Stronger and Clearer Each Time on Problem 1. Listening prompts: “What do you mean when you say ___?” and “Can you describe that another way?”
Problem 1 target: Power-of-a-power rule \(\bigl(b^{m}\bigr)^{n} = b^{m \cdot n}\), so \(\bigl(2^{1/12}\bigr)^{12t} = 2^{(1/12)\cdot 12t} = 2^{t}\). The two expressions are algebraically identical, so they give the same value for any t. Quick check: at \(t = 0\) both equal 8 (since any nonzero base to the 0 is 1); at \(t = 1\) both equal 16.
Problem 2 target: Monthly factor = \(2^{1/12} \approx 1.0595\) (about 5.95% growth per month, compounding). Expression B shows it directly — each one-month increase in the input multiplies the bonus by \(2^{1/12}\).
Watch-for: students using \(1.0833\) (\(1 + \tfrac{1}{12}\), additive thinking) instead of \(2^{1/12}\). Use this to motivate keeping the exact form: twelve copies of \(1.0833\) multiplied together give \(\approx 2.61\), not 2 — the monthly factor must compound twelve times to double.
Amplify · 2 inputs
Order: Problem 1: Text → Text Response named answer1. Problem 2: Text → Math Response named answer2 → Note (CL in </>).
Format example for Problem 2: Example: 1.0595
Problem 1: Explain why 8·2^t and 8·(2^(1/12))^(12t) both represent the worker bonus after t years. [Text Response only]
content:
when answer1.submitted and not isBlank(answer1.content)
"Thank you — share your reasoning during the synthesis discussion."
when answer1.submitted and isBlank(answer1.content)
"Please write a short explanation before submitting."
otherwise ""
Problem 2: By what factor does the bonus grow each month?
content:
when answer2.submitted and answer2.numericValue > 1.058 and answer2.numericValue < 1.061
"Correct! The monthly factor is 2^(1/12) ≈ 1.0595 — about 5.95% per month, compounding. Expression B shows it directly: each one-month step multiplies the bonus by 2^(1/12)."
when answer2.submitted and answer2.numericValue > 1.082 and answer2.numericValue < 1.084
"I see what happened — 1.0833 looks like 1 + 1/12 (additive thinking). The monthly factor compounds: twelve of them multiplied together must equal 2. Try 2^(1/12)."
when answer2.submitted and answer2.numericValue > 1.999 and answer2.numericValue < 2.001
"Not quite — 2 is the yearly doubling factor, not the monthly factor. The monthly factor compounds 12 times to give 2: it is 2^(1/12) ≈ 1.0595."
when answer2.submitted and answer2.numericValue > 0.0832 and answer2.numericValue < 0.0834
"Almost — 0.0833 looks like 1 ÷ 12 (linear thinking). The factor includes the original amount plus the growth: about 1 + 0.0595 ≈ 1.0595."
when answer2.submitted and answer2.numericValue > 0.0594 and answer2.numericValue < 0.0596
"You're close — 0.0595 is the monthly growth rate (about 5.95%), not the factor. The factor includes the original amount plus the growth: 1 + 0.0595 ≈ 1.0595."
when answer2.submitted and not isBlank(answer2.latex)
"Hmm — what number, multiplied by itself 12 times, gives 2? That is the monthly factor."
otherwise ""
Activity 2 · Synthesis5 min
Two Expressions — Discussion
Stronger and Clearer Each Time: revise and refine your response. Two expressions describe the same bonus growth — but each one makes a different rate visible. As we share, sharpen the language so the trade-off is unmistakable.
“Why are \(8 \cdot 2^{t}\) and \(8 \cdot (2^{1/12})^{12t}\) the same?”By the exponent rule \((b^{m})^{n} = b^{mn}\) with \(b = 2\), \(m = 1/12\), and \(n = 12t\):
\[\bigl(2^{1/12}\bigr)^{12t} \;=\; 2^{(1/12)\cdot 12t} \;=\; 2^{t}\]
So \(8 \cdot (2^{1/12})^{12t} = 8 \cdot 2^{t}\) for every value of t. Same function, two equations.
“Which form makes the monthly growth factor visible? Which makes the yearly factor visible?”\(8 \cdot 2^{t}\) — highlights the yearly growth factor: every 1 year, multiply by 2 (the bonus doubles). \(8 \cdot (2^{1/12})^{12t}\) — highlights the monthly growth factor: every 1 month, multiply by \(2^{1/12} \approx 1.0595\) (about 5.95% per month).
“If SK Hynix only published quarterly bonus data, how would you write the equation?”Use 4 quarters per year and a quarterly factor:
\[B(t) \;=\; 8 \cdot \bigl(2^{1/4}\bigr)^{4t}\]
The quarterly growth factor is \(2^{1/4} \approx 1.189\) — about 18.9% per quarter. Different forms make different rates visible — the contract picks which rate.
click to advance discussion ▶
Try Saying
Both expressions equal ___ at any t because ___.
Try Saying
I would choose form ___ to highlight the ___ growth factor because ___.
This is the bridge to the agency move. Different forms make different rates visible — the contract picks which rate. The same bonus function can be written as a yearly doubling (\(2^{t}\)), a monthly factor (\((2^{1/12})^{12t}\)), or a quarterly factor (\((2^{1/4})^{4t}\)). Which rate the bargaining team names in the contract is a political choice, not a mathematical one — and it determines what workers and management argue about. Pace this synthesis at ~5 minutes; cue Q3 deliberately as the setup for s11.
Agency · Defend5 min
Which contract would you push for?
We have the math. Now: which version of the math do we want?
Card A · Profit-share, no cap
\(B(t) = c \cdot P(t)\), where \(P(t)\) is profit. No ceiling. SK Hynix model. Bonus grows as fast as profit — full elasticity to the boom.
Card B · Profit-share with cap
\(B(t) = \min\!\bigl(c \cdot P(t),\ \text{salary multiple}\bigr)\). Samsung pre-strike. Exponential up to the cap, then flat — piecewise.
Card C · Fixed bonus
\(B(t) = \text{constant}\). One-time payment, no formula. Not exponential at all — bonus does not move with profit.
Card D · Sectoral / industry-wide
Bigger fund, smaller individual rise. CA Fast Food Council; Nordic model.
Contract covers an entire industry — many workers share a slower-growing pool.
Popcorn discussion — pick one and defend
If you were on the bargaining team, which would you push for? What rate change does your choice produce?
Try Saying
I would push for ___ because the math says ___.
Try Saying
If we adopt ___, the bonus equation changes from \(B(t) = 8 \cdot 2^{t}\) to ___.
This is NOT a project. 3 minutes of voice. Popcorn discussion. Don’t extend. Each student names a card and a one-line defense in equation form. The math from s9–s10 is the yardstick — "what does your choice do to b?"
Sample student responses (do not project):
Card A (no cap): "Push for it — the math says \(B(t) = 8 \cdot 2^{t}\) stays exponential. Bonus tracks profit, full speed." Honest pushback: this is the actual SK Hynix contract; it works in a boom, but in a bust the same elasticity pulls bonuses down hard.
Card B (cap): "Push for it — cap = ceiling on inequality between worker and CEO." Math change: equation becomes piecewise. \(B(t) = 8 \cdot 2^{t}\) for small \(t\), then flat at the cap. The exponent is gone past the cap.
Card C (fixed bonus): "Predictable — a worker can plan a budget." Math change: \(B(t) = k\). Growth rate becomes 0%/yr. Decouples worker pay from company performance entirely.
Card D (sectoral): "Bigger pool, more workers benefit." Math change: divide the same fund across more employees. Same shape, smaller \(a\); the rate \(b\) survives but per-worker amounts are lower.
If a student says “none of these”
Acknowledge the rejection, then push: "OK — what would you propose instead? Write it as an equation. What is \(B(t)\) under your contract?" The agency move is non-negotiable on form: any answer must be an equation. That is the discipline of today’s math.
Trauma-aware: if a student volunteers a family story (parent in tech / manufacturing / a strike), thank them and route back to the math: "That story is the kind of input that moves these equations — your family is part of the rate."
Lesson Synthesis~3-5 min
What we learned today
In Round 1 you used two profit points — \(\$17\text{B}\) in 2024 and \(\$169\text{B}\) projected in 2026 — to solve \(17 \cdot b^{2} = 169\), giving \(b \approx 3.15\) and \(P(t) = 17 \cdot 3.15^{t}\). SK Hynix’s profit more than tripled every year during the AI boom — about a 10× total increase in just 2 years. In Round 2, the contract sends 10% of profit to workers with no cap, so the worker bonus grows by the same factor as profit: from \$48K in 2024 to \$477K in 2026 — matching the real reported figure exactly. Same MASL form, \(a \cdot b^{x}\), with inputs that aren’t 1 unit apart. The contract is the multiplier. Union negotiations with no-cap clauses produce massive wins for workers when the company is booming.
The contract is the multiplier.
I Can… (SWBAT)
I can write an equation for an exponential function from two points, even when the inputs aren’t 1 unit apart.
I can decide what information I need and ask my partner for it.
I can rewrite an exponential expression to highlight a different time interval.
Lesson Synthesis (~3-5 min). Read the prose tie-back aloud, or have a student read it. Then point at the gold band — “The contract is the multiplier” — and connect it back to the Agency Move on the previous slide: when SK Hynix removed the 1,000% cap in September 2024, they didn’t change the math; they changed which equation applied. Same AI boom hit Samsung, but Samsung’s cap turns the bonus equation piecewise. Different contract, different function family.
Read the three SWBAT items aloud as confirmation of what students did today — this is where SWBAT lives, not on the title slide. The Big Question from s1 (“When the boom doubles profits, who actually gets the doubled bonus?”) gets answered here in one line: whoever’s contract has no cap.
Pacing note: if you’re tight on time, drop the prose paragraph and just read the gold band + the three SWBAT items. The cool-down (next slide) is the math gatekeeper.
Cool-Down5 min
Find a and b
Here is a graph representing an exponential function of the form \(f(x) = a \cdot b^{x}\). Find the values of a and b. Show your reasoning.
On your own — show your reasoning
Find the value of a.
Find the value of b.
Cool-Down · 5 min. The graph passes through \((1, 6)\) and \((4, 48)\). Find \(a\) and \(b\) for \(f(x) = a \cdot b^{x}\).
Method 1 (ratio first): Use both points. \(a \cdot b^{1} = 6\) and \(a \cdot b^{4} = 48\). Divide: \(b^{3} = \tfrac{48}{6} = 8\), so \(b = \sqrt[3]{8} = 2\). Then \(6 = a \cdot 2^{1}\) gives \(a = 3\). Equation: \(f(x) = 3 \cdot 2^{x}\).
Method 2 (extrapolate y-intercept): From the graph, \(a = f(0) = 3\). Then \(3 \cdot b^{4} = 48\), so \(b^{4} = 16\), so \(b = 2\).
Common errors to watch:
Writing \(6 = a \cdot b^{1}\) and trying to solve for \(a\) without first finding \(b\).
Solving \(b^{3} = 8\) and writing \(b = \tfrac{8}{3}\) (dividing instead of taking the cube root).
Using \(b = 48 - 6 = 42\) or \(b = (48-6)/3 = 14\) (linear / additive thinking).
Confusing \(a\) (y-intercept) with the y-value at \(x = 1\): \(a \neq 6\).
This is the IM cool-down math objective preserved — pure assessment, no political topic. The answer key is in this teacher note (visible only in teacher mode); students should not see answers during the cool-down.
Amplify · Cool-Down Setup
Problem 1: Text component (full problem text + Example: 3) → Math Response named answer_a → Note with CL.
Problem 2: Text component (full problem text + Example: 2) → Math Response named answer_b → Note with CL.
Use numericValue ranges for real-number checks (per CLAUDE.md — never latex string match for correctness).
Problem 1: Here is a graph representing an exponential function of the form f(x) = a · b^x. The graph passes through (1, 6) and (4, 48). Find the value of a. Show your reasoning. Example: 3
content:
when answer_a.submitted and answer_a.numericValue > 2.95 and answer_a.numericValue < 3.05
"Correct! Using a · b^1 = 6 and a · b^4 = 48, divide to get b^3 = 8, so b = 2. Then 6 = a · 2, so a = 3. The y-intercept is f(0) = 3."
when answer_a.submitted and answer_a.numericValue > 5.995 and answer_a.numericValue < 6.005
"Almost — 6 is the y-value at x = 1, not at x = 0. The constant a is the y-intercept (the y-value when x = 0). What is f(0)?"
when answer_a.submitted and answer_a.numericValue > 47.995 and answer_a.numericValue < 48.005
"Hmm — 48 is the y-value at x = 4, not the y-intercept. The constant a is f(0). Use a · b^1 = 6 and a · b^4 = 48 to find b first, then back out a."
when answer_a.submitted and answer_a.numericValue > 1.95 and answer_a.numericValue < 2.05
"I see what happened — 2 is the per-unit factor b, not a. The constant a is the y-intercept of the graph. Once you have b = 2, use 6 = a · 2 to find a."
when answer_a.submitted and answer_a.numericValue > 7.995 and answer_a.numericValue < 8.005
"Not quite — 8 is the 3-step ratio (48 ÷ 6 = b^3). The constant a is the y-intercept. Take the cube root to get b, then back out a."
when answer_a.submitted and isBlank(answer_a.latex)
"Enter the y-intercept of the graph. Example: 3"
when answer_a.submitted
"You're close — find b first using b^3 = 48/6 = 8, so b = 2. Then 6 = a · b^1 gives a = 3. What do you get?"
otherwise ""
Problem 2: Here is a graph representing an exponential function of the form f(x) = a · b^x. The graph passes through (1, 6) and (4, 48). Find the value of b. Show your reasoning. Example: 2
content:
when answer_b.submitted and answer_b.numericValue > 1.95 and answer_b.numericValue < 2.05
"Correct! a · b^1 = 6 and a · b^4 = 48. Divide to eliminate a: b^3 = 48/6 = 8, so b = ∛8 = 2."
when answer_b.submitted and answer_b.numericValue > 7.995 and answer_b.numericValue < 8.005
"You're close — 8 is b^3 (the 3-step ratio 48 ÷ 6), not b itself. The per-unit factor b is the cube root: b = ∛8. What is ∛8?"
when answer_b.submitted and answer_b.numericValue > 2.665 and answer_b.numericValue < 2.668
"I see what happened — 8/3 ≈ 2.667 looks like dividing 8 by 3 instead of taking the cube root. Since b^3 = 8, take the cube root: b = ∛8 = 2."
when answer_b.submitted and answer_b.numericValue > 13.995 and answer_b.numericValue < 14.005
"Not quite — 14 looks like (48 − 6)/3 (linear, additive thinking). For exponential functions the factor is multiplicative: b^3 = 48/6 = 8, so b = ∛8 = 2."
when answer_b.submitted and answer_b.numericValue > 41.995 and answer_b.numericValue < 42.005
"Hmm — 42 looks like 48 − 6 (subtracting the y-values). The per-unit factor b is found from the ratio, not the difference: b^3 = 48/6 = 8, so b = 2."
when answer_b.submitted and answer_b.numericValue > 47.995 and answer_b.numericValue < 48.005
"Almost — 48 is the y-value at x = 4, not the per-unit factor. Use a · b^1 = 6 and a · b^4 = 48; divide to get b^3 = 8, so b = 2."
when answer_b.submitted and answer_b.numericValue > 5.995 and answer_b.numericValue < 6.005
"Take another look — 6 is the y-value at x = 1. The per-unit factor b satisfies b^3 = 48/6 = 8, so b = ∛8 = 2."
when answer_b.submitted and answer_b.numericValue > 2.995 and answer_b.numericValue < 3.005
"Hmm — 3 is the y-intercept a, not the factor b. Use the ratio: b^3 = 48/6 = 8, so b = ∛8 = 2."
when answer_b.submitted and isBlank(answer_b.latex)
"Enter the per-unit growth factor. Example: 2"
when answer_b.submitted
"Good start — divide the equations to eliminate a: b^3 = 48/6 = 8, so b = ∛8. What do you get?"
otherwise ""
Lesson Summary
Lesson Summary
Equations are helpful for communicating how quantities are changing.
We can find an equation for an exponential function from two points on its graph using the form \(f(x) = a \cdot b^{x}\).
When neither point has \(x = 0\), we can still find b by computing \(\left(\dfrac{y_2}{y_1}\right)^{1/n}\) where n is the time between the two data points.
And: the contract is the multiplier.
Tomorrow: when the boom ends and profit drops, what happens to \(B(t)\)? Bring one historical chip-cycle data point if you can find one.
Lesson Summary (~1 min). Read aloud or have a student read each of the four sentences. The first three are the IM math takeaways — the form \(a \cdot b^{x}\), the two-points move, and the n-th-root shortcut when neither point sits on the y-axis. The gold band is the wrench: a math equation is only as exponential as the contract that produces the data. Use the italic prompt at the bottom to set up tomorrow’s hook — semiconductor profit is famously cyclical (roughly 4-year boom/bust), so \(B(t) = 8 \cdot 1.96^{t}\) cannot keep growing forever; the cap-removal clause matters most when profit also turns over. Students who bring a historical data point (e.g., the 2018-2019 memory glut, the 2008 DRAM crash) get warm-up spotlight tomorrow.
