☣️ Operation: Containment - Day 3: The Certification Protocol

You have the skills. Now you have to prove it. Welcome to Certification Day.

Change of Base Logarithms Comic Book Style

"We've intercepted the antidote formula... but it's locked behind a 4-level security protocol. To get clearance for the final mission, your team must conquer 3 of the 4 stations. Earn your stickers. Get certified." - Director Stone

📜 Mission Briefing

• 🎯 Mission Objective: Prove mastery of log skills (converting, graphing, translating) by conquering 3 of 4 "Certification Protocol" stations.
• ⏳ Class Time: 1 Class Period (80 mins)
• 📚 Subject & Level: Secondary Math 3 / Algebra 2
• 👻 The Adventure: A 4-station "Escape Room" challenge. Teams work together to earn stickers by "conquering" each protocol. Get 3 stickers to win.

🎒 The Armory

🔧 Day 3 Gear (The Stations)

• 🖥️ Main Slideshow: Get the QERC Briefing Slides
• 🎯 Station 1 (Kahoot): Protocol: Conversion & Translation
• 📝 Station 2 (Worksheet): Protocol: Spiral Review
• 🧩 Station 3 (Card Sort): Protocol: Scaffolding
• 📈 Station 4 (IDO): Protocol: Identification

Sticker Cards

Sticker Cards for Change Of Base Exponentials

🧑‍🏫 Analyst Support (Video)

• 🆘Pre-load / Post-Help: Watch the Support Briefing

👕 Costume & Prop Ideas:

• QERC Director: Same as Day 1 & 2. Your QERC ID badge is key.
• Stickers: Get some simple, "dragon worthy" stickers. Biohazard symbols, checkmarks, or even simple gold stars. Hand them out with ceremony as teams "conquer" a station.
• Mission File: Give each team a manila folder labeled "QERC MISSION FILE: [TEAM NAME]". They will collect their stickers on this file.

🗺️ The Walkthrough

✨ Introduction: The Motivational Message (Approx. 10 mins)

The Script:

"Recruits. You have proven you can identify the signals. You have mastered the 'Decryption Phrase' to convert them. But today, we've intercepted the *actual* antidote formula, and it's locked behind a 4-level security protocol. Even worse, the data is encrypted in 'alien' bases... Base 7, Base 4, Base 12. Our calculators can't read them.

Your mission is to get certified for the final hack. You will work in teams to conquer **at least 3 of the 4 Certification Protocols** and earn your stickers. Each station tests a key skill. You'll need your Day 1 graphing, your Day 2 conversion, and the new 'Universal Translator'—the Change of Base formula. Get 3 stickers, and your team is cleared for the final mission. Good luck, analysts."

⚔️ Phase 1: The Warm-Up / Daily Briefing (Approx. 10-15 mins)

QERC FIELD ASSIGNMENT: 4.B (STARTER ANALYSIS)

Recruits, analyze these incoming signals. This content is from the Starter Doc and is essential for recalling your Day 1 and Day 2 protocols. This is high-value content for all analysts.

1. Write in Exponential Form:
`log₃(243) = 5`
Answer: `3⁵ = 243`

2. Write in Log Form:
`(1/2)³ = 1/8`
Answer: `log₁/₂(1/8) = 3`

3. Graph: `y = log₂(x)` then graph `y = log₂(x + 3)`

This is a spiral review of transformations. The graph of `log₂(x + 3)` is the parent graph shifted **left 3 units**.

4. Solve: `(1/2)log₅(x) + 2 = 2`
Answer: `x = 1`

5. Match the equations with the scenarios:

This section is a critical **spiral review** of exponential models. Analysts must differentiate between standard growth (`A=P(1+r)^t`), decay (`A=P(1-r)^t`), half-life (`A=P(1/2)^(t/h)`), continuous growth (`A=Pe^rt`), and compounded growth (`A=P(1+r/n)^nt`).

• A 50g acid substance has a half-life of 2 hours. (Matches D. `A = 50(1/2)^(t/2)`)
• Fifty cells double once every 2 days. (Matches E. `A = 50(2)^(t/2)`)
• Fifty cells grow continuously at a rate of 2%. (Matches C. `A = 50e^(0.02t)`)
• Fifty dollars is placed in an account that earns 2% interest compounded semiannually. (Matches A. `A = 50(1 + 0.02/2)^(2t)`)
• They track polar bears... dying at a rate of 2% a year. (Matches B. `A = 50(0.98)^t`)
• Fifty grams of bacteria are doubling twice every hour. (Matches G. `A = 50(2)^(2t)`)
• An area of elk... growing at a rate of 2% every year. (Matches F. `A = 50(1.02)^t`)

The Story: We can't decrypt the new data until we review the old data. Recruits will access their "Daily Briefing" starter doc (or view it on the main screen).

The Activity: This is a crucial **Spiral Review (d=0.60)** to activate prior knowledge. You, as "Director Stone," will introduce the key to success: the **"Decryption Phrase"** (your mantra: "Base comes down, make the X!").

✨ Phase 1: The "Universal Translator" Protocol (Approx. 15 mins)

The Story: "Recruits. Yesterday we unlocked the variable. We got t = log_2(10,000). But we have a problem. Your calculator doesn't have a `log_2` button. It only has `LOG` (Base 10) and `LN` (Base *e*). We need a way to *translate* any 'alien' base into a language our calculator understands."

The Activity (Teacher Clarity): This is the direct instruction for Change of Base. You, as Director Stone, model this one, critical protocol.

"This is the **Change of Base** formula. It is our 'Universal Translator.' The protocol is:

logb(a) = ln(a) / ln(b)

"Note-takers, write that down. It is 'ln of the answer' divided by 'ln of the base'."

🔥Director's Pro-Tip (A Note for Teachers)

This is the "significant info" that beats the dragon. I *insist* on using **Natural Log (`ln`)** for the Change of Base formula, and I forbid my students from using the common `log` button for it. Why?

1. It Prevents Confusion: Students constantly confuse `log(a) / log(b)` with the "Quotient Property" `log(a/b)`. Since `ln` *looks* different, it creates a separate "bucket" in their brain. They see `ln` and think "Ah, this is the Change of Base protocol."
2. It's Better for Calculus: In Calculus, the natural log `ln` is the standard. Common log `log` all but disappears. Using `ln` from the start builds a bridge to their next course.

   So, to solve our Day 2 problem:
   t = log_2(10,000)
   "Translate it!"
   t = ln(10000) / ln(2)
t ≈ 9.208 / 0.693
t ≈ 13.29
   "We have 13.3 hours. That's the intel we needed. Now, it's your turn to prove you can use this protocol."

Inspiration: The 4-Island Challenge

In a VR Getaway adventure, you don't just "win." You have to conquer different "islands" or "zones," each with its own unique puzzle. That's what today's stations are. Each one tests a different skill—graphing, sorting, speed. Giving students choice and a "world" to explore (even if it's just 4 posters in a room) is the key to engagement.

The Story: Teams work in a "station" format to "conquer" 3 of 4 challenges. When they complete a station, they show you (Director Stone) their results. If they pass, they get a "Protocol Conquered" sticker for their team's Mission File.

• 🎯 Station 1: Protocol: Conversion & Translation (Kahoot)

  Task: Teams play the Day 3 Kahoot. It mixes Day 2 conversion with Day 3 translation (CoB). (Approx. 20-25 mins)

  Sticker Goal: The team must get at least **10 out of 13** questions correct.

• 📝 Station 2: Protocol: Spiral Review (Worksheet)

  Task: The "Student Choice" option. Teams complete the Spiral Review Worksheet. (Approx. 15-20 mins)

  Sticker Goal: Complete 8 Correctly on worksheet with 90% accuracy (checking the key).

🧩 Station 3: Protocol: Scaffolding (Card Sort)

This is the "boss" station for Day 3 and the direct scaffold for Day 4. Teams must sort these 16 jumbled cards into 4 correct "hack sequences." This is a high-value activity that requires analysts to see the full process. (Content from the Card Sort Doc.)

[SITUATION - A]
The "Horde" starts at `5,000` and grows annually at `8%`. The model is `A = 5000(1.08)^t`. How long until the horde reaches `20,000`?

[TRANSLATE & SOLVE - D]
`t = ln(8) / 0.08`

`t ≈ 26.0 hours`

[ISOLATE - B]
`20000 = 5000e^(0.08t)`
`4 = e^(0.08t)`

[CONVERT - A]
`t = log_1.08(4)`

[ISOLATE - C]
`40000 = 5000(1.08)^t`
`8 = (1.08)^t`

[SITUATION - B]
The "Horde" starts at `5,000` and grows continuously at `8%`. The model is `A = 5000e^(0.08t)`. How long until the horde reaches `20,000`?

[CONVERT - D]
`0.08t = ln(8)`

[TRANSLATE & SOLVE - A]
`t = log(4) / log(1.08)`

`t ≈ 18.0 years`

[CONVERT - B]
`0.08t = ln(4)`

[ISOLATE - A]
`20000 = 5000(1.08)^t`
`4 = (1.08)^t`

[SITUATION - C]
The "Horde" starts at `5,000` and grows annually at `8%`. The model is `A = 5000(1.08)^t`. How long until the horde reaches `40,000`?

[TRANSLATE & SOLVE - B]
`t = ln(4) / 0.08`

`t ≈ 17.3 hours`

[ISOLATE - D]
`40000 = 5000e^(0.08t)`
`8 = e^(0.08t)`

[TRANSLATE & SOLVE - C]
`t = log(8) / log(1.08)`

`t ≈ 27.0 years`

[CONVERT - C]
`t = log_1.08(8)`

[SITUATION - D]
The "Horde" starts at `5,000` and grows continuously at `8%`. The model is `A = 5000e^(0.08t)`. How long until the horde reaches `40,000`?

• 📈Station 4: Protocol: Identification (IDO)

• Task: The "fast feedback" review station. Teams complete the 8-question IDO quiz. (Approx. 10-15 mins)Sticker Goal: Get at least **7 out of 8** correct.

👾 The Final Boss

📝 The Final Field Certification (At Home Quiz)

The Story: Time to prove you're ready. Complete your certification exam. You must pass to be cleared for the final mission tomorrow.

The Activity: Students work individually on the Final Certification Exam (IDO). (Note for teachers: To get access to this amazing resource, please contact darcistone@alpinedistrict.org! Also, please hit "submit" to start the quiz fresh, as it generates a new version on each attempt.)

🎬 Epic Reward: Certification Granted!

The Story: "Congratulations, Analysts. I see teams with 3, and even 4, stickers. You've passed the certification. You are officially cleared to handle the antidote formula. Tomorrow, we save the city."

🏆 The Level Clear Screen: Performance Review

The Story: File your End-of-Shift Report. Which protocols are clear? Which are still 'garbled'?

The Activity: Students use their Printable "End-of-Shift" Report to complete their Self-Reported Grade and Cognitive Task Analysis.

🔟 Master Analyst (A "10" Report)

I conquered at least 3 stations and I could teach the "Card Sort" (Station 3) to another team. I understand *why* Change of Base is the necessary 'partner' to converting.

My Reflection: (Explain *why* a calculator can't solve `log_7(50)` directly, and what the two-step 'hack' is to solve it.)

My Answer: _________________________

🎱 Skilled Analyst (An "8" Report)

I conquered at least 3 stations. I am solid on the Kahoot and IDO. The "Card Sort" (Station 3) was tricky, but I see how the steps fit together now. I'm ready for tomorrow.

My Reflection: (What is the *only* reason we need to use the Change of Base formula?)

My Answer: _________________________

⭐ Analyst-in-Training (A "6" Report)

I conquered 1 or 2 stations. I'm good at the *old* skills (Graphing, Converting) but the new Change of Base formula is confusing. I'm not sure when to use it.

My Reflection: (Fill in the blank: To solve `log_3(10)` in a calculator, I would type `log(__) / log(__)`.)

My Answer: _________________________

🌱 New Recruit (A "4" Report)

I was present, but the stations were confusing and I'm not sure how Day 2 and Day 3 are connected. I need to review the support video.

My Reflection: (What was the *one* station that was the most confusing and why?)

My Answer: _________________________

🗣️ De-Brief: Cognitive Task Analysis (Class Discussion)

After the Exit Tickets are in, lead a class discussion using these prompts:

• "Which station did your team find the *easiest*? Which was the *hardest*? Why?"
• "Let's talk about Station 3 (The Card Sort). What was the *first* step you always looked for? What was the *last*?"
• "In your own words, why can't we just skip Day 2 and jump right to the Change of Base formula? Why are they 'partners'?"

About Director Stone (Shauna)

"Director Stone" is the in-class persona of Shauna, the creator of the popular **VRGetaway YouTube channel**.

Shauna brings her passion for immersive storytelling, "dragon-worthy" adventures, and inspirational messages from her virtual worlds directly into the math classroom. This blog, "Math Adventures," is the practical home for those high-efficacy, story-driven lesson plans.

🧐 Hattie Expert Debrief (For Teachers)

This lesson is a high-efficacy "dragon worthy" design that builds on the previous days. It's a "gamified" station rotation that maximizes several Hattie strategies:

• Student Choice (d=0.57): The "Conquer 3 of 4" model is the core. It gives students agency and allows for natural differentiation. Teams can target their weaknesses or start with their strengths.
• Mastery Learning (d=0.65): Each station has a clear, high-bar "Sticker Goal" (e.g., 10/13, 90%). Teams must demonstrate proficiency, not just "completion."
• Spiral Review (d=0.60): Two of the four stations (Graphing IDO, Kahoot review) are *deliberately* pulling in skills from Day 1 and Day 2. This reinforces old learning, builds confidence, and connects it to the new skill.
• Scaffolding (d=0.53): The new "Card Sort" station is a *perfect* scaffolding activity. It lowers the cognitive load by having students "recognize" the full problem-solving sequence *before* they are asked to "reproduce" it from scratch on Day 4.
• Cooperative Learning (d=0.54): The "Sticker Challenge" for teams encourages positive interdependence. Students must "talk it out" to conquer the stations, especially the puzzle-like "Card Sort."
• High-Impact CTA: As always, the "Level Clear Screen" combines **Self-Reported Grade (d=1.33)** with a **Metacognitive (d=0.69)** prompt that links the "why" to the "how."             

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