☣️ Operation: Containment - Day 2: Unlocking the Time Variable

The infection is exponential, but the solution is logarithmic.

Exponential versus Logarithms Day 2

"The outbreak is a function of *rate*, but our survival is a function of *time*. Today, we find the key to unlock 't'." - Director Stone

📜 Mission Briefing

🎯 Mission Objective: Convert between exponential and log form to "unlock" and solve for the unknown variable.
⏳ Class Time: 1 Class Period (50-70 mins)
📚 Subject & Level: Secondary Math 3 / Algebra 2
👻 The Adventure: Our infection models are locked. We need to find the inverse operation—the logarithmic key—to solve for 't' and find out *when* the breach will happen.

🎒 The Armory

📦 Unit-Level Gear

🗺️ Unit Schedule: Get the Full Mission Plan
👟 Steps to Success: View the "Conquer" Steps
📈 SMART Goal Cards: Set Your Personal Goal

🔧 Day 2 Gear

🖥️Main Slideshow: Get the QERC Briefing Slides
📄Daily Briefing (Starter): Get the "Decryption Drill" Starter
🎯Practice Drill (Kahoot): Start the "Target of 20" Drill
📝Practice Drill (Worksheet): Get the "Advanced Protocol" Worksheet
📎Printable Exit Tickets: Get the "End-of-Shift" Report

🧑‍🏫 Analyst Support (Video)

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

Stop the Breach - Logarithms is the Key

👕 Costume & Prop Ideas:

QERC Director: A simple lab coat or a black polo with your custom-printed QERC ID badge. Look professional, concerned, and in-command.
Student Badges: Have students create their own! Use the ID Badge Template here. This gets *massive* buy-in!

Exponential to Logarithmic Equations Day 2

Director Stone Badge

🗺️ The Walkthrough

✨ Introduction: The Motivational Message (Approx. 5-7 mins)

The Script:

"Recruits. Yesterday, you successfully identified the two primary signals in the data stream: the 'Infection' signal—that terrifying exponential growth—and the 'Containment' signal, the log curve showing our resources depleting.

But we have a critical problem. We can see the *rate* of infection. We know the virus is doubling, tripling, quadrupling... but we don't know *when*. We have equations like `10,000 = 2^t`, where 't'—time—is trapped in the exponent. We are flying blind.

Today, we change that. We are activating a new protocol: the **Logarithm**. It is the *inverse*. It is the *only* mathematical tool designed to 'unlock' a variable from the exponent. Your mission today is to master this "key" and convert these locked equations into a form we can actually solve. Pay attention. This one is critical."

[YOUR INTRODUCTORY VIDEO GOES HERE]

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

The Story: We can't decrypt the new data until we review the old data. Recruits will access their "Daily Briefing" starter doc.

The Activity: Students complete the "Daily Briefing" (Starter). This reviews Day 1 graphing and introduces the day's core concept. You, as Director Stone, will introduce the key to success: the **"Decryption Phrase"** (your mantra: "Base comes down, make the X!").

🔥 Main Activity: The "Target of 20" Mastery Loop (Approx. 25-35 mins)

The Story: This is the "Decryption Drill." We need to master this new skill with speed and accuracy. Recruits must correctly decrypt **20 transmissions** (problems) to prove they are field-ready.

The Activity (Mastery Learning & Student Choice):

Worked Examples: You, as "Director Stone," will present the QERC Briefing Slides and model the 3 new decryption skills, repeating the "Decryption Phrase" ("Base comes down, make the X") for every example.
Mastery Drill: Students begin the "Target of 20 CORRECT." They have a *choice* of how to practice. Usually the mix the 2.
- **Option A: The Kahoot Drill:** Use the fast-feedback Kahoot. (I go over each example if students miss one, and if they didn't miss that one, they are choosing a problem off of the paper. Feed back is happening for just those that need the support. )
- **Option B: The Paper Drill:** Use a traditional worksheet (if you have one).
Reteach Loop: This is key! After the first Kahoot attempt, you *reteach* the most-missed problems as a class. Students then try again, aiming for that "Target of 20." This cycle of practice, feedback, and reteaching continues until the class shows mastery.

Exponential to Logarithmic Form Lesson

👾 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 next mission.

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: Time Is On Our Side!

The Story: "Mission Accomplished, Analysts. You've given us the key. We can now solve for 't'. You didn't just learn math today... you bought us *time*."

[YOUR EPIC REWARD VIDEO WILL GO HERE!]

🏆 The Level Clear Screen: Performance Review

The Story: Time to file your End-of-Shift Report. We need to know what you know. Be honest. This is how we get better.

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 can convert fluently between exponential and log form. I can use the "Decryption Phrase" to solve all 3 types of problems, even if they look tricky, and I can teach it to someone else.

My Reflection: (Explain *how* you used the 'Decryption Phrase' ("Base comes down, make the X") to convert `125 = 5^t` AND `log_3(x) = 4`.)

My Answer: _________________________

🎱 Skilled Analyst (An "8" Report)

I can correctly use the "Decryption Phrase" ("Base comes down, make the X") to convert both ways. I can also use Backwards PEMDAS to solve for 'x' most of the time, but I might make a small error.

My Reflection: (Explain *how* you used the 'Decryption Phrase' to convert `64 = 4^t`.)

My Answer: _________________________

⭐ Analyst-in-Training (A "6" Report)

I understand the "Decryption Phrase" ("Base comes down, make the X") and can convert *from* exponential *to* log form. I am still a little confused about converting the other way or using Backwards PEMDAS.

My Reflection: (Fill in the blank: The "Decryption Phrase" is: "Base comes down, make the ____.")

My Answer: _________________________

🌱 New Recruit (A "4" Report)

I was present for the briefing, but I'm still confused about what the "base" is and what "log" means. I need to review the support video and ask for help tomorrow.

My Reflection: (What is the *one* thing from today that was most confusing?)

My Answer: _________________________

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

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

"What was the 'muddiest' (most confusing) part of using the 'Decryption Phrase'?"
"When you converted from Log *back* to Exponential, what was the first thing your brain looked for?"
"Why is 'Backwards PEMDAS' a good way to describe solving for 'x' *after* you've converted?"

🧐 Hattie Expert Debrief (For Teachers)

This lesson is designed to have an extremely high effect size by stacking several of Hattie's most powerful strategies. Here's the "why" behind the design:

Self-Reported Grade (d = 1.33): This is one of the *highest* effect sizes. The "Level Clear Screen" isn't just a grade; it's a tool for students to evaluate their own understanding, which is a powerful predictor of success.
Teacher Clarity (d = 0.75): The "Decryption Phrase" ("Base comes down, make the X") is the core of this. It's a simple, memorable, and repeatable success criterion that students can latch onto.
Metacognitive Strategies (d = 0.69): We achieve this with the Cognitive Task Analysis (CTA) built into the rubric. We don't just ask "Did you get it?" We ask "Explain *how* you got it using the Phrase." The class discussion prompts further this by having students reflect on their *thinking process*.
Mastery Learning (d = 0.65): The "Target of 20" loop is a perfect example. We're not moving on with a C-. We practice, get feedback (from Kahoot), you reteach, and we practice again until we demonstrate proficiency.
Worked Examples (d = 0.57) & Direct Instruction (d = 0.60): The lesson starts with you (the expert) clearly modeling the "Decryption Phrase" from the slides. This lowers cognitive load before they begin practice.
Student Choice (d = 0.57): Giving students the choice between the Kahoot drill or a paper drill gives them a sense of agency over their learning, increasing buy-in.