Handling Billions and Millions
Before we can map planetary timelines, we must fluently translate between extreme magnitudes. Writing out zeroes leads to errors; scaling mathematically is much safer.
Magnitude Definitions
- 1 Million = $1,000,000 = 10^6$
- 1 Billion = $1,000,000,000 = 10^9$ = $1,000 \times \text{Million}$
Example: Billion to Million
Convert $4.5$ billion into millions.
1. Identify the relationship:
There are $1000$ millions in a single billion.
2. Calculate:
To go from Billions (Large unit) to Millions (Small unit), we multiply.
Example: Millions to Billions
Convert $66$ million into billions.
1. Identify the relationship:
There are $1000$ millions in a single billion.
2. Calculate:
To go from Millions (Small unit) to Billions (Large unit), we divide.
Knowledge Check 1
Marking Schedule:
Q1: Billions to Millions (Large to Small) $\to \times 1000$.
$13.8 \times 1000 = 13,800$ million.
Q2: Millions to Billions (Small to Large) $\to \div 1000$.
$250 \div 1000 = 0.25$ billion.
Proportional Mapping (The Unitary Method)
Advanced problems often ask you to "map" a large sequence (like history or distance) onto a smaller model (like a clock or a ruler). The most reliable way to do this is to find out what $1$ unit of the model represents.
The "Per Model Unit" Formula
Example: Distance Model
A $4000 \text{ km}$ highway is mapped onto a $50 \text{ cm}$ piece of paper. How many kilometers are in $1 \text{ cm}$?
1. Identify Totals:
Real = $4000 \text{ km}$, Model = $50 \text{ cm}$.
2. Apply Formula:
$$ \frac{4000 \text{ km}}{50 \text{ cm}} = 80 \text{ km per cm} $$Example: Finding a Position
If $1 \text{ cm}$ represents $80 \text{ km}$, where on the paper is the $600 \text{ km}$ mark?
1. Setup:
We have $600 \text{ km}$. We know each cm holds $80 \text{ km}$.
2. Calculate:
Divide the target by the rate.
Knowledge Check 2
Marking Schedule:
Q1: $\frac{2400 \text{ years}}{60 \text{ mins}} = 40 \text{ years per minute}$.
Q2: Target is $600$. Since $1 \text{ min} = 40 \text{ yrs}$: $600 \div 40 = 15 \text{ minutes}$.
Target Synthesis (JMC 2019)
The Earth Clock Problem
"The 4.5 billion-year age of the Earth is compressed into a 24-hour clock. At what exact time on this 24-hour clock did the dinosaurs go extinct (66 million years ago)?"
Step 1: Unify the Magnitudes
Convert the total reality into millions of years to match the dinosaur metric.
Step 2: Unify the Model (Find the Rate)
Convert the $24$-hour clock entirely into seconds, then find how many seconds represent $1$ million years.
Step 3: Map the Event & Subtract
Calculate the clock duration for $66$ million years. Since this was "ago", we must subtract it from midnight ($24:00:00$).
Duration of Event:
$$ 66 \times 19.2 = 1267.2 \text{ seconds} $$Convert back to mins/secs:
$$ 1267.2 \div 60 = 21.12 \text{ minutes} $$ $$ 0.12 \text{ mins} \times 60 = 7.2 \text{ seconds} $$ $$ \text{Total time ago} = 21 \text{ mins and } 7.2 \text{ secs} $$Final Clock Time:
$$ 24:00:00 - 00:21:07.2 = \textbf{23:38:52.8} $$Knowledge Check 3
Using the exact Earth Clock logic above: Life began approximately $3.5$ billion years ago.
Marking Schedule:
Step 1: $3.5$ billion = $3500$ million years.
Step 2: $3500 \times 19.2 = 67,200$ seconds ago.
Step 3: Convert to hours: $67,200 \div 3600 = 18.666... \approx 18.67$ hours.
Synthesis Task
Combine all skills. Complete 18 mixed-context questions. You have 2 attempts. (Use exact decimals unless specified).
Section A: Large Numbers & Magnitudes
Section B: Foundational Proportions
Section C: Clock Translation Math
Section D: Advanced Chronological Scaling
Student Mastery Report
Take a screenshot of this report to submit to Mr. Shen.