Comparing Quantities Using Scientific Notation

Lesson 2 of 2

In this lesson:

• Answer "how many times as much?" by dividing in scientific notation
• Apply the quotient rule for exponents:
• Interpret comparison results in real-world context

Learning Objectives for This Deck

By the end of this lesson, you should be able to:

4. Determine how many times as much one quantity is than another by dividing two numbers in scientific notation and interpreting the result
5. Apply scientific notation estimation and comparison to solve real-world problems involving populations, distances, sizes, and quantities spanning extreme scales

How Much Bigger Is the Sun Than Earth?

• Diameter of the Sun: about 1,400,000,000 m ( m)
• Diameter of Earth: about 12,700,000 m ( m)

How many times wider is the Sun than Earth? What operation do we use?

Think before advancing...

"How Many Times" Means Division Not Subtraction

Example: 8 is how many times as much as 2?
→ times (not "6 more" from subtraction)

Division Procedure for Scientific Notation Comparisons

• Step 1: Divide the coefficients:
• Step 2: Subtract the exponents: (quotient rule from 8.EE.A.1)
• Step 3: Adjust if coefficient falls outside

World Population Divided by US Population

World population ≈ ; US population ≈ .

How many times larger is the world than the US?

The world population is more than 20 times the US population.

Check In: State the Answer in Context

From the previous slide, we found:

Complete the sentence:

"The world population is about ______ times _______ than the US population."

Verify: Does 23 make sense? The world has ~8 billion people; the US has ~330 million. Is a ratio of 23 reasonable?

Comparing Two Very Large Quantities

Milky Way ≈ stars; observable universe ≈ stars.

About 3 trillion times more stars in the universe.

Adjustment: →

Comparing Two Very Small Quantities

Water molecule ≈ m; red blood cell ≈ m.

A red blood cell is about 23,000 times wider than a water molecule.

Exponent:

Comparing a Very Large to Very Small

Universe age ≈ s; light-crossing time for an atom ≈ s.

The universe's age is about times longer.

Key step:

Visualizing Comparisons Across Orders of Magnitude

Each arrow spans the number of orders of magnitude between the two quantities.

Check In: Earth Mass versus Mosquito Mass

Earth ≈ kg; mosquito ≈ kg.

How many times more massive is Earth than a mosquito?

Divide coefficients, subtract exponents, state in context.

Answer:

Earth is times more massive than a mosquito.

Guided Practice: Sun Diameter Divided by Earth's

Sun ≈ m; Earth ≈ m.

Find the exponent and coefficient ratio.

Answer: — the Sun is about 100 times wider.

Practice: Two Comparison Problems to Solve

1. Speed of light vs. speed of sound:

   • Speed of light ≈ m/s
   • Speed of sound ≈ m/s
   • How many times faster is light than sound?

2. National debt vs. median household income:

   • National debt ≈
   • Median household income ≈
   • How many times larger is the national debt?

Show all steps and state each answer in a complete sentence.

Practice Answers: Comparing Quantities in Context

Problem 1:
→ Light travels 1,000,000 times faster than sound.

Problem 2:
→ The national debt is about (400 million) times the median household income.

Key Takeaways from Comparing Quantities

✓ "How many times" = divide (), not subtract

✓ Divide coefficients; subtract exponents:

✓ State every answer as a complete sentence with context

Watch out: Division → subtract exponents (not add)

Watch out: "Times" → divide; "more" → subtract

Next Lesson: All Four Operations in Scientific Notation

8.EE.A.4 preview:

• Multiplying:
• Adding requires aligning exponents first
• Mixed standard and scientific notation problems

Splitting coefficients and exponents transfers directly to multiplication. Addition adds one step: align powers of 10 before combining.