Scientific Notation: Definition and Estimation

Lesson 1 of 2

In this lesson:

• Express numbers in the form where
• Convert fluently between standard form and scientific notation
• Estimate real-world quantities as a single digit times a power of 10

Learning Objectives for This Deck

1. Express very large and very small numbers as (, an integer)
2. Convert fluently between standard form and scientific notation, magnitudes to
3. Estimate real-world quantities by rounding to a single digit times a power of 10

Three Numbers That Are Hard to Read

How many zeros are in each number below?

• 93,000,000 — distance from Earth to Sun, in miles
• 0.000000001 — diameter of a molecule, in meters
• 7,900,000,000 — world population

Are you sure you counted correctly? Would you bet on it?

Scientific Notation: Structure and Definition

Form: , where and is an integer

• Coefficient : captures significant digits
• Power of 10: captures the scale

Examples: , ,

Coefficient and Exponent: Labeled Parts

• Coefficient : must satisfy — exactly one non-zero digit before the decimal point
• Exponent : positive → large number ()

• Exponent : negative → small number ()

Converting Large Numbers to Scientific Notation

Procedure: Move the decimal left until coefficient is between 1 and 10. Places moved = exponent.

7 places left; number is large → exponent is positive.

3 places left → exponent is 3.

Converting Small Numbers to Scientific Notation

Procedure: Move the decimal right until coefficient is between 1 and 10. Places moved = absolute value of exponent (negative).

4 places right; small number → exponent is negative.

9 places right → exponent is −9.

Converting Scientific Notation to Standard Form

Procedure: Positive exponent → move decimal right (large number). Negative exponent → move decimal left (small number).

Move decimal 5 places right.

Move decimal 3 places left.

Which Expression Is Proper Scientific Notation?

All three equal 320,000:

Decimal Direction Depends on the Exponent Sign

• Positive exponent → large number → decimal moves right to restore standard form
• Negative exponent → small number → decimal moves left to restore standard form

Chunk One Key Rules and Conversion Summary

Form: , coefficient

Converting:

• Large number → decimal left → positive exponent
• Small number → decimal right → negative exponent

Watch out:

• Negative exponent → decimal moves left to restore
• Verify coefficient is between 1 and 10

From Exact Notation to Single-Digit Estimates

The standard requires: a single digit times an integer power of 10

• US population: (exact scientific notation)
• Single-digit estimate: ← what 8.EE.A.3 requires

Why? One significant figure enables mental math and magnitude comparison.

Three Steps for Estimating Any Quantity

Step 1: Write the number in scientific notation

Step 2: Round the coefficient to one digit

Step 3: Write the single-digit estimate

Examples: Estimating Large Real-World Quantities

Examples: Estimating Small Real-World Quantities

When the Coefficient Rounds Up to Ten

Problem: mass of a bacterium ≈ g

Step 2: Round 9.5 to one digit → rounds to 10

But violates !

Fix:

When rounding pushes the coefficient to 10, increase the exponent by 1.

Powers of Ten Reference Chart

Use this chart to check whether your estimates make sense.

Check In: Estimate This Earth-Mars Distance

The distance from Earth to Mars at its closest approach is about 55,000,000 km.

Express this as a single digit times a power of 10.

Work it out before advancing...

Answer: km

Practice: Estimate These Real-World Quantities

Express as a single digit × power of 10:

1. Speed of light: about 300 million m/s
2. Visible light wavelength: 0.00000055 m
3. National debt: 27 trillion dollars

Practice Answers: Estimation with Scientific Notation

1. Speed of light: m/s
2. Visible light: m
3. National debt:

Key Takeaways from Definition and Estimation

✓ , where , an integer

✓ Positive exponent → large; negative → small

✓ Estimate: convert → round coefficient → check

Negative exponent → decimal moves LEFT

Coefficient after rounding → exponent

Next Lesson: Comparing Quantities with Scientific Notation

Lesson 2 preview:

• "How many times as much is A than B?" → divide, not subtract
• Connection to 8.EE.A.1: the quotient rule for exponents

Prerequisite: fluency with single-digit estimation from today's lesson