Scientific Notation in Context: Part 2

Mixed Notation, Calculator Displays, and Applications

In this lesson:

• Compute with mixed-format numbers
• Read calculator displays in "E" notation
• Choose appropriate units for very large or very small quantities

What You Will Learn in Part Two

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

3. Compute with mixed-format expressions — some in scientific notation, some in decimal form
4. Interpret calculator "E" notation and convert to standard scientific notation
5. Choose units of appropriate size for very large or very small measurements

Real Problems Use Mixed Number Formats

Numbers appear in whatever format is convenient:

• Blood cell diameter: m (scientific notation)
• Sand grain diameter: m (decimal form)

Convert to a common format before computing. Scientific notation is usually the better choice.

Example: Blood Cell vs. Grain of Sand

How many times wider is a grain of sand ( m) than a blood cell ( m)?

Convert:

Divide:

About 286 times wider.

Example: Mixed Subtraction — Factory Bolts

A factory produces bolts. Rejects: .

Convert:

Subtract:

The rejected quantity barely changes the total. The vs. gap reveals that 4,500 is tiny compared to 3.2 million.

Calculator "E" Notation Means Times Ten

• 3.5E8 means
• 2.1E-5 means
• -4.7E3 means

"E" = "times 10 to the power of" — not "exponent base"

Practice: Translating Calculator E Displays

Try It: What Does 6.02E23 Mean?

A calculator displays: 6.02E23

• Write this in standard scientific notation.
• Write it in words.
• Is this number very large, very small, or ordinary?

Pause and answer before the next slide.

Answer: This Is Avogadro's Famous Number

6.02E23 =

In words: "6.02 times 10 to the twenty-third"

This is Avogadro's number — atoms per mole. Compare:

vs.

Choosing the Right Unit for Measurements

What makes a unit a good choice?

• Coefficient stays in a manageable range (roughly 1–1,000)
• Unit is familiar to the audience
• The combination is easy to visualize

Visual: Same Rate in Four Different Units

The Mid-Atlantic Ridge spreads at about 2.5 cm/year.

Which unit would you use in a geology report? Why?

Example: Seafloor Spreading Over Ten Million Years

Rate: m/yr. Time: yr. Distance?

Best unit: km — 250 km is intuitive; 250,000 m is not.

Try It: Convert a Virus Width to Nanometers

A virus is meters wide.

Express this in nanometers, where .

Which unit is better for describing the virus — meters or nanometers? Why?

Work it out, then continue.

Answer: 250 nm Is the Better Unit

Nanometers is better — puts the coefficient in a readable range. requires interpreting a large negative exponent.

Multi-Step: Computing Earth's Average Density

Mass: kg. Volume: km³.

A km³ is enormous — scientists report density in g/cm³.

Earth's average density ≈ 5.5 g/cm³.

Multi-Step Example: Computer Calculations Part One

Computer A performs calculations per second.

How many calculations in one day ( s)?

About calculations per day.

Multi-Step Example: Computer Calculations Part Two

Computer B performs calculations per day.

Computer A: per day

How do they compare? Find the difference.

Computer A outperforms Computer B by about calculations per day.

Example: How Many Nanotubes Fit Across Hair?

Hair: m. Nanotube: m.

About 70,000 nanotubes fit across one hair.

Independent Practice: Two Multi-Step Problems

1. Debt per person: National debt ≈ $; population ≈ . What is the debt per person?

2. Reservoir: Holds liters; storm adds liters. New total?

Show all steps before the next slide.

Answers: Debt per Person and Reservoir Total

Problem 1 (Division):

≈ $94,000/person

Problem 2 (Addition):

Key Takeaways: All Four Operations in One Table

Always check: is the coefficient between 1 and 10?

Up Next: Proportional Relationships and Slope

8.EE.B.5 — Proportional Relationships and Slope

You will explore:

• Graphs of proportional relationships
• Comparing unit rates using slope
• Connecting to constant rate of change

The skills you practiced here — working with numbers in any form — will support every quantitative topic ahead.