Scientific Notation: Writing and Converting Numbers

Content Standards and Learning Competencies

Content Standards

The learners should have knowledge and understanding of operations using scientific notation. (MG)

Performance Standards

By the end of the quarter, the learners are able to write numbers in scientific notation and perform operations on numbers written in scientific notation.

Learning Competencies

At the end of the lesson, the learners are expected to:

Write numbers in scientific notation to represent very large or very small numbers, and vice versa.

Perform operations on numbers expressed in scientific notation.

Activating Prior Knowledge

Activity 1: Do You Remember?

Express the following into figures and identify the number of significant figures in each item:

Examples

1. Five hundred forty-five thousand

2. Eighty-seven ten-thousandths

3. One hundred fifty million

4. One millionth

5. Seven hundred twenty-five and four hundredths

Purpose

This activity helps learners recall the proper way of writing words into figures including decimals and to recall the concept of significant figures.

Answer Key for Activity 1

Each figure shows the proper representation and significant figures for the given word form.

Five hundred forty-five thousand

Figure: 545,000

Significant Figures: 3

Eighty-seven ten-thousandths

Figure: 0.0087

Significant Figures: 2

One hundred fifty million

Figure: 150,000,000

Significant Figures: 2

One millionth

Figure: 0.000001

Significant Figures: 1

Seven hundred twenty-five and four hundredths

Figure: 725.04

Significant Figures: 5

Lesson Purpose: Too Big, Too Small?

Some quantities in our world are extremely large or extremely small. Let's visualize these examples:

Distance from Earth to Sun

150,000,000,000 meters

Mass of average human cell

0.000000000001 kilograms

Diameter of the moon

3,474,000 meters

Temperature of the core of the sun

15,700,000 Kelvin

Diameter of an electron

0.00000000000000282 meters

Guide Questions:

What do you notice with these extremely large and small numbers?

Do you think there is a way to express each number in a shorter way?

Introduction to Scientific Notation

Scientific Notation is a system used to express very large or very small numbers conveniently. It uses exponents to avoid writing many zeros, which can be confusing and lead to errors.

Purpose

Express very large or very small numbers in a compact, manageable form without writing numerous zeros

Example: 150,000,000,000 meters (Earth to Sun) becomes 1.5 × 10¹¹ meters

Format

Written as a × 10ⁿ, where 1 ≤ a < 10

The coefficient (a) must be at least 1 but less than 10, while the exponent (n) is an integer

Components

Coefficient (a): A number between 1 and 10 that provides precision

Exponent (n): Shows how many places to move the decimal point, indicating magnitude

Application

Used extensively in science, astronomy, physics, and other fields that deal with extremely large or small quantities

Examples: Mass of electrons (0.00000000000000000000000000000091 kg) or distance to stars

Why Use Scientific Notation?

Numbers can be extremely large or small in science. Scientific notation provides a concise way to express these values.

Large Numbers in Standard Form

Distance between Earth and the Sun: 149,600,000 km

Writing many zeros is inefficient and error-prone.

Large Numbers in Scientific Notation

Distance between Earth and the Sun: 1.496 × 10⁸ km

More concise and reduces the chance of errors.

Small Numbers in Standard Form

Size of a bacteria: 0.0000005 m

Counting decimal places becomes difficult.

Small Numbers in Scientific Notation

Size of a bacteria: 5 × 10⁻⁷ m

Simplifies representation of very small quantities.

Understanding Scientific Notation

Scientific notation is a way of writing numbers as the product of two factors. The coefficient "a" must be at least 1 but less than 10, while the exponent "n" indicates how many places the decimal point moves.

Basic Formula

a × 10ⁿ is the standard form for scientific notation, where "a" is multiplied by a power of 10

Coefficient (a)

A number greater than or equal to 1 but less than 10 (1 ≤ a < 10) that provides precision

Exponent (n)

An integer (positive or negative) representing the number of places the decimal moves, indicating magnitude

Converting a Large Number to Scientific Notation

Example: Write 5,600,000 in scientific notation.

Step 1: Place the decimal after the first nonzero digit

For 5,600,000, place the decimal after 5: 5.6

Step 2: Count how many places the decimal moved

The decimal moved 6 places to the left

Step 3: Write in the form a×10ⁿ

5.6 × 10⁶

Converting a Small Number to Scientific Notation

Example: Write 0.00042 in scientific notation.

Step 1: Place the decimal after the first nonzero digit

For 0.00042, place the decimal after 4: 4.2

Step 2: Count how many places the decimal moved

The decimal moved 4 places to the right

Step 3: Use a negative exponent

The result is 4.2 × 10⁻⁴, where 4.2 is the coefficient and -4 is the negative exponent indicating the decimal moved 4 places to the right.

Converting Scientific Notation Back to Standard Form

For Positive Exponents

Move the decimal right by the number of places indicated by the exponent.

Example: 3.5 × 10⁴ = 35,000

The decimal moves 4 places to the right.

For Negative Exponents

Move the decimal left by the number of places indicated by the exponent.

Example: 7.8 × 10⁻³ = 0.0078

The decimal moves 3 places to the left.

Practice Activity: Matching

Match each number in standard form with its equivalent in scientific notation.

4,500,000

Standard Form

Converts to: 4.5 × 10⁶

0.0023

Standard Form

Converts to: 2.3 × 10⁻³

67,000

Standard Form

Converts to: 6.7 × 10⁴

0.000000078

Standard Form

Converts to: 7.8 × 10⁻⁸

3,450,000

Standard Form

Converts to: 3.45 × 10⁶

Evaluation: Multiple Choice Questions

Which of the following is the correct scientific notation for 5,600,000?

A) 56 × 10⁵

B) 5.6 × 10⁶

C) 0.56 × 10⁷

D) 560 × 10⁴

What is 0.00042 written in scientific notation?

A) 4.2 × 10³

B) 4.2 × 10⁻⁴

C) 4.2 × 10⁻³

D) 4.2 × 10⁻⁵

Which number is equivalent to 6.1 × 10⁴?

A) 61,000

B) 6,100

C) 610,000

D) 610

Evaluation: Multiple Choice Questions (Continued)

Test your understanding of scientific notation with these practice problems.

Convert 2.5 × 10⁻³ to standard form.

A) 2500

B) 0.0025

C) 0.00025

D) 25

Which of the following is NOT a correct scientific notation?

A) 8.9 × 10⁵

B) 3.45 × 10⁻²

C) 12 × 10⁴

D) 6.7 × 10³

Answer: B - When converting from scientific notation with a negative exponent, move the decimal point to the left by the number of places indicated by the exponent.

Previous Question Answers:

Answer: C - In scientific notation, the coefficient must be greater than or equal to 1 and less than 10. The coefficient 12 is greater than 10, making this incorrect.

1. B 2. D

3. B

Additional Activities: Converting to Scientific Notation

Write the following numbers in scientific notation:

Step 1: Move the Decimal Point

Count the number of places the decimal needs to move to get a number between 1 and 10.

Example: 7,800,000 → 7.8 × 10^6

Step 2: Determine the Exponent

For numbers < 1, count places to the right and use negative exponent.

Example: 0.000093 → 9.3 × 10^-5

Step 3: Write in Scientific Form

Express as a coefficient (1-10) multiplied by 10 raised to an exponent.

Example: 125,000,000 → 1.25 × 10^8

Step 4: Verify Your Solution

Check that your coefficient is between 1 and 10 and your exponent is correct.

Practice on: 0.00000052 and 43,500

Problems to solve:

7,800,000

0.000093

125,000,000

0.00000052

43,500

Additional Activities: Converting from Scientific Notation

Write the following scientific notation numbers in standard form:

Step 1: Write in standard form

Identify the coefficient and exponent in scientific notation

Example: In 5.6 × 10⁵, 5.6 is the coefficient and 5 is the exponent

Step 2: Move the decimal point

For positive exponents: Move decimal right by the exponent value

For negative exponents: Move decimal left by the exponent value

Step 3: Verify your answers

Check your work by converting back to scientific notation

Ensure you've followed the correct decimal movement rules

Practice problems:

5.6 × 10⁵

3.9 × 10⁻³

2.45 × 10⁷

6.1 × 10⁻⁶

9.75 × 10³

Common Errors in Scientific Notation

Coefficient Error

The coefficient must be at least 1 but less than 10.

Incorrect: 12 × 10⁴

Correct: 1.2 × 10⁵

Exponent Error

The exponent must be positive for large numbers and negative for small numbers.

Incorrect: 5.6 × 10⁻³ (for 5,600)

Correct: 5.6 × 10³

Decimal Placement Error

The decimal must be placed after the first non-zero digit.

Incorrect: 56.0 × 10²

Correct: 5.6 × 10³

Real-World Applications of Scientific Notation

Astronomy

Distance to stars: The distance to Proxima Centauri is 4.0 × 10¹⁶ meters

Physics

Mass of an electron: 9.11 × 10⁻³¹ kilograms

Biology

Size of DNA molecule: 2.5 × 10⁻⁹ meters wide

Economics

National debt: 3.1 × 10¹² dollars

Synthesis: What Have You Learned?

Purpose of Scientific Notation

Reflect on why we use scientific notation and how it provides a standardized way to express very large or very small numbers.

Converting Between Forms

Consider the steps for converting between standard form and scientific notation, and how moving the decimal point relates to the exponent.

Real-World Applications

Think about how fields like astronomy, physics, and biology rely on scientific notation to express measurements that would be unwieldy in standard form.

Mathematical Advantage

Reflect on how scientific notation simplifies calculations and helps avoid errors when working with numbers that have many zeros.