Scientific Notation in Mathematics
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
Problem 1
Solve: (4.2 x 106) + (3.8 x 106)
Answer: 8.0 X 106
Problem 2
Solve: (7.5 × 104) − (2.3×104)
Answer: 5.2 X 104
Problem 3
Solve: (9.6 x 105) + (4.4 x 104)
Answer: 1.004 X 106
Problem 4
Multiply: (3.0 x 103) x (2.0 x 104)
Answer: 6.0 X 107
Problem 5
Multiply: (5.2 x 106) x (4.0 x 102)
Answer: 2.08 X 109
Lesson Purpose and Intention
Understand Scientific Notation
Understand the importance of scientific notation in representing extremely large and small numbers in a compact and manageable way.
Learn Operation Rules
Learn and apply the rules for addition, subtraction, multiplication, and division of numbers in scientific notation.
Master Exponent Rules
Strengthen their understanding of exponent rules to correctly manipulate powers of ten in different operations.
Develop Problem-Solving Skills
Develop accuracy and efficiency in solving real-world mathematical problems involving scientific notation.
Apply to Real-World Scenarios
Enhance problem-solving and critical thinking skills by applying scientific notation in fields like space exploration, microbiology, and economics.
Key Mathematical Terminology
Notation
A system of symbols used to represent numbers or expressions, such as scientific notation for writing large or small numbers efficiently.
Decimal Point
A dot used to separate the whole number part from the fractional part of a number. In scientific notation, the decimal point is placed after the first nonzero digit.
Order of Magnitude
A way to compare numbers based on powers of ten, indicating how many times larger or smaller one number is than another.
Positive Exponent
An exponent greater than zero, indicating multiplication by powers of ten. Example: 10³ = 1,000
Negative Exponent
An exponent less than zero, indicating division by powers of ten. Example: 10⁻³ = 0.001.
Exponentiation
The mathematical operation of raising a base to a power, such as 2⁵ = 32.
More Mathematical Terminology
Significant Figures (Sig Figs)
The digits in a number that contribute to its precision, including all nonzero digits and any zeros between them or after a decimal.
Precision
The level of detail in a numerical value, often determined by the number of significant figures in scientific notation.
Rounding
Adjusting a number to a specified degree of accuracy by removing less significant digits, often used when converting to scientific notation.
Scaling Factor
The number by which a quantity is multiplied or divided, often expressed in scientific notation as a power of ten.
Exponent Shift
The process of adjusting the exponent when converting numbers to scientific notation by moving the decimal point.
Advanced Mathematical Terminology
Key concepts and applications in scientific notation and related mathematical fields:
Astronomical Numbers
Extremely large numbers used in fields like astronomy to measure distances in space, such as the number of stars in a galaxy.
Microscopic Numbers
Extremely small numbers used in sciences like biology and chemistry to measure atomic and molecular sizes.
Dimensional Analysis
A method of converting between units using scientific notation, commonly applied in physics and engineering.
Scientific Calculator
A specialized calculator that simplifies calculations involving scientific notation, exponents, and large numbers.
Importance of Scientific Notation
Scientific notation provides an efficient way to represent and work with very large or very small numbers.
Efficiency in Representation
Numbers can be extremely large (e.g., the distance between planets) or small (e.g., the size of bacteria). Scientific notation simplifies this by expressing numbers in a compact form that makes calculations easier.
Handling Microscopic Values
Scientific notation is essential when working with microscopic measurements that would otherwise require many decimal places, making them difficult to read and calculate.
Definition of Scientific Notation
A number is written in scientific notation using the format: a × 10n
a (the coefficient) is a decimal number between 1 and 10.
10n (power of ten) indicates how many times the decimal point is moved.
n is positive for large numbers and negative for small numbers.
Examples of Scientific Notation
Earth's Mass (kg)
Standard form: 5,970,000,000,000,000,000,000,000 kg
Scientific notation: 5.97 × 10^24 kg
Scientific notation makes this enormous number manageable
Human Hair Width (m)
Standard form: 0.00008 m
Scientific notation: 8.0 × 10^-5 m
Scientific notation simplifies this tiny measurement
Addition and Subtraction in Scientific Notation
Align the Exponents
The exponents must be the same before adding or subtracting the coefficients.
If the exponents are different, adjust one number by shifting the decimal point.
Add or Subtract the Coefficients
Once the exponents are the same, add or subtract the coefficient values.
Write the Final Answer
Express the result with the coefficient between 1 and 10, adjusting the exponent if necessary.
Examples of Addition and Subtraction
Example 1: Same Exponent
(2.5 x 10⁶) + (3.7 x 10⁶)
Add the coefficients: 2.5 + 3.7 = 6.2
Final Answer: 6.2 × 10⁶
Example 2: Different Exponents
(4.2 x 10⁵) + (2.1 x 10³)
Convert 2.1 x 10³ to 0.021 x 10⁵
Add: (4.2 + 0.021) x 10⁵ = 4.221 x 10⁵
Multiplication in Scientific Notation
Follow these steps to multiply numbers in scientific notation:
Step 1: Multiply the Coefficients
Multiply the decimal parts of the numbers together.
Example: 3.0 × 2.0 = 6.0
Step 2: Add the Exponents
Add the powers of 10 together.
Example: 10^4 × 10^3 = 10^(4+3) = 10^7
Step 3: Express in Standard Form
Ensure the coefficient is between 1 and 10, adjusting the exponent if necessary.
Example: 6.0 × 10^7 is already in standard form
Division in Scientific Notation
Step 1: Divide the Coefficients
Divide the decimal parts of the numbers.
Example: 8.0 ÷ 2.0 = 4.0
Step 2: Subtract the Exponents
Subtract the powers of 10 from each other.
Example: 9 − 3 = 6
Step 3: Express in Standard Form
Ensure the coefficient is between 1 and 10, adjusting the exponent if necessary.
Example: 4.0 × 10^6
Complete Example: (8.0 × 10^9) ÷ (2.0 × 10^3) = 4.0 × 10^6
Exponent Rules in Scientific Notation
Power of a Power Rule
(10a)b = 10a×b
When raising a power to another power, multiply the exponents
Division Rule
10a ÷ 10b = 10a-b
When dividing powers with the same base, subtract the exponents
Multiplication Rule
10a × 10b = 10a+b
When multiplying powers with the same base, add the exponents
Real-World Applications of Scientific Notation
Astronomy
The distance from the Earth to the Sun is 149,600,000 km, written as 1.496×10⁸ km.
Biology
A bacterial cell's diameter is about 0.000002 m, written as 2.0 × 10⁻⁶ m.
Economics
The U.S. national debt is about $33,000,000,000,000, written as 3.3 x 10¹³ dollars.
Practice Activity: Converting to Scientific Notation
Large Numbers
56,700,000 = 5.67 × 107
Move the decimal point 7 places to the left
Small Numbers
0.000045 = 4.5 × 10-5
Move the decimal point 5 places to the right
Very Large Numbers
1,230,000,000 = 1.23 × 109
Move the decimal point 9 places to the left
Very Small Numbers
0.00000082 = 8.2 × 10-7
Move the decimal point 7 places to the right
Medium-Sized Numbers
72,500 = 7.25 × 104
Move the decimal point 4 places to the left
Reflection Questions
Key Concepts
What are the key concepts of our lesson?
Easiest Part
Which part of the lesson is the easiest for you? Why?
Challenging Part
Which part of the lesson is the hardest for you? Why?
Class Experience
How are we as a class today?
Evaluation: Multiple Choice Questions
Direction: Choose the letter of the correct answer.
Question 1: Difficulty Level 3
What is the product of (3.2 x 105) x (2.5 x 103)?
Answer: C. 8.0×107
Question 2: Difficulty Level 3
What is the quotient of (6.4 x 106) ÷ (8.0 x 102)?
Answer: A. 8.0×103
Question 3: Difficulty Level 4
What is the sum of (4.5 × 104) + (3.2 × 103)?
Answer: A. 4.82×104
Question 4: Difficulty Level 2
What is the difference of (5.6 × 106) − (2.3 × 106)?
Answer: A. 3.3×106
Question 5: Difficulty Level 4
What is the result of (9.0 × 107) × (3.0 × 102)?
Answer: A. 2.7×1010
Lesson Summary and Next Steps
Understanding Scientific Notation
Mastered the format a × 10ⁿ for representing large and small numbers
Performing Operations
Learned rules for addition, subtraction, multiplication, and division
Real-World Applications
Explored how scientific notation is used in various fields
Future Learning
Prepared for advanced applications in science and mathematics