 Scientific Notation

New York State Common Core Math Grade 8, Module 1, Lesson 9.

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Student Outcomes
• Students write, add and subtract numbers in scientific notation and understand what is meant by the term leading digit.

Discussion (Magnitude of a Measurement)

Fact 1: The number 10n, for arbitrarily large positive integer n, is a big number in the sense that given a number M (no matter how big it is) there is a power of 10 that exceeds M.

Fact 2: The number 10-n, for arbitrarily large positive integer n, is a small number in the sense that given a positive number M (no matter how small it is), there is a (negative) power of that is smaller than M.

Discussion (Scientific Notation)

A positive, finite decimal s is said to be written in scientific notation if it is expressed as a product d x 10n where d is a finite decimal so that 1 ≤ d < 10, and n is an integer.

The integer n is called the order of magnitude of the decimal d x 10n

Discussion
Consider the estimated number of stars in the universe: 6 × 1022. This is a 23-digit whole number with the leading digit (the leftmost digit) 6 followed by 22 zeroes. When it is written in the form 6 × 1022, it is said to be expressed in scientific notation.

Example 1:
The finite decimal 234.567 is equal to every one of the following:
However, only the first is a representation of in scientific notation.

Exercises 1–6
Are the following numbers written in scientific notation? If not, state the reason.

Example 2:
Let’s say we need to determine the difference in the populations of Texas and North Dakota. In 2012, Texas had a population of about 26 million people and North Dakota had a population of about 6.9 x 104.
Example 3:
Let’s say that we need to find the combined mass of two hydrogen atoms and one oxygen atom, which is normally written as H2O or otherwise known as “water.”

Use the table below to complete Exercises 7 and 8.
The table below shows the debt of the three most populous states and the three least populous states.
Exercise 7
a. What is the sum of the debts for the three most populous states? Express your answer in scientific notation.
b. What is the sum of the debt for the three least populated states? Express your answer in scientific notation.
c. How much larger is the combined debt of the three most populated states than that of the three least populous states? Express your answer in scientific notation.

Exercise 8
a. What is the sum of the population of the 3 most populous states? Express your answer in scientific notation.
b. What is the sum of the population of the 3 least populous states? Express your answer in scientific notation.
c. Approximately how many times greater is the total population of to the total population of North Dakota, Vermont, and Wyoming?

Exercise 9
All planets revolve around the sun in elliptical orbits. Uranus’s furthest distance from the sun is approximately 3.004 × 109 km, and its closest distance is approximately 2.749 × 109 km. Using this information, what is the average distance of Uranus from the sun?

Exercises

Exercise 1
Let M = 993, 456, 789, 098, 765. Find the smallest power of 10 that will exceed M.

Exercise 4
The chance of you having the same DNA as another person (other than an identical twin) is approximately 1 in 10 trillion (one trillion is a 1 followed by 12 zeros). Given the fraction, express this very small number using a negative power of 10.

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