 # Magnitude

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Lesson Plans and Worksheets for Grade 8
Lesson Plans and Worksheets for all Grades

Examples, solutions, lessons, and videos to help Grade 8 students learn that the exponent of an expression provides information about the magnitude of a number.

### Lesson 7 Student Outcomes

• Students know that positive powers of 10 are very large numbers, and negative powers of 10 are very small numbers.
• Students know that the exponent of an expression provides information about the magnitude of a number.

Lesson 7 Summary

• No matter what number is given, we can find the smallest power of 10 that exceeds that number.
• Very large numbers have a positive power of 10.
• We can use negative powers of 10 to represent very small numbers that are less than one, but greater than zero.

Classwork

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 S (no matter how small it is), there is a (negative) power of 10 that is smaller than S.
Example 1:
Let M be the world population as of March 23, 2013. Approximately, M = 7,073,981,143. It has 10 digits and is, therefore, smaller than any whole number with 11 digits, such as 10,000,000,000. But 10,000,000,000 = 1010, M < 1010 so (i.e., the 10th power of 10 exceeds this M).
Example 2:
Let M be the US national debt on March 23, 2013. M = 16,755,133,009,522 to the nearest dollar. It has 14 digits. The largest 14-digit number is 99,999,999,999,999. Therefore,
M < 99,999,999,999,999 < 100,000,000,000,000 = 1014 That is, the 14th power of exceeds M.
Exercise 1
Let M = 993,456,789,098,765. Find the smallest power of 10 that will exceed M.
Exercise 2
Let M = 78,491 899/987. Find the smallest power of 10 that will exceed M.
Exercise 3
Let be a positive integer. Explain how to find the smallest power of 10 that exceeds it.
Example 4
• The average ant weighs about 0.0003 grams.
• The mass of a neutron is 0.000 000 000 000 000 000 000 000 001 674 9 kilograms.
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.
Exercise 5
The chance of winning a big lottery prize is about 10-8, and the chance of being struck by lightning in the US in any given year is about 0.000001. Which do you have a greater chance of experiencing? Explain.
Exercise 6
There are about 100 million smartphones in the US. Your teacher has one smartphone. What share of US smartphones does your teacher have? Express your answer using a negative power of 10.

Powers of Ten
Powers of Ten takes us on an adventure in magnitudes.
Starting at a picnic by the lakeside in Chicago, this famous film transports us to the outer edges of the universe.
Every ten seconds we view the starting point from ten times farther out until our own galaxy is visible only a s a speck of light among many others.
Returning to Earth with breathtaking speed, we move inward- into the hand of the sleeping picnicker- with ten times more magnification every ten seconds.
Our journey ends inside a proton of a carbon atom within a DNA molecule in a white blood cell.
In addition to applications within mathematics, exponential notation is indispensable in science. It is used to clearly display the magnitude of a measurement (e.g., How big? How small?).
What does it mean to say that 10n for large positive integers n are big numbers and that 10-n for large positive integers n are small numbers?

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