Lesson 8: Magnitude. Students know that positive powers of 10 are very large numbers, and negative powers of 10 are very small numbers.

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1 Student Outcomes Lesson 8: Magnitude 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. Magnitude used to easily compare very large and very small numbers The order of magnitude of a number is its exponent in scientific notation. For example, 703 = 7.03 x 10 2 : order of magnitude is 2 600,000 = 6 x 10 5 : order of magnitude is = 9.5 x 10 2 : order of magnitude is 2 1

2 Every number between 10 and 100 has an order of magnitude of = 1 x = 6.5 x = 8.7 x = 4.3 x = 7.7 x = 9.9 x 10 1 Every number between 100 and 1,000 has an order of magnitude of = 1 x = 4.3 x 650 = 6.5 x = 7.7 x = 8.70 x = 9.9 x 10 2 Which is the larger value? How can you tell? 6.5 x x 10 2 Which is the larger value: 3 x 10 5 or 2 x 10 6? 2

3 ,000 10, ,000 1,000,000 Between what two powers of 10 do the numbers fall? 899 between and 78,140 between and 8.57 between and between and 245, between and Do you notice a quick way to figure this out?? What type of numbers fall between 0 and 1?? (base 10) = = = = = = The numbers keep decreasing but NEVER get as small as ZERO!!! 3

4 Example 1: Let M be the world population as of March 23, Approximately, M = 7,073,981,143. How many digits does it have? It is smaller than a whole number with digits. What power of 10 exceeds this M? Example 2: Let M be the US national debt as of March 23, M = 16,755,133,009,522 to the nearest dollar. How many digits does it have? It is smaller than a whole number with digits. What power of 10 exceeds this M? 4

5 Exercise 1 Let M = 993,456,789,098,765. Find the smallest power of 10 that will exceed M. Exercise Let M = 78,491. Find the smallest 987 power of 10 that will exceed M. back to note sheet 5

6 Example 3a: The average ant weighs about grams. We want to express this number as a power of 10. As a fraction it would be In scientific notation: Example 3b: The mass of a neutron is kg We want to express this number as a power of 10. How many places after the decimal is needed to reach the first natural number? Will the exponent be positive or negative? 6

7 In general, numbers with a value less than 1 but greater than 0 can be expressed using a negative power of 10. The closer a number is to zero, the smaller the power of 10 that will be needed to express it. Exercise 3: Let M be a positive integer. Explain how to find the smallest power of 10 that exceeds it. 7

8 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 zeroes). Given the fraction, express this very small number using a negative power of ,000,000,000,000 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 Which do you have a greater chance of experiencing? Explain. 8

9 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. Summary of the lesson: 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. 9