Introduction to Scientific Notation

Transcription

1 ACTIVITY 3.1 Introduction to Scientific Notation K anye, Corinne, and Brock wanted to know how many times their entire class has blinked in their lifetimes. Each student used a different technology device: Kanye used a basic calculator, Corinne used the calculator on her phone, and Brock used a graphing calculator. There are 25 students in the class. Kanye, Corinne, and Brock decided that, on average, the students each had blinked 98,112,000 times. 1. Analyze the display on each calculator. Corinne could also rotate her phone to see a display similar to the display on the graphing calculator. 2.45E+09 MC M+ M MR C ± = e9 c + / % = a. What was the total number of blinks for the entire class? Which display did you use to determine the total number? b. Use the total number of blinks to interpret each of the remaining displays. How are the displays similar? How are they different? LESSON 3: The Big and Small of It M5-45

2 Scientific notation is a tool to help you read and think about extremely large or small positive numbers. The num bers on the smaller displays represented the large number of blinks in scientific notation. Scienti fic notation is a notation used to express a very large or a very small number as the product of two numbers: a num ber that is greater than or equal to 1 and less than 10, and a pow er of 10. In gen e ral terms, a 3 10 n is a number written in scientific notation, where a is greater than or equal to 1 and less than 10, and n is any integer. The number a is called the mantissa, and n is the called the characteristic. Scient i fic notation makes it much easier to tell at a glance the order of magnitude. The order o f magnitude is an estimate of size expressed as a power of ten. For example, Earth s mass has an order of magnitude of kilograms. 2. Write the number of blinks from the calculator displays in scientific notation. Identify the mantissa and characteristic. What do you think the e in the displays means? M5-46 TOPIC 1: Exponents and Scientific Notation

3 3. Use your graphing or scientific calculator to explore extremely large and extremely small numbers. a. Enter each given number into your calculator and complete the table. Given Number Calculator Display What Does the Calculator Display Mean? 35,400,000,000 60,000,000,000, b. Describe the characteristics for extremely large numbers. c. Describe the characteristics for extremely small numbers. d. Describe the mantissa in each. LESSON 3: The Big and Small of It M5-47

4 4. Kanye, Corinne, Brock, and Daniel each tried to write the number 16,000,000,000 in scientific notation. Analyze each student s reasoning. Kanye s Method I start with 1.6, a number that is less than 10 and greater than 1. Next, I need a power of 10. If I multiply 1.6 by 10, I get 16. Then, if I multiply by 10 again, I get 160. Multiply by 10 again, and I get So, I can just keep multiplying by 10 until I get back to the original number. I have to multiply by 10 ten times, so my power of 10 is So, 16,000,000,000 in scientific notation is 1.6 x Brock s Method I have to write a number greater than 1 and less than 10 multiplied by a power of 10. So, I have to multiply 1.6 by a power of 10. Since there are 9 zeros, my power of 10 will be So, 16,000,000,000 is 1.6 x Daniel s Method Well, that number is 16 billion. And 16 billion is 16 times 1 billion. One billion has 9 zeros, so 16,000,000,000 in scientific notation is 16 x Corinne s Method Well, that number is 16 billion. And 16 billion is 16 times 1 billion. 16 x 1 is the same as 1.6 x 10, so 16 times 1 billion is the same as 1.6 times 10 billion. I have to multiply 10 ten times to get 10 billion, so my power of 10 is That means that 16 billion in scientific notation is 1.6 x M5-48 TOPIC 1: Exponents and Scientific Notation

5 a. Compare Brock s and Daniel s methods. NOTES b. Compare Kanye s and Brock s methods. c. Compare Daniel s and Corinne s methods. d. Of the correct methods, which method do you prefer? Why? LESSON 3: The Big and Small of It M5-49

6 ACTIVITY 3.2 Scientific Notation and Large Numbers In this activity, you will practice writing large numbers in either scientific notation or standard notation. 1. Write each number in the notation that is not given. a. There are approximately molecules in a gram of water. b. There are red blood cells in the human body. c. One light year is 5,880,000,000,000 miles. d. The speed of light is 186,000 miles per second. 2. The estimated populations, as of December 2016, of several countries are shown. Decide whether the number is written in scientific notation or standard notation. If the number is not in scientific notation, explain how you know it is not. Then, write the number in scientific notation. a. People s Republic of China: people b. Pitcairn Islands: 50 people c. Australia: people d. United States: people M5-50 TOPIC 1: Exponents and Scientific Notation

7 3. List the countries from Question 2 in order of population from least to greatest. Explain your strategy. 4. The primary U.S. currency note dispensed at an automated teller machine (ATM) is the 20-dollar bill. There are approximately 6 billion 20-dollar bills in circulation. a. Write the approximate number of 20-dollar bills in circulation in standard notation. b. Write the number of bills in scientific notation. c. Calculate the value of all the 20-dollar bills in circulation. d. Write the value you calculated in part (c) in scientific notation. LESSON 3: The Big and Small of It M5-51

8 ACTIVITY 3.3 Scientific Notation and Small Numbers Now, let s explore writing very small numbers using scientific notation. 1. A water molecule has an approximate length of 0.1 nanometer. One nanometer is 1 of a centimeter. Complete the statements 10 7 and answer the question. Write your answers as decimals. a. 1 nanometer 5 centimeter b. 0.1 nanometer 5 centimeter c. How many centimeters long is a string of 7 water molecules? Show your work. Just as with large numbers, scientific notation can be used to express very small numbers in a more compact form that requires less counting of zeros. The value of the number does not change, only how it is written. 2. Each student tried to write the number in scientific notation. Analyze each student s reasoning. Brock s Method I can start with 6.5, which is less than 10 and greater than 1. If I divide by 10, I get If I divide by 10 again, I get I just keep dividing by 10 until I get to the original number I divided by 10 four times. So, = 6.5. But in 10 4 scientific notation, I have to use multiplication, not division. That s okay because 6.5 is the same as x 1. And since 1 is 10-4, I can write in scientific notation as 6.5 x M5-52 TOPIC 1: Exponents and Scientific Notation

9 Corinne s Method I can write as a fraction less than 1. In words, that decimal is sixty-five hundred thousandths, so I could write it as ,000. If I divide both the numerator and denominator by 10, I get As a power of 10, the number 10,000 10,000 is written as So that s 6.5, which is the same 4 10 as 6.5 x 1 10, which is the same as 6.5 x That s the answer. Kanye s Method I moved the decimal point in the number to the right until I made a number greater than 1 but less than 10. So, I moved the decimal point four times to make 6.5. And since I moved the decimal point four times to the right, that s the same as multiplying 10 x 10 x 10 x 10, or So, the answer should be 6.5 x Daniel s Method I don t like decimals, so I moved the decimal point all the way to the right until I had a whole number. Because I moved the decimal point five times to make 65, that s the same as dividing by 10 five times. So, the answer in scientific notation should be 65 x a. Explain what is wrong with Kanye s reasoning. LESSON 3: The Big and Small of It M5-53

10 NOTES b. Explain what is wrong with Daniel s method. c. Of the correct methods, which method do you prefer? Why? There are names given to measurements smaller than a meter (m). You are familiar with the centimeter, the millimeter, and now the nanometer. These statements show how some small measurements relate to a meter: 1 centimeter (cm) 5 1 meter millimeter (mm) 5 1 meter micrometer (μm) 5 1 meter nanometer (nm) 5 1 meter picometer (pm) 5 1 meter Write each measurement as a power of 10. It is appropriate to have an expression with negative exponents in this question set. a. 1 centimeter b. 1 millimeter c. 1 micrometer d. 1 nanometer e. 1 picometer M5-54 TOPIC 1: Exponents and Scientific Notation

11 4. Write the radius of each type of blood vessel in standard form. a. The capillary is one of the minute blood vessels that connect arterioles and venules. The radius of a capillary is mm. b. The venule is a small blood vessel that allows deoxygenated blood to return from the capillaries to the veins. The radius of a venule is mm. People that work with very small or very large quantities, such as scientists or astronomers, use scientific notation to make numbers more reasonable to operate on and to compare. c. The arteriole is a small blood vessel that extends and branches out from an artery and leads to capillaries. The radius of an arteriole is mm. 5. Convert each measurement to meters, and then write the measurement in scientific notation. a. The diameter of a water molecule is 0.29 nanometers. b. The diameter of a red blood cell is 7 micrometers. c. The smallest microchip is 7 nanometers wide. d. A helium atom has a radius of 31 picometers. LESSON 3: The Big and Small of It M5-55

12 6. Complete the table shown. Object Measurement Measurement in Standard Form Measurement in Scientific Notation Earth Radius in meters m Brachiosaurus Mass in kilograms 77,100 kg Dust mite Length in meters m Nucleus of an atom Diameter in meters m ACTIVITY 3.4 Comparing Numbers in Scientific Notation You already know how to compare these numbers. Think about place value. Writing numbers in scientific notation is useful when comparing very large or very small numbers. 1. Compare each set of large numbers written in scientific notation using the appropriate symbol:,,., or 5. a b c d e f M5-56 TOPIC 1: Exponents and Scientific Notation

13 2. Explain how to compare two large numbers using scientific notation. NOTES 3. Compare each set of small numbers written in scientific notation using the appropriate symbol:,,., or 5. a b c d e f Explain how to compare two small numbers using scientific notation. 5. Describe the similarities and differences between the numbers and LESSON 3: The Big and Small of It M5-57