Understanding Scientific Notation and Significant Digits: A Student Exploration | Exams Nursing | Docsity (2024)

Download Understanding Scientific Notation and Significant Digits: A Student Exploration and more Exams Nursing in PDF only on Docsity! Student Exploration: Unit Conversions 2 – Scientific Notation and Significant Digits Latest Questions And Answers Update 2024 Note to teachers and students: This lesson is designed to be a follow-up to the Unit Conversions Student Exploration sheet. The same Gizmo is used for both activities.] Directions: Follow the instructions to go through the simulation. Respond to the questions and prompts in the orange boxes. Vocabulary: resolution, scientific notation, significant digits Prior Knowledge Questions (Do these BEFORE using the Gizmo.) Philip measures a room using his feet. (His feet are each about a foot long.) He estimates the room measures about ten and a half feet by thirteen and a half feet. He calculates the room’s area to be: 10.5’ × 13.5’ = 141.75 ft2. 1. Which do you think is the best description of the area of the room? Highlight your choice. A. The room’s area is exactly 141.75 ft2. B. The room’s area is about 142 ft2. Gizmo Warm-up When scientists report a value, they have to take several things into consideration. Sometimes values are very large or small. In this case, scientists can use shorthand called scientific notation to report the value. Scientists also must consider how precise their value is. The precision of a measurement can be shown by the number of significant digits in the value. To begin, check that the Burj Khalifa question is shown. Drag the three tiles shown below to determine the tower’s height in micrometers. The answer is given in scientific notation. 1. The Burj Khalifa is 828,000,000 micrometers tall. How is this value written in scientific notation in the 2. In a value, any non-zero digit is considered a significant digit. (Zeroes may or may not be significant.) 828 8.2 * 10^8 I chose that answer because it the answer but rounded up to 3 sig fig 2. Explain your choice: Gizmo™? What is the minimum number of significant digits in 828,000,000? Activity A: Scientific notation Get the Gizmo ready : ● Select Metric units only and Distance from the Conversion menu. Make sure Show result is off. ● Click Next until you reach the question about Proxima Centauri. Question: How can you convert numbers into and out of scientific notation? 1. Observe : Some of the problems in this Gizmo involve very small or very large quantities. Look at the bottom three Unit Conversion Tiles. What do you notice in the numerator? The numbers in the numerators are written in scientific notation. In scientific notation, a number is converted to the product of a number between 1 and 10 and a power of 10. For example, 1,000,000 is written as 1.0 • 106. The first part of this number is called the coefficient. The second part is called the base. 2. Convert : To convert a number written in scientific notation into a standard number, first look at the exponent on the base. If it is positive, move the decimal point on the coefficient to the right as many times as the exponent indicates, as shown below: Look at exponent Count digits Move decimal point Standard form 8.35 × 107 1 234 567 8.3 500 000 83 500 000.0 83,500,000 Practice converting the two numbers below into standard form: 1.0 • 109 = 10000000000 6.72 • 1012 = 672,000,000,000,000. You can perform this process in reverse to convert numbers in standard form into scientific notation. The number of times you move the decimal point to the left will be equal to the exponent on your base. Remember there should be only one digit to the left of the decimal point in scientific notation. Standard form Place new decimal point Count digits Scientific notation 3,700,000 3.700 000 123 456 3.700 000 3.7 × 106 Practice this with the two numbers below: 8,200,000 8.2*10^6 50,880,000,000,000 5.088*10^13 The number is low Activity B: Significant digits Get the Gizmo ready : ● Select Distance and Metric units only. ● If necessary, click Next until you reach the question about Burj Khalifa. Introduction: When you report a measurement, you are really doing two things. Not only are you communicating the value of the measurement, but you are also stating something about the resolution, or fineness, of your measurement. This is conveyed by the number of significant digits in the value. Question: What digits are significant, and why? 1. Think about it: Mark measures the volume of water in a beaker marked in 50-mL intervals. He reports the volume is 43.927 mL. What is misleading about this value? Ideally, reported measurements should reflect the resolution of the instrument used to obtain the data. With Mark’s beaker, a better value might be “about 50-mL.” This is done by specifying the number of significant digits in the data. A significant digit is any value that has been directly measured. Other digits are not significant. Here are some rules for determining whether a digit is significant: Significant Digit Rules and Examples 1. Any non-zero digit is significant: 227.4 has four significant digits. 2. Any digit that is between other significant digits is significant: 200.08 has five significant digits. 3. Zeros to the right of a significant digit and to the right of a decimal point are always significant: 6.00 has three significant digits. 4. Zeros used to space a number to the right of a decimal point are not significant: 0.000147 has only three significant digits. 5. In scientific notation, all digits in the coefficient are significant: 8.75 × 105 has three significant digits. 6. Zeros to the right of a significant digit but to the left of a decimal point may or may not be significant: 875,000 has at least three significant digits, but may have as many as six. 7. If a number ends in a decimal point, the zeros to the left of the decimal point are significant: 875,000. has six significant digits. Beaks only come in 50ml, 100ml, and 150 ml, so saying its under 50 and has decimals points is wrong. 2. Practice : Look at the quantities mentioned in the Metric units only Distance problems. For each value, state the number of significant digits in the first column and the rule(s) you used to determine the number of significant digits. The first has been done for you. Question Number of significant digits Rule(s) used The tallest building in the world, the Burj Khalifa in Dubai, is 0.828 kilometers high. 3 1, 4 The largest human cell is the egg cell, with a diameter of 121 micrometers. 3 1 On a caterpillar’s map, all distances are marked in millimeters. The caterpillar’s map shows that the distance between two milkweed plants is 4,012 millimeters. 4 2 The closest star to our Sun is Proxima Centauri, which is 4.242 light years away. 4 1 A helium atom has a diameter of 9.8 • 10- 11 meters. 2 4/5 3. Practice : The following quantities are all found in the Speed problems (both metric and mixed units). State the number of significant digits for each value and the rule(s) you used to determine the number of significant digits. Metric units only Mixed units Value Significant digits Rule(s) Value Significant digits Rule(s) 3.50 m/s 2 1 113 km/hr 3 1 72 km/hr 2 1 1.25 mi 3 1 4.25 min 3 1 2.00 min 3 3 2.0 • 103 μm/s 1 5 9.58 s 3 2 45.215 d 5 1/2 1.0 yd/d 2 3 60,004,000 km 5 2 1.02 mm/hr 3 2 Activity C: Get the Gizmo ready : ● Select Speed and Metric units only. Calculating with significant digits ● Turn off Show result. ● If necessary, click Next until you reach the question about Marcia. Question: If you multiply or divide two numbers, how many significant digits are in the answer? 1. Observe : The first metric speed question is about Marcia, who ran at a speed of 3.50 meters per second. 2. Solve : Use the tiles to convert Marcia’s speed to kilometers per hour. When you are finished, turn on Show result and click Submit to check if the answer is correct. A. What value do you get? 12.6 B. How many significant digits are in this value? 3 3. Learn a rule : In any calculation, the number of significant digits in the answer should not be greater than the number of significant digits of any measured value. For example, suppose you do the following area calculation: 5.73 cm • 2.1 cm = 12.033 cm2 In this situation, because the second measurement has the fewest significant digits, the answer must also have just two significant digits and should be reported as 12 cm2. For values that are exact, there are unlimited significant digits. For example, there are exactly 100 centimeters in a meter. For conversion factors that are exact, you do not have to worry about the significant figures in the conversion factor. The list below shows all the conversion factors in the Gizmo that are not exact values. For each inexact conversion factor, write the number of significant digits it has: 9.461 • 1012 km/1 light year 4 2.20462 pounds/1 kilogram 6 1.60934 kilometers/1 mile 6 16. 3871 cm3/1 cubic inch 6 453.6 grams/1 pound 4 3.785 liters/1 gallon 4 28.35 grams/1 ounce 4 1.308 cubic yards/1 m3 4 2How many significant digits are in this value?