Integrated General Biology

Transcription

1 Integrated General Biology A Contextualized Approach FIRST EDITION Jason E. Banks Julianna L. Johns Diane K. Vorbroker, PhD

2 Chapter 4 On-Target: Units, Accuracy, and Conversion Section 4.1 Directions for the Student: Objectives: And Then There was the U.S. This lesson is designed for you to complete, on your own or in your study group. Use your notes and follow along in the text, as you find necessary. 1. Understand and be able to properly apply the metric system in practical applications. Since civilization started, measurement has been important. Many different systems have been invented, but, today, only one system is used world-wide. The metric system, also known as SI or the International System of Units (in French, it is called Système International d'unités), is used almost exclusively in every country in the world. The one notable exception that does not use this system as its primary system is the United States. However, all scientists use the metric system, even the ones in the U.S. Metric Prefixes and Their Meanings Prefix Symbol Factor Scientific Notation tera- T x giga- G x 10 9 mega- M x 10 6 kilo- k x 10 3 hecto- h x 10 2 (none) (none) 1 1 x 10 0 = 1 deci- d 1/10 = x 10-1 centi- c 1/100 = x 10-2 milli- m 1/1000 = x 10-3 micro- μ x 10-6 nano- n x Make some observations about the metric system. Explain how the system makes its jumps from one term to the next, and how you know some of the prefixes already. It seems to use powers of ten. It jumps from one term to the next by increasing or decreasing the exponent And Then There was the U.S. 2

3 The metric system uses powers of ten. The prefixes may be familiar to you for many different reasons: computer use, running distances, or even how to classify groups of years, like a decade. Now let s take a look at how the metric system addresses the issue of determining length, and how that differs from the United Stated Customary Units approach. 2. Compare the metric system to the United Stated Customary Units for LENGTH and VOLUME, and determine some strengths and weaknesses for each. Length Metric System United States Customary System 1 meter 1 inch 10 meters = 1 decameter 12 inches = 1 foot 100 meters = 1 hectometer 3 feet = 1 yard 1,000 meters = 1 kilometer 5,280 feet = 1 mile 2. Make some observations about the metric system. Describe some strengths and weaknesses to this approach. Answers will vary; ease of conversion and manipulation. Weaknesses labels can be confusing For the last comparison, let s see how the United Stated Customary Units handles volume, and how that varies from the metric system. Volume Metric System United States Customary System 1 liter 1 US fluid ounce (= 3 tablespoons) 10 liters = 1 decaliter 8 ounces = 1 cup 100 liters = 1 hectometer 2 cups = 1 pint 1,000 liters = 1 kiloliter 2 pints = 1 quart 3. Make some observations about the U.S. system. Describe some strengths and weaknesses to this approach. Labels are not confusing but conversions seem complicated And Then There was the U.S. 3

4 It is easy to see that, since the metric system uses powers of ten, conversion from one unit to another is made very easy. However, some of the strengths of the United States Customary units may be more difficult to see. The factors of 10 are 1, 2, 5, and 10, but the factors of 12 are 1, 2, 3, 4, 6, and 12. Sometimes, having more factors can be very beneficial. An unusual aspect of the units used in the U.S. system is that other systems use the same names, but they have different measurements. A cup in Canada is not the same as a cup in the U.S., nor is it the same in the United Kingdom. Liters, however, are the same everywhere. Becoming familiar with the metric system is very important. You must be able to move smoothly from one unit to the other, and then communicate your data with accuracy. Scientists around the world communicate their information using the metric system, and communication is one of the most important parts of science. Once you learn the system, it becomes easy to immediately understand measures even when they apply to entirely different things. 4. Test your knowledge by answering the following How many? questions: How many bytes are in a gigabyte? 10 9 How many liters of oxygen are in a milliliter? How many liters of kerosene are in a deciliter? 10 How many floating point operations are in a teraflop? We know that there are ten years in a decade, but why is a deciliter one tenth, or 1, of a liter? Notice 10 the i" in the middle of the term. This small part denotes that it is a fraction of the original. Deciliter is one tenth of a liter, centiliter is one hundredth of a liter, and milliliter is one thousandth of a liter. 5. Test your knowledge by answering the following How many? questions: How many meters are in a kilometer? 1000 How many meters are in a micrometer? 10-6 How many meters are in a nanometer? 10-9 Cells are usually very small, measured in nanometers. There are exceptions, however. Some amoebas can be several centimeters in length, and an egg is actually a single cell (before it begins to develop). The solutes in blood are usually measured in milligrams per deciliter (mg/dl). Whatever you study, knowing the metric system is a must. And Then There was the U.S. 4

5 Section 4.2 Directions for the Student: Objectives: Changing the Way We Think This lesson is designed for you to complete, on your own or in your study group. Use your notes and follow along in the text, as you find necessary. 1. Use the proper units when solving problems and reporting data. 1. Explain various quantities within the metric system, including mass, temperature, and volume. 2. Convert various units, including conversions between metric and U.S. customary, and conversions of different temperature scales. Math is the language of the cosmos. As we have learned in previous chapters, mathematics allows us to reveal some of the hidden patterns that exist all around us. Some objects that we observe can be simply counted to gain a numerical understanding of the quantity. For example, we can count the number of people in a building and use a number to describe our observation. Sometimes we must describe something that cannot be simply counted. If we wish to describe the amount of matter that an object contains or the amount of space that an object takes up, we cannot simply count. We must use some standard unit if we wish to accurately communicate our observations. 1. Complete the chart by placing one of the UNITS choices listed into the proper space, matching the correct unit with the appropriate measurement. Choices Measurement Units Seconds (s) Meters (m) or centimeters (cm) Degrees Celsius (ᵒC) Milliliters (ml) or Cubic Centimeters (cm 3 ) Kilograms (kg) or Grams (g) Quantity of matter in an object (mass) Quantity of distance (length) Kinetic energy of particles (temperature) Quantity of space an object occupies (volume) Time kg or g m or cm 0 C ml or cm 3 s Units are just as important as numbers. Units and numbers together describe our observations. Imagine what would happen in a healthcare setting if a doctor s orders didn t have units. How would you handle a script that says a patient should receive 800 ibuprofen? Is that 800 pills? 800 milligrams? 800 grams? By standardizing the units that we use we can accurately communicate with others around the world. Although there have been many different units established and used throughout history, the metric system is the most widely used and easy to use system ever developed. When converting measurements from one unit to another it becomes apparent that the metric system is user-friendly. 2. Refer to section 6.1 in your textbook to answer the following questions on the metric system. Changing the Way We Think 5

6 How many centimeters are in a meter? 100 How many millimeters are in a meter? 1000 How many grams are in a kilogram? 1000 How many micrograms are in a gram? 10 6 How many microliters are in a liter? 10 6 Unit of Measure Scenario: Cynthia is working with her trainer to get ready for an all day hike up and down a mountain. Her trainer recommends that she drink at least three quarters of a gallon of water and have food containing at least three ounces of carbohydrates along the way. Cynthia compares this with a formula that she read online that calculated her amounts to be 3.0 liters of water and 80 grams of carbohydrates. She wants to be prepared for her trip but she does not know how to know which recommendations to follow. 3. What will she need to do to be able to compare the two sets of recommendations? 4. In order to compare the two, she will need to get both recommendations into the same type of units. Measurements of water include gallons and liters. What do these units measure? 5. Measurements of the amount of carbohydrate include grams and ounces. What do these units measure? 6. Of the units used described in the scenario which two are metric? She will need to convert her trainer s recommendations into metric values Volume Mass Liters and grams In order to compare the two sets of recommendations we will need to get all measurements into the same units. In order to convert the units, we will first need to understand the relationships between the two units. We will need to know how a gallon compares to a liter and how a gram compares to an ounce. Without understanding the relationship between two types of units it is impossible to convert one to the other. 1 ounce = grams 1 gallon = liters Here is a demonstration of how to convert gallons to liters. Changing the Way We Think 6

7 If we have 3/4 gallon, and we want to convert it liters then the only value that we can multiply 3/4 gallon by is one. If we multiply a measurement by anything besides one, then we would change the value of the measurement. In unit conversion, we multiply a measurement by a fancy version of one, known as a conversion rate. If 1 gallon = liters, then we can determine the following conversion rates: 1 gallon liters = liters 1 gallon Both of these rates are equal to one. Just as 5/5 equals one, anything divided by itself equals one. The next step in converting in choosing which of the conversion rates we will use in the situation. We multiply this rate times the original measurement. 3/4 gallon x liters 1 gallon Notice that gallons cancel out. If we would have multiplied by the other conversion rate then gallons would not cancel out. Multiply numbers. Units left in the answer are liters. 3/4 gallon x liters 1 gallon = 2.83 liters 3/4 gallon is the same amount of volume as 2.83 liters. Now it is your turn. Starting with the knowledge that 1 ounce = grams, write two possible conversion rates that can come from this relationship between ounces and grams. 7. Conversion rate #1: Ounce to grams, multiply ounces by , or grams/ 1 ounce 8. Conversion rate #2: Gram to ounce, divide grams by What are these conversion rates equal to? 1 1 ounce/ grams 10. How many grams are equal to three ounces? (Multiply 3 ounces by the proper conversion rate. Make sure units cancel out! If units don t cancel you made a mistake.) 11. How does the recommendations of Cynthia's trainer compare to the recommendations that she found on the internet? Explain g They are approximately the same. They are in different units. Changing the Way We Think 7

8 Unit conversion allows you to transfer a measurement form one type of unit to another. Remember, the value of the measurement does not change, only the units. Converting within the metric system Within the metric system the relationships between the units are all based on powers of ten. 12. How many millimeters are in a meter? 1000mm Create two conversion rates based on the above relationship between millimeters and meters. 13. Conversion rate #1: Meters to millimeters; Multiply by Or 1000 millimeters/ 1 meter 14. Conversion rate #2: Millimeters to meters; divide by Or 1 meter/ 1000 millimeters 15. Convert 300 millimeters to meters. (Make sure units cancel out.) 0.3 meters 16. Convert 2 meters to millimeters mm This process of unit conversion demonstrated above works for most all types of units. Converting temperature is one exception. Examine the side by side comparison of Fahrenheit, Celsius, and Kelvin scales. Changing the Way We Think 8

9 To understand what makes temperature conversion unique, you will compare the temperature scales to other measurements. 17. Zero meters is equal to how many millimeters? 0 mm 18. Zero grams is equal to how many kilograms? 0 kg 19. Zero liters is equal to how many microliters? 0 μl 20. Zero degrees Celsius is equal to how many degrees Fahrenheit? 32 F 21. Zero Kelvin is equal to how many degrees Celsius? -273 C While zero meters is the same value as zero millimeters, zero degrees Celsius is not the same value as zero degrees Fahrenheit. Since all three temperature scales have different zero points, we cannot convert in the same way as with conversions of mass, volume and length. Conversion of temperature must take into account the different zero points of each scale. Your textbook has the formulas for converting units of temperature. If we wish to convert 75 degrees Fahrenheit to degrees Celsius we must select the formula that is solved for Celsius and has Fahrenheit as the input. The appropriate formula is: Input the given temperature in for [F] and solve. [ C ] = 5/9 ( [F] - 32 ) [ C ] = 5/9 ( [75] - 32 ) [ C ] = 5/9 (43) [ C ] = degrees Fahrenheit is the same temperature as 24 degrees Celsius. Now it is your turn. 22. Convert 50 degrees Celsius to Fahrenheit. 122 F 23. Convert 50 degrees Celsius to Kelvin K Changing the Way We Think 9

10 Now let s review some unit of measure conversion concepts. 24. Which has more mass: a kilogram of feathers or a kilogram of lead? equal 25. Which has more Volume: a kilogram of feathers or a kilogram of lead? feathers 26. Is it possible to convert 500 grams to liters? Explain. it depends on the substance; water has 1000 grams to a liter. Other substances have more grams or less grams to a liter depending on their density Changing the Way We Think 10

11 Section 4.3 Directions for the Student: Objectives: Numbers with Significance This lesson is designed for you to complete, on your own or in your study group. Use your notes and follow along in the text, as you find necessary. 1. Determine which numbers in a value are significant. 2. Report the correct number of significant figures when taking measurements and performing calculations. 3. Properly round numbers when necessary. You are moving into your new home and need to know if your furniture is going to fit through the front door. You measure the door width with a tape measure and get a measurement of 95 cm. Your friend, who is helping you move, has a tape measure with more tic marks on it and gets a measurement of 94.7 cm. Who is right? It is possible to get different measurements on the same object, depending on what instrument you use. But, even with the most accurate tape measure will never be able to tell you the EXACT measurement. It is always possible to measure to one more place value, one more number beyond the decimal point. No matter how accurate your measuring tool is, the TRUE value has an infinite amount of places the true value has a measurement that goes on forever. In the example we used here, the width of the door is actually cm. (There are some exact values that we can determine, however, like the number of people in a room.) With this in mind, we can focus on the important rules that tell us how to report evidence. Science is based on evidence, and if our evidence isn t any good, then our ideas about what that evidence means are useless. But how can we make sure we are confident in our evidence? Obviously, collecting data using good practices and ethical practices is a must, but should you always report every digit of your findings? How can you be assured that each place value you report is accurate? Scientists have set out certain rules that help us only report the parts of the number we are confident in. Let s try to figure some of them out for ourselves. 1. For each value, read the description about how it was obtained and determine which parts of the number you should report (which parts you feel confident in). Write down the value the scientist should report. Original Value and How It Was Obtained Value Scientist Should Report cm from a ruler that is accurate to the tenth of a cm 527 ml from a graduated cylinder that is accurate to the nearest ml 15.4 cm 527 ml Numbers with Significance 11

12 g from adding g and g (both recorded and measured properly) cm 2 from multiplying 2.5 cm and cm (both recorded measured properly) g 17 cm 2 Numbers that we are confident in, ones we believe to be accurate, are called significant figures. These figures carry meaning. 2. You just got a new job as a nurse at a new hospital, and you need to measure the mass of your patient s fecal sample. The balance reads 0.00 g, so you put the sample on and get a reading of g. How many significant figures are there in this reading? Which ones were measured? 5 5 Since the balance gives readings to 1/100 of a gram (or two places past the decimal), it is accurate to the hundredths place. Therefore, all of the places were significant, so there are 5 sig figs (significant figures are often abbreviated this way). 3. Write the value from the balance ( g) in scientific notation x 10 2 When we write values in scientific notation, we only write numbers that are significant. We see that there are 5 sig figs in g when we write it as x 10 2 g. 4. You dropped a hammer on your toe last week, and the nail just fell off. You are wondering how much mass the nail has, so you take it into school and use a good balance, accurate to the thousandths place or three decimal places. You zero the balance, making sure it reads g. You place your nail on the balance and get a reading of g. How many significant figures does this number have? 2 Write this value (0.098 g) in scientific notation. 9.8 x 10-2 Once you write this value in scientific notation (9.8 x 10-2 ), it is easy to see that there are only two significant figures. But why was the zero in g significant when the zeros in g were not significant? Numbers with Significance 12

13 5. Continuing on with this scenario, it just so happens that when you shared your toenail story with your classmates, by complete coincidence one of your lab partners dropped his bowling ball on his toe last week and his toenail fell off too. Naturally, you want to know which toenail has more mass. His toenail reads g. How many significant figures are there in this reading? 2 Write this number in scientific notation. 9.0 x 10-2 g We saw earlier that the zeros to the left of the first non-zero number (like g) are not significant, but the value of g has a zero after a non-zero number. Because the last zero was part of the reading and within the balance s accuracy, it is significant. And when we write it in scientific notation, 9.0 x 10-2 g, we see those 2 sig figs in the 9.0 part. 6. The nearest start to Earth, besides our own Sun, is 25,284,280,000,000 miles away. Determine the number of significant figures in this value, and write it in scientific notation. 7 miles There are 14 place values in 25,284,280,000,000 miles, but how many are significant? Since the zeros are trailing (they are after the non-zero numbers) they are not significant. When we write this value in scientific notation, x miles, we only write the significant values. Trailing zeros that are before (to the left of) the decimal point do not count a significant. There is one point to be very clear about, however. If the value has zeros that seem to be trailing zeros, but there is a decimal point placed after the zeros, then the zeros are significant. For example, 34,000 m has two sig figs, but 34,000. m has five sig figs. The decimal point at the end means that those were measured values, and just happen to be zeros. You would write 34,000. m as x 10 4 m in scientific notation to show that this value is accurate to 5 significant figures. How to determine if a number is significant I. All digits that are not zero are significant. Examples: a. 568 has three significant figures. b. 4 has one significant figure. II. Any zero between two significant numbers is significant. Examples: a has four significant figures. b. 7, has six significant figures. III. Zeros that only serve to place the decimal are not significant. Examples: a. 470,000 has two significant figures. Numbers with Significance 13

14 b. If we wanted to show that some of the zeros from the previous problem were significant, we would need to use scientific notation x 10 5 has four significant figures this means that we are not confident about the last two zeros are correct. IV. Any trailing zeros (a zero to the right of the significant figure) after the decimal point are significant. Examples: a has five significant figures. The zeros written on the left are not significant. Also, please notice that there is a zero written to the left of the decimal point this is an important practice because it reduces confusion. b x 10 9 has four significant figures. V. Exact numbers, like the number of people in a classroom, have an infinite number of significant figures and should not serve as the limiting value for significant figures. 7. Determine how many significant figures the number has, and write it in scientific notation. Value Number of Sig Figs Scientific Notation g x m x ,034,690,000 miles x ,000,030. L x people x 10 1 Addition with Significant Figures You have two samples of urine from a patient, and you need to add them together to find the total amount excreted. But the samples were measured using different instruments, and they have a different number of significant figures: Sample A: ml Sample B: 98.6 ml 8. What should you report as the total amount of urine? Justify your answer with a brief explanation ml. Sample B is accurate only to the tenth of a decimal. We can see that Sample A has 5 significant figures, and Sample B only has 3 sig figs. How many significant figures should our answer have? It turns out that the total number of significant figure doesn t matter when you are adding or subtracting. The only thing that really matters is the last place Numbers with Significance 14

15 where ALL of the values have a significant figure. In this case, Sample A goes to the hundredths, but Sample B only goes to the tenths so our answer cannot go past the tenths Notice that we do not round any value until the last step! Look at the number just before the cut off space (in this case, the tenths place is the cut off space) and round up if it is 5 or higher. The final answer with the proper number of significant figures is ml. Rounding your answer to the correct number of significant figures can only be done as a final step. If you round your values any earlier than that, you risk getting incorrect answers. There are three steps for rounding numbers. Example: Round to hundredths place ) Underline the place value that you wish to stop at ) Highlight or circle the number just to the right of the place value where you wish to stop 3) If the highlighted number is 5 or higher, round up. If it is 4 or less, keep the value the same (round down). 9. Record the answer with the proper number of significant figures. Remember not to round until the final step. Problem Preliminary Answer Last Shared Place Value Final Answer Tenths nes ,800 10,250 Hundreds 10, Hundredths Multiplication with Significant Figures You are trying to find the area of a rectangle with your group. You will measure Side A and your partner will record Side B. However, your partner is using a ruler that is less precise than yours. What is the area of the rectangle? Side A Side B cm 26.7 cm Numbers with Significance 15

16 This problem is different from addition because now we need to use multiplication. Before, with addition, we were only concerned about the lowest place value that all the values shared. Now, we must look at the entire number and count the number of significant figures for each value. The number with the least number of significant figures determines how many significant figures there can be in the final answer cm x 26.7 cm = cm 2 Preliminary answer 4 sig figs 3 sig figs 625 cm 2 Round answer to 3 sig figs cm 2 rounds to 625 cm 2 (with 3 sig figs) as the final answer We see that we always go with the least number of significant figures when multiplying. But, be careful when using exact numbers. An exact number, like the number of people in this room, is considered to have an infinite number of significant figures and we would never use it as the limiting value. 10. Complete the table with the correct number of significant figures in your final answer. Remember not to round until the final step. Problem Preliminary Answer Lowest Number of Sig Figs Final Answer x Two x Two x 9,800 4,410,000 Two 4,400, x Two 74 There were questions earlier in this activity that asked you to complete problems before you may have known how to do them. Review your answers from these questions before you move on and identify any errors. Numbers with Significance 16

17 Section 4.4 Directions for the Student: Objectives: The Rate Equals ONE This lesson is designed for you to complete, on your own or in your study group. Use your notes and follow along in the text, as you find necessary. 1. Explain how to use conversion rates and how they work. 2. Perform conversions on a variety of units. When converting a value in one unit into another unit, people often say some unusual things. For example, when converting meters to centimeters you may often hear someone say, Just multiply by 100. Let s see what that actually would do: Convert 8 m into centimeters. 8 m 100 = 800 m We have clearly changed the length here, not the units. Multiplying by 100 simply makes the value 100 times larger, and does nothing to convert the units. Instead, we need to multiply by something that does not change the value, but changes the units. 1. Think about what you could multiply something by that would not change its value. Write you re your thoughts. One, rest of answer will vary: multiplying by the identity Let s look at the relationship between meters and centimeters. Let s assume this line is 1 meter in length. How many centimeters is it? 1 meter line 100 centimeter line We can see that 1 meter in length is the same length as 100 cm. They are identical. And what happens when you divide something by itself? 7 = 1 53 = x x = 1 1 meter = centimeters = centimeters 1 meter Now that we understand the relationship between meters and centimeters can be written as a 1, think about what happens when you multiply a value by 1. The Rate Equals ONE 17

18 9 1 = 9 x 1 = x 38 1 = 38 Converting values from one unit to another does not change the value, even though the numbers seem to change. Instead, conversion rates are really just a big ONE in disguise, and multiplying something by ONE does not change its value. 2. Write the two possible conversion rates for each situation. Remember, conversion rates equal 1. Unit Equivalencies 1 st Possible Conversion Rate 2 nd Possible Conversion Rate 1 m = 100 cm 1 m 100 cm 100 cm 1 m 1 mile = 5,280 feet 1 mile/5,280 feet 5,280 feet/ 1 mile 1 in = 2.54 cm 1 inch/ 2.54 cm 2.54 cm/ 1 inch 1,000 ml = 1 L 1000 ml/ 1 L 1L / 1000mL 10 dl = 1 L 10dL/1L 1L / 10 dl 1 hour = 60 minutes 1 hour/ 60 minutes 60 minutes/ 1 hour Conversion does, however, change the units. And for this to happen, the unwanted units must cancel out, leaving only the desired units. So which conversion rate do we use? 3. Complete the conversions using the proper conversion rate. Make sure the unwanted units cancel out, leaving only the desired units in the numerator. Problem Conversion Work Space Answer Convert 75 meters to centimeters 100 cm 75 m ( 1 m ) = 7500 cm 7,500 cm Convert 2.3 miles into feet 5,280 ft 2.3 mi ( ) = 1 mi 12,144 ft Convert 7.29 centimeters in to inches Convert 0.93 liters into milliliters Convert 1.38 deciliters into liters Convert 289 minutes into hours 1 inch / 2.54 cm 2.87 in 1000 ml / 1 L 930 ml 10 L / 1 dl 13.8 L 1 hour / 60 minutes 4.82 hours Using this method, you can even do multiply conversions at the same time. Convert inches into meters. The Rate Equals ONE 18

19 2.54 cm in ( ) ( 1 m ) = m 1 in 100 cm Since the original value only has 5 sig figs, the answer must also only have 5 sig figs. (The conversion rates are exact values, so they are counted as having an infinite number of sig figs.) Applying what we learned about significant digits, that means the answer rounds to m for the final answer. The Rate Equals ONE 19