Appendix B: Skills Handbook

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1 Appendix B: Skills Handbook Effective communication is an important part of science. To avoid confusion when measuring and doing mathematical calculations, there are accepted conventions and practices regarding units of measurement, working with very large and very small numbers, and acknowledging uncertainty. The following appendices explain these conventions. B1: SI Units The scientific communities of many countries, including Canada, have agreed on a system of measurement called SI (Système international d unités). In this system, all physical quantities can be expressed as a combination of seven fundamental SI units, called base units (for example, length, mass, and time). The seven SI base units are listed in Table 1. For example, the unit of force that causes a mass of 1 kg to accelerate at a rate of 1 metre per second per second is known as a newton (N). In base units, the newton is m kg/s 2. The dot between m and kg means multiplied by, but m kg is simply read as metre kilogram. The slash means divided by and is read per. The whole unit is read metre kilogram per second squared. You can see why a special name and symbol are given to some derived units. Some common quantities and their units are listed in Table 2. Note that the symbols representing the quantities are italicized while the unit symbols are not. Table 2 Quantity name Common Quantities and Units Quantity symbol Unit name Unit symbol Table 1 The Seven SI Base Units distance d metre m Quantity name Unit name Unit symbol length metre m mass kilogram kg* time second s electric current ampere A temperature kelvin K** amount of substance mole mol light intensity candela cd * The kilogram is the only base unit that contains a prefix. ** Although the base unit for temperature (T) is a kelvin (K), the common unit for temperature (t) is a degree Celsius ( C). For example, the speed of an object is relative to the distance travelled during a specified time period. The unit for speed is metres (distance) per second (time). Units that are formed using two or more base units are called derived units. Some derived units have special names and symbols. area A square metre m 2 volume V cubic metre m 3 litre time t minute min hour year speed v metre per second m/s acceleration a meter per second per second L h a m/s 2 concentration c gram per litre g/l temperature t degree Celsius C pressure p pascal Pa heat q joule J energy E joule J work W joule J 546 Appendices

2 An important feature of SI is the use of a common set of prefixes to express small or large sizes of any quantity conveniently. SI prefixes (Table 3) act as multipliers or factors to increase or reduce the size, in multiples of 10. The most common prefixes change the size in multiples of 1000 (10 3 or 10 3 ), except for centi, as in centimetre. Table 3 Prefix Common SI Prefixes Symbol Factor by which unit is multiplied Example giga G m 1 Gm mega M m 1 Mm kilo k m 1 km hecto h m 1 hm deca da m 1 dam 1 deci d m 1 dm centi c m 1 cm milli m m 1 mm micro m m 1 mm nano n m 1 nm Converting Units 1 km 1000 m 1 km Therefore, m m 1 km Multiplying by a conversion factor is like multiplying by 1. The conversion factor does not change the size of the quantity, only the unit in which it is expressed. Sample Problem 1 illustrates how to convert from one unit to another. SAMPLE PROBLEM 1 Convert Units A chemistry class of 30 students is performing an experiment in the laboratory. Each pair of students will use a volume of ml of silver nitrate solution for a chemical reaction. Silver nitrate costs $56.25 per litre. Determine the cost of the silver nitrate for the class. Solution The first step is to determine the total volume required. Each pair of students requires ml so the whole class will require ml 1875 ml. The second step is to convert the volume from millilitres to litres. There are two possible conversion factors between ml and L. They are 1 L and ml ml 1 L You should always choose the form of the conversion factor that cancels the original unit. In this case, the original unit is ml, so the correct conversion factor is 1 L ml 1L 1875 ml L 1000 ml Notice how the initial units, ml, cancel (divide to give 1), leaving L as the new unit. Now that you have determined the volume in L, the solution to the problem is simple. Multiply the price per L by the volume in L. $56.25/L L $ The cost of the silver nitrate for the class is $ Practice Make the following conversions. (a) Write 3.5 s in ms. (b) Convert 3.5 cm to µm. (c) Change 5.2 L to cl. (d) Convert 7.5 µg to ng. Appendix B Skills Handbook 547

3 When creating a conversion factor for prefixes that represent fractions of a unit, you may find it easier to avoid fractions and use integers. For example, 1 1 mm m, which means mm 1 m Therefore, convenient conversion factors to convert between millimetres and metres are mm 1 m and 1 m mm Conversion factors can be used for any unit equality, such as 1 h 60 min and 1 min 60 s. Sometimes, conversion factors are combined to convert several units in one step of a calculation. The problem below shows this multiple conversion method to convert a speed from metres per second to kilometres per hour. SAMPLE PROBLEM 2 Convert Units: Multiple Step John and his friends walked around the school at a speed of 1 m/s. Convert this speed to an equivalent speed in km/h. Solution v 1 m s In this problem, three conversions are necessary: metres to kilometres, seconds to minutes, and minutes to hours. Remember, always choose the form of the conversion factor that cancels the original unit. 1 m s 1 km 60 s 60 m 1000 m 1 min 1 in h 3.6 k m h 1 m/s is equivalent to 3.6 km/h. Practice Use the appropriate conversion method to solve each of the following problems 1. A space shuttle travels in orbit around Earth at a speed of approximately km/h. Convert this speed to m/s. 2. A DVD is capable of storing 4.7 GB of information. A CD can store 700 MB of information. How many CDs would be required to store the same amount of information as a DVD? (Hint: First determine how many MB equal one GB.) B2: Scientific Notation Scientists often work with very large or very small numbers. Such numbers are difficult to work with when they are written in common decimal notation. For example, the speed of light is about m/s. There are many zeros to keep track of, if you have to multiply or divide this number by another number. Sometimes it is possible to change a very large or very small number, so that the number falls between 0.1 and 1000, by changing the SI prefix. For example, mm can be converted to 237 km, and kg can be expressed as 895 mg. A prefix change is not always possible, however, because an appropriate prefix may not exist or because the given prefix is essential if we want to use a particular unit of measurement. In such cases, it is best to deal with very large or very small numbers by using scientific notation. Scientific notation expresses a number by writing it in the form a 10 n, where the letter a, referred to as the coefficient, is a value between 1 and 10. The number 10 is the base, and n represents the exponent. The base and the exponent are read as 10 to the power of n. Powers of 10 and their decimal equivalents are shown in Table 4. To write a large number in scientific notation, follow these steps: 1. To form the coefficient, place the decimal after the first digit and drop all the trailing zeros. If all the numbers after the decimal are zeros, keep one zero. For example, when writing the speed of light ( m/s) in scientific notation, the coefficient becomes To find the exponent, count the number of places to the right of the decimal. In the speed of light example, there are eight places after the decimal, so the exponent is Combine the coefficient with the base and exponent. For example, the speed of light can be expressed in scientific notation as m/s. Very small numbers (less than 1) can also be expressed in scientific notation. For very small numbers, the base (10) must be given a negative exponent. 548 Appendices

4 For example, a millionth of a second, s, can be written in scientific notation as s. Note that the number of the exponent is the number of places after the decimal that includes the first non-zero number. Table 5 shows several examples of large and small numbers expressed in scientific notation. Table 4 Powers of 10 and Decimal Equivalents Power of 10 Decimal equivalent Table 5 Large or small number Numbers Expressed in Scientific Notation Common decimal notation Scientific notation million km km km 154 thousand nm nm nm 753 trillionths of a kg kg kg 315 billionths of a m m To multiply numbers in scientific notation, multiply the coefficients and add the exponents. Express the answer in scientific notation. Look at the following examples: ( km)( km) = km 2 = km 2 ( m)( m) = m 2 = m 2 When dividing numbers in scientific notation, divide the coefficients and subtract the exponents. ( m) ( s) = m/s ( N) ( m) = N/m = N/m Note that, when writing a number in scientific notation, the coefficient should be between 1 and 10. In the first example above, the product of 3.5 and 7.4 is This can be expressed in scientific notation as The answer can be combined as km 2. Adding the exponents gives us km 2. The coefficient should be rounded to the same certainty (number of significant digits) as the measurement with the least certainty (fewest number of significant digits). In this example, both measurements have only two significant digits, so the coefficient 2.59 should be rounded to 2.6 to give a final answer of km 2. B3: Uncertainty in Measurement There are two types of quantities that are used in science: exact values and measurements. Quantities that are exact values include defined quantities, such as those obtained from SI prefix definitions (e.g., 1 km 1000 m) and from other definitions (e.g., 1 h 60 min). Counted values, such as 5 beakers or 10 cells, are also exact values. All exact values are considered to be completely certain. In other words, 1 km is exactly 1000 m, not m or m. Similarly, 5 beakers could not be 4.9 or 5.1 beakers; 5 beakers are exactly 5 beakers. Appendix B Skills Handbook 549

5 Every measurement, however, has some uncertainty or error. No measurement is exact. The uncertainty depends on the limitations of the particular measuring instrument used and the technological skill of the person making the measurement. The certainty of any measurement is communicated by the number of significant digits in the measurement. In a measured or calculated value, significant digits are the digits that are certain, plus one estimated (uncertain) digit. Significant digits include all the digits that are correctly reported from a measurement. Significant Digits Table 6 provides the guidelines for determining the number of significant digits, along with examples to illustrate each guideline. Table 6 Guideline Count from left to right, beginning with the first non-zero digit. Zeros at the beginning of a number are never significant. All non-zero digits in a number are significant. Zeros between digits are significant. Zeros at the end of a number with a decimal point are significant. Zeros at the end of a number without a decimal point are not significant. All digits in the coefficient of a number written in scientific notation are significant. Number Example Number of significant digits Rounding Use these rules when rounding answers: 1. When the first digit discarded is less than 5, the last digit kept should not be changed. Example: rounded to four digits is When the first digit discarded is greater than 5, or when it is 5 followed by at least one digit other than zero, the last digit kept is increased by one unit. Examples: rounded to five digits is rounded to four digits is When the first digit discarded is 5 followed by only zeros, the last digit kept is increased by 1 if it is odd, but not changed if it is even. Examples: 2.35 rounded to two digits is rounded to two digits is rounded to two digits is When adding or subtracting, look for the quantity with the fewest number of digits to the right of the decimal point. The answer can have no more digits to the right of the decimal point than this quantity has. Example: = Because is the quantity with the fewest digits to the right of the decimal point, the answer must be rounded to Example: = Because 32.4 has the fewest digits to the right of the decimal point, the answer must be rounded to When multiplying or dividing, the answer must contain no more significant digits than the quantity with the fewest number of significant digits. Example: This answer must be rounded to 2.6 because 0.34 has only two significant digits. Example: This answer must be rounded to 0.29 because 8.4 has only two significant digits. 550 Appendices

6 6. When performing a series of calculations, do not round each calculated value before carrying out the next calculation. The final answer should be rounded to the same number of significant digits that are in the quantity with the fewest number of significant digits. Example: ( )( ) Three calculations are required: (Do not round.) ( ) (Do not round.) Because the smallest number of significant digits among the quantities is three, the answer must be rounded to Measurement Errors There are two types of error that can occur when measurements are taken: random and systematic. Random error results when an estimate is made to obtain the last significant digit for a measurement. The size of the random error is determined by the precision of the measuring instrument. For example, when measuring length, it is necessary to estimate between the marks on the measuring tape. If these marks are 1 cm apart, the random error is greater and the precision is less than if the marks were 1 mm apart. Systematic error is caused by a problem with the measuring system itself, such as the presence of an interfering substance, incorrect calibration, or room conditions. For example, if a balance is not zeroed at the beginning, all the measurements taken with the balance will have a systematic error. The precision of measurements depends on the gradations of the measuring device. Precision is the place value of the last measurable digit. For example, a measurement of cm is more precise than a measurement of of cm because was measured to hundredths of a centimetre whereas was measured to tenths of a centimetre. When adding or subtracting measurements of different precision, round the answer to the same precision as the least precise measurement. Consider the following example: 11.7 cm 3.29 cm cm cm The first measurement, 11.7 cm, is measured to the nearest tenth of a centimetre and is the least precise. The answer must then be rounded to the nearest tenth of a centimetre, or 15.5 cm. No matter how precise a measurement is, it still may not be accurate. Accuracy refers to how close a value is to its accepted value. Figure 1 presents an analogy that uses the results of horseshoe tosses to explain precision and accuracy. (a) precise and accurate (c) accurate but not precise (b) precise but not accurate (d) neither accurate nor precise Figure 1 The patterns of the horseshoes illustrate the comparison between accuracy and precision. Appendix B Skills Handbook 551

7 How certain you are about a measurement depends on two factors: the precision of the instrument and the size of the measured quantity. More precise instruments give more certain values. For example, a mass measurement of 13 g is less precise than a mass measurement of g you are more certain about the second measurement than the first. Certainty also depends on the size of the measurement. For example, consider the measurements 0.4 cm and 15.9 cm. Both have the same precision that is, they are measured to the nearest tenth of a centimetre. If the measuring instrument is precise to ± 0.1 cm, however, the first measurement could be between 0.3 cm and 0.5 cm. The second measurement could be between 15.8 cm and 16.0 cm. An error of 0.1 cm is much more significant for the 0.4 cm measurement than it is for the 15.9 cm measurement, because the second measurement is much larger than the first. For both factors the precision of the instrument used and the value of the measured quantity the more digits there are in a measurement, the more certain you are about the measurement. Estimating Measurement All measurements are our best estimates of the actual values. The accuracy of a measuring device and the skill of the investigator determine how certain and precise a measurement will be. The usual rule is to estimate a measurement between the smallest divisions on the scale of the instrument. If the smallest divisions on the scale are fairly far apart (for example, greater than 1 mm), then you should estimate to one tenth, (± 0.1) of a division (for example, 34.3 ml, 13.8 ml and 87.1 ml). If the divisions are closer together (for example, around 1 mm), then you should estimate to two tenths (± 0.2) of a division (for example, 12.6 C, 11.2 C, and 35.8 C). If the divisions are very close together, then you should estimate to five tenths (± 0.5) or half of a division (for example, 13.0 g, 33.5 g, and 42.0 g). B4: Creating Data Tables Data tables are an effective way to record both qualitative and quantitative observations. Making a data table should be one of your first steps when conducting an investigation. You may decide that a data table is enough to communicate your data, or you may decide to use your data to draw a graph. A graph will help you analyze your data. Sometimes you may use a data table to record your observations in words, as shown in Table 7. Table 7 Aquatic system Fertilizer 1 Fertilizer 2 No fertilizer Effect of Different Fertilizers on Algae Growth Observations Day 1 Day 2 Day 3 Day 4 Day 5 Sometimes you may use a data table to record the values of the independent variable (the cause) and the dependent variables (the effects), as shown in Table 8. (Remember that there can be more than one dependent variable in an investigation.) Table 8 Average Monthly Temperatures in Cities A and B Temperature Temperature Month ( C) in City A ( C) in City B January 7 6 February 6 6 March 1 2 April 6 4 May 12 9 June Follow these guidelines to make a data table: Use a ruler to make your table. Write a title that precisely describes your data. Include the units of measurement for each variable, when appropriate. List the values of the independent variable in the left-hand column of your table. List the values of the dependent variable(s) in the column(s) to the right of the column for the independent variable. 552 Appendices

8 B5: Graphing Data We organize the data collected from investigations so that we can identify a trend or pattern in the data, which will indicate a relationship between the data. Trends or patterns in data are usually easier to see if you graph the data. A graph is a visual representation of numerical or quantitative data. There are many types of graphs that you can use to organize your data. Three of the most useful kinds of graphs are bar graphs, circle graphs, and point-and-line graphs. Each kind of graph has its own special uses. You need to identify which type of graph is most appropriate for your data before you graph your data. Bar Graphs When at least one of the variables is qualitative, use a bar graph to organize your data (Table 9 and Figure 2). For example, a bar graph is a good way to present the data collected from a study of the number of births (quantitative) during each month of the year (qualitative). Each bar represents a different category, such as the month of the year. In this type of bar graph, the qualitative data is usually placed on the x- axis and the quantitative data is placed on the y-axis, so that differences among the numbers of births per month are more easily observed. Table 9 Births per Month in 2006 Month Number of births January 2037 February 1863 March 1597 April 1698 May 1436 June 1752 July 1648 August 1871 September 2283 October 2562 November 2749 December 2624 Number of births Figure 2 Births by Month, 2006 J F M A M J J A S O N D Month Circle Graphs Circle graphs and bar graphs are used for similar types of data. If your quantitative variable can be changed to a percentage of the total quantity, then a circle graph is useful. A circle graph (sometimes called a pie graph) can show the whole of something divided into all of its parts. For example, a circle graph can show the proportion of the gases that are found in air (Figure 3). trace gases 1 % nitrogen 78 % oxygen 21 % Figure 3 The gaseous components of air and their percentages. Appendix B Skills Handbook 553

9 Point-and-Line Graphs When both variables in the data are quantitative, use a point-and-line graph. For example, we can use the following guidelines and the data in Table 10 to construct the point-and-line graph in Figure 4. Table 10 Distance (m) Figure 4 A Running White-Tailed Deer Time (s) Distance (m) A Running White-Tailed Deer Time (s) Making Point-and-Line Graphs 1. Use grid paper to construct your graph. Use the horizontal edge on the bottom of the grid as the x-axis and the vertical edge on the left as the y- axis. The larger the graph is, the easier it is to read and interpret. 2. Decide which variable goes on which axis. Label each axis, including the units of measurement. The independent variable is generally plotted along the x-axis, and the dependent variable is plotted along the y-axis. The exception is when you plot a variable against time. Regardless of which variable is the independent variable, always plot time on the x-axis. This convention ensures that the slope of the graph always represents a rate. 3. Title your graph. The title should be a short, accurate description of the data represented by the graph. 4. Determine the range of values for each variable. The range is the difference between the largest and smallest values. Graphs often include a little extra length on each axis, to make the axes less cramped. For example, the time in Table 10 ranges from 0 s to 6.0 s, but the x-axis in the graph in Figure 4 ranges from 0 to 7.0 s. 5. Choose a scale for each axis. The scale will depend on how much space you have and the range of values for each axis. Each line on the grid usually increases steadily in value by equal increments, such as 1, 2, 5, 10, 50, or 100. In Figure 4, one line is used for every 1 s on the x-axis, and for every 10 m on the y-axis. 6. Plot the points. Start with the first pair of values, which may or may not be at the origin of the graph. The origin of the graph is the point at which the x-axis and y-axis intersect. In the graph in Figure 4, the first set of points is 0 on the x-axis and 0 on the y-axis. 7. After all the points are plotted, draw a line through the points to show the relationship between the variables, if possible. Not all points may lie exactly on a line; small errors in each measurement may have occurred and moved the points away from the perfect line. 554 Appendices

10 Average temperature ( o C) Draw the line of best fit a smooth line that passes through or between the points so that there are about the same number of points on each side of the line. The line of best fit, which may be a straight or curved line, attempts to minimize the effect of random measurement errors. 8. If you are plotting more than one set of data on the same graph, use different colours or symbols to indicate the different sets and include a legend. You can use a point-and-line graph to make predictions. For example, you can use the graph in Figure 4 to predict that at 4.5 s the deer will have ran 60 m. Predicting values that lie between known values is called interpolation. You can also predict outside the plotted values. From the graph in Figure 4 you can predict that at 7.0 s the deer will have ran 90 m. Predicting values that lie outside known values is called extrapolation. You should be careful when extrapolating. The farther you are from the known values plotted on the graph, the less reliable your prediction will be. Climatographs Some graphs, such as climatographs, combine a bar graph and a point-and-line graph. In Figure 5, both precipitation and temperature are measured for each month of the year. Average precipitation (mm) Vancouver, B.C J F M A M J J Month A S O N D 40 Figure 5 Appendix B Skills Handbook 555

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