Numbers in Science Exploring Measurements, Significant Digits, and Dimensional Analysis

Transcription

1 Foundation Lessons Numbers in Science Exploring Measurements, Significant Digits, and Dimensional Analysis About this Lesson This lesson is an introductory activity for proper measuring techniques, the correct use of significant digits, and dimensional analysis. Students are asked to gather data on a cube and a sphere using proper metric measuring techniques and significant digits. The students use the data to calculate volume, circumference, diameter, and density. This lesson is included in the LTF Middle Grades Module 2. Objectives Students will: Be introduced to proper measurement techniques, the correct use of significant digits, and dimensional analysis Take dimensions of and identify significant digits for a cube and a sphere Calculate the volume and density of a cube and a sphere Calculate the circumference and diameter of a sphere Use dimensional analysis to make conversions Level All Common Core State Standards for Science Content LTF Science lessons will be aligned with the next generation of multi-state science standards that are currently in development. These standards are said to be developed around the anchor document, A Framework for K 12 Science Education, which was produced by the National Research Council. Where applicable, the LTF Science lessons are also aligned to the Common Core Standards for Mathematical Content as well as the Common Core Literacy Standards for Science and Technical Subjects. T E A C H E R Code (LITERACY) RST (MATH) A-CED.4 Standard Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Level of Thinking Apply Apply Depth of Knowledge II II Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at i

2 Teacher Overview Numbers in Science Code (MATH) N-Q.1 Standard Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Level of Thinking Apply Depth of Knowledge II Connections to AP* Students are expected to report measurements and perform calculations with the correct number of significant digits. *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Materials and Resources Each lab group will need the following: aprons balance beaker, 250 ml goggles graduated cylinder, 100 ml, plastic paper towels die marble ruler, clear metric string Assessments The following types of formative assessments are embedded in this lesson: Visual assessment of measuring techniques used within the lesson The following assessments are located on the LTF website: Short Lesson Assessment: Numbers in Science Introduction to the Science Classroom Assessment th Grade Posttest, Free Response Question 1 T E A C H E R Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at ii

3 Teacher Overview Numbers in Science Teaching Suggestions This lesson is designed to introduce or reinforce accurate measurement techniques, the correct use of significant digits, and dimensional analysis. Dimensional analysis is also called the Factor-Label method or Unit-Label method, and is a technique for setting up problems based on unit cancellations. Lecture as well as guided and independent practice of these topics should precede this activity. Students should be provided with reference tables containing metric and standard conversion factors. The purpose of significant digits is to communicate the accuracy of a measurement as well as the measuring capacity of the instrument used. Remind students repeatedly to take measurements including an estimated digit and to perform their calculations with the correct number of significant digits. Emphasize that points will be deducted for answers containing too many or too few significant digits. The correct number of significant digits to be reported by your students will depend entirely upon your equipment. Small wooden alphabet blocks or dice should be inexpensive and easy to obtain. Be sure to find a cube/graduated cylinder combination that ensures total submersion of the cube because its volume will be determined by water displacement. If the chosen cube or sphere floats, forceps can be used to gently submerge the o bject just under the surface of the water. Spherical objects could be a marble or small rubber ball. Again, be sure to check the sphere/ cylinder size to ensure that total submersion of the sphere is possible. Provide students with a length of string and metric ruler or a flexible tape measure. The string can be wrapped around the sphere, marked, and then removed and measured. T E A C H E R v. 2.0, 2.0, 2.0 Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at iii

4 Teacher Overview Numbers in Science Answer Key Data and Observations Mass (g) Table 1. Measurements and Significant Digits Cube Data (4 sd) Dimensions (cm) Length Width Height.68 ( sd).65 ( sd).67 ( sd) Volume (ml) Initial Final Beaker 100 (1 sd) 150 (2 sd) Graduated cylinder (4 sd) (4 sd) Mass (g) Dimensions (cm) Sphere Data 19.8 (4 sd) Circumference 7.62 ( sd) Volume (ml) Initial Final Beaker 100 (1 sd) 110 (2 sd) T E A C H E R Graduated cylinder (4 sd) 182. (4 sd) Volume of a cube Formulae for Calculating V = length width height Circumference of a circle C = πd Diameter of a circle d = 2r Volume of a sphere 4 V r Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at iv

5 Answer Key (continued) Teacher Overview Numbers in Science Exercise 1 Left: Middle: Right: 5.75 ml.0 ml 0. ml Analysis 1. The number of significant digits will be determined by the equipment you are using mg 2. a g 15,050 mg 1g b. 1lb 16oz g oz 454 g 1lb. V = l w h =.68 cm.65 cm.67 cm = 49. cm 4. 1m 1m 1m 49. cm m 100 cm 100 cm 100 cm 5 5. V = V final V initial = 150 ml 100 ml = 50 ml = 50 cm 6. V = V final V initial = ml ml = 50.1 ml = 50.1 cm g 7. a. D 50.1 cm 0.00 g/cm T E A C H E R b. c g D 0. g/cm 50 cm g D 49. cm 0.05 g/cm Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at v

6 Answer Key (continued) Teacher Overview Numbers in Science 8. a. b. c. g 1 kg 100 cm 100 cm 100 cm kg/m cm 1000 g 1 m 1 m 1 m g 1 kg 100 cm 100 cm 100 cm kg/m cm 1000 g 1 m 1 m 1 m g 1 kg 100 cm 100 cm 100 cm kg/m cm 1000 g 1 m 1 m 1 m 9. a. b. 10. C = πd The bar above the last zero of the number 00 communicates it is a significant zero, transforming the recorded answer from one significant digit to three. It is equally appropriate to teach your students to use scientific notation to effectively communicate three significant digits. The number could be correctly written as Another way to communicate a number accurate to the one s position is to use a decimal at the end of the number. The number could be written as 00., representing that this measurement is accurate to the last digit. 1kg 19.8 g kg 1000 g 1lb 19.8 g lbs 454 g C 7.62 cm d 2.4 cm.14 T E A C H E R 11. d = 2r d 2.4 cm r 1.22 cm V 4 r 4 ( )(1.22) 7.61cm Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at vi

7 Answer Key (continued) Teacher Overview Numbers in Science 1. V = V final V initial = 110 ml 100 ml = 10 ml = 10 cm 14. V = V final V initial = 182. ml ml = 7. ml = 7. cm 15. a g D 7.60 cm 2.55 g/cm b g D 2g/cm 10 cm c g D 7. cm 2.7 g/cm 16. a. b. c g 1lb 2.54 cm 12 in 159 lbs/ft 1cm 454g 1in 1ft 2 g 1lb 2.54 cm 12 in 100 lbs/ft 1cm 454g 1in 1ft 2.7 g 1lb 2.54 cm 12 in 170 lbs/ft 1cm 454g 1in 1ft T E A C H E R Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at vii

8 Answer Key (continued) Teacher Overview Numbers in Science Conclusion Questions 1. The density of the cube has three significant digits when measured with the ruler. After subtracting to find the difference between the initial and final water levels in the graduated cylinder and beaker, there are two significant digits when measured with the graduated cylinder but only one significant digit when measured with the beaker. The ruler is the more accurate measure of the volume when compared to the volume obtained by water displacement using the graduated cylinder. Any instrument used to submerge the cube will contribute a small amount to the volume recorded because it contributes to the total amount of water displaced. See if your students can discover this concept. Student answers may vary in significant digits depending on the equipment used. 2. The calculated density of the cube would increase. Measuring a wet block will make the mass appear greater. Because mass is in the numerator of the equation mass D volume the density value reported will be too great.. The density of the sphere would increase. If the student measured the circumference at any point other than the center, the circumference would be reported as too small. If the diameter is reported as too small, C d d the radius will thus be reported as too small. If the radius is reported as too small, d r r 2 T E A C H E R the volume will thus be reported as too small. If the volume is reported as too small, 4 V r V the density will thus b e reported as too great, m D D V Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at viii

9 Foundation Lessons Numbers in Science Exploring Measurements, Significant Digits, and Dimensional Analysis The accuracy of a measurement depends on two factors: the skill of the individual taking the measurement and the capacity of the measuring instrument. When taking measurements, you should always read to the smallest mark on the instrument and then estimate another digit beyond that. Centimeters Centimeters Figure 1. Measuring a steel pellet For example, if you are reading the length of the steel pellet pictured in Figure 1 using only the ruler shown to the left of the pellet, you can confidently say that the measurement is between 1 and 2 centimeters. However, you must also include one additional digit estimating the distance between the 1 and 2 centimeter marks. The correct measurement for this ruler should be reported as 1.4 or 1.5 centimeters. It would be incorrect to report this measurement as 1 centimeter or even 1.45 centimeters given the scale of this ruler. What if you are using the ruler shown on the right of the pellet? What is the correct measurement of the steel pellet using this ruler: 1.4 centimeters, 1.5 centimeters, 1.40 centimeters, or 1.45 centimeters? The correct answer would be 1.45 centimeters. Because the smallest markings on this ruler are in the tenths place, convention states we carry our measurement out to the hundredths place. If the measured value falls exactly on a scale marking, the estimated digit should be zero. The temperature on the thermometer shown in Figure 2 should read 0.0 C. A value of 0 C would imply this measurement had been taken on a thermometer with markings that were 10 apart, not 1 apart. The value 0 C represents anything that will round to the value 0. This means a value could fall between 29.5 C to 0.4 C, or a full 1 of possible error. Yet by including an additional digit, the number 0.0 C indicates a value between C to 0.04 C, or a possible error of only 0.1. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 1

10 Student Activity Numbers in Science Figure 2. Reading a thermometer Accuracy is important, so remember to always report measurements one decimal place past the accuracy of the measuring device. When using instruments with digital readouts, you should record all the digits shown. The instrument has done the estimating for you. When measuring liquids in narrow glass graduated cylinders, most liquids form a slight dip in the middle. This dip is called a meniscus. Your measurement should be read from the bottom of the meniscus. Plastic graduated cylinders do not usually have a meniscus. In this case, you should read the cylinder from the top of the liquid surface. Practice reading the volume contained in the three cylinders shown in Figure. Record your values in the space provided. Left: Middle: Right: Figure. Reading graduated cylinders Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 2

11 Student Activity Numbers in Science Significant Digits There are two types of numbers you will encounter in science, exact numbers and measured numbers. Exact numbers are known to be absolutely correct, and are obtained by counting or by definition. Counting a stack of 12 pennies is an exact number. Defining a day as 24 hours is an exact number. Exact numbers have an infinite number of significant digits. As we have seen previously, measured numbers involve some estimation. Significant digits are digits believed to be correct by the person making and recording a measurement. (We assume that the person is competent in their use of the measuring device.) To count the number of significant digits represented in a measurement, follow some basic rules: 1. If the digit is not a zero, it is significant. 2. If the digit is a zero, it is significant only if: a. It is sandwiched between two other significant digits; or b. It terminates a number containing a decimal place. Examples:.57 ml has three significant digits (Rule 1) 288 ml has three significant digits (Rule 1) 20.8 ml has three significant digits (Rule 1, 2a) ml has four significant digits (Rules 1, 2a, 2b) 0.01 ml has only one significant digit (Rule 1) ml has two significant digits (Rule 1, 2b) ml has three significant digits (Rule 1, 2a, 2b) kg has three significant digits (Rule 1, 2b) Significant Digits in Calculations A calculated number can never contain more significant digits than the measurements used to calculate it. Calculation rules for significant digits fall into two categories: 1. Addition and Subtraction: Answers must be rounded up or down to match the measurement with the least number of decimal places. Example: 7.24 ml ml = ml (calculator value), report as 47.5 ml 2. Multiplication and Division: Answers must be rounded up or down to match the measurement with the least number of significant digits. Example: 1.2 cm 12.4 cm = cm 2 (calculator value), report as 15.2 cm 2 Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at

12 Student Activity Numbers in Science Dimensional Analysis Throughout your study of science, it is important that a unit accompanies all measurements. Keeping track of the units in problems can help you convert one measured quantity into its equivalent quantity of a different unit, or help set up a calculation without the need for a formula. In conversion problems, equality statements such as 1 foot = 12 inches are made into fractions and then strung together in such a way that all units except the one desired are canceled out of the expression. Remember that defined numbers, such as 1 foot or 12 inches, are exact numbers and thus will not affect the number of significant digits in your answer. This method is also known as the Factor-Label method or the Unit-Label method. To set up a conversion problem, follow these steps: 1. Think about and write down all the = statements you know that will help you get from your current unit to the new unit. 2. Make fractions out of your = statements. There should be two fractions for each = and they will be reciprocals of each other.. Begin solving the problem by writing the given amount with units on the left and then choose the fractions that will let a numerator unit be canceled with a denominator unit, and vice versa. 4. Using your calculator, read from left to right and enter the numerator and denominator numbers in order. Precede each numerator number with a multiplication sign and each denominator number with a division sign. Alternatively, you could enter all of the numerators, each separated by a multiplication sign, and then all of the denominators, each separated by a division sign. 5. Round your calculator s answer to the correct number of significant digits based on the number with the least number of significant digits in your original problem. Example 1 How many inches are in 1.25 miles? 1ft 1ft 12in. or 12in. 12in. 1ft 5280 ft 1mile 5280 ft 1mile or 1mile 5280 ft 5280 ft 12in. 1.25mile 79,200 in. 1mile 1ft Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 4

13 Student Activity Numbers in Science As problems get more complex, the measurements may contain fractional units or exponential units. To handle these situations, treat each unit independently. Structure your conversion factors to ensure that all the given units cancel out with a numerator or denominator as appropriate and that your answer ends with the appropriate unit. Sometimes information given in the problem is an equality that will be used as a conversion factor. Squared and cubed units are potentially tricky. Remember that a square centimeter (cm 2 ) is really cm cm. If we need to convert square centimeters to square millimeters (mm 2 ), we need to use the conversion factor of 1 cm = 10 mm twice so that both centimeter units cancel out. Example 2 Suppose your automobile tank holds 2 gallons and the price of gasoline is.5 per liter. How many dollars will it cost you to fill your tank? From a reference table, we find 1 L = 1.06 qt and 4 qt = 1 gal. We should recognize from the problem that the price is also an equality (.5 = 1 L) and we should know that 100 = $1. Setting up the factors, we find 4 qt 1L.5 $1 2gal. $29 1gal qt 1L 100 In your calculator, enter = However, because the given value of 2 gallons has only two significant digits, your answer should be rounded to $29. Example One liter is exactly 1000 cm. How many cubic inches are there in 1.0 liters? We should know that 1000 cm = 1 L, and from a reference table we find that 1 in. = 2.54 cm. Setting up the factors, we find 1000(cm cm cm) 1in. 1in. 1in. 1.0L 61 in. 1L 2.54 cm 2.54 cm 2.54 cm (The answer must have two significant digits because our given value 1.0 L contains two significant digits.) Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 5

14 Student Activity Numbers in Science As you become more comfortable with the concept of unit cancellation, you will find that it is a very handy tool for solving problems. By knowing the units of your given measurements and by focusing on the units of the desired answer, you can derive a formula and correctly calculate an answer. This is especially useful when you have forgotten (or never knew) the formula. Even though you may not know the exact formula for solving this problem, you should be able to match the units up in such a way that only your desired unit does not cancel out. Example 4 What is the volume in liters of 1.5 moles of gas at 29 K and 1.10 atm of pressure? The ideal gas constant is L atm. mol K It is not necessary to know the formula for the ideal gas law to solve this problem correctly. Working from the constant (because it sets the units), we must cancel out every unit except liters. Doing this shows us that moles and Kelvin must be in the numerator and atmospheres in the denominator: atm L (1.5 mol) (29 K) mol K V 2.8 L, or L 1.10 atm The answer is reported to two significant digits because our least accurate measurement (1.5 mol) has only two significant digits. Note: Never rely on the number of significant digits in a constant to determine the number of significant digits for reporting your answer. Consider only the number of significant digits in given or measured quantities. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 6

15 Student Activity Numbers in Science Purpose In this activity, you will review some important aspects of numbers in science and then apply those number handling skills to your own measurements and calculations. Materials Each lab group will need the following: aprons balance beaker, 250 ml goggles graduated cylinder, 100 ml, plastic paper towels die marble ruler, clear metric string Procedure Remember that when taking measurements, it is your responsibility to estimate a digit between the two smallest marks on the measuring instrument. 1. Determine the mass of the small cube on a balance and record your measurement in Table 1 on your student answer page. 2. Measure dimensions (the length, width, and height) of the small cube in centimeters, being careful to use the full measuring capacity of your ruler. Record the lengths of each dimension.. Fill the 250 ml beaker with water to the 100 ml line. Carefully place the cube in the beaker. Record the new, final volume of water. Remove and dry the cube. 4. Fill the large graduated cylinder three fourths full with water and record this initial water volume. While holding the graduated cylinder at an angle, gently slide the cube down the length of the graduated cylinder to submerge the cube. Record the final water volume. 5. Measure the mass of the spherical object on a balance and record your measurement in Table Use the string to measure the widest part, or circumference, of the sphere. Mark the circumference on the string with a pen and the use the ruler to determine the value of the circumference in centimeters. Be careful to use the full measuring capacity of the ruler. 7. Fill the 250 ml beaker with water to the 100 ml line. Carefully place the spherical object in the beaker. Record the new, final volume of water. Remove and dry the spherical object. 8. Fill the large graduated cylinder three fourths full with water. Record this initial water volume. While holding the graduated cylinder at an angle, gently roll the sphere down the length of the graduated cylinder to submerge the sphere. Record the final water volume. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 7

16 Student Activity Numbers in Science Data and Observations Mass (g) Table 1. Measurements and Significant Digits Cube Data Dimensions (cm) Length Width Height Volume (ml) Initial Final Beaker Graduated cylinder Mass (g) Sphere Data Dimensions (cm) Circumference Volume (ml) Initial Final Beaker Graduated cylinder Volume of a cube Formulae for Calculating Circumference of a circle Diameter of a circle Volume of a sphere Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 8

17 Analysis Student Activity Numbers in Science Show your organized work on a separate piece of paper. Transfer your final answers to the blanks beside each question. Staple your work to your answer sheet before turning it in to your teacher. Remember to follow the rules for reporting all data and calculated answers with the correct number of significant digits. Table 2. Common Conversions Length Mass Volume Standard 1 in. = 2.54 cm 1 lb = 16 oz 1 gal. = 4 qts 1 ft = 12 in. 1 qt = 2 pints 1 mile = 5280 ft 1 pint = 2 cups Standard to Metric 1 mile = 1.61 km 1 lb = 454 g 1 L = 1.06 qts 1 m = 1.09 yds 1 kg = 2.21 lbs 1 tsp = 5 ml Metric 1 m = 100 cm 1 g = 1000 mg 1 cm = 1 ml 1 m = 1000 mm 1 kg = 1000 g 1 L = 1000 ml 1 km = 1000 m 1. For each of the measurements recorded in Table 1, indicate the number of significant digits in parentheses after the measurement. For example, 15.7 cm ( sd). 2. Use dimensional analysis to convert the mass of the cube to: a. Milligrams (mg) b. Ounces (oz) Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 9

18 Analysis (continued) Student Activity Numbers in Science. Calculate the volume of the cube in cubic centimeters (cm ). 4. Use dimensional analysis to convert the volume of the cube found in Question from cubic centimeters (cm ) to cubic meters (m ). 5. Calculate the volume of the cube in ml as measured in the beaker. Convert the volume to cubic centimeters (cm ) using 1 cm = 1 ml. 6. Calculate the volume of the cube in ml as measured in the graduated cylinder. Convert the volume to cubic centimeters (cm ) using 1 cm = 1 ml. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 10

19 Analysis (continued) Student Activity Numbers in Science 7. Using the density formula mass D volume calculate the density of the cube as determined by the following instruments: a. Ruler b. Beaker c. Graduated cylinder 8. Use dimensional analysis to convert the three densities found in Question 7 into kg/m. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 11

20 Analysis (continued) Student Activity Numbers in Science 9. Convert the mass of the sphere to the following units: a. Kilograms (kg) b. Pounds (lbs) 10. Using the measured circumference, calculate the diameter of the sphere in centimeters. 11. Calculate the radius of the sphere in centimeters. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 12

21 Analysis (continued) Student Activity Numbers in Science 12. Calculate the volume of the sphere in cubic centimeters (cm ) from its calculated radius. 1. Calculate the volume of the sphere in ml as measured in the beaker. Convert this volume to cubic centimeters (cm ) using 1 cm = 1 ml. 14. Calculate the volume of the sphere in ml as measured in the graduated cylinder. Convert this volume to cubic centimeters (cm ) using 1 cm = 1 ml. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 1

22 Analysis (continued) Student Activity Numbers in Science 15. Using the density formula mass D volume calculate the density of the sphere as determined by the following instruments: a. Tape measure b. Beaker c. Graduated cylinder 16. Use dimensional analysis to convert the three densities found in Question 15 into lbs/ft. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 14

23 Conclusion Questions Student Activity Numbers in Science 1. Compare the densities of the cube when the volume is measured by the ruler, beaker, and graduated cylinder. Which of these instruments gave the most accurate density value? Use the concept of significant digits to explain your answer. 2. A student first measures the volume of the cube by water displacement using the graduated cylinder. Next, the student measures the mass of the cube before drying it. How will this error affect the calculated density of the cube? Your answer must be justified and should state clearly whether the calculated density will increase, decrease, or remain the same.. A student measures the circumference of a sphere at a point slightly above the middle of the sphere. How will this error affect the calculated density of the sphere? Your answer must be justified and should state clearly whether the calculated density will increase, decrease, o r remain the same. Copyright 2012 Laying the Foundation, Inc., Dallas, Texas. All rights reserved. Visit us online at 15