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How Are Units of Measurement Related to One Another?

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How Are Units of Measurement Related to One Another? UNIT 1 Measurement How are Units of Measurement Related to One Another? I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind... Lord Kelvin (1824-1907), developer of the absolute scale of temperature measurement Engage: Is Your Locker Big Enough for Your Lunch and Your Galoshes? A. Construct a list of ten units of measurement. Explain the numeric relationship among any three of the ten units you have listed. Before Studying this Unit After Studying this Unit Unit 1 Page 1 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach 1. Use the measuring instrument provided to you by your teacher to measure your locker (or other rectangular three-dimensional object, if assigned) in meters. Table 1: Locker Measurements Measurement (in meters) Uncertainty in Measurement (in meters) Width Height Depth (optional) Area of Locker Door or Volume of Locker Show Your Work! Pool class data as instructed by your teacher. Table 2: Class Data Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Width Height Depth Area of Locker Door or Volume of Locker Unit 1 Page 2 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. Unit 1 Measurement 2. Did all class members report their measurement data to the same number of decimal places? Did all class members report the same uncertainty in their measured quantities? Explain any differences. 3. Summarize your class discussion about uncertainty in measurement. What determines the number of digits that should be expressed in a measured quantity? How many digits should be expressed in reporting your locker’s (or other object’s) volume? Use the terminology from the class discussion where appropriate. Unit 1 Page 3 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach Explore 1: How do Scientists Express Uncertainty in Measurement? 4. Consider the illustration below. Figure from: Cracolice, M.S., & Peters, E. I. (2007). Introductory Chemistry: An Active Learning Approach (4th Ed.). Belmont, CA: Brooks/Cole Cengage Learning. Each part of the illustration shows a board being measured with meter sticks that have different graduation marks. Using a ± value to express uncertainty, state the length of the board in meters based on the accuracy of each meter stick. Explain 1 5. Scientists sometimes use the ± convention to express the uncertainty in measured values. Another convention is called significant figures. The measured quantity is expressed as the number of digits known accurately plus one estimated or uncertain digit. Express each measurement using the significant figure convention. Unit 1 Page 4 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. Unit 1 Measurement 6. If you count the number of digits in a quantity expressed using the significant figures convention, you are counting the number of significant figures in that quantity. You begin with the first nonzero digit and end with the uncertain digit. How many significant figures are in each measurement? Explain. 7. Express the length of the board in each case in centimeters. How do the number of significant figures in each measured quantity compare when the quantity is expressed in meters? in centimeters? Why do the number of significant figures compare as they do? 8. What does the location of a decimal point have to do with significant figures? (Hint: Consider the measurements in meters in Item 5 with measurements of the same object in centimeters in Item 7.) 9. If you carefully count the number of paper clips in a box and determine that the box contains 14 clips, how many significant figures are in the counted quantity? Explain the role of the significant figure convention when applied to exact numbers. 10. The masses of two different items were measured on two different balances. Their masses were recorded and correctly expressed according to the significant figure convention: 43.1 g, 234.99 g. What is the total mass of the two objects? Explain how you decided upon the number of significant figures in the sum. Unit 1 Page 5 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach Explore 2: Are You a Fat Head? 11. Imagine living long ago in a medieval kingdom. The standard measurement units were often based on anatomical parts of the ruler of the time hence the origin of the foot. You are the King or Queen of your kingdom and the new unit of measure will be based on the circumference of your royal cranium. Construct a standard length unit based on the circumference of your cranium by wrapping a string around the head, along the eyebrows and above the ears. Carefully cut the string so that it is matches the distance around your royal cranium. Name your unit after your royal highness, such as 1 Jennyhead or 1 Tommyhead. Develop a method of improving the precision and accuracy of your measuring device. Note: you may not use a calibrated measuring device such as a ruler. Using your personal measurement unit, determine the lengths of the objects provided by your teacher. For example, how many JennyHeads is a paperclip, textbook, or a desk? Record your measurements in the table below. Object Description Length Length (Personal Measuring (Centimeters) Unit) a) b) c) d) 12. Explain how you improved your measuring device and why it was improved as a result of your modifications. Unit 1 Page 6 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. Unit 1 Measurement 13. Things are going along well in your little kingdom until the evil French invader, Marquis de’ Centimeter arrives and forces your kingdom to adopt their standard units, called the metric system. The Marquis is ordering you to re-measure objects A-D using a meter stick and record the measurements, in centimeters, in the data table. 14. On the grid below, construct a plot of your unit of measurement with your improved device on the y-axis and length in centimeters on the x-axis. Use the grid on the next page to construct the opposite plot: length in centimeters on the y-axis and your unit of measurement on the x-axis. Unit 1 Page 7 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach Unit 1 Page 8 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. Unit 1 Measurement Explain 2 15. (a) Determine the slope of the line of best fit for the personal unit vs. metric length graph. (b) Is (0,0) a valid point for these data? Why or why not? (c) Determine the equation of the best-fit line. (d) Determine the equation of the best-fit line for the metric length vs. personal unit graph. 16. What do the equations of the lines tell you? 17. What is the reciprocal of the slope of the line determined in the first graph? How does it compare with the slope of the line in the second graph? Why does this relationship exist? 18. Determine the length in centimeters of your personal unit of measure by using a meter stick calibrated in millimeters. Unit 1 Page 9 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction. High School Chemistry: An Inquiry Approach 19. Combine with another lab group to analyze your data and discuss the relationship between the two graphs, the slopes of the two best-fit lines, and the value found when your personal unit of measure was compared to meters. Summarize your findings. Thinking About Your Thinking Proportional Reasoning 20. In the margin next to your answer to the previous question, there is a thinker icon and the words “Thinking About Your Thinking: Proportional Reasoning.” Why did your thinking about the answer to that question involve proportional reasoning? What is proportional reasoning? 21. Explain how the relationship between your personal unit of measure and the meter involves a proportionality. 22. Calculate the length of the following in your personal unit of measure and also in meters (the objects A–D that you measured now become a unti of measure.) You must document and support the reasoning or proportionalities you used to determine your specific values. (a) 7.5 object A’s (b) 0.75 object B’s (c) 5.25 object C’s (d) 7.65 object D’s Unit 1 Page 10 Copyright © 2012 Montana Partners This project was largely funded by an ESEA, Title II Part B Mathematics and Science Partnership grant through the Montana Office of Public Instruction.
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