Statistical Process Control

Activity 1 (Teacher Copy)
Measuring and Means

• Precision measuring devices-see second bullet under Introduction [optional]

A framework for the unit will be laid by having students discuss quality assurance in industry, precision measurement and measuring devices, and the use of metric units for some measurement. Students will exercise the prerequisite skills of rounding decimals, measuring length in tenths of centimeters, and finding the mean of decimals in tenths. Use with student copy.

• To become familiar with the role of quality assurance in industry.
• To list items that would be measured in small units or precise measures involving small units or partial units, and to name appropriate devices for making such measurements.
• To name items that might be measured in metric units, and to give reasons why some items might be manufactured in metric units.
• To round a decimal to the nearest tenth or hundredth.
• To measure length to the nearest tenth of a centimeter.
• To find the mean of several decimals in tenths to the nearest tenth.

Introduction

• give examples of industrially produced items that would be measured in small units of length and/or to a very precise degree of measurement (nails, bolts, tongue-and-groove hardwood flooring ["close tolerance" required], computer chips, ball bearings, etc.)
• name measuring devices that can be used for finding the length of items in very small units or fractional parts of units (rulers/yardsticks/meter sticks/tape measures ruled in fractions of an inch or in centimeters or millimeters, micrometers, calipers, measuring pins, devices that use sonar/radio waves or laser beams); if possible, have some on hand for students to examine and, perhaps, to use to measure some items
• tell why the length of some items might be given in metric measurements, for example, centimeters, millimeters, or fractional parts of these given as decimals (for items that will be exported to countries that use the metric system, replacement parts for items made overseas, etc.)
• name some items that are likely to be or might be measured in small metric units (examples: metric bolts, used in the auto industry and in some electronics equipment; medical supplies-catheters, wire guides, needles)
  
• [You might also want to discuss why the United States has not converted from the English or customary measurement system to the metric system in light of its minority status in using its traditional system, the importance of a standardized system for international trade in an increasingly "shrinking globe," and the greater ease of using the metric system (you might also want to compare the relative ease of using the two systems where pertinent in this unit). (Reasons might include adherence to tradition and difficulty in effecting change among people in such an entrenched and familiar way of doing things in everyday life, and historical self sufficiency because of the size of the nation as well as its physical/geographical separation from others.)]

Part A

1) 3.6 (mean). The first drawing uses a number line to show that 3.56 is closer to 3.6 (3.60) than to 3.5 (3.50). The second shows the same by examining the fourth box in which each row represents one tenth (note that the first three boxes are optional because the wholes may be assumed).

Click for the image

2) Decimals rounded to the nearest tenth:

A. 12.719 cm	12.7 cm	D. 8.02 cm	8.0 cm
B. 0.97 cm	1.0 cm	E. 7 cm	7.0 cm
C. 6.9 cm	6.9 cm	F. 0.6239 cm	0.6 cm

3) Decimals rounded to the nearest hundredth:

A. 0.098 cm	0.10 cm	C. 0.19 cm	0.19 cm
B. 4.63501 cm	4.64 cm	D. 15.6827 cm	15.68 cm

4) Find the place value to which you are rounding (e.g., tenths). Look at the next place value to the right (hundredths) to determine whether to round the tenths digit up one or to keep it the same. If the hundredths digit is 5-9, the tenths digit is raised one; if the hundredths digit is 0-4, the tenths digit stays the same. All digits after the place to which you are rounding are dropped, but there should be a number in each place up through the place value to which you are rounding, even if the last few digits are zeros.

Part B

1) (You might want to allow answers that are off by one-tenth of a centimeter. Also, photocopy machines often reduce images slightly. This could change the answers given below.)

1. 6.5 cm
2. 5.3 cm
3. 7.8 cm
4. 4.2 cm
5. 6.0 cm

1. coils longer than the mean-A, C; shorter-B, D; close/same-E
   
   
2. All coils would extend exactly (or, almost exactly, because of the possibility of rounding error) to the vertical line.

1. • estimated mean: 6 to 7 cm; reasons will vary but might include finding a point where total points above and below it are about the same
   • actual mean: 6.4 cm (rounded from 6.425)
   • lengths longer than the mean-7.0, 8.3; shorter-4.9, 5.5; closest- 7.0
   [You might also want to ask which length is next closest-5.5, and if the two closest numbers to the mean must always include one above and one below the mean (no) and have students support their answers.]
   
   
2. • estimated mean: 6 to 6.5 cm
   • actual mean: 6.2 cm (rounded from 6.24)
   • lengths longer than the mean-6.6, 9.1; shorter-3.6, 5.7 (might include 6.2 because mean rounded down); closest/same-6.2

Extensions/Discussion Questions

1. See drawings in Part A #1 for examples.
   
   
2. If the measurement falls between two of the designated units or partial units to which you are finding a measurement, the measurement used is the one to which the actual measurement falls closest (rounding up if halfway).
   
   
3. Two (e.g., see the previous item).
   
   
4. Because we have to agree on a procedure that is universal/standard among people performing mathematics everywhere for better communication and exchange, and there are probably more cases in which it is practical to round up and have too much/too many than to round down and have too little/too few.
   
   
5. • To reduce the possibility of error in reading written decimals.
   • To reduce the possibility of error/interpretation-for example, combining other numbers written too closely to the whole number or being uncertain that a number had been rounded to the tenths place (rather than to the nearest whole number).
   The zero could be omitted in both cases, and the numbers are still correct. However, chances of accuracy are raised through the enhanced clarity provided by using the zeros (important to precise fields such as math and science).