Practice Reading Between the Lines
Take a look at Example 2 in your workbook to look at some length measurements. (These examples will also be shown on-line but may be distorted due to digital reconfiguration.) The scales shown here are arbitrary, so we will just work with the numbers and not with the units.
Take a look at the lines drawn above each scale. If you can measure each of the lines shown in the workbook and get the same value that is shown below the scale, feel free to move on to the next topic (Significant Digits). If you want additional explanation of these measurements, continue with the examples shown below (Example 2).
Example 2 - parts a, b, c.
On this scale the numbers mark 1, 2 and 3 units. Each numbered division is divided into 10 parts, so each marked division is 0.1 unit. You are expected to read between the lines, so you need to report measurements to 0.01 unit. This is not hard but it is important.
Look at line a. It extends past the 1.8 mark on the scale. By my estimate it goes to 1.84 so that is what I would write down. You might see it the same way or estimate it at 1.83 or 1.85. Either of those would be a reasonable measurement.
Line b presents a bit of a problem. It extends to the 1.5 mark on the scale and no further. I could write down 1.5 but I'm supposed to read between the lines. Since it seems to me to line up with the mark I would write 1.50. That gives the measurement to the nearest 0.01 unit or says that the measurement is right on the line as far as I can tell. In this situation that 0 is a measured value and should be included in the measurement.
Line c is somewhat similar to line b. Here the length of the line matches a numbered mark. So I needed to put in two 0's in order to show that the measurement was measured to 0.01 unit.
Also note that I didn't extend any of these numbers by using any extra unmeasured digits. When dealing with precision, 1.50 does not mean the same thing as 1.500. When writing down a measurement, put down all measured numbers including the estimate between the lines, or zero if on the line, but no more.
Example 2 - Parts d and e.
On this scale, the numbers mark 10, 20 and 30 units. Again each numbered division is divided into 10 parts, so each mark is 1 unit. Since we read between the lines, measurements should be made to 0.1 unit. Thus line d is 14.3 rather than 14 or 14.30. Occasionally in cases like this, people will forget that each mark is 1 unit instead of 0.1 unit and read the length as 10.43. Try to avoid that mistake if you can.
Line e measures at more than 20 and less than 21 and is estimated to be 20.8. Be careful to note that the value is 20.8 and not 28.
Before going on to the next scale I would like you to review the correct and incorrect values given in the workbook for each of the lines a-e. You should be able to say what is right and wrong about each of those values. If you have any trouble check with your instructor to get squared away.
Example 2 - Parts f and g.
On this scale the numbers mark 1000, 2000 and 3000 units. Each numbered division is divided into 10 parts, so each marked division is 100 units. Since we can read between the lines the measurements can be made to a precision of 10 units.
Line f extends more than 7 marks (but less than 8 marks) past 1000. So its length is between 1700 and 1800. We can estimate between the marks to get 1740. Note that even though there is a zero at the end of this number it was never measured. We couldn't get that much precision from this scale. It is there to show size (1740 rather than 174) but not precision (1740 rather than 1741 or 1738). There are ways of dealing with this dilemma. Most important right now is that you recognize it as a dilemma, that properly showing the precision of a measurement is important enough that that "extra" zero is a problem.
Line g extends to the 500 mark but not beyond, so I would write 500 rather than 490 or 510. Because this scale allows a precision of +10 units, the zero in the ten's place is a measured (estimated) digit. Note that the one's place has not been measured. Measuring line g creates the additional problem of having one zero that does measure precision and another one that does not. Study the examples of lines f and g to make sure that you understand which digits (especially zeros) represent measured values and which zeros are placeholders.
Example 2 - Parts h and i.
On this scale the numbers mark 0.1, 0.2, and 0.3 units. Each division represents 0.01 units. By reading between the lines we should be able to make measurements with a precision of 0.001 units.
The problem with lines h and i is that they are smaller than even the first number on the scale. Because of this our measurements are going to have leading zeros.
Line h is less than 0.1 units so it will be 0.0 something. It extends past the sixth mark so it is 0.06 something. I estimate it to be seven tenths of the way to the next mark so its length is written as 0.067 units.
Line i not only doesn't make it to the first number, it doesn't even make it to the first mark so its length is less than 0.01. I estimate it to be 0.006 units.
E-mail instructor: Eden Francis
Clackamas Community College