The concept of Measurement and Significant Figures can induce discomfort.^{1} This article and the associated resources take an empirical, logical approach to this important concept, which your students will understand, and dare I say, enjoy.

As a disclaimer, this is not a scholarly treatment. But it will resonate with high school students and will provide them with a solid, conceptual foundation, with virtually no reliance on rote memorizing.

I suggest teaching Measurement and Significant Figures immediately *before* it is necessary. That is, Measurement and Significant Figures should not be the first topic in 11 Chemistry, to be set aside until the Mole Concept rolls around a month or two or three later.

Before teaching anything about measurement and significant figures, I recommend that students perform the following two activities,^{2} followed by a thorough post-mortem. This will establish an empirical—and logical—foundation for this important concept.

While these activities can be completed in an 80-minute period, a more leisurely approach might not be a bad idea. Regardless, a fulsome post-lab discussion is called for.

**1. Precision—and confidence level—of Glassware—empirically determined**

Each group (pair) of students requires:

Five pieces of volumetric glassware:

- 50 mL Erlenmeyer flask
- 50 mL beaker
- 10 mL graduated cylinder
- buret filled with water, mounted in a stand
- 10 mL volumetric pipet with pipet bulb (optional; these are not graduated)
- plastic Beral pipet
- extra 50 or 100 mL beaker
- large beaker of tap water
- access to a centigram electronic balance

Students are to use each piece of glassware __one time only__ to measure 10 mL of water as precisely as they can. They then pour the water into the tared “extra” beaker and weigh it. Since the density of water is 1.00 g·mL^{–1}, the mass of water delivered in grams may be converted directly to volume in mL.

Students will see that a 50 mL Erlenmeyer flask and a 50 mL beaker will not deliver anywhere near 10 mL of water, in spite of this being indicated on the (imprecise) graduations. More precise volumetric instruments—with more finely marked graduations—are the graduated cylinder, and, of course, the buret. The most precise piece of volumetric glassware, when used correctly, is the volumetric pipet.^{3}

This allows you to start a conversation about *confidence level*. A beaker or flask, whose graduations are few and imprecise, has a low confidence level, while a finely graduated device, such as a buret is much more precise—and affords a higher *confidence level*. Similarly, a millimetre ruler is more precise than a centimetre ruler.

This is what it’s about: the precision of the measuring device determines *confidence level.*

**2. Determination of the thickness of Aluminium Foil**^{4}

Each pair of students requires:

- plastic millimetre ruler more than 10 cm long
- a teacher-supplied ca 10 cm x 10 cm square of Aluminium foil—not folded.
^{5} - centigram electronic balance

Without folding the Al foil, students are to determine its thickness, in millimetres. It won’t take long to realize that the foil is a thin rectangular prism. Students will need access to the density of Al (ρ_{Al} = 2.70 g·cm^{–3}).^{6}

The rectangular prism below represents the square of Al foil.

**Figure 1.** A square of Al foil is a rectangular prism

mass of foil = (volume) ∙ (density)

= (*l ∙ w ∙ h*) ∙ density

Solving for the height (thickness), *h*, gives:

Have each pair of students write their value for the thickness of Al foil, in mm, on the board.

Here’s a sample of what you’ll get from your class:^{7}

**Table 1.*** Sample class data for the thickness of a 10 cm x 10 cm square of Al foil*

a) Students will report __varying quantities of significant digits__ (column 5 in Table 1) which, at this point, is exactly what we want. This is where the lesson begins. It begs the question: How can a modest electronic balance and a one-dollar plastic ruler determine the thickness of a piece of aluminium foil precise to one-hundred-millionth of a millimetre?

b) Students will invariably under-report the precision of the length and width of the foil—they don’t know how to use an analogue measuring device, such as a mm ruler, the way a scientist does. Students likely don’t know that every *measured* quantity must have one estimated (uncertain) digit, which is obtained by __estimating between the two smallest scale divisions__. For example, the top-most length value from Table 1 must be reported to the tenth of a millimetre, not 10.1 cm as shown, but, say 10.10 cm or 10.16 cm, depending on the length. We are certain about the tenth of a centimetre (or mm)–it is the tenth of a millimetre that must be estimated. This ties-in nicely with the volumetric glassware activity: the confidence level of a measurement has a lot to do with the precision of the graduations and how the device is read. And so an inexpensive plastic millimetre ruler, when measuring something over 10 cm in length, will have four significant figure precision, or a *confidence level* of four sig figs.

Estimating between the two smallest scale divisions on an analogue device will be a major paradigm shift for students. Expect to nag your students, pretty much until the end of AP Chemistry...

For measuring devices with a digital readout, such as an electronic balance, keep it simple.^{8} Assume the rightmost digit to be uncertain. This will strike students as reasonable; they often see the final digit fluctuate. In high school, there is no need to go into the +/- thing. Stick to the underlying concept of *confidence level*.

All that said, students may be surprised that a one-dollar plastic millimetre ruler is more precise than a centigram digital balance that costs a heckuva lot more.

c) Students now see that they under-reported the length and width of the foil—these measurements are precise to four significant figure and that the mass of the foil is precise to *two significant figures*. Keeping in mind the fact that a measured quantity—and a quantity that is calculated based on measured data—__must__ have one estimated (uncertain) digit, look at the data in the final column of Table 1. When rounded to two significant figures, every group reported the same thickness—0.02__3__ mm, which, to two significant digits, is the same as 0.02__4__ mm.^{9} I make it a point to ALWAYS underline the estimated (uncertain) digit; this helps students keep track of confidence level. This discussion will also show students that anything beyond two significant figures is scientifically meaningless; there is no confidence in the value. The values are mathematically correct, but physically meaningless.

**3. Formal Lesson on Measurement and Significant Figures, followed by Practice Questions**

After these introductory laboratory activities, students are ready for a formal lesson on Measurement and Significant Figures, followed by some decent practice problems^{10}. I suggest that you consider wrapping up with a data analysis activity on the Determination of the Density of Ethanol^{11}. Student-determined data, given in the activity, are provided in Table 2.

**Table 2.*** Student-determined data for the determination of the density of ethanol.*

The real value of this exercise is in the data analysis using spreadsheet software.

**Figure 2.** Graphical analysis (EXCEL) of student-determined data for the determination of the density of Ethanol

The best-fit lines and the accompanying R^{2} values are given in figure 2. From the given data, the fewest quantity of significant figures is in the volume of ethanol: 5.0 mL is the smallest volume, which determines the confidence limit of these data: two significant figures. This means that the density of ethanol, represented by the slope of the lines, should be reported as 0.81 g·mL^{–1}, the same as what students will report when they work through the Analysis Questions. Graphing software will calculate the slope of the line—*mathematically*. But the chemist must adjust this according to the confidence level of his or her measured data. Further, according to the precision of the electronic balance, with is 0.01 g, the y-axis (mass axis) intercept should be 43.37 g for the data that includes the mass of the graduated cylinder; 0.06 g for the data that excludes this. This will make for an interesting, eye-opening discussion.

Although I *ask* students to submit the assignment before we discuss it in class, I typically don’t collect it until the class *after* we have the post-lab discussion. This allows students to reflect on what was talked about, and to redo the assignment as necessary. Only then do I grade it. This approach, which I use frequently, results in deeper learning—and higher grades.^{12}

As a final piece, you may want to ask students, as part of the written test, to do a quick lab component: the determination of the density of water, given a specific volumetric device and balance.

And there you have it, gentle readers. An empirically based approach to measurement and significant figures, accompanied by a formal lesson and a set of practice problems. The emphasis throughout all of this is *confidence* *limit*—in measurements and in calculated values based on these measurements. (You may direct you students to this website.^{13)}

Now students will be ready to immediately begin the Mole Concept, or any quantitative aspect of Chemistry.

See the **Supporting Information** to access several documents and PowerPoint slides. (Readers must log into their account to access. Not a member? Register for free!)

- I speak from experience ~ as a student and as a teacher.
- Please refer to the accompanying materials. These are NOT my ideas; the Determination of the Thickness of Aluminium foil I got from an NSTA conference in Las Vegas (!)—back in the ‘90s; the precision of glassware reference is anybody’s guess. I am indebted to the teaching master who thought of these excellent activities.
- Since a volumetric pipet is not graduated, you may omit it. If you choose to include the volumetric pipet, this will give you the opportunity to instruct your students in its proper use, which will come up later in the course. (Repetition is a good thing.) When you view this video, note that the chemist is pipetting with his index finger—not his thumb—atop the pipet—not his thumb. This is the appropriate technique, said Karen Henderson, my top-notch Analytical Chemistry laboratory instructor at the U of Toronto, back in the day.
- While the accompanying handout is not essential, you may find it useful.
- a) Make sure that each side measures at least 10.00 cm—not under. This will give four significant figures. b) All groups to use foil taken from the same roll. This facilitates comparison of thickness values.
- Aluminium - Wikipedia
- Heavy duty foil will be thicker than regular foil.
- This is high school, not a IUPAC lab.
- The right-most (ie uncertain) digit in a measurement provides some “wiggle” room. That’s why 0.02
__3__mm is the same as 0.02__4__mm. I make it a point to always underline the estimated, or uncertain, digit. - Please refer to the accompanying PPT and student handout.
- See the accompanying activity. This idea is courtesy of Herb DeRuyter, Chem 13 News, February, 2008.
- I call this “win-win”.
- Significant Figures (purdue.edu)

Arrange materials listed.

Herb DeRuyter, Chem 13 News, February, 2008.

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