Every year when the day came to discuss the rules for significant figures in measurements with my classes I would write the rules on the board, we’d work through a couple examples, and I’d try to find a way to explain why we needed to use them when reporting measurements. This has never been my favorite topic to teach, mostly because I had a difficult time helping students see why these rules for measurement and reporting uncertainty were important.
My students came to believe that sig figs were only important because there would be a grade penalty for not reporting to the right number of digits.
This year I was determined to do a better job. The modeling curriculum has a sequence of activities I used to help illustrate how and why we report uncertainty in measurement.
The sequence begins with practicing reading various devices – rulers, graduated cylinders, beakers, and burets. Students work in groups to make the readings and report them to the class. Answers are compared. Ideally, some students will report the measurements to different degrees of precision, some will only report the numbers they can read, some will estimate an additional digit, some will estimate several additional digits. This will lead into a class discussion on the need to have a universal way for all scientists to report their measurements. (I reminded them of the Build-A-Boat activity when they didn’t all report their measurements the same way resulting in unfair advantages to some and disadvantages to others.) We discuss how all measurements have some degree of uncertainty and some devices have different levels of precision. We talk about the meaning of precision and accuracy. Some questions that came up in my classes were:
Why is a buret more precise than a beaker?
What determines a device’s precision?
How can you determine the precision of a digital device when there are no estimated digits?
When reading a device, how can we make sure that the values we report are accurate?
How can we express our uncertainty in the measurement?
To wrap up, we establish that a good rule to follow is to report all the values we can read with certainty and one estimated value, which is uncertain. Except on digital balances and thermometers because we can’t estimate a value on digital devices.
Now, this type of activity isn’t novel or unique. Many of us do this or something similar every year. This year, after being inspired by one of my colleagues, I made one subtle change in my presentation and vocabulary and I believe it laid the foundation for a more meaningful understanding of significant figures. The change? I don’t ever say “significant figures”. After establishing that we all agree to read every digit and estimate one more, I simply refer to this as reporting to the appropriate level of precision. This phrasing is now anchored to the experiences students had in class. This phrasing is more meaningful to the students than establishing this as the foundation for the “rules for significant figures”. I now refer to this sequence of instruction as “Making Measurements Meaningful”.
The next activity is one that reinforces the ideas of precision and uncertainty and demonstrates how we establish the rules for calculating with significant figures the appropriate level of precision. This activity is provided in Unit 1 of the Chemistry Modeling teacher’s notes.
I passed out a paint stirring stick to each group. I told them we were going to use the new, non-conventional measuring device. (The activity calls this device a “glug”, but you can name it anything you want.) The stir stick is longer than a foot so it’s not comparable to using inches or centimeters. The glug is not calibrated, so the only certain number we can read on the glug is “1 glug”. I asked my students to measure the length and width of their whiteboards and then determine the area. I told them to assign meaning to the values they report (this is my phrasing for “label your answer with appropriate units”). They also needed to justify the value they reported for area since it was calculated and not measured, this is my phrasing for “show your work”. They were asked to follow the standard measurement rules and estimate one beyond what they could read when reporting the measurement. I asked the following questions.
If we assume all the whiteboards are the same size, should we expect everyone to report the exact same values?
If not, can you guess on what digit (ones, tenths, something else) the groups will agree and disagree?
I then put all the class data on the board so everyone could see and compare the range of values we obtained. Generally, the groups will only differ in the digit reported in the tenths place. I asked the following questions.
How can everyone agree on the whole number value reported, but disagree on the next value?
If someone reported a value to the hundredths place, is it appropriate to do so?
Which reported digit is uncertain?
What is our range of uncertainty?
What is the precision of this device?
How precisely can we report the calculated area?
If our device is uncertain in the tenths place, does it make sense to report an area – a value calculated using measurements from this device – to the hundredths (or beyond) place?
Next, we calibrated our glugs to the tenths place. I passed out masking tape so students could cover their glugs – this saves me from having to go the Home Depot every year and getting a whole new set of glugs. I then followed the these instructions taken from the teacher’s notes in Unit 1 of the Chemistry Modeling Materials:
“Tape a couple of sheets of (college ruled) lined notebook paper side-by-side. Lay the glug on the paper so that one end touches one line and the other end touches the 20th line below the first. Mark the glug where every 2nd line on the notebook paper touches the side of the glug. You have now divided the glug into 10 equal intervals, so that it is calibrated to 10ths.”
I then had the students measure the whiteboard again with our more precise glug. When I recorded the new values on the board the students saw that our uncertainty was greatly reduced. We now had agreement in the ones and tenths places and we were only uncertain in the hundredths place. We talked about how to report the area value now. How many digits could be included, now that the device was more precise? We talked about how many digits we knew for certain when we made the measurements and how it wouldn’t be appropriate to report a calculated value with more certain digits than we obtained form the original device. The students could now tie our rules for reporting precision when multiplying or dividing to an experience then had in class.
Do you use any guided-inquiry activities to help students understand precision, accuracy, and significant figures?