*In this post, Chemistry Teacher Scott Milam discusses how he helps is students to develop rich understanding of measurment that goes beyond memorized sig fig rules. *

Do we need to round using significant figures? I’ve been asked this question so many times over my career that my answer is automated in my memory. “Try your best to round properly, but don’t stress about it.” There are many examples of significant figures that do not always lead to a universal agreement about the final result. For example, if I mass out 0.01 g on a scale 100 times, how many grams will I have? Will I have 1.00 g? Will I have 1 g? Some rules would suggest 1.00 g is appropriate. But I would disagree and I’d like to use an experiment to justify why.

The Geosolids experiment has students measure the volume of water in a specific Geoslid. The student must determine the area of the base along with the height of the water. Some shapes are more challenging than others. But students should eventually determine the volume in units of cm3. Then the student pours the water from the Geosolid into a graduated cylinder. They measure the volume in mL. This experiment is one of our new activities in PivotPIVOT Interactive focused on Mmodeling Iinstruction.

Students put their data into a whiteboard where they put a table of their data, a plot of their data, and a line of best fit (y=mx+b). In a circle, we make some observations about any patterns that students notice about the numerical data. We then analyze the graphs. Students are asked about the y-intercept values. Should they be 0?

We then spend some time on the slope. Which set of units (mL or cm3) should be the independent variable? What would happen if two groups had the same data, but switched the axes? And most importantly, we construct our very first set of “For every” statements. A “For every” statement must include three components.

- Begin with the two words “For every”
- Include the slope value
- Describe the physical meaning of the slope

Some example (I’m going to use 0.9 as the slope for these) statements provided were:

- For every increase in cm3 of water, there was an increase of 0.9 mL.
- For every 1 cm3, there was an increase of 0.9 mL.
- For every volume unit in cm3, the mL are 0.9 times bigger
- For every increase in cm3, the slope goes up by 0.9 mL

“For every” statements are challenging for students to construct initially. They do not always notice small distinctions between them. Whether or not the number 1 is explicit is critical, but students will often omit or not notice the omission. Does the word increase change the meaning? Is the statement better or worse with “increase” included?

After this analysis, it’s time to get to a point of consensus. Why don’t students come up with a slope of 1 and an intercept of 0? No group has ever gotten such a result even though 1 mL is by definition equivalent to 1 cm3. Students initially think that they did the lab incorrectly. They did not. This is not an error issue. This is a measurement issue.

When we measure something to be 6.2 cm, that measurement is different from a number. The number 6.2 in a math class is a single point. But a measurement of 6.2 cm is a range of possible values. Students have spent their entire lives working with numbers. But in measurement, we are going to ask them to consider a new perspective. When we make a measurement, the precision is how narrow the range of values are. If we use a precise measuring apparatus, we should end up with a precise measurement that has a narrow range of values. And if we manipulate that measurement somehow (multiplying, adding, etc.) then we must communicate the final result in a way that represents the ranges of each measurement.All measurements, no matter how precise, are a range of possible values, not a single number.

Let’s return to the highly unusual example of measuring 0.01 g 100 times. Each of those is probably somewhere around the range of 0.005-0.015 g. It is possible then that our final result will be somewhere between 0.5-1.5 g. If we state that the result is 1.00 g, we are misrepresenting the precision of our range. Instead an answer of 1 g is more appropriate.

The concept of measurement uncertainty is more important than the application of significant figure rules. And it is very difficult to convert students from a 10 year tradition of mathematical numbers to measurements being something new and different. When I was in high school, our chemistry teacher provided us the rules for significant figures without justification. I asked her if 6x4 would be 20, since according to her rules it would be rounded to 1 significant digit. She responded that it would be 20 and I remember other students getting upset. They considered this to be lying. But they didn’t understand that 6 was not the number 6. It was a range about 6. And this particular range is not precise. To specify the result as 24 would not match the wide range of results that 6x4 represents.

In this set of lessons I’m trying to get students to learn three things.

- Measurements are fundamentally different from numbers.
- We need to record measurements in a way that matches the precision of the instrument.
- We need to communicate our level of uncertainty or range of measurements correctly when we manipulate measurements by doing calculations.

Significant figure rules fail at communicating all of these things.

To help tie these ideas together, we have students make measurements using the wooden boards shown below. When students measure with the first one, they should come up with 60 or 70 cm for the red dot. The second ruler should produce a measurement of 63 or 64 cm for the same red dot. The third ruler has the most precise measurement with a result between 63.2-63.5 cm depending on where on the red dot the measurement is taken (what assumptions do we make when we measure?). This shows the student how the number of lines influences are ability to narrow the range. This also shows students how a measurement such as 100 mL is a poor measurement with a wide range of possible values. 100.0 mL is a very different measurement even though the numerical value appears the same.