This activity is designed to provide students with many types of scales on measuring devices in order to reinforce the idea that the physical graduations on an instrument determine the number of digits which need to be recorded. This activity also helps students address how to choose appropriately precise instruments when measuring and how to properly physically measure masses, volumes, and lengths of objects.

Very often students ask, “How many decimals should I write down?” when completing a laboratory investigation. Although frustrating for science teachers, the significant figures misconceptions teachers address in the lab exist because of common practices in math classes. For instance, most students have been told in the past to record two decimal places for every answer on math worksheets. Students have a difficult time overcoming these practices from their math disciplines when moved into a science class because in one subject they deal with exact numbers, while in the other they deal with measured numbers. Problematically, the difference between a measured number and an exact number is not often addressed in math classes. Chemistry class is often the very first place that the issue of inherent uncertainty present in a measurement due to a tool is necessary for success at a task, such as measuring the density of an object, or to determine the percent yield of a reaction.

Moreover, understanding that the precision of the tool dictates the number of recorded digits in a measurement poses a challenge due to the limited number of rulers and other measuring devices available in a typical classroom. Even after learning simple rules for determining how many decimals or digits to record, students struggle to apply these rules to physical examples.

significant figures, precision, accuracy

Approximately 35-45 minutes of active in class time, and approximately 10 minutes of review once activity is completed.

Lowes paint stirring sticks are available in packs of 10 for $0.98, and are made of soft pine. Available here: https://www.lowes.com/pd/SHLA-1-Gal-Paint-Stick-10-Pack/1001036790

Home Depot paint stirring sticks are available in packs of 10 for $1.10, and are less flexible, and thicker than the Lowe’s type. Use if you are concerned about durability. Available here: https://www.homedepot.com/p/1-Gal-Paint-Mixing-Craft-Sticks-10-Pack-HDPS-10/206137830

I find this activity best used in person, but due to the COVID-19 pandemic was forced to convert it to a virtual format to limit physical sharing of materials between students, as well as to provide a format for virtual students to complete. Both the virtual copy of the activity, which includes pictures students can use to make measurements with (see figure 1), and the in-person copy of this activity are included in the Supporting Information. This exercise in measuring and significant figures uses what is available in a normal chemistry classroom, as well as inexpensive materials which can be found in any hardware store.

**F igure 1: **

*Examples of images for virtual assignment*

In my classroom, we complete a challenge activity at the start of this unit which introduces a need for knowing how to correctly measure in order to be successful at a given task. I highly recommend this as an introduction to error and measurement. This task can be as simple as asking students to measure an object and recording their answers in a common location, to as complicated as providing students with a small piece of unknown metal and asking them to match its density to a known provided list. As teachers, we know that they will all arrive at slightly different answers when not using proper measuring techniques. Students find it especially difficult to match the density of an unknown metal to a known list when the sample is very small (1.0 grams or less). Additionally, they often select poor choices in graduated cylinders (ones with scales that are not precise enough) when determining their sample volumes. The problem, we can point out, is that things like densities and lengths are not opinions, and therefore we should be arriving at the same answer within reason!

You can then have a discussion on error in measurement and learn basic measuring rules. It is after learning these rules and seeing a few examples of how to measure that students can then be asked to complete this activity to further reinforce how to measure correctly. After this activity, I usually have my students repeat the measuring challenge we began the learning with, to ensure that now as a class, we all get similar values for lengths, densities, etc.. They nearly all can measure their object or determine their metal sample’s density without issue after having completed this activity.

**Virtual Activity Instructions for Teachers:**

Students should complete the prelab questions either alone or in pairs of two, which are about precision, accuracy, and the dependence or independence of the two words before they begin the lab (5 minutes) This is because you want students to use these terms correctly in the course of the activity. You should review this as a class before you allow them to complete the measuring section. (<5 minutes)

Since students will not have access to most of the measurement tools found in the lab, the student document provides images for their use. As the students complete the measuring section independently, you should be prepared to address questions about how to measure a tiny object with a ruler that contains very few graduations. The copper wire is a great example of an object that would probably be best measured by a more precise ruler. (25-30 minutes)

After students are finished, you should especially review as a class the section with 25 mL graduated cylinders, as these have odd scales that run by 0.2 mL or 0.5 mL. You should discuss which digits students can know with certainty! (5-10 minutes)

*The Word version of this student document and the associated teacher resource are available in the Supporting Information below this post. (Log into your ChemEd X account for access.)*

**In Person Activity Instructions:**

The in-person version of this activity is almost identical in procedure to the virtual version, except the actual objects are staged around the classroom at different stations for students to practice independently. The following stations are set up: one with three triple beam balances, one with three 10 mL graduated cylinders, one with three 25 mL graduated cylinders, and one with three 50 mL graduated cylinders. In addition, three copies of each object of choice (three marbles, three copper wires, or other things you have around the room) are put out against the three types of rulers (figure 2). For measuring lengths, choose very small (<1.0 cm) and very large objects (>60 cm) found around your classroom. It is also helpful to watch students as they practice measuring lengths to ensure they do not line up their object with the end of the ruler, and instead line their object up using the mark for zero.

**Figure 2:** Three rulers all have different scales.

Note that parafilm on the top of the graduated cylinders is recommended so that in the event that a student knocks over the glassware, you do not have to precisely refill them. Normally it’s also recommended to include examples where the top level of the liquid’s meniscus, and not the bottom, rises to a whole mL value to check whether students are using correct volume measurement techniques. A drop of food dye helps students to see the liquid as well.

*The Word version of this student document and the associated teacher resource are available in the Supporting Information below this post. (Log into your ChemEd X account for access.)*

### Extension:

The activity can be expanded to include even more examples of measuring devices, such as thermometers, barometers, protractors, and even digital measuring devices. It is a great introduction into deeper discussion for higher level students regarding “atypical” scales, which do not rise by normal values of 1, 0.1, etc, error on digital devices, error values listed on glassware (100 mL ± 1 mL), the difference between TD and TC glassware, and discussions into why chemists choose certain glassware (i.e. volumetric flasks) and not others (beakers, larger graduated cylinders) for some lab procedures requiring high degrees of precision. It’s also a great way to introduce propagation of error, and prepare students for learning arithmetic with significant digits.

**For In-Person Activity:** To create different rulers, obtain one-gallon paint stirring sticks from any hardware store. Then, use a “real” ruler as a reference and mark on one set of rulers every 1 cm, from 0 to about 30 cm. On another set, mark only every 10 cm. You can also create your own scales of 0.50 cm should you choose to challenge higher level students. Using a ballpoint, non-gel type pen is best for this.

A special thank you to Ms. Robin DeShong for her idea of using paint stirring sticks to create new rulers with unique scales. I am indebted to her support and creativity.

## Safety

### General Safety

General Safety

**For Laboratory Work:** Please refer to the ACS Guidelines for Chemical Laboratory Safety in Secondary Schools (2016).

**For Demonstrations:** Please refer to the ACS Division of Chemical Education Safety Guidelines for Chemical Demonstrations.

#### Other Safety resources

RAMP: Recognize hazards; Assess the risks of hazards; Minimize the risks of hazards; Prepare for emergencies

## NGSS

Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. Use mathematical representations of phenomena to support claims.

Mathematical and computational thinking at the 9–12 level builds on K–8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. Use mathematical representations of phenomena to support claims.