March 2016 | Resource Feature
Bringing Real-Life Context to Chemical Math
By W. Patrick Cunningham
Not all students who enroll in pre-AP chemistry have the mathematical background to attain a mastery of the subject without an intense math review. To address this challenge, for the past several years, the pre-AP chemistry team on our campus has positioned a stout math and measurement unit during the first three weeks of school. This accomplishes a number of objectives: it positions chemistry as math- and measurement-driven, it compels students to either demonstrate and hone their skills or transfer to regular chemistry, and it prepares students to use algebra, dimensional analysis, graphing, and data-collection skills.
If there is a limitation to this strategy, it may be that while students are happy to see their math skills are useful in physical science, they may not see their utility in daily life. Most of our students, even those in advanced chemistry, will not spend their lives doing chemistry. They ought to be able to take away an appreciation for the value of their classwork in their real life.
Coffee with too many zeroes
Let’s begin by considering one of the most useful skills in chemistry: using scientific notation to report data. According to Texas standards, students should have acquired capability in scientific notation before or during their last year in middle school. Yet experience and testing tell us that even if students have shown mastery of this skill in eighth grade, many of them have lost it by their first chemistry class. The loss of this skill is probably due to underuse, even though the typical ninth grade science courses in Texas (biology, physical science, or Earth and space) are supposed to use it.
|tera||T||1012 = 1,000,000,000,000|
|giga||G||109 = 1,000,000,000|
|mega||M||106 = 1,000,000|
|kilo||k||103 = 1,000|
|centi||c||10-2 = 1/100|
|milli||m||10-3 = 1/1,000|
|micro||μ||10-6 = 1/1,000,000|
|nano||n||10-9 = 1/1,000,000,000|
The way I bring chemistry and scientific notation together is probably familiar to most chemistry teachers: Remind the students about exponential notation and the rules for manipulating numbers with exponents, show a table of conversion values (usually with the prefixes as in Figure 1), display some example problems converting customary notation to scientific and vice-versa, and then offer time for directed practice with practice problems.
A critical look at this lesson would note the absence of an effective engagement phase, according to the 5E lesson plan model. One can capture some students’ attention by saying, “this is important” or “this will be tested,” but the teacher’s task is to first elicit students’ prior understanding and then connect with their interests in such a way that they want to be engaged with the skill (1).
One way to engage students would be to begin class with a (non-graded) quiz asking some pointed questions about applying exponential notation in their daily life. I have found it helpful to use a formative assessment tool that has students solve problems about themselves, but using scientific notation. For example, in one exercise, they calculate how much of the national debt they owe; in another, after listening to a convoluted conversation on the radio about coffee, they evaluate and recalculate the figures from the conversation. The follow-up video shows how using scientific notation facilitates the solution to the problem. The math becomes “just a cup of joe.”
In the other direction—the very small quantity—scientific notation can be just as useful. In fact, we find that most teachers and students would rank multiplying and dividing numbers less than one to be a bigger irritation and stumbling block than doing the same with large numbers. The quiz provided with this lesson can help the teacher diagnose a student’s ability with very small quantities. A small group discussion during the teacher’s presentation might elicit students’ frustration with multiplying and dividing with values such as 0.000356 g/L.
AP and college textbooks that we surveyed placed information regarding scientific notation in either an appendix or in a review chapter in the very beginning of the book. Only one out of the five AP edition textbooks we reviewed included information about scientific notation in the first chapter of the book (2); however, the phrasing implied that scientific notation is material that AP students should have already covered prior to enrolling in AP chemistry. The other four books placed their scientific notation material as review tools at the end of the books (3)(4)(5)(6) and included information about adding and subtracting, multiplying and dividing, and handling powers and roots in scientific notation. Quantities throughout the textbooks, such as sample calculations, were expressed in scientific notation. For example, a solution to a problem was stated as “8.75 x 10-3 mol OH-.”
Thoroughly modern milli-
Using this method, we can consider how to calculate the molar mass of an ideal gas from the following data:
Pressure = 151.99kPA
Volume = 100. mL
Temperature = 27.0 ºC
Mass (gas only) = 0.195g
Here is a common way of presenting and setting up the problem, using the formula (7)
and since density (d) has the following formula
the formula for molar mass can also be found
MM = (0.195 g) x 8.3145 L-kPa-mol-1-K-1 x 300.K
(0.100 L) x 151.99 kPa
The molar mass is 32.0 g-mol-1.
This calculation requires the conversion from a laboratory-familiar dimensional unit (milliliters) to a less familiar unit (liters). After all, we nearly always measure lab volumes in mL, do we not?
From students’ earliest days in the lab, I try to encourage them to think of masses in milligram quantities. This mirrors the use of milliliters for volume. Students are asked to record masses in milligrams or convert to that unit as soon as possible after making a measurement. With this experience, then, it makes sense to restate the solution to the molar mass calculation above in this way:
MM = (195 mg) x 8.3145 mL-kPa-mmol-1-K-1 x 300.K
(100. mL) x 151.99 kPa
The molar mass is the same in number, 32.0, but the unit is now expressed in the unit mg-mmol-1. This may seem awkward at first to both teacher and student, but it inevitably leads to the conclusion that the molar mass is the same, whether stated in grams per mole or milligrams per millimole. We rarely deal in mole quantities in the laboratory, so it is sensible to use millimoles whenever possible. Students then rarely have to divide by numbers less than 1, and that reduces the incidence of decimal point errors.
The use of milligrams, milliliters, and millimoles is particularly helpful when calculating the molarity of solutions. Volumes remain in the dimensional unit actually measured in the lab throughout the calculation process.
There is another way in which we can use this “thoroughly modern milli-” approach in learning calculations. Since it is efficient to use smaller quantities when working with acids and bases in laboratories, teaching students to calculate molarity through the use of millimoles and milliliters is enormously beneficial. The calculation of molarity of hydrochloric acid in the accompanying slides demonstrates the practicality of working with milli- units. Students can understand that whether expressed in moles per liter or in millimoles per milliliter, the numerical value of the concentration of a solution is the same.
The use of short assessment tools such as we provide in the accompanying file, even on a daily basis, has been found to be a very adequate substitute for daily homework assignments. These can be used as either pre-lesson quizzes or post-lesson exit tickets.
The skills involved in chemical mathematics are critical parts of the early scaffolding needed for a successful chemical education experience. By incorporating a gripping set of engagement and eliciting experiences at the very beginning, teachers can help motivate students to want to learn those skills. By using both scientific notation and the small units milligram and millimole throughout the chemistry program, students practice these skills in ways that can be useful throughout their scientific studies, even to the graduate chemistry level.
Taylor Brown, Evan Jose, Jeff Tomes, Emily Zagrzecki, and Subhash Venigalla are members of the Scientific Research and Design class at Johnson High School in San Antonio and contributed to this article.
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