In the Classroom
Cross-Proportions: A Conceptual Method for Developing Quantitative Problem-Solving Skills
W
Elzbieta Cook* and Robert L. Cook Department of Chemistry, Louisiana State University, Baton Rouge, LA 70803 and Department of Chemistry, Southern University and A&M College, Baton Rouge, LA 70813; *
[email protected] In the last decade or so a viewpoint that chemical problems must be solved from a concept-based approach rather than an algorithm approach has emerged (1–6). The major thrust of this work has been that students must first understand and appreciate the problem they are trying to solve rather than simply arrive at the correct answer by a currently accepted “plug and chug” algorithm. For these algorithmic methods to be useful in the context of learning, an understanding of the concepts is mandatory and, in fact, the algorithms are derived based on a thorough understanding of the problem. However, once an algorithm has been formulated, its application by others does not ensure an understanding of the problem being solved by it (7, 3–4). One such algorithm that has emerged almost universally in contemporary introductory and general chemistry textbooks (8–9) is the so-called dimensional analysis (DA) method. The DA method itself is a powerful and highly efficient method, however, as an initial teaching tool it is simply unsuitable because this method can yield the correct answer by perfunctory unit cancellation rather than understanding why and how the units are being cancelled due to scientific principles. A parallel example would be the use of the automatic dial option on an already programmed phone without the need to understand the concept of area codes. This weakness has been described in the literature (2, 10–16). The cross-proportion (C-P) method, on the other hand, requires that students understand the underlying chemical principles in order to set up the correct cross proportion at each step of the problem solving exercise. The C-P method
uses well understood and practiced mathematical skills, thus allowing the student to focus on understanding the chemistry at hand. This paper presents the C-P method, demonstrating its usefulness and effectiveness in teaching concepts via problem solving. We are reintroducing this method as an alternative to the omnipresent DA method as well as providing teachers and students with support to use the C-P method. The Cross-Proportion Method The method discussed below (and its several mutations: proportions, ratios, ratio and proportion, cross-multiplications) is not new (17, 1–2). Since students are not always able to articulate the meaning of various operations encountered during problem solving (1), the idea is to get students to talk about the chemical problem, draw a mental map with clear directions toward a solution, and punctuate the necessary steps. Each step corresponds to a distinct chemical or physical principle students have already learned, such as: unit or magnitude conversion, the concept of a mole, Avogadro’s number, molar mass, molar ratio, and so on. In the initial stages of learning this technique, every time such a principle is used students are asked to either write it down or say it out loud. This eventually is replaced by silently repeating the concept and making symbolic notes on paper. Since as a whole, students read and write from left to right (as opposed to from top to bottom) of the page, it makes sense (and may be more intuitive) to set up a cross proportion horizontally. Example 1 represents a commonly performed conversion between moles and mass.
Example 1 What is the mass of 3.21 mol of water (MM 18.02 g/mol)? Here, it is essential that students understand the meaning of molar mass, which is the mass of one mole of a substance. The upper “line” in the cross-proportion utilizes this principle: If Then
1 mole H2O has mass of
18.02 g
3.21 moles H2O have mass of
x
The “moles” are placed under the “mole”. The above can be written into a 4-compartment box: 1 mol H2O 3.21 mol H2O
18.02 g x
To solve for x, multiply the contents of the two known compartments that lie diagonally across from each other, and divide the result by the contents of the compartment that lies diagonally across from the unknown. Hence,
x
3.21 mol H2O 18.02 g 1 mol H2O
57.8 g
Notice that “mol H2O” cancels out, and the answer is in grams.
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In the Classroom
As chemical questions become more involved, several cross proportions have to be set up and evaluated. In these cases students must use their knowledge of principles and how they are connected. Which principles are to be exploited may not be clear initially, yet it becomes intuitive with practice. Conventionally, stoichiometry is one of the hardest concepts for students to grasp, as it often demands applying more than one principle at a time. By doing this students can clearly see the application of and connectivities between the individual principles they have learned and used in single-concept calculations before (i.e., they are learning at the interpreta-
tion rather than the lower levels of recall and translation outlined in Bloom’s Taxonomy: see ref 18 ). Example 2 shows a systematic C-P approach to solving a typical stoichiometry problem, including the limiting reagent issue and percent yield. Note that each next step in Example 2 uses information from the previous step, allowing students to connect concepts by having the number just calculated (and displayed on their calculators’ screens) ready to be used in the next cross proportion. It is necessary to remind students to keep track of the “full” unit that includes the unit —mol, g, L—and the “of what” part—moles of HCl, g of NaOH, L of
Example 2 What is the volume at STP conditions of hydrogen gas produced in the reaction between 3.00 g of Zn metal with 0.100 L of a 1.00 M solution of HCl, if the percent yield of the reaction is 97.6%? Zn(s) 2HCl(aq) → ZnCl2(aq) H2(g) Step Step Step Step Step
1: 2: 3: 4: 5:
Find moles of the reacting species: Zn(s) and HCl(aq). Find the molar ratio between Zn(s) and HCl(aq) and determine the limiting reagent. Find the molar ratio between the limiting reagent and H2(g) and find the moles of H2(g). Find volume of H2 at STP conditions. Determine the actual yield of H2 based on the percent yield.
Step 1: Principle: MM(Zn) = 65.39 means that 1 mole of Zn weighs 65.39 g.
1 mol Zn
65.39 g
x
3.00 g
1 mol Zn 3.00 g
⇒
x
4.588 102 mol Zn
65.39 g
Principle: 1.00 M HCl means that 1.00 L of solution contains 1.00 mole of HCl solute. 1.00 L 0.100 L
0.100 L 1.00 mol HCl
1.00 mol HCl
⇒
y
y
0.100 mol HCl
1.00 L
Step 2: The molar ratio between Zn and HCl is 1:2. 1 mol Zn
2 mol HCl
4.588 102 mol Zn
z
⇒
4.588 102 mol Zn 2 mol HCl
z
1 mol Zn
9.176 102 mol HCl
Since 9.176 102 moles HCl are needed to completely react with 3.00 g of Zn, and 0.100 moles of HCl are actually available, HCl is in small excess. Consequently, Zn is the limiting reagent. Step 3: The molar ratio between Zn (the limiting reagent) and H2(g) is 1:1
1 mol Zn
1 mol H2
2
4.588 10 mol Zn
⇒
t
4.588 102 mol Zn 1 mol H2
t
1 mol Zn
4.588 102 mol H2
Step 4: Principle: 1 mole of any gas at STP conditions takes up volume of 22.4 L.
1 mol H2
22.4 L
4.588 102 mol H2
s
⇒
4.588 102 mol H2 22.4 L
s
1 mol H2
1.028 L
(Alternatively, step 4 could be replaced by the ideal gas law calculation utilizing the pV nRT formula, especially for other than STP conditions.) Step 5: Principle: 100% yield corresponds to the amount of product that can be theoretically produced (1.028 L).
100% 97.6%
1188
1.028 L H2
⇒
u
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97.6% 1.028 L H2
u
100%
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1.003 L H2
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In the Classroom
H2. In particular, at the core of any stoichiometric problem where molar ratios are used (steps 2 and 3 in Example 2), it is critical that moles of A are easily distinguished from moles of B, so that the wrong moles are not cancelled out. The C-P method can also be applied in calculations involving percent composition, stoichiometry of redox reactions and electrolytic processes, stoichiometry of heat effects, standard additions (used in analytical chemistry) and any instance where two quantities maintain the same ratio to each other.
5. 6. 7. 8.
Conclusions As can be seen from Examples 1 and 2, the C-P method promotes student learning of the underlying chemical problem, the underlying chemical principle needed to solve the problem, the concept of units and their cancellation, and the connectivity of chemical principles and concepts and the application or discovery of these connectivities in complex problem solving. Students using the C-P method can more easily visualize the connectivities between chemical concepts, as shown by the box notation used in the problems presented here. In addition, the C-P method allows for both the instructor and the student to easily determine where an error has been made during problem solving, unlike the problem layout students use with the DA method. Finally, using the C-P method supports a strong cognitive foundation upon which DA and other diagnostic methods can be developed by students as they advance in chemistry and scientific careers, subsequent to internalizing fundamental concepts and their connectivities. Several other efforts to show students connections between concepts have been previously presented in this Journal (19–21). Also, just as there are different learning styles (22–23; 10–12), we should be prepared to offer students a variety of learning experiences (24–27), including different numerical problem solving techniques. W
Supplemental Material
Several additional examples from general chemistry are available in this issue of JCE Online. Acknowledgments The authors would like to thank the reviewers, the editor of this Journal and Maria Graça Vicente (LSU) for careful reading of this paper and valuable suggestions, as well as Fran Frost of Baton Rouge Magnet High School for sharing her experiences, encouragement, and most importantly, brave application of the C-P method in her high school teaching career of more than 30 years. Literature Cited 1. Arons, A. In New Directions in Teaching and Learning; Wiley: New York, 1990; Ch. 1. 2. Cohen, J.; Kennedy-Justice, M.; Pai, S.; Torres, C.; Toomey, R.; DePierro, E.; Garafalo, F. J. Chem. Educ. 2000, 77, 1166– 1173. 3. Nurrenbern, S. C.; Robinson, W. R. J. Chem. Educ. 1998, 75, 1502–1503. 4. Lyle, K. S.; Robinson, W. R. J. Chem. Educ. 2001, 78, 1162– 1163.
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