A Comparison of Techniques:

Proportion Analysis vs Dimensional Analysis

Avoiding Medication Errors: Examining the Policy for Testing the Proficiency of Student Nurses by Theresa Pietsch

Rice and Bell (2005) also discussed nursing students as classified poor performers in mathematics, expended high anxiety levels when dealing with calculations, and exposed to inconsistent teaching strategies in medication calculations. Mathematical formulas have been identified as problematic due to misapplication of formulas or miscalculations. Best teaching practices have been elusive and results for nursing students were dismal.

Drawing on previous skills taught in Chemistry courses, Rice and Bell (2005) studied the application of dimensional analysis as an alternative learning strategy for medication calculations. Working from the assumption that students successfully completed science courses, the authors studied a total of 30 nursing students over two semesters. The authors concluded the learning strategy, dimensional analysis, was successful in improving conceptual skills, thereby reducing the number of incorrect responses to test items.

Introduction to Pharmaceutical Calculation by Michael C. Brown American Journal of Pharmaceutical Medicine

· “Proportion analysis and dimensional analysis are both heavily dependent on ratios.”

·“The use of proportions to solve calculations has an important consideration.  A single proportion can only solve 1 step of a calculation.”

· “Dimensional analysis has the ability to “cancel units” to help verify the setup of the problem.”

· “Dimensional analysis has the ability to handle multiple steps of a calculation all at once.”

Scaling the Natural World Using Dimensional Analysis by Stephen Kass Yale University

· The concept of proportions (proportional reasoning) is seen as fundamental to understanding many scientific applications as well as consumer problems, advanced science and math courses, and intellectual development in general (Inhelder and Piaget have studied intellectual development in relationship to students’ ability to deal with science concepts. They regard proportionality (proportional reasoning) as a primary acquisition at the stage of formal operations). Rates can be found in most aspects of life including cooking, navigation, physics, earth science, economics, electronics, business, and industry. Since a large percentage of adolescents are lacking this critical skill, the determination of possible ways of successfully teaching the concept is an important issue.

Sci-Math is an interdisciplinary curriculum designed to address these issues of teaching proportionality (proportional reasoning) in science and math courses while using large or very small numbers. Its development and field testing were funded by the National Science Foundation. Sci-Math was cited by the U.S. Office of Education as an exemplary educational innovation worthy of national dissemination within the National Diffusion Network (N.D.N.).

Sci-Math focuses on the understanding of the concept of proportions and on the use of proportions in word problem-solving. Specifically, Sci-Math uses the rate concept and dimensional analysis used in introductory physics and chemistry courses to solve proportions (problems involving proportional reasoning). This rate and dimensional analysis method has slowly moved into textbooks and has completely replaced the method of ratio-and-proportions taught exclusively in junior and senior high school mathematics textbooks.

There appears to be good reason for dimensional analysis to have replaced the ratio-and-proportion method in advanced science courses. Dimensional analysis is a simple, problem-solving, error-reducing procedure which seems to require less conceptual reasoning power to understand than does the ratio. Furthermore, it can condense multi-step problems into one orderly extended solution.

http://www.psybox.com/web_dictionary/proportionalreasoning.htm

· Proportional reasoning - The realization that the relationship between two variables remain constant despite their changing in values and, according to Piaget, a skill emerging as a result of the cognitive development of formal operational thinking.

http://members.tripod.com/susanp3/snurse/id28.htm

The Factor Label method of drug calculation (sometimes known as dimensional analysis math for nurses) has the following advantages over the methods currently being taught for drug calculation:

1) It is simpler. All calculations are performed in one step rather than in separate stages.

2) It is more accurate. Since there is only one step, there are no intermediate steps at which answers are rounded off before proceeding. When using conversion factors separately with the proportion method, for instance, for ease in calculation, a weight in kilograms will be rounded off to the nearest tenth before proceeding. With the factor label method, all intermediate results are carried in the calculator memory and are accurate to the capacity of the calculator itself. Rounding off is not done until the last step, which is accepted mathematical practice.

3) It is easy to keep track of the units.

a) The unit of measure is included in the answer automatically.

b) All units other than the answer you are looking for will cancel out automatically. If they do not, you know that you have made an error and are alerted to go back and find it.

c) When done, the units on the left of the equation must match the units that remain on the right of the equation, indicating that the answer found is in the form of the units desired.

Calculate with Confidence by Deborah Gray Morris

· Several methods are used for calculating dosages.  The most common methods are ratio and proportion, and use of a formula.

· The advantage of dimensional analysis is that because only one equation is needed, it eliminates memorization of formulas.  Dimensional analysis can be used for all calculations you may encounter once you become comfortable with the process. … Although some may find the formalism of the term “dimensional analysis” intimidating at first, you will find ti’s quite simple once you have worked a few problems.

  

The following pages show three different problems worked out by both the proportion analysis technique and the dimensional analysis technique for comparison.

Problem #1: A suspension is labeled to contain 250 mg of amoxicillin per 5 mL of product.  If a patient’s dose is 150 mg, what volume of suspension should the patient receive?

By Proportion Analysis

250 mg	=	150 mg
5 mL		?

250 mg x ? = 5 mL x 150 mg

? =	5 mL x 150 mg
	250 mg

? = 3 mL

By Dimensional Analysis

5 mL	x	150 mg	=	3 mL
250 mg		1

Comments: In simple problem, there is not much different between the work involved in the two analysis techniques.  Dimensional analysis, however, avoids the “cross-multiplication” and the need to solve for the unknown variable of the proportion analysis technique.

Problem #2: Administer 2000 mL NS intravenous at 115 mL/hr.  The available drip chamber drop factor is 10 gtt/mL.  Calculate the infusion rate in gtt/min

By Proportion Analysis

10 gtt	=	?
1 mL		2000 mL

10 gtt x 2000 mL = 1 mL x ?

? =	10 gtt x 2000 mL
	1 mL

? = 20,000 gtt

115 mL	=	2000 mL
1 hr		?

115 mL x ? = 1 hr x 2000 mL

? =	1 hr x 2000 mL
	115 mL

? = 17.3913 hr

60 min	=	?
1 hr		17.3913 hr

60 min x 17.3913 hr = 1 hr x ?

? =	60 min x 17.3913 hr
	1 hr

? = 1043.4780 min

? =	20,000 gtt	=	19.1667	gtt	=	19	gtt
	1043.4780 min			min			min

By Dimensional Analysis

10 gtt	x	115 mL	x	1 hr	=	19.1666	gtt	=	19	gtt
1 mL		1 hr		60 min			min			min

Comments: The proportion analysis technique requires setting up, cross-multiplying, and solving for the unknown variable 3 separate times.  In the end two intermediate results must be set as a ratio to get the final answer.  Intermediate results introduce the risk of inappropriate rounding that may introduce significant error into the final calculation.  With the dimensional analysis technique a single equation is set up and all intermediate results are automatically held in the calculator’s memory to at least 8 decimal places so that a chance for error is eliminated.  Also, a final ratio does not need to be set up, as with the proportion analysis technique.  Instead a single universal approach is used.

Problem #3: A 55 pound patient is prescribed 5 mg/kg/day of diphenhydramine HCl.  The daily dose it to be divided into 4 doses, each administered 6 hours apart.  Available medication is a 12.5 mg/5 mL solution.  How many teaspoons need to be given to the child at one time.

By Proportion Analysis

1 kg	=	?
2.2 lb		55 lb

1kg x 55 lb = 2.2 lb x ?

? =	1 kg x 55 lb
	2.2 lb

? = 25 kg

5 mg	=	?
1 kg		25 kg

5 mg x 25 kg = 1 kg x ?

? =	5 mg x 25 kg
	1 kg

? = 125 mg

5 mL	=	?
12.5 mg		125 mg

5 mL x 125 mg = 12.5 mg x ?

? =	5 mL x 125 mg
	12.5 mg

? = 50 mL

1 tsp	=	?
5 mL		50 mL

1 tsp x 50 mL = 5 mL x ?

? =	1 tsp x 50 mL
	5 mL

? = 10 tsp

? =	10 tsp	=	2.5 tsp
	4

By Dimensional Analysis

1 tsp

x

5 mL

x

5 mg

x

1 kg

x

55 lb

x

1 day

=

2.5

tsp

5 mL

12.5 mg

1 kg ∙ day

2.2 lb

1

4 “6 hr interval”

“6 hr interval”

 

 

Comments: The proportion analysis technique setting requires up, cross-multiplying, and solving for the unknown variable 4 separate times.  In the end the student nurse must remember “10 tsp” is a full day’s dose, not the amount given every 6 hours.  To get the final answer the “10 tsp” should be divided by 4 (a 5^th proportion could have been set up to arrive at the same result).  Intermediate results introduce the risk of inappropriate rounding—not the case in this specific example because intermediate results were whole numbers—that may introduce significant error into the final calculation.  With the dimensional analysis technique a single equation is set up and all intermediate results are automatically held in the calculator’s memory to at least 8 decimal places so that a chance for error is eliminated.  Also, a final ratio does not need to be set up, as with the proportion analysis technique.  Instead a single universal approach is used.