Rationale and Introduction
My friend Eric told me, "I've been on this great diet. I started off at 220 pounds, ate nothing but hamburgers, fries, and milkshakes for a month, and now I'm down to 120 kilograms."
Joking aside, in December 1998 NASA launched the $125 million Mars Climate Orbiter. Nine months and 416 million miles later, NASA lost that spacecraft because the navigation team trying to insert the craft into orbit was making calculations in meters and kilograms while the computer guiding the operation had been programmed in feet and pounds. 1 Some of the smartest people in the world were undone by a mistake a fourteen year old could have corrected.
Units matter. Units make numbers make sense.
My unit will explore the use of dimensional analysis, the process of using units properly in calculations. Dimensional analysis lends itself to nearly every type of real word situation where math is used because numbers are associated with the measure of quantity in nearly all real world problems.
When students work on word problems, they frequently write down only the numbers without the units that describe what the number measures. The failure to write units removes meaning from the problems, with the result that students frequently put the numbers together using one of the four basic operations without any regard to whether their calculation makes sense. This technique is rarely successful and students soon learn to hate mathematics word problems of all types. That dislike also spills over to problems with math content in the sciences.
Dimensional analysis requires students to always include the units when they write down a number. The process of dimensional analysis takes advantage of the fact that units interact in a mathematically rational way such that the result of a calculation carries units as well. If the units of a calculation do not match the units that the question sought, the students know that their calculation requires re-examination. Further, knowing the units of the input data and the units of the desired output frequently gives clues on how to solve a problem by suggesting how those inputs might interact with each other mathematically in a calculation.
I will use the human circulatory system as the source of problems to investigate with dimensional analysis with an eye towards the development of artificial replacement organs. However, the techniques demonstrated can also be used to attack any type of word problem that involves numbers that carry units. I also introduce a technique of using units without numbers that students can use to pre-screen data for informative relationships between input variables before performing detailed calculations.
Comments: