Big Numbers and Scientific Notation
by Jennifer M. Wenner, Geology Department, University of Wisconsin-OshkoshJump down to: Teaching strategies | Materials & Exercises | Student Resources
What is scientific notation?
The concept of very large or very small numbers is something that is difficult for many students to comprehend. In general, students have difficulty with two things when dealing with numbers that have more zeros (either before OR after the decimal point) than they are used to. They often do not understand:
- that "big" and "small" are relative terms,
- the concept of "order of magnitude"
Scientific notation is a way to assess the order of magnitude and to visually decrease the zeros that the student sees. It also may help students to compare very large (or very small numbers) But students still have little intuition about scientific notation. Teaching them to recognize that scientific notation is a short hand way to better understand big and small numbers can be useful to them in all aspects of their academic career.
In science, often we work with very large or very small numbers. For example, in geology,
Scientific notation allows us to reduce the number of zeros that we see while still keeping track of them for us. For example the age of the Earth (see above) can be written as 4.6 X 109 years. This means that this number has 9 places after the decimal place - filled with zeros unless a number comes after the decimal when writing scientific notation. So 4.6 X 109 years represents 4600000000 years.
Very small numbers use the same type of notation only the exponent on the 10 is usually a negative number. For example, 0.00000000000000000000000000166 kg (the weight of one atomic mass unit (a.m.u.)) would be written 1.66 x 10-27 using scientific notation. A negative number after the 10 means that we count places before the decimal point in the scientific notation. You can count how many numbers are between the decimal point in the first number and the second number and it should equal 27.
The nice thing about scientific notation is that it also tells us something about significant figures.
Teaching Strategies: Ideas from Math Education
Put quantitative concepts in context
There are a number of geologic contexts in which big (and small) numbers and scientific notation come up. A few of them include:
Use multiple representations
Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that big numbers and scientific notation can be represented.
- Verbal representation
- Numerical representation
- Graphical representation
- Symbolic representation
Use technology appropriately
Students have a number of tools that can be used to help them become more familiar with both scientific notation and very large and very small numbers. Calculators can be set up to display scientific notation. Spreadsheet programs can tabulate data, expressing it in a number of forms.
Work in groups to do multiple day, in-depth problems
Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.
Very big and very small numbers come up over and over in introductory geoscience: Geologic time, radiometric dating, unit conversions (especially metric), etc. When each new topic is introduced, make sure to point out that this is not new material, it's just a matter of remembering.
Teaching Materials and Exercises
- Jerry Johnson of University of Nevada at Reno has an excellent example of Understanding Big Numbers that applies to population growth and government spending. This file includes student exercises and instructor notes.
- In the Quantitative Skills site, Barb Tewksbury discusses Back of the Envelope Calculations and has a number of exercises that give students practice in quantitative skills and in estimating numbers.
- A View from the Back of the Envelope has a number of links to interesting ways to explain (and estimate) big numbers and orders of magnitude.