National Benchmarks, Project 2061 for Grades 6-8

The Nature of Mathematics

2a Patterns and Relationships:

• Usually there is no one right way to solve a mathematical problem; different methods have different advantages and disadvantages.

The Mathematical World

9a Numbers:

• Comparison of numbers of very different size can be made approximately by expressing them as nearest powers of 10.

9b Symbolic Relationship:

• Mathematical statements can be used to describe how one quantity changes when another changes. Rates of change can be computed from differences in magnitudes and and vice versa.

• Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease indefinitely, increase or decrease in steps, or do something different from any of these.

9c Shapes:

• The graphic display of numbers may help to show patterns such as trends, varying rates of change, gaps or clusters. Such patterns sometimes can be used to makepredictions about phenomena being graphed.

Common Themes

11d Scale:

• Representing large numbers in terms of powers of ten makes it easier to think about them and to compare things that are greatly different.

Habits of Mind

12b Computation and Estimation:

• Express number like 100, 1,000, and 1,000,000 as powers of 10.

• Express and compare very small and very large numbers using powers-of-ten notation.

• Recall immediately the relations among 10, 100, 1000, 1 million and 1 billion.

12d Communication Skills:

• Organize information in simple tables and graphs and identify relationships they reveal.

National Science Education Standards grades 5-8

A Scientific Inquiry

• Use appropriate tools and techniques to gather, analyze, and interpret data. Use mathematics in all aspects of science inquiry. Develop descriptions, explanations, predictions, and models using evidence.

Scientists are always faced with the manipulation of very large numbers. Whether we talk about stars in the galaxy, cells in our bodies, or atoms in a cell we are dealing with numbers that dwarf those of everyday experience. The mathematical notation we use in describing everyday situations quickly becomes unwieldy when dealing with very large numbers.

These labs will help your students to understand the manipulation of scientific notation, and to begin to graph and interpret exponential behavior. This experience will be useful in all further scientific investigations, in other ASPIRE lessons and beyond.

Activity 1 - Scientific Notation I: The Exponent

In the first of two lessons on interpreting and manipulating scientific notation, students are introduced to the concept of the "exponent" or "power-of-ten" in a number expressed in S.N. Two short JAVA labs show, first by demonstration and then by practice, how the exponent relates to the number of powers of ten and the number of "zeros" of a number expressed in conventional notation. A third lab - a step-through tutorial - then illustrates how numbers expressed in exponent form are multiplied and divided.

Activity 2 - Scientific Notation II: The Mantissa

In the second of two lessons on interpreting and manipulating scientific notation, students are introduced to the concept of the "mantissa" of a number expressed in S.N. Two short JAVA labs show, first by demonstration and then by practice, how the mantissa together with the exponent complete the description of a number expressed in scientific notation, and how these can be related to conventional notation. A third lab - a step-through tutorial - then illustrates how numbers expressed in mantissa/exponent form are multiplied and divided.

Activity 3 - Linear Versus Exponential:

As an application of the scientific notation skills obtained in Activities 1 and 2, students are introduced to the concept of exponential (as opposed to linear) behavior of a system using the example of bacterial growth on a petri dish. The relevance of this to human population growth is also described. Students first perform an interactive lab in which they "seed" a petri dish with initial populations of bacteria and growth parameters, and record population data as a function of time. After graphing this data - by which the students should appreciate the inadequacy of "linear" graphing techniques to express exponential growth - students are guided through a log-paper graphing exercise. Finally, students plot their bacterial "data" and answer a series of questions on the relationship of the various growth curves.

Technical consultation and assistance is freely available to teachers and schools interested in using the ASPIRE website, on-line labs, and curriculum materials. In time resources will be made available to assist schools lacking the computer technology required to access the labs. For technical assistance and resource information please contact the following:

Teacher materials and information:
Julie Callahan, julie@cosmic.utah.edu