Project SkyMath
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Project SkyMath: Making Mathematical Connections

Using the Science and Language of Patterns to Explore the Weather

1996 University Corporation for Atmospheric Research

This Web Page is designed for middle school mathematics teachers. It contains all of the information needed for teachers to use the 16 classroom activities of SkyMath, including the module itself. We believe that it is an effective and innovative way to present elements of the middle school mathematics curriculum.

Index(Click on to move to subject and to return to Index)
SkyMath: Mathematics in Context
SkyMath and the NCTM Standards
Module Design
SkyMath and the National Science Education Standards
Education Development Center
The SkyMath Module
End Of Unit Assessment Instrument
Finding a Partner Class
SkyMath Mailing List
Field Tests
Insites' External Evaluation of the Program
SkyMath Pamphlets
SkyMath Research Teachers
Mathemtmatics for a Blue Planet


The University Corporation for Atmospheric Research (UCAR) received funding from the National Science Foundation to prepare a middle school mathematics module incorporating real-time weather data. The goal of the pilot project is to demonstrate that acquiring and using current environmental and real-time weather data in middle school classrooms, in ways that embrace the dynamic and the uncertain natures of these data, will promote the teaching and learning of significant mathematics, consistent with the standards set by the National Council of Teachers of Mathematics.

The module may be freely downloaded from this page and we hope that middle-school mathematics teachers will find it interesting enough to use in their classrooms. The module can be incorporated as a replacement unit that uses connections to mathematical concepts in data analysis, graphing, number and number relationship, patterns and functions, and statistics and measurement.

The project is led by a Design Team of mathematicians, math educators, scientists, technology experts, and teachers; this group reflects the interdisciplinary nature of the program and its emphasis on the use of technology in a science/math school environment. SkyMath's Organization Chart illustrates the contributions made by its affiliated groups.

Project SkyMath has profited by the close collaboration of Dr. Nancy Songer, Program Director for Kids as Global Scientists. Her research into the learning opportunities of Internet-based weather studies has encouraged us to apply her pioneering work to mathematics.

SkyMath: Mathematics in Context

The SkyMath Module, "Using the Science and Language of Patterns to Explore Temperature", calls for the development of several mathematical concepts using a single central concept from weather --- temperature. The SkyMath curriculum uses real-time weather data, involves classrooms in hands-on mathematics, elicits higher-level thinking, engages students in purposeful projects, and calls for reflection and communication.

The SkyMath module's focus is on doing -- rather than hearing, reading, seeing, or saying -- and is consistent with what is known about learning retention rates associated with these different levels of involvement. One overall goal of this design is to provide guidance to student learning while allowing increasing degrees of student independence in the development of ideas and activities. The 16 activities of the SkyMath module reflect this development of student-initiated learning style.

The students are organized in groups; they collect and analyze data, exchange data and messages electronically with distant peers, identify and solve problems that emerge from classroom activities and that may have many possible solutions, and present or publish information acquired during the unit.

The activities lead students to develop methods of representing change; how temperature changes with time and with location. Students are challenged to measure, represent, and analyze these changes. Activities include developing symbol sets, preparing graphs for median-based analysis, predicting magnitudes of changes, and learning about the correctness of their work by seeing what really happens!

The sixteen activities, which take at least six weeks to cover, include:

  1. Brainstorming: Temperature and Temperature Changes
  2. Be a Weather Watcher
  3. Introduction to Blue Skies
  4. Reading Celsius and Fahrenheit
  5. Make Your Own Temperature Scale
  6. Converting from One Unit of Measurement to Another
  7. Making a Rule to Convert Between Fahrenheit and Celsius
  8. Is Our Room All One Temperature?
  9. How Can We Describe Our Room Temperature?
  10. Sampling and Comparing Temperatures
  11. Line Graphs
  12. Exploring the StowAway Data
  13. What's Been Happening?
  14. Were the Predictions Correct?
  15. Answers and Questions
  16. Presentations and Reflections

SkyMath and the NCTM Standards

At the 5-8 level, the NCTM Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) identifies 4 process standards (1-4) and nine content standards. These are all addressed in the SkyMath materials. Those standards which are given explicit conceptual development are shown in the chart below. A tenth column has been added, (use of) Technology, since this aspect of the work in SkyMath deserves to be highlighted as well. Number work is always present, and the nature of the module infuses Problem Solving into every activity.

The Context-Matrix

NCTM Standard 1 2 345891013 Technology
Activity #1xxxx
#2xx xxx*xx
#5 xxxx xx
#6 xxxxx xxx
#7 xxxxx xxx
#8xx xx
#9 x x xx
#10 x x xx xx
#11 x xx xxxx x
#12 xx xxx xx
#13xxxx x x x
#14 x x x x xx
#15 x xx x xx x x x x
#16 ** xx

* The work with patterns and functions is implicit

** Activity 16 will most likely include all of the areas - and perhaps others, as students reflect on their work and share results. Since communication is so central here, a xx has been placed in that column to show that emphasis.

At the beginning of each of the 16 activities, the exercises that address the mathematics standards identified above (mathematical goals of the activity) are described:

  1. Students identify questions and pose problems that can be solved using mathematics. They are introduced to the Celsius and Fahrenheit temperature scales.
  2. Students review what state, national and world weather data are available and how those data are collected. They design a process for collecting data to answer local weather-related questions, and decide on a common set of data to be collected. Students learn how to read a thermometer in Fahrenheit and Celsius scales and how to use a weather log, read and record temperatures, and calculate changes.
  3. Students read a max-min thermometer. They record minimum and maximum temperatures and analyze differences and use maps to identify locations. Students look for patterns of differences across the United States. They learn to use Blue Skies to access and upload temperature and weather data.
  4. Students learn to read Celsius and Fahrenheit thermometers with intervals representing 1 and 2 degrees, respectively. They match Fahrenheit and Celsius temperatures. They add and subtract signed numbers (integers) and look for patterns and discuss rules.
  5. Students list statements that are measured by temperature and rank order them. They review how Fahrenheit and Celsius created their temperature scales and make and calibrate their own temperature scales by identifying fixed points. Students find differences and use other operations. They measure distances and divide them into equal intervals and assign temperatures to events.
  6. Students describe how units of measurement are mathematically related. They develop formulas for converting between units of measurement.
  7. Students identify and describe mathematically significant patterns in data sets. They use ratios to describe the relationship between two scales. They graph number pairs on a coordinate system and discuss properties of their graph, such as slope and intercepts. Students use the slope of the line to informally develop and describe a rule for converting Fahrenheit and Celsius scales. Students translate an informal rule into a formula using a spreadsheet. They create spreadsheets for converting between Celsius and Fahrenheit and describe the rule.
  8. Students read analog thermometers and examine the concept of calibration (equating scales so that all instruments read the same at the same location and the same time). They make conjectures about room temperature. They make bar graphs and consider maximum, minimum, and range of temperature readings in the room.
  9. Students examine the concept of "typical", or average, and explain their reasoning. They are introduced to measures of central tendency: mode, median, and mean. They discuss which measure of central tendency best describes the "typical" temperature in the room and in a given situation. They explore the effects of extreme data points on the mode, median, and mean of a data set.
  10. Students discuss ways to gather data to determine the "average temperature" at a location. They report on and critique their plans. They are encouraged to consider how the choice of measure of central tendency is affected by sample size, placement of recording devices, timing, range, and repetition of specific readings. Students identify extremes, compute the range, and find the mean, meadian, and mode of given sets of data. They make decisions based on an analysis of data.
  11. Students read and interpret line graphs; they discuss scales, axes, and labels. They construct line graphs to show changes in temperature over time and use line graphs to informally compare rates of change and they informally discuss slope.
  12. Students interpret line graphs and discuss patterns in the data. They use the LogBook software to construct a line graph and import data into a spreadsheet. They use the data from the StowAway to create a table of temperature highs, lows, and range. They identify patterns and trends in line graphs and tables. Students compare weather log data with StowAway data.
  13. Students calculate the rate of change between high and low temperatures. They explore the meaning of a positive (or negative) rate of change and relate rate of change to the slope of a line in a line graph. They compare rates of change. They write about data collected outside the classroom using StowAway data, noting patterns in the data.
  14. Students read and interpret a temperature map and discuss legend, labels, patterns, and differences between predicted and actual high temperatures. They use computers to gather data about high temperatures in selected cities and compare the actual high temperatures with the predicted highs. They represent national highs as color bands on a map and construct a temperature map of the United States and compare actual to predicted highs for the day. They discuss reasons for differences between predicted and actual temperatures.
  15. Students consolidate knowledge and skills acquired in previous sections. They pose and solve authentic problems and apply reasoning and problem-solving skills to independent projects.
  16. Students use mathematical communication to explain the results of their work on a final project. They explain how they solved problems and reflect on what they have learned. They assess their own progress and achievements.
  17. (This analysis was prepared by Dr. Fernand J. Prevost. )

Module Design
In an article published in the Mathematical Sciences Education Board's On the Shoulders of Giants, Ian Stewart discusses mathematical concepts related to "change." Noting that every natural phenomenon is a manifestation of change, Stewart stresses that mathematics is the most effective tool for understanding patterns of change. He notes that to master the concept of change, one must be able to "represent changes in a comprehensible form, to understand the fundamental types of change, and to recognize these types of changes when they occur".

The SkyMath Design Team identified the mathematics of change as the most natural concept to address in a module that utilizes current weather data. The module articulates a mathematical study of the changing weather, focusing on temperature. It leads students to develop methods of representing change; how temperature changes with time and with location. Students are challenged to measure, represent, and analyze these changes.

Thinking about how the real world changes with time and recording real-time data to accumulate a time series are activities that will evoke student ownership of the data and will provide experiences that retrospective data cannot.

The module makes use of the KWL design, a research-based feature that is commonly used in reading instruction.

The 16 activities of the module reflect this development of student-initiated learning style.

The module design calls for three levels of performance assessment. Two of these (lesson-by-lesson assessment to guide instruction and summary assessment through an extended project) are internal to the module; note that KWL is also a form of imbedded assessment. The third level is an external assessment with a comprehensive test of the module's key mathematical concepts.

Also included in the module design is an array of "teacher inservice" features that are intended to make the module self-standing, i.e., usable without formal inservice by outside experts. For instance, it provides initial inservice or support for teachers by providing "stories" from teachers who have used the module. Stories that anticipate problems, suggest solutions, and provide background lead to enriched classroom discussions and offer teachers tips alerting them to challenges and opportunities. Step-by-step descriptions of the use of technological tools are amplified by diagrams and screen images. Internet mail groups are established early in the module for use by teachers and students.

The module is designed to employ technology as a tool in the service of the learning of mathematics. The level of technology used is appropriate to the tasks at hand. The pursuit of questions leads students through a progression of resources: newspapers, television, thermometers, graphs, max- min thermometers, an electronic device (coupled with a computer) for the organized collection/display of data over extended periods of time, e-mail, a spreadsheet, and software for displaying and exploring real-time surface and satellite weather data. The rule for the use of technology is that its form follows an identified need.

A typical lesson begins with an overview page that describes the math goals, the ongoing assessment, and the necessary advanced preparation. Following this are descriptions of the student activities with associated "teacher stories". Background information and explanations of the technology used are included, followed by assessment guidelines.

Examples of the kinds of mathematics addressed include the use of scales, statistics, and data representation. One example of the richness of the mathematics involved in reading a temperature graph is given here in the story of Euclid's tour of the University, and an assessment profile is also provided.

SkyMath and the National Science Education Standards
Although the modules are mathematics modules -- instructional units focused on the development of mathematical concepts and procedures found in the NCTM Curriculum and Evaluation Standards -- they are also rich in the science content and procedures set forth in the National Science Education Standards. According to these guidelines, the fundamental abilities and concepts that underlie the science standards include the ability to identify questions that can be answered through scientific investigations:

The activities in the SkyMath temperature module follow this scientific method and develop all of the abilities described in the Science Education Standards by applying a scientific approach to the investigation of temperature.

The physical science standards for grades 5-8 focus on the characteristic properties of substances, such as boiling and melting points; thus, the concept of temperature must be included. The Earth and Space Science content standards for grades 5-8 stress the development of an understanding of earth and the solar system as a set of closely coupled systems, the atmosphere being one of the four major interacting components of the earth system. SkyMath clearly incorporates elements of the quantitative study of the temperature of the atmosphere with associated weather patterns.

The Science and Technology Science standard for grades 5-8 suggests that students investigate simple, familiar objects (in the case of SkyMath, it is a thermometer or temperature probe) through which students can develop powers of observation and analysis. In SkyMath, the technology is introduced only as needed to aid a specific investigation. The students can see the value of technology in providing instruments, techniques, and communication capabilities that help them master their SkyMath projects. The history of the development of the thermometer, provided as background material for the teachers in the SkyMath module, and further developed on Web page About Temperature, serves as a specific example of the History and Nature of Science standard for grades 5-8. The pedagogy of the SkyMath module is consistent with the Science Teaching Standards that call for inquiry-based science programs.

Education Development Center The SkyMath educational strategies and materials have been prepared by the Education Development Center,Inc. (EDC) of Newton, MA, a nonprofit research and development organization (see their Skymath page), in a flexible curriculum module that includes guidance and tools for exploration plus a collection of resources and activities that use weather and real-time data to teach math concepts.

The SkyMath Module
[Get Acrobat Reader]The second draft of the module, which was used in our April 96 field tests, can be downloaded from this page. These files are in pdf format and you will need to first download a copy of the Adobe Acrobat, which can be accessed here free of charge. You also may need (for a Mac) to download a Stuffit Expander program to decompress the Acrobat file.

Once you have installed this reader on your computer (read the text read me file for help in this), then click on the files listed below to download the module. Instruct your browser to save the files to your harddrive or to floppy disks. You can then open them with your Acrobat reader and find the SkyMath module. When you open the SkyMath files in the Acroread program, you will find that the hand cursor will change to a hand with index finger extended if there is a link to somewhere else. A solid red border opens up the WWW link to which the text inside it refers, but you will need another program to do this. Acrobat requires Plug-Ins to link to the WWW. They can be accessed here.

Bookmarks have been added that allow easy navigation within the documents. The Bookmarks must be turned on (a button that allows bookmarks to be turned on appears second from left in the toolbar of the reader). They appear on the left-hand side of the screen and are titled. Clicking on a bookmark takes the reader to the position described.

The module is copyright 1996 University Corporation for Atmospheric Research; permission for limited reproduction of the module or portions of it for educational purposes but not for sale may be obtained upon request from UCAR. Contact The entire module can be downloaded in the following sections:

We suggest that you send a letter (or in Spanish) to the parents of the children explaining the activities of the module.
To do SkyMath in its entirety, you will need the following materials and equipment:

The last three items cost about $200 in 1996.One good source for the StowAway and for less expensive data loggers may be found at the Science Education Solutions web page. If you have no StowAway data logger (the most expensive of the last three items), limited or no access to computers, or no connection to the Internet, you can do a scaled down version of SkyMath as follows:

SkyMath utilizes software and data dissemination methods of the University of Michigan's Blue Skies program, an endeavor developed under a National Science Foundation grant. Blue Skies software, which can be downloaded from its homepage, offers an extraordinarily simple interface so that users with a minimal computer experience can easily and quickly obtain needed information. Access is provided to hundreds of real-time weather and environmental images by using the client-server protocol developed for the University of Minnesota's "gopher" system (though Michigan dubs the information-providing components "groundhog" servers). If a teacher registers with Blue Skies, the classroom students can upload their data to his/her site.

End of Unit Assessment Instrument

The SkyMath Assessment Team, headed by Dominic Peressini, University of Colorado at Boulder, has prepared a two-part assessment instrument that can be used at the end of the unit as a tool to evaluate student achievements in mathematical skills, abilities, and understanding. The Team refined a set of assessment items that map directly back to the goals and objectives of the SkyMath curriculum. The Assessment Instrument is designed around three primary constructs: (1) National Assessment of Educational Progress (NAEP) items, (2) SkyMath short-response items, and (3) performance-based items. The short-response items are based on the SkyMath curriculum, classroom observations, collections of student work, and teacher and student interviews.

The performance-based items are based on the SkyMath curriculum and have been developed using the same process as was used for the short-response items. These tasks allow students to demonstrate their ability to apply mathematics in meaningful problem situations. They ask students to demonstrate their problem solving, reasoning, and communication abilities as well as their ability to make connections within the discipline of mathematics and across content areas. Dominic's team has also prepared a General Scoring Rubric for the open-ended items and a Scoring Profile Sheets for each of the open-ended problems.

These performance-based items are available in .pdf files here:

Finding a Partner Class
One of the SkyMath activities most popular with the students is corresponding via e-mail with a partner class. Teachers wanting to use the module will have to pair up with another classroom and coordinate the timing of the activities so that students can effectively share their weather studies with each other. In order to find a partner class, please register here. Review the list and contact a possible teacher for collaboration. When the two of you have agreed to be partner classes, remove your names from the list by scrolling down the form to the "remove your name from the list" form (you have to remember your password!). It is important that the paired classes coordinate the scheduling of activities so that students will be corresponding while participating in the same SkyMath activities.

SkyMath gives special thanks to Matt Hicks of Unidata for preparing this SkyMath form for us!
SkyMath Mailing List
We have established a mailing list for all educators interested in or using SkyMath. We hope that teachers will share their experiences with others and that any helpful supplementary materials can be referenced here. To subscribe, go to the Unidata mailing list registration and scroll down to skymath in the "select list" option. Then register and participate in the SkyMath correspondence!

This mailing list is not for students; to set up an e-mail exchange for students, teachers must find a partner class using the "Finding a Partner Class" procedure.

Field Tests
The schools that took part in the Spring 96 Field Test were Hanscom Middle School, Bedford MA; Centennial Middle School, Boulder CO; Gilbert School, Gilbert IA; Long Middle School, Atlanta GA; Sherman Indian High School, Riverside CA; John Muir Middle School, San Jose CA; Gilroy School, Gilroy CA. Centennial Middle school was assisted by University of Colorado at Boulder.Long Middle school was assisted by Clark Atlanta University Earth System Sciences Program; Gilbert School was assisted by Iowa State University; the Gilroy and San Jose schools are assisted by San Jose State University.

The fall 1996 field test was conducted at

The final field test in the Spring of 1997 added the following sites:

We have prepared a questionnaire that we would like to have all teachers who have used the module to complete and send their answers to
Beverly T. Lynds
P.O. Box 3000
Boulder CO 80307-300

The SkyMath module is now complete. The materials are available on the Internet so that anyone wishing to use the module can do so with Internet access. All feedback is welcomed and should be sent to the Program Director, Beverly Lynds (

Insites' External Evaluation of the Program
We began classroom evaluations during the 95/96 school year and continued through 1997. Initial results indicated that the students learned the mathematical concepts featured and that they and their teachers liked the activities.

Insites' final report focused chiefly on instructional design and development considerations for learning, teaching, and technology. Findings in these areas provided evidence that SkyMath is a viable method for teaching and learning mathematics in the middle grades, and that technology supports the integration of weather data and mathematics. Together, weather data, mathematics, and technology are appealing to students. We quote here the summary of their report:


"Over the course of development and field testing, the SkyMath module has proven to be a highly effective tool for teaching and learning middle school level mathematics. Some key contributing features are:

Some key findings that emerged from the evaluation of SkyMath are:

Challenges to implementing the SkyMath module are:

Insites has provided us with some anecdotal data they obtained.

One teacher conducted the Skymath pilot with a group of students who have low math skills. She said that the value of Skymath for this group is that they are getting math from a completely different angle. "They have all struggled with math in the past. With Skymath, they don't realize it's math they're learning because they're not struggling." She was impressed that some of the students in her group said that learning the mean, median, and mode was easy. She thinks these are generally difficult concepts for 6th graders to master.

Skymath fits well with one school's strong overall focus on developing writing skills.

"With the old way of teaching mathematics, kids could do the mechanics, but they didn't know what it meant. With Skymath they really understand the concept."

"When using Skymath I am covering the curriculum and doing it better."

Two teachers comment that they are able to fit in all of the required 6th grade curriculum and Skymath because they are complementary.

One very positive aspect of the module is its flexibility. Teachers adjust use of the module to their context and schedule and gear instruction to their students' levels of experience and skill. One teacher has her kids working in mixed ability groups. She sees evidence that all are learning math because, "I give very little directives to students and they take what I say and know what to do with it." "I am amazed the groups have worked and have had very little trouble."

"Skymath is very motivational. The kids get excited."

"Kids will work with numbers where they never would before because they have a reason for doing it."

"Skymath has been very enjoyable for my students and for myself. We have become collectors of information and have learned how to record items in a useful manner. My students have become more aware of their environment and the changes that take place in and around school."

SkyMath Pamphlets
Building on their experiences with teachers, administrators, students, and parents, Insites, Inc. has prepared four pamphlets for SkyMath to use in publicizing its program. These pamphlets target select groups who should be informed about the program if it is to be incorporated into a school curriculum.

  • SkyMath Research Teachers
    During the course of the development of the module, we have been assisted by a group of very talented and dedicated teachers, without whose participation the project would not have succeeded. We have designated these educators as SkyMath Research Teachers. They have agreed to serve as resources for other teachers interested in using the SkyMath module. We would like to hear from any teachers interested in trying out this module in math classrooms. Internet connectivity is necessary for certain of the SkyMath activities. Please contact if interested.

    A tutorial, About Temperature, has been prepared for the use of the SkyMath (and other) teachers. It covers the basic physical concept of temperature, temperature scales, thermometers, thermodynamics, kinetic theory, and radiation temperature.

    There are many sites on the World Wide Web that hold -- or will route you to -- useful resources in mathematics education. Here's a starter set.