Using Internet with a Mathematics Replacement Unit: "Growth Patterns"



by Joan Carlson

Mendocino Middle School
Mendocino, CA

Subject Area:
Mathematics: Growth Patterns Unit

Grade Levels: 8th

Length of Project: Variable, depending on the time allowed for independent work by students. "Predicting Population and Services for a Country" takes about 3 hours in class and 2 more hours outside class time.

For information and/or additional copies, contact: EQUALS Lawrence Hall of Science University of California Berkeley, CA 94720 Attn: Investigations Units

Copyright © 1994 by the Regents of the University of California ISBN 0-912511-57-5

Produced in conjunction with the NASA NREN K-12 Partner School Program and the California Telemation Project

I. Introduction

One of the strategies used by the California Department of Education to help teachers change the way they are currently teaching mathematics is the use ofreplacement units. These are sets of learning activities which include whole Êgroup lessons, individual lessons, assessment, homework, and some long Êterm projects or investigations. They address a large piece of the mathematics curriculum and might last from four to twelve weeks. They are designed to Êmeet the goals of the NCTM Standards, so students are focused on solving Êproblem, making conjectures, investigating mathematical relationships and Êconnections, and communicating about their thinking both verbally and in Êwriting. These units are not designed as supplementary or enrichment Êmaterials, but are alternative ways to teach topics which normally occur at Êvarious grade levels. The focus is on having the students work on what the ÊCalifornia Mathematics Framework calls "large assignments," or assignments Êwhich may need days or even weeks to complete. My goal is to incorporate Êresearch done by students on the Internet with one of these projects.Ê

In the past few years, a number of interesting replacement units have been Êcreated using NSF grants especially for middle school students. These units Êare appealing to students because they stimulate their thinking and Êimaginations and make math interesting. I believe that the addition of a Êtelecommunications component to the units will only increase students' Êinterest in these projects, as well as help them get ready for the use of this Êpowerful tool in the real world. One of my favorites is a unit called Growth ÊPatterns by Dan Brutlag and now published by EQUALS.Ê

The use of the Internet along with the original investigation project will Êallow students more flexibility in how they collect, interpret, and present Êtheir information. They will have access to current statistics reflecting Êpopulation growth and decrease in various countries, as well as various Êregions of the United States. They will also have access to a population Êsimulation which allows them to let the population grow in any way they Êwant, adjust birth and death rates, and see the results graphed in a variety of Êways. The Internet can be a powerful addition to an already interesting unit!Ê

II. Student Outcomes

-- Strengthen proportional reasoning abilities by exploring the different effects of growing by adding vs. growing by multiplying on various populations.
-- Develop approaches to tackle new problems, including technology.
-- Contextualize math by using the Internet to find various sources of population data such as the Census Bureau, Bureau of Labor Statistics, World Population Bureau, ECIS, etc.
-- Use the data they find to create a final project which may be presented to the class either in written form or as a multi-media presentation.
-- Use a population simulation to evaluate the validity of the conjectures students are making about how populations grow.

III. Activities

It is assumed that students will participate in Sessions 0-6 of the Growth ÊPatterns Unit. This would mean approximately 12 hours of classroom work Êexploring various geometric, numeric, and graphical models for growing by Êadding and growing by multiplying. Students would be working in groups, Êusing calculators, making scatter plots by hand or with computer graphing Êprograms, and doing various hands- on activities which will support their Êgrowing understanding of linear and exponential growth.Ê

At Session 7, students work in groups to predict the growth of world Êpopulation given data for 1970, 1980, and 1990. The information intentionally Êincludes values for population for three years even though only two values Êcan be used to find the add-on or multiply-by number. Which pair should be Êused is not a simple question, and these are the kinds of ambiguities which Êneed to be discussed with students and resolved by them. In the real world Êdata doesn't always come out in nice even numbers, and the same problem Êwill occur for them in their individual projects.Ê

Students are given two class periods to work on the World Population ÊInvestigation given the numbers in the unit. Each group then reports orally Êto the class either using overhead transparencies prepared ahead of time or Êlarge chart paper. The presentations are discussed in class and then posted for Êthem to see while they are working on their individual projects.Ê

At this point, students get a short presentation about population growth using Êthe videotape "World Population" from Zero Population Growth, Inc. and ÊSouthern Illinois University, and some graphical information from The ÊSearch for Solutions TEACHING NOTES #12. Ê
They are also introduced to HumPop, IntlPop and Migmodel population simulations which are downloaded from the GeoSim Project. These are at the following URL: http://geosim.cs.vt.edu or by gopher at geosim.cs.vt.edu

Brief descriptions of the simulations written by their authors appears below:
HumPop is a multimedia tutorial program. HumPop introduces and illustrates population concepts and issues. Users unfamiliar with such concepts should follow the tutorial before running simulations with IntlPop.

IntlPop is a population change simulation program. Factors such as Birth ÊRate, Life Expectancy, and Migration Rate affect population growth. IntlPop Êallows you to manipulate these variables and simulate population growth Êbased on your values. Ê

MigModel is a program that provides a student with the means to develop Êand explore models relating to migration. The student selects factors that he Êor she thinks were significant to migration between counties of the United ÊStates during a decade. The student can then compare his or her model with Êthe actual migration that occurred during that period.

Using these simulations, students select a country, grow or shrink the Êpopulation in any way they see fit, adjust both birth and death rates, and Êpredict into the future. Various graphical displays can then be printed and Êincluded in their reports. They would spend at least one or two days Êexploring the simulation, and do a short written report about their findings.Ê

Once they are familiar with the simulation, students are given the Êassignment for their final individual project and will then begin to explore Êother sources of population statistics on the Internet. These might include:Ê

Gopher Sources
Florida State Population Gopher Server: gopher.fsu.edu

Population Studies Gopher at The University of Pennsylvania
lexis.pop.upenn.edu

United States Census Data
gopher.census.gov

Basic Demographic Information
bigcat.missouri.edu /reference/census/us/trend

Mosaic Resources:
The Population Studies Center at the University of Michigan http://www.psc.lsa.umich.edu

The final project involves predicting the size of a population for a time in the Êfuture. The project offers the opportunity for consolidating ideas in the unit, Êand gives the student an opportunity to demonstrate what he/she has Êlearned about linear and exponential growth. It also allows for an in-depth Êexploration of a mathematical idea in a open- ended format, and allows the Êstudent to choose how he/she will present their information.Ê

"Predicting Population and Services for a Country:"
A country is selected and a source of data found. My additional Êrequirement is that at least one source of the students' data be on the Internet. Ê(This could also be modified to be done for a state using data from the Census ÊBureau on the Internet.)

An appropriate government service must also be chosen, as some Êservices provided by government seem to be independent of population. The Êsize and wealth of the population also determine the tax base from which Êgovernments can get income to provide these services. Some examples Êmight be schools (they have already used the linear and exponential growth Êmodel to predict how many school would be needed earlier in the unit), Êmedical care facilities, public transportation, parks and recreational facilities, Êetc.Ê
Students then prepare a report to be sent to the appropriate Êgovernment agency of the country they have chosen. In the report, they must Êmake a prediction about population changes over the next 20 years and make Êrecommendations about a particular government service that depends on Êpopulation size.Ê
In order to do the report, they get data on how the population has Êchanged in the past using statistical information from the Internet. They Êthen use the data and the mathematical methods they have learned in the Êunit to make population predictions. At this point, they might also use the Êpopulation simulation to see what it might predict for the same period of Êtime. The students' projects can be written or oral, or a multi-media Êpresentation using slide shows with the information gathered from the Êpopulation simulation.Ê

IV. Assessment

"Assessment is accomplished by having each student keep a folder of all work Êdone during the unit. The contents of the folder provide a wide range of Êcontexts for examining student thinking including notes, written reports, Êmodels, diagrams, group work, experiments, and projects." (Growth Patterns) ÊA rubric is included in the unit (see addendum)

After individual projects have been handed in, a day is set aside for students Êto share their projects with each other. Oral presentations are given, and sets Êof written projects are passed around from group to group with specific tasks Êfor the groups to address such as: -- find an example of a project which was Êorganized clearly; describe the format and explain why you think it is clear; -- Êfind an example of a method of predicting growth other than the add-on or Êmultiply-by method; describe the method and tell why you think it is or is not Êvalid; -- find and example of a report that used information from the ÊInternet effectively; what are some interesting things other people learned Êfrom the Internet? Each group can then report to the class about the reports Êthey read. This allows for the work done in each class to be shared with all Êthe other sections of eighth grade math.

Projects are graded using a rubric of Well Done, Acceptable, Ready for ÊRevision, and Restart/Reteach. In order to complete a high quality project a Êstudent must include everything in the assignment as well as reflect on the Êideas learned in an active and purposeful manner. Students would also be Êasked to reflect on their use of the Internet to find information for this Êparticular project. Project work enables students to use their new Êmathematical knowledge and to show what they have learned in a creative, Êopen-ended way. The projects will become part of the students' work folders Êfor the entire Growth Patterns Unit, and may be chosen to be included in Êtheir portfolios for the trimester grade.Ê

Addendum 1
Growth Patterns - Judging Quality
Well Done

Mathematics Content

Add-on and multiply-by methods of prediction are used and compared.

Graphs of each are shown.

Report describes events that may make predictions invalid.

Estimates of probabilities are included.

Effects of each event on predicted numbers are considered.

Mathematics is careful, accurate, and appropriate.

Many or all possible variations are considered.

Multiple representations (diagrams, words, formulas, etc.) are used.

Analysis is thorough.

Processes:

A third method of predicting is devised and used.

Specific recommendations are made based on results and predictions of final investigation.

All work is completed.

There is thoughtful organization (of data, information, presentations, etc.)

Generalizations are made.

Presentation of Report:

Sources of data contributing to predictions are given.

Written report is well-organized.

Language is clear and appropriate (complete and understandable sentences).

Format is neat and orderly.

Appropriate audience is addressed.

Acceptable

Mathematics Content:

Mathematics is generally careful and accurate, but with some errors in the work or in the logic applied.

Limited number of possible variations are considered.

Single representation is used.

Analysis may need more care.

Processes:

Organization may need to be improved.

All work is completed.

Recommendations may not relate completely to results or findings.

Generalizations and predictions or further questions may be missing.

Presentation of Report:

Language is generally clear and complete.

Work is accurate and neat.

Audience may not be clearly defined.

Sources of information may not be complete.

Ready for Revision

Mathematics Content:

Work may be the beginning of an acceptable report, but mathematics needs careful review and correction or refining.

Processes:

The processes used should be reviewed and amended as needed.

Focus questions from the teacher will lead to revisions.

Presentation of Report:

Report provides a start but needs major revisions.

Restart/Reteach

Mathematics Content:

Quality of work needs improvement or the topic is not useful-- the project needs to be started over with further instruction.

Processes:

More instruction is needed to find processes that will work for this investigation.

Presentation of Report:

Report needs more direction in applying format, language, or other standards.