# MSP:MiddleSchoolPortal/Connections: Math History as a Teaching and Learning Tool

### From Middle School Portal

## Connections! Math History as a Teaching and Learning Tool - Introduction

*This is the second publication in our Connections series, which identifies resources that not only support math teaching but also illustrate connections to other subjects in the middle school curriculum. The first publication in the series connected math to social studies, art, and science.*

How does history fit into the math curriculum? Not, we believe, as an extra unit, but as context that enhances the standard curriculum. Students find it hard to believe that mathematics is a human endeavor — created and invented by real people to solve real problems. Teaching how such mathematical ideas as numerals and number systems, scale and measurement, geometry and probability developed over time presents the material on a wider stage and goes a long way to explaining its relevance.

## Contents

The resources featured here can deepen student understanding as well as enrich classroom teaching of topics covered at the middle school level. The Connections Standard of the Principles and Standards for School Mathematics is the overall linkage among the resources. The resources are grouped in sections aligned with the content standards of Number and Operations, Measurement, Geometry, and Data Analysis and Probability.

## Background Information for Teachers

If you are looking for problems in a historical context, the first three web sites below offer several. The last site is the mother lode of mathematics history, an overabundance of trustworthy, researched information on people and topics that have shaped the field of mathematics.

**Famous Problems in the History of Mathematics**
The purpose of this site is to present a small portion of the history of mathematics through an investigation of some of the great problems that have inspired mathematicians throughout the ages. Included are problems that are suitable for middle school and high school math students, with links to solutions, as well as links to mathematicians' biographies and other math history sites.

**Using Historical Problems in the Middle School**
This is a collection of historical problems drawn from medieval times, from a 19th-century American textbook and from a 19th-century Armenian textbook, and from other sources. Included are answers and, most often, complete solutions. The problems can be solved through arithmetic, measurement, and algebra skills covered in middle school.

**Completing the Square**
A major goal for algebra students is to understand, solve, and apply the quadratic equation. Useful as is factoring, it is not the original way of solving quadratic equations. The quadratic equation, as we know it today, was first discussed and taught by Muhammed ibn Musa al-Khwarizmi (790-850). He actually solved quadratic equations by the method we now call "completing the square." This site offers a visual explanation of the method.

**The MacTutor History of Mathematics Archive**
This site has topical articles, short biographies of more than 1,300 mathematicians, and timelines. Helpful for student projects! Also particularly interesting are the overview of Indian mathematics, a history of zero, and biographies of female mathematicians.

## Number Systems and Number Patterns

The first part of this section, Number Systems, examines the number systems devised by different cultures. Students should become aware that all peoples invented systems of counting as needed for trade, tax collection, and other activities. Scroll down to the second part, Number Patterns, to find activities that are, perhaps surprisingly, based on Pascal’s triangle.

**Number Systems**

**Egyptian Mathematics: Numbers**
This web site has a brief introduction to the ancient Egyptian way of writing numerals. It includes graphics of what the hieroglyphics looked like, problems written using Egyptian numerals, and a downloadable worksheet creator, which is also available as a CD.

**History Topics: Babylonian Mathematics**
This web site contains an overview of Babylonian mathematics, with links to in-depth analyses of some topics. All students will be interested in the Babylonian numerals and their sexigesimal (versus our decimal) system. Older students who have studied the Pythagorean theorem will be surprised that knowledge of this theorem, so thoroughly linked to Greek mathematics, appears in a Babylonian clay tablet written between 1900 B.C. and 1600 B.C.

**History Topics: Chinese Numerals**
Chinese number symbols from ancient times (14th century BC) are shown here, along with ideas on why particular symbols were chosen to represent certain numbers. A second set of symbols appeared after the abacus came into use. A great opportunity for discussion on the evolution of place value notation!

**Indian Numerals**
This article begins with a quote from the mathematician Pierre-Simon Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India." Theories of how the Indians developed this method are described in detail, including the evolution of the numerals themselves and the invention of the decimal place value system we use today.

**Links to Information on Number Systems**
If you are looking for more resources about different numbering systems, you will find them here. The site includes links to Arabic, Chinese, Mayan, Roman, Greek, Egyptian, and Babylonian numbering system resources.

**A Creative Encounter of the Numerical Kind**
In this WebQuest, students help an imaginary civilization develop a number system. They work in teams to explore place value, counting, and different number systems. After this preparation, they create and name a set of original number symbols for a base four number system and explain it in a formal presentation.

**Number Patterns in Pascal’s Triangle**

**Sierpinski Meets Pascal**
This activity opens with students constructing Pascal’s triangle on a special grid. It continues with their creating a pattern in Pascal’s triangle as they shade in all the triangles except the odd-numbered ones. A surprise connection to the Sierpinski fractal results! (

**Coloring Multiples in Pascal's Triangle**
Teachers can assign this applet and discussion materials to small groups to help students identify multiples of numbers. As each multiple of the selected number is correctly identified, it changes color. The patterns created in this way are both surprising and satisfying. Since the triangle can be increased to as many as 15 rows, finding all the multiples can become quite a challenge!

**Which Way? Oh, Which Way Do I Go?**
In this math challenge, the student looks for different ways to go from his home to the video store. The challenge page contains links to hints, the solution, and other similar investigations. The solution describes and illustrates how a number pattern is embedded in the question. The connection to Pascal’s triangle is shown under the "Did You Know?" link.

**Pascal's Triangle: Number Patterns**
How to construct the triangle, notes on its history, and a link to several patterns to explore — triangular numbers, Fibonacci numbers, hexagonal numbers, and so forth — are found here. Among these patterns, you will find a concise but clear explanation of the connection between Pascal’s triangle and the coefficients of a binomial expansion An unexpected pattern for your algebra students!

## Measurement

Measurement is a topic your students have been exposed to since the early elementary grades. These sites offer either more advanced work or in-depth historical information for middle grades students.

**Approximating Pi**
Mathematics and science owe a great deal to the Greek mathematician Archimedes, including a way to approximate pi. Here is historical information along with an applet that allows users to replicate his approach. Students can approximate pi as a number between the lengths of the perimeters of two polygons, one inscribed inside a circle and one circumscribed around the circle. The number of sides for the polygons may be increased to 96 with the value for pi always lying between the two approximations.

**Making a Pi Necklace**
A seventh-grade teacher uses this activity as a way to introduce irrational numbers. Diana Funke has her students make a pi necklace for Pi Day to reinforce the idea that the decimal part of some numbers never repeats or ends. They assign a color to each digit (including 0) and then string beads of those colors into a necklace, using the digits of pi as their guide.

**Measuring the Circumference of the Earth**
This Internet project is hands-on, real-world, and historical. Students join with classes around the world to repeat the experiment of Eratosthenes — measuring the shadow of a meter stick and making calculations to approximate the circumference of Earth.

**Eratosthenes and the Mystery of the Stades**
This interesting and clearly written article explains Eratosthenes' famous measurement of the circumference of the Earth and discusses the mystery surrounding the accuracy of that measurement. A key element in the mystery is the ancient unit of length used in the measurement: the stade.

**Meter (Metre)**
Here you will find detailed scientific information related to the creation of the meter. The text does a fine job connecting the scientific theory behind the metric system to the practical efforts of Mechain and Delambre. The development of the meter, first as a bar and then as a specific distance measured by light, is covered in an easy-to-understand format, well suited to both teachers and students.

**Archimedes’ Puzzle**
The stomachion is an ancient tangram-type puzzle. Believed by some to have been created by Archimedes, it consists of 14 pieces cut from a square. In this activity on measurement, students learn about the history of the stomachion, use the pieces to create other figures, and investigate the areas of the pieces.

**Weights and Measures**
This part of a virtual museum produced by the National Institute of Standards and Technology explores weights and measures and is divided into museum "rooms." Of special interest, Room 1 focuses on the American colonies before there were standard weights and measures. Room 4 shows a platinum-iridium bar, cast sometime between 1875 and 1889, that was used as a standard of measure for the meter in the United States. Room 8 is filled with interesting information and objects that offer a social history of weights and measures, including the part weights and measures play when we buy gas at the service station or milk at the supermarket.

## Geometry

These resources deal with geometric solids and the Pythagorean theorem, two topics covered in middle school classrooms. These topics can be enriched by introducing them in their historical contexts.

**Platonic Solids (Grades 6-8)**
The Greeks saw the world of mathematics through geometry, through shapes and the relationships among them. Here students use a virtual manipulative to examine in detail the five Platonic solids. They can rotate each solid, view it from every angle, change its size, and then use the transparent mode to see only its skeletal structure.

**Geometric Solids and Their Properties**
Europeans examined three-dimensional geometric shapes and developed mathematical arguments about geometric relationships. In this lesson, students analyze characteristics and properties of several solids, counting the number of faces, edges, and vertices. Eventually, they discover Euler’s theorem for themselves.

**Pythagoras Theorem**
Images taken from ancient Chinese mathematics texts depict a proof of the Pythagorean theorem, as well as 3rd-century problems and solutions familiar to today’s older middle school students.

## SMARTR: Virtual Learning Experiences for Students

Visit our student site **SMARTR** to find related math-focused virtual learning experiences for your students! The **SMARTR** learning experiences were designed both for and by middle school aged students. Students from around the country participated in every stage of SMARTR’s development and each of the learning experiences includes multimedia content including videos, simulations, games and virtual activities.

## Careers

**The FunWorks**
Visit the FunWorks STEM career website to learn more about a variety of math-related careers (click on the Math link at the bottom of the home page).

## Proability

## NCTM Standards

These resources deal with geometric solids and the Pythagorean theorem, two topics covered in middle school classrooms. These topics can be enriched by introducing them in their historical contexts.

**Platonic Solids (Grades 6-8)**
The Greeks saw the world of mathematics through geometry, through shapes and the relationships among them. Here students use a virtual manipulative to examine in detail the five Platonic solids. They can rotate each solid, view it from every angle, change its size, and then use the transparent mode to see only its skeletal structure.

**Geometric Solids and Their Properties**
Europeans examined three-dimensional geometric shapes and developed mathematical arguments about geometric relationships. In this lesson, students analyze characteristics and properties of several solids, counting the number of faces, edges, and vertices. Eventually, they discover Euler’s theorem for themselves.

**Pythagoras Theorem**
Images taken from ancient Chinese mathematics texts depict a proof of the Pythagorean theorem, as well as 3rd-century problems and solutions familiar to today’s older middle school students.

## Author and Copyright

Terry Herrera taught math several years at middle and high school levels, then earned a Ph.D. in mathematics education. She is a resource specialist for the Middle School Portal 2: Math & Science Pathways project.

Please email any comments to msp@msteacher.org.

Connect with colleagues at our social network for middle school math and science teachers at http://msteacher2.org.

Copyright January 2008 - The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671 and since September 1, 2009 Grant No. 0840824. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.