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Math and Games of Ancient Civilizations 

Tamara Werner

Academic Setting

For readers that are hoping to use my unit, I want to explain some statistics.  These statistics have everything to do with why a lot of my students are having a hard time with academics in school.        

The majority of parents in the West Gate Community make below $25,000 annually.  Actually 43% of the parents make this low wage.  I hope that you can see how these children come from poor backgrounds.  For a family of five or more, they make less than a teacher does.  I feel that I make a low wage, but in the end I make more than most of my students’ parents. 

Parents who send their children to Truman make a low wage because most of them have only a high school diploma.  Exactly 42% have such a Diploma.  This seems low to me, because I come from a background where everyone at least gets through high school and graduates.  Most even go and get some college education.  But with the West Gate Community, many of them do not have that opportunity to be able to finish all their schooling.  There are about 5% who do get a bachelor’s degree.  Then again we go back down with 19% not even finishing high school. 

            Also because of the parent’s low income most of our students are on a free or reduced lunch.  The last statistics that I was quoted for the previous school year was 78.8%.  This is almost our total school population that does not pay for their lunches.  During the 2000-2001 school year, the free and reduced lunches were 76.7%.  It has increased over just one year.  I do not have the exact percentage for this past school year, but it is very similar. 

            Truman’s enrollment has always been higher than the average middle school in this district.  In 1999-2000 Truman had 1018 students compared to 812 with other middle schools.  In 2000-2001 Truman had 808 students, where the other middle schools had 781.  Truman’s enrollment went down because of the opening of a new middle school.  But our enrollment is on the rise again, due to the development of many new houses near us.  We are expected to be about 950-1000 students again this coming year.  This is hard on the students and the teachers.  We are working with larger then optimum class sizes and being crowded into one school. 

            One good thing about Truman is that our drop out rate is a lot lower than the APS district’s average.  In 1999-2000 Truman had 0.97% drop out rate, where the district had 1.42% drop out rate.   I always thought that this was weird, but I found out that our counseling follows up with all students to find out where they go.   There never is a student that they do not try to find. 

            Truman’s attendance rate has been in the 90’s the last five years.  This is outstanding.  Our students come to school, instead of ditching.  In 2000-2001 our attendance was at 93%.  This has not been our highest, but this is still very good.

            At Truman we have a large majority of Hispanic students.  We have 83.7% of these students.  The next one is White with only 9.3% of these students at Truman.  There is not a higher percentage for any other ethnicity.  African American is 3.3%, Native American is 2.6%, Asian is 0.6%, and other is 0.5%.  We have a skewed look at the different ethnicity that we have.   

            I wanted to give you the current status of Truman’s Terra Nova scores.   This was just given to the school at the end of May.  This past year Albuquerque Public Schools took a different form of the Terra Nova.  Overall we showed a small improvement in our scores.  But we are still in probationary status.  This means that we might be taken over in a few years by the state.  If we continue to show improvement, we will be taken off the probationary list.   

            Our scores for the 2001-2002 school year are as follows: 

Grade	Subject	Percentile
6th	Reading	32%
6th	Language	36%
6th	Math	28%
7th	Reading	33%
7th	Language	44%
7th	Math	30%
8th	Reading	38%
8th	Language	34%
8th	Math	32%

As you can see we are almost all below the 40 percentile.  There was one major rise with the seventh grade in Language.  These percentiles may not look like we are really getting better, but we are.  At one point, our eighth graders were close to being below the 20^th percentile.  I have not included Science and Social Studies.  Our students got 29-38% on the Terra Nova.  I did not put them in because when the administrators and the teachers are talking about the test scores, we are always worried about the Language and Math scores.  The others are very important also, but we as a school are trying to improve our Math and our Reading and Writing scores on the tests.

Context and Background 

Rationale 

The rationale of the unit titled “Math and Games of Ancient Civilizations” is something that I am still thinking and rethinking in my head.  This unit is the hardest one that I have done so far.  I keep thinking of trying to implement games into the unit, but a lot of the games that I have found from the ancient civilizations that I am looking at do not involve mathematical games.  So again, I had some thinking to do about it.  I decided to try mathematicians and math problems from their civilization.  I wanted to have the students do some mathematics from the original people that found out these answers.  But I thought that would be a little boring for my students. 

            The final unit will be based on mathematicians and mathematics from their time (ex. Egyptian writing).  It will be more meaningful to the students if they can find out and know where the things that they have to learn came from.  Plus each lesson will include a mathematical game from their civilization that will help bring home that math was everywhere.  Most of the games will be board games, because of all the mathematical thinking that goes into trying to win.   The games will be seen again with the lesson plans. 

            Many of my students have asked me where certain concepts come from.   An example of this is one of my students asked where and when did decimals start.  He said that who ever started this part of mathematics was “stupid.”  He did not want to do the work and he was having a hard time understanding this concept.  I am hoping that when I am able to explain why decimals were conceived, my students will understand why they need to learn how to do the concept and understand why they are doing it. 

            Many of my other students always wonder what mathematics has to do with their lives.  Again with the decimal example, I could tell that student more about decimals than just using it with money.   I feel there must be more examples out there that would help my students to realize that mathematics is important, and that it does not always have to be something that they just have to do. 

            The last part of my unit will involve a research project that my students will have to complete.  They will be researching one mathematician.  They will be looking for who these people were, what they did in their life, and more.  I want my students to really get a good idea about one mathematician one that they can relate to!  They can present this research project in a couple different ways.  The students can write a one-page typed paper about their mathematician or they can do a report on the mathematician.  This will include a poster board with pictures or drawings, and information about that mathematician.  The last selection that the students can pick from is to become that mathematician for the day.  They will dress up like they were that person and come into the class to introduce themselves.  They have to be prepared to answer all questions that the other students will have for that mathematician.   So they can not slack on the research of their person. 

The main reason that I want my students to do this research is because Truman is working on becoming more literate.  We want our students to read more and write more.  This includes my mathematics class.  I felt that this was one way to involve my students in reading and writing, without it feeling like I am cracking down on them.   

This unit will be given over the length of the year.  Each new mathematician and mathematics from their time will be introduced when the students are studying each civilization that the mathematicians came from.  I want this unit to also be cross-curricular for the social studies class.  This will bring home that all subjects in school relate to each other.   

Background 

For the background of this unit I wanted to talk about some of the mathematicians that my students would be studying.  It is important that the teacher knows what to expect before starting to teach the unit.  So I will be giving a brief biography on each mathematician. This will hopefully give a sufficient amount of background.   

            I will also be including a female mathematician in my unit.  I feel that we can not talk about the history of mathematics, or the mathematicians from history, without talking about the first women mathematician.  I will also be allowing my students to research some of these women mathematicians, since most of these women came in more recent times than from ancient times.  I know that I am almost contradicting myself, but I want my girls to feel that they can have someone to look up to in mathematics.  If I only allow them to do mathematicians from the ancient civilizations then they will more then likely have only one female to research.  I want my students to each have a different mathematician.  I have a problem with my students just copying each other’s work.  They will be more likely do their work when they cannot copy from their friends. 

            The civilizations that the mathematicians will be from are the ones that sixth graders are expected to study.  The civilizations are Babylonian, Egyptian, Greek, Roman, and Persian.  They will also be studying ancient civilizations from the Orient.  These Oriental civilizations are the Indian and Chinese.  All the civilizations that my students study in sixth grade are from 5000 BC to about AD 500.   

            Mathematics started in about 4000 BC in Babylon.  Mathematics first started with the use of numbers and counting their livestock to make sure they knew how many and what it was worth.  In Babylonian times, livestock was their livelihood.  They needed to know how many they could barter for food or anything else.  But as with many civilizations from around that time, they did not leave a good written record of their transactions.  What we do know is “that many of the business methods we use today, such as rates of interest and credit, were used in principle some 4000 years ago, and had a great influence upon the lives of the people” (Freebury 8).  But we do not have mathematicians from these ancient times that wrote their thinking down for later use. Apprenticeship programs educated people by passing down learning by word of mouth. 

            The next civilization that came about that used mathematics extensively was the Egyptians.  Egyptian mathematics is better understood by historians today because there were two rhinds or papyruses that have mathematical problems on them.  Also we know more about Egypt because they were better record keepers, leaving uncounted documents for later civilizations to decipher.   

            After the rise of Egypt came the Greeks.  Their first great mathematicians were Thales and Pythagoras.  “No mathematical masterpiece from either one has survived, nor is it even established that either Thales or Pythagoras ever composed such a work.”  (Boyer 49).  The greatest works in mathematics from Greece are in geometry.     Thales measured the height of a pyramid.  Pythagoras wanted to know the area of a triangle.  This is just the beginning of Greek mathematics, there is much more.   Euclid and many others were the great mathematicians of their time.  Many of them came from Greece and helped Greece become one of the major beginnings of mathematics. 

            “The Romans contributed very little to the development of mathematics, or to any other science, with a few slight improvements here and there, over what had gone before”  (Freebury 63).   However, the Romans were very interested in mathematics and what it could do for them.  Julius Caesar was one of the notable differences with this.  Caesar was one of the few Romans that was interested in mathematics.   He helped change the calendar, and also had his empire surveyed.  The major contribution of the Roman Empire to math is their Roman Numerals.  “You know that these are really letters, I, V, X, L, C, D, M, and you many have read that these originated as hand signs”  (Freebury, 63): Also with that the C and M have to do with century and millennium.   

            The Persian Empire brought about our own number system.  Also another great gift from them was our decimal place system by Alkarismi or, in Latin, Alggorismus.  It was also this empire that translated the different works from other parts of the world.  They would translate it and then it would be taken to Europe where it was this new thing to the people.   Many other things came from this part of the world.  A different way of medical science came about with the new discoveries. 

            In China, the first work that was produced was done in about 1200 BC.   The work that was thought a better piece was a book that “includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and properties of right triangles”  (Boyer, Pg. 218).   The Chinese actually liked patterns.  They were the first to have the magic squares. 

            India became a mathematics center in about AD 500.  Before then, Indian mathematics was not progressing very fast.   At the end of the Roman rule, more and more people started to write down their discoveries.  The first known mathematician was Arya-Bhata, but most of his work had to do with astronomy.  The second did a lot of work with mathematics.  This was Brahmagupta.  He wrote about arithmetic, algebra, and geometry.  A lot of their work was used in astronomy.  The last one that I would like to mention is Mahavira.  He did a lot of work with number theory.  The concept of zero came from this part of the world. 

            One of the mathematicians from Egypt that I am going to have my students study is Hypatia.  She was born in Alexandria, Egypt in AD 360.  Her father Theon brought her up.  He wanted to have a boy so he raised Hypatia the same way he would have raised a boy.  “Theon was determined to produce a perfect human being”   (Osen, 23).  He had her in many different subjects at the University of Alexandria. Hypatia was into learning and exploring.  She learned to question everything that came her way.  She also played a lot of sports.  “Her father made sure she knew about the arts, the sciences, literature, philosophy, swimming, rowing, horseback riding, speech, and mountain climbing”  (Morrow, 94).              

            Soon she surpassed even her father.  It was once thought that he sent her to Athens to learn more in mathematics, but there was never any proof that she actually studied there.  What is known though is that Hypatia corresponded with many different mathematicians of her time.  She was well thought of by her peers.  She even was able to obtain a position at the University of Alexandria.  She was brought in to teach geometry and astronomy, but her love was for teaching mathematics, in particular algebra.  This was very different for the women during her time because women were their husbands’ property.  They had to obey everything that their husband told them to do.  

            Hypatia was different and she did not take a husband, but she did take the position at Alexandria.  It was there that she became a well-know mathematician.  Other mathematicians and students came from all over to study and attend lectures given by Hypatia.  Hypatia “wanted to help her students understand the mathematics she was teaching, so she wrote books that gave explanations that were easier to understand that the original books” (Morrow 95).  These were called commentaries.  Hypatia wrote one for a book that was written by Diophantus.   

            Hypatia was well known for her commentaries.  They were so well written, that most of her students were able to understand them.  That was what made Hypatia so different from other teachers during that time.  She wanted her students to learn and succeed with what they were doing.   She did not try to mess them up or mislead them.  Hypatia tried to help all that were willing to learn. 

              These commentaries were the first of a kind.  Never before had anyone tried to show students and other scholars how to do mathematical problems from examples.  That is what these commentaries were.  They would show the problem that other mathematicians were working on, and she would write a step-by-step guide on how to solve them. 

                Hypatia was so well-liked and well-thought of that she became a threat to the Romans who were annexing Alexandria.  When Cyril came to power, he started rumors that Hypatia was a witch and used black magic.  Cyril did so because he thought that Hypatia could stop the spread of Christianity.  The citizens of Alexandria finally believed in these lies and sent a mob out to find her.  When they did, they dragged her out of her chariot by her hair.   Then they dragged her behind a chariot through the streets of Alexandria, where she was finally killed in front of a Christian church. 

            The next mathematician is also from Alexandria.  Euclid was the first professor of mathematics at the University of Alexandria.  Euclid was born in about 330 BC. He died in 275 BC.  Proclus writes almost all that we know of Euclid.  

Euclid studied under the great Greek Philosopher Plato.  Euclid wrote the book called Elements that was used for over 2000 years in the mathematical community.  Euclid was a great teacher for others to follow his work for so long.  The book of Elements was based on the laws of geometry.  This book gave logical reasoning why different parts of geometry are true.  There has been a rumor that Elements was written by a group of mathematicians at the University of Alexandria, but this is no longer thought to be so. 

            The Egyptian ruler, Ptolemy, founded the University of Alexandria.   Ptolemy invited Euclid to teach at the university.  “He was a modest, gentle man, and a patient teacher who loved his work at the university.  He enjoyed his students, and they responded by studying diligently” (Reimer, 4).  Euclid noticed that his students did not understand how the concepts were related to each other.   So Euclid put together other peoples’ mathematical problems into a text.  They he gave step-by-step instructions, more like reasoning, on how to do the problems.  “Euclid was most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself”  (www-gap.dcs.st-and.ac.uk). There were many different people who thought that Euclid was a great teacher.  He and his work are well-documented. 

            Thales of Miletus “was the first known Greek philosopher and scientist.  He is the first person with whom the use of deductive methods in mathematics is associated”   (Reimer, 1). Thales was thought to measure the height of the pyramids.  Even though he was the first, Thales really was an engineer by profession.  Thales was born in and died in Asia Minor around 636 BC – 546 BC.   

            A later writer known as Proclus wrote this about Thales:  “Thales first went to Egypt and thence introduced this study [geometry] into Greece.  He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attacking problems has greater generality in some cases and was more in the nature of simple inspection and observation in other cases” (www-gap.dcs.st-and.ac.uk). 

            Pythagoras of Samos was born in Greece.  He was alive around 560 BC – 480 BC.  “Pythagoras was a Greek philosopher and religious leader, responsible for important developments in the history of mathematics, astronomy, and the theory of music.   Pythagoras is most famous for the theorem on right triangles that bears his name”  (Reimer, 9).            

            Pythagoras was well traveled.  He studied under Thales and Anaximander, who influenced him in mathematics.  Anaximander told Pythagoras that he should visit Egypt to see the pyramids.  Pythagoras decided to visit there.  While he was there Pythagoras went to the different religious temples.  It was at the last one that he was allowed to join after the rite.   This temple was Diospolis.  It was there that he learned what he later had his own religious group follow.  “(T)he secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt”  (www-gap.dcs.st-and.ac.uk). It was this religious group with whom Pythagoras did most of this work.  Even though none of his actual work is left, it was the group’s work that we most remember.  They believed in the principles of mathematics.  

            One of the few mathematicians that are not from ancient times but is very important to my students is Blaise Pascal.  He lived in France from 1623 – 1662.  Pascal “was a French thinker, mathematicians, and scientist.  He made many invaluable contributions to mathematics and physics, and is also remembered for his religious and philosophical writings” (Reimer, 53). 

            His father raised him; his mother died when he was three.  Pascal’s father had very different beliefs.  His father also home schooled him.  When Pascal’s father decided to teach him, he told Pascal that he would not teach him mathematics until he was 15.  It was because of this that Pascal started to look into mathematics.  At the age of 12 Pascal figured out that the angles of a triangle equals 2 right angles.  “At the age of sixteen, Pascal presented a single piece of paper to one of Mersenne’s (a religious group) meeting in June 1639.  It contained a number of projective geometry theorems, including Pascal’s mystic hexagon”   (www-gap.dcs.st-and.ac.uk).  Pascal is best known for his work on Pascal’s triangle. 

            For the civilizations that I could not find a representative mathematician, I will be having the students work on mathematics from that civilization.   Example, the Romans did not have any mathematician but they did have Roman Numerals that the students need to know.  Also I could not find any mathematicians or mathematics for the Persian Empire.  I will be supplementing it with a mathematics game for the day.  They will still have to work with the game to understand how it is played. 

Implementation 

Each lesson is based on a sixth grade classroom, with some LEP students and special education students.  The time limit for each lesson is one hour.  The extensions will be games that come from the civilizations that each mathematician comes from.  You may use the games as extensions or they can be used as a lesson on their own. 

Lesson:  1                                 Title of Lesson:   Babylonian Multiplying

Objective:  To introduce the civilization of Babylon to the students.  The students will be using Babylon wedges to decode tablets and use the wedges to multiply.   This lesson is based on Exploring Mathematics through History.

Standards and Benchmarks:  Standard 1:  Number Sense and Operations

Materials:  Worksheet 1

Anticipatory Set:  Introduce the civilization of Babylon.  Let the students know how old this civilization really is.  Babylonians use Cuneiform numerals for their numbers.  If any pictures are available use them to show the students what their writing was like.

Guided Practice:  Hand out Worksheet 1 to the students.  Let the students look over them in pairs or in groups.  Tell them to try to figure out what is written on the worksheet.  Hopefully they will start to decode them to know that they are multiplication tables.  If the students do not decode this on their own, give them suggestions to lead them to the correct answer.   Let the students try to figure out what each symbol means.  Once they have figured it out or you have helped them along, have the students write their age, date, birth date, or height in Babylonian numerals.  This is in Base 60.  Students will need to figure out the different base from their own.

         60                                             10                                     1

 They will have to learn to put their base 10 into base sixty.  For example, if someone is 72 years old, a student will use one 60, one 10, and two 1’s.

Independent Practice:  Have the students finish making the multiplication tables.  They can work on 4x with no problem.  For some more work with Babylonian numbers, the students can work with 7x, 10x, 12x, and for your advanced students they can work on 15x.

Extension:  Royal Game if Yr   (Games of the World)

Materials:  2 sets of 7 counters, red and white; 2 sets of 3 pyramid shaped dice, with 2 marked corners; Board that has 20 squares

Rules:  This is a race game.  Try to move all seven counters on and off the board along a route of 20 sq. as shown in the diagram.  Throw the dice, 3 marked tips is 5 points, 3 unmarked is 4, 2 unmarked is 0, and 1 marked is 1.  Throw the dice, move the counter onto the board with a 5 or 1 only.  A counter can be added to the game if a counter lands on a square with a dot.  Once on the board move the counters to the number on the dice.  There can be any number of counters on a square.  But after the counters are at the center or the top track they may be attacked.  If a person’s counter lands on a square that has another person’s counters then they can be taken off and these counters have to start all over again.  The squares with dots and the last square are safety squares where they cannot be attacked.  To go off the board, the dice must hit the number of the exit square.

Lesson:  2                                 Title of Lesson:   Hypatia Number Patterns

Objective:  To be introduced to a historical mathematician from the Ancient Egyptian Civilization and to work with some of the mathematics that she is known to have invented.  Also to have the students work with a mathematical game that first came about during the Egyptian civilization.  This lesson is based on Multicultural Science and Math Connections.

Standards and Benchmarks:  Standard 4:   Patterns, functions, and algebraic concepts.   4.1, 4.11

Materials:         12 games of Wari, overhead, counters/beans, and bulletin board paper, Worksheet 2

Anticipatory Set: In the beginning of class introduce Hypatia as a person.  Give a biography on how her life was.  If you have any pictures of her, go ahead and show them to the class.   The biography of Hypatia can be found in the background information.  More information and the picture can be found on the web at www-gap.dcs.st-and.ac.uk.

Guided Practice: Introduce the number patterns that Hypatia came up with.  She thought up the Triangular Numbers, Square Numbers, and more.  Pass out the counters/beans to the students.  Use the overhead or a bulletin board paper (with the beans taped on to the paper) to show the different patterns.  Place one counter/bean on your sheet of paper or overhead, and have the students do the same.  Keep adding beans to the original one to start with a triangle figure.  Talk to the students about the pattern that is happening.  Make sure that they understand what is going on.

Triangular Numbers

            Start     0           1          3          6          10        __        . . .

            Add     1           2          3          4          __        __

            Result   1           3          6          10        __        __

 Then do the same with the Square Numbers.  These numbers will all be consecutive odd numbers.

Square Numbers

            Start     0           1          4          9          16        25        . . .

            Add     1           3          5          7          9          11        . . .

            Result   1           4          __        __        __        __

 Soon the students will run out of beans/counters.  When they really understand it you can write a general formula to predict the result.  If the students are not ready for formulas, try using just a pattern that they understand.

Here is the pattern:

            The first sum is the square of 1.

            The second sum is 1 + 3, the square of 2.

            The third sum is 1 + 3 + 5, the square of 3.

            The fourth sum is . . .

Continue with the rest of the formula in the independent practice.

Independent Practice:  Have the students work on worksheet 1.  They will be continuing the tables from above.  They will also be working on the formula of the squares.  Also for your higher end students there will be time for them to work on the Pentagonal and Hexagonal numbers.  Also they will need to show the patterns.

Extension:  The students will be working with the game Wari.  This game can be purchased at any store.  The rules are with the game.  This game uses mathematical logical thinking.

Lesson:  3                                 Title of Lesson:   Euclid’s Primes

Objective:  To learn a little about Euclid and some of his work.  Also they will be learning a new game from Egypt.  This activity is based from Exploring Mathematics through History.

Standard and Benchmarks:  Standard 4:   Patterns, Functions, and algebraic concepts.   4.1, 4.1.1

Materials:         Bulletin board paper, worksheet 3

Anticipatory Set:   Introduce the biography on Euclid.  If you have any pictures of him or more information let the students know.  The more they know the more they will understand Euclid.  For some of this go to www-gap.dcs.st-and.ac.uk.

Guided Practice:          In this unit you will need to introduce composite and prime numbers.  If the students already know this, just have a review and the students can work independently.  Introduce composite numbers to the students.  Example, a composite number is a number that is a multiple of any other number (4 is a multiple of 2, so 4 is a composite).  Give a lot more examples for the students who do not understand.  Also you can point out that a composite number is a number that can be divided by another number.  Then talk about prime numbers.  These are numbers that have no multiples and no other number can be divided into them except for 1.  Use the bulletin board paper to write down the numbers to 25.   Start working through them, taking out the multiples of 2, 3, 5, and so on.  Make sure that the students understand the concept of primes.

            Primes               from 1 to 10                  2, 3, 5, 7

                                    from 11-20                    11, 13,  17,   19

                                    from 21-30                    23, 29

                                    from 31-40                    31, 37

                                    from 41-50                    41, 43, 47

                                    from 51-60                    53, 59

                                    from 61-70                    61, 67

                                    from 71-80                    71, 73, 79

                                    from 81-90                    83, 59

                                    from 91-100                  97

By 300 BC Euclid know that primes did not come to an end.  Encourage your students to think on if there is an end to the primes.

Independent Practice:  Have students work on worksheet 2 on finding all the primes from 1 to 100.   Make sure to walk around, making sure that the students understand what they need to do.  Allow you higher end students to look for the primes up to 200.

Extension:  Nine Men’s Morris  (Math around the World)

Materials:  2 sets of 9 counters, board game

Rules:  One player takes 9 counters and the other also 9 with a different color.  Take turns placing your pieces on the board onto any point that is not occupied.  If you place a row of three pieces of your color along any line, you have a mill.  Each time you form a mill, you can remove one of your opponent’s piece form the board, as long as it is not part of a mill.  Any piece in a mill is safe.  When all pieces are placed on the board, take turns moving your pieces one space along any line to an empty point.  You win when your opponent has only two pieces left or when they cannot be moved.

  

Lesson:  4                                 Title of Lesson:  Pythagoras Puzzle

Objective:  To have the students be introduced to a historical mathematician.  To be introduced to his main formula that is named after him.  To work with a mathematical game that comes from his country.  This lesson is based from A Peak into Math from the Past.

Standards and Benchmarks:  Standard 2:   Geometry, Spatial Sense, and Measurement, 2.3, 2.3.1,

Materials:  Worksheet 4a, 4b, and 4c, (for more accurate worksheets go to A Peak into Math from the Past page 118-119) scissors, glue stick

Anticipatory Set:   Introduce the mathematician Pythagoras.   Include that he was not the first person to come up this equation, but he was the person that it was named after. Guided Practice:  Hand out copies of Worksheet 4a and 4b.  Give students one triangle from worksheet 4a, and one section from worksheet b.  Let the students color worksheet 4a green, and 4b yellow.   Have the students compare if they are equal.   Now the students are going to prove that they are equal.  Have students glue their triangle on the middle of their paper.  Have the students cut out and glue the pieces of a, b, c, together from worksheet 4b.   They need to be put together in a square.   The students may help each other out with it.   As the students are gluing their squares to the triangle, make sure that the students understand that there is a length a, b, c (the sides).  Plus that the area of a square is a^2 and the same for squares b and c. 

Independent Practice:  Have your students summarize what the triangle and the squares are supposed to be.   Make sure that the students have answered the rest of the questions on worksheet 4c.

Extension:  Dog Eat Dog:  Polis Board Game (Ancient Greece)

Materials:  An 8x8 square, 2 sets of 8 counters

Rules:  This game is played in pairs.  Line up counters on each side of the board.  A player can move their dog any number of spaces in any one direction, except diagonally, and must stop when they are blocked by another dog.  To capture a dog, surround that dog with 2 counters.  To capture two or more a player must surround a line of the others counters with dogs at both ends.  The first player to take all of the others dogs wins, or by corning the other dogs so they cannot move.

Lesson:  5                                 Title of Lesson:   Thales Triangles

Objective:  To be introduced to a historical mathematician and some of his work, also to be introduced to a game from that civilization.  This lesson is based on A Peak into Math from the Past.

Standards and Benchmarks:  Standard 2:   Geometry, Spatial Sense, and Measurement

Materials:  Masking tape,

Anticipatory Set:   Introduce the mathematician.  Make sure to explain that Thales was the first person, after the builders were long dead, to find the height of the pyramids.  Thales was also the first person to come up with rules for geometry.   Give this mathematician life.  These students need to understand that they had more than mathematics, they had a life.

Guided Practice:   During Thales travel to Egypt, he was able to figure out the height of the pyramids.  Have the students act this out by getting into groups of three.  If you do not have enough room, have a couple of groups work on it with the class helping them out.  One student will be a column.  The student should stand straight up, standing on a piece of masking tape.  The other student will stand on the other side of the room.  This student will use his/her thumb and index finger holding them straight up and down in an L shape.  The student is to make his/her finger the same height as the student that is standing across the room.  Do this by sight.   Then the second student rotates the hand 90 degrees.  Make sure they do not change the distance between the fingers.  Lining up their thumb with the feet of the first student on the other side of the room.   Then a third student moves to be with the sighting finger.  The first student lies down, feet touching the third student stood and their head towards the tape.   The height of the student/column is the distance on the floor. 

Independent Practice:  Have the students write on a piece of paper explaining what they did to find out the height of the student/pyramid.  Have the students also explain what they learned from the activity.

Extension:  Number Bow (Ancient Egypt)

Materials:  Board game, colored pens, pair of dice.

Rules:  Two players each with a copy of the game.  Roll the dice and add the two dice together to find the total.  Find the Egyptian symbol for the number.  Color that part of the number bow.  First one to color the whole thing wins.

Lesson:  6                                 Title of Lesson:   Pascal’s Triangle

Objective:  To introduce the students with a well-known mathematician that is not from an ancient civilization, but is very important to middle school mathematics.  To have the students work with the triangle with hands-on and with actually doing the triangle.  Plus there will be a game that the students can then work with.

Standards and Benchmarks:  Standard 4:  Patterns, Functions, and Algebraic  Concepts, 4.1, 4.1.1

Materials:  Blue and mint colored spray can caps (or anything similar to it), Worksheet 5

Anticipatory Set:   Introduce Pascal to the students.  Include a picture and information that you can find.  Let the students know that Pascal was a leading mathematician in his time.  Really let your students get into the mathematicians.  They are going to have to research one themselves.

Guided Practice:   Split the class up into two groups.  They are going to work as a group to contract Pascal’s Triangle.  The rules are as follows:  1.  Mint is on the outside of the triangle.  2.  A mint and a mint make a blue.  3.  A blue and a mint make a mint.  Have the students construct a triangle using these rules.  They are trying to make the biggest triangle to win the game.   Ask the students if they started to notice the pattern.  The pattern is:   mint is odd, blue is even.

Independent Practice:  Have the students work on the Pascal’s triangle in worksheet 5.  The students are to work with the rules that are on the paper.  You may want to go back over the rules to make sure that the students understand.  Have them complete this to finish up the actual Pascal’s Triangle.

Extension:  Tower of Hanoi  (Math Around the World)

Materials:  A board with 3 vertical poles on it.  Can be purchased at Dale Seymour Publications 1-800-321-3106, Part #07090-08666.

Rules:  Transfer all the disks from one pole to another pole.  You may move only one disk at a time.  A large disk may not rest on top of a smaller one at any time.

Lesson:   7                                 Title of Lesson:   Roman Numbers

Objective:  To be introduced to the Roman civilization.  To learn about Roman numbers and understand them better.   Also to play a game with these Roman numerals and learn to add and subtract them.  This is lesson is based on Ancient Rome: A Comprehensive Resource for the Active Study of Ancient Rome.

Standards and Benchmarks:  Standard 1:   Number sense and Operations  1.5,  1.5.1  Standard 3:  Data Analysis, Statistics, and Probability  3.1, 3.1.1, 3.1.2, 3.2, 3.2.1

Materials:  Worksheet 6a and 6b

Anticipatory Set:   Introduce to the students the Roman Civilization.  You can get more information than I have provided on the web or from the social studies teacher.  Tell the students that the Romans did not believe in mathematics, they just used what they needed to, to build their buildings.  Also they had very little use for large numbers, and they particularly avoided the use of fractions.  Any middle school student’s dream!

Guided Practice:   Explain to the students what each of the Roman numerals stands for.  Have the students practice understanding what each letter means.  Example, put on the board the letter C.  Have the students write down what it means.  Then explain to the students what IV means with addition and subtraction.  IV  (5-1)=4  The students will need to work with a few of these to help them out with the activity. 

Roman Numerals:

            I  =  1                V  =  5              X =  10             L  =   50           C  =  100                    
            D  =   500        M  =  1000             V  =  5000       D  =  500,000

Independent Practice:  After you have explained all about the numerals, have the students work on the worksheet about the Roman numerals.

Extension:  The students will be playing a game based on Roman Numerals.   This game is to help the students understand and get used to adding the numerals together.  This game is to be with partners.  Rules:  Both players need to estimate what their score will be after 10 spins.  Place a ballpoint pen into the center of the wheel.  Player 1 spins the wheel and writes down the amount opposite the pen.  Then player 2 spins the wheel and records his or her own score.  Players then add up their score and the difference between their estimate to actual score.  Then they are to post it on the board.  After the whole class has finished, have a class discussion on it.  The students can graph the scores.  Have the students answer these questions.  Are there any small differences showing good estimates?  Did any get their estimate?  Could any one be really sure that their estimates would be right?  Explain?

Lesson:   8                                 Title of Lesson:   Persian Backgammon

Objective:  To learn about and play a game that was played in ancient Persia. 

Standards and Benchmarks:  Standard 3:   Data analysis, statistics, and Probability   3.1, 3.1.1, 3.1.2

Materials:  12 sets of the game Backgammon

Anticipatory Set:  Introduce the Persian Empire to the class.  They need to know about it since I did not find out many things from there.

Guided Practice:   Introduce the rules of the game to the students.   To start the game each player throws the die to see who will start with the highest number.  The player moves using the combined score of the two dice.   The black men move from their home (the black) into the outer board (the white).  The white men start in the white and go to the black.   Follow the rules of the game Backgammon.  There are many rules, so make sure that the rules have been looked over before you introduce the game.

Independent Practice:  Have the students pair up to play the game.  You may need more than one class period to do this.  Have the students make a little game out of it.  This will involve the students more in the game.

Extension:  Have the students find out more about the Persian Empire.  They can either research it or they may ask the librarian, search on the computer, or go to a public library to find out more about it.

Lesson:  9                                 Title of Lesson:   Suan Pan from China

Objective:  To have students learn more information about China from a mathematics point of view.   Also to have the students learn to count with a Suan Pan like the ancient Chinese did.  The students will learn how to play magic squares from China.  This lesson is based on Multicultural Science and Math Connections.

Standards and Benchmarks:  Standard 1:   Number sense and operations, 1.1, 1.1.2,

Materials:  Chinese suan pan  (also known as abacus, can be found in a Chinese store and restaurant.  The instructions are included with the suan pan.), red toothpicks, and black toothpicks

Anticipatory Set:   Introduce the Chinese culture to the students.  Make sure to include many facts on how different their culture is from ours.  This is very important for the lesson that the students will be doing.   There was no major mathematician that would be helpful for 6^th grade mathematics.

Guided Practice:   Using the red toothpicks show the students regular additions.  Show the same steps with the Suan Pan.  You are trying to show the students that there are different ways to do mathematics that will end up with same answer, and that people do not always do things the same way.  Then show the students addition and subtraction with the red and black toothpicks,  red to show the larger number and black to show how many you are taking away.  Also there is a game that you can use with this.  It is red vs. black toothpicks.  The rules are:  Red stands for positive, black for the negative numbers.  When a red and a black rod meet, they are both annihilated, they are taken away (they cross each other out).  The rods that are left are winners.  Show the students how to do a battle (–7 + 7 = 0).  Have the students answered these questions:  Can you predict which side will win?  When are there no winners?  To subtract, one rod of the opposite color is put in its place.  Example:  --3 = +3.  Then play the game as normal.

Independent Practice:  Have the students play some battles with each other.  You may want to think up some of the questions that the students will need to answer.

Extension:  Magic Squares  (Math Around the World)

Materials:  3x3 square, Tiles numbered one through nine.

Rules:  Students are to place the tiles in each cell of the square in such a way that the sum for every row, column, and diagonal will be the same.

Lesson:  10                               Title of Lesson:   India’s Right Triangle

Objective:  To learn more about India, because our students do not know much about this culture.  The students will learn to use Pythagora’s theorem also using the Indian way of doing it.  Students will learn to play the game of chess that originated in India.  This lesson is based on Multicultural Science and Math Connections.

Standards and Benchmarks:  Standard 2:   Geometry, Spatial Sense, and Measurement,   Standard 1:  Number sense and Operations, 1.1, 1.1.1

Materials:  Graph paper, ruler, calculator, and 12 chessboards

Anticipatory Set:   Go over the background information about India.  Students need to know how these people lived and that our numbers came from them.  Also again there is no mathematician that is useful in the sixth grade from India.

Guided Practice:   With the students go back over what an exponent means.  The students will have been introduced to this during order of operations.  3^2 means 3x3 not 3x2.  Once you are sure that the students understand exponents, have the students put together their triangles on the graph paper.  They will be putting together 12 right triangles.  One square on the graph paper will represent one Indian foot called pada.  All work will be done in pada. Draw on graph paper a vertical line with the points OP, which is 36 padas long.  On the line OP mark the points OQ=5, OR=28, OS=35.  Draw a perpendicular to OP at O.  On your graph paper this is the horizontal line that passes through O.  On this perpendicular, locate T and U each 12 padas from O.  Connect this line.  Draw a perpendicular to OP at P.  This is the horizontal  line through P.   On this perpendicular locate V and W each 15 padas from P.  Connect  these lines.  Carefully connect points for a triangle TQU, TPV, UPW, VRW, and VSW.  Draw diagonals TW and VU. 

Independent Practice:  Have the students use a calculator to find out the lengths of the side c in each triangle using the theorem.  Students are to finish worksheet 7.

Extension:  Students will be learning to play and playing the game of chess that originated in India.  Have students get comfortable with this game.  There is a lot of logical thinking involved with this game and it is very useful in mathematics.

Lesson: 11-?                             Title of Lesson:   Mathematicians

Objective:  To get to know and understand better one mathematician from history. 

Standards and Benchmarks:  Language Arts:   C.  Reading, 1., d., 1, 2, 3, 4, D.  Writing, 1.

Materials:  Use of computers, use of library materials, and use of any materials that the students can use for research

Anticipatory Set:   Reintroduce the different mathematicians from this unit.  Make sure that the students have a good idea of these different mathematicians.

Guided Practice:   Go over a different mathematician than was previously used.  Show the students what you are wanting with their research.  An example would be to show Einstein and give clues into his life and then show different mathematics that he did.  Give the students the different options that they have to do the research and the way that you will be grading it.

Independent Practice:  Students will be researching their different mathematicians.  I will not be allowing students in the same class to do the same mathematician.   This may take up to a week for the students to finish researching it.  The students will then need a few days to put together their work.

Extension:  The students then can present their research on their mathematician to the class.  This can be done several ways that I have presented above in the paper. 

Benchmarks and Standards 

APS Standards 

Standard 1:  Number Sense and Operations

            Learners will demonstrate number sense through experiences with meaningful mathematical problems while focusing on number meaning, number relationships, relative effect of operations, and multiple representations to communicate sound mathematical thinking.

1.1    Use order of operations and grouping symbols to simplify expressions. 

1.1.2        Understand that order of operations applies consistently across all math topics. 

1.5    Use estimation strategies in context.

1.5.1        Select and use the appropriate estimation strategies in a variety of situations.

 Standard 2:  Geometry, Spatial Sense, and Measurement

            Learners will demonstrate an understanding of concepts, properties, and relationships of geometry and measurement through experiences with meaningful mathematical problems, while focusing on identifying, describing, classifying, visualizing, comparing, estimating, and measuring various aspects of shapes and objects.

2.3       Investigate the congruence between, and symmetry of, 2-dimensional figures.
2.3.1    Create and explain congruent 2-dimensional figures using mathematical terminology.

Standard 3:  Data Analysis, Statistics, and Probability

            Learners will identify patterns and special features of data and events of chance through experiences with meaningful mathematical problems while focusing on comparing, predicting, representing data, and making decisions to communicate mathematical understanding.

3.1    Consider and design a process to investigate a statistical question.

3.1.1        Collect data using a variety of appropriate data collection instruments.

3.1.2        Organize data using appropriate tools.

3.2    Analyze data and present conclusions.

Standard 4:  Patterns, Functions, and algebraic concepts

            Learners will demonstrate an understanding of algebraic concepts through experiences with meaningful mathematical problems while focusing on discovering, describing, modeling, and generalizing patterns and functions, representing and analyzing relationships, and finding and supporting solutions.

4.1               Analyze, create, and generalize numeric and visual patterns paying particular attention to patterns that have a recursive nature.

4.1.1        Investigate and predict sequences and patterns involving varying rates of change.

Language Arts

C.                  Reading

1.       Learners read and gather information from a variety of printed material, literature and own written language.

d.       Research

1)        Use appropriate sections of text structure.

2)        Gather and apply information from a variety of resources.   

3)        Use library resources to locate, research, and collect information on a topic or theme.

4)        Use technology to access and apply information.

D.                  Writing

1.       Learners write to convey information and to express individual ideas and understandings.   

Documentation 

Boyer, Carl B.  A History of Mathematics.  Princeton, New Jersey:  Princeton University Press,              1968.

Eagle, Ruth M.  Exploring Mathematics Through History.   New York, NY:  Cambridge
            University Press, 1995.                

Freebury, H. A.  A History of Mathematics.  New York, NY:  The Macmillan Company, 1961.

GEMS, LHS.  Math Around the World.  University of California at Berkeley, California:  LHS                GEMS, 1995.

Grunfeld, FredericV.  Games of the World: How to Make Them How to Play Them How They                  Came to Be.  New York, NY:   Plenary Publications International, Inc., 1975.

Hamilton, Robyn.  Ancient Egypt Activity Book.  Dana Point, CA:   Edupress, Inc, 1994.

Krause, Marina C.  Multicultural Mathematics Materials 2^nd Edition.  Reston, Virginia:  The                  National Council of Teachers of Mathematics, Inc., 2000.

Lumpkin, Beatrice and D. Strong.  Multicultural Science and Math Connections  Middle School                  Projects and Activities.  Portland, Maine:  J. Weston Walch, 1995.

Moore, George.  Ancient Rome  A Comprehensive Resource for the Active Study of Ancient                  Rome.  Rowley, MA:  World Teachers Press, 2001.

Morrow, Charlene and T. Perl.  Notable Women in Mathematics A Biographical Dictionary.                  Westport, Connecticut:  Greenwood Press, 1998.

O’Connor, J. J. and E. F. Robertson.  Articles on Mathematicians.  January 1999.   School of                    Mathematics and Statistics University of St. Andrews, Scotland. 
                 <http://www-gap.dcs.st-and.ac.uk/>

Osen, Lynn M.  Women in Mathematics.  Cambridge, Massachusetts:  The Massachusetts Institute                    of Technology, 1974.

Price, Sean Stewart.  Ancient Greece  A complete resource filled with background information,                    cross-curricular activities and games, library and internet links, art projects & a                    play.  New York, NY:   Scholastic, 2000.

Reimer, Luetta and Reimer W.  Mathematicians are People, Too.  Stories form the Lives of                  Great Mathematicians.  Parsippany, New Jersey:  Dale Seymour Publications, 1990.

Reimer, Luetta and Reimer W.  Mathematicians are People, Too.  Stories from the Lives of                   Great Mathematicians, Volume Two.  Palo Alto, CA.:  Dale Seymour Publications,                   1995.

Voolich, Erica Dakin.  A Peek Into Math of the Past.  Parsippany, NJ:   Dale Seymour
                Publications, 2001.

List of Mathematicians 

Neils Henrik Abel                                              Heron
Maria Agnesi                                                     Caroline Lucretia Herschel
Aryabhata al-Khwarizmi                                    Grace Murray Hopper
Archimedes                                                       Hypatia
Charles Babbage                                               Omar Khayyam
Benjamin Banneker                                            Sonya Kovalevskaya
Lenore Blum                                                      Joseph Louis Lagrange
Mary Everest Boole                                           Ada Byron Lovelace
Girolamo Cardano                                             Fanya Montalvo
Emilie du Chatelet                                              Emmy Noether
Rene Descartes                                                 Isaac Newton  
Charles Lutwidge Dodgson                                Edna Lee Paisano
Albert Einstein                                                   Blaise Pascal
Euclid                                                                George Polya
Leonhard Euler                                                  Pythagoras
Eratostenes                                                        Srinivasa Ramanujan
Pierre de Fermat                                                Julia Bowman Robinson
Fibonacci (Leonardo of Pisa)                             Mary Ellen Rudin
Galileo Galilei                                                    Mary Fairfax Somerville
Evariste Galois                                                   Thales
Carl Frederich Gauss                                         John Venn
Sophie Germain                                                 John von Neumann
Evelyn Boyd Granville                                        Norbert Wiener
Mary Gray

List of Female Mathematicians

Agnesi, Maria                                                      Hay, Louise Szmir                    
Neumann, Hanna                                                 Antonelli, Kathleen M   Hayes, Ellen               Nicolson, Phyllis                                                   Bari, Nina                                 
Hazlett, Olive                                                       Nightingale, Florence
Baxter, Agnes                                                      Herschel, Caroline                    
Noether, Emmy                                                    Blum, Lenore                            
Hopper, Grace                                                     Oleinik, Olga
Browne, Marjorie                                                 Hypatia of Alexandria               
Perer, Rozsa                                                         Cannell, Doris                           
Janovskaja, Sofja                                                  Pless, Vera
Cartwright, Mary                                                   Karp, Carol                              
Rasiowa, Helena                                                    Chang, Sun-Yung Alice            
Keen, Linda Goldway                                            Rees, Mina   
Chatelet                                                                 Kochina, Pelageia                     
Robinson, Julia                                                       Chisholm Young, Grace            
Kovalevskaya, Sofia                                               Rudin, Mary Ellen
Chung, Fan Graham                                                Kramer, Edna                           
Schafer, Alice T                                                      Cox, Gertrude                          
Krieger, Cecilia                                                       Scott, Charlotte
Daubechies, Ingrid                                                   Kuperberg, Krystyna                
Somerville, Mary                                                     Falconer, Etta                           
Ladd-Franklin, Christine                                          Stott, Alicia Boole
Fasenmyer, Mary                                                    Lovelace, Augusta Ada Swain, LornaFlugge-Lotz                                                             Macintyre, Sheila Scott Taussky-Todd, Olga
Freitag, Herta                                                          Malone-Mayes, Vivienne          
Uhlenbeck, Karen                                                   Geiringer, Hilda von Mises       
McDuff, Dusa                                                         Velez-Rodrigues, Argelia
Gentry, Ruth                                                            Merrill, Winifred                       
Wheeler, Anna J Pell                                               Germain, Sophine                     
Morawetz, Cathleen                                                Young, Lai-Sang
Granville, Evelyn                                                      Moufang, Ruth
Hamill, Christine                                                       Nelson, Evelyn Roden

 

 
Worksheet 2:   Hypatia’s Number Patterns

Complete this table of triangular numbers.

Start                  0          1          3          6          __        __        __        __        __

Add                  1          2          3          4          5          6          7          8          __

Result                1          3          6          10        __        __        __        __        __

Complete this table of square numbers. 

Start                  0          1          4          9          16        25        __ 

Add                  1          3          5          7          9          11        __ 

Result                1          4          __        __        __        __        __ 

Term                 1^st        2^nd        3^rd        4^th        5^th        6^th        7^th

 Look for the pattern in the row of results in the square numbers.  Use this pattern to find the 12^th.

 

 

 

Explain the pattern in a couple of sentences.  Use examples if you need to.

 

 

Worksheet 3:  Euclid’s Primes 

Find all the primes from 1 to 100. 

1          2          3          4          5          6          7          8          9          10        11        12

 

13        14        15        16        17        18        19        20        21        22        23        24

 

25        26        27        28        29        30        31        32        33        34        35        36

 

37        38        39        40        41        42        43        44        45        46        47        48

 

49        50        51        52        53        54        55        56        57        58        59        60

 

61        62        62        63        64        65        66        67        68        69        70        71

 

72        73        74        75        76        77        78        79        80        81        82        83

 

84        85        86        87        88        89        90        91        92        93        94        95

 

96        97        98        99        100

Find the primes up to 200!

From what you have done with primes, do you think that there is a prime that is the last prime number?  Yes or No.  Explain your reasoning.

 

 

Worksheet 4a:  Pythagorean Triangles  

	c		c
a		a
	b		b

 

 

 

Worksheet 4b:  Pythagorean Squares

                                                   b                                                  b 
                                                    c                                                   c

          Worksheet 4c:  Pythagorean Theorm

You will be given a triangle with the sides a, b, c.  Label these the way that your teacher shows you.
You will then be given a grid with squares and triangles in it.  Glue your triangle to the middle of your
paper.

Look at the areas of the rectangles with a pieces, then look at the areas of the rectangles with b and c.  Do they have equal areas?_______________________________________

Cut out the triangle.  Cut out the a square, then the b pieces, and then the c pieces.  Put all the pieces togother to make an a square, b square, and c square.

Using the three squares, tell about the relationship that you found with the equal areas.

 

Bring back out the triangle.  Glue the a square onto the side of the a triangle.  Glue the b square onto the b side of the triangle, and do the same with c.

What is the area of the squares?

(a) square_____________       b square______________       c square______________

In your own words tell what the triangle and the squares are showing.

 

Worksheet 5:  Pascal’s Triangle

The number 1 is repeated along the two sides of the triangle.  All the other numbers equal the sum of the two numbers diagonally above it.

Activity

Complete the entries in the triangle below.  Finish the next 8 lines that would be in the triangle.

				1
				1	1
			1	2		1
			1	3	3	1
		1	4	6	4	1

Worksheet 6a:  Roman Numerals 

Use addition and subtraction of the numbers to solve the problems. 

Example:  addition:  VII = 5+1+1=   7  subtraction:  VI = 5-1= 4 

Complete the following Roman numerals.  Put them into our numbers.

1.      XI  = _______________________  =  _______________  (__________)

2.      CL = _______________________  =  _______________   (__________)

3.      XL  =  ____________________  =  _______________  (__________)

4.      CDM  =  ____________________  =  ________________  (_________)

5.      MMDLI  =  __________________  =  _______________  (__________)

6.      XXLLC  =  __________________  =  ________________  (_________)

7.      MDCLXVII  =  _______________  =  ________________  (_________)

8.      LXX  =  _____________________  =  ________________  (_________)

9.      CVII  =  _____________________  =  ________________  (_________)

10.  DM  =  ______________________  =  ________________  (_________)

Write the meaning for the following Roman Numerals.   Remember that a line over any number multiplies it by 5000!

11.  I   =  ________________________          12.  V  =  ______________________

13.  X   =  _______________________          14.  L  =  ______________________

15.  C   =  _______________________          16.  D  =  ______________________

17.  M =   _______________________           18.  CL  =  _____________________

19.  DX   =  ______________________         20.  LV  =  _____________________

 Worksheet 7:  Indian Triangles 

 

 

 

 

 

Use a calculator to fill in the chart.  While fill in the chart remember to use the theorem  a^2 + b^2 = c^2 to find the missing side c.  Make sure to use a ruler made from your graph paper to check these numbers to make sure that they are correct.  Each box stands for 1 panas.  If you are correct all, but two sides, will have whole numbers.

                                    A         b          a^2      b^2      a^2 + b^2 = c^2          c, whole #?

Triangle TOQ                12        5          144      25        169                               13 yes

Triangle UOQ               12        5          ____    ____     ____                             _____

Triangle TOX                12        16

Triangle UOX                12        16

Triangle TOP                 12        36

Triangle UOP                12        36

Triangle TSO                 35        12

Triangle USO                35        12

Triangle VPX                15        20

Triangle WPX               15        20

Triangle VPR                 15        8

Triangle a                      15        8