Objectives

Students will look into notations used in number theory during this lesson. Students will:  - trace the origins and history of numbers and symbols. - examine specific types of numbers and historical documents.

Core Questions

- What notations are widely accepted in mathematics? How do notations contribute to mathematical understanding? - How does the process of solving problems involve the usage of mathematical notations?

Vocabulary

- Cardinal Number: A number that represents “how many” by counting. - Converge: To approach one specific number. - Diverge: Not converging; for a series, one that has no bounded sum. - Divisibility: The characteristic of a quantity that it can be divided evenly by general or specific divisors; the property of an integer dividing another integer with no remainder. For example, a number is divisible by 3 or 9 when and only when the sum of its digits is divisible by 3 or 9. - Hexagonal Number: One that represents the number of points in a hexagon with n regularly spaced points on one side. The first few hexagonal numbers are 1, 6, 15, 28, 45,…. The formula for the nth hexagonal number is \(h_n\) = 2\(n^2\) – n. - Limit: The value that a function or sequence approaches as the input or index approaches some value. - Number Base: The number of units in a given digit’s place or decimal place, which must be taken to denote 1 in the next higher place. For example, if the base is ten, ten units in the units place are denoted by 1 in the next higher place. - Ordinal Number: A number that represents the position of the number, relative to other numbers, i.e., first, second, third, etc. - Pentagonal Number: The number of distinct points in a pattern of points in the sides of regular pentagons whose sides contain 1 to n points, overlaid so they share one vertex. The first few pentagonal numbers are 1, 5, 12, 22, 35,…. The formula for the nth pentagonal number for n ³ 1 is \(p_n\) = \(3{n^2}-n \over 2\). - Perfect Number: A number that matches the sum of its divisors. - Pi: A number that represents the ratio of a circle’s circumference to its diameter, typically accepted as approximately 3.14. - Prime Number: A number that is only divisible by 1 and itself; a positive integer greater than 1 whose only integral factors are 1 and itself. - Square Number: An integer that is the square of an integer; For example, 121 is a square number since 11² = 121. A number is a square number only if one can arrange m points uniformly in a square. - Triangular Number: The number of points in the sides of an equilateral triangle uniformly filled with points. The nth triangular number is the number of points in an equilateral triangle with n points on a side. \(T_n\) = \(n(n+1) \over 2\).

Materials

- copies of the Symbols Organizer handout (M-A1-2-2_Symbols Organizer) - To view pi to the first 1,000 decimal places, <a href="http://www.factmonster.com/ipka/A0876705.html">http://www.factmonster.com/ipka/A0876705.html</a>  - Dover, 1990. A History of Pi by Petr Beckmann. . - NCTM’s pi applet available at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=161">http://illuminations.nctm.org/ActivityDetail.aspx?ID=161</a>

Assignment

- Observe/evaluate class discussions. Question students individually and collectively, both orally and in writing. Direct individual students to offer solutions in front of the class and ask them to self-critique their presentations.  - Evaluate students' performance on: + creating the Eratosthenes Sieve. + demonstrating alternate methods/procedures for determining primes.  + using a PowerPoint presentation, illustrate, describe, and portray pi.  + discovering two perfect numbers ranging from 0 to 100.  + determine the convergence or divergence of a function.  + discovery of corresponding fractions to infinite decimals.  + create a short PowerPoint presentation on the history of counting.  - Examine the textual answers for converting repeated decimals to fractions. For these exercises, reducing the solution to the fewest terms is less critical than determining the suitable place value to begin the computation.

Supports

Scaffolding, Active Engagement, Modeling, Explicit Instruction W: This lesson explores the history and theology of numbers. Activities encourage students to conduct studies on special numbers, investigate number representations and operations as they have evolved, and make deductions/discoveries about number notations.  H: Asking students to locate the first 100 prime numbers is a very difficult assignment. Students may or may not be familiar with Eratosthenes' sieve. The activity of creating the Sieve is sure to pique students' interest.  E: The lesson is separated into two parts. Part 1 delves into the study of numbers, while Part 2 examines the evolution of symbols and representations.  R: Our open-ended and exploratory exercises provide many chances for students to reflect, return, revise, and rethink. This lesson is mainly based on group work. Thus, group discussion and debate encourage introspection and revision.  E: The lesson promotes group reflection and self-evaluation. Students must analyze what they currently know and comprehend before moving on to the next level of the debate. The approach necessitates the use of precise questions that direct students to seek specific results.  T: Group work supports all learners. The variety of representations and methods of number investigations provides success for learners with different learning styles.  O: The lesson is quite abstract in nature. Students progress towards deductions and discoveries.

Procedures

Part 1: Study of Numbers Prime Numbers "What is a prime number? How may a prime number be expressed? What does a prime number look like?" Engage students in an open discussion. Ask students to come to the front and share their prime number representations. "A prime number is a number that is only divisible by 1 and itself." Note: The number 1 is not prime or composite. Ask students to explain what they mean when they say that a number is divisible in general, and ask them to provide examples of divisibility. Ask students to search for the first 100 prime numbers. They are free to use any approach they like. When frustration sets in, start the first activity mentioned below, which will shorten the search process and provide a structured technique for finding the first 100 primes. Activity 1 Students work in groups of three or four to investigate the method of determining prime numbers using the Eratosthenes Sieve. They should be prepared to present an example of the procedure, outlining each step while removing composite numbers. Students should be instructed to cross out any number between 1 and 50 that may be divided equally by any other number than 1. After the activity, distribute the NLVM Sieve of Eratosthenes applet (<a href="http://nlvm.usu.edu/en/nav/frames_asid_158_g_2_t_1.html">http://nlvm.usu.edu/en/nav/frames_asid_158_g_2_t_1.html</a>) with students. Maybe they found this tool already while searching for something else. (The pronunciation is air-a-TOS-then-eez.)<figure class="image"><img style="aspect-ratio:521/316;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_37.png" width="521" height="316"></figure>Activity 2 Have students present any methods/procedures for determining primes that they thought or discovered during their investigation. "Is there a pattern in the prime numbers? If there is no pattern, how would you approach the problem of detecting whether a number is prime? How did other mathematicians do this?" After students have had the opportunity to investigate and examine whether there is a precise method for discovering prime numbers, reconvene and explain that there is no equation for determining all prime numbers. Several mathematicians have come incredibly close. However, no single equation has captured all prime numbers. "Fermat famously developed the function \(2^{2^n}\)+ 1 with n ≥ 1. However, the function only elicits prime numbers for the first 4 natural integers (1, 2, 3, and 4). The output for the fifth natural number, 5, or f(5), is 4,294,967,297, which is not prime!" Invite students to learn more about Fermat's theorem and other mathematicians' theorems. "What is the largest prime number found thus far?" Invite students to decide whether the set of prime numbers is finite or infinite. After students have thought the question, give them the following resource, which includes Euclid’s proof of infinite primes at <a href="http://mathforum.org/isaac/problems/prime1.html">http://mathforum.org/isaac/problems/prime1.html</a>.  pi Many methods have been used over the years to find the most precise number for pi. This section of the lecture aims to encourage students to investigate and research these processes. One of the characteristics of pi that makes it unique and interesting to mathematicians is that it cannot be the root of any algebraic equation. Begin the topic with an open dialogue about pi. Ask students, "What is pi?" Allow students time to think. Many will give a decimal representation. "What exactly does pi represent? Where did pi come from? Why did we need Pi? How do we use pi in everyday life?" "Pi to the first 10 decimal places is 3.1415926535." "Can you go further than that? Check out <a href="http://www.factmonster.com/ipka/A0876705.html">http://www.factmonster.com/ipka/A0876705.html</a> to see pi to the first 1,000 decimal digits!" A History of Pi by Petr Beckmann includes a table showing pi to the first 10,000 decimal places! It also provides a chronological chart that illustrates the history of pi-related findings. "Both the Babylonians and the Egyptians discovered Pi in about 2,000 BC. The Babylonians obtained π = \(3 {1 \over 8} \), while the Egyptians obtained π = \(4 ({8 \over 9})^2\). The methods used to arrive at these calculations are mere speculation. They could have simply measured a circle's circumference, diameter, and ratio. The problem was that they didn't have any exact or calibrated measuring instruments. They used ropes and stakes in sand to get very close estimations, with an initial approximation of π = 3. Petr Beckmann's book, A History of Pi, is an excellent resource on the subject." "The Chinese applied a different approximation method. Around 264 AD, the Chinese employed inscribed polygons within a circle to calculate a pi-related inequality. For example, using an inscribed polygon with 192 sides, Liu Hui discovered: 3.141024 < π < 3.14270 Using an inscribed polygon of 3,072 sides, he discovered  π = 3.14159 The fraction \(22 \over 7\) is commonly employed for calculating pi with approximations since its value, 3.\(\overline{142857}\), is fairly near pi." "When more precise fractional calculations are necessary, \(355 \over 113\) may be employed due to its value of 3.141592…, which is even closer to pi." Finish with a measurement activity that includes items such as a basketball, soccer ball, lids, bottle caps, and more. Using common measurement equipment like rulers and measuring tapes, students can determine how close they can get to an accurate value of pi by dividing the circumference by the diameter. Use π = c ÷ d. Students can utilize NCTM's pi applet, available at <a href="http://illuminations.nctm.org/ActivityDetail.aspx?ID=161">http://illuminations.nctm.org/ActivityDetail.aspx?ID=161</a>, to examine two different approaches for computing pi. Students will see that as the number of sides of the polygon increases, the area and perimeter get closer and closer to the true estimate for pi. Activity 3 The goal is for students to build a table or other graphic organizer that compares different representations of pi. Students will be responsible for determining which representation is most useful. Tell students that: "You must trace the progression of the discovery of pi, beginning with the earliest records and ending with any current discoveries. Please include pictures, numerical representations, descriptions of procedures used, and any other relevant material. Compare the many ways that pi is represented or written using at least one graphic organizer, such as a chart. Create a PowerPoint presentation to use as a teaching aid for the topic of pi. Choose the method that makes the most sense to you and provide supporting explanations. Include at least one slide that discusses Pi's real-world applications. Feel free to consult A History of Pi or any other reliable sources." Perfect Numbers "What is the perfect number? Have you heard of this term before? Where did you hear this? In what context?" "A perfect number is one that matches the sum of its divisors. Can you think of any numbers that meet this criteria? Note that the addens of the sum will not include the number itself." Activity 4 "In groups of three or four, identify at least two perfect numbers ranging from 1 to 100. Be ready to share your process." "Euclid's Elements contains an entire chapter devoted to perfect numbers. Euclid taught how to build a perfect number from a specific type of prime number, as well as how the formula produced the first four perfect numbers. \(2^{p-1}\)(\(2^p\) - 1).  Use your calculator to replace the first four prime numbers (p), then refer to this table to see how you did." (Answers/table:)<figure class="image"><img style="aspect-ratio:179/97;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_46.png" width="179" height="97"></figure> After students have done the activity and (hopefully) determined that 6 and 28 are perfect numbers, make the table below. Give the students the factors and confirm that their sum equals the number. The illustration's purpose is to help students gain a better conceptual understanding of the issue. “The first four perfect numbers are 6, 28, 496, and 8128.”<figure class="image"><img style="aspect-ratio:602/417;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_38.png" width="602" height="417"></figure>"Are there any other perfect number patterns that you've noticed?" Students may have noticed that the sum of the reciprocals of the divisors of a perfect number equals 2. Use the perfect number 6 to illustrate this example: The divisors of 6 are 1, 2, 3, and 6. The reciprocals are 1/1, 1/2, 1/3, and 1/6. These fractions add up to 2.  "Try it with the next perfect number: 28. Does it work? Why, or why not?" Zero Explain to students, "We must pay tribute to another special number, zero, perhaps arguably the most important special number." "Why was zero invented? Can you imagine what the need was? Does anyone know for how many years the number zero has evolved? Who invented zero?" "Zero was created to represent 'no things,' as well as to serve as a placeholder for 'no things.' This number originated between 3000 BC and AD 1000. The Indians were the first to acknowledge 0 as a number. As early as the \(4^{th}\) century BCE, the Mayan civilizations of Mexico and Central America advanced the use of zero in their arithmetic and calendar. Neither the Greeks nor the Hindus desired to see a vacant representation as a number. Early Greek mathematicians and philosophers expressed their uncertainty regarding the concept of zero as a number in their writings. They repeatedly wondered, 'How can nothing be something?'" Slice an apple in half, then into quarters, eighths, and so on. Explain how an infinite number of cuts could cause the apple's portion to become indefinitely small. The Idea of Convergence "When we think of the concept of 'convergence,' we often correctly think of approaching a specific number as n (or the cases) increases. For example, consider our prior investigation into the universe of pi. It was discovered that as the sides of the polygon increased, the value of pi became closer and closer to the real approximation of 3.14. We can state that the value ‘converged’ towards 3.14 as n (the polygon's number of sides) increased." "Convergence is diametrically opposed to divergence, in which a number either expands and grows without bound, or declines without bound. You may suppose that many real-world examples of data in spreadsheets show a certain convergence." Activity 5 Determine and demonstrate a convergence of some sort. Highlight the formula used. Ask the students, "Is there a general form that is needed to illustrate a converging sequence?" Does y = \(1 \over 8\)x converge? Why or why not? What about a series of numbers? Can you think of a sum that converges?” Infinite Decimal Fractions Ask students, "Have you considered repeating decimals as an infinite convergence? Did you know that repeated decimals can be written as fractions that yield the precise decimal value?" "Let's take the decimal as an example. 4747474747… What fraction corresponds to this decimal? How can we determine this? What method can be used?" Activity 6 Divide the students into groups of three or four. Allow them to generate suggestions for determining the fraction comparable to the decimal. 4747474747…. They will likely get extremely close and continue to enter numbers. Ask students to share their results. Review the following structured way for determining the precise fraction following the presentations and discussions. "We already knew that .4747474747… This can be expressed as 47/100 + 47/10,000 + 47/1,000,000. The ratio is 1/100. So, we can write:<figure class="image"><img style="aspect-ratio:146/140;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_39.png" width="146" height="140"></figure>Notice that if we write the decimal in greater detail, its value converges to \(47 \over 99\). This is another illustration of the concept of convergence." "Repeating decimals are commonly represented with a bar over the repeating section of the decimal. For example, 1.\(\overline{567}\) means 1.567567567…, 1.5\(\overline{67}\) means 1.5676767…, and 1.56\(\overline{7}\) means 1.56777…. It is vital to notice that the repeat symbol only applies to the decimal's repeating section."  "Suppose we want to convert 1.\(\overline{567}\) to an equivalent fraction. To handle the decimal component, we can easily build up the following problem:   <figure class="image"><img style="aspect-ratio:175/131;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_40.png" width="175" height="131"></figure>Remember that the decimal had a 1 before it. Consequently, the equivalent fraction is represented as \(1 {567 \over 999} \). We can reduce this fraction to \(1 {21 \over 37} \)." "There is another solution to this problem, however. The repeating decimal is approaching its limit. Consider the decimal as an endless geometric series. If we approached the conversion this way, we would write the sum of the repeated portion as:  . 567 + . 000567 + …. Notice that the ratio is \(1 \over 1000\), or .001. The repeating portion, or \(a_1\) is .567.” "Now, applying the sum of an infinite geometric series formula, we have: S = \(a \over 1 - r\) Substituting our values, we find:  S = \(.567 \over 1 - .001\) = \(.567 \over .999\) To eliminate the decimals, multiply the numerator and denominator by 1000. = \(567 \over 999\) To get to the lowest terms, we can add the 1 back in front of the fraction \(1 {567 \over 999} \). Once more, we discover that \(1 {21 \over 37} \) is the equal fraction to the repeating decimal, 1.\(\overline{567}\). "Now assume we want to do something more fascinating and hard. We wish to convert 1.5\(\overline{67}\) into an equivalent fraction. This time, just the numbers 6 and 7 repeat! To solve this problem, we will need to use two equations. First, we'd like to move the repeating portion to the left of the decimal point. We can do this by moving the decimal point three positions to the right. Moving the decimal three places to the right necessitates multiplication by 1000. Therefore, we write: 1000a = 1567. 676767 We must now consider that just the numbers 6 and 7 repeat; therefore, we only need to move the decimal one position to the right. Moving the decimal one point to the right requires a multiplication by 10. So, we are now writing: 10a = 15. 676767 Now we are in a well-known scenario where we just need to subtract our two equations. This gives:<figure class="image"><img style="aspect-ratio:178/132;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_41.png" width="178" height="132"></figure>"As a result, 1.5\(\overline{67}\) expressed as a fraction is \(1552 \over 990\). This fraction can be reduced to its lowest terms." "What if we wanted to convert to an equivalent fraction? Take note that just seven repetitions occur. What would we do?" Divide the students into groups of three or four. Ask them to find the equivalent fraction. (The steps are outlined below.)<figure class="image"><img style="aspect-ratio:160/134;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_42.png" width="160" height="134"></figure>"The formula for this is \(1 {511 \over 900} \)." "Finally, let's take a look at one where only one digit is repeated. In this case, we do not have a whole number piece. Let's discover the fraction equivalent of .89\(\overline{5}\)." "We'll begin by relocating the repeated part to the left of the decimal point. We shall consequently relocate the decimal point three places to the right. Moving the decimal three places to the right requires multiplying by 1000. Therefore, we write: 1000a = 895. 5555 Because there are only five repetitions, we need to move the decimal two spaces to the right. Moving the decimal two places to the right requires multiplying by 100. So, we are now writing: 100a = 89. 5555  We now just subtract the two equations. Doing so gives:<figure class="image"><img style="aspect-ratio:171/117;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_43.png" width="171" height="117"></figure>This corresponding fraction can, of course, be reduced to its lowest form." Gauss "We frequently do the following with mathematics: 'think harder, not smarter.' " We should 'think smarter, not harder.' The well-known story of Gauss solving the following problem when he was a young boy serves as an illustration of this adage." "Gauss's teacher instructed students to calculate the sum of the first 100 numbers, which included 1 and 100. While other students laboriously calculated the sum, Gauss created an organized list of two columns of integers. He wrote as follows:<figure class="image"><img style="aspect-ratio:145/449;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_44.png" width="145" height="449"></figure>He saw that there were 50 pairs, with each pair totaling 101. Thus, the total of the first 100 numbers was 50(101) = 5,050. How inventive is that?" Part 2: Evolution of Symbols and Representations Number "What's a number? How do you define a number? Who was the first person to think of numbers? How do the ideas of numbers and counting differ? Do they? Is knowing numbers an innate or learned ability? These are all crucial questions to consider. Let's start with some basic knowledge and concentrate on the concept of the 'representation of numbers.'" Cardinal Numbers "An abacus, numerals, hieroglyphics, figures, notches, scratches, marks, finger symbols, and modern number symbols have all been used to represent numbers throughout history and beyond." "What is a cardinal number? A cardinal number is a number that indicates 'how many.' Mathematicians have analyzed and argued the subtle discrepancies between cardinal number representations and counting for centuries, including today." "Numbers are represented in a variety of bases, including base ten, five, and two. The decimal system uses base ten. The other two bases continue to play essential roles, particularly base two, where the binary system is used for various codes, computer equipment, and so on. In base two, the place values are \(2^1\), \(2^2\), \(2^3\), \(2^4\), ..., \(2^n\). The well-known character 5 represents the number five in base ten. In base 2, five equals 110 because the first place indicates how many \(2^2\)s, the second place indicates how many \(2^1\)s, and the third place indicates how many \(2^0\)s. Counting from left to right, that is one \(2^2\), one \(2^1\), and zero \(2^0\)s; 4 + 1 + 0 = 5." Ordinal Numbers "What is an ordinal number? Ordinal numbers represent a number's position with other numbers. For example, 1 comes before 2, 2 comes before 3, and so on." "Consider the following question: Was the discovery of cardinal numbers prior to the discovery of ordinal numbers?" Lead a class discussion about this question, presenting supporting ideas and rationales for each possible solution. For example, one of the challenges in answering the topic is determining how far back in human development one must explore for evidence. It is possible that less developed tribes in the past had to measure their basic necessities, such as animals, logs, tools, fruits, and vegetables. Similarly, order and priority are believed to have evolved from fundamental concepts such as first child, second child, full moon, new moon, or choosing which fruit to harvest first. Activity 7 Tell students: "Your task is to create a short PowerPoint presentation outlining the history of counting. Answer questions like, 'When and how did counting originate?' "What was your exact need?' Include presentations and methods employed since the beginning of counting. You might incorporate finger signals, an abacus, pictures, manipulatives, and today's number symbol. You will present your PowerPoint presentation to the class."  Symbols for Operations and Other Mathematical Representations Have students conduct a comparable investigation into agreed-upon symbols for operations and other essential mathematical representations. For example, students can consider the best notations for addition, subtraction, multiplication, division, pi, e, and f(x). As part of the problem-solving and discovery process, students must decide whether or not notations should differ at different levels of schooling. To put it another way, pose the following question: "Is the most commonly used multiplication symbol at the middle school level different from that used at the elementary level?" This activity requires a Symbols graphic organizer (M-A1-2-2_Symbols Organizer). Hold a class discussion to review the lesson. Ask each student to describe the most essential item they learned about the history of numbers and/or number theory. Extension: Perfect numbers are connected to prime numbers. “If the sum of a series of numbers is prime, then the product of that sum and (\(2^n\) - 1) is a perfect number. Suppose we have the following table:<figure class="image"><img style="aspect-ratio:202/263;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_45.png" width="202" height="263"></figure>Consider where the n - 1 comes from.... For example, \(2^3\) = \(2^{4-1}\)." "Let's add the values of the first four terms. We have 1 + 2 + \(2^2\) + \(2^3\) = 15. How can we represent this total with a formula or function?" Give students time to investigate the pattern and draw conclusions. "Because we were summing through the fourth term, we got n = 4. Note that \(2^4\) - 1 = 15. Thus, the sum equals \(2^n\) -1. Let's check several more summations before making a deduction." Ask students to verify the formula with three additional summations. "Consider the second term, n = 2, which has a value of 2 because \(2^{n-1}\) or \(2^n - 1\) is a perfect number. \(2^{n-1}(2^n -1)\) = \(2^1\)(\(2^2-1)\) = 2(3) = 6 And 6 is the perfect number!" Show students that for n > 1, if \(2^{n-1}\) is prime, then \(2^{n-1}(2^n -1)\) is a perfect number. Students can learn more about the Rhind Papyrus and Euclid's Elements, as well as continue their study of perfect numbers. Perfect numbers equal the sum of their proper divisors. Numbers with proper divisor sums less than the number are referred to as deficient numbers, whereas numbers with proper divisor sums more than the number are known as abundant numbers. Assign a categorization to numbers in various ranges (20-30, 40-50, 60-70, etc.) and explore possible patterns and frequencies of abundant and deficient numbers.

Layla Nguyen

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History of Numbers and Symbols (M-A1-2-2)

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Layla Nguyen

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Description

Students will look into notations used in number theory during this lesson. Students will:
- trace the origins and history of numbers and symbols.
- examine specific types of numbers and historical documents.

Grade

Grade 9 Grade 10