• Observe/evaluate class discussion. Question students individually and collectively—orally and in writing. Direct individual students to present solutions in front of the class and ask them to self-critique their presentations.
• Evaluate student performance on:
  • creation of the Sieve of Eratosthenes.
  • showcasing other methods/procedures for determining primes.
  • providing illustrations, descriptions, and representations of pi via a PowerPoint presentation.
  • discovery of two perfect numbers from 0 to 100.
  • determination of the convergence/divergence of a function.
  • discovery of equivalent fractions to infinite decimals.
  • creation of a short PowerPoint presentation on the history of counting.
• Examine the written solutions of conversions of repeating decimals to fractions. For these exercises, reducing the solution to lowest terms is less important than finding the appropriate place value with which to begin the computation.

Part 1: Study of Numbers

Prime Numbers

“What is a prime number? How can a prime number be represented? What does a prime number look like?” [IS.5 - Struggling Learners] Engage students in an open discussion. Ask students to come to the front and share representations of prime numbers. “A prime number is a number that is only divisible by 1 and itself.” Note: The number 1 is neither prime nor composite. Solicit divisibility examples from students and have them express their understanding of what divisibility is as a general characteristic of a number.

Ask students to work toward finding the first 100 prime numbers. [IS.6 - Struggling Learners] They may use any method they like. After frustration sets in, [IS.7 - Struggling Learners] set up the first activity listed below, which will alleviate the search process and provide a structured method for finding the first 100 primes.

Activity 1

In groups of three or four, students research the process of using the Sieve of Eratosthenes to determine prime numbers. [IS.8 - Struggling Learners] They should be prepared to share an illustration of the process, describing each step used while eliminating composite numbers.
For each number 1 through 50, direct students to cross out each number that can be divided evenly by another number other than 1 and itself.

Following the activity, share the NLVM Sieve of Eratosthenes applet (http://nlvm.usu.edu/en/nav/frames_asid_158_g_2_t_1.html) with students. They may have already discovered this tool in their own search. (The pronunciation is air-a-TOS-then-eez.)

Activity 2

Have students demonstrate any methods/procedures for determining primes they thought of or came across in their research. Ask, “Is there a pattern to the prime numbers? If there is not a pattern, how would you handle the problem of determining if a number is a prime number? How did other mathematicians do this?” [IS.9 - Struggling Learners]

After students have had a chance to explore and research whether there is a precise method for determining prime numbers, reconvene and explain that there is not an equation for finding all prime numbers. Several mathematicians have come very close. However, there has not been one single equation to capture all prime numbers.

“Fermat famously developed the function . However, the function only elicits prime numbers for the first 4 natural numbers (1, 2, 3, and 4). The output for the fifth natural number, 5, or f(5), equals 4,294,967,297, which is not prime!” [IS.10 - Struggling Learners]

Invite students to explore more about Fermat’s theorem and theorems provided by other mathematicians.

“What is the largest prime number found thus far?”

Invite students to determine whether the set of prime numbers is finite or infinite. [IS.11 - Struggling Learners] After students have pondered the question, provide the following resource, which includes Euclid’s proof of infinite primes at http://mathforum.org/isaac/problems/prime1.html.

pi

There have been many methods used over the years to determine the most precise value for pi. The intent of this portion of the lesson is to invite students to explore 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.

Start the discussion with an open dialogue, regarding pi. Ask students, “What is pi?” Allow students time to ponder. Many will give the decimal representation. “What does pi represent? Where did pi come from? Why was there a need for pi? How do we use pi in everyday life?”

“Pi to the first 10 decimal places is 3.1415926535.”

“Can you go beyond that? Check out http://www.factmonster.com/ipka/A0876705.html to view pi to the first 1,000 decimal places!” A History of Pi by Petr Beckmann provides a table with pi to the first 10,000 decimal places! It also includes a chronological chart, outlining the progression of discoveries related to pi.

“Pi was discovered around 2,000 BC by both Babylonians and Egyptians. The Babylonians arrived at ; the Egyptians arrived at . The methods employed to arrive at these calculations are mere speculation. They could have easily measured the circumference of a circle, measured the diameter of the circle, and calculated the ratio of circumference to diameter. The problem is that they did not have any precise or calibrated measuring tools. Using ropes and stakes placed in sand, they were able to determine very close approximations, with a first approximation of . A History of Pi, written by Petr Beckmann, is a wonderful resource on the topic of pi.”

“The Chinese used a different approximation method. Around 264 AD, the Chinese used inscribed polygons within a circle to determine an inequality involving pi. For example, using an inscribed polygon of 192 sides, Liu Hui found:

Using an inscribed polygon of 3,072 sides, he found:

In calculating with approximations for pi, the fraction  is frequently used because its value, , is reasonably close to pi.”

“When more accurate calculations with a fraction are required,  may be used because its value, 3.141592… is even closer to pi.”

Conclude with a measurement activity using, for example, a basketball, soccer ball, lids, bottle caps, etc. [IS.12 - Struggling Learners] Using ordinary measuring tools, such as rulers and measuring tapes, see how close students can come to an accurate value of pi by dividing the circumference by the diameter. Use π = c ¸ d.

Students may use NCTM’s pi applet, found at http://illuminations.nctm.org/ActivityDetail.aspx?ID=161, to compare two different methods for computing pi. Students will realize that as the number of sides of the polygon increases, both the area and perimeter get closer and closer to the actual approximation for pi.

Activity 3

The intent is to have students create a table or other graphic organizer, comparing representations of pi. It will be the responsibility of students to determine the most useful representation.

Tell students, “You are to trace the evolution of the discovery of pi, starting from the earliest records and ending with any present-day findings. Provide illustrations, numerical representations, descriptions of methods employed, and any other pertinent information. [IS.13 - Struggling Learners] Make comparisons across representations/notations of pi, using at least one graphic organizer, i.e., a chart. Produce a PowerPoint presentation to serve as a teaching tool on the topic of pi. Highlight the method that makes the most sense to you and provide supporting reasons. Include at least one slide, related to the real-world application(s) of pi. Feel free to use A History of Pi or any other reputable resources.”

Perfect Numbers

“What is a Perfect Number? Have you ever heard this term? Where have you heard it? In what context?”

“A perfect number is a number that matches the sum of its divisors. Can you think of a number that fits this criteria? Note: The addends of the sum will not include the number itself.”

Activity 4

“In groups of three or four, try to find at least two perfect numbers, from 1―100. Be prepared to describe your process.”

“In Euclid’s Elements, there is an entire chapter devoted to perfect numbers. Euclid showed how to construct a perfect number from a special type of prime number and proved that the first four perfect numbers are generated by the formula
Using your calculator, substitute the first four primes (p) and check your result with this table:”

(Answers/table:)

After students have completed the activity and (hopefully) discovered that 6 and 28 are indeed perfect numbers, create the table below. Give students the factors and confirm that their sum does indeed equal the number. The point of the illustration is to increase and support students’ conceptual understanding of the topic.

“The first four perfect numbers are 6, 28, 496, and 8128.”

“Do you notice any other patterns with perfect numbers?” Students might notice that the sum of the reciprocals of the divisors of a perfect number equals 2. Show this example with the perfect number 6:

The divisors of 6 are 1, 2, 3, 6.

The reciprocals are 1/1, 1/2, 1/3, 1/6. These fractions sum to 2. [IS.14 - Struggling Learners]

“Try it with the next perfect number, 28. Does it work? Why or why not?”

Zero

Tell students, “We must pay tribute to another special number, zero, perhaps arguably the most important special number.”

“Why was the number zero invented? Can you imagine what the need was? Does anyone know the years of evolution for the number zero? Who invented the zero?”

“Zero was invented as both a means to represent ‘no things,’ as well as to be a placeholder for ‘no things.’ The number was invented around the years 3000 BC––AD 1000. The Indians were the first to accept zero as a number. The use of zero was also advanced in the arithmetic and calendars of the Mayan civilizations of Mexico and Central America as far back as the 4^th century BCE. Neither the Greeks nor Hindus wanted to recognize a void representation as being a number. The writings of early Greek mathematicians and philosophers showed how uncertain they were about recognizing zero as a number. They frequently asked, ‘How can nothing be something?’”

Demonstrate cutting an apple in half, then into quarters, eighths, and so on. Explain how an unlimited number of cuts could lead to the fraction of the apple becoming infinitely small.

The Idea of Convergence

“When we think of the idea of ‘convergence,’ we often correctly think of approaching a particular number, as n (or the cases) increase. [IS.15 - All Students] For example, take our previous exploration into the world of pi. It was noted that as the sides of the polygon increased, the value of pi got closer and closer to the actual approximation of 3.14. We can say that as n increased (the number of sides of the polygon increased), the value ‘converged’ towards 3.14.”

“Convergence is completely opposite from divergence, whereby a number either grows and grows without bound, or decreases without bound. You can imagine that many real-world occurrences of data in spreadsheets reveal a certain convergence.”

Activity 5

Determine and illustrate a convergence of some sort. Highlight the formula used. Ask students, “Is there a general form that is needed to illustrate a converging sequence?”

“Does  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 thought of repeating decimals as an infinite convergence? Did you know that repeating decimals can be represented as fractions that reveal the exact decimal value?”

“Take, for instance, the decimal .4747474747… What fraction is equivalent to this decimal? How can we determine that? What process can be used?”

Activity 6

Break students into groups of three or four. Have them brainstorm ideas for determining the fraction equivalent to the decimal .4747474747…. They will likely get very close and continue to plug in numbers. Ask students to share their findings.

After the presentations and discussion, review the following structured way to find the precise fraction. “We know that .4747474747… can be written as 47/100 + 47/10,000 + 47/1,000,000,… The ratio is 1/100. Thus, we can write:

Notice, that as we write the decimal in more and more detail, the value of the decimal converges to . This is another example of the idea of convergence!”

“The common notation for repeating decimals is a bar over the portion of the decimal that repeats. For example,  means 1.567567567…; means 1.5676767…;  means 1.56777…. It is important to note that the repeat symbol covers only the repeating portion of the decimal.” [IS.16 - Struggling Learners]

“Now, suppose we wish to change  to an equivalent fraction. We can simply set up the following problem to deal with the decimal portion:   

Recall that we had a 1 in front of the decimal. Thus, the equivalent fraction is written as . We can reduce this fraction to .”

“There is another way to solve this problem, however. The repeating decimal is actually approaching a limit. We can think of the decimal as an infinite geometric series. If approaching the conversion in this manner, we would write the sum of the repeating portion as:

Notice that the ratio is , or .001. The repeating portion, or   is .567.”

“Now, using the formula for the sum of an infinite geometric series, we have:

Substituting our values, we find:

Multiplying the numerator and denominator by 1000 eliminates the decimals.

We can add the 1 back in front of the fraction and reduce to lowest terms. Again, we find  to be an equivalent fraction to the repeating decimal, .”

“Now, suppose we wish to do something a bit more interesting and challenging. We want to change  to an equivalent fraction. This time, only the 6 and 7 repeat! We will need to set up two equations to deal with this problem. First, we wish to move the repeating portion to the left of the decimal point. We can do so by moving the decimal point three places to the right. Moving the decimal three places to the right involves multiplication by 1000. Thus, we write:

Now, we must consider that only the 6 and 7 repeat, so we need to move the decimal only one place to the right. Moving the decimal one place to the right involves multiplication by 10. So, we now write:

We are now in a familiar situation, whereby we simply subtract our two equations. Doing so gives:

Therefore, , written as a fraction, is . This fraction can be reduced to lowest terms.”

“What if we wanted to change  to an equivalent fraction? Notice that only the 7 repeats. What would we do?” Divide students into groups of three or four. Have them find the equivalent fraction. (The steps are shown below.)

“This can be written as .”

“Finally, let’s look at one more, where just a single digit repeats. In this example, we don’t have a whole number piece. Let’s find the fraction equivalent for .”

“We will start by moving the repeating portion to the left of the decimal point. We will thus move the decimal point three places to the right. Moving the decimal three places to the right involves multiplication by 1000. Thus, we write:

Since only the 5 repeats, we need to move the decimal two places to the right. Moving the decimal two places to the right involves multiplication by 100. So, we now write:

We now simply subtract our two equations. Doing so gives:

This equivalent fraction can, of course, be reduced to lowest terms.”

Gauss

“We often do the following with mathematics: ‘think harder, not smarter.’ We should actually ‘think smarter, not harder.’ This adage is illustrated by the common story of Gauss solving the following problem as a young boy.”

“Gauss’s teacher asked students to find the sum of the first 100 numbers, including 1 and 100. While other students laboriously calculated the sum, Gauss simply made a structured list of two columns of numbers. He wrote the following:

He noticed there were 50 pairs, with each pair summing to 101. Thus, the sum of the first 100 numbers was 50(101) = 5,050. How ingenious is that?”

Part 2: Evolution of Symbols and Representations [IS.17 - Struggling Learners]

Number

“What is a number? How would you define number? Who first thought of number? How do the concepts of number and counting differ? Do they? Is the understanding of number an innate or learned ability? These are all important questions to ponder. Let’s look at some background information and focus on the idea of ‘representations of number.’”

Cardinal Numbers

“Throughout history, numbers have been represented using a variety of methods and tools, including an abacus, numerals, hieroglyphics, figures, notches/scratches/ marks, finger symbols, and the current number symbols used today.”

“What is a cardinal number? A cardinal number is a number that represents ‘how many.’ The nuanced differences between cardinal number representations and counting have been discussed and debated by mathematicians for centuries, including today.”

“Numbers are represented in different bases, including base ten, base five, and base two. Base ten is the system used with the decimal system. The other two bases do hold their place in important arenas, especially base two, with the binary system applying to various codes, computer circuitry, and so on. In base two, the place values are 2¹, 2², 2³, 2⁴,… 2ⁿ. In base 10, the number five is the familiar character 5. In base 2, five is 110, because the first one tells how many 2²s, the second place tells how many 2¹s, and the third place tells how many 2⁰s. Counting left to right, that is one 2², one 2¹, and zero 2⁰s, 4 + 1 + 0 = 5.”

Ordinal Numbers

“What is an ordinal number? Ordinal numbers represent the position of the number relative to other numbers. For example, 1 comes before 2, 2 comes before 3, and so on.”

“Here is a question to ponder: Did the discovery of cardinal numbers precede that of ordinal numbers?” Lead a class discussion of this question, providing supporting ideas and reasons for each possible answer. For example, one of the difficulties in answering the question is how far back into the history of human development one must look for evidence. It is likely that less developed societies in the past needed to quantify their basic necessities, such as animals, logs, tools, fruits, and vegetables. Similarly, it is also likely that order and priority developed as well from such basics as first child, second child, full moon, new moon, or selecting which berries to pick first.

Activity 7

Say to students: “Your job is to create a short PowerPoint presentation, delineating the history of counting. Answer such questions as, ‘When and how did counting originate?’ ‘What was the specific need?’ Include representations and methods used since the origins of counting. You might include finger symbols, abacus, drawings, manipulatives, and the number symbol of today. You will share your PowerPoint presentation with the class.” [IS.18 - Struggling Learners]

Symbols for Operations and Other Mathematical Representations

Have students do a similar exploration into agreed-upon symbols for operations and other important mathematics representations. For example, students can think about the most appropriate notations for addition, subtraction, multiplication, division, pi, e, f(x), and so on. As part of the problem-solving and discovery, students will need to determine whether or not notations should be different at varying levels of education. In other words, ask: “Is the most widely accepted symbol for multiplication at the middle-school level different from that used at the elementary level?” A Symbols graphic organizer is required for this activity (M-A1-2-2_Symbols Organizer.docx).

As a review of the lesson, hold a class discussion: Have each student describe the most important thing s/he 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 is a perfect number. Suppose we have the following table:

Notice where the n – 1 comes from… Take for example  .”

“Let’s sum the values for the first four terms. We have  . How could we possibly represent this sum using a formula/function?” Give students time to explore the pattern and make deductions.

“Since we were summing through the fourth term, we have n = 4. Notice that . We could thus say that the sum equals . Let’s check some more summations, first, before making a deduction.” Ask students to check the formula with three more summations.

“Let’s take an example: If we use the 2^nd term, n = 2, the value of the term equals 2 because , or and   is a perfect number.

And 6 is a perfect number!”

Lead students to see that for n > 1, if  is prime, then  is a perfect number.

• Students can research the Rhind Papyrus and Euclid’s Elements, and continue their investigation into perfect numbers.
• Perfect numbers are equal to the sum of their proper divisors. Numbers whose proper divisors’ sums are less than the number are called deficient numbers and numbers whose proper divisors’ sums are greater than the number are called abundant numbers. Assign the classification of numbers within various ranges (20-30, 40-50, 60-70, etc.) and investigate possible patterns and frequencies of abundant and deficient numbers.