Primes and Tests for Divisibility

1. Primes and Tests for Divisibility Chapter 5 Section 1 By: Tiffany Fey

2. Prime Numbers Can anyone tell me what a Prime Number is in your own words? And how do you know this?

3. So What is a Prime Number? The definition of a Prime Number from our book is: A counting number with exactly two different factors. (This excludes 1 because a Prime must be greater than 1.) For Example: 2x1=2 or 3x1=3 But to clearly understand what a Prime number is, you must also understand what a Composite Number is as well. The Definition of a Composite Number from our book is: A counting number with more than two factors. For Example: 2x2=4 or 3x2=6

4. My Reaction Now that you’ve seen the definition you are probably like, “Oh yeah! I remember that!” Well when I picked my topic, Primes and Tests for Divisibility, I completely drew a blank when I tried to think of what a Prime Number was. In the back of my mind I was pretty sure that I knew what one was, but I just couldn’t remember exactly what the word “prime” meant for a number. If you’re like me (when you remember what a Prime Number is :P) you realize that you’re really not sure how you know that a number is a Prime Number, you just automatically know and can say “Oh, well that’s a Prime Number” from memory or by thinking about it for a second or two. But how did we learn Prime Numbers?

5. Why This Is Difficult For Teachers and Students • As I did research trying to remember how I learned them, I discovered that you can’t learn Prime Numbers. You just have to memorize them. • This is very hard for teachers because when they teach a student something they must memorize, the student normally asks, “Why?” And as you can imagine it is hard for a teacher to explain exactly why a student must know something without saying “Just because”. Prime Numbers are difficult for students to learn because they must first know how to multiply and divide, and they must also know what a Least Common Multiple is (Definition: The smallest number that is a multiple of two or more numbers.) This is also very frustrating for students, especially since they are already learning about so many different things that apply to numbers, like place-value, decimals, adding, subtracting, and all of the other things they are being exposed too. But luckily, most students do not have to learn about Prime Numbers until they are in 5th grade, so they have a little bit more experience with numbers by this time.

6. First of All…Why do we need to know about Prime Numbers? Some students might ask their teachers “Why do I need to know what a Prime Number is?” And it is often very hard for a teacher to explain exactly why, so most tend to just say “It’s just something you have to know.” But why do students really need to know Prime Numbers? According to Dr. Ian from the Math Forum, Prime Numbers are used in life by chemists and people who go into technical fields of work, but for every day math users Dr. Ian says: • “Mostly, prime numbers are good for quickly transforming a situation with zillions of possible outcomes into an equivalent situation with only a handful of possible outcomes.” • “Here is another way to think about it: If you're looking for some needles in a haystack, you can start picking up each piece of straw, checking to see if it's a needle, and then tossing it over your shoulder. Or you can use a magnet to find the needles right away.” –Dr. Ian, The Math Forum This shows you that Prime Numbers are used in the world, but they are mostly just helping hands.

7. So How Do We Figure Out if a Number is Prime? There are many different ways to figure out if a number is Prime or Composite; this is where “Tests for Divisibility” comes into play in my presentation. But today I am just going to show you 2 different ways. The first way to find Prime Numbers and Composite Numbers that I will show you is called the “Sieve of Eratosthenes” The Second way that I will show you is the rectangular array approach that we have seen in class many times. But remember to keep in mind when we become teachers that there are many different ways to find Prime and Composite numbers and that there are no right or wrong ways to do this. Just as long as students can understand the concept. 

8. The Sieve of Eratosthenes The Sieve of Eratosthenes was created by the ancient Greek mathematician Eratosthenes. The definition of what the Sieve is from Wikipedia states that it: “[Is] a simple, ancient algorithm for finding all prime numbers up to a specified integer. It works efficiently for the smaller primes (below 10 million).” When you first see the Sieve it looks very confusing, but after you know exactly what is happening it is actually quite simple. I’ll go ahead and let you watch it first and then I will explain what is happening. http://upload.wikimedia.org/wikipedia/commons/8/8c/New_Animation_Sieve_of_Eratosthenes.gif

9. The Explanation! So as you can see, when you first look at the Sieves of Eratosthenes it makes no sense, but here is a step-by-step explanation from Wikipedia: To find all the prime numbers less than or equal to a given integer n by Eratosthenes' method: Create a list of consecutive integers from two to n: (2, 3, 4, ..., n), (On the one I showed you it was listed to #120.) Initially, let p equal 2, the first prime number, While enumerating all multiples of p starting from p2, strike them off from the original list, Find the first number remaining on the list after p (it's the next prime); let p equal this number, Repeat steps 3 and 4 until p2 is greater than n. All the remaining numbers in the list are prime. Let me show it to you one more time now that you know how it works. http://upload.wikimedia.org/wikipedia/commons/8/8c/New_Animation_Sieve_of_Eratosthenes.gif

10. The Rectangular Array Approach For some people the Sieve of Eratosthenes is hard to understand or just too complicated. But the second approach that I will show you is a good way of teaching younger children in elementary grades what a Prime Number is. The Rectangular Array Approach is what Dr. Byrne taught us in class, but when you are trying to figure out if a number is a Prime Number with this approach it is slightly different than what Dr. Byrne has shown us to this point. Time for some fun! :P Everyone take out a sheet of paper please. Ok, pretend that you have single (or one’s) base 10 blocks. With your “blocks”, see if you can make a perfect rectangle with 13 blocks.

11. This is what you should have on your paper: With one left over. OR: Or something similar to the ones above.

12. Why We Use This Method The point of the Rectangular Array Approach is to show you that if a number is a Prime Number then you cannot form a perfect rectangle with the blocks. While on the other hand if a number is a Composite Number then you can form a perfect rectangle with the blocks. This approach is a good way for children to visibly see whether a number is Prime or Composite, and as we have previously learned, children understand things better if they can actually see and touch the answers to their questions.

13. In Conclusion… As you can see, Prime Numbers and Composite Numbers can often be confusing for students to fully understand and difficult for teachers to teach, but if you use simple tricks, like the Sieves of Eratosthenes or the Rectangular Array Approach, to show your students how Primes and Composites work, then you should have no problem teaching it yourself. Thank you!

14. Works Cited Least Common Multiple Definition: http://www.mathsisfun.com/definitions/least-common-multiple.html Sieves of Eratosthenes: http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes Why 1 is not a Prime: http://primes.utm.edu/notes/faq/one.html Dr. Ian, The Math Forum http://mathforum.org/library/drmath/view/57182.html And our book: Mathematics For Elementary Teachers.