The more I teach math, the more I love numbers, especially prime numbers. When I learned math in high school, I never really understood why prime numbers mattered. It wasn’t until I started teaching students whose experience with math was less intuitive than my own that I really saw the benefits of recognizing prime numbers.

My favourite discovery so far has been prime factorization: writing numbers as the product of their prime factors. I never learned this in high school or elementary school, but I teach it to my students early in the year and refer back to it regularly. You can use it to identify perfect squares and perfect cubes. You can use it to find the factors of a number and the greatest common factor and lowest common multiple of sets of numbers. You can use it to simplify fractions or even just to make mental multiplication easier to do.

Because I use prime factorization with my students so often, I’ve really slowed down how I teach factors, multiples, divisibility, primes and composites. Having taught (or reviewed) the vocabulary earlier this week, and having gone through a couple of examples, today we worked on visual representations.

In class, students created a multiples chart. Starting with a hundreds chart, we circled all the multiples of 2 in red. Students knew a divisibility rule for 2: any numbers that end in 2, 4, 6, 8 or 0 are divisible by two (and are multiples of 2 and have a factor or 2). As we circled numbers in the chart, they also saw that the multiples of 2 are in vertical columns. Then, we circled the multiples of 3 in blue. We found the first few multiples by skip-counting and then another visual pattern emerged: they saw that the multiples of 3 make diagonal lines. They were also able to observe a divisibility rule: the the sum of the digits in multiples of 3 are also divisible by 3. As we marked the multiples of subsequent prime numbers, students were able to identify the next prime number by finding the next unmarked square. They also figured out how to find common multiples and identify factors by looking at the symbols attached to each number. We continued until we had marked all the multiples of the first 7 prime numbers (2, 3, 5, 7, 11, 13 and 17)

There was a lot that I liked about this exercise:

- students liked that we were using markers and pencil crayons in math class
- students were able to review what they already knew and use it to see patterns
- students came at factors, multiples and divisibility from another angle to reinforce their prior learning
- students were able to use what they knew to make predictions about how to find more prime numbers

My plan is for students to continue using the multiple charts that they have created. I will introduce prime factorization next week, and they will be able to use their charts to write numbers as a product of prime factors. And then, when they use prime factorization for other purposes, they can continue to refer back to the chart for a visualization that complements and illustrates the expression.

This is the best idea about integer factorization, written here is to let more people know and participate.

A New Way of the integer factorization

1+2+3+4+……+k=Ny,(k<N/2),"k" and "y" are unknown integer,"N" is known Large integer.

True gold fears fire, you can test 1+2+3+…+k=Ny(k<N/2).

How do I know "k" and "y"?

"P" is a factor of "N",GCD(k,N)=P.

Two Special Presentation:

N=5287

1+2+3+…k=Ny

Using the dichotomy

1+2+3+…k=Nrm

"r" are parameter(1;1.25;1.5;1.75;2;2.25;2.5;2.75;3;3.25;3.5;3.75)

"m" is Square

(K^2+k)/(2*4)=5287*1.75 k=271.5629(Error)

(K^2+k)/(2*16)=5287*1.75 k=543.6252(Error)

(K^2+k)/(2*64)=5287*1.75 k=1087.7500(Error)

(K^2+k)/(2*256)=5287*1.75 k=2176(OK)

K=2176,y=448

GCD(2176,5287)=17

5287=17*311

N=13717421

1+2+3+…+k=13717421y

K=4689099,y=801450

GCD(4689099,13717421)=3803

13717421=3803*3607

The idea may be a more simple way faster than Fermat's factorization method(x^2-N=y^2)!

True gold fears fire, you can test 1+2+3+…+k=Ny(k<N/2).

More details of the process in my G+ and BLOG.

My G+ :https://plus.google.com/u/0/108286853661218386235/posts

My BLOG:http://hi.baidu.com/s_wanfu/item/00cd4d3c5a2fd089f5e4ad0a

Email:wanfu.sun@gmail.com