Divisibility Rules

     I have always known that a number is divisible by two if it ends in 0,2,4,6 or 8. I have also used the divisibility test for 3 by adding the digits and finding out if the sum of those digits is divisible by three. I did not remember, however, that a number is divisible by 11 if the sum of the even-powered places less the odd-powered places is divisible by 11. Multiples of 11 were not focused on as intensely in my elementary years, so I find this divisibility rule beneficial.  In looking at these rules, I found myself wondering about the benefits of  these divisibility tests for students.

     I think of strategies concerning video and board games and the elation from a child when they figure out a new "trick" relating to a game. Students can benefit from learning these divisibility rules because they give students the power to solve these division problems. I know I personally keep trying to solve problems when I know the algorithm or formula that assists in "cracking" the puzzle presented. Knowing that there is more than one method to solve math problems can also benefit students as they choose the methods that work the best for them individually. Here are some of the divisibility rules:

 
Here is a fun divisibility song:

Here are a couple of game sites involving division or divisibility rules:

http://www.mathplayground.com/game_directory.html

http://www.oswego.org/ocsd-web/match/dragflip.asp?filename=slanedivrules

     In summary, I can personally understand the benefits of learning these divisibility rules. When given a large number, the task of finding factors is less daunting because I know the "tricks" for determining if the number is divisible by the numbers on my divisibility chart. I have been given a special "compass" to help me navigate this sometimes tricky terrain of division. My excitement and desire to solve these problems, proving the validity of the rules as I go along, can be shared by my future students. Math instruction can be fun and enjoyable for students when we take the time to teach strategies and tricks!

The Benefit of Estimation

     I have always loved and excelled in the area of estimation. I learned the rules of rounding numbers quickly and applied them to many aspects of my life. I was taught one simple strategy and was able to apply it to every situation I encountered. If the number following the number to be rounded was 4 or less, the number stayed the same. If that same number was 5 or more, the number being considered for rounding went up by 1. A simple strategy that works every time....or does it? I have recently been introduced to clustering, front-end estimation with or without adjustment, and substitution of compatible numbers. While clustering and the substitution of compatible numbers are similar to my rounding technique, front-end estimation without adjustment made me pause. How does this form of estimation work and how does it benefit the student?
     As I began my search for the answer, I found that even Dr. Math questions the technique. However,I also found that having a good sense of estimation and the use of different techniques leads to better estimation in school and in life. So how does front-end estimation benefit students? It gives students another method to use and builds the foundation for adding the adjustment later. Here is an example of front-end estimation:

This technique then leads to front-end estimation with adjustment.

    1560
+    821
________

To solve, first apply the rules of front-end adjustment to add 1,000 plus 800 to get 1,800. This leaves about 600 (560) and 20 (21), or 620 left to add. The total amount is about 2,420. With rounding the first number, the answer would be about 2800. The actual answer is 2381, making our front-end estimation with adjustment the closer estimate in this case.
    The lesson learned in the exploration of estimation is that each math problem is unique and one technique may not fit every problem. Also, students benefit from using different estimation techniques to find which method makes the most sense to them and fits the problem they are trying to estimate . This skill is beneficial as we need to be able to estimate in order to double check the totals on our calculators and computers. I may still use my rounding technique, but now I know that there are alternate estimation techniques that may arrive at a closer answer. Here are some fun sites with estimation activities:

http://www.mathsisfun.com/numbers/estimation-game.php

http://www.gamequarium.com/estimation.html

Base Conversions

     It is the beginning of summer, which is my busiest season of the year. Baseball, soccer, extended child care days and event planning keep me on my toes. On top of the added responsibilities, I am also taking summer courses that will aid me in realizing my goal of becoming a teacher. I often find myself wishing that I could build an exact replica of myself to accomplish half of my tasks or to offer another set of arms and a lap on those days that the children in my care need more one-on-one time. A computer for a brain would also aid in remembering important dates and keeping a tight schedule. All of this thinking ties into the base ten conversions I encountered in my course this week.

     The first time I took a math for elementary teacher course, I was attending Saint Cloud State University for Special Education. I remember struggling through the course and acquiring a D despite my efforts to understand the base conversions. When I realized I needed to tackle this concept again, I was nervous that I would fall short again. When I first studied base conversions, we did not have a clear system defined for us in order to convert the bases and it left me wondering why this was even important. Fast forward 19 years and the need for base conversion is apparent in the binary and hexadecimal systems required for computing and digital technologies. 

     

     In using the binary codes, a zero represents "off" or no current and a one represents "on" or to allow current. The use of base two in computer science and electronics is reason enough for students to acquire the ability to perform base conversions. The bit system that is built from these binary digits determines the transmission and storage available in our computers and other digital electronic devices.  The hexadecimal system (base 16) lists memory space in a two digit number versus several binary digits. As I work from my laptop to create this post, I not only see the value of the programmers responsible for my computer and software, but also the students who have a future in programming. By teaching base conversions, students will have a better understanding of the technology that surrounds them and some may find a future occupation in programming. Maybe one of these students will be able build and program the "robot replica" I could so badly use this summer!

Bring on the conversions!