Week 5

The latest concept explored in Math 251 would be the "base" concept. So to explain, when we add, and the sum of two placeholders is greater than 10, we carry the one to the next placeholder to be added there. Its something drilled into our heads at a young age and becomes second nature later in life. The issue with these "second nature" equations is that when trying to explain why we do this to children, our only answer is "because thats how you do it." So to better understand why, and how, this concept occurs, we experimented with different bases.

For example, base of four

  231     Why is this? Because when two numbers of a placeholder reached 4, we had to carry one
+121     and add from there, just like we do with ten! So, in the 2nd placeholder, since 3+2=5, we had
_____   lower the one leftover, and carry what would be the base four to the next placeholder.
  1012

Changing the rules to better understand your own understanding of a concept, will heighten your ability to be a better teacher in that concept. Its like seeing things from many perspectives to understand it better as a whole! Which is a great way to teach.

Week 4

Addition. A simple enough concept that we are taught at a young age, though the way we are taught doesn't completely allow us to understand what addition means. The coming together of multiple things to create a new sum, a sum in which can be discovered in so many ways!
This week in class we explored some of those ways.

Traditional:
This form is the method we were taught from day one of formal education, the stacking of the two numbers like so, not much explanation needed.

http://everydaymath.concordnhschools.net/modules/cms/pages.phtml?sessionid=df8d994395a2e08afb0842c63b74e11e&sessionid=&pageid=158683

Partial Sums:
This form is not farm from the traditional form, but it paints a clearer picture of what it is the traditional concept entails. By separately adding each placeholder, for example 34+98 would be 30+90 and 4+8.

http://teacher.sheboyganfalls.k12.wi.us/staff/laschwab/EveryDayMath.htm

Decomposing:
Another more creative visual of addition involves using a number-line to show the growth involved in addition. Similar to the regular number-line we worked on earlier in this year, though you are able to start at one of the numbers involved in the equation.

This example goes a step further to also implement compensation to make the jump easier.

http://www.kcptech.com/dynamicnumber/elementary_number_properties.html

Compensation:
This form of addition makes numbers that are hard to add without the traditional form and "carrying" much more manageable, because as a teacher of young math students the key to most problems are finding a "friendly" or "nice" number.
For example, if we were to be adding 34+28, not many of us would be able to solve this without counting. If we were to round the 28 to a "nicer" number like 30, we would have 34+30. Which you can easily do in your head, you come to 34+30=64 then COMPENSATE for the numbers you rounded to by subtracting the 2 (64-2) coming to the actual answer 62.

http://mindfull.wordpress.com/tag/compensation-addition/

Give and Take:
Similar to the compensation concept, give and take turns difficult numbers into nice numbers, but instead of rounding on its own, you give and take numbers from the numbers in the problem. Therefore you don't have to compensate in the end, because you never added or subtracted numbers that weren't there to begin with!

http://mathcoachscorner.blogspot.com/2012/10/more-mental-math-strategies.html

Addition can be as complicated or as simple as you want! Here are a few ways to do either.