Saturday, January 5, 2013
DBoyz Playing With 3rd Power Algebra
So here is a post but if you are new to base ten blocks and using manipulatives to teach algebra you may be a little lost. A little background can found at The House Of Math that leads to the algebra tab. You might also search Crewton Ramone 3rd Power Algebra...lots of videos that start at the beginning will come back. You can see that I use the algebra to teach other math concepts.
Basically all we are doing is giving a visual representation to the symbols, giving them geometric form makes them much easier to deal with and count. These boys have lots of experience playing with manipulatives so this drawing is just in black and white but you certainly wouldn't want to start here...this is several years of playing around coming to fruition.
But as you can see it's paying off, they can see what they are doing and the symbols have some meaning for them. They understand factoring means form a rectangle and count the sides and it's more akin to a puzzle than a formal math problem. Sometimes they make up their own, sometimes I tell them the symbols and they have to build it and count the sides (factoring), other times I tell them the sides and they have to build it (multiplication), and sometimes if it's a tough one I give them the whole rectangle AND one side they have to count the other side (division), they know division is the easiest because you get the most information. Here is a popular blog post covering division using base ten blocks...there are others too, just search for them. (Crewton Ramone Division.)
Hopefully you can see the benefits to you kids of learning math this way. Just because they can do some of this doesn't mean they have all the multiplication tables mastered and can do large subtraction problems in their heads...it does mean that they understand quite a few math and specifically algebra CONCEPTS that other children their age haven't learned yet and may never learn. We just play around with algebra concepts. This simple problem can be drawn three ways, because it can be factored three ways. It can look daunting to a high school algebra student, but we know it's "easy peezee lemon squeezee."
x3 + 6x2 + 11x + 6 = (x2 + 3x + 2)(x + 3) = (x+1)(x+2)(x+3)
This video had several takes in this take we miss them counting each rectangle carefully before they realize they all have the same amount, they are just shaped differently, that is they have different factors...
Keeping it in two dimensions makes the arithmetic easy. You will note I did not write out all the different symbols for all three because I wanted to keep it simple but we did talk about them and the side that can be factored is drawn again to the right. These are easy for them to factor and they can see that x2 can have two shapes, the one they are used to which is indeed square and the one where it's "hiding."
At this age we are most interested in counting, addition and multiplication and addends than we are in the actual algebra.
I make no attempt to set it equal to zero, don't talk about roots, or graphing we are just playing and counting more will be added later after we have done lots of problems with sides that can be factored and ones that can't.
Then when we return to this simple one and we add new concepts they will be easy and unclouded by concepts that we have already mastered. This is what we mean by degree of difficulty or baby stepping our way to the "higher" mathematics. If you have to try and learn all of it at once it can be overwhelming. Better to build a firm foundation. Then when I talk about it having 3 real roots and hero zero that's the only part they have to focus on, the rest already being understood so it doesn't add to the confusion.
Here we begin to see why Mortensen Math is head and shoulders above other manipulative teaching systems, and how Jerry took the Montessori method and ran with it. Remember, hese boys are 6 and 7...