Here is a gem. Don't tell me they have to have base 10 blocks or they will become block dependent using this method. Watch the older boy picture this problem
x³ + 9x² + 23x + 16 =
in his head...counting out the parts as he goes and then give it to me to solve.
This is a lesson on factoring and counting and multiplication...and how to be cool and have fun when you do math. The reason they don't need base 10 blocks is because they have been using base ten blocks...and we are learning by drawing and visualizing. Here is a post on 3rd power algebra where you can get an idea of what it is he is "looking at" in his head. We baby stepped our way here but now you begin to see the POWERFUL results of using this method.
We watched this together. The youngest boy pointed out that he drew 6x² + 10x + 4 and didn't get very much attention for it. He also told me the factors (you can see the drawing in black in front of him) (3x + 2)(2x + 2). "Come on, that was pretty cool dad." So we are going to make a new vid where he gets as much attention as his brother...
Here we play around with algebra concepts. This "simple" problem can be drawn three ways.
x³ + 6x² + 11x + 6 =
(x+1)(x+2)(x+3) =
(x+1)(x+2)*(x+3) =
(x+1)(x+3)*(x+2) =
(x+2)(x+3)*(x+1)
We stay in two dimensions and simply change the sides because in this case the long side can be factored.
In the video we play with it a little and talk about x to the fourth too.
This video had several takes in this take we miss them counting each rectangle carefully before they realize all three have the same amount, they are just shaped differently, that is they have different factors...too bad too because you could really see them using their skills with multiplication and addends. I have written entire articles about the importance of addends and how they need to be mastered. Here they add to their mastery of addends subconsciously.
Keeping it in two dimensions makes the arithmetic easy. You will note I did not write out all the different symbols for all three but we did talk about them and the side that can be factored is drawn again to the right. At this age we are more interested in counting, addition and multiplication and addends than we are in the actual algebra. We 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 when we add new concepts they will be easy, unclouded by concepts that we have already mastered. Math the regular way introduces everything at once and it can be daunting. We use degree of difficulty to baby step 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 is already understood.
Here is an older student working on factoring by grouping. She never got to see these as a kid so it gave her a little trouble at first.
And here we play with a 10, 000 square, just talking about it and understanding the dimensions it represents.
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. These boys are 6 and 7...and as I explain in the video we get to see a synthesis of counting all in one lesson. The algebra is just along for the ride.
