Math Lesson 29 Sept 2009

This is the house that math built. Looks like a pile of blocks but it was quite a feat of engineering...lots and lots of math went into it on the subconscious level but also on the very conscious level, as he figured out how to make the roof, how many he needed to make it even on both sides etc. A few questions from me also helped make this a learning experience as well as a reward for doing lots of math problems. NOTE the square root problems under it...we of course counted the sides but before this it was a simple lesson on multiplication and counting by 2s and 5s...




This shows how we don't stop at 9 nor even 12...we have multiplication to 16 here...and these squares were counted in many ways. First we counted to 32 by 2s, then by 8's...and to 48 by 3s and then by 12's. He found counting by 12 was actually pretty easy because "it's just like counting by ones and twos...12, 24, 36, 48!"

Then we counted square roots. The symbols are sloppy but readable and again the child is but SEVEN...! Writing out that the square root of 32 is 4 square roots of 2 was easy and redundant...same with the square root of 48...4 square roots of 3. "This math stuff is easy."

Before all this we built a full top tray of addends, then did some 100.00 minus xx.xx and then some two digit subtraction...we did a whole lot of problems in a short period of time and the reward was the house...Crewton Ramone's House of Math.

6hrs 9 students....

A new student looks at Algebra in a new way for the first time. "Here is the expression build me a rectangle." However long it takes them to discover how to do it is how long I wait. Over the years I have learned that I can tell one hell of a lot about a student (or a person) by the way they do this. It is crucially important that they do this for themselves; directed discovery, not memorization and formulae.

Ran the full gamut yesterday, from Counting thru Calculus. Made me come up with a few lessons for little kids to prepare them for Calc and even teach them some basic calc concepts...like distance formula and derivatives via building squares and Pythagoras. Always comes back to the five concepts...and their application. ALWAYS.

Also have an article about critical thinking skills and their development with regard the mathematics banging around in my head.

Am I ever gonna finish a post? Hopefully starting them will cause me to get around to finishing them...need links and more pics...

Playing Math For Fun And Profit

Since that square root video I made in the last post many classes have taken place. In this post, I will focus on the classes that used algebra for the primary cross concept method. I use it to teach addition, multiplication, division, factoring and "algebra." Occasionally I use it to teach other concepts as well, but today we focus on basic operations.


The work above was done by a seven year old male student. Our focus was on addition and then on multiplication and lastly on algebra.

The idea is simple. Algebra is abstract mathematics. This method makes the abstract concrete, therefore we can start teaching concepts using algebra. We start simple and work our way up to the complex, keeping the 5 basic concepts in mind and using three period lessons as needed to make sure everyone is on the same page as it were. This is CONCEPT based teaching NOT memorization or formula based teaching. I teach by example with the understanding that the first three times are the baby steps. First time is new. Second time is familiar. Third time is "we did this already, let me do it..."


These two "simple" factoring problems are easy to solve once they have te blocks to work with. The point is not to amaze and astonish the stodgy, well entrenched teachers of mathematics by training monkeys to factor polynomials. The point is to make impressions on the child's memory. It takes about 18 impressions to get a fact from the short term to the long term memory. Some methods use wrote drills, where the child is subjected to seemingly endless worksheets. Here, when they factor

x² + 7x + 12

They learn that 4 + 3 = 7 and that 4 x 3 = 12 simple enough. They also learn that if the whole rectangle is 12 and one side is 3 the other side must be 4...lastly they can see that if the whole thing is x² + 7x + 12 the factors are (x + 3)(x +4), which means that if the whole thing is x² + 7x + 12 and one side is (x + 3) the other side must be (x + 4).

There is a whole lot of math going on in one problem...and they learn and discover all by themselves. Further the trial and error teaches them quite a bit also, and in this method they are not wrong just acquiring more information, because the model is self-correcting due to the fact that success means a rectangle is formed. Hence the slogan "if you can count to nine, tell if something is same or different or not and identify a rectangle, we can teach you math."


Here we have some team work going on. 4 students, 2 teams. The students are racing to try and finish this battery of problems:

x2 +  7x + 12                 x2 + 9x + 18
x2 + 9x + 14                  x2 + 10x + 25
x2 + 10x + 30                 x2 + 12x + 32

They have to factor them and draw a picture in their notebooks. There is an age range of 8 to 14 in this group. Each students is getting a little something different from this exercise, the youngest are still working on addition and multiplication facts, the older students are "doing algebra" all of them are building self esteem and confidence because they "know" algebra is supposed to be hard.

These problems got done a lot faster because they were racing, which is why I like a group as compared to solo lessons.


This post is FAR from finished...

Square Numbers

This is a quick video I did on square numbers.




This was a "one take Eddie" but I think it was good enough to get the point across that when we talk about square numbers and expressing the terms as a radical we are simply counting the sides of squares.

This is the beginning of Crewton Ramone's Supremely Simple Square Roots. I plan on making a simple concise Manual with a DVD available that covers square roots, how to square numbers and etc. It will contain a few "tricks" but mostly show the concepts. It will also contain the cross teaching techniques for multiplication...

Algebra is child's play.

So much to do so little time. I just wanted to put this down before it slips away. I did a class with a 7 year old boy. When I met him he couldn't count to 20.

I put

x²+5x+6

on a white board. He didn't get out blocks, he didn't draw pictures he just looked at it for a moment and said "that's gonna be x+2 across and x+3 up...."

And he looked at me funny when he saw that I was a little choked up...now these are the kind of tears that should be associated with learning algebra not the kind that you get in public schools where kids literally jump out of buildings because they hate their math...