The sample lessons are here. Look for the one called "Difference of 2 Squares." The vid is 34 minutes long. Scroll to the bottom for this one. As you may or may not know your password opens up a bunch of pages not just that one. Several people have told me that one page is worth a lot more than 3 bucks but so far only my friends in Namibia have been kind enough to actually put it writing.
Here we get into some very simple analysis of the blocks:
These very same questions have gone on in teacher trainings and it has at times taken me a lot longer to get a room full of adults to have the same light bulb go on as it did with the 6 year old after just a minute...literally. Watch. Sadly the camera was not pointed at his face when he had his ah-ha moment. YOU have to explain and show them and ask questions that they answer themselves NOT just tell them.
"Here: memorize this. Here is the formula. Now go do it." DOES NOT WORK. 20 years later and not much has changed except things have gotten worse...
Here is an opportunity to change. Start by going to the house of math and watching the vids or if you are more advanced break out three bucks and get a password...
x2 - 25 = (x + 5)(x - 5)
Is obvious to these kids: they can SEE it. No big deal. The concept of zero is being introduced and later when they see zero identities it won't be scary or confusing just really, really obvious.
x2 - 1 = (x + 1)(x - 1) and x2 - y2 = (x + y)(x - y) are also easy once the concept is understood. Several more impressions and it will START to be internalized. One of the main problems with high school and college courses is the fact that they don't go over concepts enough times (if at all) so the formula is stuffed into the short term memory recalled for a test and then forgotten. You also get to pay thousands per credit for the privilege.
Also added to their quiver will be adding zero to help with problem solving. Math concepts are never beyond a child if presented at the child's level. The next thing is negative expression and very shortly I was able and will be able to let them build and count problems that high school teachers will consider advanced. And we got there in baby steps...by the ripe old age of 4 in one case and 6 in the other...I have done this with HUNDREDS of students over the years. I know it works...we are past the testing stage. Soon there will be some pdf's with more evidence if somehow you can't believe your eyes here. Newspaper articles from 20 years ago and such.