Vid From The Vaults

A year later at Sweet 16, but now Sweet 16 is about to turn 21.

This is a fun video covering some basic concepts in geometry. Basically, I demystify distance formula for a giggly girl.  Pythagorean theorem.  I always looked forward to fooling around with math with her because we always ended up laughing our butts off.  (See second video.) And this usually led to sloppy notation on my part. The point is she was understanding the fundamental concepts. There is a point in the video where I write out distance formula and fail to put the 2's in for the squares...very annoying looking back but she fully understood they were squares. Other than that, I hope you see how teaching her the fundamental concept of a² + b² = c² improved her understanding dramatically and therefore her grade.

Then she could see that a was equal to the difference of x's and b was equal to the difference of y's.

a = (x₂ - x₁)  b = (y₂ - y₁)

Look at how many less symbols I need to say it mathematically.

Then we have to square them and then square root them AFTER we add them together.

                                                       _______________
(x₂ - x₁)² + (y₂ - y₁)²  = D²  so  D = √(x₂ - x₁)² + (y₂ - y₁)²         which if you think about it a little is just

a² + b² = c²

a little "gussied up" as my grandma would say.   Anyhow there should be 2's there...but as I said the lesson was understood.


Anyhow I dug this out of the archives. The first video hasn't been seen by anyone but me. The second one as I type this has about 50 hits.




We always just laughed and laughed and laughed. Before me, math made her cry. It was those tears that made her dad break down and call me.




These are typical high school math lessons right out of the textbook. I claim exemption from copyright under the education purposes only clause. This blog is free to view and so is the video. Besides these types of questions and horrid explanations certainly are not unique to this text book manufacturer. Often I barely have time to develop concepts because the book covers so much in just a few pages. And people say I skip around. What a joke.

Take a look at the topics covered in that first video...the way they are presented the kids can't figure out it's all basically the same concept...just variation of applying algebra and Pythagorean theorem to get the job done whether it was distance or mid-points.  Learn how to use base ten blocks.

Third Power Algebra With Little Kids

Base ten blocks make math easy and fun, even third power algebra. Left off an equals sign but you get the point. He factored it two ways and then realized that factoring it down to (3x-2)(x+1)(x+2) would cover all of them…he knew without thinking very hard that x² + 3x + 2 = (x+1)(x+2) and that 3x² + 4x + 4 was just (3x-2)(x+2) so the “last way” was just going to be (3x-2)(x+1) and the other factor of (x+2). So basically he discovered on his own why "we factor it down as far as we can go."

Take 18 for example, 3x3x2 covers all the bases. 2x9, 3x6 same thing, 18.  Just depends on how you multiply the factors.  I use algebra to teach all manner of math concepts. Not only that, you teach more faster because you can compound the lessons.

What you aren’t seeing is the base ten blocks and drawings we used to make this EASY.  So here they are.  My notes for the days class. This was just over an hour's worth of work. But these two are aces. This was FUN for them. Might take you more or less time depending on all manner of factors. (Get it? I crack myself up.)

Started off with the difference of two cubes and they very quickly saw some patterns there...
I leave x³ - 64 for you the reader to do on your own time...might want to start at x³ - 1 with basic ops pieces and work your way up...drawing as you go. Click here for more third power algebra.

We did a few more problems not shown here but they were variations on the drawings you do see.  If these drawings make no sense to you or you are new here go explore over at my house of math. 

The other student was 11. But as he says, "once you get to 10, nobody is impressed you can do algebra."  I'm pretty sure there are still plenty of people who are impressed when all you see is symbols...but what these kids do is see the symbols and translate them into geometric figures that can be easily counted and manipulated in their heads or if need be with the aid of sketches on paper.

They have been factoring polynomials since they were little, just like these two.



Algebra is important. It is the greatest labor saving device ever devised by the mind of man and yet it is currently the most failed subject out there. If you work in retail or fast food, you won't need the algebra much unless you get into top management.  But if you are in the sciences or engineering or architecture you will use algebra EVERYDAY. How do I know? Because when ever I am around engineers or scientists or architects I ask them, "how often do you use the algebra?" And the answer is always: "EVERYDAY."

Little girls can do this just as easily as little boys by the way:


Third (and 4th) power algebra, negative expressions, integers and more are covered in depth on the advanced algebra page. You'll need a password though.

The lessons go like this...I give them the symbols for example x³ - 27 and they have to give me the drawings or just give me the factors...this is exceptionally effective for teaching much more than just factoring polynomials, it can also be used to teach remedial math.

Concept based teaching with base ten blocks works. I can show you how, Find me on facebook, twitter, tumblr, instagram etc.

If "advanced algebra" for your 8 year old is too much for you maybe you could start with subtraction...here is a FREE SUBTRACTION BOOK to get you started, soon stuff like you see here will be child's play.

3rd Power In His Head

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.




More algebra at CRHOM. If you want to see more "advanced" algebra click on the "advanced algebra" tab.

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...

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Simple 3rd Power Algebra With Dboyz

Happy New Year!

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.

More algebra at CRHOM.

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Algebra its many uses.

I have said it before, algebra is just generic math. It has many uses for teaching math at various levels. Here is the same exact problem with two very different students:

Note how it is nicely color coded but this is NOT sufficient, even the drawing is not sufficient many reading this may still not "get it" completely. I found as I traveled the USA that the only time we got 100% comprehension in workshops was when everybody had blocks or manipulatives. If you only have symbols or drawings some people get lost. They have to be able to get there hands on it. Now we can start off with manipulatives and then stop sing them over time but the concrete is ALWAYS where you start off. And one or two examples does not cut it it takes MANY before you can move to drawings only and then symbols only and this process often takes months and years.

Here is vid one with an older student:



Here is a much younger student doing the same exact problem notice it takes much longer and more patience with the younger child:



Note the difference in emphasis between the students. The older student's focus is the distributive theory of multiplication in preparation for problems that have negative coefficients. The younger student is learning to count up things that are SAME and as you can see because the x² are not the same shape even though they are color coded the same he has a bit of trouble and because the x are different color again he has a bit of trouble...but he can see what he is doing and it makes sense. Later more understanding can be added. Meantime adding up and counting in algebra isn't scary or hard and it's kind of fun this is enhanced because he knows older kids do this stuff and I tell him some of them find it scary and hard and he thinks I'm teasing him...but then they met some students that assured them that this isn't how they were taught in school and indeed algebra can be quite scary and hard.

Look for a future post where I go into a little more detail on adding layers of understanding. Hopefully I will remember to link it here.

Now some people comment that having the little kids do algebra shames the older students. Only if they have poor self esteem brought about by poor parenting skills. My students know that the point of showing them that little kids can do it is not to shame them but to emphasize that algebra is so EASY it is child's play. Even advanced algebra is no big deal, just more counting. And that if a little kid can get it they can get it. And so can YOU.

Algebra can be quite useful for teaching basic operations, that is addition, subtraction, multiplication AND division. As can be seen with the younger students they get experience with adding or combining like terms and counting as well as multiplication when the factors are positive and combining like terms that are negative and positive as well as multiplication when they have negative factors or coefficients. Further how do you teach multiplication without teaching http://www.crewtonramoneshouseofmath.com/division.html at the same time.

This is a lot more fun than memorizing multiplication tables or doing endless worksheets on addition or multiplication...yes they need practice and LOTS of repetition but don't turn it into drudgery. Be careful with worksheets. I tell my students as with sports you don't throw the ball a few times and become a pro you throw the ball a many times a day for many days and you still will need practice. That's why in textbooks you often see the problem sets labeled practice.

Here is another video with young students where I use algebra to teach counting skills like multiplication and addition:



And more "higher algebra" for the same purpose:



If you want to see the negatives you need a password.

Also people say I don't put enough posts up on my blog. That may be. But i can tell you that I post on my math page over on facebook everyday. If you want a steady stream of vids and math etc like this page:

http://www.facebook.com/Crewton.Ramone

Also you may want to do some searches with crewton ramone in them because there are now MANY videos showing how to use algebra to teach math. Too many teach algebra all by itself as a subject unto itself thus adding to the confusion by cutting the mathematics up into too many segments that seem unrelated to each other.

Algebra Is Counting and Not Really That Complex

Algebra is just generic math. We count stuff. That stuff has names. We can use it for lots of things. Problem solving. Honing our critical thinking skills. Learning to count. Simple stuff like that.

This nine year old can already count, so all I'm doing is giving him new stuff to count. That stuff has names, like "x to the third" and "x squared" and "x", you can also teach kids who are just learning the names of numbers these too: it's just more vocabulary. Mostly I find that the teachers themselves are the ones who have a bias against counting "x to the thirds" due to their own ill experiences with math, but really, to little kids it's just counting.

Little kids actually enjoy algebra because it's like learning with puzzles...the teacher can add more or less meaning as needed, depending on age and skill level.

This little snippet may look impressive but if you watch the whole thing which is on the "advanced" algebra page you can see it was easy and fun and it was a natural progression. The child even made up the problem himself.

I watched the long version of the video with a certain pair of students who were four and six and they made me stop the video so they could answer before the child in the video did. You can also do this with your students. I know three bucks is a lot to pay for a home made 22 minute video but you also get pdf's and access to literally 50+ more videos. Go crazy and spend 12 bucks and you get that access for a year and you'll see how much stuff I add weekly is worth a buck a month.

It will be a while before we move into negative expressions and we still have to play around quite a bit before we start setting these equal to zero and solving them and then move on to graphing them. One step at a time.

Also for homework you could take that last problem we did and draw it so that one side is x² + 2x + 1 and the other is 3x + 2. You'll get the same thing just in a different shaped rectangle. There's often more than one way to factor third and fourth degree problems.

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3rd Power Algebra? It's Just Complex Counting.

Here is an excerpt from a lesson where we played around with some third degree algebra made rectangles and counted the sides.

You can see from the vid we are in the drawing stage, but we still get out manipulatives so we can get a solid handle on what we are doing which is counting. We start off with all positive expressions and basically factor them. Then we move on to expressions with negative coefficients and do division too. When you are first starting out just doing one problem can take up 15 minutes or more...but then as the understanding increases it will go faster and faster.

The way to increase understanding is not to do the same thing over and over again just using symbols and changing the problems, the way to get them to "get it" is to start in the concrete move to drawings and increase the degree of difficulty gradually as concepts are understood and mastered. This also shows how important neat writing is...lol...sloppy symbols can make the difference between an A and a B, or worse a C and an F. Also, for those with a password you will see on the follow up vids on the Advanced Algebra Page. Password costs all of 3 bucks or $12 for a year pass. Buck a month is hard to beat.

You will note sometimes I say one thing and mean another in these vids when I read the problems and say "x-squared" instead of "x to the third" and so on. I watched one vid where I did it over and over again...but eventually I say it right. Painful for me to watch.

Problems like x³-4x²+x+6/x-2 take a long time just to say but they are quite instructive. They are also intimidating to some students who find them confusing and scary just because of all the symbols, but once we get into it and it becomes familiar, they see that actually it's pretty easy...just have to count and keep track. Here we are going over the positive ones:



As usual I forget to turn the camera on until we've done a lot of them. Remember always start in the concrete, use three period lessons and make sure drawing is part of the lesson too, don't just use symbols.

Here we see that base x also has place value and answer the question "why do they go where they go?"...you'll see this on the next video look for "P2 Crewton Ramone and Syd Play with the 3rd Powers" on the Advanced Algebra page. Scroll down to the part that covers 3rd and 4th power.

My friends in Africa have shown that the passwords are valuable enough to steal, and enjoy putting jibberish in the email list box to let me know they have gotten into the password pages without paying. So far the ISP's trace back to Ghana and Nigeria. It's good to know they have enough value to some that they are worth stealing. Passwords will change soon so put your name in when you get in, that way you can be kept up to date on the new password and the new stuff that's being added so you can get the most out of your password while it works. Also Next year prices will go up a little again as more and more people find out about how cool this is. Those who have passwords or have bought passwords will get special discounts on renewals. If you buy kits or products from me or take tutoring you automatically get a password gratis.

Be sure to find me on Facebook or follow along on Twitter. There's only one Crewton Ramone so searching out vids and pages should be pretty easy.