Flying Blind: Factoring by Grouping Without The Pictures

Here is a fast lesson where I show a 15 year old what her text book is trying to teach her when they talk about factoring by grouping.  You can see the distributive theory if you have mastered the concept, but if you haven't trying to take these third power polynomials apart and factor them can be intimidating. But once yo know how, it's easy and once you can actually SEE what you are doing it's easier still.

Here you see no pictures, which can make it very difficult to do the math, most students (and by most I mean 99%) like it when they can actually see what they are doing rather than just using rules and process to factor these problems. Most teachers and professors I show this to are absolutely STUNNED.

For problems like these you need a password. They are covered on password protected pages at CRHOM click on Advanced Algebra. (You need a password). There's plenty of 3rd power algebra on this blog and on my website for free though. The term advanced algebra is amusing when you see how easy it is.

Once you understand the concepts, bigger is just funner.

Here is a quick video I made covering basic factoring by grouping...staying positive, I put three short videos together and the last one shows what the picture looks like. We spent lots of time going over the pictures and suddenly the rules made sense. This is the second example and by the time we got here it wasn't very hard. If you look around on this blog you'll see an 11 year old doing this and more.
And of course little kids can get in on the act too, since all math is is counting once you know what to count and how to count it...here are 4 and 5 year olds playing with "advanced" algebra. For them they are basically learning how to build rectangles and add and do some simple multiplication. 3rd power algebra is a "by-product" of the lesson not the focus of the lesson.

More algebra at CRHOM.

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Algebra For Addends and Multiplication.

By now you have seen me use algebra for teaching more than just factoring to little kids.  Here we see the boys playing with one problem, using their critical thinking skills to make rectangles. They learn many math concepts in addition to addition and multiplication, they also get to practice writing their symbols which is a source of a great sense of accomplishment for youngsters. The 6 year old again much more impressed with his 5 and 2 than his ability to factor x² +  7x  + 10.  People don't think of algebra as one of the fun preschool activities but it is...if presented correctly.

I cannot stress enough that you must be patient and let them do it. Don't do it for them and don't give hints...just let them fool around with it. AFTER they have done the work talk about their discoveries and emphasize the fact that they weren't/aren't wrong just getting more information. Also when the older boy starts building his with the blocks on the wrong side he was building for you the viewer, being a silly adult I thought he was building it the "wrong" way. He wanted to make so "the people could understand it"...so he was going to try to build it up side down and backwards for your benefit. Next time I'll let him....



Here are the problems we did in order. At about 5 minutes a problem this is an hours worth of work. They had to get out the blocks, and form them into rectangles and count the sides. I talked about the partial products and made them point to them but that is ancillary to learning the addends and the multiplication of the factors.  I also talked about the distributive theory and what that meant but again the main lesson was about counting out x and figuring out how to make the numbers fit in the corner. 

x² +  2x  + 1.
x² +  3x  + 2.
x² +  4x  + 3.
x² +  4x  + 4.
x² +  5x  + 4.
x² +  5x  + 6.
x² +  6x  + 5.
x² +  6x  + 8.
x² +  6x  + 9.
x² +  7x  + 6.
x² +  7x  + 10.
x² +  7x  + 12.

The discoveries made here are too numerous to mention but a few I noted were, you can't build a 10 with 3's. 3 times x and x times 3 are the same thing, 5 twos and 2 fives are the same thing but it's easier to get out 2 fives than fooling around with all those twos...adding x's is easy...x² +  2x  + 1, x² +  4x  + 4 and x² +  6x  + 9 are square numbers. Again these are their discoveries I didn't tell them these things...

You may note that there are 45 addends and thus 45 problems you can build and master. Ending with
x² +  18x  + 81.  Soon  there will be worksheets on the Password Protected Pdfs Page with all of these problems on them. You should get comfortable with these and positive problems with two or more x² BEFORE you move on to negative expressions.

Here is the Password Protected PDF's Page. You need a password. Buy one, they are cheap.

Speaking of PDFs here is a one that covers the practice polynomials from x² +  2x  + 1 to
x² +  18x  + 81...

POLYNOMIAL PDF

You may want to down load this and use it as a guide. DO NOT do each problem in order. The natural inclination will be to start at the beginning and work your way up. START with x² +  3x  + 2. DO a few and and skip around, keep to the lower ones with the younger students who are just starting to skip count. In the first session I usually end around 7 or 8x...then next time do a few more (some of them the same as the first time...and then end with 10 or 11x...and then next time do a few more. This pdf represents several lessons even if the child is "older"...for more you may want to check out my Parent Teacher Training.

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Cabbage For Your Brain

The lowly cabbage is a super food.  Excellent for the beneficial flora in your stomach and also good for your brain.  Find out more about cabbage with the help of your search engine.

Gas however can be a problem, so if you let it ferment for a few days you get salted cabbage and many of the benefits of cabbage without the gas...there is nothing like raw cabbage though, packed with enzymes and phyto-nutrients...but really this couldn't be easier.  Sun, Salt, Cabbage a Sterilized Jar. Put the lid on loosely after you pack it with cabbage, so the gas can escape and unwanted stuff doesn't get in while you are waiting...not pictured a wooden spoon to pack it with and the sun that makes it all happen.

Factoring Positive 3rd Degree Polynomials

Factoring 3rd degree polynomials can be intimidating not to mention frustrating if all you have to go by is some rules in a textbook, but when you can see it it's EASY. This student is autistic and we just made a game of it once we had done a few examples and were familiar with the names of the pieces. Refer to the concepts page and use the 3 period lesson to make sure we know the names of the symbols and and what the symbols represent. The use of color markers helps but isn't necessary.

"Whoa! That's big," she exclaimed upon seeing the symbols.

Just count the sides and you're done.
"Well, that's EASY!" She laughs and writes it down. We do a few not pictured here and then she has a chance to make one up of her own. Of course she makes a big one:

The picture and the symbols are fast and easy to write. Much faster than drawing it, which is the point.

Now we make a trickier one with some square x²'s...

But of course that easy too because she can SEE it. Factoring polynomials is a snap, and kind of fun and builds the child's math confidence and self esteem.

Then we start making a game of it. I draw a picture give her the symbols and she has to make a drawing that matches mine. We have fun and a few laffs as she tries to figure it out...in no time she gets the hang of it. "My mind is warmed up now." I made some for her so it's only fair that she gets to make some for me.  This helps her with mastery and makes the game more fun.

So she reads me her drawing next:

I start out like this because of all the clues in the terms.

Factoring polynomials: pretty easy once you are used to it and have had some practice. One hour does not a pro make. Also it gets more complicated when the terms get negative, so lets get comfortable with the basic concepts and stay positive. Children under 5 can also do this! All we do is count.

Then the rest just falls together. A better explanation is given in the screencast below, because for some of you it can't be this easy so you need more than just pictures.

Then we move on to a little problem solving with hero zero and no fun get back to one...

Then a few problems with fractions which she says are harder and not as much fun as the algebra!!!

After a few problems she remembers what to do and why we have to make them SAME kind before we can add them. Again it's just rectangles and counting.

And there it is done.

Here is the screencast:


Here is a much better explanation of factoring by grouping on YouTube. The one on the sccreencast is verbal only and lame...the YouTube vid is an actual lesson with a 15 year old cheerleader who is freaking out because of all the symbols...

Find me on Facebook and be sure to dig around at the house of math and on this blog for more. There is and will be a lot more on this on the Advanced Algebra Page which is password protected; how to do negative expressions for example is well covered as are many other topics.. Passwords now cost 3 bucks for a month, and 12 for a year pass. And soon they will cost more than that.

Slightly More Advanced Problem Solving, 3 Variables.

We see Slightly More Advanced Systems Of Equations in this video...three variable problem solving is easy. The math concepts are simple. Use the information you have to find the information you don't have. Isolate a variable and the whole thing comes into focus. Hero zero and no fun get back to one...done.

Soon he will be doing systems of equations where all three equations have all three variables and he has to eliminate to solve them but we aren't there yet; however, this is where we are going:

3x + 2y + 4z = 9
-x + 3y + 2z = 1
4x - 5y - z = 3

For now he is doing problems like this:

L + T +3D = 8
T + 3D = 14
L + 2D = 0

There are a lot of intermediary steps between those two sets of problems. Steps that are ENTIRELY ABSENT in modern day text books. Now they give you three or four examples a few rules and some process and "off you go...and hey graph it while you're at it even though this is your first exposure to a "Z" axis."

Here are several vids strung together with my poor (but improving) editing skills:



For those pressed for time here is a short video covering just the one problem shown above:



To see how we started off and why I'm using L, T and D check out this post Simple Systems Of Equations.


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Playing With Nines

Here is a simple vid where two little boys learn about nines and multiplication using stuff they already know: addends and counting...

The older boy who BTW turns six today said upon review of the video..."I thought they were supposed to be hard."
"Really?"
"Yeah, but that was almost TOO easy."
"Yeah, too easy," chimes in the four year old...

Now it's just practice and more games and more play until the nines are part of their instant recall memory.

I was working with a 14 year old and discovered her multiplication was severely lacking and nines in particular were not there. This same lesson was used to similar effect: she found nines to be easy. I also showed her "one less make a ten."

In other words 3 x 9 is one less than three which is 2 and 3 needs seven to be ten so 27. 6 x 9 is one less than 6 which is 5 and six needs four to be ten so 54. OR you can just take 30 minus 3 or 60 minus 6...unless of course you suck at subtraction because they taught you to count backwards and you're 14 and still 100% dependent on fingers.



That day we also watched Multiplication Rock vids, which upon reflection is where he might have gotten the idea that nines were supposed to be hard...lol...and counted by threes and sixes which they are getting very good at, and talked a little about what numbers they heard that were the same on three (read 3, 6 and 9) multiplication tables.

Those two boys are never going to have the problems with math that I commonly make 40.00 an hour to "fix" with children who are 13 thru 18. Further by the time they are 7 or 8 there will be very little math that high school kids can do that they can't do.