The Distributive Theory of Multiplication.

Is visually obvious when you use base ten blocks.

Base ten blocks make understanding he Distributive Theory of Multiplication easy. My students have no issue with it at all. Obviously.

Do it get easier than that? The Distributive Theory of Multiplication is almost taken for granted when you use base ten blocks or simple diagrams or drawings.  Four 5 times, is the same as four three times, plus four two times.  If they would have explained it to you this way you might have gotten a better grade in algebra.

i would prefer to read it left to right, three four times, plus two four times is five four times.  You could get out five yellow fours or you could get out four 3s and four 2s. Either way you can see it. 

Build it and stack the blocks on top of each other so there can be no doubt. The concept is available to kinesthetic learners at a very young age.  It is absolutely no big deal. You are pointing out the obvious. 

If we rotate now we can see 4 x 5 = 4 x 2 + 4 x 3 and the fours are stacked on top of each other with a little break in between.  Visually obvious, nothing confusing. Any little boy or girl can SEE that. Distributive Theory is super easy. 


3 x 4 + 2 x 4 = 5 x 4 

(4 x 2) + (4 x 3) = 4 x 5

The drawing next to the blocks...uncomplicated. We can add some symbols to describe the blocks .

12
+8

Lets get bigger because bigger is funner.  

In our consistent model the multiplicand comes first and then the multiplyer. Across and up for multiplication, across and down for division.  

Complex numbers are easy too. Binomials are clearly two kinds of numbers the "x" kind and the units kind. Remember, numbers have two parts the "how many" part and the "what kind" part. 

3(x + 3) = 3x + 9

On the left you get the dimensions, 3 by (x + 3).

On the right you have a rectangle over a rectangle, 3x + 9. It is visually obvious that the three is multiplied by the x and the three. And again we just stacked them on top of each other. You can also clearly see 9 is a square number. 

You can SEE the distribution.  You can see the distributive theory of multiplication is EASY. 

Let's get bigger:

(x + 1)²


(x + 1)(x + 1) = x² + x + x + 2 = x² + 2x + 2

Here it is in color. One red block, two blue blocks, one green unit block. Simple. We made a square. You can see  the dreaded F.O.I.L method.  We have to count four things but two of the things are same so we can count them together, x and another x is 2x.  One big red we call that x squared, x². The blue ones are called x, one times x and x times one, and he green one is one times one, 

 if you want to get complicated about it. x² + 2x + 2 which is what we get when we multiply

(x + 1) by (x + 1)

using the Distributive theory Of Multiplication.  Base ten blocks make this concept child's play.  You can also see that this is a polynomial, poly more than one, many, nomial: number. It has three kinds of numbers.  

x² + 2x + 2.

And i think now you can SEE why they are called quadratics. Four parts.

You want to see third power? Go get M0AR training.

Bonus: (Ex)Sample Lesson

Example is the lesson. Click the image for video but his could just as easily be the next thing you do after a lesson on distribution which may have started off as a lesson on counting by fours. FLOW. One thing leads to anoher...let's build 12s, which are the same as the 3 fours, and build 8s...which was 2 fours. And we talked about distribution.  If the students are older you could keep getting bigger with distribution and then *factor a bunch of polynomials or if they are younger you could move on to building stuff and counting it.

You could build and eight pyramid and a twelve pyramid, why? Because back in the beginning we had 12 + 8...so why not play with eights? FLOW. No need to bore them with a long confusing lesson on distributive rules using symbols only with no pictures. the way i was taught in high school.  Five minutes or less they "get it", now do something else.  They can understand distribution but still need to practice addends and counting by 12s out to 240. You can SEE the addends, 9 + 3, 8 + 4, 7 + 5, 6 + 6, and cooperating one child puts a 7, the other puts a 5, or they grab one addend in each hand and add it. The bigger the pyramid the funner it is. Now we compound the lesson by counting the 12s we made up of addends...12, 24, 36...thats a lot of 12s. But they'll see an easy pattern.  Just flow naturally from one thing to another. 

You could also work a lesson on prime factors in 4 x 5 = 2 x 2 x 5 by making orange squares out of your yellow fours. 

There's 12s, now how about 8's, build a pyramid with combinations for 8 and the count eights and build stuff with eights. Then we could watch a couple of multiplication rock videos and play toss across. FLOW. Then we could build rectangles out of 12s, ten 12s is 120, twenty 12s is 240 and we built lots more 12s than that. Counting high is fun too.  You could talk about distribution again (10 + 2)(10 + 10)  and you could build that with two red ones and four blue ones. And you could stack the 12s you made out of addends from the pyramid on top of the hundreds and tens. Like icing on a cake. *Then we could play a game where i make the symbols and they have to make a rectangle out of the blocks. (Factoring.) These four problems:

x² + 12x + 36
x² + 12x + 35
x² + 12x + 32
x² + 12x + 27

I give them the rectangle they give me the sides. 

Why? Because those are all the addends for 12 when factored.  More will get done faster in group learning situations. They often race each other, or help each other as he case may be. he playing field is leveled. Sex and age should be no barrier to learning math when you know how to do it right. Little boys and girls get algebra right along with trig and multiplication. But there are so many things you can do in an hour.  Notice i turn the camera off so we can practice subtraction. Once you get it you have to practice. Make a game of it.  If you can't understand how we can introduce algebra to very young pupils get M0AR Training.

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As it turns out people who i thought were using base ten blocks to teach algebra AREN'T. 
https://www.crewtonramoneshouseofmath.com/Algebra-with-base-10-blocks.html