Here you will see students as young as 4 and 5 years old doing algebra and "advanced" math, without ever knowing it's supposed to be hard.
You are invited to learn how to use this method...

Monday, June 25, 2012

Long Division With Base Ten Blocks

UPDATE: Here is yet another post on division due to the fact I see people showing how to use base ten blocks that are less effective than they could be. This one includes the #1 video hit for division with base ten blocks.

UPDATE: Divinely Dandy non Difficult Division is DONE! (See below.)

There is now an in-depth post at The House Of Math about Long Division With Base Ten Blocks.

Long Division. Not well loved among students. It's EASY if you understand the concepts. And here we will add a little about square roots too....just for fun. Just because we can. Just because you should cross teach whenever you get the chance so your students can see how it all fits together into one language.

Start simple and work your way up. Remember degree of difficulty...and the other 5 basic concepts. We are fooling around with rectangles and counting. Some call this multiplication and subtraction and others call it division. Bottom line: we are counting. And one more time computation is not mathematics. Computation is how we DO mathematics. Today I found myself with an older student and I told him I don't care about the computation at the moment...I want you to understand the concepts...we'll do the computation later...we trying to find the area of a shaded region with a square and a circle inscribed where the only information was the radius of the circle. But step one was knowing I had to do subtraction. But I digress.

Observe. Here is a very simple rectangle:

The question arises why don't you use this pattern (above) instead of this pattern:

Well, because we want to keep track of subtraction. Long Division With Base Ten Blocks allows us to see what we are doing.

long division with base ten blocks, math manipulatives, division

We are counting a rectangle that is 4 across and contains 12.  4 is contained in 12, 3 times. The number inside the rectangle is the dividend the thing being divided, and the number outside (the 4) is the divisor and the 3 is the quotient.  We are humans we name everything. "I think I'll call this place Golgatha and move on."  Kids have a hard time reading division because of this. This is solved quite simply by saying 4 is contained in 12 how many times? We read left to right so saying 12 divided by 4 is confusing. Besides we denote that this way: 12 ÷ 4 = 3.

Dividend. Divisor. Quotient.  Simple no argument.  You be surprised at how many people can't figure out the syntax of Multiplier, Multiplicand, Product or think it doesn't matter. To little kids you are right it doesn't matter and you will note you do not hear these math terms in the video below because at the moment this is extraneous information. Later when they are comfortable with the concepts we can start naming names. Meantime we do Long Division With Base Ten Blocks so that it is readily understandable and visually obvious. Once we start getting it down they will be able to do long division in their heads with no blocks and no paper and pencil either. This is called mastery.

Note the lovely blend of symbol and manipulatives. It completely makes sense to he who is but 5.
Now because this is a demo video more or less and because my students are familiar with the blocks we jumped a bunch of steps to a much bigger problem. YOU would NOT do this with your students unless they were quite familiar with the blocks and even then. Do many smaller ones and have them might also do a few where they have to give you parameter just for fun.  Many. More than three.

Here we are going to count how many times 12 is contained in 132. The obvious answer is 11. These are easy static problems. I used train teachers all the time and the first thing out of their mouths after their initial excitement subsided was how do do a problem like 7 is contained in 132. Pat answer: "you don't." You do lots of easy ones where it works out perfectly and then a few where we have remainders like 4 is contained in 13 how many times and 12 is contained in 133 how many times BEFORE you even think of moving to dynamic problems. By then they have the concept and they realize paper and pencil is MUCH FASTER than playing with blocks. Besides that's all the algorithms do is make counting fast. But long division is a bitch if you don't understand the basic concepts and you can't multiply.  Further let them figure out the algorythim for themselves as you direct their discovery in a math rich environment.

So now lets get even bigger and do one that just happens to be square. AGAIN, YOU would NOT do this until you had done many smaller ones and worked your way up here.

But I am going to illustrate and explain each step.  You could do this with 12 is contained in 132 or 13 is contained in 156 etc.
Because Mrs. Irma Hardbottom would accuse your little genius child of cheating if all he they did was write 19 and be done with it, we have to show our work which is again why we pattern with the blocks this way. Across and down.
And you can see other videos on Long Division With Base Ten Blocks where I talk about Hiram the Ant and use little men or animals or dinosaurs to walk along the edges and count. But you should point out that ten 19's are 190 and that one is one 10. Don't get confused with the edges, we are talking about the distance from one side to the other. 
Once we take 190 from 361 we had to do a little work to figure out what was left on this problem. After some figuring they counted 171 which with the blocks was 90 and 81 which was 19, 9 times. They were counting the blocks NOT doing subtraction, if they did subtraction it might have been even easier because all they would have to do is take 0 from 1 and get 0, add 1 ten to 6 tens to get 7 tens (see vids on subtraction)  leaving us with 2 hundreds because we had to take the 9 tens out of one of the hundreds which is how we end up adding 1 ten to the 6 tens and then taking 1 hundred from 2 hundred is EASY.

It's also easy to see that the square root of 361 is 19 and when it comes to notation this is much easier.
So we did 20 just for fun.
And then we were basically done. Here is all of the above in one fairly concise video:

Anyhow go check out the division page at the House of Math for a little more...long division shouldn't be hard. Look for another post about more advanced problems where the rectangles are more dynamic...these are best drawn or done with symbols AFTER the concepts are mastered.

Here is another GREAT post on long division that even has scans of pages from the smiley face books.

People get excited and ask me what they should get when it comes to blocks and stuff...go here for the simple answer.

Here are some more division worksheets (you need a password) and the video on that page teaches you how to use them. YOU could use that vid as a primer and instead of using pencil and paper use the BLOCKS.

Learn to use your base ten blocks.

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Divinely Dandy Non Difficult Division

 Get Divinely Dandy Non Difficult Division for just $19.99.  This book will show you everything you learned here and MORE laid out step by step with links to videos and pages that give simple concise explanations for how to use the rectangle to organize thought,  how to introduce division concepts at a very young age, and how to make fun while you are doing it.  I guarantee that video alone will expand your thinking when it comes to division and math.

Watch the video on the Preview and Purchase page that gives you a page by page over view of the PDF so you can "try before you buy", see exactly what you are getting and be confident it will be money well spent. 

"Can you do Division? Divide a loaf by a knife - what's the answer to that?" ~Lewis Carroll, Through the Looking Glass 

 “We divided ourselves among caste, creed, culture and countries but what is undivided remains most valuable: a mere smile and the love.” ~Santosh Kalwar

PDF Prices

The Curious Counter's Compendium.

math manipulative book, Base Ten Block Book

Get this book if you have children 7 and under...find out more about it, and a look inside here.  You can get it without a password for just $2.99

"Great book for teaching how to use the blocks! Colorful, clear pictures and cute rhymes make the book fun to read and play around with. We printed the book out, and my 5yo loves how many of the block pictures are big enough to put his blocks directly on top of the pictures. The text plays fast and loose with niceties like punctuation, but is engaging when read aloud.

Most of the book focuses on playing with addition facts up to 10, which gives a solid foundation. But it also delves briefly into such topics as square roots, place value, addition of multi-digit numbers, and a glimpse at multiplication. And in true Crewton Ramone fashion, problem solving with 'x' (basic algebra) is sprinkled throughout. A great intro to playing with math." ~CS, GA.


  1. Wow what an amazing explanation to teach and learn long division. Solid practical examples are in front of the kids. 95% of kids at our tutoring center claimed advancement in their division skills after watching the videos on this site and using base ten blocks by themselves. Most of them can solve all the long division practice sheets printed online or drills from their text books.

    Great Thanks for loading such a nice math resource online
    Guru Angad Educational

    1. LOL. Those worksheets show exactly how NOT to do it. Before you spam my page you should read the posts and the division page at the house of math which lambasts those very type of worksheets. I will leave you post because I know you want the link so your crappy page ranks higher and I am in favor of efforts to improve math performance even when they are sometimes lame and ineffectual.

  2. I have consumed myself for days now with reading, watching, and thinking about all of the information you have shared on your website (which I am no where near done with reading, watching, nor thinking.) I quickly realized I am not as clever as I'd like to be with "counting." I am hoping by 'rewiring' my brain to learn this method you are sharing that I can help in spreading the word.

    I like how you related it to learning a language because math is a language that must be communicated with careful wording.

    I am personally still struggling with the "complete the square" pictoral part. :( I can do it (sometimes), but it's slow. I would like to be able to "master" it so that I can teach someone else to "master" it and other skills based on the base 10 blocks. Although, I admittedly have not done it with the manipulatives and went straight to attempting it with pictures. Which my issue.

    Question: Any possibilities of the books becoming pdf files to print?

    1. You NEED blocks that will clear things up quickly and yes, check it out! 3 ears later pdfs are being kicked out...and you can get them.