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.

Homeschool Math. Base Ten Block Method Founded In Montessori.

"The teacher must take her materials from the school, and her principles from what she has learnt; and then she must face practically, for herself, the question of this recall. Only her own intelligence can solve the problem, which will be different in every individual case." ~Maria Montessori

I remember Jerry being quite amused that I wanted to be a master trainer yet had no concept or knowledge of Maria's work. I went home to escape winter and I had been reading "The Men of Mathematics" to try and figure out what the hell these guys were thinking, Newton did not develop the calculus to torture high school seniors, and college students. I also read "Innumeracy" by Paulos. He was surprised by this, so he took me down into his basement where he had a library set up and directed me to READ...particularly works by Mario and Maria. It was a large collection of books; I have an affinity for them, so I was quite content to spend hours down there, I found a book on Vedic mathematics, quite a few classics (Socrates, Plato, Aristotle etc) and many more. Jerry was content to have me down there because while I was down there I wasn't bothering him, and occasionally I'd come up with a topic of conversation "worth talking about."

Try to remember my lineage is direct. Me, Jerry. Mario. Maria. I don't recall Jerry (or Maria or Mario) talking about a standard curriculum that was to be applied to every child. Quite the opposite. The teacher was to be trained in such manner that each child's individual interests could be tapped, nurtured and developed. Discovery was directed by the teacher with regard to the child's individual interests.

Therefore lessons were developed for each class and each child in the class on subject matter at hand, in this case math. There was no cookie cutter approach. Even the books he developed were made with the understanding that the student picked where to start and the order. All books were to be completed but how and in what order was up to the individual student. And yes it's a hell of a lot more work, AND the education is far more effective and superior to what currently passes for education in the public school system of the USA. This year's class is not the same as last years class. How is it possible that the lessons would be the same?

"A teacher, by his passive attitude, removes from the children the obstacle that is created by his own activity and authority.". ~Maria Montessori

For example, today we are going to learn trigonometry concepts. The teacher introduces the CONCEPTS then directs the student to discover more and thru socratic method the child discovers axiom and postulate...and the formulas, they aren't hit over the head with them. Once concepts are understood mastery can be attained. Mastery is demonstrated by being able to make up and solve one's own problems. Again: once they understand the concepts all we can do is change the numbers.

"Education is a natural process carried out by the human individual, and is acquired not by listening to words, but by experiences in the environment." ~Maria Montessori

Your job is to create a math rich environment. Place the child in a math rich environment and they will learn math, in their own way and in their own time. This is the education of individuals in groups...how can the lessons be the same, in the same order at the same time. If you were educated in a public school yea verily this is completely foreign to you as everybody pledges allegiance and then opens their text book to page 123, or does the same worksheet in preparation for the standardized test. BTW it has become clear that homeschooled and Montessori students do incredibly well on standardized tests, because again, once the concepts are mastered all the givers of the test can do is change the numbers trying to ascertain if the math is understood or not. This has been circumvented by students and teachers who memorize answers to the test questions, rather than understanding the subject matter well enough to answer any questions. Once they understand addition they can add any two numbers. The principle is the same with regard to "advanced mathematics". The practice is another thing entirely: memorize long enough to take the test.

"Our care of the children should be governed not by the desire to 'make them learn things', but by the endeavor always to keep burning within them the light which is called intelligence." ~Maria Montessori

"The essential thing is to arouse such an interest that it engages the child's whole personality." ~Maria Montessori

Once interest has been pricked discipline and concentration flow as naturally as water downhill, time stands still, and learning takes place.

"The first essential for the child's development is concentration. The child who concentrates is immensely happy." ~Maria Montessori

"Discipline must come through liberty. . . . We do not consider an individual disciplined only when he has been rendered as artificially silent as a mute and as immovable as a paralytic. He is an individual annihilated, not disciplined." ~Maria Montessori

"Teaching is a sacred trust." ~Joyce A. Rogers.

Base Ten Blocks For Fractions Success

Base ten blocks make fractions much easier to understand. The Fractions page on my website is in serious need of an over haul. Eventually I will get to it. Meantime there Are posts like this.

Here we packed more into the hour (but not too much more from the stand point of CONCEPTS), going over the same concepts using fractions tiles and multi-tens, and drawing them with symbols and practicing problems adding up to less than one. Also did a couple subtraction problems. Then ended with a few that were one and some more...next lesson will contain more symbols less blocks and more on improper fractions. All of this and all we learned was adding and subtracting fractions. Multiplication and division of fractions will come another da, and they are just as easy to present and understand using manipulatives.

Please read this from the POV of how you would present it to your students or kids, and note the common stumbling blocks to be avoided. Base ten blocks and manipulatives allow you to avoid the misconceptions in the first place. Starting with the most common misconception, that some how the rules have changed when we get to fractions. They haven't: we can only combine same kind with addition or subtraction.

I have this one kid I would really like to video tape but I don't want to embarrass him, because he consistently shows a basic lack of understanding of CONCEPTS. This is because he has never been exposed to them.

Asked him to add 1/2 +1/3

His answer 1/5. "Because if the tops are the same you just leave them alone and add the bottoms...or is it the other way around?"

When I showed it was 5/6 using manipulatives there was a lot of "ohh!" and "OK" and "I get it" so we went over the 5 basic concepts focusing on they have to be SAME before we can add them...and because he is still lacking in multiplication skills, I used fractions manipulatives to help him practice multiplication (and addition).  We could skip count this by 3's or by 4's...put the next over lay on and then we will be counting 5's...etc...you don't have to know your multiplication tables in order to do fractions. Use the fractions to help learn multiplication.

I also asked him during the course of adding fractions which was bigger 1/4 or 1/3...of course he said 1/4.

Visually Obvious.

Then before I showed him any of the fractions tiles and asked how many to make a whole one if I have 1/3...he said 3 and how many to make a whole one if I have 4ths?...4 so four is more than three. TRUE. But the more pieces you need the smaller each piece is...puzzled look. But as soon as I get the factions pieces out one look and it's VISUALLY OBVIOUS that 1/4 is the smaller one. "OH!" This child is a teenager.

Not stupid, just never been exposed to math concepts and manipulatives just rules, formula and stuff to memorize long enough to take a test and forget. He is a Classic American Math Student. Had he never met me he could probably have a bright career in climate "science."

We talked about 1/5 and 1/5...how many? 2/5. EASY because they are same. Once we make them the same kind we can add them. x + x is easy because they are same kind...you get 2x. One of something plus another one of the same thing gives you two of something.  Easy.  But if they aren't same kind we better make them same kind so we can add them. Then we talked about the numerator and the denominator. He had forgotten the name of "the bottom one"...de nom, de name, da kind of it...but he did remember the numerator which tells you how numerous the kind is...

"What kind of math is this?"

"Adding and subtracting fractions. About fourth grade level in most public schools..."

"But for your kids it's like first grade."

"Right."

We did several more problems until he got the hang of it using manipulatives...THEN I moved to drawings...and we stopped there. Tonight his sub-conscious gets a chance to put some things together and then we will work on going from blocks to drawings to symbols. Because he is a teenager I can go much more rapidly than if he were 6 or 7.

Then we talked about improper fractions and did about three of those just to give him a taste, ending with...5/6 + 4/9...could you think of a common denominator?

Symbols, Drawings and Blocks

"NO."

So he plays with his sliders on his multi-tens blocks and gets the answer. But still couldn't convert it to 1 and 5/18ths on his own...that's a concept for another day for now one and some more is good enough. Like remembering numbers if you try to remember too many at a time some of them are going to get lost so just a few concepts at a time is best.

Right now the concept of SAME and doing the multiplication is enough. Next I'll add more to that...which will further cement the foundational concepts of making them same before we add them. The rules didn't change we have to understand equivalent fractions and common denominators. With young kids first we play with the tiles then we play with the multi-tens, having multiple lessons in both. With this student he got it all in one lesson because he has more experience and is older and can handle it...if he was 5 we would have done less. Just the one concept of making them same and then practicing adding them using tiles; then maybe some drawing and then some symbols. That might take THREE HOURS. Then on a different day a lesson with multi tens...again another hour or two. Five or six separate classes, spaced out over time. 

Using "sliders" is the fun way to do it. Over all kids like using sliders more. If you don't have multi-tens don't worry you can just use base ten blocks out of your combo kit and get them out a pair at a time,  kids like this too and it's especially good for the hands on experience, it takes longer though which isn't a problem if you are homeschooling. Just for fun I uploaded this to blogger so it's a video you won't find anywhere else.

Can you also see how easy it is to talk about equivalent fractions and learn skip counting while we do?

For those with dyslexic students not that the image is reversed because I used my lap top's camera for all of these pics, but I am able to "flip' the photos but not the video. And it doesn't matter...base ten blocks level the playing field.

Want some Fractions Worksheets made for base ten blocks?

Percentages usually come next after fraction. Showing how to do percentages with base ten blocks is EZ.

BTW if you want to learn how to use the blocks to teach your kids I have SEVEN Hours up so far, but you need a special PT password. Go here for Parent Teacher Training. $15 bucks such a deal....

“A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction.” ~Leo Tolstoy

Teaching Second Graders With Manipulatives.

Crewton Ramone in the classroom! Here are a few videos and shots where second graders use manipulatives and do math that is supposedly beyond their years. These videos should give you a good idea how to incorporate the blocks into the classroom. Manipulatives make math fun and easy. They make math concepts visually obvious and you can get a lot more math done in a shorter amount of time once the students become familiar with the base ten blocks. Look how many addends get made when they build a tower. It's much better than writing them down over and over again--and more effective. Be careful with worksheets.

But don't be fooled into thinking all these kids can do with manipulatives is add and subtract and do some place value...they can learn all kinds of math concepts. The basic operations (addition subtraction, multiplication, division) and much, much more. Use your search engine I have tons of vids and posts up about math from counting thru high school math.

I often forget to turn my camera on during the lessons. You can look at the board and tell we covered a lot of math concepts, from square numbers to square roots to multiplying by nine. Here is a post lesson short:



Here they are doing second grade math, just learning their addends. I can't stress enough how important it is to learn and master all 45 addends.  We have broken into teams and they are racing to build pyramids of  10's and 13's...or you could use worksheets and get less done in more time. Note how we go from addends to factoring polynomials without skipping a beat. I expect you can see how it's all one lesson. Even when we are doing algebra, the algebra is just along for the ride what we are focused on at the second grade level is multiplication and addends and having fun.



You can hear the fun.  Building towers is a favorite fun second grade math activity. I use it as a reward. They practice addends and have fun doing it.  But they can do much more than just addends, they can do algebra, understand problem solving fool around with Pythagoras' math concepts regarding triangles and much more. As you can see addends in algebra go together all in one breath as it were, factoring cements the addends and gives them some multiplication and factoring of numbers and algebraic expressions.

Second Graders Factoring x² + 9x +18 with EASE!


This video is unlisted and won't come up in searches. It is on the sample lessons page along with a whole lot more. Gives you a little taste of the stuff you can get with a password. The password protected pages have hours and hours of vids. Here is a sample addends lesson with very young students.

Second Graders Tell ME Pythagorean Theorem

Breaking them into teams makes the whole thing much more manageable, creates a little competition but also lets kids know as they are working on the problems and other teams jump up and yell "done!" that they aren't impossible and that if their peers can do it, they can do it. It also adds a little excitement.

This little group of girls is working together trying to figure out what fits in the corner...not only do they get math skills but they learn team work and cooperation. I can't tell you how many times the kids who are supposedly the top of the class in math end up "losing" to other kids who are not as good at math as they are (my own son included) because they couldn't get their group to work as a team. And by "losing" I mean finishing after the other "winning" group, sometimes by mere seconds.

Keep it fun and keep it moving. Just play. Play is very important...it makes learning easier and faster.


I have several posts showing the use of base ten blocks in classroom situations plug Crewton Ramone In the Classroom into your favorite search engine and see what comes back...besides these:

http://crewtonramoneshouseofmath.blogspot.com/2013/04/crewton-ramone-in-classroom.html

http://crewtonramoneshouseofmath.blogspot.com/2013/05/crewton-ramone-using-base-ten-blocks-in.html

And now you know why in the world of Mortensen math this is no big deal, astonishing to some but not me or other who have spent the time to learn to use this effective method for teaching.

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Don't forget to find me on FB.

Simple Math In Chinese Using Manipulatives With DBoyz.

What's annoying to me is a lot of the kids their age can barelyadd numbers in English. I had a interesting conversation with a Chinese teacher who pointed out that yes there may be as many honor students in China as the USA has students but they also have as many failing students as we have students as well.



"Only the smart one comes to USA, no failing students get out..."

I had a similar conversation with a friend who is from Hong Kong. She did "okay" in math but a lot of her friends failed...and what did they fail? Algebra. Base ten blocks make math easy to see in any language. They are quite useful with the deaf and blind too.

Practice and patience is pretty much all it takes to learn another language...and like Chinese they aren't going to learn it all at once. It's the same with math...you didn't get bad at math in one day....you won't get good at math in one day.  Chinese, English and Spanish with those three Languages you can communicate with most of the inhabitants on this planet. They got English by default, Chinese is proving to be a challenge but it's FUN and after Chinese Spanish will seem like a breeze. I was fortunate to go to a high school where I got Latin,  5 years of Spanish and a year of French JFSAG. But no Asian languages were offered...helping them learn Spanish will be MUCH easier than helping them learn Chinese. And Yes there are plans to get some of the Math Materials translated into Spanish. Right now the house of math is mono-lingual.

Anyway we are just having fun playing around with math in Chinese. They still have lots of work to do when it comes to writing symbols...lots of people have already noted that their stroke order is incorrect. Considering their age at that they only get an hour a week I'd say they are doing pretty well.



There is always room for impoverishment, and I have no doubt that they will get better as they progress. Right now they are finding fun that they can write a little bit in "code." I am told they need to master about 3,000 symbols...numbers are pretty straight forward in Chinese. Very simple rules. More uniform than English.  Eleven is ten and one literally in Chinese...twelve is ten and two. Twenty is two tens...lol...in English we have a word for two tens: twenty.

The Mathematics is a Universal Language, learning some in Chinese is fun and will pay dividends in the future. I was going to attempt to do some simple math in Pinyin here but it's too much work...I think I will let the videos do the talking...and people who speak Chinese can cringe all they want.

Trying to write the inflections for even simple stuff like jia (add or plus) jian (subtract or minus) would be a little more effort than I want to put into a simple post like this. Eventually, I think I will discover free software that does some translating that I can just cut and paste...I'm sure it exists I just don't know where at the moment.

The Kit shown is called the numeral identification  kit, or Number ID for short. It is great for REALLY young kids. like two and a half...if they are learning to talk they can learn the names of numbers just like they learn other words. It's just vocabulary. The how many part can be learned later and the fact that the six can be made up of a combination of other numbers can be learned later too.

Use the basic concepts and teaching techniques like the three period lesson and degree of difficulty to make the learning easier and fun...also don't forget to play, they learned a lot of Chinese playing Chutes and Ladders in Chinese and you can make up other games as well even if they aren't learning Chinese. Games and Play: that's HOW they learn.

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More Fun With Addends And Symbols For Little Kids

More from the make math fun file. Two brothers playing and learning together.

Here is a clip from a session with a couple of little boys who are just getting the hang of addends. One is 5 and has had very little exposure to math concepts or numbers and the other is older but has a few 'challenges' but is eager to learn.

The basic concept here is numbers are made out of other numbers and we also get some addends practice by solving for x (which is of course "the trick") where they figure out what's under the cloth...listen to the excitement and the laughter. IT'S FUN.



Math can be fun if you let it.

Also note that like a lot of kids the one child is still not sure of the symbols and what they mean so we do a quick digression into what the symbols mean but then return to the fun part where we are basically playing what's under the cup but have a cloth instead. It's not magic it's math. The younger boy still has to count because he doesn't know his addends. The older boy has more experience and it builds his confidence as he gets answers right.

They both laugh and have fun and although there is always competition and sibling rivalry we use it to best effect as motivation, and we keep it lively, light and fun.

This is a fun way teach addition using addends. As the younger boy gets some practice he won't have to count one by one he'll be able to use the addend to make it faster.

At the end the older boy has made a problem for me...that's fun too. It also tells you if they understand the problems well enough to create one. It helps them attain mastery to make up their own problems and is an extremely effective teaching tool. The children are placed in a math rich environment where they can not fail, and then encouraged to do some self directed learning. Note how we can see the associative property of addition but we never even mention it, but when we do it will be easy to understand because they have SEEN it already.

Obviously I don't have permission to show their faces, you are forewarned, starting Jan 1, 2013 those who don't give permission to use their kids in vids and on the websites are going to get 86'd as 2013 will be all work on the web all the time. My one on one tutoring days are drawing to a close. Will be focusing on groups and if it is solo then the sessions will be used to train others too via youtube vids and web pages and maybe even video products.


http://www.crewtonramoneshouseofmath.com/

Yet More Manipulative Multiplication.

Make learning multiplication fun and easy playing with base ten blocks.

How to get multiplication mastered. You will have practice. There is no way around it. But you can make this practice fun. And you can make it easier by using some basic memory tools. The concept of a peg. And working in a more natural progression that complements the knowledge as it is acquired instead of just a jumble of facts in ordinal order.

Screencast with boys


Starting with Square Numbers and Pairs a lot of the multiplication can be understood. Then it is easy to add another for three's and of course 10's and 5's are easy so then it's easy to add one more or one less. For example fours can be thought of as one less than what ever number you were counting by fives...is of course four so 5 x 6 is thirty and one less six is 30 - 6 = 4 x 6 which is 24. Etc. This needs to be re-cut because the fan messed with the microphone but I STILL haven't had time.

Multiplication


You can use this same thinking with 9's and you can go up AND down with the square numbers...6 x 6 is 36 6 x 5 is 30 one less six and 6 x 7 is 36 + 6 which is 42 and the way you get there easily is with wanna be a ten, six needs 4 to be ten so it takes the 4 out of the six and we get another ten with two left over for 42...4 tens and two. Here is a screencast where I use addends to make multiplication easy for an Autistic student. If it works for her it will probably work for you.  Multiplication by nine is EASY btw. So easy FIRST graders can do it. Click that link and watch the second vid. More explanation:

Using squares as a peg


Multiplication is THE milestone in mathematics and should be taught early and mastered so that all the rest of the math becomes easy...

There are books and worksheets that can help but more than anything you want to PLAY and have fun not beat them to death with math facts that make them hate math and turn them off at an early age. Play with blocks, sing songs do a little each day starting when they are very young and by the time they are 6 or seven most of it will be out of the way. Waiting until 4th grade is INSANITY.

Here is a web page of the books Mortensen uses. The facts mastery books are simply more practice and to be used to reinforce the PLAY you do, which should amount to much more time spent than time you spend on the books.

Also do some web searches for "Crewton Ramone Multiplication"...there are tons of pages and videos on this extremely important concept.  And of course here is the Multiplication page at the House Of Math.

I strongly recommend you follow a lot of the links on this page. I know it's a lot but I can't stress how important multiplication is for making "the rest" of math easy and understandable because it allows you to see patterns and count very quickly. Then search "Crewton Ramone Multiplication" and see the plethora of pages and vids I've made.

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.

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 count...you 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

CCCCC $19.99 USD SSS Prinable $9.99 USD DDnDD $19.99 USD PDF Page Only $39.99 USD

The Curious Counter's Compendium.

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.

TERC = Lame and ineffectual way to teach math.

Here is an interesting thread from Facebook that I got involved with briefly you might enjoy reading it.  

DJ: wrote: I found this woman several years ago when looking for help retaining algebra. Math Education: An Inconvenient Truth M.J. McDermott is speaking about the current state of math education, as a private citizen . KCPQ does not endorse this video.

 

KB: This is fascinating. How could any group of mathematics teachers agree to the TERC Investigations method? Cluster problems = cluster f***. No one would as they grew up even bother to calculate a problem. Can you imagine going to the grocery store and trying to figure out which size can of tomatoes was the better buy in your head? Hahahaha I loved the part where the teacher's manual says that it's just too hard to teach kids how to do division and that they're better off really using a calculator because it would take too long to teach them! OMG Have you ever seen this Crewton?  

Crewton Ramone: Yep. Deliberate dumbing down. Notice her explanations of the standard algorithms are not exactly sterling, nor does she make it clear what is going on...but she is correct that TERC sucks ass. (Comment got 2 likes)  

KB: Hey at least she spoke out. Besides she's not the genius you are. She's just the weather lady. *lmao*  

DJ I tried to find this since. This is exactly what happened to me. I hate them.

 KR: this video alone should convince people to homeschool their kids.  

SSK: Ok. My daughter's school adopted this "Investigations" math program this year and I have to say I hate it. I also have friends who teach at the school and they hate it as well. When you hear the kids try and explain it, or read the manual, you are thinking WTF? At the beginning of the year, I and my son were having to re-teach the concepts at home. But as everyone has discovered how confusing it is, and so many teachers and parents have complained, they are using it less. It won't last long. Good example of consumer preferences making a shitty product go away. I think the premise is that there are many ways to learn a math concept, and some kids don't get traditional forms. I'm not certain of their research and development, don't care, don't like it.

 SSK: Myself and my son...

 KB: The TERC method she showed I can't imagine how anyone could do even grocery store math in their head with it.  

Crewton Ramone: I have to write a bit of a dissertation on this and I will make a blog post out of it. This is blog post week for me I have lots of math vids piled up for crewton ramones blog of math and even a couple for math genius making. Please allow me to use this thread in my blogpost. IF you would like your name withheld or your comments withheld let me know.  

SSK: I would have to see your blog before I could decide. Debbie who is "them" that you hate?  

Crewton Ramone: First off it is amusing to me how many "scientists" can't do math. And when I say math I mean statistics and critical thinking using computation. I have seen this with biologists, meteorologists, botanists etc. They took the math required and then promptly forgot it because for the most part our math instruction has become a meaningless dance where most participants can't see how it applies to their field of study memorize whatever the fuck it is long enough to pass the test and then forget it. I just heard this recently from a student that was in Calc II which was a requirement for him becoming a doctor why do I need to learn this? I want to be a Pediatrician.

 "Oh I dunno, maybe so when the pharma companies use stats to lie about the effects of mercury on child development you can look at the study and see the bullshit?

Would that be helpful?"

Silence.

"Now let's take the integrate sin⁵...which I will admit does little more than test your ability to follow complex rules; however, it will as we proceed force you to think critically and make you pick and choose formulae and when to apply them."

I also have heard it from a more than a few "math moms" (parents of my students) one of whom actually had a Masters in Mathematics. Here is a cut and paste from an email:

"By the way I have a masters degree in maths but dont ask me any question b/c i don't remember any thing which i learned. :)"

This is not uncommon. At all.

TERC would be great as an adjunct, that is in addition to or as a preamble to learning the "standard algorithms"...if you watch her do the multiplication you can actually see where the rules come from, same with division you are actually keeping track of subtraction with division...which is why there are quotes to the effect that you can do ALL mathematics with just addition and subtraction, it just takes longer. Mathematics when it comes to computation is just counting, multiplication is counting very quickly, summation (using that goofy sigma symbol) quicker than that.

BUT mathematics is not just computation, computation is what we use to do mathematics. And mastering basic operations that is addition subtraction multiplication and division is indeed laborious and time consuming AND NOT OPTIONAL.

As with all languages earlier is better.

The human mind goes through development stages. A two year old can barely reason...a seven year old can, but even then reasoning must be developed. And sure there are outliers on the bell curve but over all the bulk of two year olds have almost no reasoning skills which is why I am always amused at parents who try to reason with kids that age, just because they have developed some language. At the early stages of development repetition is quite normal as evidenced by the child's ability to sing a song 1000 times over and over again and never tire of it, they can watch a video over and over again or a TV show. This is the time to introduce the memorization of addends and multiplication facts and the algorithms for their effective use. If they can memorize nursery rhymes why not certain formulae too? Simple things like 1/2bh=a or a² + b² = c² or the quadratic equation...which is every bit as fun as "Barney is a dinosaur...." Or "Sunny day chasing the clouds away..."

I'm not saying instead of to be clear I'm saying in addition to, right along with...math is fun for little kids. The way you teach basic operations that is addition, subtraction, multiplication and division is with endless repetition. You don't just teach multiplication in 4th grade and be done with it. It should start early and be repeated YEAR AFTER YEAR. Further it can't be "just memorize this multiplication table"...it has to be fun, involve play, songs movement, writing, drawing etc.

I can't stress enough that play is the primary way children learn. Here is a simple story about teaching two bright little boys to count by sevens: (stay tuned)  

Crewton Ramone: http://crewtonramoneshouseofmath.blogspot.com/ Crewton Ramone's Blog of Math  

KB: My g/f the doctor has to ask me to do math problems for her. I've seen masters prepared pharmacists not be able to do drug calculations for IV titration. I will never forget the time I called the pharmacy department to have a pharmacist double check my calculations for a heavy duty drug that had to be administered in micrograms per kg per minute and he flat out said, "I'm gonna have to go with what you've got because I can't do that calculation".

DJ: Teacher told mom I just wasn't trying in math, as I was a pretty good student otherwise. I've felt I had a math learning disability for decades. Who knows?  

SSK: Interesting, Crewton. I am a speech pathologist and I work with disabled preschoolers. I agree, learning must be fun and I use play therapy we well. I did the same with my own children, except with language. I taught them sign language at 9 months. They each had a vocabulary of 50+ words by 12 months, and they were fully conversational by 18 months. Just so you know, I am not defending TERC. However, I do not believe it is part of some conspiracy to dumb down math. You have to understand, these curriculum programs are sold much like pharmaceuticals. The reps come and explain their research and devel. as well as their field tests and success, etc. School districts decide what to adopt (buy) from year to year. http://investigations.terc.edu/index.cfm  

SSK: Of course they are going to make it sound wonderful, they want the sale. Perhaps our school district chose it because we have a diverse population, I don't know. Teachers and parents here were quick to see it's problems, and move away from it.
 
RA: ur right SSK, sold just like drugs. 4 some reason people were never taught how 2 break things down to understand them. when i was early n school i noticed that lots of students didn't want 2 take time 2 learn the procedure 4 getting the answer 2 a problem ---they wanted to memorize the answer & can't do that 4 there r infinite problems w/ infinite answers. U MUST LEARN 2 REASON LEARN THE PROCEDURES.

  KB: The whole idea is to force the child (especially with math) to preform cumbersome and unpleasant regimented operations. How many times have you watched a child be scolded for figuring out math by a different and I might add more organic method? My father was a mathematic whiz and he saw all kinds of problems with the taught methods, he had his own. He could do calculations in his head that most people can't do on paper. He was horribly frustrated watching my brother and myself be castigated in school for using better methods he showed us at home or ones which we thought up ourselves. The whole point of their exercises is conditioning. You WILL do what you are told even if it doesn't make sense, no matter whether you like it or not, you will preform. End of story. You cannot deviate from the plan.

Now if a child had done this in the Soviet Union they would have been plucked out of the normal classes and fast tracked to advanced mathematics, chemistry and physics just like their kids who twirled and danced around were sent off to ballet and gymnastics school or if they plinked around on the piano pretty good at age 3 they went to music conservatory. Now I'm not supporting the dogmatic development of talent as you saw in the USSR or in the current People's Republic of China but it does go to show the depth of talent. For every concert pianist on the circuit they've produced there are 100's more who are just about as good. Where they are anomalies in the West, they're a dime a dozen in those systems. We're wasting a lot of talent by making everyone the same.

Why should the kid good at certain times be forced to spend inordinate amounts of time at things they neither like nor enjoy? Why do we make people learn subjects just long enough to pass a class and then later in life they remember nothing of it, not even people with masters level math degrees? It's insane.  

GH: Absolutely true. As I posted in your other thread, we need to get away from the Machine, as bad as it was to start it's completely atavistic at this point. Forcing kids into the public school system is like tossing them into a meat grinder and hoping they'll miss the whirling blades.  

SSK: KB, when was the last time you saw a child scolded in school for finding the right answer in the wrong way? From what you describe, I would think Investigations would be your cup of tea since it embraces the idea that there are many ways to solve a problem and allows many forms of operation. For my own daughter it did not work as she was shown 4 different ways to do long division. She picked up bits a pieces of each, put them together, and it did not make sense. I have dedicated the last 20 years of my professional career working with special needs kids in the public schools. My 3 children attend public school. I will be leaving this thread now. Have fun bashing.  

KB: My nephew experienced the same thing in Huntington public schools. Same scenario of finding the answers by alternative methods and being told he can't do it that way. My brother took him out of school and put him in private Christian school by third grade which ended up being a mistake because he ended up further behind than he would have been in the public school. If my brother and his wife didn't both have to work they would have chosen to home school. It just wasn't an option for them.

Crewton Ramone: I don't see them scolded but I see kids get marked wrong even though they got the right answer because they didn't do it "the teacher's way." And often that way is the hardest most confusing way possible. I use manipulatives to teach addends, then use those addends to teach multiplication. The process ends in cross multiplication for two digits which is just a faster way of doing the same standard algorithm she talks about in the video with proper verbiage understanding you are multiplying tens in the tens places... Lets take the example of 7's because they don't repeat their pattern until you have added ten times. Other numbers have an easy pattern you can see after just five such as 6:

 6/36/66/96 etc
12/42/72/102
18/48/78/108
24/54/84/114
30/60/90/120

So the emphasis can be on the first five and then repeat the pattern...and at the house of math we learn the "tables" out to 20 not just 9 or 12. First we sing songs, and whisper count and listen to patterns and count by ones, then we use addends for seven it goes like this: Once you understand numbers are made up of other numbers and all numbers want to be ten un less they are tens the they want to be hundreds, unless they are hundreds because then they want to be thousands....etc.

7 plus 7, the 7 takes 3 from the 7 and we get one 10 and 4, 14
add another 7, 4 takes 6 out of the 7 two 10s and 1, 21
add another 7, easy 1 + 7, 28
add another 7, 8 takes 2 out of 7, 5 left: three 10s and 5, 35
add another 7, 5 takes 5 out of 7, 2 left: four 10s and 2, 42
add another 7, easy 2 + 7, 49, we stop to observe 49 is square
add another 7, 9 takes 1 out of 7, six left: 56
add another 7, 6 takes 4 out of 7, three left: 63
add another 7, 3 wants 7 and gets it! 70.

Now they have all ten and can start again...it is a long process but it gives the child an algorithm for adding them all up and not just memorizing a string of digits. The lesson is repeated MANY times in may different ways. As the months go by more understanding is added and we are covering many math concepts besides multiplication.

Nobody wants to do multiplication over and over again by the time they are in high school. As young children repetition is natural but as they get older it is not unless it is music. My two little boys have done this many times and we hadn't done it for a while but now it's easier than before and when they watched a multiplication rock video "Lucky Seven Sampson" I could see it made a lot more sense to them. Especially when he gets past ten and they could see 7 x 13 was indeed 70 plus 21...I could literally see them getting more out of a vid they had seen many times because they had done the work of adding one seven at a time.

They are well on their way to having instant recall when asked what is 7 x 4 they can say 28 just as easily as they can say 2 x 2 is 4. It takes lots of time but multiplication IS THE CRUCIAL MILESTONE in the mathematics because it allows us to count very, very quickly and see and discover factors and patterns in algebra, fractions, and percentages and more. Once single digit multiplication is mastered moving up to numbers like 6 x 16 is easy it's just 60 + 36 = 96, then 13 x 13 is also easy because we have manipulatives where they can see it is

100 + 30 + 30 + 9 = 100 + 60 + 9 = 169

and the standard algorithm works and they can see it works and WHY it works. The next step is just doing it all at once and keeping track of tens.



Two Digit (Cross) Multilpication With Crewton Ramone



I was going to make comments and add more but just formatting this and changing the names to protect te innocent took a lot of time perhaps I will work on this post more in the future but don't count on it...the thread continued after this too...needles to say base 10 blocks are better.