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...
Using base 10 blocks for a multisensory math experience is crucial to understanding. Using manipulatives to teach multiplication should be both easy and fun for student and teacher. Some children get "a little big for their britches" as my grandmother used to say. They want to prove that they're a big girl or a big boy by not playing with blocks, because Base 10 Blocks are for little kids...In fact quite a few high school teachers believe the same fallacy.
However, it's finally becoming recognized that adding the tactile and visual components the mathematics makes it much easier to understand. I have PDFs at Crewton Ramone's House Of Math that you can download that say the same, from prestigious schools like Stanford and Cornell. It's very simple. Maria Montessori figured it out decades ago. Manipulatives make math easier to understand, because it allows for all the senses (Learning Modalities) to get involved.
I have these. I can see now why they are inferior.
Visual auditory and kinesthetic learners polite benefit from using Base 10 Blocks. On this blog and website I show you how to use your base 10 blocks to best effect. Any-base 10 blocks will work, but some are better than others. I got this picture and note from a person that just recently purchased a password.
The reason people are so enthusiastic about my website, Is that I don't just give you boring theory, you get actual lessons with actual students, non-rehearsed in real-time.
They still work, it's just that it's going to take a lot longer to build simple problems if all you have is units, rods and flats.
The primary reason these aren't used more or more widely adopted in public schools is it takes too long to build problems...& using base 10 blocks becomes tedious when you have to get units or tens out one by one.
Further, most of them use these only for Addition Subtraction & Place Value & have no idea how to use them to teach algebra, precalculus or calculus Concepts.
All we are doing is counting quickly and accurately, counting quickly without being accurate is no good and counting accurately but taking forever is also no good. Multiplication is the absolute milestone for counting quickly. At the house of math, We don't just teach the multiplication tables out to 9 x 9, or even 12 x 12, you want your students to have an easy math life? Help them memorized there multiplication tables out to 20 x 20.
The preferred methodology which is an absolute failure currently in public schools is using worksheets and flashcards ad nauseam. How about we play and have fun instead? Multiplication tables will be memorized easily and quickly By building, playing and singing songs.
All I am showing you are better algorithms. which make Multiplication fast and easy and therefore fun. Rather than telling you I'm showing you.
Count the big ones first.
The entire video along with "a lot" of other content, is available with a password. Those that have joined and paid for the superduper supersecret math group already have access on Facebook. This is not the same as the regular Facebook page. This lesson will be found on the Sample Lessons Page. & on the password-protected Multiplication Page. Scroll down, under the green arrow you will see a link that simply says, "more two digit cross multiplication" right above "Playing with Macaroni."
There's a difference between theory and practice. Here is a little bit of theory. on my website you will see theory in practice. The trainings were for August 4 and 11th...If you want to see the long version of the second video Consider a password, or hit my for a buck or two & I'll let you into the Facebook Page.
On my website, I explain the importance of starting the concrete blocks, moving to pictures, and then finally to symbols. I also explain that this is the ideal, but practicality may dictate otherwise. You can see that I start in the concrete (Manipulatives) & then have the symbols right alongside the Base 10 Blocks...Soon I am only drawing pictures and she is building them...Above you can see her doing 15×13. This is for her to get the idea or concept of it. You don't want to build every problem. Why would you want to build 59 × 37? Drawing it is fast and easy, and easier than that is just using symbols which is the goal. If you catch them early enough they'll be able to do it in their heads, which answers the age old question, "What if they don't have the blocks?"
Please take a moment to follow some of the links on this page. Sharing this and liking this post helps out. You might consider donating a dollar a month to https://www.patreon.com/CrewtonRamone
People have been telling me to do this for years now. And I agree patreon makes it much easier to donate your time. And if you were so inclined please leave a comment below. However, if you want to contact me, use the information on the contact tab, because I might not see your Facebook or other comment for months.
This looks like a mess because what you see here is a the end of a Trig lesson but it started off with just the triangle the jet and the numbers that you see in blue the 36 degrees and the 7,000. But as we talked about it and all the different ways we could go about solving for the numbers we didn't know more and more got written down so it looks complicated. It isn't. Once concepts are understood students generally go from F to A in short order. Here is a page on trig showing one such student.
What you see are actually THREE ways to go about finding answers, using Sin, Cos and Tan depending on the angle you use what function you use to give you the side you are trying to find.
Trigonometry is just playing with triangles instead of rectangles.
In fact trig is so easy that I built a page with lessons on it designed to show you how to teach trig to eight year-olds. Yes you read that correctly. Eight. Years. Old.
In this video they are doing trigonometry and having FUN doing it. Here is a video showing a lesson with two students...who don't know this is supposed to be hard and that quite few students twice their age fail this pretty regularly in high schools and colleges around the nation. If you are intimidated or lost then maybe you need to take a few steps back and check some of my other pages, but the point is any kid can do this even a grown up kid like you...and I show you how starting from step one. Here you see us AFTER we've been playing for a while because I want to show you what's possible.
Once we've played for a while and have gotten comfortable with the concepts and know when to divide and when to multiply and what function to use and the definitions of those functions the math itself is pretty easy, especially if we employ a calculator, but--and this is rather important, it's crucial to have enough "number sense" and understanding to know whether the number you got after you punched your calculator makes sense and isn't ridiculous. Sometimes you hit the wrong key...sometimes to divide when you are supposed to be multiplying and sometimes you may set the problem up incorrectly...so using a rectangle to clarify your thought at first can be quite useful.
These students are past that point. But I drew it in so you can see it applied to these problems. The students in the video already have played enough with these concepts so that these rectangles are understood. Algebra and trigonometry go hand in hand because algebra is man's greatest labor saving device. Coupled with multiplication math becomes easy to understand and learn because we aren't bogged down in computation or slowed down by basic problem solving. This way we can focus on CONCEPTS, and trigonometry becomes child's play.
Then we can ask very simple questions like how fast in the UFO flying in miles per hour and what is it's ground speed in miles per hour starting with just this much information:
A UFO takes 5 seconds to reach a height of 1 mile taking off at a 58 degree angle in a straight line. How fast is the UFO going and what is it's ground speed?
Remember I gave this problem to an eight year old. Now I will grant you that this is a "gifted" eight year old but I've taught this to other not so gifted eight year olds also, just takes a little longer which means we have to play more. So take a moment and do the math...if an eight year old can do it so can you or your teens. But start at the beginning...it has taken us multiple lessons to get this far...this didn't happen on day one. The same way they won't be speaking in Spanish sentences after a lesson or two in Spanish.
For a measly forty bucks I can teach you how to teach (just about) any "regular non-gifted" eight year old to do this too. No tears. No fear. Just fooling with math. If you actually buy a LIFETIME password you will find the trig page alone is pretty much worth the price of the password. Seriously.
And with a lifetime pass you not only get "How To Teach Trig To An Eight Year Old" you also get The Trig Page and a bunch of other pages that make teaching and learning math EZ too. How To Teach Trig To An Eight Year Olds is a "stand alone page" that comes with some added bonuses that you get with a full blown password and it also comes with some pdfs to get you going. So if you Don't have 350.00 bucks for a password or want to pay 37 bucks a month for 10 months and just want help with Trig just pay 40 bucks once and you are into that page for life. No expiration. And I think you will see that for 40 bucks you get a whole lot more than you are used to getting for 40 bucks.
What I have found over the years is that a lot of kids that come to me for trig help don't know basic algebra concepts and don't even have their multiplication tables down pat. Of course they are having trouble with Trigonometry and are confused. Couple with that with text books and their proclivity for starting in the middles and expecting you to have some serious perquisites under your belt and you end up with a lot of anxiety confusing and failing grades. You need to be able to identify a rectangle count to nine and tell if something is same or different or not to get started with me.
There is no need for fear or failing grades. Forty bucks will make all the difference. Don't believe me though, no brochure makes the hotel look bad. Go read what some other people have got to say. Skip Starbucks 8 times and you can change your kid's life forever.
Click on the links to the trig pages and you will find FREE lessons there.
Math is fun when you use base ten blocks. Here is a young girl who isn't so very keen on math, she'd rather be playing with her bows and across or her sword but playing with nines and building walls can be fun too. Measuring out how many nines in 27 turn out to be pretty easy.
Building walls of 24 can also be fun...here she sees quite clearly four sixes and six fours are the same also three eights and eight threes....and of course we counted by 3's, 4's, 6's and 8's...and we see counting by eights is the fastest...no need for worksheets. No muss no fuss. We talk about it she hears it she sees it.
She gets a math experience using base ten blocks that evolves more than just paper and pencil and memorization. She can get her hands on it. She is in a math rich environment. Everything we do is math during this lesson and counting. Here we are doing multiplication and division and skip counting. It's visually obvious that the bigger the number is the fewer we need to count to 24...we see a relationship between numbers. Six 4's is the same four 6's...which is pretty cool when you are six. But lets not get stuck doing the same thing over and over again...let's do something else.
Remove the no from the lesson.
So lets make some puzzles. Build me x² + 7x + 12...and then count the sides...basically here the algebra is just along for the ride mostly we are cementing in the addends and doing some multiplication facts. Then we also see x² + 8x + 16 is (x+4)² which is just a shorter way to write (x+4)(x+4)...this also gives us a better understanding of just plain ol' x²...
10 to 15 Minutes per Topic.
Factoring is fun and easy. And it builds confidence because every kids knows algebra is supposed to be hard. So if they can do it with ease they must be just as smart as grandma says they are.
That moment when you see the answer.
Keeping it small and easy makes it fun but there is plenty of math going on here none the less. Dividing by two, squaring a number, addends, multiplication she gets exposed to all of it in one simple problem. If you put the child in a situation where they can not fail and feel like they are making their own discoveries rather than you just telling them math facts and practicing with drills and memorization you will find lower stress and more fun which leads to more learning and happy memories and associations. Or you can bust out your work sheets.., and see how much fun that isn't.
It is a very simple concept, the making of positive associations with the mathematics instead of negatives ones. It should be pretty clear from the pictures that she is having a good time. If you have doubts listen to her excitement in the video.
Once you get the thing factored counting is the easy part. While she is building it fine motor skills are being developed. Also recognition of spatial relationships...
Algebra is fun and easy.
Factoring Polynomials is Child's play.
Here is a picture that's fun because we are seeing quite clearly that 3 sixes and six 3's is the same thing. We made a purple sponge cake with pink frosting and as we frosted the cake we counted by threes. We put them on top of each other and counted the sides...
Multiplication doesn't have to be drudgery, if you do it right the child see's that actually it's a time saver and learning to multiply makes life easier not harder because it allows us to count quickly which allows us to do more math with less pain. Discovering constancy of numbers is fun when you are six.
It's these little repeated exposures using base ten blocks that over time add up to a deeper understanding of how math fits together.
So here is a sample for you, showing us "jumping around" from topic to topic. The topic is math. so we are just playing math...we start off counting 11's...and then fool around with multiplication concepts...then we solved for x playing what's under the cup, told some stories...did some thinking...then we factored some polynomials and we were done...moved along in lively fashion didn't get bored doing the same thing over and over again even though we were doing the same thing over and over again...which is math...
Not just Theory. Theory in Practice. I have HUNDREDS of FREE videos that show you how.
I don't want to tutor your kids I want you to tutor your kids. Play math and have fun. Spend quality time with your kids and learn together. This method will transform your math time into something your kids look forward to instead of dread. But don't take my word for it, here are testimonials, you can see many others have found that this is a better way to go. All you have to do is dive in. Christmas is coming and it will be a great time to get some blocks and GET STARTED. Get a set of blocks and a password and you are good to go.
The points of this post are manifold.
One, for those who are thinking they might want to start a tutoring business, DO IT. NO ONE is out there teaching kids math like this...well, I can't really say that anymore because actually quite a few people are now using these techniques to teach math...more than six years ago when I started this nonsense anyway, but you know what I mean. The number of people doing it this way is very, very SMALL. You should have no trouble finding a few parents who understand that giving their child, particularly if the child happens to be female, a real head start and are willing to pay well you for it. Don't undercut yourself.
You can charge decent money because of this, especially if you have been to several Friday trainings and/or have gotten passwords and watched all the videos (and played along with your blocks) on the Parent Teacher Training Page (there are quite a few more Hours about to added there BTW). Ask yourself where else are they going to get this? From me? I can only do so many students at a time and my time for teaching students is just about come to an end...I am interested in training teachers and ultimately training trainers and master trainers to replace myself.
Two, when teaching older students who have massive holes like this child in their math knowledge and basic math facts it is important to preserve the child's self esteem. With younger remedial students playing like this is cool because in kid culture "everybody knows algebra is HARD" so although we are factoring polynomials not so much to learn algebra and distributive theory but more to learn addends and multiplication, the older students don't feel "dumb" because after all we are still doing algebra, and the younger students feel smart because after all we are still doing algebra. Get it? I think you will find this is a recurring theme here on this blog. I use third and fourth power algebra to accomplish this too.
Three, it should become abundantly clear after watching this video that having a calculator in hand wouldn't help him factor an expression like x² + 9x + 18...you have to know your addends, which will make repetitive addition (which some call multiplication) easy and fast and will bring home the concept that numbers are made up of other numbers and that there are many ways to express any number...as the sum of two (or more) numbers as the difference of two numbers as the product of two (or more) numbers or as the quotient of two numbers...
I also have another video with a 7 year old where you can see having played with square numbers and being familiar with addends and ratios and fractions makes trig EASY. But if you can't even recognize a square number when you see one...well...an "F" is much more likely than an "A".
Get a password, get a couple of my books and get going. Trainings on Fridays has become "a thing", cost is 20 bucks for 1 to 4 hours, lately they have been going for 4...if you come for an hour it's $20 if you come for all for hours it's $20, starts at 8am HST...you need gmail vid chat and you need to have the video plug in ALREADY INSTALLED. Email me for more info. I know you think 4 hours is a long time but trust me: time flies when you are having math...ask anybody that has come it won't feel like four hours.
This video is pretty self explanatory, so I won't put too much here in the way of words. People don't read these posts much anyway just look at the pictures and watch the videos...so:
But for those interested in a little more in the way of training for parents or teaches, with little kids you don't need to write symbols to start. (By little kids I mean 6 and under.) Just do what you see here and talk about it. Later the symbols will make sense when we write them. Don't forget to spend some time DRAWING them too. (Look at the second excerpt for more on this on the training entry page.) Lastly, do symbols...but this kid is 17 so we went directly from manipulatives to symbols...because he's paying by the hour and it takes too long to draw and label them although it is a very worthwhile step, I told him to do it at home. I had other things I wanted to do in the hour. The idea is to teach concepts but this child needs work on multiplication...multiplication will make all of his math easier to understand because there is no time lost on computation (if multiplication is easy and fast, if not the math problem is compounded because they have to think about multiplication AND whatever the problem is too...once they have the multiplication down as Forest would say, "that's one less thing."
Here again I want to drive home the point that all we are doing is counting and basically I am teaching him to count and count quickly using fractions manipulatives. Counting quickly like this is called multiplication. The fractions are also discovered...and equivalent fractions understood and reducing fractions will not be difficult when we introduce that concept later...only then we will be using division instead of multiplication. I hope you can see the rectangles...
Introduce skip counting early and sing songs and play and this whole problem can be avoided...too many teen-agers have this exact same problem, they are having trouble with math (particularly algebra) because they can't even multiply and having a calculator doesn't help. In order to see relationships, ratios and multiples it is imperative that they know their multiplication tables bare ass minimum out to 9x9, out to 12x12 better than that and out to 20x20 (yes, 400 facts) best of all. We went thru a little period there where well meaning but horribly misguided math teachers tried to say that with the advent of cheap calculators students didn't need to know multiplication by rote memory. OPPS.
They were trying to spare students the pain of memorization and drill. I agree about taking the pain out of mathematics but not with taking the learning of multiplication out of the mathematics, which is why we didn't spend the whole hour on this...I assure you drill, worksheets, and flash cards will turn the kids off faster than you can say "modern mathematics teaching in America is an utter failure." But if you play games, sing songs, skip count, and do fun math activities starting at an EARLY AGE, multiplication can be mastered without tears, muss or fuss.
I have a book (my Curious Counter's Compendium) you can get for practically NO MONEY that will help you start teaching math to your toddlers, and another one (my Subtraction Manual) you can get for NO MONEY at all. More manuals are being worked on as we speak, there will one day be 11 in all. Side note: We measured the distance he walks using the odometer on my car and it was just over two and a half miles, one way. Math is either important or it isn't.
More side note:AFTER you go thru a lot of posts on this blog, videos on youtube and of course my website you might want to get a password...
Nice, now change the game and make it so he can only use one three...then the other ones have to be a two and a one or three ones, you might mention right now they are all three plus zero and it's more fun to make combinations...same with the four, only use one four and the others have to be addends for four and so on also nothing says you only have to use 4 on the corners, you can use 5 or 6 or 7 or more pillars...then you can play "jenga" an see if you can take pillars out without knocking the whole thing down...then when does fall down just build it again and see who can take the most pillars out before it falls...Also, make a video of it and maybe you could win something.
Many math concepts can be learned with just this one simple exercise. You can slide a 2 into the level that he built with three's and the concept that 2 is "smaller than" three is easily seen...in math we could say less than...and we could show the symbols on a white board if we wanted to but at this age playing blocks and verbal reinforcement is enough. You are creating a math rich environment, where several of the senses are used to learn math and basic math concepts like more and less and addition and of course subtraction.
The mind absorbs all this then later (days, weeks, months or even years) this experience can be drawn upon to make sense of these symbols: 2 < 3...
You could put on the skip count CD or multiplication rock vids on youtube in the background and sing along as you skip count the threes he has used. PLAY, make teaching addition fun; little kids love to sing and they don't much care if you can carry a tune or not as long as you are making a joyful noise.
Anyhow make a vid of your kids playing math and you might win some cool stuff. UPDATE: this part is over but you should still check out primal math for some cool apps...Zombie Fish Bits is fun....
If you'd like a FREE manual on showing how to teach even very young children a super easy way to subtract click this link to my Supremely Simple Subtraction manual. A new Games and Activities Manual is in the works but it will be a while so meantime use the old one. (Takes a while to load so give it a minute...)
My response:
Excellent...! Now you can talk to him about the 9 in the corner...you could flip the blocks over and count 1 hundred, 6 tens and 9 units...you could talk about square roots and square numbers playfully with the 9 and the 169...and of course the polynomial...x² + 6x + 9...the square root is one side which is x + 3....and of course (x+3)² = (x+3)(x+3) = x² + 6x + 9 and you talk about economy of symbol...that we don't need to write it twice...you could talk about 13² = (13)(13) = 169...you could have him draw a simple picture of it...and talk about these things while he's making his drawing...note I say could you don't have to...you could write some symbols and show him how those symbols describe his creation the same way words can describe a brown cow...and you do it casually, not drilling him not formal instruction just, "hey...look at these green ones there--they make a square and we can count the sides or one side...hey look at that! You made all those blocks into a square can you count the sides? What's the blue one called?"
If you are on the base ten side he should say, "10...and three units." Or something like that. And you could say, "what's the name for one ten and 3 units...?"
If they are on the smooth side and you are in base x, the blue one's name is x and three units...
A ton of math concepts can be taught off of this one rectangle...you don't need to cover all of them at once...just a few at a time. Check this post out, it talks about all the things you can use this simple rectangle (square) for, when it comes to teaching math concepts.
Now build one that's 14 or x+4 squared...and then 15...or maybe he wants to make a different shape with his 6x...and make it (x+2)(x+3) or some other combination for 6, 5+1, 4+2, 3+3...still 6x but now a different number fits in the corner...just play math, bigger is funner.
Here are a few recent vids. The point of them and this post is to illustrate and bring home basic concept number three: we form rectangles to facilitate counting. Rectangles make it easy to count and we can reduce many seemingly complex math problems to basic counting with the help of our friend the rectangle. In the first vid they make percentages EASY by using a rectangle.
What basic concepts do you see in these three vids?
Rudimentary Editing Skills Level 101: ACHIEVED. Mastery w/ percentages is being attained. THEN we can move on to ones that end in 5. Baby steps. Note the block serves as a "post" or memory anchor on what to do. They understand the CONCEPT so now I'm just changing the numbers...and we also get practice with multiplication and division.
Here is their little sister playing and learning addends...she had asked very nicely if I would play what's under the cup with her since her brothers were playing outside with two neighbor boys...and she wasn't being included. So I brought DBoyz in and shot that short vid and then sent them back out and played with her...we played what's under the cup and built tens and then she made a house for little ponies to stay in and then we cleaned up and it was time to go. She made the comment, "Time always goes so fast when I come to your house." Because time flies when your having math.
This next video is also edited. The full version is on Dboyz page at the house of math. All three of these videos can be used for educational and training purposes...lets use them to focus on basic concepts.
You can stop here. The rest of this post is long and aimed at those who are interested in learning more from the perspective of a trainer or an aspiring master trainer. A link to this post will appear on the teacher training page too. Here is the whole converstion on Crewton Ramone's FaceBook page.
I highly recommend you click that link and read the thread, what follows are some (slightly edited) excerpts lifted from that thread starting with this: First things first, I don't write or make videos for Unicorns. Celeste is a unicorn. Female engineer. Statistically speaking they don't even exist. There are few Unicorns who have become interested in the house of math...Staci was another one: black female electrical engineer...Irene is another one I'm confident she knows more math and certainly more chemistry than I do...these are the exception not the rule...and it must turn tables and become the rule not the exception. I have been paid large sums by men who fully understood the mathematics well enough to apply it to physics or engineering or economics but could not explain why you invert and multiply to a seven year old's satisfaction...much less why the x's don't change when you add them...what has to happen is people who actually understand math have to be given the tools to be able to explain it to others.
Most parents statistically speaking failed algebra. Many teachers got a teaching degree because it "didn't involve taking a lot of math..." Teachers currently still think there's a trick to it:
Question: if you understand math and math concepts will you still need special techniques? Answer: once you understand the concepts all they can do is change the numbers.
I'm starting to break lessons into small pieces for a couple of reasons, First: because we know the attention span of the average viewer is extremely limited...two minutes is optimal video length...what a laff: people want to learn math in two minutes....if not screw it...girls can become strippers and guys can always cut grass--wealthy people rarely do their own yard work. Second: because too many parents and teachers can't wrap their head around switching from topic to topic because they can't see how the lessons relate...we are learning about Pascal and summation but all we are doing basically speaking is addends and multiplication....the percentages practice was practice on multiplication and division. Once we get the CONCEPT down all we can do is change the numbers and then it just becomes practice with doing whichever of the basic operations are required because AGAIN computation is how we DO math but it's not THE mathematics.
You can do all the math with just addition and subtraction but it takes longer...further if you cannot DO computations quickly and easily mathematics is HARD because you have to contend with two things layered on top of each other: trying to remember the rules and process for the problem at hand and trying to remember the rules and process for the computation required itself. This overloads the system and results in failure, anxiety and incorrect answers which leads to negative feed back, which leads to more fear and anxiety, which leads to an aversion for the mathematics which leads to more of the preceding and a feed back loop is created that includes actual pain, mental anguish and damaged self esteem which leads to hating math, which leads to the dark side.
The concepts are very basic here. We form rectangles to facilitate counting. There are more as an exercise for those who are thinking about tutoring or maybe one day training others...what other basic concepts are covered in these three short videos?...you might also go over them in your mind even if you are "just a mom" teaching your own kids. You can put comments below if you like...but there are several more at work here. I mention some of them in that FB thread.
Now, the reason that this method is so effective is because if all math is counting and we can reduce any problem to simple COUNTING, then anybody can do math that can count. The trick, if there is one, is knowing how to reduce a problem to simple counting, which as it turns out is the genius of it all and which is why we say if you can count to nine form a rectangle and tell if something is the same or different or not we can teach you math...I add that they must already be able to speak a language generally it's English and if you can speak that language math should be EASY. Anyhow what basic concepts do you see in these vids?
Mathematics is the study of relationships and we describe these relationships with symbols. pie is not just 3.14....it is the relationship between the circumference and the diameter, it's not just sin30° = .5, sin is the relationship between the side opposite 30° and the hypotenuse...addition and subtraction are inverse functions but basically the same thing in order to do one you need to do the other basically you are combining, the same with multiplication and division, things look scattered and random (even though the tab bar on my website basically goes in the order you will see it presented in the public schools) because you don't get how the language goes together or the gestalt of it...you can't study one part without studying some other part...using this method. You can certainly compartmentalize the mathematics and they do for an example think back to or visit your nearest public education institution.
"Just show me how to do the problems" is the problem. My son is currently suffering under this line of thinking where he gets to do timed worksheets for multiplication facts. He doesn't do as well as other kids on these because he doesn't write quickly, therefore he must not be very good at math. They are majoring in minors.
When asked to be able to use and apply this knowledge to something like 28 is what percent of 70 or how many 4's in 72 they either can't do it at all or take forever and certainly can't do it in their heads...but they can whip out a 50 problem worksheet of single digit multiplication facts in under 3 minutes...but then ask them to figure out how much change they get if they buy 7 pounds of rice at 2.78 per pound and pay with a $20...no pencil, no paper...and they can't do it, because they can't organize and keep track of their thoughts.
Concept #3: Rectangles facilitate counting.
A = LW This gets totally LOST if you just memorize (n² + n)/2 which is what most of my students have done if they even know about this at all....they use summations on SATs and other standardized test to "separate the men from the boys..."
In the example for 14 we see various addends. a couple for 14 and also for 4, we also see the number 105 is 70 + 35 again numbers are made up of other numbers and that part will come in handy when doing division but also trainers should see the distributive theory of multiplication as we distribute the 7 across the 10 + 5 so the 70 = (7)(10) and the 35 = (7)(5)...students some how are mystified by this when they get to algebra even though they have been doing it all along, where the most common grade is "F" and when asked to multiply 7(x + 5) often have a very hard time even though it's easier than doing it in base ten because all we get is 7x + 35...whereas in base ten we have to combine like terms (the tens) which is a basic Concept #2: we only count things that are the same. If you were going to be a trainer and were asked to delineate the concepts on these vids and you came back with counting, forming rectangles and "The blocks help you see the patterns." I would FAIL you...because you failed to see the the basic concepts...
If you were being tested under Jerry or another Master Trainer named Brian you would fail and more than likely receive a serious dressing down to go along with it...I remember one woman in particular in tears after she got done being schooled...although Brian would try to be a little nicer about it. I'm going to (try and) develop some Master Trainers...if I get 7 in the next 9 years I will have done phenomenally. When you are done you should be able to take any textbook and then take just about any problem and either show how to do it with the blocks or step back and show the basic concepts needed to do the problem because by now it should be clear that you don't want to model every problem with the blocks nor should you need to...and you don't need to hit the student over the head with them or take away the aha moments by just telling them....let them explore and discover for themselves. You nudge them in the right direction or point them down the path that will lead to the discovery you want them to make....but they walk down there alone. You are not far off, just in case and to catch them if the fall---control of error is important but failing your way to success is also important...not wrong just getting more information...that attitude leads to discovery and opens up the wonder. Control of error does not mean don't let them make mistakes ever or avoid making mistakes: it means learn from the mistakes, take that information and then use it to get it right.
Fear of failure or frustration of getting it wrong or worse being put in a position where they can not succeed shuts down learning decreases wonder and joy to zero or less--usually less, and makes the math what it is today. I see math memes like this:
And like this:
If you understood what you were doing how could you NOT know? Yet this is a completely common response from many of my students who are not used to getting A's. Try it some time ask your child or students how they did on their math test...
I would encourage to click this link and read the thread. The point is not to "talk down" or to deride other people who don't quite get it...entire companies have been built that purport to use base ten blocks to help you see the math...but some methodologies with base ten blocks are better than others. I can swing a hammer and work a saw...and my trig is strong, but I know quite a few carpenters who using the exact same tools, would get whatever it was built faster, better and prettier than I would. My aim is to make you, if you so choose into a person who can teach math using base ten blocks and other math manipulatives, faster better and prettier than the majority of your peers. Unfortunately, I can't do it in an hour or even in 50 five minute videos...but I do have some parent teacher training more than 7 hours worth at VERY reasonable rates which many have found to be very, very useful.
If you would like to enter into a dialogue regarding concept number three, or make a comment about the teacher training (many have taken it now) please do so in one or the other of the comment boxes or ask questions about concept number three not 4 not 5 but 3 that would be concept number three the one about rectangles and how they facilitate counting.
I do not recommend my Tumblr because there's sex, and "naughty" pics, and swearing and all manner of content for grown ups, along with controversial topics...but the other ones are fairly tame, although still not tame enough for the Christian Homeschool crowd.
Here is a lesson that uses bases for practicing math.
This is again cross teaching or a compound lesson. I am not interested so much in the student memorizing what one number in base ten is in another base, I am interested in the student being able to grasp some basic concepts like numbers are made out of other numbers, a deeper understanding of place value, a deeper understanding of division concepts...how many times is a number contained in another number--that's basic division.
When we convert from base ten to another base like 2 or 3 IT'S FUN to do. Kids enjoy it if you present it correctly.
This blog post is just to give you some ideas. I'll add more video on this later...I made this video on a lark because a parent was having some trouble explaining division. Play with this a little bit and long division will be a snap. I have lots of other videos on long division and using algebra to teach not only division but the other basic operations as well. Algebra is just generic math.
This video stops before we get to the repeating decimal. As I was doing this I was thinking too many concepts at a time. Keep it simple...next time we come back to this we can take it a step further and see what happens when we divide by sevens or nines...for not finding out how many time one number is contained in a another number with division should become much easier after converting to bases where you have to think about multiplication and subtraction with different numbers. For example converting 100 into base two...we don't need a 128, but we do 1-64, 1-32, no 16's, no 8's one 4 and no 2's or 1's...so 1100100. When you are a little kid that's a lot of thinking. But it makes sense and they can see that what they are basically doing is adding 64 to 32 to 4 to get 100...and keeping track with symbols in certain places...that one in the middle of the zeros means one four. Converting it into base three or 4 is also FUN.
I didn't get hung up in the details of notation, but instead focused on focused on the basic concepts. I assure you when they go do base ten or base x division it will be easier...and again doing division in base x makes "regular" long division easier too. Repetition is the mother of skill and memory but that doesn't mean you repeat the same thing over and over again. We want them to learn the 45 addends and their multiplication tables....this exercise practices all of that. For the little kids stat out with base two and maybe later do some base three. Do a few easy ones and work your way up.
I often start with base two and then make a table...from x⁰ to x10 across the top (largest on the left to smallest on the right) and then going down the right hand side binary, 3 to 10, 12 and then 16. Just making the table is a great exercise...do as much as you can without a calculator. Then it's fun to punch a calculator and get the answers, unless you are interested in the students practicing the computation. You might want to do it in stages over several lessons so it doesn't get tedious.
Here are the answers...
This gave me an idea for a password protected page called "solutions."
This post is mostly done, check back there may be more...Winter break is proving very challenging for getting things done....lol...
Part of what makes this method so effective is the idea of compound lessons. Why learn "one thing at a time"? One thing at a time is great when introducing concepts...but once concepts are understood it's more fun to practice more than one thing at a time, which is why I emphasize compound lessons. And as you can see using base ten blocks to do it makes math fun.
A six year old factors x² + 11x + 30 all by himself and celebrates with his magical math microphone.
People ask me what they should get. Get a combo kit. A set of Multi-Tens and a password or both passwords...and you are good to go. Seriously. That's it.
In this hour we practiced mostly addition and multiplication but we did activities that were fun to practice them rather than doing worksheets over and over again. Using worksheets to memorize multiplication facts becomes counter productive at some point.
Just building addends over and over again would make math anything but fun. So even with this activity which by itself many kids find fun, we spice it up by racing your brother or mom or who ever happens to be around. Then we build some pyramids. All kids can benefit from building pyramids, the benefits are manifold. You get fine motor skills, addends which naturally include addition and subtraction skills, and the all important repetition which puts the facts in the memory for instant recall later...
Here the older boy is feeding his mother and brother combinations for their pyramids. and supervises the construction. They are building 11's, later we could build and find patterns with 11's.
Pretty soon he gets in on the base ten block building action. And his little brother hands over some combinations...sometimes we build all 45 addends at one sitting but today we just did all the ones from 8 to 10 and then 11's and 12's. How many math facts is that?
Children are always quite pleased when they complete one of these...or several of these. Sometimes we build towns or cities or temple complexes or moon bases (or whatever is in their imagination), made entirely out of pyramids built using base ten blocks, but today we played algebra which will give us addends and multiplication too. Started out with problems like x² + 7x +12 and worked our way up. We started there because he already has some experience factoring polynomials and I wanted to start easy...then we went on to x² + 8x + 12, and x² + 8x + 15, x² + 8x +16...Went thru several with 9x then on to 10x, those polynomials have lots of combinations.
x² + 9x + 8 you get addends for 9, 1 and 8, and 1 x 8 = 8.
x² + 9x + 14 you get addends for 9, 2 and 7, and 2 x 7 = 14.
x² + 9x + 18 you get addends for 9, 3 and 6, and 3 x 6 = 18.
x² + 9x + 20 you get addends for 9, 4 and 5, and 4 x 5 = 20.
You don't have to do every single one in order...mix it up and have fun. I did a couple with 10x like x² + 10x + 16 and then x² + 10x + 24...didn't do x² + 10x + 25...which is a good place to start if you want to go down the square numbers and square roots path which are made plain and EASY with base ten blocks. I had a mom tell me they completed the square with 24 once. I said go back and take a look at that...24 will complete a rectangle but certainly doesn't complete a square. 25 would complete a square...and completing the square is quite a useful skill to have later on when you are graphing polynomials and want to easily convert to vertex form.
Playing with squares and radicals are also fun math activities where you can compound the lessons and learn algebra, addends, multiplication, square roots and more. You decide the emphasis depending on the student. For some older students who are supposedly past addition and multiplication because of their age you can use these techniques for remedial math without them even knowing it.
Instead I gave him x² + 10x + 24 which he got pretty easily but then I gave him x² + 11x + 24, which is a challenge at this age when they are still learning their times tables. You can see he figured out he needed 8 and 3 to make it work.
Then I gave him a challenge. x² + 11x + 30...the dilemma after he figured out it was going to be (x+5)(x+6) was finding enough fives or sixes....they were being used in the pyramid...and he found that he could count by 5's but not by sixes as easily...remember this boy is 6 years old, he hasn't mastered his 6 times tables yet but he will....and there won't be tears or frustration. It will be part of fun memories spent playing with his mom instead of being angry at his teacher because of yet another worksheet.
After a little fooling around he decided to make 30 with 6's...which meant he was going to have to bust up his pyramid.
Completing the corner was important to him, later you will find students especially older students who know their times tables skip putting the units in because they already know 5x6...and that 5x6 is the same as 6x5...when they are younger this is still a fun discovery.
Challenge completed! It's time for some celebration which is where we came in. Also some high fives with mom never hurt nothin'.
This lesson took place way back in the beginning of October. Just got around to finishing it here. In this hour we basically practiced math facts in a very disorderly fashion...we didn't just do a bunch of addition or subtraction for an hour. We did both AND we did multiplication and some division AND we factored polynomials but the algebra was just along for the ride...as you can see it wasn't scary or hard and I used it to teach concepts and facts...most teen-agers I know FAIL algebra right at this point because they don't understand the distributive theory and never saw a polynomial until they were 12 or 13 years old...
Positive memories and positive associations with math are so important i don't have words. Wouldn't you rather have your child's or student's early childhood math experience be like this instead of rife with tears and frustration? They won't all be diamonds but more often than not you will find YOU the parent or teacher can make math fun and teach a whole lot of math all at once in compound lessons like this.
Want to get the materials you see in this blog post? GO HERE.
Distributive theory EASY. Especially if you've been doing it since you were four. But even if you haven't base ten blocks make it easy to understand. This is an explanation the THE explanation. Makes the idea easy to see, the symbols become visually obvious and we take the concept from the abstract to the concrete with base ten blocks.
You could use the blocks to show 3(2 + 7), use three's and you'd have two rectangles one with 6 and the other with 21, one above the other...three across and then 2 up and then 7 more up...
The Algebra is just as easy to get. Using the blocks and building bigger, funner problems makes it simple child's play.
More explanation can be done by flipping the blocks over and showing (10 + 3)(10 + 2), now instead of x squared and x's and units we have a hundred tens and units. Same blocks we could do percentages too. Imagination is powerful and these tools help unlock it and put it to use...also helps with visualization.
Just that little line drawing can make such a difference for understanding but they have to get their hands on the blocks first. Just watching this video isn't enough if they don't have blocks it can still be confusing for some. SO make sure they get their hands on the blocks...and don't start here you start with x² + 3x + 2 and if they have never seen it before let them discover how to build it don't just show them. Then after building a few and explanation like this can be helpful. This is just one example do several more where the child has to build it draw it and make symbols while you talk about it and factoring and division will make a lot more sense.
The distributive property is easy with base ten blocks and so it the associative property. That page BTW is far and away the most popular webpage I've ever built getting thousands of hits per month, month after month year after year. Why? Because base ten blocks make math concepts easy to understand.
