Fractions Teach Multiplication Using Base 10 Blocks

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...

Why Invert and Multiply.

Used to be to get this info on teaching fractions using manipulatives you had to come to a seminar or training, usually held in Idaho...now I think the next one will be in LA or maybe Las Vegas...if we ever do one again. It's on my list but it could be a while. Nothing like in person training but it's costly and you have to travel and all that goes with it and blah, blah, blah.  Much easier to just learn how to use base ten blocks by going the the house of math. Yes I know my fraction page needs serious work.

Now you can learn to use your base ten blocks for much more than addition, subtraction, and place value plus you have all the pages on this blog. First we see fraction multiplication and why nothing is getting smaller, we are just counting parts of a part and if you insist yes a third of a half is smaller but we were counting not doing magic and with 1/2 ÷ 1/4 = 2 really nothing got bigger we were counting quarters. Two of them. Observe:



Here is a quick lesson with the symbols no blocks for why invert and multiply. Division of fractions. Also easy to understand. If you want to see the whole lesson go to Sarah's page. There you will see us use manipulatives to make fractions clear and easy. And you will get the joke about "clear"...



If you actually take some time to study the picture above you can see that the symbols are quite confusing to some students particularly to deaf students who couldn't for the life of them figure out the relationship between those numbers if they didn't understand division, or even if they did. How the heck does it turn into 2/9?!?

Well, it doesn't turn into anything: we are counting. The rectangle we see has the one sixth in lowest terms instead of 6/36 which is what you get if you multiply 3/4 x 2/9...which is equal to 1/6...but getting there with symbols only and no concepts is impossible and modern math teaching doesn't even attempt to clear this up they just give you the rule INVERT AND MULTIPLY.

WHY?

And it's right about here many students make the fatal decision that math doesn't make any sense...I mean the rules changed for addition and subtraction, you can just add 1/2 and one 1/3 and get 2/5, it works for multiplication though, and then it's some crazy rule for division and they get bigger when you divide: this math is arbitrary crazy stuff.

No. It's not. It is consistent and beautiful and simple. The five basic concepts are alive in well in our fractions games...knowing what one is, same, rectangles, the how many part the what kind part, and economy of symbol...but the problem with that last part is 95% to include the teachers don't understand what the short cut is doing. It's economy of symbol: fewest symbols possible, and it's a short cut and work saver. Now, if you want to see more with manipulatives to make it clear why we invert and multiply a 15 minute video on this (and a lot more vids) is on Sarah's page you need a password. Just get a month and I think you will see an annual pass is worth it because you won't get through everything I have up in a month and as the months go by I add more and more. I will be offering a special where you can apply your monthly to the annual if enough people get a monthly pass.

One example with symbols only may not have been enough, a couple more examples with manipulatives should clear the whole thing up. The idea being if an autistic student can glean the concepts SO CAN YOU. Same when you see me using 5 and 6 year olds as students for algebra...if they can "get it" so can teen-agers.

Passwords used to cost a buck. Now they cost 5 times as much...inflation and the fact that there's 10 times as much stuff on the password protected pages and then someone sending me a $24.99 vid that had basically 30 minutes on it, and not even a good 30 minutes just 4 math concepts poorly explained with lots of special effects and production. Here you get ZERO special effects and very, very low production value but very heavy on the information and concepts, well explained...or so I am told.  AND HOURS AND HOURS ARE NOW UP. It's worth the 5 bucks. Here is a free page on addition and subtraction using manipulatives. But I digress.

What we are doing is finding how many times one number is contained in another number. In order to do this we have to make them same or we can avoid all that work with the rule...basically we are just skipping a lot of steps and in the process skipping a lot of students who throw in the towel and change their belief to math is hard and arbitrary. How do teachers teach students who already know they can't learn whatever math it is later on in high school? This will be discussed in depth with another little girl who got a convenient label put on her. She loves playing math but as soon as we bust out the school work, math is no longer fun and she knows it's hard even though we were just doing it with ease a minute before. That's how powerful beliefs and paradigms are. Her page is here. Both of these pages will grow over the coming months, and are especially good for home schoolers and teachers.

Also now lots of Mortensen Math Materials available on my site, if you buy something there, a password used to be free. As of 20014 this is no longer the case. The fractions Page was going to go up in it's new form Monday but it's MLK day so I will have DBoyz and I doubt it will get done...although it is on the agenda for Wednesday...fractions from beginning to end.

People on face book and twitter got the vid FREE for a day...it has since been taken down...joining us there has it's benefits.

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Learn how to use base ten blocks at my website.

More Fractions Fun.

You have to do a lot of preparation before you dive in and start adding problems like these.

You need to understand basic concepts like SAME and ONE. But basically all you have to be able to do is count. You don't have to be able to add and multiply although it helps.

Fractions concepts are easy. Any little kid can learn them and understand them, it's that much better if you let them fool around and DISCOVER concepts and how things work for themselves while you give copious amounts of encouragement and direction.

Working with very young students is similar to working with SPED and other "learning disabled" or "special needs" children because often what you have is "developmental retardation" or to be plain many Downs Syndrome and Autistic students who are in their teens are in about the same developmental stage as very young children. Very young children can learn math concepts as proved consistantly on this blog, ergo: SPED kids CAN learn math.

This isn't rocket science. Simple logic.

Here are a five and six year old playing math. Fractions. They have very limited experience with fractions and still can't multiply easily because the don't know their multiplication tables completely yet, but they can learn fractions and begin to understand the concepts. Now if SPED kids and little kids can learn fractions then certainly high school kids and adults can also learn it, even the "problem" children...who aren't particularly disabled or SPED but may be special needs because of circumstances at home, like poverty or parents who use drugs or alcohol etc. Sadly most of you know what I'm talking about because it's more and more common in public schools.

Here they are playing a game where they try to discover the fraction from the disguise it's wearing and they literally get to pull off the disguise once they figure it out.



They do not have enough experience to go about it methodically, but they do have natural thinking skills...and with me as a guide they can be directed to the answers by simple questions. Remember remove the "no" from the lesson. Allow them to explore and discover for themselves. Ask "what if" questions, count, see what they have.

Once we get through some of this we can do some simple addition and see how that works. Also we are headed for story problems and these activities get them ready for that too. There are also everyday math activities that help them make sense of the symbols and numbers when they see them. Also check out this post where they fool around with food and no symbols to help them understand what's going on with fractions.

Also note how the younger boy is very much a participant in this and often gets the answer before his older sibling.



Let them discover and have fun! You will even hear them say "Don't tell us!" The joy of discovery makes learning math (or anything else) FUN.

The whole video in one piece is available on youtube and on my website here. [Link not built yet.]

To get a fractions kit,

check my expanding product exchange:
http://www.crewtonramoneshouseofmath.com/product-exchange.html

Also don't forget these free fractions worksheets.

Fractions Despite SPED

People often ask me if there is a special way to do it for SPED students, the answer is no.

I've had conversations with students that went along these lines. "I know everybody calls you special, but in my class your just like any other kid that's going to learn math. I know you can learn math because you understand English. So quit telling me how special you are and count this."

95% of the time they shock everyone except me...they even shock themselves sometimes. Bear in mind when I met Sarah they weren't even teaching her math because there was "no point to it." they knew she couldn't learn it, and as we say in Hawaii they were wrong but true. She would never have learned math with just paper and pencil and rules and process out of a traditional text book.

(There was a time in my life when I was shocked. After I got over it, I said to myself more people need to see this. And, since no one else has stepped up to the plate it seems to have fallen to me. Hence this blog and my website and the many things that will be made manifest in the next 7 years or so...word is getting out.)

If you try to teach SPED students (or even "regular" students) rules and process the results will be poor. There was a time when I used to have to argue the point but the dismal performance of American math students as reflected by ANY metric saves me the trouble; not to mention that in the 22 years I've been doing this the results have gotten WORSE not better....Fractions and long division still top the list of leaste favorite math subjects pre-algera.

But I digress.

This is an excerpt from an email exchange between myself and a parent of a child with Downs Syndrome.

First off:

Treat them like "normal" kids.

How is her speech, does she speak English well? If she speaks English she can learn math. Period. There are no special needs lessons plans. Do the same lessons you see on the blog and website. You may have to go slower and for shorter periods...but that's it. MAKE SURE SHE gets her hands on the blocks and isn't just watching you play with the blocks. I have had success with downs syndrome, deaf kids, blind kids, autistic kids, kids who climb on rocks...kids is kids no matter what "the experts" tell you. Use all the senses or as many as possible. Play, have fun, learn math. Don't approach it as math time so much as play time. Sing songs. Play games. Watch vids and then DO the stuff in my vids...don't worry about getting it wrong. You can't play wrong.

Sarah is Autistic use your password and check out her page: http://www.crewtonramoneshouseofmath.com/Sample-Lessons-Sarah.html

Here Sarah is doing fractions (again):



After about half an hour she starts "getting it." We have covered fractions before in different ways...we started from scratch, like she had never seen them before and went from there. And this is the result:



As you can see it's becoming easier and easier. You do the same thing with "normal" kids especially younger children. Teachers lament that they have forgotten everything they were taught over long summer breaks. This is normal and natural and often shows mastery has not been attained, although sometimes the are doing as fast as you can write a problem looking very much like mastery has been attained and then a week later they can't remember how to do it. This shows it takes many exposures to put information in long term memory and more than that to make knowledge available for instant recall. Expect this.

She did this one in her head as soon as I wrote it down...however mastery has not been attained, one small change in the process and she gets flustered. As the picture where the problem is scratched out above shows...watch the vid and you'll see what happens when a change is made to the process.

I do the same lessons several times separated by time, they seem new for some, but usually I say the first time it's new, and might seem strange and maybe even a little hard. The next it's I've seen this before, the third time it's I got it already lets do something else. With SPED students it may take an extra time or two beyond that.

If you want to see the whole lesson (you need a password)...just go to Sarah's page. The vid is 24 minutes long. You will see the ups and downs. Here you just see the success. There you will see how we got there...and future vids will make even more sense.

I recommend this for people who don't have SPED students too. The lessons are basically the same...now that she has this lesson mostly mastered and I have watched the lessons, I will emphasize different things, next time we will focus on "filling up the one" and work on improper fractions and then we will be able to do more fractions where we add and then subtract and eventually have to borrow out of that one...we can also go back and forth with reducing fractions and equivalent fractions....then multiplying fractions will be easy and at last division will be covered and you can be assured she will discover WHY we invert and multiply. Again.

If you have teen age students that are doing algebra you might go over fractions again with them because consistantly I see kids in algebra that can't do fractions...or forget how or can sort of do fractions but hate them which is silly when they're so easy.

One last note. Several parents have told me they get lost watching the vids and sometimes don't understand what they are seeing...although it looks fun and easy they don't get it. "Do you have a set of blocks?"

"No."

Therein lies the problem. I can't stress enough that in all the time I've been doing this the ONLY time we lost people is when they didn't have a set of blocks to play with in front of them or wouldn't put their hands on the blocks...the blocks get you through your mental blocks. That's why Jerry used to say "Math is NOT a spectator sport." You have to participate.

If your are an extremely visual learner just watching vids may be enough, but if your are kinesthetic or auditory or a mix where visual is not the primary learning style then hands on is a must. Not saying this to sell you blocks. I'm telling you this because my experience has proven beyond any shadow of a doubt that this is TRUE.

Crewton Ramone's House Of Math

Search this blog for "fractions." (There's like 8 pages of results worth.)

BTW if you want a set of fractions tiles like the ones you see here it's $105.00 shipped to your door and is available on my site at Products and Passwords now. Or order at right. ⇒

Fractions Are Easy Too

This is a simple introduction to fractions tiles. Fractions concepts are great for teaching counting and multiplication to young students.

Most students hate fractions because after four years of learning math one way suddenly the rules change. You can't add

1/2 + 1/3 and get 2/5...some kids will put 1/5

showing an utter lack of understanding and confusion about the rules. But the rules haven't changed. They just didn't understand them in the first place or more likely were never even taught the rules properly to begin with.

If you go back to the 5 basic concepts, you will find under concept 2, we only count things that are the SAME. Also it's key to understand that numbers have two parts the what kind part and the how many part. Fractions are cool because they tell you the how many and what kind all in one place, whereas with whole numbers you need to understand place value.

Note how in this introduction we don't cram in adding fractions in the first lessons all we do is play with simple concepts. We see how many make up a one, and we count and we learn their names. Later we will learn about numerator and denominator and build on the concepts of "same" or equivalent fractions. Also we will use other base ten blocks to represent fractions and fractions concepts. One crucial concept is the idea that x/x = 1. Right now they are seeing 2/2 = 1 and 3/3 = 1 and so fourth on up to 9/9...let them discover that anything over itself is 1...except for our hero. Plus they will discover this concept from multiple angles. Right now it's counting (three of three is one), but later it will also be division, (three is contained in three one time).

But for now we just play a little and tell stories about fractions wearing disguises.



In this fractions intro I cover the same info but without video. Be sure to get the free fractions worksheet. It's free. Note with me "free" doesn't mean "give me your email address." It means free. If you want more, get a password if not I don't want to waste time with you and an email list.

It should be quite obvious you can use fractions to teach counting and addition and multiplication. These students are advanced so the lessons are easy and fast but if you had a student that hadn't even mastered counting to nine these same fractions tiles would help them learn that. Compare and contrast with the thinking that says they have to have mastered multiplication before they learn fractions which is still ridiculously prevalent in math instruction in the USA and elsewhere.

The fractions page at the house of math has been in disrepair for about a year now...but that is getting fixed. It is very near the top of the to do list now. Look for more updates and improvements to that page over the next 30 days as I cut videos and add lessons and hopefully some more pdf's. Right now there are some videos there and a list of terms but it will become much more cohesive shortly.

Next there will be a video where we add simple fractions and find out what 1/2 plus 1/3 is and more importantly WHY, and we won't just learn some silly rule about multiply this, this and this...and then add this and this...but keep this the same...stay tuned.

BTW if you want a set of fractions tiles like the ones you see here it's $105.00 shipped to your door and will be available on my site at Products and Passwords shortly.

90 Minute Addtion & Multiplication Engagement

Here is a short screencast covering a lesson with two young students of 7 and 8, we managed to spend the whole time just playing addition and multiplication, everything we did was addition and multiplication...even though we did fractions and division...it was still just addition and multiplication.

We practiced concepts we already knew and practiced writing...this is the only way to attain mastery: practice. So this class was pretty easy for them no stretches nothing new really although each time we do multiplication some of the facts SEEM new event though they've seen them before.

Here they have threes on their fingers, and had to work together to show me eight threes...we get practice with threes AND we get practice with addends because they have to decide who will will show what, three and five or four and four or whatever...

Here we took our threes and laid them out and put them in groups of 9,  talked about 3x + 3 and x being 9 in this case, also talked about  9 times 3 plus 3...and of course 10 threes all being 30.

Writing out what they see in symbol form and beginning to get what equivalent fractions mean, we also talk about common multiples.

Here we build rectangles and count the sides, we count them several ways the sides only for example 4x5 and 5x4 and then what if I have a rectangle where one side is 4 and the whole thing is twenty? What's the other side. What if te whole thing is 35 and one side is 7? What's the other side...? Etc...

This sheet shows the fractions and a pause to write out sixes in a matrix where they CLEARLY see the pattern with the symbols...

Once in a great while I give them a worksheet to take home, I am making webpages at Crewton Ramone's House of Math that have free math worksheets, starting with some first grade math worksheets and fraction worksheets...I also have a kindergarten math worksheet for them to practice writing their numbers and learning their addends, of course it can be used with older students too, it takes lots of practice to write neatly. Right now there's only a few worksheets but I remember when I only had a few videos too, last count I was over 70...

One day there will be that many and more printable math worksheets available there. Not all of them are free but you can get a password for a buck and tat lets you get into a page that has all of them PLUS you can get into the advanced algebra page and my password protected screencast channel that shows how to factor negative expressions...a buck goes a long way.  Here is the open channel, I made this as an alternative to my YouTube channel so teachers and students could watch my vids at school.


Anyhow here is the screencast covering the 90 minute session.




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Notation Really Does Tell The Story

The point of this post isn't so much the mathematics that the students learned; there are plenty of "how to" posts on Equivalent Fractions, as well as other fractions concepts, Card Games and Algebra on this blog and on my website.

The point of this point is to draw your attention to the student's notation as a big clue to how well you are doing at getting whatever concept you are teaching across. Here it starts out "all hamajang" as we say in Hawaii and quickly turns into clear, neat notation. You can actually see the confidence build in the notation.

You can also see it here in the drawing and notation if you know what you are looking for, as he becomes more comfortable with the concepts and just drawing and counting the sides his pictures and notation get "neater" or "nicer", as I often admonish my students "Write neatly. Neatness counts."  And we laugh at how punny that is.

Here is a quick screencast covering the lesson and the concept that the notation is an important clue for you as the teacher. I use these because the example is dramatic and you can't miss it you may find it more subtle with your own children or students.



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Playing Math, Story Problems Fractions and More.

More work with my Autistic student. She was happy to report her school work with fractions was "too easy."

I treat her pretty much like any other kid and use the same basic lessons to provide powerful understanding of math concepts. After the concepts sink in math itself is pretty easy.



It was late when I made this screencast and my verbalization of the story problem in the middle is quite screwed up, lol!! One day will learn how to edit. I left it instead of re-cutting it because even with the verbal errors I think the concept comes through at last, and you can see even if you make a mistake you can still do a lesson.  So many parents and teachers never get started for fear of "doing it wrong", some of you may have even gotten it better due to my errors because you thought to yourself "what he's trying to say is..." Anyhow here it is correctly:

If each student got one bottle of water we could give 120 students one bottle each.

If each student got 2 bottles each we could give 60 students 2 bottles each.

If each student got 3 bottles only 40 students would get water, if we gave each student 4 then we could only give 30 students water...when I was doing it with her I was saying correctly and she understood what I meant. In the picture we see 120 broken into 4 groups of 30 each...we could also form a rectangle and count the sides.

Here is a good reason to use rectangles instead of circles or pies when presenting fraction concepts. being able to draw it demonstrates comprehension and moves the student from the concrete to pictures to symbols naturally.  For more on what that picture represents and how we develop the concept of equivalent fractions go here.


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If you donate a buck you get a password...the password is worth more than a buck...unlocks "advanced algebra" page and soon a screencast channel that will have all manner of instructive scereen casts on topics you can't get anywhere else...the videos on the password pages contain less errata.

“The authority of those who teach is often an obstacle to those who want to learn.” ~
Marcus Tullius Cicero

Moving Towards Mastery

Here is another enrichment session with a 6th grader That flew by.

"It felt like 5 minutes..."

That's when you know you are doing it right. Learning is fun and time flies when you are having fun.

First we looked over the left over lesson from the 4 and 5 year old and talked about equivalent fractions.

Then we wrote out a few equivalent fractions all the way to 12...1/2, 1/3, 1/4, 3/4, 5/7's...later will do other combinations.

Then we built all 45 addends. He tried to stand them up and goofed around a bit while he made them. I encourage this. Then we did "drills" verbally what's 7 + 5, what's 37 + 5, what's 25 + 7, what's 97 + 5? etc...did lots of them for various addends the idea is to get him off his fingers. Using your fingers is great! But you need to get past that stage if you are going to be fast and attain mastery.

Using fingers is a step on the way to mastery, unfortunately many students remain stuck at this step all the way into adulthood. For kinesthetic learners using fingers and hands IS IMPORTANT, that's HOW they learn, and you need to help them move past this, manipulatives are a great way to move them into "doing it their heads." For young students using fingers and hands is just natural...you can also spot the kinesthetic learners because they will rely more on their fingers and be slower to move on. This does not mean they are slow or any less able then visual or auditory learners, they grasp concepts just as fast or faster than those with other learning styles. We also find when it comes to sports and other activities requiring hand eye coordination (like arts and crafts) they often excel.

Many speed reading courses incorporate the use of the finger to guide the eye along the page, some use this to start, and then drop it for other courses this is the main stay of the course. Adding more sensory input increases learning, and in the case of reading the hand and the eye are integrally connected. The point is you want to encourage students to move through this step when it comes to the mathematics NOT discourage or skip the step all together. Some students will naturally NOT use their fingers when doing mental calculations...

After addends we played cards. Not poker, but war. We played multiplication facts war where we each put down two cards and multiplied them together; as an added bonus we couldn't start another round until he told me the difference between our scores. It was fun although I beat him.

Then because he had behaved will wrote neatly and did all the work I asked him without a single complaint I let him play Timez Attack for about 10 minutes...he really had fun with it and when it was time for class to be over he wasn't ready to leave...

"Always leave them wanting more..." as some famous circus guy once said...



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“If I am walking with two other men, each of them will serve as my teacher. I will pick out the good points of the one and imitate them, and the bad points of the other and correct them in myself.” ~Kung Fu-tzu Confucius