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...
Showing posts with label Problem solving with Manipulatives. Show all posts
Showing posts with label Problem solving with Manipulatives. Show all posts
Here is a short video that might change the way you look at equations and your understanding of the equals sign. =
This symbols never lies. = means same. Here is what many consider a novel approach to problem solving. Hero zero makes an appearance and over time problem solving this way will be second nature. You job is to get them to understand equals means SAME and introduce the concept of hero zero and sometimes same is NOT visually obvious and we have to get help from our hero zero.
In the problem 5x + 16 = 4x +12 we are going to end up with NOTHING on one side. We have a symbol for nothing ZERO.
We start off with a low degree of difficulty like this:
3x + 2 = 2x + 6
then
5x + 16 = 4x +12
then
2x - 3 = x + 5
a few more and these kinds of problems won't be scary at all.
Get my book on subtraction, Crewton Ramone's Supremely Simple Subtraction it also covers the basics of this kind of problem solving. These concepts can be taught to and understood by even very young students. Hopefully you begin to see how it all goes together...check out this what's under the cup video and you can see problem solving introduced in the most basic way with what's under the cup. When we bust out hero zero we have moved down the road a little. But 5 + x = 10 and 8 + x = 10 and 3 + x = 8 are great ways to start with the CONCEPT of making same and then finding same on both sides is a natural progression. Moving from whats under the cup to seeing 8 + x = 10 = 8 + 2 therefore x must be two is just another more complicated way of playing what's under the cup, but to kids who have played what's under the cup at an early age the symbols make sense because they have SEEN what the symbols represent from an early age.
"Sometimes it is useful to know how large your zero is." ~Anon
"The hardest arithmetic to master is that which enables us to count our blessings." ~Eric Hoffer, Reflections On The Human Condition
I had a student talk about cross multiplication like it was some kind of dark magic. Here is an explanation of WHY cross multiplication works.
You should be able to see it. The relationship is one to three. in the second set the relationship is still one to three...but it's two to six. You can play games where you get one and the student gets three. In that case you might have put a two down and the student might get out 3 twos.
But here we are looking at the relationship in a slightly different way.
You can see how it works: if I have the rectangle and one side or two sides of the rectangle...and this is what we are employing to problem solve. Concept three: we form rectangles to facilitate counting.
Now from there we can do problem solving in Geometry with triangles and other shapes..."this is to this as x is to this..."
For example 6 : 15 as 14 : x...of course you could cross multiply and divide but there is an easier way. If you can start using rectangles in you mind problems like this will be easy to see. And even if you can't, the concepts are still easier to understand. This also points out the need for mastering multiplication tables. People will say who need it now that we have calculators...well being able to punch a calculator will not help you recognize 6 and 15 are on the threes multiplication table. And people often lament that calculators don't help with fractions very much...
They do but only for computation not for understanding the math. Some kids can "see" right off the bat that on top we have 2 threes and 2 sevens and if we have 5 threes on the bottom we must have 5 sevens too. Others take more explaining. Adults always take more explaining.
Here I drew in the rectangles. Use you imagination to see the sides. The whole thing is six and one side is two the other side MUST be 3. The whole thing is fourteen and one side is 2 the other side HAS TO BE 7. If the "top" is 3 the bottom is 3, if the top is 7 the bottom has to be seven because the relationships are the same.
So if I know the sides are 7 and 5 the whole thing HAS TO BE 35. We love math. It isn't going to be different tomorrow. No exceptions. And you should be able to see the relationships are the same two to five...doesn't matter what it is.
2x/5x in this case we have 3 and 7...the logic also works no matter where we put the x.
Then we the student sees this, it makes perfect sense. It's not some kind of magic or a stupid rule to follow that just is.
Students need to see where the rules come from and that makes them easier to understand and then we can work on applications with triangles. We can also work on applications with triangles or rectangles and show how these rules work, but...
It's been my experience that having some experience here first makes a big difference there.
And here is some more covering this topic where we can do it the easy way or the hard way. And if we HAVE to do it the hard way at leaste we understand why it works. Taking the mystery out of the math is important.
This student has just about gotten the concepts mastered. He is a quiet boy so he doesn't say much but he did exclaim, "Ok I get it!" at one point and "that makes it easier to understand..." at another. You will find this response almost universal among students who just learned it the other way...using cross multiplication.
Here is a slightly different way of looking at it using different symbols:
In this method it becomes visually obvious that we are talking about multiples.
There is lots more we can do with ratios but that's enough for now...
Here is a second post about how easy it is to incorporate base ten blocks into your classroom situation. Here is the other post I did called Crewton Ramone In the Classroom.
This is also great for stoking their imaginations as they tell stories about the ruins of Egypt and Great MATH Pyramids.
I've gotten into contentious debates about whether little kids can do math like this. What is so hard about it? They use their ability to discover SAME on both sides to discover x and then they have to do some more thinking to discover what the number is that is disguised on either side. You start off easy and work your way up...in both of the cases I cover here some students got the answers instantly.
You can hear one little girl in the class throughout that vid. She has been given lots of labels, I wanted her to stay in the class even though she is disruptive because like all the other kids given the chance she proves to be pretty good at math and this is valuable at repairing her damaged self esteem. Anyhow, there is a myth that employing base ten blocks will cause mayhem in the classroom. This doesn't have to be the case as seen here:
I will be the first to admit that it CAN get loud and out of control if you let it. Here some of the class is gone because they have remedial reading, but groups of four per combo kit works well and there could easily have been all 24 of them. I never really got any of the lessons on Pythagorean theorem I did on vid but factoring, fractions and percents are all easy to do with manipulatives. Here is an additional page on how to teach addition. (Get it additional!! C'mon--that's funny right there.) You could spend quite a bit of time there if you follow all the links on that page.
At the same time you introduce those concepts you play and build addends learn about integers and subtraction and practice multiplication so they can see the whole picture not just tiny compartmentalized parts.
This one shows them being creative, developing their fine motor skills and more. They not only built addends they had to work at getting the sevens fence to work using the posts...which are addends for 5...younger students would have a hard time making that fence work...these activities are great for preschool thru 3rd grade although older kids like to play too and some who never mastered their addends could also benefit.
Building and playing gives them a positive math experience.
They are in second grade. It shouldn't be work. It should be fun. These kids looked forward to Mondays with Crewton Ramone. They go home and tell their parents about what they did and the parents are amazed at their little second graders explaining factoring or Pythagorean theorem to them, I was at a birthday party and basically got mobbed by the parents there who were amazed that I was the math guy. They thanked me for teaching their kids math beyond adding and subtracting which is about the extent of the accepted 2nd grade curriculum... Adding subtracting a little measurement money and time telling.
This one can get out of hand because if they aren't careful the towers crash and there is a lot of excitement. Note that the point is to build addends NOT to build towers. I have seen teachers have them build using all the same blocks instead of the combinations...which defeats the purpose. Also you can count each level by the 4 or five blocks they used to get some skip counting in...in other words count the all the 7's on the 7 level...7, 14, 21, 28, 35...all the nines on the nines level etc just to get the patterns for the first few in.
For older kids (third or fourth graders) I make them build it and then tell me how many units they used...they have to do a lot of counting. That means count every block...including the hundreds...so lessee that's 100 plus 5 tens, plus another 100 plus 6 nines plus another hundred etc...some kids count all the hundreds then the pillars...no matter how they do t it's a great exercise in addition and multiplication.
Here is another fun exercise making a cube (it takes two top trays however) and I didn't make this vid in the classroom but at home with three home schoolers but there's nothing stopping you from doing in the classroom if you have the blocks. I did this with the first graders but never got it on vid. I made them line up and each kid built a combination and they went around and around their tables as fast as they could trying to build theirs faster than the other teams. A teacher could help keep the blocks straight and from falling down as it gets ten high. When you are done you can talk about cubes and notation if you want.
All in all it's EASY to use blocks in the classroom. You will find that just by playing they learn their addends and a lot more all on their own. Some direction from you can help but it's more fun and effective than worksheets and book work, the students learn more faster and you will earn a reputation as a good teacher and a fun teacher that students look forward to coming to rather than not.
Here is a page at the house of math that has a little more about using base ten blocks in the classroom and an instructional video with a million dollar idea in it...it basically links back to this blog post and the other one but it also contains a little more info and a couple of other vids to help you.
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.
Ratios are easy. All you do is cross multiply. Why? And what if you for get the rules? Here is a series of videos staring Sarah who is studying for her GED. This series gets cut off before we finish but what is on vid is worth sharing. Remember this kid is autistic...but I don't treat her with kid gloves (anymore), and she is doing quite well with math. Look for other blog posts where she is featured and prepare to be amazed. Also look for a future post where the autistic savant rears it's head...she is beginning to recall math facts and remember seemingly random facts that she has seen before.
{I'll put a link in when I make the post. Note to self.}
In these vid we begin to see what do when presented with ratio problems and rather than give rules on when to multiply or divide we think about the relationships. Once we understand that then we can start making up rules to describe what's happening and what to do to get the answer. Note I give her no rules. Later she tells me the rules...unfortunately I did not get it on video. But by the last video you can see she understands it. Then and only then do we move on.
Note I put the P1 in front to make it easy to search and find the next vid...P1 Crewton Ramone and Sarah Playing Ratios and Relationships.
Note when I say tricky I am using her verbage because there's nothing tricky about these it's just math.
As they understand it they do it easier and faster.
And here is the final one where she shows she understands it.
After these we move on to problems where it's not just whole numbers but fractions but we didn't get any of it on vid...you can see more with Sarah on Sarah's page.
Basically, we are playing advanced "what's under the cup?" but the name of this game is "finding stuff that's same on both sides."
It's fun and even easier than "what's under the cup?" if you do it right. Start off easy and all positive and no fractional answers.
The answers to problems like these are visually obvious. Once you remove the stuff that's same on both sides, this is a slightly more advanced problem because it involves hero zero AND no fun get back to one. The boys can see the answers and what to do. It's FUN when you get it right. Some observant people have noted there are 5x worth of manipulatives on the one side instead of 4, don't worry we sort it out in the longer video.
These two had been couped-up in the house for a couple of days with runny noses and fever but were still able to play math a little bit. Along with several hellacious games of Chutes and Ladders...
Here is the short version where all they do is set up this problem, in the full video we solve this one and several more. The boys say they are easy-peezy. I know several high school students who would beg to differ...as well as a few home school moms.
4x + 1 = 2x + 9
Child's play. This problem is easy if you can see it. Many people I know tend to have their eyes glaze over as soon as they see the algebra. Once they see the problem solving page at Crewton Ramone's House Of Math they can't believe it's that easy. Really, most of the math is child's play and you can make games of it if you are willing to get creative. There are math teachers who get quite excited when they see this way of teaching. Parents who thought they couldn't do math also get excited but they lack the knowledge to make up problems easily on their own, this can be remedied with practice. Here are a couple of story problems that can be represented with these symbols:
4x + 1 = 2x + 9
Bob and Jay are out of town. Bob is trying to catch Jay. Bob is on his skate board going 4 miles an hour, Jay is walking along at 2 miles an hour. Bob is one mile out of town and Jay is 9 miles out of town. How long does it take Bob to catch Jay and how far out of town are they when he does?
Or you could make it a little more difficult: Jay is walking at 2 miles an hour and is 9 miles out of town. Bob is only 1 mile out of town but is walking twice as fast. How long does it take Bob to catch Jay and how far out of town are they when he does?
Same Problem different story. Bob and Jay are going to have a snow ball fight. It the middle of summer in Utah and there is still over 8 feet of snow on the ground in places that are usually green with grass growing on the ground. How long before Jay and Bob learn enough math to debunk global warming claims? No wait...let's try that again:
Bob and Jay are going to have a snow ball fight. Bob has 1 snow ball and Jay has 9, which isn't fair. So they each build more snowballs. Jay works for 2 minutes and Bob works for 4 minutes, they build the same amount of snowballs per minute and when they are done they each have the same amount of snow balls. How many snow balls do they have and how many did they make each minute?
Math is more fun if you make a game of it. But eventually games are no fun if they are too easy and present no challenge. They are also no fun if they are too hard. There is a game called "what's under the cup?" that many think is just for little kids.
The idea is to start easy then advance by degrees. You will find that pretty soon your students are doing some pretty "serious" algebra. The idea is to understand the concepts and the way you do that is obviously not to memorize a lot of rules and formula. In fact, you will see students begin to make up their own rules as they get better at playing "what's under the cup?"
At first problems like these pose a challenge as the students are learning to count, and can pose a problem for some students when these are first introduced in 5th or 6th grade. They get confused with "inverse addends" and "inverse functions", as well as rules about walking across the equals sign...just tell me what has to be under the cup.
Once this game becomes easy, you can move on to more advanced what's under the cup. Problem Solving is easy and you will see their thinking skills improve as they use their computation skills to start doing math via algebra. I thought this lesson was about 10 minutes long, turns out it's 25 minutes long. Times flies when your having math.
This was their first time playing advanced what's under the cup like this. The next time will be easier and the time after that easier still. Each time we play another concept can be introduced and other concepts reinforced. Look for more vids with other kids in the coming weeks where we start adding negative numbers...they will be on the advanced algebra page...
The video above will be added to the Problem Solving page. Which is about to undergo a revamp...or I may have to add a another page or two. Meantime you can search "Crewton Ramone Problem Solving" in your favorite search engine or on Youtube.
Here is a video that gives an overview for older students who don't have the luxury of taking one concept at a time. This is really advanced what's under the cup:
And the little boys can do this also...the game is called find stuff that's same on both sides...
So nice to have the luxury of time! When I made that video I had 10 minutes to get the point across and Youtube gave you one minute grace and I used every second of it. Now I can pause and blab about food and whatever bounces into their heads...and eventually you will see the boys doing some of the exact same problems you see in the second video. If you keep it simple and make it so the answer ends up being x equals a number they are actually easier than some of the problems in the first video...
Look us up on Facebook and Twitter...you might consider a password as a donation. This project is pretty much 100% powered by passwords and some larger donations I have received. People are quick to send me emails asking questions, advice, and for videos on specific topics but pretty slow to send even three bucks. Those who buy combo kits or take lessons get free passwords and those who donate 25 bucks or more have found they get vids made for them pretty quickly and everybody gets a response to email eventually...some quicker than others, but I generally answer them all. If you already have base ten blocks or know somebody who does, make sure you let them know about this site and encourage them to get a password.
I always enjoy the emails that basically say, "what can I do to help?" Some people have some great and creative ideas. The best thing you can do is spread the word. That doesn't mean sell passwords for me. That means just post on your Facebook page, share posts or vids with friends, hit the "like" button, or "tweet" button, make comments, that's the best help. Eventually I will take some of you up on some of the offers...but for now just spread the word...oh, and get a password. Even three bucks helps...
Start off easy and it will stay easy. What are we doing? Understanding basic concepts and ten seeing how we come up with te rules and process they teach for solving equations. This student is 11. This is his second exposure.
l + d + t = 12
l + t = 8
l + d = 7
Solve for l, d, t...
Now make up some of your own...
Then lets go back to some "regular" problem solving, which seems extra easy now that we only have one variable to figure out...later once the concepts are fully understood we can go back and have multiple lions and tigers and dogs, oh my. (I couldn't resist.) Later we can introduce two lions, then maybe two lions and three tigers...etc...
Right now what to do when a negative number shows up?...understand same, hero zero and no fun get back to one.
Look they practically give the thing away...get one if you haven't already got one tucked away somewhere. Great for making highskool kids see how easy it can be...
Here is a quick screencast, less than five minutes showing two problems.
6x + 2 = 14
There are 6 Girl scouts with some cookies. I have two cookies. Together we have 14 cookies do we all have the same amount? (...or how many does each girl scout get?)
9x + 3 = 21
Nine kids have 21 apples divided among themselves, they have three left over...how many apples does each kid get...?
There's a lot more I would have said on the first one by way of explanation...but it was on the fly unrehearsed one take...the problem solving page goes into more detail because I'm sure after watching these which are NOT beginner problems you will have questions.
For a more complete explanation on problem solving go to Crewton Ramones House of Math.
Soon there will be a few more pages devoted to this and more "story problems" on my site. Right now you can go here for a little bit more: CRHOM Problem Solving.
That page is just the tip of the iceberg. So much to do so little time...
First we did a quick math lesson on same. I just brought a tiny white board into bed. Often we watch Multiplication Rock vids...but today we did this instead.
= means same.
Then we did 3x + 2 = 2x + 5
They love erasing same on both sides.
It looks like this:
||| : = || ::.
The could easily see same because I had color pens, the x's are blue lines, the units green dots.
So they erased same on both sides
| : = :::.
First we erased x's.
One little boy got to erase two on one side the other little boy got to erase two on the other.
We talked about HERO ZERO! (2x - 2x) older students hear hero zero no fun get back to one repeatedly.
3x + 2 = 2x + 5 -2x -2x
Then we erased two green ones from each side. One boy got to erase two green ones from one side and the other got to erase two from the other.
| = :.
x + 2 = 5 -2 -2
x = 3
This took a lot of counting and thinking for the 4 year old using his fingers to take away 2 from three 2 from 2 and 2 from 5 BUT ALL WE DID WAS COUNT and understand the CONCEPT of same.
The we made sure we were right replacing each x with a 3 because we found that
x = 3!
So if x =3 then each blue bar is a THREE.
:. :. :. .. = :. :. :. ..
3 +3+3 +2 = 3+3+5
11 = 11
Counting to 11 was a big deal...and lots of fun.
This is what the white board looked like when we were done:
This is even clearer with the right manipulatives...
THEN
we read "Ten Apples Up on Top". We counted the apples just about every way possible including skip counting and counting by tens...
Then we read The Popcorn Dragon which has no math in it at all...
