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 math problem solving. Show all posts
Showing posts with label math problem solving. Show all posts
You need more than one example but this is a quick 3 and a half minute lesson on how to find the inverse of a function.
In this case what is the inverse of
y = -4x² + 2 ?
Due to my math experience I looked at the multiple choice answers and knew which one it had to be...however due to my carelessness when I tried to show him why I got the wrong answer because I left off a negative when I copied down the problem. We did the problem more than once and this vid shows the last time we did it.
Two principals here:
One: NEVER TRUST THE TEACHER.
Two: You aren't wrong you are just getting more information.
Too many parents (and some teachers) are afraid to work with their kids because they might get it wrong and be embarrassed or what have you. Just get to work and see if you can get the answers...sometimes it's good to get the answer first and then look at the problem. I have had more than one student say it's easy when you know the answer. That's true and then with the thought this is easy you can see how to solve the problem and you know where you are going.
Another thing with some multiple choice tests: you can see answers that reflect common mistakes or misunderstandings of concepts that would lead you to pick the wrong answer. Talk about them. Talk about why they would put that answer as a choice on the test. This often helps deepen understanding.
Now in order to learn how to do these it requires more than on 3 and a half minute video. They can get the rule "just switch the variables and solve" but they will forget the rule in just a few weeks...
You need at leaste three examples of which this would be one of the last ones. They should have also spent some time understanding the basic concepts of Hero Zero and No Fun Get Back To One. (HZ & NFGBT1)
Here is the first example I showed this particular student:
y = x² therefore the inverse would be y = ± √x
First we "swap" the variables: x = y² which is the same as y² = x and solve (NFGBT1) by "square rooting" both sides: y = ± √x
Then you just make them slightly more complex: y = x² + 1 etc...
Then you get two thumbs up because they "get it."
"Example isn't another way to teach, it is the only way to teach." ~AE
Note that depending on age the emphasis changes...for the older girls it's about the concepts and seeing the rules they have been taught, for the younger students it's about finding same and building addends.
The basic concepts are hero zero and no fun get back to one...but it's all about using those to find same and SEE what x is...the equals sign tells us what x is...and x can be anything depending on the problem.
For all of them it's about making it easy and fun. Let's start with older students:
Now as is typical of many high school math students, they have yet to master multiplication and resort to fingers when adding. I will get them off their fingers and they will have to master multiplication on their own...
Then Watch a six year old doing basically the same thing, CONCEPTS are the same. (BTW it's a piece of organic all natural endangered species chocolate which he picked out from the health food store.)
And the older kids (begin to) SEE the rules:
They are hampered by a mental paradigm that math is hard even when it's this easy...it takes a while for them to realize really math is easy; but when that shift happens the math becomes easy and with or without me A's can be achieved. In the vid we did not evaluate for x = 6 but I did so with them and with Dboyz who figured out that 30 plus 3 is the same as 18 plus 15...and they were quite proud of their skip counting skills...counting by 6's all the way to 30 with the younger boy in the lead. When doing 18 plus 15 they knew five wants to be ten so it takes a five from the 8 leaving 3 tens and three...they also did 8 takes the 2 from the five and again we get 3 tens and 3...lots of DO-able math for kids of all ages. 10 years from now this will be old hat for some students. More to come on the problem solving page at the house of math...the question will come up, how do you do a problem like this?
3x - 2 = 2x + 5...there's no -2 in five. True. You need hero zero again, none of this balancing stuff...you just see same on both sides.
Or
5x + 3 = 2x - 12 again many people can't see same on both sides...that's because of hero zero again. Want a detailed explanation? You need a password. PPPSP = The Password Protected Problem Solving Page.
There are tons of different problems on the PPPS and vid that make them ALL super easy. Remember those boat and current problems? Or how about constant rate problems? AND of course the ones you see here where you might think it's not same on both sides because things got negative...just a few minutes and you will see your way clear...
I have heard from a student that has since moved to the mainland that she is getting A's for the first time in math...all I did was adjust her thinking and hand her some concepts she can use no matter what the topic...beliefs are powerful things. She now believes math is easy...she knows it...so of course she's getting A's. I don't need to be there. And that's the point of all this.
I used to keep a overhead slide of a Far Side Cartoon By Gary Larsen with me as I introduced my two hour lecture on Problem Solving. It showed a picture of the devil and a bunch of books; on closer inspection all the books were story problems and the caption read "Libraries in Hell".
It was worth a chuckle. The point of learning the language of mathematics is for problem solving. Counting and computation as I have said many times are for doing math but the point of being able to count quickly (using basic operations) is to solve problems. Story problems show some of the uses for algebra and computation, build up problem solving skills. Problem solving skills include logic and reasoning, which together help form critical thinking skills. Some of the skills learned along the way include using mathematics to express reality, that is, translating real world situations into mathematical symbols and then knowing what those symbols mean. Turns out the same symbols can be used to express lots of different things...
Of course the vid gets cut off at the very end. If you watch it we will see that Spaceman Spiff catches him after 3 minutes. How far away from the moon base are they when he does? An astounding 90+% of students can NOT do two stage problems, that is solve a problem to get the answer to another problem. Once I find X, I can evaluate the equations and see if they are same. In mathematics once we find one variable we can usually find other variables easily. If we understand the concepts.
In calculus we study one and several variables. I used to start my pre-calc lecture where we just did linear algebra by asking the people in the room especially college graduates and teachers to give me a short definition of calculus anybody could understand. 99.99999% of the time none of them could. They might come up with some convoluted definition that even they were somewhat unsure of or come up with a very complex definition that they understood but when asked the rest of the room was not able to make sense of it...and I we would all laugh. This happened in cities from Maine to Maui and everywhere in between. The stress test was always could a little kid "get it." I have a simple definition. Look for it in an upcoming blog post meantime think of one for yourself if you've studied calculus.
In linear algebra we just begin to delve into calculus concepts and the most basic concept of all: the idea of a variable and how two variables interact. For now we are just fooling around with X. Later, as we see at the bottom of the page we can get into Y. (Make up your own joke.)
"The most powerful single idea in mathematics is the notion of a variable." ~K. Dewdney
So we start off with X.
Lets take this problem for example. 3x + 4 = 2x + 9 we can tell a story about Spaceman Spiff, or snowball fights, or we can just play with the blocks and add more meaning later. The idea is not to just give them a set of rules. Give them concepts and algorithms that make sense and that they can see in action.
The basic concepts in use here are Hero Zero, No Fun Get Back To One and of course the rectangle.
They can then use those concepts and algorithms to DO math, rather than just memorizing rules and process...which we have seen DOES NOT WORK. But who is in charge of math education? The 5% who easily memorized rules and process...see the problem? Good, because they don't.
The problem solving page is starting to grow. There is also a Password Protected Problem Solving Page now with more stuff on it. There are tons of common, perennial classic, story problems that students get exposed to during their journey through mathematics. Rate and Distance, Boat and Stream, Percents, Mixture and Solution, (I have some 5% solution and some 12% solution, how much 12% solution do I have to mix with 5% solution to get 8% solution)...there also some like in the case of ordered pairs that they don't get exposed to and never get to understand. I know a ton of students that could not tell you that an ordered pair could be a story about a water tank or a doughnut factory and the resulting equation could be a graph of the production or amount of water as it fills the tank. All they "know" is slope intercept form, y = mx + b, and points on a graph. Half the time they can't remember which is the x and y axis. How this relates to a story about a tank of water or factory or anything else.
In that short 60 second clip I don't tell the story that went with the symbols but in the longer video OF COURSE I do. Here is a story:
I have a doughnut factory. After 3 hours, there were 16 cases of doughnuts, and after 8 hours, there were 31 cases of doughnuts how many cases where there to begin with and how many cases do they make per hour? That's the most basic story. You can do the same with a tank of water, after 3 hours 16 feet of water after 8 hours, 31 feet of water in the tank. How fast is the tank filling up and and how much water must have been in there to start?
Then you can throw extra bells and whistles like the tank holds 40 feet of water how long does it take to fill? Go negative and drain the tank...use 39 and make them do fractions...but make sure the concepts are understood before you do any of that.
You can find many amazing and easy ways to explain these in the Series A Manuals from Mortensen Math. Eventually, I will make videos of all of these kinds of problems but it takes time because I take the many steps in the degrees of difficulty...and don't just dive in and do the problems. Parents and Home schoolers seem to appreciate this, but it's driving some of the math teachers crazy.
This method is slower at first but pretty soon it goes faster and faster until it surpasses traditional methods by far and you see little ten year old kids that seem like geniuses...it all starts with a game for 4 year olds called "what's under the cup?"
There is also some problem solving going up 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...
