Playing Trig With Young Students.

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.


GET How To Teach Trig To An Eight Year Old
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.

Trig With Base Ten Blocks

Base ten blocks are well suited for teaching trig and trig concepts in a very concrete manner.  Starting off simply with Pythagorean Theorem it becomes quite clear that the blocks make things much more tangible than even the simple formula a² + b² = c².

Basically we take a rectangle and cut it diagonally into a triangle and then study that and those relationships.  We have names for those relationships. SOH CAH TOA will help you remember those names but it's important to understand what those relationships actually mean.  As I have said about 1,000 times probably more, if all you know is sin30° = .5 and cos60° = sin30° you are going to have a bad time.

Even the entry page to the Trig page has valuable lessons you can use with your young students on it.  And judging from the instant and huge response there is some demand for a method that makes trig simple and easy. One the trig page you get hours of video including lessons that actual students were given that allowed them to get 100% scores when they were formally getting less than 50% of tests right.

Base Ten Blocks make Trig assessable.

Once you understand the concepts all they can do is change the numbers. Click this link to go see what I'm talking about. Then get yourself a password. The Trig page is worth the price all by itself, but you get 14 other password protected pages all for just 6 bucks...plus for a two and a half bucks more you get parent teacher training and that is also getting good reviews and blowing people's mind's when it comes to how easy math can be. If you let it.

The lessons on building squares are going to make Pythagoras easy to understand, Pythagoras made Trigonometry easy to understand and you will see the base ten block method makes it all understandable. On the entry page there is now a lesson about the definitions and how to use them for parents and teachers but students could benefit from it too.  There is also some quick info on what to expact on the password protected page and another vid showing simple definitions. Like any language you have to know what the words mean before you can use them properly. When you teach math try to think about it the way you would teaching any other language, lighten up and play math.

Base Ten Blocks Make AP Calc EZ Too

I was working with an AP calc student. They are just getting started. As usual the student did not see the relationships between what we had done before and the formulas he was getting. When I showed him these side by side with a brief explanation the tell tale "OHHHHHH!" was exclaimed (the light bulb going on / the rush of endorphins when you understand something) and he said, "well, that's easy then."  Base ten blocks give a solid foundation so that when we get to this point many obstacles to understanding have already been mastered and are not issues.  Understanding square numbers and square root for example and trig which is basically the study of how triangles work is made plain using base ten blocks...no one ever explained to me in highs chool that math was a language and how all those formulas were basically saying the same thing and how and when to apply them. It was basically just memorize this, this and this...and I never got the relationships between them until years later.

Desmos.

This student comes once in a blue moon. If he came on a regular basis I wouldn't have to hit him over the head with it as it were and he would see that we are just finding the slope of a line, as well as see Pythagoras not only in distance formula but in the basic trig identity too.  In the last formula we talked about h being more than just h it's the distance  between the a and a+h expressed using economy of symbol.  (The thick black line below the red lines in the last pic below.)

He might also see our old friend from the doughnut factory and playing with ordered pairs rise over run for rate and slope and the "m" in y = mx + b or as he will now see it f(x) = mx + b where m = dy/dx....but all these triangles do go together....and it's pretty easy to see that even the complicated looking formula with a's and h's and stuff isn't so hard after all. The basics are covered on vids and on password protected pages like this. Look for Water tank, doughnut factory constant rate problems.

Then tangent lines and secant lines are no problem either and with a little more explanation of economy of symbol and that y = f(x) and how we just labeled the points with letters, the anxiety subsides rather rapidly and learning can take place. It's hard to learn anything while in flight or fight response.  We have proven this time and time again. It's pretty obvious that the secant and the tangent lines are going to need different formulas. Average slope between those two points is going to be less accurate the further the points are from each other and more accurate the closer those to points are together.  His online textbook actually had some cool gifs and animations that really helped explain things.  That link goes to a page that has an animation on the bottom give it a few seconds to load.

Now this and the formulas that came with aren't thought of as hard anymore and we can play and fool around with the concepts. If I want to know the distance between those points of course I use distance formula, if I want to know the slope of that line then I need to know dy/dx but now instead of just x and y, I have x and f(x) or in this case the points are labeled [a, f(a)] and [a+h, f(a+h)] but it's really just x and y...and secant and tangent and limits and intervals are all just language to describe what's going on here. But what if I don't want an average? What if I don't want the "as a crow flies" answer but the actual length of that curved line? Well I need more math and this guy named Newton came along and gave us more math, we call it calculus.

We can also find the tangent line to point P or any point along the curve.  Tangent Line = Instantaneous Rate of Change = Derivative.  That line would be "outside" the curve not under it....speaking generally.

Some more explanations and drawing make the general concepts understood. Here we can see the curve being analyzed a little further. See the triangle shaping up in the corner there?

We are going to use symbols to describe that triangle and the part we are interested in is the hypotenuse, but in order to find the hypotenuse which is the average slope of the curve between those two points we need to know the sides which are the changes or distances between  x and y.  

Secant line = Average Rate of Change = Slope. Simple.

 Add more information and it looks more complicated but actually there is MORE understanding taking place, not less...but you don't want to come in at the end...or in the middle which I find is where most text books start, instead of starting at the beginning.  I can also usually find the "holes" in the students mathematics that are making it hard.  Sometimes those holes go all the way back to not understanding division or fractions. Lots of times a review of triangles and Pythagoras is in order.

 Now I've added labels for the axis and put more info in blue the tan line and the secant line and the basic formula for it, the axis of symmetry is the dotted line and we can see that the tan line and secant line are not the same thing so we will need different formulas to describe them.

At last it becomes absolutely clear that we are just playing with triangles and those fancy formulas are just talking about the sides of the triangle...and all of this is in very general terms we are just playing with concepts. Next we will add numbers instead of just having the distance between a and a +h we'll have numbers that tell us what (where) a is and how far it is to a + h.  We'll get an quadratic expression  that describes the curve and from that we do lots of math. We can also find the area under that curve between a and a+h and we can be quite exact about it.    But that's a lesson for another day...and the blocks will come in really handy here too. Because how are we going to do that? By making and summing up rectangles...

Meantime we talked about limits and derivatives and how tan and secant lines differ and how to think about things in general and specific terms depending on the symbols.  We  talked about the power rule a lot and the chain rule a little bit and I showed him how the power rule works and what it means.

Going from f '(x²) = 2x and f '(x³) = 3x² isn't hard but the WHY of it can be. Then a little more discussion of dy/dx and d/dx and time was up...

A Little Geometry, Triangles, and counting to 180.

Some simple concepts regarding triangles and 180...This student up until quite recently found triangles to be confusing and stupid.  But with a little directed discovery and explanation things very rapidly started making sense.  Geometry is often easy for visual learners because they can see it...this is not a primarily a visual learner,  this is a kinesthetic learner. He is good at sports and the reason he showed up in the first place was because if he got an F they wouldn't let him play.

"If you can count to nine tell me if something is same or different or not and identify a rectangle I can teach you math if you want to learn it...and you need to be able to speak a language preferably English." Is basically how our lessons started and at first he was incredulous. After an hour triangles were EASY. Now we just have to work on the negative associations he has built up over the last two years...he got a D in algebra and hated it. I showed him problem solving and factoring and he was hooked.

Next we will mix in a little algebra...this vid also brings home the fact that knowing your addends by heart makes a BIG difference. Concepts build on concepts and skill sets build on skill sets...soon instead of "2" they will put something like 4x - 20 there. And the student will have to figure out that 4x - 20 = 80.   Which is a snap if they understand problem solving concepts but just adds to the confusion if they just got a "D" in algebra and hate it and none of makes sense. Hero Zero and No Fun Get Back To One to the rescue.

In this video a student I am tutoring discovers that the stuff he thought was hard might not be so hard as long as he can count to 180. The angles of triangles always add up to 180 and using a little logic and reasoning we see why the exterior angle and the two non-supplementary angles add up to 180 degrees. I did a lesson already using directed discovery so that he was able to pretty much figure that out for himself he just couldn't elucidate it well so I showed him using some symbols and it really started making sense. (See also vid below.)

You can't see the x but it is the other angle in the triangle.

This student is a little camera shy and as with all students, tends to behave a little bit differently when the camera is on which adds a little pressure. This is another student that was in danger of not being able to play sports because of his math grade and apparently he's pretty good at baseball...and does "ok" in his other classes except, of course, math.

From what I can see he is indeed a kinesthetic learner and once he gets his hands on the blocks a lot of math problems fade away. (Do you see what I did there? I crack myself up.)

Now we are ready for a "complicated" problem like this on to the left. Looks like it might be tricky but again if you can count to 180 it's not very hard. Turns out it's just two problems stuck together.  Suddenly it's pretty easy except for the part that he sees a problem like this and automatically runs a program in his head that this is hard...but then it turns out to be easy and the program is modified. Soon he will be able to understand even poor explanations because he will be running a program called "this is easy I can do this."

The geometry wasn't making any sense to him nor were the explanations he was getting from his teacher, so he quit paying attention and this turns into a loop that ends with an "F." BTW he tells me probably 90% of his peers are bombing tests and getting poor grades. I'm pretty sure the teacher is just as frustrated if not more than the students.

Anyhow, once he starts understanding basics concepts and the generic ideas all they can do is switch the numbers...and test scores go up dramatically....and people think Crewton Ramone is magic. But it's not magic it's just math....and knowing when to start counting to 180....



Oh and BTW Happy New Year.

Sample Lesson Math Vocab

Here is a sample lesson where we learn a little math vocabulary. It's important to know what the words mean...in the mathematics the words sound funny to little kids because they come to us from the Greeks and from Latin...but the concepts are easily understood.

You should also be able to see how the concepts of area and perimeter NATURALLY lead to practice with addends and multiplication and division. This goes well with basic lessons about addition.



I employ the three period lesson and you will note the repetition and lack of the word "no". If they get point to radius when I ask for diameter I tell them that's the radius show me the diameter...no need for the word NO.

The vid is a little long but it's a useful as a sample lesson and will appear on the Sample Lessons page.

Here is a short vid from a little later in the lesson where we are just playing with blocks and learning and reinforcing addends: