Trigonometery For Little Kids with base ten blocks.

In this pic we see our little man looking from 60° so we can figure out the relationships and put them on the table.

Quick lesson on Trig. This was maybe 15 minutes of an hour lesson where we practiced addends, algebra, multiplication, subtraction, division and more. If it was a little kid we would have drawn a generic triangle and talked about SOH CAH TOA. And that would have been it.  See: How To Teach Trig To Eight Year Olds.

This child was 10, so we did a little more. Talked about two special triangles and made a little trig table, NOTE we did not rationalize the denominators, that will be another lesson. Baby steps. Concept 5: no fun get back to one...

One concept at a time. Keep the cognitive load LOW. Right now most important is understanding what SIN means, which is simply the relationship between the opposite side and the hippopotamus...but you need to understand the what the opposite side means and that it depends on where you are. But look at it as vocabulary. SIN COS AND TAN are words that have meaning and are the basis (along with
Pythagorean theorem) of trigonometry.

We talked about Pythagorean theorem (one or more lessons in and of itself), the fact that the angles add up o 180 another lesson in and of itself and a great way to practice some subtraction...subtract from 100, 100.00 and then subtract from 90 and 90.00.

I hope you can see how after a little bit of practice figuring out angles, where we subtract from 90 you could easily segue into subtraction or algebra lessons x + 60 = 90...as opposed to dumping it all on them all at once and then going into the unit circle...sin = y and cos = x?? What? Just memorize SIN30° = .5

We talked about that too...if all I did was memorize SIN30° = .5  and didn't even know that meant the relationship between the opposite side and the hypotenuse was one to two, trig is going to be hard and NO FUN...and no fun. And if I play on the unit circle without even having been exposed to basic concepts the cognitive load is too high, not to mention they probably aren't even trying to teach to my learning style, and I get an "F" and then I have self esteem problems...and $35 bucks a month is too much. OK.

There are a leaste SIX separate lessons here...all of which can be EASILY understood by a six year old if you keep the numbers small while explaining the concepts. Get a password. For $35 bucks a month for 10 months you can have these lessons and more FOR LIFE and AVOID the tears that typically come with math when they are teens...most people wait until their kids are 13 or 14 or even 15 and 16 and confused instead of avoiding the problem in the first place. 

Soh Cah Toa Again.

Here is a fast lesson on the special triangle that is 30, 60, 90 with a hypotenuse of 2. Later, when I talk about the unit circle and dividing all the sides by 2 it will already be somewhat familiar. In the picture above you can see the lesson on Pythagorean Theorem using the blocks to show a 3,4,5 triangle. And bring home the point that we have to take the square root to find the side it's not 25 it's 5 just as it's not 4 on the triangle we are studying it's 2.

During the lesson which I failed to film I used the three period lesson to drive home the names of the sides and we used Pythagoras to help figure out what the hypotenuse HAD to be if the sides were 1 and √3...later I will take an equilateral triangle with sides of 1 cut it in half and talk about the special triangle that is formed. I did this but due to equipment malfunction the little boy accidentally ejected the battery while filming me we lost the lesson. I will record it again another day.



It was good for me as a teacher to spend the time with this student because he asked the exact same questions I get from high school and college students as if he were reading from a script...why does the opposite side change when the hypotenuse stays the same? I usually get them to answer their own questions by asking them questions...

I also think you should approach a lot of this as vocabulary, but not all of it. Many "just memorize it" when in fact you can figure it out if you know the definitions of the words, like hypotenuse: the longest side. We had a discussion about how not all triangles have a hypotenuse, and more.

We already knew x/x = 1 so square root of three over square root of three (√3/√3 =1) being one wasn't a scary concept or hard to understand...he also understand the identity that when you multiply by one you don't really change the thing you multiplied although the symbols may change. Concept based teaching begins to compound.

He also understood that √4 = 2...a common question is where does the 2 come from if the student doesn't fully understand Pythagorean Theorem...I made him do it on the white board (it got erased) at first we had √4 for the hypotenuse but that just a complicated way of saying 2....but the step should not be skipped.

We also applied this same lesson to the 3,4,5 triangle...found Soh Cah Toa for angles and looked at the fractions as RELATIONSHIPS of the sides...for example 3/5 HAD TO BE the Sin 30 on that triangle and he discovered that the opposite side of the small angle was of course shorter than the opposite side of the larger angle (60°) and of course the longest side was opposite the largest angle...the 90. I didn't tell him he figured it out...important concept. You can learn a lot by studying a triangle instead of just memorizing the sides and sin cos or tan for the angles...

Classic Algebra Problems

Here are some videos I've made covering those Perennial Algebra Problems...the ones you'll find in every Algebra Text Book or course.

These "exercises" are supposed to teach you about fog, substitution/evaluation and how to express one thing in terms of another. Instead of f(x) for the function they want f(h)...in order to do that you have to put x in terms of h.

In this case you have to figure out that x = h/2 and you have to know how to find the area of a triangle...

It helps to understand rectangles and that all it is is A = (1/2)LW but instead of Length and Width we are talking about Base and Height so we get A = (1/2)bh and all we have is a special triangle that is 30-60-90 so the base is h/2 and the (√3)(h/2) or vise versa depending on how you look at your triangle. So then all you do is substitute terms and you're done. Somehow they usually don't make it that easy.

Express The Area Of A 30-60-90 Triangle In terms Of Hypotenuse:


Crewton Ramone Express Hypotenuse In Terms Of Perimeter of a Triangle.



Explaining via Mortensen Math...this is basically the same as this vid except I explain it with the emphasis for teachers and trainers who are using the Mortensen method...here is regular explanation: http://youtu.be/7NvnLPwkZVw



In this vid I talk about the concepts of Hero Zero and No Fun Get Back To One as well as the rectangle and SAME.

I left out the concept of the rectangle and squares...and that

(a+b)² = a² + 2ab + b² which should be familiar. (x+1)² = x² + 2x + 1 or x² + 2x + 1²...

Here are little boys playing with the concept of (a+b)² or (x+1)² or (x+y)²:



Important concepts to understand here. They stumble in exactly the same places high school and college kids do...but by the time they are teens these concepts will be second nature.

It got cut off at the end, but basically I was explaining that I give him a hard time because he's my son...but that I was very pleased with his work today. And then they did their usual Crewton Ramone's house of math sign off...but the camera died. Happily the lesson was over so we didn't really lose anything...less than 30 seconds got cut off.

Also we talked about it afterwards and the reason he said 14 when adding 6 + 5 is that he was looking at it upside down and the 6 does indeed look like a 9 when you are looking at it upside down so he automatically decoded the symbols as 9 + 5 and got 14 even though he heard 6 + 5..."I know what 6 + 5 is dad...I've been doing it my whole life!"

A note to those with dyslexic kids...what they "see" and the decoding going on there is usually more "powerful" than what they hear.

More videos and sample lessons you won't find anywhere else with these two are here:
http://www.crewtonramoneshouseofmath.com/Dboyz.html There are several hours worth of video there now. You need a password.

Here is some more typical high school algebra:

These are screencasts.



Yet more:



And last but not leaste a problem where we put Pythagoras to the test:



Here a blogpost on using Pythagorean theorem.

All of these problems can be fun if presented properly and if the students have a firm foundation in mathematical concepts. If not they tend to strike fear and cause confusion and delay to quote Thomas and Friends. Just remember rectangles are easy to count and triangles can be a bitch to count so we turn them into rectangles. We also do this with circles and areas under curves.

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