a2 + b2 = C2
I Introduced a 4 and a 5 year old to Pythagorean Theorem. Really it's that easy. The videos below show an introduction to this concept, skip ahead to the 5 min mark on the first one if you are pressed for time. There will be more here soon, and by more I mean pictures, videos and links. Check back often and be sure to hit REFRESH on your browser.
The graphic above though poorly labeled shows the basic idea of Pythagorean Theorem: the two smaller squares add up to the bigger square.
The first part of the first video HAS NOTHING TO DO WITH Pythagorean Theorem, so you may want to FF to the 5 min mark, because watching a 5 year old factor a polynomial he's never seen before like a high school student three times is age could bore you to tears. Or skip it and just watch the second one, you are 5 minutes away from complete utter total understanding of what that Greek guy was saying all those years ago. Pythagoras did not discover it but he gave it his name and made it famous...or it made him famous.
In the first part of the video you see a five year old factor a problem he's never seen before:
x2 + 8x + 12
The day and week before, he has already done x2 + 8x + 16 a couple of ways once as a problem for completing the square:
x2 + 8x + ??
?? = What completes the square?
and once as a problem for factoring where he got all the info:
x2 + 8x + 16 and we talked about (x+4)2.
To make it a bit of a challenge I changed it to x2 + 8x + 12, but he's almost at the point where they are all EASY.
Note how his brother builds squares and the quietly arranges them according to size while I am focused on his brother...he received a lot of praise off camera when we reviewed this video together.
Then we played with the concept of squaring the sides of a triangle...Pythagorean Theorem. If a 4 year old can get it and note how amazed he is when he sees proof with his own eyes, so can you.
If you build the lesson correctly where your students have to play with the triangles and tell YOU about the relationships (instead of the other way around) you will find 90% of your students will come back with
a2 + b2 = C2!!!
They will tell you the Pythagorean Theorem instead of you telling them. That's called Directed Discovery.
We can also see that later when they see
a2 = c2 - b2
b2 = c2 - a2
it will make total sense to them.
This was just the first time they saw Pythagorean Theorem and they will see it many times again before they reach the ripe old age of 7...note how in the video I didn't even write out the equation. We just played with the blocks and got an experience of the concepts. Next time when we do write the symbols that go along with the blocks they will understand what it means instead of just memorizing a formula.
They will also see that a + b > c ALWAYS, although they may make the joke 3 + 4 = 5 they will know it's a joke and that 32 + 42 = 52 is true but 3 + 4 = 5 isn't.
Also note how much faster and easier this lesson is now that they each can skip count by 3's 4's and 5's...and this lesson gave then another chance to practice skip counting.
I made an additional video that is a more formal lesson for YOU on Pythagorean Theorem sans little children where we can see how to use simple "Pythagorean Triples" to understand more about the theorem before we ever have to get out our square rooting skills for numbers that aren't perfect squares.
ALWAYS start simple and use Pythagorean Triples to make the concept easily understood. Soon I'll have more on how to generate Pythagorean Triples...and a list of them.
I've talked to a few teachers who never saw a proof of the Pythagorean Theorem, I mean any proof, not just this one...they just got the formula and memorized it. I talk to teachers all the time that are amazed by this little demonstration...INCLUDE it in your lessons with little kids.
Below you will find a short screencast for adults or older students talking about Pythagorian Theorem in general terms. We haven't really gotten down to the nitty-gritty of solving for "C" by square rooting but it becomes visually obvious that if I know two sides of the rectangle I know the third side. Then you'll be ready to click the link to the pre-calc page for more on it's uses.
Search "Pythagorean Triples" and use those to start off don't just pull funky numbers out of your butt. It blows my mind that people will use 2 and 3 as sides which gives the square root 13 for "c" or 4 and 5 which gives the square root of 41 for "c" and then wonder why their students don't "get it." Make sure the student can easily find the square roots without a calculator to start. AFTER they understand the simple concept then you can start showing them that if we know ANY two sides we know the third side; at last we can put in any two numbers and find the third side, BUT START WITH PYTHAGOREAN TRIPLES.
Here are a couple comments I pulled off YouTube for those who think I'm kidding:
ok. The Pythagorean Thm is pretty simple for me but something continues to haunt me. How exactly does squaring a and b give you the length of c? What's the relationship?***
I don't get it, why does it 13 become the square root? Help me plz, my exam is tommorow morning
The video they watched on Pythagorean Theorem had a triangle where one side was 2 and the other 3...which obfuscates the fact that you add the two rectangles together and then take the square root to get the third side, which is "c". Square root 13 DOES NOT MAKE ANYTHING OBVIOUS.
Again any student can see that "C" isn't 25, it's the SQUARE ROOT of 25. 25 is the whole square! C2. "C" is just once side...they can also SEE that the two little squares have the same area as the big square...and to get one side you need to square root. Click the square #'s tab for more.
And then we can start figuring out how to find the other side if we know any two sides by using simple subtraction.
If I take out one of the squares what is left is the other square either "a" or "b"...
Once you get 9 + 16 = 25 down and have played with it you can get out another one and in fact this one is a good one to start with when you have younger students...but it might be hard for them to count that high so you could just use the bigger squares for the concept that the two small squares add up to the big one...note you can but 36 in the middle and the eights around the outside just like with 9 and 16...also you might play with concepts around squaring and exponents...try and get them to tell you that although we doubled the side the area is FOUR TIMES as big...but that's another lesson.
Watch this, it will clear things up in 5 minutes.
Here is a short screen cast on Pythagorean Theorem talking about the above graphics...
Now go to the pre-calc page here at the house of math and watch the screencasts so that we can apply this knowledge to find the distance between two points...and it should be simple to find the sides of triangles too...allow a few seconds for those screncasts to load...the page goes white for a few seconds and it seems like nothing is happening.
You will note in one of those screencasts I talk about a student who was familiar with Pythagorean Theorem but could not apply it because he didn't understand some basic concepts...
Later on we can play with more proofs and again they will make sense. Here is a famous one showing the area in white INSIDE the two big squares is the same:
The House of Math on FaceBook.
GO from The Pythagorean Theorem back HOME.
Learn more algebra for a buck.
SEE how to derive Πr² (pi r squared). Also has a vid showing some "advanced" use of Pythagorean Theorem.
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