More fun with Geometry

So i like to visit the elementary school every so often to see what they are doing in the elementary levels. The teachers all know i love math and especiall geometry.

So this 4th grade teachers instructs her class to take you thier rules, pencils and scissors. I LOVE it already. This is how you REALLY learn geometry. You have to get your hands dirty. "Show me and i will learn it for the test, but let me discover it and i will know it forever" (hopefully for a while anyways).

So the teacher tells the class to draw a 1-2-3 triangle. A Triangle with 1 inch on one side, two inched on another and three inches for the third. And then she gives them a hint: put the three inches side first, across the bottom of your paper.

Now i know a little about geometry and one thing i do know is that you can NOT construct a 1-2-3 triangle.

I call it the "Drawbridge" theorem. If you bring the 1 inch and then the 2 inches down like a drawbridge, they will meet on the 3 inch line at the bottom of your triangle. And OOPS there is NO triangle because 1 + 2 = 3 !!

Now this would be a neat trick to pull on students in maybe 10th grade geometry
(by the way students should NEVER take algreba in the 8th grade, big mistake made in the 1990's and early 2000's....only 20% of the children are ready for "real" algrebra in the 8th grade...the rest just memorize and then struggle in algebra two or precalculus later on).

But why would a 4th grade teacher ask her students to do this ?

I watched as the 20 students tried over and over again to construct a 1-2-3 triangle.
(Ok, get your paper and pencil and ruler and YOU try it.)

What i saw was amazing (remember i am a high school teacher and not used to the "out of the box" thinking that goes on in the elementry schools....although i must admit i LOVE creative spelling....and when they let you color OUTSIDE the lines).

One kid looked around and just filled in the missing space at the top of the triangle. Teach said "make a triangle" and that is exactly what i am going to do.
Even if the lines do NOT meet...i will make them meet. Several kids did that (mainly the girls) and then the other just copied (mainly the boys). Now there were 20 kids cutting out 20 skinny triangles and everyone was showing off their production.

The teacher panic because what she had hoped to do with the triangles, did not work out. So she said, ok put away your rulers and pencils and scissors and lets do a "mad minute" where they try to do as many arithmetic problems as fast as you can in one minute.

Now to be fair to this teacher, she is an excellent teacher and both of my boys had her in school. She is the kind that brings in a cow's heart when studying about the heart. That is the kind of thing that gets kids excited about learning.

So i waited until after class to ask her about the 1-2-3 triangle.

She said that she remembered doing this exercise with a 3-4-5 triangle, but that she wanted to use "nicer" numbers with the kids.

WOW !! Yes a 3-4-5 would have been great. Remember the "drawbridege" theorem that states: the Sum of any Two sides of a Triangle MUST be greater then the Third side.

Therefore 3 + 4 > 5, and 3 + 5 > 4 and finally 4 + 5 > 3.

But a 1-2-3 triangle will NOT work since 1 + 2 = 3 (The DRAWBRIDGE)

Also 1-1-2 triangle will not work either....sorry.

So i was crossing the draw bridge one time going to Largo Beach just south of Clearwater Beach. And the door was open to the control house. The man was sitting there watching the ball game on a small TV and i struck up a conversation (my sons tell me i would talk to a telephone pole, if it would talk back...lol).

Anyways i was hoping to see the drawbridge in operation from the INSIDE. We talked briefly and he told me he had just gotten his GED (high school equivalency diploma), and that he was called a bridge engineer making $17.50 and hour (and this was back in the 1980's when min wage was $3 and hour and most people only made $5 an hour working in an office all day).

So here comes a beautiful 40 foot sailboat and the captain gives us a TOOT-TOOT with his blow horn....the "bridge engineer" does NOT move...(I almost jumped our of my pants)....then we hear the TOOT-TOOT again...and nothing.

Now the sailboat has to make a 180 degree (hey a little geometry here) and go back up the inter coastal waterway....and then make another 180 degree turn to come back.

Have you ever seen one of these 40 foot sailboats make these turns ? It is about 3 or 4 people pulling on ropes and the captain shouting out orders like, "DUCK !! DUCK"

And then they have to do it all over again, to come back. And it is a beautiful day out there with plenty of sunshine and about 95 degrees...

So they come back and we hear the TOOT-TOOT....and the man does NOT move. So i have to intervene and ask, "when are you planning on opening up this drawbridge ?"

And he says, "isn't it fun watching rich people work so hard ?"

And then he opens up the bridge. He pushes a red BUTTON and that is it !! so disappointing for me.

Later he tells me they are only allowed to open his bridge every 15 mins starting at the top of the hour. But boaters do not like to wait and try to rush the bridge operators with their TOOT-TOOT.

Now what is the "drawbridge" theorem ?

Another way to remember this theorem in geometry is to draw two sidewalks crossing each other.

And every college student knows the shortest distance between two points is to walk ACROSS the grass...therefore going along the sidewalks (two sides of the triangle) is always longer than cutting across the grass.

That is why some grounds keepers wait a semester or two BEFORE putting in the sidewalks. The students will tell you by the path in the grass they make, where the sidewalks SHOULD be !!!

Now if that teacher had used the 3-4-5 like she was supposed to, the class would have seen that no matter how everyone drew the triangle, they would ALL be the same size and same shape. We call this "congruent triangles". And the "Side-Side-Side" Theorem (SSS) assures us that this will always be true.

In addition, our 3-4-5 triangles would all be "Right" triangles, because of the Pythagorean Theorem (discussed in another one of my blogs).

And believe it or not some 4th graders would be able to discover that
a^2 + b^2 = c^2 !!! (3^2 + 4^2 = 5^2 or 9 + 16 = 25 or 3-4-5 triangle)

Why do we allow some 4th graders who read at the 10th grade level, to read 10th grade level books, but not expose them to 10th grade math ? Not fair !!