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More Hands-On Math--Using the Pythagorean theorem
Hands On Math/Real Life Applications..
In Middle School Math, another topic that needs to be learned is the Pythagorean theorem. Before I explain what the Pythagorean Theorem is for those not familiar with it, I will give the reader a bit of background on why I planned two different lessons for this topic.
I taught this lesson using two different real life models. First, I taught it using the Super Bowl and how to solve for a side of a triangle and the other example utilized the Baseball World Series and solving for the hypotenuse of the triangle (say what..no worries I am about to explain this in detail)!
Thus, I designed two very distinct lessons that I used these scenarios to motivate my students knowing most kids love baseball and football. My classroom came equipped with a SmartBoard and Wifi internet. So I utilized both in teaching this lesson.
What is the Pythagorean Theorem?
Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together!
It is called "Pythagoras' Theorem" and can be written in one short equation:
a2 + b2 = c2.
- c is the longest side of the triangle
- a and b are the other two sides
Now, we also must redefine the word hypotenuse, because when teaching this topic, this word does come up quite often. So the hypotenuse is the longest side of the triangle or in the pythagorean theorem "C" is the hypotenuse.
The simplest example when first demonstrating this...
I put this example up first, because when teaching this topic, this truly is the basic example shown universally.
Example: A 3,4,5 triangle has a right angle in it.
Let's check if the areas are the same:
32 + 42 = 52
Calculating this becomes:
9 + 16 = 25
It works ... like Magic!
Now to Use It, Lesson 1--The Super Bowl..
a2 + b2 = c2
So earlier, I stated that I taught two separate lessons on this topic to teach how to solve for a side or to solve for the hypotenuse. The first one teaches how to solve for a side.
My first lesson was actually taught using the Super Bowl and how most of the country is obsessed with watching the big game on a big screen TV. So I set up the lesson that the students wanted to each buy a TV for their home and also needed to purchase a cabinet/stand for the TV to sit on.
So now on my Smartboard, I went to Best Buy's website to look at the different TVs and compare a few models.
Best Buy HDTV's
Samsung - 55" Class - LED - 1080p - 120Hz - Smart - HDTV
So, the TV that was decided upon is a Samsung 55" LED 1080p. Gorgeous and definitely a very nice size to watch the big game on. But now even though this TV is 55", is it 55" long where it sits on the actual TV stand? The answer is no. So how do we find out how long it is so we purchase the right size stand to put this TV on?
Well think of the TV as two right triangles put together. If the diagonal or hypotenuse (remember that word from earlier) is 55", then how will we find the length. Well on Best Buy's site, we are given the height, which is 30".
How will we find the length then? You guessed it we will use the pythagorean theorem!!
a2 + b2 = c2
(30)2 + (b)2 = (55)2
900 + (b)2 = 3025
(b)2 = 3025-900
(b)2 = 2125
√b = √2125
b = 46.0977 ≃ 47"
So, we need a TV stand that is at least 47" long.
Now to Use It, Lesson 2--The Baseball World Series..
So for the second, we are supposedly in the middle of baseball playoff season and we have already learned in the past about setting up and solving one step equations. In New York, we all seem to be fans of the Mets or the Yankees. Say the Mets and Yankees end up playing each other in the World Series. If we are at Yankee Stadium and the total distance around a baseball diamond is 360 ft. What is the distance from third base to home plate? I put up a visual of the diamond on the smartboard and show that that there are 3 bases, plus home plate that gives us a total of 4 bases. Also, I will then illicit that the diamond is really a square drawn on an angle and 360 is the measurement of the outside of the square.
Next let's think about what we have been talking about so far with formulas for with geometric shapes, what formula measures only the outside of a square or rectangle? Perimeter!!!! 360 is the perimeter!!!! Then, how can I find what one side measures. Think about what you know about a square and the sides. Students should recall that are all 4 sides are equal. Now, have students, recall the formula for perimeter, which is L + L + W + W or 2L + 2W. However, all sides are equal, so how do you think we can set up the equation? Students should realize that 4S = 360 or 4 times the side = 360. Now, have them solve for s, to find out s = 90 ft. (All of this so far should be review, but what a great review to set up for our current scenario).
Now let's say it is the bottom of the ninth, two men are on base at 1st and 3rd. The batter has got a full count and the guy at first is trying to steal second. How can we use the Pythagorean Theorem to figure how the distance from the catcher at home plate to throw out the the player that is trying to steal 2nd base? Well the baseball field is a square as we said earlier and if you were to draw and imaginary line down from 2nd base to home plate, you would have two right triangles, now we know the length of both sides of the square, but are missing the diagonal or the hypotenuse.
So here we go:
a2 + b2 = c2
(90)2 + (90)2 = (c)2
8100 + 8100 = (c)2
16200 = (c)2
√c = √16200
c = 127.279 ≃ 127.3"
So the hypotenuse or the distance from 2nd base to home plate to throw the runner out is 127.3"!
Summing This Lesson Up..
I love this lesson, because it is another hands-on lesson using two sports that most middle school students are not only aware of, but interested in. Now, when they watch both maybe they will remember a bit about the pythagorean theorem even after the topic and test are long over and behind them. And as a teacher that is truly something you hope for that the student takes away something from your lesson and doesn't just forget it once the test is long gone! Anyone who has read any of my other math teaching articles previously will know from them that I am a strong believer in using stuff that interests middle school students to help them learn many of the basic concepts that are a requirement. I truly enjoy engaging my students and showing them how we can use math in everyday life and believe this lesson is another one that does just that.
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