Length, Area, and Volume

17 May

Them-PosterDid you know… that if you DOUBLE up the size of a chair by doubling up its dimensions in every direction, you will need FOUR times as much paint to paint that chair and that it will weight EIGHT times as much?

Did you know… that if a gorilla was to really proportionally increase to be the size of King Kong, that his guts would just explode and gush out of him from the sheer weight of itself? (The same concept prevents an elephant from ever getting a c-section!)

Did you know… that if an ant was really the size of the ants in “Them!”, not only would they not be able to stand up on their legs, they would also suffocate immediately?

Did you know… that if you proportionally double up in size, you’ll only be able to jump the same height as you did when you were normal size? Likewise, if you got 100 times smaller, you will also be able to jump the same height as you did when you were normal size! (So how long should it have taken for the kids in “Honey, I Shrank the Kids” to get home?) 

tumblr_m78krxad8d1qiceiuo1_500Interesting, right?  All these facts  take into consideration that very first “Did you know…,” that changing a linear dimension changes the area and volume differently. Intuitively we all have some understanding of this. I mean, imagine making an itty-bitty model of a wee little ant out of Play-Doh. How easily will the ant stand on its tiny legs? Now can you imagine building a 2-foot model of the same ant? You’re going to have some difficulty making it stand on the same proportional legs.

So I found it all the more interesting when I went online to search for great lessons on this concept and found NONE. I had never been able to teach this topic to very much detail due to our crazy pacing plans and, frankly, due to my hazy knowledge of the science behind it. However, this year, I had a couple of advantages that allowed me to actually try to plan a lesson- time, of course, and an awesome student teacher who is ridiculously skilled in the sciences!

Anyways, Day 1, I started with a lesson on the basic concept* and followed it up with a follow-up to an exit ticket from the area chapter. It was pretty interesting to see how many students didn’t make the connection to the worksheet that was previously given. I’d say it was about 50/50. Half who figured it out and half who simply multiplied the 52 square foot tiles by 12 inches per foot… (I posted this student’s work because she had asked me if she could simply multiply by 12 or if she had to convert all the numbers on the page first and re-work it. Like a good teacher should, I said, “Exactly! Test it!”)

tiles

The next day was the fun day though. I had almost an entire block period to now talk about the implications of same concept in reality.


 

Again, looking online, I found almost nothing fun, nothing very deep, and nothing very useful to me, except these notes… which I devoured. And from these, we came up with this lesson

__________________________________________________________________________________________

FACT: If you double up the dimensions in every direction, then the area multiplies by ____four____________ and the volume multiplies by ______eight________.

QUESTION: If you have a chandelier barely hanging from a rope and you doubled up the size, including the rope that it hangs on, will the chandelier still hang or will the rope snap?

Why or why not? (Make your argument here. You will not be graded based upon whether your answer is right or wrong, but on how well you make your case.)

 By this point, most students understand that although the dimensions are doubled, the weight of the chandelier has now become 8 times heavier. However, italicizing the part about the rope has them also consider the fact that the rope has also gotten 8 times heavier proportionally. A very true fact. I have them in groups and ask them to talk about it and then write out their argument, emphasizing that they will NOT be graded on correctness. 

I then take a vote. The vote for all 3 of my classes was close to 50/50. I say awesome, and before I give away anything else, I have one person from each camp make their case. I then take out a piece of rope to begin my “Consider…” lecture:

Consider… a really heavy object hanging from this piece of rope. I want to make it stronger and so I double up the length, like so, because I know that this will increase the VOLUME by twice as much. Will that make the rope stronger? [Class agrees, no way!] So then, how do I make the rope stronger?? … exactly! I need to double up the rope by making it thicker. So what does that mean? What does the strength of the rope depend on? The volume? (no) The thickness (yes!). And how do we measure thickness?? … the cross-sectional AREA!

So will the rope snap of the chandelier snap or not?! Discuss…

So yes, the rope definitely snaps. The cross-sectional AREA only increased four-fold (which is what the strength of the rope depends on) while the weight increased eight-fold. In this case, two chords of the thicker rope will hold the larger chandelier.

__________________________________________________________________________________________

FACT: The height that a body can jump vertically is directly proportional to the ratio of muscle mass to the total mass of the body.

jump height

, where C is a constant (enough).

QUESTION: If a human being who can jump 1.5 feet is doubled up in size (pretending like it’s actually possible), how high will the new larger human be able to jump? Explain.

 This was the section that my student teacher had helped me out the most in. He came back w/ an equation that we boiled down and simplified to the one above (apologies to any physicist who might get offended by this). I helped to explain what this equation meant and how it made sense. I made sure they all knew that working out actually made your muscles heavier, which makes total sense in this equation. I had them discuss and then take a vote. 

Their logic behind this one was so adorable. Most of them completely ignored the equation and went straight to guesses. “If twice as tall, muscles get bigger so they’re stronger… they can jump higher.” But the winner of all arguments…

He jumps shorter! Like Mario! When Mario grows, he can’t jump as high! 

The group cracked up and then voted, yes, he jumps less than 1.5 feet… like Mario. 

Final answer? This person will jump the same height of 1.5 feet. His/her muscle weight will definitely increase eight-fold, but so will the person’s total weight, canceling each other out. 

QUESTION: Now reverse the idea. If the same human being is shrunk to half the size, how high can the new small human jump?

Likewise, the smaller human can jump 1.5 feet.

(So how long would it have taken the kids in “Honey, I Shrunk the Kids” to go home?!)

It would’ve taken the same amount of jumps as when they were full size! This was an awesome discovery after reading that one professor’s notes.

__________________________________________________________________________________________

FACT: Ants don’t have lungs. Air is received through “blind tubes” that cover the surface of the ants’ bodies where air enters through.

QUESTION: If an ant was to be blown up as big as the ants in the movie, “Them!”
(pretending the physical structure is possible), what can you conclude about the
air intake of the giant ant?

 We didn’t have time in class to go over these, but these should be easier after all that we’ve learned. I should’ve given it to them for homework… 

 FACT: With larger animals came the development of blood circulation and lungs as a more efficient way of oxygen distribution. Lungs increase the surface area available for oxygen absorption by the blood.

QUESTION: If a human was blown up twice as big (again pretending like the physical structure is possible), how do the lungs have to increase along with the body in order to support the oxygen level needed to survive?

__________________________________________________________________________________________

Anyways, if there are any other lessons/facts out there that you think are good for this concept, let me know! I didn’t have this much fun lecturing in awhile!

*Some of the problems from that worksheet were either directly taken or modified from the NCTM Illuminations site.

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2 Responses to “Length, Area, and Volume”

  1. suevanhattum May 17, 2013 at 11:41 am #

    Nice! There’s a great article about this in The World of Mathematics, a four-volume set by James R. Newman. It’s ‘On Being the Right Size’, by J.B.S. Haldane. I found it! Here.

  2. www.youtube.com May 17, 2013 at 6:05 pm #

    It’s impressive that you are getting ideas from this paragraph as well as from our dialogue made at this time.

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