Around the World: Surprise! (Part 2)

Before you read part 2 of this blog, I suggest you read part 1.  The problem I gave my students was as follows:

Imagine that the Earth is a perfect sphere and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

Students worked together to find an answer of about 15.9 meters.  Groups had to explain their solution to me before they could continue on to the next phase - finding the solution for a second spherical object.  

On the front board I had written the following:

1 - baseball

2 - moon

3 - Jupiter

4 - beach ball

5 - basketball

Groups selected an index card at random and were assigned the object based on what they selected.  They needed to research the dimensions for their particular object and do the calculations again.  They submitted their results through a google form and when I displayed the results to the entire class, this is what they saw. (Note: These results are for all 3 of my classes combined.)

WHAT????  

No matter how big the object, the wire would always be 15.9 meters above its surface. 

Wait...WHAT????  But, why???

That was the reaction I was hoping for.  But these are the reactions that I got.

Period A - Huh. Interseting, I guess. (I think they were still on spring break.  It was Monday and 8:20 AM.)

Period B - That's cool.  Why does that work?

Period F - We are not surprised at all.  You do stuff like that with us all the time.  

So...now you may be asking Wait...WHAT Why does this work?  The derivation I did with one of the classes appears below.  The reason it works is because there is a direct linear relationship between the radius of an object and its circumference.  (Students know this direct linear relationship as C = 2*pi*r.) When the circumference increases by x, the radius increases by (x/2) divided by pi.

I love problems like this that are a SURPRISE to students and go against what student think should actually happen.  I would love to have more problems like this to share with my students.  If you have any problems like this, please send them my way.

Around the World: Surprise! (Part 1)

Back in the late 90's I worked with The Math Forum Problems of the Week. (POWs) Specifically I worked on the Trig/Calculus POW
and the Discrete Math POW.  Submissions came in from around the globe and I enjoyed seeing the variety of methods students used to solve the problems. There was even one time when a student used calculus to prove she found the minimum number of moves needed to solve the discrete math problem for that week!

This past week I was on spring break and I pulled out a wonderful GeoPOW problem from my files.  I plan to use it on Monday, because we often miss students on the first day back from spring break due to families extending their spring break.  Rather than having several students miss new material, I thought it would be fun to look at this surprising problem.  In the GeoPOW files, this problem was called "All Around the World".

Imagine that the Earth is a perfect sphere*, and that a metal wire is snugly wrapped around its equator. Now imagine that we cut this wire in one spot and splice in an additional 100 meters of wire.  We take up the slack by using posts to raise the wire an equal distance all the way around the Earth.  How high above the surface of the earth will the wire be?

*many students are surprised to learn that the earth is not a perfect sphere*

Collaboration is a big part of learning math in my classroom; students will work on this problem in groups.  After each group gives me an explanation that convinces me that they have the correct answer (Note: The answer is about 15.9 meters.), the group will reach into a hat and pull out a slip of paper that will have one of the following items written on it: basketball, moon, Jupiter, beach ball or Mars.  The groups will then repeat the problem that they just did with the Earth, but with their new spherical object.  After students have finished the problem, they will go to the tinyurl written at the front of the room to enter data into the following form:

After students click the Submit button, they will see the message "We will review the results from the class shortly."  The surprise will come when I turn off the AV Mute button on my projector and we see the results at the same time!  Come back for part 2 of this blog when I share how the lesson went and my students' reactions.  But before then, do the math and predict why my students will be surprised.

Teaching Trig Ratios in Geometry

Although I have probably taught trig ratios (sin, cos and tangent) 2 or 3 times each year for about 15 years, I was never really satisfied with how it turned out.  Yes, students could parrot the definitions and find sides and angles until the cows came home, but did they really understand why the sine ratio was always a certain number for a specific angle?  I was doubtful that the big picture was being understood...until today!  OK...I am still not satified with the entire lesson, but I think it was better.

As much as possible, I want students to construct their own knowledge/understanding of concepts with my guidance at the side.  In the past, I would have them draw right triangles that were similar, measure segments and calculate various ratios. (See handout below.)  But guess what the problem was.  Yes, that is right.  Inaccurate measurements led to the ratios not being equal and me saying things like "Well, the ratio 0.75 is pretty close to 0.76.  So, close that if you had measured accurately, I bet they would be equal."

Bleh!!!

The world would be a utopia if the students could measure accurately with 25-cent rulers and 25-cent protractors.  But they haven't and can't.  How could I get more accurate measurements?

ANSWER = Geogebra  

I was going to create a worksheet for them to do, but didn't quite have enough time.  (Both enough time in class and enough time to create the worksheet.) So, then I decided to create a screencast; it is uploaded to my youtube channel and I was planning on showing the 6 minute screencast in class. But since I really wanted students to experience the discovery for themselves, I led them through the activity as a whole class with each student working on his or her own laptop.

As long as students did exactly as I did and clicked in the exact same order, we were ok.  If they clicked in a different order, then we had different outcomes; their segment d was the hypotenuse and my segment d was the leg.  Luckily, only one or two students clicked in a different order and they figured out how to make adjustments for what they had on their own computer.

Students in my one class were initially surprised that we all got the same values, and when I asked them why this made sense, they understood!  Students chimed in that all the ratios must me equal,  because the triangles we had on our screens were all similar to each other by Angle-Angle similarity!

I think some students didn't find this too amazing, because as a class we decided what angle we would use and we all used the same angle.  Their feeling was "So, what?" Next year I still plan to lead the students through this activity rather than have them follow a set of typed directions.  However, I would have the data collected by table. I would have each table group choose what acute angles their right triangle would have.  This means there would be 5 sets of similar right triangles and they would see that it is not just one specific set of triangles where the sin ratio is the same, but the sin ratio is the same for any one given angle.