3-D Printing -- Project #2

For our second 3-D design project, I had students make game tokens for games they were designing.   They have been using systems of inequalities in 3-variables along with Mathematica to design and print 3-D objects.  There are no shortcuts -- everything is defined by an equation.

I told them that one of my regrets as a parent is that I didn't have my children (who are now out of college) mathematically design and 3-D print their own game pieces that they could carry with them wherever they went.  That way, while other kids could fight about who was going to be the race car and who had to be the thimble, my well-prepared children could pull out a personalized way-cool game token and say, "I'm good."  Ah, opportunity lost.

We began by learning about the equations to circles, and I gave them some challenges to complete -- create a cylinder, intersecting cylinders, a cylinder with a hole, and the car-looking-thing below.

 
 

Most were successful pretty quickly, as they'd become used to thinking in more than just the x-y axis from our previous work in Mathematica.

The next set of challenges were more difficult.  I had them think about how a cone and a cylinder are related.  They could see that a cone is a cylinder whose radius changes as you move "up."  That is, if the cylinder has its base in the x-y plane, the radius of the cone changes as z changes -- r is a function of z.

For example, if we want our cone to have a base radius of 5 and a height of 10, we take the inequality of the cylinder:

x^2 + y^2 < 5^2

and replace the radius (5) with a function of z.  When z = 0, r = 5 and when  z = 10, r = 0.  We're really looking for the equation to the line between these two points, which is r = -1/2 z +5.

So, the new inequality for the cone is:

x^2 +y^2 < (-1/2 z + 5)^2

Many mathematical tangents to talk about here. 

• How would we write the equation to a cone with the relevant variables (Height, base radius, and center of base)?
• We've used composition of functions in a productive way.
• What if we wanted the sides to be a parabola, for example, or a Hershey's kiss?  This would lead us to fitting quadratics and higher degree polynomials to data.
• How would we think of a (hemi-)sphere this same way, and develop the formula for a sphere?

Two further challenges followed.  The rocket ship reviewed how to place two objects in one design.  The leaning tower keeps the radius constant, but makes the x-coordinate of the center a function of z.

 


Again, many more paths to pursue.  In particular, what would happen if both the x and y coordinates were functions of z?  What if they were different functions, say a quadratic and a line?  We're hinting toward parametric functions here and are also begging for periodic functions, neither of which we've covered yet.  Maybe next year I'll rearrange things.

The only requirements for the game piece was that it had to have at least one quadratic, at least one circle and at least one changing dimension (like the cone).  The results were varied and excellent.  Some students were pretty literal and made actual game pieces, with appropriate detail.  Some were abstract, and others were representational.  Particularly impressive was Jake's dragon, which used around 200 inequalities, including lines, parabolas, circles, spheres and a very cool third-degree polynomial neck.  (Photos of a sampling the student work are below.)

Once again, I heard all the good classroom sounds, particularly frustration turned through persistence into joy.  Music to my ears.

Next year, I will rearrange some things so that we can be more creative in our "changing radius" possibilities.  I'm considering, too, adding in some parametric graphing which will open up new possibilities as well.  That's another thing on my list to think about this summer.

9^(1/2) isn't Purple - The Value of Wrong Answers

Last week, as I was getting ready to introduce rational exponents to my honors Algebra II classes, for some reason I flashed on a session offered by Ed Burger at last year's NCTM conference.  He spoke to the importance of productive failure - practicing being wrong and recovering.  He talked about asking kids to give him answers that they knew were wrong and how learning would follow.

So, I gave it a try.  The question I posed was, "What does 9^(1/2) mean?"  Their job was to give me either a correct answer and reason why it was correct, or a wrong answer and a reason why it was wrong.

I asked first for the wrong answers and reasons.

Student 1: "It's not purple."
Me:  "Why not?"
Student 1:  "Because the answer is a number."

Back in my early teaching days, I probably would have been irritated by that answer.  Experience, however, allowed me to see that even though there's a wise-crack element to the response, at its heart is real truth -- the answer is a number!  And the student is engaged!

Me:  "Okay, that's good.  So we've narrowed the answer down to a number.  Give me another wrong answer and why it's wrong."
Student 2: "It can't be 70 million."
Me:  "Okay, why not?"
S2: "It's way too big."
Me:  "But why 70 million?"
S2:  "Alright, 30,000.  Or 500.  It just can't be bigger than 9."

Bingo.  Had I asked the question "what is this?" I don't believe the student would have seen this fact.  By asking "what isn't this?" the student found it on his own.

Student 3:  "It has to be bigger than one, because 9^0 is one."

Student 4:  "It can't be 4.5, because it's not multiplication."

This was my favorite moment, because I know that if I had asked them right away what they thought 9^1/2 was, many of them would have said 4.5 for lack of a better answer, even if a little reflection would have told them that it was wrong.  By approaching the question from the negative, it forced them to reflect and it made that wrong answer a correct answer.

The conversation continued as we narrowed down the possibilities -- it probably was closer to 1 than it was to 9 because that's the pattern we saw with higher integer powers.

Eventually, I asked for a correct answer.  Some students knew what it was, but it took a little prompting from me for them to see the why.  I also helped them see that we were really "defining" rather than "figuring out."

Approaching an idea through wrong answers was effective for every student in the room.  The students who had been taught the right answer in another class were challenged with explaining why.  Some of them chose to find wrong answers with an explanation because it was easier.  For student who didn't know what the answer was, the entry bar was low -- there are a lot of wrong answers, yet they ended up making critical contributions to the discussion.  In the end, I believe everyone had a better understanding of the idea than they would have had I just allowed those who knew to tell those who didn't what the answer was.

Plus, it was way more fun this way. 

Travel to Ghana

For the past several years, my school has been part of a program based in Ghana called Right to Dream.  We've taken several students from the program who attend our school and then go on to US colleges.  They are true scholar-athletes, outstanding soccer players with tremendous drive and dedication in the classroom.

This summer, I am travelling with my wife and a group of students to visit the Right to Dream Academy in Ghana, where the students train and learn in preparation for their move to the States.  We will be doing some service learning while we're there, and among other things we'll be visiting a local school where the Academy students volunteer, as their own way of giving back.

I've been tasked with bringing some "math manipulatives or equipment" with me to leave at the school we're visiting.  It's a lower and middle school, and I am a high school teacher.

So:  Help!  I don't have much more information other than the school has very little.  What would you bring?  I'm sure that a few decks of Set are a good idea.  What else?  Any help would be mighty much appreciated.  

Thanks!