I finished a working prototype set of gears this afternoon. The gears in the video were fabricated from a single sheet of card stock and fastened to a folded card stock base with two brass fasteners. I fabricated the gears and base using a computerized 2D die-cut machine and basic software that is included with the machine. The intricacy of the gears almost necessitates computer-controlled precision versus the varying results of scissors.

Total Materials: Two sheets of card stock (gears and base) and two brass fasteners

Questions:

1. Do the gears spin in the same direction? Kind of obvious.
2. How many different ratio relationships can you identify? Not so obvious. You need to think about more than the gears’ teeth.
3. The blades on a wind turbine produce energy that is converted to electricity through a generator. Would you attach the blades to gear A or gear B? Not obvious without the context of wind turbines. You probably need some additional information. Nevertheless, logic will likely lead you to a reasonable guess.
4. If it took me 12 seconds to spin gear A in the video, gear A spins at what RPM?
5. If it took me 12 seconds to spin gear A in the video, gear B spins at what RPM?

My Drawdio kit just arrived via snail mail, and I am thinking that I will wait until tomorrow morning to begin playing with it. The kit, mind you, came with no instructions and a bunch of parts despite the claim that it was a working, tested unit. (Cough) Engineers (Cough). Luckily ladyada has a relatively helpful manual…

I am thinking that I might craft a couple of lessons about measurement that use a drawdio-equipped pencil. Measurement is a consistently problematic area of math for the teachers/students with whom I work. I am imagining a mixture of art, music, and math! Science would be the glue.

Print this template and watch the following video to make the design.

As is the case with any activity or lesson in this day and age, it must align with some standards or curricular objectives.  I chose ratios, and I produced the design with that in mind.  Where are the ratios?  Thoughts?  What is the follow up activity?  Other mathematical or artistic connections?

The artistry of 3D design is ripe for mathematical exploration, especially if you have design constraints (like paper size).

“The Fraction Ruler” is a useful manipulative for helping students understand measurement and the lines on a ruler. This fraction ruler can also serve as a resource for practicing addition and subtraction of fractions.

It’s pretty cool that I can create a physical prototype of an instructional manipulative the very same afternoon after having a conversation with an advisor.  From idea-to-interpretation in less than five hours using nothing but common materials and some design software/hardware is kind of remarkable.  I am not sure if I am at the point where I can make anything, but I know that I can produce a fraction ruler…

Actually, I have to admit that the following manipulative wasn’t created in one afternoon. It wasn’t really the same afternoon because I had some follow-up questions, but if you count the time that it took me to make and fabricate “The Fraction Ruler,” it was pretty much a single afternoon.

Creating the instructions that follow is another story.  How do I tell teachers how to make what I made?  That’s a lot of work.  It’s too much information to fit in a single .pdf!  Here are the instructions and design files.  I did my best.

• Instructions
• Ruler Base & Template
• Ruler Fraction Overlays
• Ruler Fraction Overlays 2

I just finished some final design tweaks on a paper airplane launcher. The launcher (see video above) is likely going to be a part of a larger mathematics/modeling unit for upper elementary students.

The launcher is made out of eight sheets of standard card stock (65 lb). I used Silhouette Studio, a free download from the creators of ModelMaker, to design the various components and a Silhouette SD to digitally fabricate the parts.

I am including the files below in case anyone might like to download the software, purchase a Silhouette SD, and build a personal paper airplane launcher. I am in the process of working on assembly instructions…

• Base Supports
• Brace Supports
• Left Base
• Right Base
• Protractor
• Protractor Supports
• Ruler
• Vertical Arms

The Paper Airplane

Here are the design files for one of the paper airplanes that appears in the video.  A fellow graduate student and I are still testing various designs so there are no promises…

• Fuselage
• Inverted Wings

It is not uncommon to hear a fantasy football expert make the following statement:

Start PLAYER X because the opposing team is not very good against the RUN/PASS.

Watch the following video and listen to the predictions made by Yahoo’s Brad Evans. Note how he uses NFL stats to bolster his claims to “start” or “sit” a particular player.

Here are the two big picks that Brad Evans makes (summarized):

1. Matt Schaub is going against a Tennessee Titans team that is allowing 278 passing yards per game this season. Play him.
2. You don’t want anything to do with Jay Cutler! He is going against one of the most aggressive defensive lines in the league in the Philadelphia Eagles. They have limited opposing passer to just 1.6 touchdowns per game. Jay Cutler is going to be swarmed over!

I recently began exploring the mathematical realities behind fantasy football experts’ predictions for a seminar course on mathematical problem solving. Why? The topic interests me in a Moneyball-type of way, and I think that there is the potential to connect NFL data to an emerging personal line of research: creating engaging modeling activities for K-12 students. Plus, I might be able to eventually make my own informed decisions about fantasy football roster decisions!

My exploration into the mathematical realities of so-called fantasy football experts use of NFL stats arose from an observation: I listen to what they say, their advice seems plausible and informed, and I act upon their statements by changing my fantasy football lineup. Yet, more often than not, my decisions don’t pan out and my final point total for the week is lower than not making a change. At least that is the way that it seems to me when I lose- and, honestly, I only begin to question the experts when I fall to an opponent. Last week I lost.

Taking my loss in stride, I began wondering whether or not mathematics might be more informative than the experts. For example, does a quarterback like Matt Schaub actually throw for more yards when playing a team that gives up more passing yards on average for a given week? Is there a correlation? One would expect there to be a significant, positive correlation across all teams based on common sense. It’s a logical rationale: if a team’s defense averages more “passing yards allowed” then it would make sense that opposing quarterbacks would throw for a roughly equivalent amount of yards or greater (disregarding all of the other variables like home field advantage, injuries, and the like). This is the logic used by Brad Evans when he made his prediction. It makes sense, but is it mathematically verifiable?

Using this as my research guide, I began combing the past week’s NFL box scores and stats summaries. The spreadsheet below contains stats based on defenses.

Take Matt Schaub as an example. Schaub, the quarterback for the Houston Texans (HOU), played the Tennessee Titans (TEN). Looking at the Tennessee defense, the average passing yards allowed per game given up by the Titans is 278 prior to the game against the Texans. Schaub threw for approximately 158 yards (assuming he was the primary quarterback) against the Titans, a value that is well under Tennessee’s cumulative average to date.

Is Matt Schaub’s passing yards significantly (statistically and mathematically) different from the opposing team’s passing yards allowed? Can we draw any conclusions about all quarterbacks in the NFL based on the variables of (1) defensive passing yard average and (2) actual passing yards allowed? I crunched the numbers in SPSS and came up with the following correlations for passing in week 12 in the NFL.

The table shows a statistically insignificant relationship between defensive passing yards allowed and passing yard average at any common level of significance. Even if it was significant, the correlation is both negative and so small that we can’t draw any conclusions about a quarterback’s passing yard performance based on the opposing team’s defense! Quarterbacks are just as likely to throw for fewer yards as more yards when facing a team that allows a certain average.

Brad Evans, using the NFL stat of defensive passing yards allowed to suggest predicted passing yards allowed by a quarterback just did not work for week 12. Sorry.

I wonder how middle or high school students might use this data to interpret and model a solution that is mathematically-based?

A couple of notes:

1. A player’s point total in fantasy football is calculated by using more than just total yards accumulated. Touchdowns and fumbles also factor into the final tally, although this varies in different leagues.
2. The correlation culls data from week 12 only. Incorporating additional weeks and looking at an entire season would provide more information about the connection between passing/rushing yards and defensive averages.
3. There is probably some interesting longitudinal connections that could be made by examining correlations throughout the NFL season. For example, Week 1 statistics would be significantly positive. I imagine that this would fluctuate and drop as the season progresses.
4. I did not consider home field advantage, injuries, and other relevant/influential variables that likely impact a team’s weekly yardage totals.

Although I disagree with some of Conrad’s claims (i.e., computers are a silver bullet, programming hasn’t been used to teach mathematics- LOGO and Scratch), much of what he says resonates with me. A lot of what he talks about are the underpinnings of a models-and-modeling approach to mathematical problem solving.

You are offered a 20% discount on an item but you must pay a 15% sales tax. What would you prefer to have calculated first, the discount or the tax?

I am somewhat comforted by the fact that, when I pose this question to fellow graduate students and various people with fancy degrees, everyone has a definite opinion about what should be calculated first. Including me- I was quite surprised when I began doing the actual math to arrive at a more informed conclusion.

I am going to guesstimate that approximately 80% of my informally surveyed audience opts for the discount before taxation- and this makes rational sense. Take the discount, reduce the total cost that must be paid, and compute the tax based upon the revised amount that is now much lower than the original price. My mental reasoning for an unknown item that costs $100 looked something like this:

If I take 20% off of 100, then my price would be $80. Adding 15% tax to this revised amount HAS TO BE lower than reducing $115 ($100 + 15% sales tax) by 20%.

It became glaringly apparent after “doing the math” that the computational order is inconsequential as are the actual percentages involved in the equation. Why?

100 x 1.15 x .80 = $92
100 x .80 x 1.15 = $92

If you think of the problem in terms of just multiplication, then it becomes evident that neither the order nor the percentages matter.

Why does this misconception exist? Why did I believe that the order influenced the results?

This post is about how various professionals are using digital fabrication techniques to extend, enhance, and redefine their respective fields.* The sections that follow highlight real life examples of crazy ways that people are using the process to do something different- and making breakthroughs. These examples don’t necessarily shed light on the educational implications of digital fabrication; just what is possible in the most far reaching possibilities, many years away for the target audience (upper elementary and middle school students). At the same time, I wonder if the groundwork established by teachers at Lovett, Crozet, and Punahou aren’t the foundation for future architects, doctors, and filmmakers.

Aside: Most of the examples involve some variation of 3D printing, a component of digital fabrication that relates to the production of physical objects. 3D printing involves machines similar to conventional printers that precisely apply materials other than ink. Think of squeezing frosting through a tube onto a cake in an additive manner that builds the depth and volume of an object and you have a very rough idea of 3D printing. The difference between this metaphor and actual 3D printing is that 3D printing involves the use of computers that add both precision and control to the process. Still confused? Here is a video of me creating a miniature version of UVA’s Rotunda on elementary-appropriate software that is then fabricated using a 3D printer (see 2:48 of the video).

Cinema: Film & Video Game Prototypes

It’s a bit hard to believe that the creators of such hit titles as Ironman 2 and Halo 3 really need physical models. After all, the final medium is entirely 2D (for the most part). Yet, making and creating models is an important facet of the visual reality of video games and movies.

Automobile: Rims, Intake Valves & Other Customizations

Steeda Autosports creates Ford accessories of all types. Regardless of whether it is a “blinging” rim or a new set of shocks and springs, Steeda uses digital fabrication to create quick models (also known as rapid prototyping) that appeal to their clients’ wants and needs.

Read more at 3D printing revs up Steeda Autosports’ R&D.

Architecture: Concrete Printing

Making scale models and printed diagrams have always been a part of the “architect’s handbook.” Both contextual and illustrative, architects often use models and diagrams to convey thoughts and ideas. However, what happens when the architect is able to create with cutting edge processes?

National Security: Surveillance

I wonder how digitally fabricating insect wings and flying objects like those in the video that follows might further my safety. And my love of remote controlled objects.

Medicine: Building Organs

Much of this is Greek to me, but I kind of see it… If you can 3D print the building blocks of an organ, where can you go?

Read more at Building body parts with 3D printing.

*I am of two minds about workforce development topics as it pertains to education. Despite my conflicting thoughts, I thought that it might be helpful for the attendees to understand how various fields are using digital fabrication in the workplace. I feel like this post needs this caveat because it might appear that workforce rationales are the primary justifications for engaging in age/context-appropriate forms of digital fabrication. There are more reasons that pertain to engineering, mathematics, and 21st Century Skills, but I am not going to get into it within this post.

Dan Willingham resonates with me and my line of thinking.  It has something to do with the way that he can frame cognitive psychobabble in layperson speak that clearly explains teaching, learning, memory, and the human mind.  Plus, Willingham is a cognitive psychologist at the University of Virginia, and his Wahoo-roots make him alright in my book.

In his book Why Don’t Students Like School? Willingham writes that teachers who help students think about the real meaning behind an activity use storytelling as an organizing principle.  This is true for teacher types (the comedic teacher, the showman teacher, the nurturing teacher) and content areas (math, science, social studies, reading/language arts, etc…).  According to Willingham, a lesson that uses storytelling as an organizing principle contains four C’s:

1. Causality: Events are causally related to each other.
2. Conflict: The inherent struggle for the protagonist to reach a goal(s).
3. Complications: Sub-problems that arise within the conflict as the protagonist struggles to reach a goal (s).
4. Character: A strong cast of characters, whether real “organisms,” places, or ideas, are essential (p. 52).

Dan Meyer’s recent WCYDWT post begins to unpack these principles in an effort to explain why his “don’t tell” philosophy works in mathematics education and student learning.  Using storytelling and film as a vehicle for explanation, Meyer’s analogy and examples seem to align with what Willingham believes are the cognitive strengths of a storytelling platform for lessons and instruction.  Three such congruences are both individuals’ belief in setting up the story/learning experience, the “uninterestingness” of strictly focusing on an answer in mathematics, and the importance of a clear question (but not a wordy one).

Consider the following quotes from Willingham (I replace Willingham’s words with Meyer’s examples in the italicized words):

A couple of things are worth noticing.  A good deal of time- often ten or fifteen minutes of a class- is spent setting up the goal [when using a storytelling approach], or to put it another way, persuading students that it’s important to know how to [answer a question].  The material covered during this setup is only peripherally related to the lesson.  Watching The Italian Job isn’t related to decimals and measurement. It’s all about elucidating the central conflict of the story.

Spending a lot of time clarifying the conflict follows a formula for storytelling from, of all places, Hollywood.  The central conflict in a Hollywood film starts about twenty minutes into the the standard one-hundred-minute movie.  The screenwriter uses that twenty minutes to acquaint you with the characters and their situation so that when the main conflict arises, you’re already involved and you care what happens to the characters.  A film may start with an action sequence [ala James Bond films], but that sequence is seldom related to what will be the main story line of the movie.  It’s kind of like playing Gimme Friction Baby and creating a class leaderboard as a mechanism for learning about angles and tangent lines.  In this respect, game play is symbolic of the setup for the conflict or question.

When it comes to teaching, I think of it this way: The material I want students to learn is actually the answer to a question.  On its own, the answer is almost never interesting [by itself and in isolation].   And the question is very important (p. 57-58).

The final paragraph seems to be a falling down point for teachers, including me.  I struggled to come up with the proper question for the cereal box challenge which I did not include in the original post: What is the most cost-effective design for cereal boxes?  While Meyer may struggle to come up with the formative assessments, I struggle to come up with the right question to ask.