Creative Pattern

So, like most semesters in most of my teacher prep classes, we started out by watching Sir Ken pose the question, "Do Schools Kill Creativity?" Especially for preservice elementary creatures, who often have trouble seeing themselves as math teachers, who often have had very negative math school experiences, and will even sometimes bust out with "I hate math" in front of their math teacher.

This semester's group got pretty into it: the story of Gillian Lynne was high impact, the idea that things need to change had traction, several recognized that they had been subject to this, and the desire to incorporate movement really resonated. (We have a drummer in class, so that might happen.) Some students wrote about creativity for their weekly work: Lauren and Kyrstin, for example.

One of the ways I'm trying to encourage creativity is a work structure (syllabus) like this:

Daily Work: I’m asking you for 1 hour per class. Document what you did somehow and keep in a binder. It is not evaluated on correctness, but on percent completed. Keep an index/table of contents for which days you have work for. This work should either be doing math or learning about the teaching of math.  It is okay to double dip - use daily time for Family Math or weekly work. Just keep track of getting in your hours. I will offer suggestions, but this is your responsibility. It’s a good opportunity to practice generating ways to meaningfully work, which will be an important part of your work as a teacher.

Creating: from our work each week I am asking you to put an additional hour or two into deeper work of your choice. Revise or extend a daily work, play or make a math game, make some math art, find and read something in an area of interest, work on a math problem of interest or create a mathematical task… there is so much different work that teachers do. If you can connect it to our course work, it’s probably okay. Each week’s work will get feedback in terms of our rubric and qualitative.  But those aren’t grades. At the end of the semester this weekly work will be evaluated ⅓ on completion (did you complete work for each week) and ⅔ on exemplars. You will pick two examplars of your doing math, and two examples of your preparing to teach math.

There's a urli.st of their weekly blogs The list helps me in finding them all for giving feedback, but I ask them to link posts to our Facebook group as well. That gets more readership amongst the class than I've ever had before. One of the purposes of blogging their work is to increase their sense of audience. So if you do take a peek, please comment!

The math content we paired with this is patterning. Our first activity (close to this previously blogged one) got us playing with the appropriately named pattern blocks, trying to get at the idea of what makes a pattern a pattern instead of a design. Our ultimate idea was that it needs to be extendable. Not necessarily predictable, but when you see what comes next it should make sense with what came before. They built and then we talked about repeating patterns and growing patterns and then sequential patterns. To emphasize the extendable idea, we built patterns, then rotated to have someone else add on. Clearly - time for pictures.

Clear to everyone

No discussion

People accepted extension,
but felt like 3rd red block
could go "anywhere"

Generated interest because
the start was in a line, and the
pattern was extended 2-dimensionally

Patter creator admitted they
didn't know what came next, but
liked the extension. Next: 3 blues
top and bottom.

Arguments! Pattern creator wanted the trapezoids
double each step, extender focused on blues
"adding one" each time.

Is this a pattern? Designer claimed it was just a design.
Extender felt like the red-blue-green were lines
extending out each direction. All agreed: lovely!

Here's the handout, if you're interested.

The next day I wanted to build on the idea of the sequential growing patterns with explicit connections to algebra. My colleague Pam Wells has the best activity I know for this, adapted from a Mathscape activity. Here's my version. (As a Word doc, if you want to edit. Wasn't displaying correctly...)


Everytime I've used it the lesson has been engaging, provoking discussion, and very supportive of symbolic representation with the visual. Students wanted to work through all the letters on the front, though I only asked them to pick a couple. Many wanted to jump to building their own pattern immediately. Most glossed over the verbal description, so I pushed for that. In general with our pattern work, visual to verbal has been uncomfortable. This is a good activity for the connection between rate of change and the symbolic rule, as several students made that jump. Some students went from data to rule, and some from the visual.

A couple students extended this for their weekly work. I based my sample weekly work on the letter patterns, so I expected more, actually; but that's why we give students choice. Brett extended the letter idea to his whole name, which is actually a pretty nice context for adding functions. (File that one away!) Emily did a really interesting project, making some mathart that  had layers of patterns.

The lesson after this was dominoes - but that's clearly a story for another day. Later in the semester we'll do more patterns using ideas of perimeter, area and volume.

Sequential Circular Reasoning

I've been wanting to get better with the Sequence command in GeoGebra. It's a powerful tool for repetitive computation or construction, and math, of course, is full of the patterns. And then I saw...

#410 - Eights

From the perpetually fascinating Geometry Daily. Perfect opportunity for Sequencing.

First I dug into the geometry a bit. It seemed to me like the interesting bits were the 90º turns and the constant (looking) increase in scale. That means you could do them as a series of dilations - but it was complicated to think about the centers of dilation. Probably better to just figure out the radius and center of each circle.  It seems important that the circles osculate - the kissing is a big part of the visual effect.

I was also thinking about what you could generalize, as that is the point of making it dynamic.It's also easier to build those things in as sliders at the beginning of a sketch than editing them in later. (Though it's not that much harder.) The angle between circles, the number of circles, the dilation ratio... pretty good start.

To build in the angle I just rotated the the circle defining point by the slider and the opposite of that angle. And then I made vectors in those two directions from the center of the circle.  I was thinking I needed those directions to build the new circles.  The radii of the new circles would just be a, a^2, a^3 if a is the scale.

Then came the messy thinking. The center of the first circle is a* the original radius in the direction of one of the points. Then the 2nd circle is.. the 1st radius plus a times that radius in the 2nd direction. I was working it out symbolically, but now that we've got a picture...

So thinking about the centers, I had to organize my data. Usually I scribble on an envelope, but didn't have one handy, so I used Word.

Pretty neat once I sorted it out. I decided to separate the circle centers by the two directions since it was really massy trying to come up with a single sequence to describe the pattern. The key here was the Sequence command.  (Quotes are from the GeoGebra Wiki.)

Sequence[ <Expression>, <Variable i>, <Number a>, <Number b>]

Yields a list of objects created using the given expression and the index i that ranges from number a to number b.

Example:
L = Sequence[(2, i), i, 1, 5] creates a list of points whose y-coordinates range from 1 to 5: L = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}.

A list is what you would expect:

Lists
Using curly braces you can create a list of several objects (e. g. points, segments, circles).

Example:

• L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C.
• L = {(0, 0), (1, 1), (2, 2)} produces a list that consists of the entered points, as well as these nameless points.

Note: By default, the elements of this list are not shown in the Graphics View.

But I couldn't get it to work. I often try to do too much at once, so I decided to make a list of the powers of a, and then the coefficients of u and v. The relevant commands in GeoGebra are:

Element[ <List>, <Number n> ]

Yields the n^th element of the list.

Example:

Element[{1, 3, 2}, 2] yields 3, the second element of {1, 3, 2}.

Append[ <List>, >Object> ]

Appends the object to the list.

Example: Append[{1, 2, 3}, 4] creates the list {1, 2, 3, 4}.

Append[ <Object>, <List> ]

Appends the list to the object.

Example: Append[4, {1, 2, 3}] creates the list {4, 1, 2, 3}.

 There is also Insert if you need more control than that.

So I defined my lists

• ays = Sequence[a^(i), i, 1, N]
• cofs = Append[0, Sequence[Sum[ays, k], k, 1, Length[ays]]] 

 It took a little monkeying around to get these next commands to work. I'm not sure if copying from Word was introducing extra characters or what. 

• Sequence[B'_1 + Element[cofs, 2n] u + Element[cofs, 2n - 1] v, n, 1, 10] (list1)
• Sequence[B'_1 + Element[cofs, 2n + 1] v + Element[cofs, 2n] u, n, 1, 10] (list2)

Then the circles are easy!

• CirclesList1 = Sequence[Circle[Element[list1, n], r Element[ays, 2n - 1]], n, 1, N]
• CirclesList2 = Sequence[Circle[Element[list2, n], r Element[ays, 2n]], n, 1, N]

I'm pretty happy with the results. If you'd like to play with the result or remix it for yourself, it's on GeoGebraTube for download or as a mobile-ready applet.

Geometry Daily is a good source for GeoGebra inspiration, as well as good geometry, and just beautiful art. (Usually I put them on my Tumblr.)

Pyth On

Mel Bochner, Pythagoras (4)
from wikipaintings

Arithmetical Design (quite a fun tumblr) posted this beauty today...

I thought that this was something that screamed to be dynamic. Off to the GeoGebra Cave, old chum!
















The sketch started with a right triangle, and then the regular polygon tool to make the squares on the side. I wanted the triangle connecting the next squares to be similar to the original, so I made the side of a square to be the new hypotenuse, rotated it by one of the non-right angles then used the perpendicular tool to make the similar right triangle. Finally, I constructed  the first two additional squares.

Clearly too much work to repeat in the dozens. To use the Create New Tool command you select item or items in the sketch. Then select the command from the tools menu. My first try I forgot that I would need the points to make subsequent squares. Delete the bad tool from the Tool Manager. (Can also rename there if you're trying for something more pythy than Tool 1.)

When I had the squares and vertices selected, the second step of the Create New Tool dialogue was to determine the inputs. GeoGebra will select some ancestors to start, but you can modify the inputs. In this case, GeoGebra selected my first two free points, which doesn't suit. I wanted the inputs to be the the endpoints of the hypotenuse. At the last step you select a name and can attach a custom icon if you're being tricksy.

Once I had the tool it was quick to construct the spirals, and then aesthetics like a coloring scheme and positioning. From the GeoGebra color dialogue you can click the plus, which brings up an RGB color input. (For those times when you need beige, 255-245-235.)

I was going to stop there, but decided that people needed to be able to make their own spirals how they wanted, so added a checkbox to go back to the beginning. (If you make something send me the pic and I'll add it to the post.) Sadly the new points show up with labels - I don't know how to turn that off. Maybe if the labels are off before I make the tool? Tried that and it works!

Here's the finished sketch at GeoGebraTube: teacher page or applet. Sadly, the custom tools don't seem to show up in the HTML5 mobile applets yet.

Bochner has several mathematically influenced paintings, as well as the first three Pythagoras painitings. Check them out at wikipaintings.

Exponential Potential

It really struck me listening to Shawn Cornally in this week's #globalmath session on SBG (click on the Recording tab) how his perspective as a physics teacher leads him to approach his math lessons as experiments. Starting with an experience that makes us want to model or makes modeling useful is definitely the start of some of my favorite math lessons. (While I'm writing this he tweets: "Leaf-Blower soccer went over *really* well today in physics. (vectors, f=ma)")

Starting exponential functions with my preservice teachers, I love to use this lesson adapted from a 5th grade Math in Art lesson. (From my pre-blog webpage.) The idea is the multiplicative patterns present in a Sierpinski Carpet.


(Also in Word format if you want to edit.)

One of the interesting discussions in the initial exploration is the 9 or 17 issue. 9 squares if we count the number of squares as distinct shapes, 17 if we unitize to the smallest level square. For algebra students there's some good opportunities for equivalent expressions, regression and even deduction of function rules. This is a good opportunity for sharing how recording how you're getting your answer can be more powerful than recording answers. The 17, for example, is 1·9+8, then the next level is 9·(17)+8·8. But later, most write it as \( 9^n–8^n \) - which can lead to a pretty neat binomial expansion. Maybe even more interesting and accessible is the 9=1+8, so the next step is 73=1+8+8·8, and gives them a way to generalize this pattern besides recursion.

Once we get to the design your own carpet, there are so many new patterns to find. Here are some samples from this week:

Note that these last two aren't really Sierpinski patterns - but they still raise interesting patterning questions that are extensions of what we already noticed.

Probably easy to see why this is one of my favorite lessons. I also have seen the power of adding in places where students who have not traditionally been strong in math class can do amazing work.

The next lesson to follow up this one has some other opportunities to gather/generate multiplicative data.



It always strikes me how even college math majors find things to be surprised about in this data. Especially the penny balancing one. This class made some neat displays of their data - but I haven't taken the pictures yet. (Didn't know I would be blogging this, as I thought I already had! Maybe I was thinking of the quadratic simulations?) I'll add them at first opportunity.

First Opportunity:

Next we'll look at modeling this data symbolically using technology, and asking questions that raise the need for logarithms. Since, of course, every exponential data set is logarithmic when seen through the looking glass.

Some Sum to One

Another quick activity. Put this version of it together for the 5th graders, but didn't get a chance to do it with them. This would be suitable 5-8, and my preservice teachers got a lot out of it also. I added some more support (in structure) on the square for middle school age kids. The grid divides well into thirds, fourths, twelfths and the like.

The idea was probably Mondrian inspired, mixed with a geometry activity that I like: divide a geoboard into four non-congruent equal parts.

The task for this one is to make different fractions that add, or fit together, to one, or a whole square. Had very interesting conversation with Dave Coffey today about fractions and introducing operations, and representation... and then that conversation spread to Lisa Kasmer.  Nice to work with such interesting colleagues.

EDIT: oops! had the permission set incorrectly on this. If it's not appearing below, here's the link to the pdf.




Two examples from my preservice teachers. The second one was made with flaps that lifted up to reveal the value of the fraction shown. Nice!

You can have your students simplify or not - both create interesting situations.

A nice extension challenge is to do this Egyptian style, with all unit fractions.

To Do + Math on Vacation

The Mathniños at the Modern Wing

Calder Mobile

Math Teachers at Play 22
I'm hosting this months blog-carnival, and I'd love your submissions from your own blog or nominations from blogs you read. Submit nominations at the blogcarnival site, or using the widget at the bottom of the right column on this page.

I never had a chance to write about Carnival 21 at Math Mama Writes... but Jonathan (JD2718) had a nice post about a subtle shift that produced a good effect.

An aside: I got to meet Math Mama over break and Sue was as nice and interesting as you'd think from her blog. We were going to meet up with Maria from Teaching College Math, but the weather prevented it (sorta).

Biggest Math Story of the Year
Time Magazine has submitted that the proof of Langlands' Fundamental Lemma was the 7th Biggest Scientific Discovery of 2009. A quite technical exposition of the Lemma (not the proof) is here. (Hat tip Peter Woit of Not Even Wrong). I do not have an understanding of the math, but it concerns a connection between Galois group theory (about solutions to algebraic equations) and general linear algebra representation theory (think matrices on steroids). One of the most famous theorems that uses the Lemma has Femat's Last Theorem as a corollary. Powerful stuff.

It's nice for students to know that math is ongoing, and also to hear terms like lemma, theorem and corollary used genuinely. I also like how this emphasizes the power of connections in mathematics, and how even (or especially) the world's most powerful mathematician's start problem solving by solving simpler problems or finding another way to put the problem. It also should be noted that Ngo's proof was submitted in 2008, but took until this year to verify.

Math (Art) on Vacation
My wife and I recently won tickets to the Cirque de Soleil show Banana Shpeel (which I would recommend.) On the quick trip to Chicago this week sans kids (hat tip: grandparents!) we had more time in the Art Institute than we would have had otherwise.

In the geometry class I've been writing about, we do one constructive project in which students build a polyhedron for the following context:

Hoity Toity, the upscale chain of Haute Couture for the masses, is having a competition to design new and original knick-knack boxes. Being an accomplished mathematician, you have an unfair advantage, which you intend to exploit to the maximum. Boxes must have a volume of at least 1 liter. (1000 cm3).

Students do a terrific job in general. One student this semester made a dog-shaped polyhedron. Cool. Some classic Archimedean solids, as well as prisms and antiprisms and such.

But at the Art Institute I had two ideas for new variations. The Modern Wing in itself is an inspiration as it is filled with beautiful proportions and rectangles with more connections and relationships than you could ever hope to count.

The Chess Set

This wonderfully sculpted set filled me with visions of two people working together to make a set of polyhedra vs non polyhedra. They could do found or constructed objects. It will require connections (what makes the difference between the king and queen, how will players know this is a rook, what's the difference in the bishops between the sides...) and analysis of the solids and their volume.

The Building

I so wanted to see the isometric drawings for this. I wouldn't insist that the building be built, which gives me pause, but I would want two different representations that would help someone build it. Students should design their own advanced building and think about it in scale

I'm curious to see where these ideas go, and what students make of them. I would like to hear more from other teachers about from where their ideas come, and the process they follow to implement them and refine them in the classroom. I think we teachers do a lot of our sharing as a product exchange.

Mobile Math

For the Math in Art Festival I did with Susan Walborn (an amazing teacher who's moved on to becoming an amazing retailer - must just be amazing, eh?), one of my favorite lessons was a mobile lesson (link leads to a pdf of a verrry complete 3rd grade lesson plan) based on the art of Alexander Calder and the math of area and average.

The emphasis is on the average as a balance. In calculus today, we covered center of mass, and built the connections among the moment, the weighted average and the idea of balance. Students used the ideas to create cardboard cutouts of curves and find the balancing point. They did a great job. Mine is the unimpressive cubic.