Looking at My Inventory

I shared before about the math learning inventory I sometimes give. The greater the diversity of learners, the more inclined I am to give it. It surveys how students think they learn mathematics.

A basic break down of the learning preferences assessed follows from a Silver, Strong, Perrini and Tuculescu article, The Thoughtful Classroom:Making Students as Important as Standards.

• Mastery learners want to learn practical information and procedures (what)
• Interpersonal learners want to dialogue and collaborate (how)
• Self-expressive learners want to use their imagination to explore (how)
• Understanding learners want to learn why things work (what)

I do not think learning styles are determined for a student, but finding out what students think they prefer seems like a worthwhile preassessment to me. If I want to be a culture changer, I need to understand from what I wish to progress. The form I use is at Scribd, but I'm happy to email the Word document, too.

I thought I'd share how this semester's group of preservice elementary teachers look, and how that affects my planning. Mostly it doesn't change what I want them to experience, but it has a reasonably big impact on how I implement it. Also, with an education class, it's a chance to point out the tension between surface and core beliefs.

 

Yes, I put it in GeoGebra to help visualize. (I may have a problem. Here's the sketch if you want to use it to visualize your student information.)

 

Here's the most consistency. Students who feel like "tell me then I practice" is how to learn.  The challenge for this group is abstract or open-ended problems. Also known as all my favorite stuff in math. How I'm responding is to be explicit about what kind of problem we're working on, and support with structural suggestions. Like in working on trying to find all the pentominoes this week, I interrupted several times, solicited progress reports from groups ("They have 16?!"), and got them to share method and develop methods as a whole group.

Especially worrisome is when you compare my inventory with their composite. At the conference where I first saw this inventory I had a number of students.  They divided us up by table based on the results and I was alone! So I'm aware of my distance from students on this and how it can affect my persective.

Whoa. All over the place. This is a caution to me to not require group work always, and to look for ways to make space for some individual problem solving. Since relationship with teacher is part of this, I'm trying to be more personal to supplement for the people at the top end of this scale.

Very encouraging. Many times when learners have had negative math experiences this category can be quite low. I have to think about how to support those four students, though. So much of our class revolves around this kind of why thinking. And I feel that it is crucial for future teachers. I hope to have experiences that justify this approach to math. Maybe the way conceptual understanding furthers skill is the entry point?

 

Another one where the students are well spread out.  We started with a week long focus on creativity, that really got pretty good buy in from my perspective.  Again the students who do not favor this kind of learning are a challenge for me. Choice in assessment is part of the course structure, but that's probably not enough. What more can I do for them? Is it important for all teachers to be at least comfortable with this mode? 

Teacher-Student Alignment

Here's my graph compared with theirs. 

 

This probably comes through in the Mastery section above, but it's clear where I don't align with students. In some math classes I give choices for practicing content that can satisfy Mastery learners, but there's just not much of that for this class.  I'd welcome any suggestions here, is what I'm saying.

The view of all their regions at once also raises a concern about students who seem to have low responses across the board, as I think that's just overall discomfort with mathematics.

The goal with this post was to share what use I make of this information. If that didn't come through, or if you have ideas for more uses, please let me know!

Post Script: went through the survey with my 8th grader - very interesting discussion.  Discussing why he was answering what he did on the individual questions was enlightening. It makes me wonder about using this as an interview tool. Even for a student I know pretty well, we got to some new ground.

Post Post Script:  forgot to add the other questions' data. Even if it's mostly for fun, some illumination is to be had.

Motto

• 16 - Show me how and let me practice.
• 2 - I want to know why.
• 1 - Let me play with it. (My choice)
• 0 - Let’s talk about it and hear everyone’s ideas.

Assignment

• 10 - p. 101: 1-39 odd
• 6 - Draw a picture that shows the ideas. (My choice)
• 3 - Work with a group to make a math skit.
• 0 - Report on a math controversy.

I have to work to understand the mindset of 1-39 odd. Other than comfort with the situation you know.

Flip Flop

Jennifer Silverman made this cool motions maze the other day. (More here.) We collaborated on it a little bit, after she did all the heavy lifting. I added buttons. (I may have a button problem.) It put me in mind of these motion puzzles I used to make in Geometer's Sketchpad, and I got to thinking how much better I could make them now. So I started, with the main new feature I wanted being the ability to generate new puzzles instead of being one static dynamic puzzle.

The user moves points A and B to try to find the line of reflection between the two flip flops. When you hit the Check button, it shows you the reflection over the line you're trying. 

What follows is my GeoGebra geek out over trying to make it look right. Here's the puzzle if you want to skip that: Flip Flop.

One thing I love about Jennifer's sketches are her excellent images. So I tried to step it up with some nice flip flops from openclipart.org.

It turns out the trickiest part was getting both sandals to always show up. That's why I'm writing this post. (A lot of my individual sketches I post at the tumblr.) The key to being able to do this is that in the graphics window you can put variables in for the window dimensions. Define those from the objects in the sketch, and,  voilà, you can see both the sandals. So I defined xmin, xmax, ymin and ymax from the two sandals. E.g.,

xmin=floor(Min[{0, x(F1'), x(F2'), x(F3'), x(F4')}]) - 1
But there's a problem then - the graphics won't be in 1:1 scale, which is always nice, but especially important for motions where the two objects should look congruent!

The Corner[ ] command is my new best friend. Corner[n] for 1, 2, 3 & 4 return the coordinates of those corners. Corner[5] returns a point with (width, height). Corner[image name, number] returns the corners of an image. This was handy for finding the corners of the reflection, F1' etc. in the command above. 

So using Corner[5] I could find out the aspect ratio of the graphics window. It took me a few minutes, but I hit on the idea of making the bottom left corner steady, and then altering the top right corner based on the aspect ratio. I defined r = y(Corner[5])/x(Corner[5]) (so, height:width) and:

• xm = If[(ymax - ymin) / (xmax - xmin) < r, xmax, xmin + (ymax - ymin) / r]
• ym = If[(ymax - ymin) / (xmax - xmin) < r, ymin + r (xmax - xmin), ymax]

If the sandals give a screen that's not wide enough, it uses the aspect ratio to find a suitable width. If the sandals give a screen that's not tall enough, it uses the aspect ratio to find a good height. (The If[ ] command works like If[condition, then, else], where the else is optional.

The puzzle, as it turns out, is pretty challenging. Give it a try, and let me know what you think. Or let me know an easier better way to do my GeoGebra graphics hacking.

Here's the teacher page for download or the mobile page.

EDIT: Bonus! Jennifer created an assignment to give it more structure as a lesson. (PDF in dropbox.)

Percent Game Remixing

In yesterday's post on a percent game, I shared two great GeoGebra sketches that students found. I remixed each of them a little, so I thought I'd share them here.

dhabecker's neat rational number arranging sketch lets students place arrows to try to put fractions, decimals, and percents in order. He has a very clever way to check if the randomly generated numbers are in the right spots.  It notifies students when they've got all 4 right, and I wanted to them to be able to check their answer along the way. So - thinking of Mastermind - I thought about what if it can give the number correctly placed? Since I was doing that I added a reset button and a bit of color. Next I would add a fifth number, as that makes so many more permutations possible.








On GeoGebraTube:
download or applet. (Unfortunately doesn't seem to work in HTML5, because the polygons won't move.)









The other sketch I modified was David Cox's great percent estimation sketch.  Almost immediately on trying it on the Smartboard, the students turned it into a game.

So I turned it into a game with turns and scoring. There's 6 rounds. I thought about forcing players to take turns going first but ultimately just decided to ask them to.

Next  I would be interested in seeing it go from a percent number line to another quantity. So the game asks you to find 38%, but the number line goes from 0 to 630. The percent and the whole would change each time. Worth a go? Probably that's in my head because of David's nice double numberline percent sketch.

On GeoGebraTube:
download or mobile-ready applet.


As always, I'd be interested in feedback on either one of these.

But I'd also be interested in what could help develop a remix culture in GeoGebra.  I learned it (am learning it) mostly on my own by experimentation, from suggestions on Twitter, and googling stuff from the online help. But in the Learning Creative Learning class they put a big emphasis on remixing as a way to learn that gives a lot of support to learners. With my middle school GeoGebraists, they are struggling to do work of value all on their own.

Are you a remixer by nature? What would it take to get you trying it in GeoGebra?

Percent Game

I was thinking about percents and ways to gain experience with them, in preparation for GeoGebra work with middle school students.

Found a couple of neat percent sketches on GeoGebra...

Nice visualization from jholcomb.
(By the way, GeoGebra has gotten input boxes to work in the HTML5/mobile device. Good going!)

Slick discount problem visualization from Anthony Or (orchiming).

Nice double numberline visualization from David Cox.

But as I looked, the idea for a game came to me, just to give percent experiences. There's no context, really, it's just a race game. No strategy, just rolling random percentages. But the mechanic of smaller roll goes first creates some nice percentage situations, and a lot of games wind up surprisingly tight.

I debated having the students find the percentages to subtract, but decided to make it optional.  There's a short video of how the game works below.

Here it is on GeoGebraTube, for download or for mobile applet.






The test game came out extremely close - most games will be shorter than that. It can be surprisingly suspenseful, though.  All students found it pretty playable, and some got very into it. I think the best benefit might be from playing and then using as a context for problems.

Searching through GeoGebraTube, students found two sketches particularly of interest.

Arrange Fractions, Decimals and Percents by dhabecker (who has quite a few rational number sketches), which lets students arrange form numbers of different form from least to greatest. A few students got quite engrossed in this sketch, and the feel of the sketch is terrific - very much like pieces snapping in place.

Estimating Percents by David Cox, which lets students make a first estimate and then a second estimate with the tens showing. Students were happy to see their % error improve from 1st to second guess, and got better quickly through playing. They also made an impromptu game out of it, and I think that would be fun.



Do you know a GeoGebra Percents sketch that you think supports students' understanding?

Here's the results collected so far. Thanks!

PS> there's a follow up post to this one with two more GeoGebra percent activities.

Penrose Tiles

From Wikipedia,
Professor Penrose

Quasiperiodic tessellations are my favorite bit of recreational math. They came up in my thesis non-recreationally, and the best bit of Mathematica I ever wrote was to generate them for various different rotational symmetries.

Recently the Penrose magnets came from the successful kickstarter. (I can't tell from their website if they are available anywhere now.) It was fun to see tweeps sharing getting theirs. @MrHonner posted a picture that started the jealousy.  This is a picture of my son's attempt to tile with them.


A recent GeoGebra project got me using the RigidPolygon[] tool for the first time, and later I realized that this would enable me to make them in GeoGebra.  The construction of the kites and darts is easy with the regular polygon tool. I couldn't make a tool to make copies, because it wanted too much information. I remembered the advice Kathryn Peake was giving to David Wees on Twitter for a sketch: make a button.

So the first sketch was the tiles without matching rules.

Purists, like Edmund Harriss (@gelada) in that Twitter conversation,  will correctly point out that these are not Penrose tiles but they can be used to make a Penrose tiling.

Without the matching rules it's really easy to get yourself in trouble.

Here's the GeoGebraTube page. Unfortunately the applet doesn't work well in the HTML5 version, since you can make new tiles, but they can't be moved. Fine in the Java applet, though.



It took me a bit to figure out how to construct the tile alterations to provide the matching rules, but the upside is that it is customizable, so that now you can make the tiles into shapes that are pleasing to you. I can see how Dr. Penrose wound up with chickens, though.

Here's the GeoGebraTube page.

I did get to meet Dr. Penrose when he was working with my advisor - very fun and charming man. As well as obviously brilliant; nice when things work out that way.

From David Austin's fun slides
Playing Penrose's Tile Game

In the sketch on GGBtube, I share some places to learn more about these:

• My favorite: my colleague David Austin's two articles, Penrose Tiles Talk Across Miles and Penrose Tilings Tied Up in Ribbons.
• Don't miss Dave's whiz-bang Java Rhomb Tiler. It is serious fun.
• Quick basics and images at http://intendo.com/penrose 
• A GeoGebra sketch to look at the motions in the rhomb tiling from jwelker.

If you get a chance to try either of my sketches, I'd love to know what you think! 

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.)

Turn Me Right Round

What makes for these mathematical mini-obsessions?

Whenever I put something up on 101qs.com, I make sure that I pose questions on
at least 10 posts from other people. (I also never skip. [So will I ever have questioned everything? #anyqs] Though for some it can be hard to find mathematical questions. [Orange.]) On a recent trip I saw Alex Shum's cool revolving door picture.

It reminded me of the problems where you're trying to take a sofa around a corner. (Which reminded me of my very first favorite screensaver, inspired by Douglas Adams, of trying to fit a sofa down a staircase.) Whatever the reason, I immediately wanted to make a GeoGebra sketch.

I made a first pass. That got me wondering how big are those openings? In particular, would a revolving door ever have and opening straight through like the one in the diagram to the left? (Seems to defeat the purpose of a revolving door.)


What are the standards for revolving doors? Thanks to Google and the International Revolving Door Company, I now know more than I ever knew I wanted to know.

In particular:

Data!

So much to wonder about these. I love the idea of describing circles as circumscribing a rectangle and it makes perfect sense for construction. But now I want to know about this data. What kind of function is it? Does it make it so the opening is always more narrow the one sector of the door? Is the angle of the opening from the center constant? How do you choose between a 3 wing and 4 wing?

First pass on the data was pretty curious.

Good for making the sketch, but weird. Why would the three wing doors have narrower openings?

GeoGebra sketch for
download or mobile applet

I made a weird little function to give the door openings, 2*radius (0.475 + 0.21 (doors - 3)), so that the ratio is .475 for 3 wings, .685 for four wings.

I liked the opportunity for modeling that this turned into, and it's also a good problem to show where modeling either supports calculation or is more efficient than calculations. Constructing the model in GeoGebra required data fitting, and some algebra to find the appropriate boundaries for the geometric objects.

And it helped me understand why 3 wing doors have narrower opening than 4 wing doors.

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.

Spirograph 2 - GeoGebra Animation

Okay, I've played with cycloids before. But when Guillermo recently updated his hypocycloid tutorial, it gave me the push to play again, since I'm always trying to get better at the GGB. Making it helped me understand GeoGebra animation a bit better, so I thought I'd share. I also think the resulting sketch could be the basis of a pretty nice open ended activity.

Obviously, having too much fun.

As good as Guillermo's instructions are, I'm the student who wants to figure it out for himself. One of my easiest teaching mistakes is to assume that my students are like me, and to provide too little support. Although I can overcompensate and then dictate too much, too. To provide student choice is the ticket.

So I started with the circle. I debated about making the controls be the radii for the boundary circle and the rolling circle, but finally decided to make it the boundary radius and a fraction of that for the rolling circle. Originally just a decimal, 0 to 1 incrementing by .05, but eventually I decided to make it a proper fraction, and on the slider control made the increment 1/60. It will show as a decimal approximation, but GGB stores it as the fraction for all practical purposes.

Then the rolling. The usual thing to do animation is to make a slider for time. My first take was to have the slider just go from 0 to 2\(\pi \). (Or 0 to 360 degrees, but I didn't want the units hassle.) Then I increased it to 10\(\pi \), but finally decided that it's neat to choose the number of rotations, so I made a 'circuits' slider for how many revolutions and defined the time slider to go to circuits*2\(\pi \). I use sliders instead of input boxes when possible because the input boxes don't work in HTML5/mobile devices yet.

Now the geometry. Really the rolling circle slides around the boundary at a contact point, rotated by the time slider away from some arbitrary starting point. The rolling you simulate by rotating the sliding contact point around the small circle. But how far does it rotate? My naive first take was that it should be as far as you've slid around the larger circle, thinking about it as a distance, like a string wrapping around. When you've gone angle \(\alpha\) around, you've gone distance \(\alpha\)*radius. Putting that in...

... pretty! But not a trochoid. It took me a few minutes to realize that I wanted the angle to rotate, not the distance. So if the circumference you've traveled is \(\alpha\)*big radius, then that's
$$
\frac{\alpha * \text{big radius}}{ \text{small circumference} }
= \frac{\alpha *a}{2\pi*\text {small radius}}
= \frac{\alpha *a}{2\pi*b*a}= \frac{\alpha}{2\pi*b}$$
So the angle is \(\frac{\alpha}{2\pi*b}*2\pi\) or just my time&angle variable divided by my radius ratio, t/b. I still have to think about what happened with the first try - obviously there's something mathy there.

It didn't look a lot like it was rolling, so I added the spokes to the small wheel by rotating a segment from the center to the rolling point around the small circle center.

Since I added the number of circuits as a variable, it made the speed of the sketch animation change and I couldn't find a value that was good for any number of circuits. But when I added a slider for the speed of the animation, I lost the play/pause button for the time. You can still control it with a right-click context menu, but that's not very user friendly.

So I started digging around the GeoGebra manual for animation controls, and finally found StartAnimation.  In particular, the boolean version,

StartAnimation[ point, slider, ... , boolean]

animates whatever is listed if the boolean is true, and stops it if false. Now I could make a boolean variable for animation (called 'animated' here), and control it with a button.

I'm using SetValue more than variable assignment lately because it avoids some weird issues that come from fixing a variable. ("!" is the text version of \(\not\) ) This also allows me to make the animation of t just a once through instead of repeating, which makes choosing the number of circuits more relevant. Here's the final version of the time slider:

To make the sketch more Spirography, I  took out some of the erasing and timer resetting from the scripts for sliders and buttons, and added in the color controls for the pen. The color values in GeoGebra are between 0 and 1, as opposed to 0 to 255 in some programming.

Activities/Problem ideas:

• Given a ratio, how many circuits to completely draw the hypotrochoid? How many vertices will it have?
• How are the hypotrochoids for ratios with common denominators similar and different? Why does that happen?
• Make an image that is aesthetically appealing to you. Document your process. What did you have to figure out to make it? What math can you see in the final image?
• What kind of mathematical curve is one side of a hypotrochoid? How do you know? Can you prove it? Why would it be that way?
• Give students a challenge image, and ask them to duplicate it or investigate the mathematical properties. Or ask them to make a challenge image for another group then swap.

How else might you use a sketch like this?

Here's the sketch at GeoGebraTube or as an applet (works on mobiles).  Hope you have at least 11/60 as much fun as I did with this.

Post Script:
With that work done, it's been easy to add features. I added a mirror point on an outside circle so that the sketch could do epicycloids. Then I dilated the pen points from the center of the rolling circles to get full on trochoid glory. Here's the final sketch on GGBT and mobile app.

Have fun! Send me a cool image!