Intermediate GeoGebra

So we had a small group in for GeoGebra learning this week, and I want to share what I've learned. One of the best ways to learn anything is to try and teach it, and this was true this week.  This post is a review of our approach with links to the materials, and some description of what the intermediate users were interested in: images, buttons, input boxes, and more.

http://bit.ly/ggbgv

First up is our revamped website. I'd really like this to be a resource for anyone learning or teaching GeoGebra to others, so please send feedback. The old website had gotten pretty clunky, as we modified and added to our materials, and it was time to start fresh. The website also includes a page with all of our handouts, and a nice Resource section.  Besides the website, we started a local GeoGebra Facebook page that we could use for links and resources.

This site has a clearer direction through it. Whether you are introducing GeoGebra to people with some time - say a two or three hour block - or in a quick one hour presentation, this is how I'd recommend starting:

The program is so intuitive to use, that many teachers can go from there. GeoGebraTube is such a huge sell, that I tried starting with it, but it turns out that it's even bigger when people know how easy and free the program is to get. They can explore the sketches on the tube better with just 5 minutes on using the program first. I separated that out into three pages at the website: Start Up, GeoGebraTube, and the Quick Introduction.  I asked participants to record the Tube finds they liked the best on a Google doc, but I've also just asked them to write the Tube number on the whiteboard.

If you'd like to add to the Google doc with your favorite Tube Finds, be my guest! (Here's the link.)






We covered search, tags and user profiles, all of which enable finding cool stuff.

For the Basics, the list of intro tasks has gone through several iterations and seems to work pretty well. Getting through those gave users a fair amount of control. The first non-basic topic most people are interested in is usually is using images in sketches (often inspired by the Dan Meyer inspired Dan and the Ball sketch). If it's not images, teachers are interested in the spreadsheet view. So we've got a guide to using images and a starter for the spreadsheet.  It's also important to cover the export menu.

Once users have gotten to making some pretty impressive setches, most of what the teacher or trainer has to provide are the commands to help in what they want to do. The two that have come up the most for me are:

• RandomBetween[ ⟨ minimum integer ⟩ , ⟨ maximum integer ⟩ ] - does just what you'd think it would. Very useful for generating targets or random polynomials using this for coefficients. You get new values by choosing Recompute All Objects from the View menu.
• Function[ ⟨ function ⟩ , ⟨ start x-value ⟩ , ⟨ end x-value ⟩ ] - limits the domain of a function. Best technique for this is often to define a function f(x)=... then make a limited version of the function using the Function[] command.

It seems the next matter for most users is to get experience with the Action Objects Tools,  sliders, input boxes, check boxes, and buttons. (That's the intuitive order for me.)

Richard Wade shared with me a neat challenge that provoked some good work as well as some instructions on animation, etc. This is my entry for a dynamic Olympic logo. I also think they should include a purple and orange ring, but I understand the importance of tradition. I wanted an interlocking rings effect, but couldn't figure out an over/under trick.

A note about embedding animated gifs: some services are weird about it. If you're having trouble, upload it to a photo sharing site (I use the mildly annoying photobucket) and put in the image by URL.

The main feature that's come up that we don't have a guide for yet is the dynamic text. Text is next.

Whenever you get to work in a group with GeoGebra you learn something. The two tidbits I found most useful this time was the Corner[ ] command, which we found in an Anthony Or (orchiming on GGBT) sketch.   Someone asked "Well how did he make two sets of axes?" and the answer was clever use of the corners.

The other was instigated by a participant: looking into what that student option is in a GeoGebraTube collection. When you click Create version for students, you get a menu of choices.

You get a palette of sketches for students, like this.

That's going to be a great feature.

Hope this can be of use to you for your GeoGebra learning, or with fellow teachers or with your students.  I'm happy to share any materials electronically, or in person if you're within driving distance.  What would be helpful for you?

Mayim Bialik

Might as well go for Controversy Week on the blog, eh?

I like Mayim Bialik as an actor, totally think she improves Big Bang Theory, and think she has obvious science chops. I personally get pretty geared up about the immunization "controversy" (which I only think is a controversy in the same way as the climate change "controversy"), and was disappointed over her Science Friday brouhaha about immunization.

Then I got a promo from NCTM giving her top billing at a meeting.

I wrote:

To: NCTM Summing Up
Cc: NCTM
Subject: Re: NCTM Summing Up: June 5, 2012

Do you all know that Mayim Bialik is a supporter of the anti-immunization movement? Pretty disastrous for teachers, should she have her way. I don't think we as an organization should be legitimizing her as science-related, let alone promoting her. This was in the news recently because of her appearance on Science Friday.

-John Golden

The NCTM was nice enough to write back...

Mr. Golden,

I am sending the attached letter on behalf of NCTM President Linda Gojak, who is copied on this message, in response to your message below about Mayim Bialik.

Regards,
Ken Krehbiel
Associate Executive Director for Communications

Hmmm. So I wrote back,

Thanks for addressing this and for responding. I understand the tensions involved.

I do think there's a difference between having a Ph. D. and qualification in math ed. For example, the many Ph. D.s in mathematics who support Mathematically Correct would probably not be advertised by the NCTM in promotional materials.

May I share this letter with others?

Thanks,
John Golden

To which they quickly replied:

Hi John

Thank you for your thoughts.

Actually Mayim has been doing some work in mathematics education with a supportive stance on the importance of mathematics as fundamental to the foundation needed in STEM areas.

I think her message around mathematics is important as we move forward. Her personal stances on personal issues are exactly that...personal

I think the CCSSM provides us the opportunity to work together from a variety of viewpoints to work together to do what is best for our students. So it is no longer a divide between mathematically correct or not correct, rather it is about what prepares our students with multiples options in the workforce.

You can share the letter as you see appropriate.

Regards,

Linda Gojak
Nctm president

I'm very happy to be part of a group that attends to its members thoughts, and I do understand that decisions are not going to be by consensus most of the time.  I don't want uniformity, relish diversity, and think I'm probably being over-sensitive here because of how I feel about immunization.

What do you think? Would you put her on stage at your conference?

Mystery Teacher Theatre 2000

Welcome to our first episode. Recently, in the halls of School University, some teachers attempted some selfdirected professional development, encouraged by their principal and given hope by Sal Khan, a man described as "Bill Gate's favorite teacher" (from a TED endorsement), "very popular," "extremely popular," "an educational revolutionary," and being like "like a nerdy, South Asian-American Seinfeld." (I'm not making any of those up. The last is from Wired.)

Here's what happened.





So, obviously, as comedy improv actors we're a couple of math teachers.

We're very interested in your comments. What do you think of this video? Of the teaching? What did we miss in our commentary? 

Dave was the instigator here, but is too smart to put it up at Deltascape.

Quadratic GeoGebra Day

I'm teaching a summer intermediate algebra course, and we recently had a full day of GeoGebra for quadratic functions. With GVSU's GeoGebra training coming up, it seemed like a good time to document what all we did.

I'm using a Bring Your Own Device philosophy for the class, so students are bringing TIs (82, 83, 84, 86 so far), smart mobiles, and laptops to run Desmos or GeoGebra. When we have a particularly heavy computer day, I let them know ahead, and a few extra people bring in laptops. The university runs a pretty good wireless service accessible to all students and staff.

Here's our agenda for the day:

Agenda – Class 9
Objective: quadratic regression and connections between the vertex form and standard form.

5 start up: multiplying 2 digit numbers with place value sense.
15 HW
30 Vertex investigation (GGBT - GeoGebraTube)
50 Regression opportunities (W|A, GGB, TI)

• Dan Ball  (GGBT)
• Bouncing Ball  (GGBT)
• Catenary vs Quadratic  (GGBT)
• Our rolling data 
• Curve photos 

30 Projectiles

• Constant Acceleration
• Projectiles with Targets (GGBT) 

5 Classroom concerns 

To do: project 1 – linear and/or quadratic math in life.

The way the class panned out, we left off projectiles for another day. The point of that is to make a strong connection between constant second difference and constant acceleration.

Some of what we were doing was to support them in the project. Here's the handout:








It helped engagement that they knew there was something for class that this supported.

The multiplication of 2 digit numbers was a connect forward piece to multiplication of binomials, by means of an area model that shows partial products and leads to the box method for multiplying polynomials.  There were no homework questions, so we spent a while on the multiplying, and then someone wanted to know why we were doing this, so we got into the box method right then and there.

I designed this sketch to have just a and b on sliders, to focus attention on those. The class had previously noticed that c in the standard form was the y-intercept, so I made that literal in the sketch to emphasize it. I gave it a dynamic window (defining x- and y-min and max interms of function values) so that the parabola would be in view. (GeoGebra note: that means that the program is constantly recomputing when values are changed, so you can't really use the trace.) The sketch has these questions on GGBTube.

This sketch helps you investigate the location of the vertex for a parabola in standard form, y=ax^2+bx+c. Since the equation will tell us y if we know x, we will concentrate on the x-coordinate of the vertex.

1) There are controls for a and b, separate from c. Why doesn't c change the left-right location of the vertex?

2) Change a and b and notice the effect on whether the vertex moves left or right. What do you notice?

3) Our goal is to find a rule or formula for the vertex's x-coordinate in terms of a and b. How can you collect data that will help us determine the formula?

Extension: what else could you investigate with this sketch?

The vertex investigation led to a good bit of mathematical reasoning. They noticed that when a increased, the vertex moved to the right, unless b was negative in which case it moved to the left. Some discussion led them to decide that it moved closer to zero.  They noticed when b=0 that the vertex is always at x=0. They also noticed a cool parabola pattern that the vertex makes when you move the b slider.

I suggested that maybe we needed some hard data, and started recording on the board a and b vs. the x-coordinate of the vertex for what they had tried. No pattern was visible. So I suggested fixing a or b and varying the other. Now there's some interesting things to see.

(It wasn't this neat on the board.) They noticed that as b increased it got bigger ("Bigger?" I ask. "Well, more negative.") As a increased it got smaller. When a was steady, it was like a line. They noticed the -1/2 slope for when a=1. One student noticed that when b doubled, the x-coordinate doubled. Then another noticed that when a doubled, it went down by half. Then a student had the insight, "oh, I have it! I think... can I say?" (We've been working a bit on being a no-spoilers class.) Is it -b/2a?  Seemed to work with all the data we had.

Steve's Rule: x-coordinate of the vertex = -b/2a.

When students reflected on this investigation (jot down one thing you want to remember) many noted the formula, but some remarked on collecting the data and the trends in a and b as it affected the vertex.

Next I wanted to give them some practice on regression. Starting with the sketch Dan and the Ball,

 

which, of course, is from Dan Meyer's photo composite.  Students pretty naturally want to know if it goes in. Most think it does when you take a vote.

On this sketch you can use the red point to gather data. We talked about entering the data in the spreadsheet, from which you can use the two-variable data tool. 

Or make points from the list and use the FitPoly[ ] command.

You can also use Wolfram|Alpha to do the regression, or a TI. Unfortunately Desmos doesn't have regression yet. W|A is surprisingly finicky about the phrasing you use. For example it prefers "fit" over "regression."

 Students were very successful with this, and I had them move on to their choice of activity.

Several did the bridge problem, which looks at whether a bridge is a parabola or a catenary. They were bold about exploring, despite the many overheard comments about not having any idea what was going on with the catenary. Several also gave the bouncing ball sketch a try, which allowed them to see the effect of constant a on different parabolas. (OK, I'm also just pretty happy with the simulation.)

By that point many were looking for other things on their own. Someone found this great world record motocross time lapse shot, among others.

 

From tifr

 I made up a guide to importing and using images in GeoGebra.




One GeoGebraTube tip that I picked up from this much student use in class was to remember to change the vertical display size of a sketch if you are adding the menu bar, tool bar or input bar. About 200 pixels is enough for all three.

You get to that screen by being logged in and choosing the Edit option on the teacher page for your sketch.

This day seemed to make (by subsequent assessment) a difference in their technical ability to do regression, their understanding of the vertex formula and use of the quadratic formula, understanding of some of the features of parabolas. They also have been working with a great amount of independence on their projects.  Which I am excited to see on Monday!

Size the Day

It was time for my last game with the fifth graders, and the content was multiplication and division of fractions.  Having just been critical of a game that was very computation focused (see my Math Evolve review) I was very wary of doing the same thing.

I'm at the point now where I've designed more games than I can remember easily, so one of my first steps in game design is to search my own stuff. Failing finding a game to use, maybe there's one to revise. Failing that, maybe one to revise. I did stumble across my post on multiplying fractions. Oh.

I have many lessons that get at the meaning of the operations, can be used to start discovering, exploring and justifying the rules... but only the Product Game (adapted for fractions) for practice. But that post closes with an Ant Man and Wasp cartoon, and we'd been talking a lot about them because of the Avengers movie.  My son is a comic book fan (sounds better than monomaniacally obsessed, right?) so this was quite a debate. He loved the movie maximally, but would have loved it even more with the original comic book Avengers crew. (Joss Whedon wanted characters to whom non-super powered audiences could relate as well as the super powerful ones.)

By Xavier Golden, Super Hero Squad style.
"The hexagons are Pym Particles."

So I was struck with the idea of a size-changing game. But why would our heroes have to constantly change size?  To get past obstacles! Sometimes they'd have to grow, sometimes to shrink. I tried to think of a battle game because there were several boys always interested in that, but for battle it seemed like you'd almost always want to be giant-sized instead of ant-sized. So a kind of maze... so it could be a race game.

I tried to think of a way to turn dice rolling into fractions for multiplying or dividing, or to roll three and choose two, but that didn't feel appropriate for such new content. I wanted actual fractions to see and think about. 






If you have transparent spinners, this is a good place to use them; I just used bobby pins, which make excellent spinner needles.  I experimented with the spinner entries and maze heights to find settings that were not too immediate but not too difficult either. Thinking about the framework I've been using...

1. Goal(s) - solid. I wanted students to get the understanding of the effect of multiplying and dividing by fractions, so contrary to their expectations. I wanted to get some sense of estimation, and some experience with calculations that would lead to support the symbolic rule they'll learn later. I'd also noticed that they were very interested in calculators, but had little experience with using them. This put everyone on an equal footing, as the numbers were messy and required a calculator.
2. Structure - like the stretching/shrinking as a context for multiplication/division. The spinners allowed a lot of flexibility in getting values to be used. Makes the game highly adaptable. And the intention of having to choose a multiplying or dividing spinner helped get across the stretching or shrinking effect.
3. Strategy - weak as it was.
4. Interaction - typically weak in race games, though .
5. Surprise - spinners help here and with...
6. Catch-Up .
7. Inertia - not meant to be a game that requires a lot of replay.
8. Rules - basic premise, spin and change your height. Move forward when you fit.
9. Context - thought this was strong, plus pop culture tie ins to a heavily advertised movie. Kids were interested and engaged, though I sold it a bit explaining about Ant Man and the Avengers. There's a little suspension of disbelief, as Wasp could just shrink and fly through all the obstacles, and it's rare that she grows in the book.

I added the Spin Again option to help with catch-up, strategy and interaction. But most students were so immersed in their own spins that they rarely used them! The other idea that I like quite a bit was the customizable board. Most of the fifth graders were happy to use the board as printed, but a few experimented with rearranging the board.  Maybe with middle school students, more would be interested in giving it a go. Designing a board for your opponents is a great opportunity for some open-ended problem solving.  I picked up a couple packs of mini post-its, and they were perfect for keeping track of the players' heights.

It was a good last game of the year. In the debrief, they definitely got the point that multiplying and dividing by fractions did not just have the same effect as multiplying by whole numbers, and a few kids were noticing that dividing by unit fractions was like multiplying by the denominator. I also saw considerably increased skill with the calculators, and some sensible rounding of the decimals involved.  (Parentheses were almost entirely new to them.) They asked me to leave the supplies so they could play later, and it got almost 100% thumbs up for keep or dump - both good signs.

Hopefully you can get a chance to give this one a try. It has some interesting features, and I think the choice of spinners and rearrangeable board will show up again - good game mechanic features. I'm always interested in your feedback, if you have any ideas or get a chance to use it with students. One dramatic need: the name is a terrible joke, and of absolutely no use with middle school.  Ideas?

Math Evolve

Math Evolve - a basic operations fluency game for iOS. Free Lite version to try out, full version $1.99.

Somehow Math Evolve came across my radar again this week; it's an iOS app for elementary math. I'm always willing to give those things a try. Tired, frustrated with other stuff, I tweet:

And didn't think any more about it.

The next day, Adam tweets back, very politely. Immediately I feel like a heel. If I was speaking to someone in person, I wouldn't express myself that way. So I want to apologize to Adam here.

I had high expectations because the app store page has the following:

★ Winner of BEST EDUCATIONAL GAME OF 2011 (2nd Place) in the Best App Ever Awards.
★ “The holy grail of edutainment math apps.” Editors Choice, 5/5 Score -Best Apps for Kids 

In Math Evolve you are an alien organism, completing number sentences on a quest to grow and return to your people. At the beginning of a level you fill in the first blank, then the second blank after the operation, then finally the result. Later you pick the first blank and the computer provides the second. (I always get the first one of those wrong as I don't notice when the switch occurs.)  All the while you are dodging combatants and trying to shoot them out of the way. You control your position on the screen by moving your character with a touch. Finally the boss appears and asks you to compute particulars.

Another part of why I expected/hoped for more is that it is slick. Trying to develop ParabolaX (or rather, watch while Alejo and Kevin program ParabolaX) has given me an appreciation for the work that goes into slickness in iOS.  Is there a role for a practice app? I think so, but I don't value it anywhere nearly as highly as one that helps students learn. After all, if they need practice, typically there's room for greater understanding.

I'd love a practice game that gave more for students to notice. Like asking 7+3, 6+4, 5+5, ... or 4x5, 5x5, 6x5... or 5x6, 6x6, 6x7... Noticing and generalizing are things that I value more about math.

Given the context, the practice mode could have meaning of the operation.  Gathering or eliminating to add or subtract, repeated groups for multiplication... lots of possibilities. I do like how at first you're gathering all the equation elements, then the game fills in the second spot, then the boss poses a computation. The incorrect answers are not particularly problematic, and the boss doesn't seem to use missed problems to pose problems. Maybe at higher levels? Ways for added difficulty could be including choices that are not computable, such as choices between 3, 6 and 7 when you have 21÷___.  I did only play the Lite version, so possibly some of these things come later.

Boss level.

I love how open and interested Adam is in improving the game, the context and game mechanics are well developed and suitable, and the game is fun enough that students will be happy to play. And he is obviously a far better internet citizen than I am, but hopefully he's taught me to watch my oafish responses.

As I think about evaluating math games and apps
Objective: facts practice, four operations, including negative numbers.
Gameplay: fun enough. Think Galaga/shoot 'em up.
Aesthetic: great. Cute characters, good effects.
Pedagogy: beyond difficulty selection, no visible fact strategies or support for learning. Minimal math practices.

I'd love to hear what you think of the app, and what do you feel like would make for a good facts learning game.

Guess My Rule

Some activities seem ageless. They work in elementary, middle and beyond. Guess my rule I have played with very young students, and this week I got to try it with my summer intermediate algebra course.

Guess My Rule
Any number of players

A rulekeeper makes up a rule that gives a number output for a number input. (It should be a function.) Players take turns giving an input, and the rulekeeper tells them the output. If a player wants to guess the rule, they tell an input and what the output would be. If they're right, they guess the rule. Then that player makes the next rule.

I don't know who originated this game; it feels ancient and right, so kudos to whomever developed it. I let rulekeepers use a calculator, and I usually start out with a few rules until the guesser feels comfortable making up a rule.

Today I started with the number times two minus three.  (Highlight to see it, or guess from the table.) Inputs came from all over. 12,  5, 10, 1... some shock at a negative answer. I encourage the students to record the results, hoping that it will lead to some ideas about what inputs to suggest. Organizing data is not a natural tendency. At one point they started asking 30, 40, 50...

inputs	12	5	10	1	30	40	50
outputs	21	7	17	-1	57	77	97

Oooh! That's a pattern. It goes up 20 when the input goes up 10. So they could predict the answer for multiples of ten. Thinking about that, they suggested a rule that worked. We looked at the rule to see from where the up by 20 pattern came.

The solver was not comfortable coming up with his own rule, so I chose another: eight minus the number. 

inputs	7	1	10	12	20	30	40
outputs	1	7	-2	-4	-12	-22	-32

\(7 \to 1 \) then \(1 \to 7\) drew an audible gasp/hmm. There was one student who loved asking for 12. The third multiple of 10 got the rule again. Subtract eight and then take the opposite of your answer. One of the reasons I love this game is that it always brings up equivalent expressions. I shared my original rule and we talked out the equivalence.

 

Now they were ready to suggest a few rules. A couple like my rules, then input times 5 divided by 2, which got a nice check of "oh, that is the same as the number plus half of it," and "I had the number times one and a half." Then one student came up with a stumper. When guesses were not focusing, I started taking notes on the board.










I also asked for what patterns they noticed. Some good stuff.  We got to the idea of asking inputs in a pattern, and I asked them to predict the output from 2 before Edras gave us the actual. (There's some really nice research on the power of prediction in math, some by my colleague Lisa Kasmer.) We spent some time on different expressions of the rule, like ___ x ___ + \(\frac{1}{2}\) ___ x ___, (___)^2 + ( ___)^2/2 and \( 1.5 x^2\).


For a last question, I asked them to find what input would give 100, and got unexpected riches. Students really worked hard, consulting group members. Several tried to solve symbolically, but didn't know what to do with the \( x^2 \). One student had an excellent guess and check. We spent some time discussing that, including what an excellent problem solving method it is, and how when I use it, using numbers gives me a much better feel of what's going on. Maybe they've been discouraged to use it by previous teachers, but it's a great strategy. We did discuss how to make our guess more accurate and efficient. Like was 8 or 9 closer to 100 in output?

Another then shared a traditional symbolic solution. One student asked, "why did you divide by 1.5 before you took the square root?"



"Because you have to."
"It has something to do with the order of operations..."
"Doesn't PEMDAS tell you what to do?"
"But you have to reverse it maybe..."

"Let's try it." (OK that was me.) They told me the steps and I wrote it down. Amazed to find the same answer. A good moment to point out to them that are almost always multiple methods in math. Teachers may have told them that there is only one way to do things in the past, but that is bullsh*t.

I should not swear in class. (In response to my usual 'jot down one thing you want to remember' there were many repetitions of that statement. In my defense, it is a college class. And the comedic potential was ripe.)

The next activity is adapted from Pam Wells' adaptation of an activity from the excellent Mathscape curriculum.




(Here's the Word file if you want to edit - it wasn't appearing correctly in the embed.)

It dovetailed very well into the Guess My Rule, allowing us time and grist to find many connections amongst tabular, symbolic and visual representations, reiterating the equivalent expressions and multiple solutions themes, and giving us a launching point for a definition. Using the Y pattern rule, I made a table for S=1, 4, 7 and 11. The changes in output were 9, 9 and 12... what went wrong? They found that the difference in the inputs was not constant, and when we changed 10 to 11, the output pattern worked. To me, that's the moment for the generalization.

We went on to launch Linear War, which gave me a chance to share why we use the term linear and some of what this stuff has to do with lines. Next class, we play!

All in all, it was a fun class. I was impressed with their willingness to try non-traditional problems, and they're gaining rapidly in ability to work cooperatively and make conjectures. The meta-messages about mathematics seem to be gaining some traction, too.