GeoGebra: Leveled Applets

This stuff is crazy.  We can actually make leveled applets that allow students to move on only after they've been successful with the previous level.  I saw this applet the other day and was blown away.  The applet itself is pretty simple, but the fact that it requires students to complete a specified number of exercises perfectly before moving on is the part that really interests me.  The problem is that the thing is in German and there are a bunch of unnecessary steps.  So, looking through the construction protocol proved to be fruitless.  I'm pretty sure the guy who built it is way smarter than I am, so I'll try to simplify this the best I can.

Keeping track of student success pretty much requires three things. 

True or False

Conditions must be set to determine whether the student's answer agrees with the target answer.  This part made my head hurt.  Having different levels made setting the conditions tough at first, but once I got a feel for what I was doing, the work started to flow. 

Let's take a look at my level 1 problem.  

In order for a level 1 problem to be considered correct, two conditions had to be met:

1.  The line graphed by the student (h) had to be the same as the line generated by the applet (e). 

2.  The "Check Answer" button had to be clicked.  The button was tied to boolean value g.  

I entered the conditions for each problem type's correctness into the GGB spreadsheet and this what was entered into cell C2:

=If[e ≟ h ∧ g, true, false]

Each subsequent cell was used for the next level.  (ie.  C2 -> Level 1, C3-> Level 2, etc.)

Each individual condition for correctness was tied to a global correct boolean value named AnswerCorrect.

The condition for AnswerCorrect to be true is below.   

If[C2 ≟ true ∧ ActualLevel ≟ 1 ∨ C3 ≟ true ∧ ActualLevel ≟ 2 ∨ C4 ≟ true ∧ ActualLevel ≟ 3 ∨ C5 ≟ true ∧ ActualLevel ≟ 4 ∨ C6 ≟ true ∧ ActualLevel ≟ 5 ∨ C7 ≟ true ∧ ActualLevel ≟ 6 ∨ C8 ≟ true ∧ ActualLevel ≟ 7, true, false]

The blue text represents the condition for a Level 1 problem.  

Buttons

The AnswerCorrect and AnswerWrong booleans were tied to two buttons:  ButAnswerCorrect and ButAnswerWrong.  These show up with the basic condition under the advanced tab.  

Scripts

This is where the magic happens.  I'm still learning how to use the scripts, but this is where the levels advance, construction is reset and a new problem is generated.  Both buttons have scripts, but the ButAnswerCorrect button is the most complex.  These scripts can be used as a template for future applets.  This is a good thing because there is no way I could create this on my own.  

The applet I created is here.  Double click the applet to open it in a GeoGebra window.  You can then save it and play around with making your own.  

I'd really appreciate feedback on this.  If you have any questions, leave them in the comments and I'll do my best to answer them.  

Big thanks to Linda for helping me weed through the junk on this.  

The Un-Lecture

Pretest

3x - 5y = 15

x intercept:  ______

y intercept:  ______

Results

0% correct

The Non-Khan Academy Un-Lecture Prompt  

Me:  Tell me what to do.

Them:  Put the red dot on ____ and the blue dot on _____.

Gentle Feedback for Misconceptions 

Encouragement 

Scaffolding

And Finally

Hit the refresh icon and Repeat




Exit Slip

7x - 3y = 21

x intercept: ______

y intercept: ______

Results

90% correct

 The applet.  (includes original .ggb file as well as jar files.)

Time to Stretch

The thing that convinced me to leave the high school classroom was the chance to work with a bunch of precocious pre-teens and follow them through middle school, hopefully sending them off to the big bad world of high school a little better off than they were when we met.  This opportunity, coupled with meeting y'all has created a professional development explosion.  It was a perfect storm of honestly asking the question: "how can I make this matter?", a group of math educators willing to push back on ideas while simultaneously offering unconditional support, an administrator willing to let go of the leash and a group of kids who constantly push me to be better.

The tough part has been the assumed disqualification when it comes to conversations about pedagogy because the stuff I do only works for "my kids." Let's just say, I feel Shawn on this front. This year, the gloves have come off.  I have two completely heterogeneous classes with skills ranging from Advanced to Far Below Basic (see Jason for explanation) and 2/3 of the kids are less than proficient based on previous years' scores. So, basically, this year I have to put up or shut up.  And some things have to change.

Let me be clear:  I'm not changing my expectations; I still believe that all kids can do math. But my planning has to change.

In my experience, "advanced" kids fall into one of two categories:  Advanced duplicators and advanced thinkers.  Advanced duplicators are the kids who take copious notes, ask if "this is going in the grade book", want to retake tests minutes after turning the original test in and will do absolutely anything the teacher asks.  They are compliant.  Advanced thinkers will often be the kid who gets labeled as lazy and distracted because, well, they are lazy and have distracted.  Problem is, they aren't lazy, the lesson just sucked.  We ride the backs of these advanced duplicators because they are good for test scores.  They make up for everything that a teacher lacks because they don't necessarily need a teacher to learn skill duplication (see:  Khan Academy) and they are willing to do it because, well, that's what good students do, right?

I've spend the better part of that past five years trying to find ways to have students explore, invent and discover while maintaining fidelity to our state standards.  My focus has always been on pushing kids beyond--but I've ended up learning as much from them as they have from me.   My goals have been singular in that my planning has been framed by the question "how far can we go with this?"  I've really learned a lot about exploring the right side of the bell shaped curve.  Keep going until maybe only three kids get it. Forget Madeline Hunter, I'm using the Daniel Tosh lesson planning model.

Now it's time to look at the other side of the curve.

Popping Popcorn in a Popcorn Popper

Popcorn Question from David Cox on Vimeo.

Most of my previous attempts with these story problems have resulted in me slapping on a timecode and cutting the video.  This one had me thinking a bit.  I'm not sure I got it.

Your assignment:

1.  What question does this provoke?

2.  If you have a tough time answering #1, what question do you think I was after?  And what can I do to help that question along?

Act 2's a Killer

I need a little help.  I think I've nailed the question:


Barbecue Q2 from David Cox on Vimeo.

But I can't figure out what to give students to help them through Act 2.

Here's the raw footage and the current conversation and Greg's run at the data.  (Thanks to @maxmathforum for archiving)

Any help would be appreciated.

Now What?

Paul Lockhart:

In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

Don't be too quick to write this off as impossible given our current system.  

Exponent Rules

I've grown tired of kids blindly following rules.  Mine have the tendency to be the worst because they have always been really good at playing school.  So we went for a different approach to exponent rules this week. 

Prerequisites:  Basic understanding of exponents

Instructions:  Choose a rule that you would like to prove (read: demonstrate why it is a rule).  As you are able to demonstrate an exponent rule, move downstream to the next group and help with that rule finally working on rule #8. 

Here's the list...

...that culminates with the question:

8. What do you think x^1/2 represents?

I'll let them tell you._[1]

_[1]This conversation took about 12 minutes with the camera being shut off a couple of times. I limited the editing so as to try to capture the classroom vibe as naturally as possible. I asked a few questions that I'd like to take back, but... you live and learn.

Proper-tays

I don't usually enjoy teaching properties because they seem so math-y.  I like asking my kids to justify what they do, but for many, the properties just seem to be vocabulary that is forced upon them.  Necessary evil, I guess?  They are great for doing mental math tricks and kids use them without thinking of them, so I suppose there is no harm in giving a name to the stuff they already do. 

Raise your hand if your kids mix up associative and commutative properties? 

No more.

The Process

1.  Give example of property with respect to addition.

ex: Associative: (2 + 3) + 4 = 2 + (3 + 4)

2.  Ask students to write another example of their own. 

3.  Ask for a rule using a, b and c.

4.  Ask students to write down the key characteristics of the associative property in Tweet form. (very few words)

Now here's the kicker:

5.  Can you guess what the property for multiplication is going to look like?

This worked great for the associative, commutative and identity properties.  A great discussion on the inverse property ensued and I ended up telling them that we want a multiplication problem that equals 1. 

Done.


None of these properties are worth anything if we don't apply them. Next step is getting them to put words to all that stuff they "just do in my head."


6.

Let's synthesize this a bit more. 

7. Now write a similar problem using the multiplicative properties. 

I told the class to keep an eye out for times when we will use the inverse and identity properties--which will happen daily once we start solving equations. 

Toy Cars

Prompt:  How far would you have to pull the car back in order to get it to go 100'?

Materials:  Toy cars, meter sticks.

Hand them the cars, ask the question and get out of the way.

Question #1
But Mr. Cox, the farthest we can get the car to go is around 10'.  We can only pull it back so far until it starts clicking.

Right.  So if you could build a car that could be pulled back farther, how far would you need in order for the car to go 100'?


Question #2
Mr Cox, what do we do if our car keeps turning?

Yeah, I'm a cheapskate a father of 5 on a single teacher's income.  Give a guy a break will ya? very thrifty. So what can we do to estimate the distance the car travels?


The two groups that had problems with the car came up with two different solutions. One group decided to tie a piece of string to the spoiler and measured the amount of string the car pulled past the starting line and the other group simply estimated by breaking the curve down into short line segments. (Oh man, do you guys just realized you set me up for a lesson plan in May?  Can you say calculus?)



Question #3


Mr. Cox, if I pull the car back 3", it goes 30", but if she pulls it back 3", it goes 36".  Why?

Turns out that one kid pushed down on the car harder than the other which kept the tires from sliding.



Our Findings
I'm not sure if it is supposed to be or not, but the data was pretty linear.  One student wondered why it would be linear since the car takes time to speed up and slow down and the shorter distance it travels, the more energy it is using to simply get up to speed.



Reason #421 Twitter is awesome

I tweet some pics of the activity and Frank asks me if I'm going to have a contest to see which group can get their car closest to the line.

*ahem* Of course I'm going to have a contest at the end to see who can best predict the distance their car travels.

Three groups were able to get within 1.5". (Two of them were the groups whose cars turned).  The best was this:

Adventures in Pedagogy: Experiments

Homeschooling has been interesting.  Suffice it to say, my kids have developed some bad habits very quickly. It's hard to admit that I've let my own children become answer chasers right under my nose.  I ask questions and we have a lot of conversations, but in a typical school day, my kids are gone from 8:00am-3:30pm and then have busywork waste-of-time-packet-work, er, homework to do.  By the time they were done, there wasn't really any desire to talk about the hoops they were jumping through stuff they were learning.

This morning was fantastic.  Nevan, my 8-year-old, was stoked that he had a science experiment to do.  He was looking through his text and saw the experiments and just lit up when my wife told him we were actually going to do them.

"Mom, my book at school had all kinds of experiments, but we never got to do them."

Wait, what?

Now, I'm no science teacher, but I couldn't agree more with Shawn's assessment of the situation.

I think the look on this kid's face speaks for itself.

Let 'em Play

Sometimes you just have to cut the kids loose and see what they come up with.  The year's winding down and I have kids reassessing (today's the last day) like crazy.  But some kids are proficient on all of our skills so I gave them a short GeoGebra lab on Drawing vs. Construction.  A student came over and showed me this:

The little boogers were playing around with sliders, defined 'a' as a variable, plugged in the equation:

y = x² + a

and proceeded to create any variation of that equation they could find.  Opposite functions, inverse functions, opposite inverses, etc. 

They marveled at the symmetry and how the whole picture changed as they adjusted the slider 'a.'

I've had kids each year ask, "How do you make a parabola lie on its side?"

I've always responded with, "I don't know. Let's see what we can figure out."

No one's ever pursued it.

These kids discovered it by "playing."

You Want Iteration?

...I'll give you iteration.

If the best thing is for students to actually encounter a situation that provokes a question  and the next best thing is to give them an image (static or dynamic) of said situation, then is the third best thing to give them an applet that models the situation?

Dan:

I only know I shot this because I felt like I needed it, because the alternative is a problem involving savings accounts with different principals and different monthly deposits and none of my kids have savings accounts.

Yeah, I'm not a big fan of the bank account problems either.  But I like race cars.  Lots and lots of race cars.  My students like 'em too.  Red ones.  And green ones. 

Since I don't know how to make the same guy show up on a picture/video in two different places and I don't really feel like going out to the park to run (a bunch of times), I figured I'd let the computer give me some iteration. 

So...

 What Can You Do With This?

Slide 1

Slide 2

Slide 3

                   

Applet's here.

So Will It?

In my previous post, I described how we developed the Standard Form of a linear equation.  Students are already comfortable with using slope (rate of change) and the y-intercept (initial condition) to determine the equation of a line.  We are currently spending a lot of time recognizing the rule of 4 and how each representation simply tells the same story from a different perspective.  So, next I wanted to see what students would do with trying to decide whether or not a point is on a line.

Prompt:

Question:  Will the point go through the line?

Students made their guesses and we split the room into the two camps:  Yes and No. 

Cheaters were cast out!

Alright, I didn't really cast him out.  But we did have a short conversation

on how reliable the paper would be in determining whether or not the point is on the line. 

We need more information:

Not quite enough to determine slope intercept form, but enough to get a fairly good guess as to the slope of the line.  Students wrote their guesses for the equation and all groups came up with: 

y = -3/4 x + 4

Are you sure about this equation? 

Nah, we need a bit more information. 

Now we have both intercepts; enough to verify our slope and enough to write the equation in Standard Form.

But, do we have enough info to determine whether or not the point is on the line? 

So we verified two different ways: 

1. 
   
    Continuing with the pattern determined by the slope.
2. 
   
    Plugging the point into both equations to see if they are satisfied.

And finally:

 

The coolest part about this is that a colleague and I developed this lesson in about 10 minutes.  It's pretty rewarding when minds get together and focus on interesting ways to deliver instruction centered around a simple question.  Have to give credit to Dan for stoking the fire.

Why Didn't I Think of This Before?

Now that we are finished with 7th grade standards, we start to take the concepts that are considered pre-algebra and stretch them into algebra.  My students have a solid understanding of slope as rate-of-change and I have been really emphasizing multiple representations.  They can handle an equation in slope-intercept form pretty well. 

I wanted to introduce them to standard form and have usually done this by giving the equation and having them graph.  In years past, I found many students really struggled with For some reason, I decided on a different approach this year.  This year I gave students the x and y intercepts and asked them if they could figure out an equation that would fit.  I introduced this equation as:

____ x + ____ y = ______

It became a puzzle and eventually kids nailed the idea that if we have the points (2, 0) and (0, 3), we can write the equation as: 3x + 2y = 6.  And after 5 or 6 examples, I gave them the points: (e, 0) and (0, f) to which they responded with: fx + ey = fe. 

Now, given the equation, can we find the x and y intercepts? 
Not a problem. 

The game play at the beginning of the lesson really opened them up to the idea which made any actual instruction I had to do much easier. 

In the end, I say it was a win.

Triangle Ratios

The Prompt:

A car is traveling NE at a rate of 60 mph.  What is the rate of the car from the perspectives of Persons A and B?

Students pretty quickly figured that in one hour, the car had traveled 60 miles and used the Pythagorean Theorem to determine the Northern and Eastern rates.

Where we are going

                         What happens if the car is traveling more North than East? 

How We're Getting There

I made this worksheet allowing students to construct right triangles containing an angle of their choosing, checked the first construction and had them gather data.  They manipulated each triangle so they could gather three sets of lengths for similar right triangles.  (Note:  Be sure students construct the triangles by defining the angles, perpendicular lines, etc. so that the triangles don't lose their integrity as they are manipulated.)

Draw Conclusions
After data gathering, I had them find the Sine, Cosine and Tangent for their chosen angles and find a connection between those values and the ratios they had already calculated.

No joke: every kid made the SohCahToa connection--now I don't have to tell that stupid story.

Problem Solving
I've really been looking at how I encourage problem solving in my classes and hoped they'd be able to find all the information about a right triangle given one angle and one side.  I really liked how some of them dove in even though they had no idea (or at least they thought that) where to start. 

Are you sure your right? 
Yeah, pretty sure.

How could you verify.
By doing this:

Extension Tomorrow we will go full circle and answer the original question regarding a car traveling a given rate at a particular angle towards the north. 

Limited Tech Version
Project GeoGebra up front and have class work on the same triangles.  It's still quicker than drawing the darn things yourself. 

Low Tech Version
Bust out the old protractors and build yourself some triangles the old fashioned way.  This still may not be a bad idea for some students who prefer the hands-on approach, although many of my tactile kids like having the ability to interact with the shapes in GeoGebra. 

Identify Your Opponent

My students have had a really tough time with factoring this year. It's definitely been a different group than in years past, but I haven't been able to put a finger on the problem. If I give them a context (ie. perfect square trinomial, difference of two squares, etc.), then they do fine, but when we got into simplifying polynomial fractions, the wheels fell off.

It became pretty clear that my students were not very good at identifying what they were up against. Are we factoring monomials, binomials or trinomials? Do I use distributive property, diamond problems or "bottom's up?"

You have to know what you're dealing with before you can pick a strategy to defeat it.

So I put on my cool shoes, gave a few hugs and went with the ELA approach:




I'm already noticing that identifying the opponent and choosing the strategy has helped with simplifying poly fractions. Let's hope it sticks.

Quadratics Unit

My introduction to quadratics has evolved over the years. Recently I settled on having my students do as much graphing as possible and allowing them to make the connections they need to make in regards to the equation of a quadratic and it's resulting graph. I have found that allowing my students to roll their sleeves up and start graphing allows them to become more comfortable with quadratics. This pays dividends when it comes time to solve quadratic equations by factoring, completing the square or by using the quadratic formula.

The order of things goes something like this:

Intro to Quadratics
This worksheet usually comes after we have exhausted ourselves on linears and introduces the idea that there are other relationships and, no, they don't all form a line. We do this by measuring various circular objects and calculating the area and circumference. We then graph circumference vs. radius and area vs. radius.

Graph the Magic Number
This worksheet is a little less organic, but still results in a quadratic relationship.

Stretch Factor
What happens when we graph change the "a" value? I give students a series of "Silent Board Games" (borrowed from CPM), where students are given incomplete input/output tables and are required to find the patterns to complete them. Once they are completed, they graph the parabolas on the given coordinate plane. The idea is for them to make the connections between the input/output table, the equation and the graph. Usually students are pretty quick at recognizing they can describe what the graph would look like just by observing the equation.

Graph Given the Vertex
In this worksheet, I give students random points on the plane and have them graph a parabola with a stretch factor of 1 or -1. By this time I expect them to have a grasp of the relationship between the points on a parabola and it's vertex. This allows them to see that if they can graph one, they can graph them all.

Vertical Shift/Horizontal Shift
Both of these worksheets are in the same format as the stretch factor worksheet. I have students complete input output tables and graph. By this time, they are looking for relationships between the equation and the visual interpretation of the graph.

GeoGebra Investigation
This is pretty intensive and I probably need to break it up into smaller more manageable labs. I would like to throw that out there for discussion. The main ideas that I wanted my students to explore are:

• parabolic symmetry
• line of symmetry is the average of the x intercepts
• x value of the vertex and the l.o.s. are the same
• relationship between a, b and the l.o.s.
• similarities and differences between quadratic functions and equations
• solve quadratic equations by graphing
• determining the number of solutions by using the discriminant

Find the vertex
This last worksheet has students take different quadratics and use the equation to find the vertex and line of symmetry. From there they will graph using only the vertex and the stretch factor. I then allow them to use GeoGebra to check their answers.

Worksheets (if you're interested)

Personal Responsibility vs. Learning?

Yesterday I had a few students absent and we did a lot of examples involving multiplying binomials, factoring and solving quadratics by completing the square. It was one of those lessons that "just happened." I had one idea I wanted to nail down and it kinda morphed into a bunch of examples. I made up most of the examples on the fly because I was just gauging their reaction and taking what they gave me. So believe me when I say, "they wasn't the pertiest lookin' notes ya ever did see."

Apparently, they were effective, though. The countenance of the class went from chin-on-hand-it's-Friday-I'm-tired-here-we-are-now-entertain-us to thank-you-sir-may-we-try-another-cuz-this-is-some-cool-stuff-and-I'm-gettin'-it.

Its tough to reproduce lessons like that so I exported the notes to .pdf and emailed them to the absent students.

I just received this email from one of the recipients:

"Thanks for the notes! They will really help. I do have one question though; did you have to take time specifically out of the lesson to take the pictures or some other program that did them for you? I’m asking this because I think that if you did this every time we learned something new and posted it on your website[s], it would be a good resource."

So I told him I slaved over my computer all of 30 seconds to export and email as an attachment. Which leads me to my question:

I have always taken a "students gotta take responsibility for their notes and review them regularly" kind of approach which has prevented my from exporting and posting the chicken-scratch covered slides from class. But if posting them is going to help them learn, should I care about the personal responsibility they take on (or don't take on) in regards to their own note taking?

Whatdaya think?


Note: if you're interested in what a spur-of-the-moment-ugly-as-heck-yet-equally-effective lesson in my class looks like on static slides, hit me up in the comments and I'll update the post with a link. I'm posting this from my phone and won't have access to the notes until Monday.

Triangle Centers Lab

The other day I made up a triangle centers lab for my 8th graders. 

Here is how it went:

Day 0: Homework for tonight is to make a triangle larger than your hand out of some material heavier than paper.  Cardstock or cardboard are ideal.

Day 1: Open GeoGebra and get to work.  Kids got after it.  Some slowed themselves down by not reading directions very well.  The nice thing about GeoGebra is that it's easy to erase. 

Day 2: Most finished the lab and went onto extension activity.  Those who didn't finish had a difficult time managing time.  They could do the work, but staying focused was the issue. 

Extension:  Now that you know how to find the circumcenter and incenter, construct an inscribed and circumscribed circle using only compass and straight edge.  These students haven't done anything with a compass, so I offered a 6th point (assignment was worth 5) for those who could figure out how to do the constructions on their own.  If they chose to look up the "how to" of constructions, they then would have to prove that the method of angle bisecting works. 

David came up with his own extension.  He asked, "why does the centroid allow you to balance the triangle?" 

"Nice question.  Now go away and come back with an answer."  He's figured out that the three medians divide the triangles into six smaller triangles with equal area and that would account for equal weight distribution from the centroid.  He can see it in GeoGebra, but is working on a formal proof.  Brandon tried to backdoor me with a proof by contradiction, "well the six triangles have to have equivalent areas because if they weren't, the large triangle wouldn't balance."

Don't try to beat me at my own game, son. 

Chris' reflection:  "Hey Mr. Cox, you just kinda gave us a test without giving us instructions."

"Yeeeeaah kinda, huh?

For Next Time:  Stamp each page after students have demonstrated the correct constructions.  Then allow them to go to the next page.  Take a little more time discussing the difference between "drawing" and "constructing."

Speaking Mathanese

Kids butcher the Mathanese language.  I'm just sayin'.  We have all these kids who speak text just fine.  It seems to me that Mathanese should be right up their alley.  All we are doing is taking a bunch of words and converting it to symbols.  Should be easy, right?  Not so much. 

I find that kids have a tough time translating algebraic expressions to English and vice versa.  Am I alone? 

Yeah, didn't think so. 

One of the things that I have been trying to focus on this year is to convey to students the universality of the things they are learning.  For example, cause/effect in language arts becomes input/output in math.  Conflict resolution is the same as problem solving.  Language arts has expressions and sentences, so does math.  Scientific method can compare to making a conjecture in geometry, testing it out and then using inductive logic to arrive at a conclusion (read: rule). 

So what happens when you tell them to translate: the product of 3 and the sum of x and 2?

You get: 3x+2, right? 

Not quite. 

Well I figured we needed to develop a mashup of English and Mathanese; Mathglish, if you will.  Here is what we came up with:

English to Mathanese:

[caption id="attachment_458" align="aligncenter" width="500" caption="This should read: The product of 2 and the sum of the product of 4 and x and 3. "][/caption]

Mathanese to English:



The key this time was to allow the mashup.  I live in a rural area where the Spanish speaking population is very large.  Many of my kids speak and understand Spanglish.  I have never done it this way before and the kids nailed it. 

How do you do it?

Update:  Just did a quick check for understanding 2nd period and  26/28 kids circled the bases.