The Monument Project

Background

In my Algebra 2 class, students use Wolfram Mathematica to design 3-dimensional objects using different functions and transformations.  In the past few years, I've had students create scale models of the Great Pyramid of Giza and the Washington Monument before I have them work on a design of their own.

Given the protests surrounding monuments this past summer, I decided to have students research a monument and then design their own monument as their first project.  

The Assignment

It was important to me not to proscribe what kind of monument the students should create.  I wanted for them to be empowered to think about something they cared about.  I did do one thing, perhaps to prime their thinking a little -- I asked them to read a short article from history.com about the history of Confederate monuments in the United States.  

My assignment had three parts:

• Choose a monument that exists and do some research.  Why was it built?  Are there people who object to the monument?  What do you think?

• Design a monument meant for a public place that commemorates something you care about.  Why did you choose your subject?  Are there people who might object to your monument?  How would you respond to their objections?

• Summarize all of the above in written, essay form.

My Goals

• To allow my students to bring themselves and what they care about to a math assignment.
• To begin using Mathematica, a program we'll be using through the year.
• To review graphing lines in a challenging way.

The Results

Simply, I was blown away by what my students were thinking about.  Not surprised, but blown away; these can co-exist, ask any teacher.   Written responses demonstrated thoughtfulness and passion for the topics being discussed. Mathematically, students were challenged and worked on the skills I wanted them to practice.

Essentially, each object below is defined by a system of inequalities in 3 variables.   Students were limited by the newness of thinking in 3 dimensions, so the projects ranged in their complexity depending on their comfort with the technology and with the math.

Here are a few examples:

"This is called 'Our Lives'
The top of the monument is a balance, but one side is longer than the other meaning that the the balance in our society might seem equal but the harsh reality is that it isn't."

"I want to display something that resembles something that is all over the place and abstract to display how the Black Lives Matter Movement is all over the place. I do believe that there are a few people who would object to my monument."

"The monument I built represents the ongoing fight against climate change. At least, it serves as motivation that we can come back from the brink."

"I made a monument for the people who have died from Covid-19."

"My monument is called 'The Fight of the Saints'.  Over 1.5 million people died during the Armenian genocide, and they all fought for the survival of their families and friends.  Creating a monument won’t bring these people back but it may remind people of the first genocide of the 20th century and stop it from happening again.
My monument's base is the outline of the present-day Armenia borders, which historically was almost ten times the size. I then created a model of The Cathedral of the Holy Cross, an Armenian Apostolic church located on Akhtamar Island in Turkey. During the Armenian genocide, this church met a very horrible fate. They forced all members of the clergy on a death march out of the country.  They then took everything out of the church and left only the structure. This cathedral met the same fate as almost all other Armenian cathedrals in Turkey and symbolizes a strong cultural factor of Armenians. We hold a lot of pride in having one of the oldest religions in the world. Memorializing this cathedral represents not only the lives that were lost in the genocide, but also the fact that we are still standing today."

Here's a photo of a 3-D printed version of the monument above:

Conclusion

Simply, this was wonderful.  It made a project that I would have done anyway have meaning, allowed students to think about something that they cared about, and let them create something they could be proud of.  They learned some math as well, and I learned something about each of them.

The Chaos Game

This is an activity that I'll do with my students once in a while.  It's called the Chaos game, and I hear of it years ago from Robert Devaney,  back when the technology in my classroom consisted of an overhead projector.  So, this is my overhead projector version.  The same activity can be done very quickly using computers, but there's something to be said for hands-on point plotting.

Here's a link to the sheets I print out onto transparency paper.  (They're a little fuzzy, as they are scans of transparencies.)

Here's how it goes.

1.  Students are put in groups of three.  Each group is given a fine point maker (I prefer permanent in this case), a transparency sheet with the three vertices of an equilateral triangle on it, a plain white sheet of backing paper and a die.



2.  Students are instructed to choose a point at random within the triangle and to simply keep track of it with a pencil point, trying not to mark the transparency.  Let's call this point P₀. (In truth, the point can be chosen anywhere on the infinite plane, but to have the result emerge more quickly, the point should be within the triangle.)

3.  Students roll the die.  If it is a 1 or a 2, they find the midpoint between P₀ and A. If they roll a 3 or a 4, they find the midpoint between P₀ and B.  If they roll a 5 or a 6, they find the midpoint between P₀ and C.  They move their pencil tip to the midpoint they found.  Let's call that point P₁.

4.  They repeat step three, rolling the die and finding the midpoint between P₁ and the selected vertex.

5.  Keep repeating.  Once they have found P₄, they start marking successive points with a small dot using the permanent marker.

6.  After about 15-20 points, they can stop.

7.  Here is one group's result:


What do you notice?  What do you wonder?

8.  Take the old overhead projector that you managed to scrape up from a closet somewhere, and hope the bulb hasn't failed since you checked it.  Place all the transparencies on the projector, lining up the vertices.  Here's one class's result:

Now what do you notice and wonder?

You can see the general shape of what the result is, but it's not crisp like on a computer.  Sometimes students will recognize the pattern, especially if they've seen the Sierpinski triangle before.

9.  Now I place the pre-made Sierpinski triangle pile on top of their diagram.  I then remove the stages one-by-one.

You can see above that there are a lot of stray marks when you put them all together, and some of the points do sneak out into the smaller triangles, but given the number of points that are plotted, it's pretty impressive.  Peeling back the last Sierpinski triangle makes this clear.

That's it.  

The only question remains:  Why does it work?  I'll leave it to Robert Devaney to explain here.

Building Soap Film Slides

I've been playing around with distance minimization problems using soap films.  It's cool stuff, as you can see in this video.

I'm still figuring out exactly how to use it in my classes.  It has application to all of the courses I teach -- geometry, algebra 2 and calculus.  I made a Desmos activity that gives an idea of one way I used it last year.

My biggest excitement came from figuring out how to make the "Soap Slides" (to coin a phrase.)  Here's what I did.

1.   The slides are actually cases for 15 collectible cards (baseball, Pokemon, etc.) I found them for 25-cents each at a local store.  They can also be purchased online for a little more.

2.  You need some posts.  There are two options.  I was able to 3-D print mine.  I made cylinders with holes in the middle.  I figured this would give the glue more surface area to grab onto.  If you have a 3-D printer my .stl file can be downloaded here.  Important:  It should be rescaled first to your liking -- I used a vertical size (z-axis) of 28mm and x-y axis sizes of 7mm.  Alternately, you can use small wooden dowels that can be found at hardware stores and online.  The ones I found were "fluted" and they tapered toward the ends, both of which made them a little more difficult to use.  There are probably better ones out there.  Still, these did the job.

With the wooden dowels (and bad glue)

My 3-D printed posts

3.   Pull the two halves of the card case apart.  (It can be fun to break things!). Using Desmos, I printed a template of the figure I wanted to make.  In this case, it was a square.  Place one half of the case on the template.  This will tell you where to glue the posts.  IMPORTANT:  The posts should be glued to what was originally the outside of the case.  If the curved edges turn to the inside, the soap film gets stuck to it, which means we don't get the desired result.

                 

3.  I used Gorilla Glue Gel (pictured below).  Earlier, I'd used a type of Gorilla Glue that expands as it dries.  This was not ideal.  Place a dot of the Gorilla Glue in the spot where each post will go.

           

4.  Place a post on each spot.

5.  Place a dot of glue at the top of each post.

6.  Put the second half of the card box on top of the posts.  Remember to have what was the outside of the box facing down (so the edges are curving up.

Let it dry, and you're done.

Last advice:  Find a container for your soap solution that is deep enough for you to dip the entire slide into the solution vertically as well as horizontally.  This will allow you to get different formations (aka local minima) when you play.

I'd love to know if anyone does this.  Let me know!  Meanwhile, I'm going to figure out where it fits in my year.

A Year-Long Geometry Problem: Barney the Ant

Many years ago, I taught from an old geometry textbook -- I think it was Sunburst Publishing -- that used a single problem as a theme through the text.  It was the classic shortest path problem:

The hiker needs to get from point A, to the river, then to the tent at point B.  What is the shortest path?

Last year, at NCTM 2018, I attended a session run by the amazing Ron Lancaster

in which we solved the closely related "laying cable" problem.  

Here, the cable is costlier to lay underwater than on land, so the goal is to minimize cost.  Ron had us solve it about 9 different ways if I recall, using everything from algebra through calculus.

I can honestly say that I had neither of these consciously in mind (but obviously both had an impact on me) when, one day in September, I decided to have my geometry class work on what we took to calling "The Box Problem."

Barney the Ant wants to travel from point A to point B along the surface of the box.  What is the shortest path?

We revisited this problem through the year, and solved it a few different ways.  Here's what I did.  (Note: the photos and data are from the work of different students working with different sized boxes, so they may not match each other.)

#1:  Introducing the problem

There is a very simple solution to the problem, if you see it.  (I won't give it away yet.) While one of my mantras this year (borrowed from David Butler) was "The End is Not the End," it would have been difficult to convince my students that we should keep working on a problem that we'd already solved.  (Spoiler Alert:  They bought into this approach eventually.)  Thus, my first goal was to have them not see the quick way.

So:  I got some actual boxes from our dining hall.  When I introduced the problem in class, I used these to explain the question, thinking that perhaps it might be more difficult to see the easy answer on an actual box, rather than the diagram above.  Plus, I could use my teacherly skills of distraction to manipulate the situation.  

It worked.  (One of the few times where I was rooting against a student having an insight.)  We reduced the problem to the following question:  To what point should Barney walk to on the top front edge in order to minimize distance.  That is, find x in the diagram below.

#2:  First approach: Data Gathering

Each group was given a box.  They measured the dimensions of their box.  Then, they took the top front edge of the box and divided it into 10 equal parts.  Students drew the paths via each of the eleven points, then measured each distance using a meter stick.

From the table of data they made a guess as to what the shortest path is.  Plotting the data in Desmos gave them a visual representation of the data and a possible solution.

#3: Second approach:  The Pythagorean Theorem

Looking at the boxes, students recognized (through discussions) that they could compute the length of each path by using the Pythagorean theorem to find the hypotenuses of two triangles and add them together.  Importantly, I had students create a table that showed the x-value, the equation used to find the computed distance via each x-value, the measured distance and the computed distance.

They graphed the computed data on Desmos, which made the estimated solution more obvious.

Both sets of data on the same graph led to a discussion about what the differences were and what might have caused them.

#4:  Making some art

The next part of our exploration was taken directly from Ron Lancaster's workshop.  The students used their boxes to cut pieces of string that were the actual length of each path they had drawn on the box.  They tied one end of each string to a dowel, and hung a clothespin on the other end.  Strings were spaced apart to match the divisions on the top edge of the box.   This created a nice visual representation of the problem that we could hang on the wall.

Full disclosure:  they initially cut the string by actually holding it along the path they had drawn on the box.  This resulted in a little too much randomness (aka human error.)  Instead, we redid the measurements using our calculated distances.  This made for much cleaner and appealing art.

#5:  Summarizing

I had students write a summary of all that we had done, in narrative form.  I had them do this at this point in order to slow them down a little and let them reflect and consolidate what they'd done thus far.

#6:  Graphing a function

I posed the question: What if you didn't know the actual value of x?  What is the expression that would give the total distance?   That is, can you create a formula?

It is with this question in mind that I'd had them show their work in the table column "equation used to find your distance."  They had re-written it so many times that they very quickly saw the solution: 

I'll pause here to point out how cool this is.  In a regular geometry class, students created a function that represented the sum of two radicals, generated by them,  from something they understood.  Nice.

They graphed the equation on Desmos, and now were able to find the answer to the question (from a graph, at least.)

#7:  Another approach - Similar Triangles

We put the box problem aside for a while, until we were studying similar triangles.  At this point, I introduced the original shortest path problem:

We played with this for a bit, then we copied it onto patty paper.  I had students find B', the reflection of point B in the line, and then they could see how to find the shortest path - travel directly from A to B'.  To find the shortest path, we needed to solve a similar triangle problem. 

At some point,  a student said that it reminded them of the box problem.  So we connected the two, and discovered that it's really the box problem in disguise.  (Seeing the patty paper reflection was very helpful in making this clear.)  If we fold up the top of the box, we can just draw the line between A and B, then solve a similar triangle problem.  Here's my work for the box used in the Desmos graph above:

Very satisfying to see the same answer appear.  

#8:  A third approach:  Trigonometry

Again, we put the box problem aside for a bit until we turned to right triangle trigonometry.  I had students create a Geometer's Sketchpad diagram to model the box problem, with a free moving point along the middle line (top edge).  They measured the angles in the middle:

The question: what is the relationship between the two angles measured at the point where the distance from A to B is the shortest?  It wasn't hard to see that they would be equal -- we kind of knew that already from the similar triangle solution.  But it means we can answer the question this way:  Find the point F where angle DFA and angle  BFC are equal.  So going back to our diagram, we want to find what value of x makes y1 equal to y2.

We have two equations -- let's go to Desmos!

It's the same answer!  This was very satisfying.  

Notice, too, that geometry students are now graphing an arctan function that they created, with a purpose to solve a problem.  Double cool.  

#9:  Epilogue:  The Final Exam

Because we had become so familiar with this problem, I told them that it would be a large part of the final exam.  After the standard part of they exam, they'd be able to use their computers if they wanted to type or use Desmos or Geometer's Sketchpad.  They'd have plenty of time to work on it. Here's what I sent them before the exam. Their task was, essentially, to show me what they knew.

The results were great.  To a student, they communicated something meaningful from the problem.  Some only solved it using similar triangles, a few solved it in all the ways we solved it through the year.  Some tried things we hadn't thought about.  

I did do something either dumb or brilliant:  In making up numbers, I accidentally made the opened up box a square.  This allowed for some calculations to be done more easily.  In fact, I was really pleased when some of the students recognized that their diagonal created an isosceles right triangle, which meant they knew the hypotenuse without using Pythagoras.  Nice. 

Also, the solution was an integer.  Is this because the opened box was a square?  Don't think so, but I'll have to think about that more -- but it points out again what a rich problem this is.

One student decided to try something new -- he looked at the area of the two triangles.  His conclusion:  "it didn't really help solve the problem, so never mind."  Nice!  

My favorite feedback:  I asked one student about how difficult she found getting ready for the exam, and she told me, "At first, I was panicked about the box problem.  But then when I realized I didn't have to memorize anything, I just had to understand it, I calmed down."

Here's one student's work (he also submitted a Desmos graph):

#10: My Reflections

I was teaching geometry for the first time in about 15 years, and I was kind of making up this problem as I went along.  My experience tells me that often, making it up as you go along is the way to go.  Too much structure might kill the spontaneity, and thus make the problem feel more routine.  So, I will give my thoughts in the form of questions.

• Should I spray paint the boxes and have students draw nicer paths?  The boxes sit around our classroom all year, so it might do to have them look good.
• Should I change from clothespins to something else?  The way we did it this year, the sculptures were easily wrecked and didn't actually end up being displayed.  The clothespins were nice, though, because they were easily adjustable.  
• (Okay, I know the answer to this one.)  I had students make a google doc for their initial response to the problem.  I had meant to have them build on that document through the year, but I did not.  Next year, I will.
• At some point while reading this, you might have thought, "Couldn't it be shorter if Barney went up the side of the box to the top, rather than the front?"  This is a question I had hoped a student would raise at some point, but nobody did.  I raised it late in the year.  Should I raise it earlier?
• I had planned on having students potentially ask related questions, like:  What if Barney is slower climbing up than climbing across?  What could you deduce about the dimensions of the box if the solution to the problem was the midpoint of the top edge?  How can we know quickly which route (front first or side first) is shorter? And so on.  How can I work this in next year?
• Are there even more ways to solve the problem (that are accessible to geometry students.)?
• Is this a problem to repeat at all next year, or should I find another one?

My main takeaway is that I'm really pleased with the entire project.  Initially, students found it strange that we were going to keep working on a problem we'd solved.  But this is the generation that watches the same movie 140 times, so it wasn't much of a challenge to get them to think about the same thing over and over.  On the final exam, instead of rushing through like I've usually seen, they were eager to show what they knew and even enjoyed the process.  The work they did demonstrated (from my reading) good understanding of geometric concepts, good problem solving skills and most importantly, confidence in themselves as mathematicians.

Any suggestions or observations would be welcomed!

Conway's Rational Tangles

I had a bonus long block period (85 minutes) with one of my Algebra II - Honors sections this week.  The other section lost their long block due to the holiday.  So, I decided to have them work with Conway's Rational Tangles.  I won't re-invent the re-explaining wheel here, but I will point you to some links.

I first saw this through a YouTube video by Tom Davis.  I adapted it to my classroom and I've done the activity four or five times with different groups of students.

After my first day at Twitter Math Camp this summer, I was back in my hotel room thinking about how I was surrounded by amazing educators, and it occurred to me that I'd be a fool not to ask questions when I had the opportunity. I remembered the Tangles activity, and how there were some pieces that hadn't ever quite worked.  I figured somebody in Cleveland with me had written something about this, so I set to Google and --- BAM! -- there's a great article by Amie Albrecht describing the activity and reflecting on her own experience with it.  (This is where you should start if you haven't seen the activity before.)

In a moment that sums up the joy of TMC, Amie was waiting for the bus the next morning when I walked outside the hotel.  I accosted her with my questions, and so as we walked to find some food, she told me about her experience with the activity. 

The most interesting thing Amie added to the activity was a Notice-and-Wonder exercise that allowed students to at least try to discover the rules of the activity themselves.

Which brings me (at last!) to my experience on this round.

I began by showing Amie's "Notice and Wonder" video.  It's a sped-up video of students doing the final Tangles exercise.  Students noticed and wondered the same sorts of things Amie's students did:  They only seem to rotate in one direction, sometimes two of the students switch places, what's the deal with the bag, etc.  The video was an addition to my usual presentation of the problem, and I liked it.  It engaged students in the exercise right away, and made them curious immediately.  I liked too that they were seeing other students in another classroom doing what they were about to do.

Next, I clarified the rules they had noticed to them -- we rotate clockwise, and can only "twist" from two positions in one direction.  I had four students demonstrate with our ropes, following my commands and creating a tangle.

Then I made a careless error in my approach.  I had divided them into two groups, and I was going to send them off to figure out how to untangle a tangle formed by a series of twists -- how would you untangle one twist?  Two twists? Three twists?  And so on.  To make myself clear, I carelessly wrote "T", "TT" and "TTT" to demonstrate what I meant.  Here's why that's a mistake:  one of the things I love about this exercise is that it starts out in a very tangible and physical way -- ropes are being tangled.  However, to keep track of what's going on as you try to solve the problems, it becomes necessary to come up with an abstract way of expressing what's being done.  In the past students have written out the words "Twist" and "Rotate", used arrows to indicate that one follows the other, and made up other sensible but cumbersome methods of recording their progress.  Eventually, they settle on the simple string of R's and T's.  I think it's valuable for them to have that experience because it mimics a lot of what mathematical notation does -- it takes a complicated situation and makes it simpler to express.

And I robbed them of this part of the exercise, because I was distracted by difficulties with the projector and my hurry to get to the project.  Dang.

So, I sent them off to solve the Problem of the Twisted Tangles.  I wandered between groups, and I was pleased that they each solved the problem in about 15 minutes.  What I love about this part of the process is that they discover that to undo a T, they do an RT.  To undo a TT, they guess it will be RTRT, only to discover that they need an extra T.  Finally, they guess that to undo a TTT, it'll be RTRTTRTTT.  When they try this, they discover that they've untangled it without that last T.  Then they feel confident that to untangle TTTT it will be RTRTTRTTRTT.  And it is.  The solution kind of finds itself.

We returned to the classroom, and summarized what they'd found.  A student wrote on the board:
T = RT
TT=RTRTT
TTT=RTRTTRTT
TTTT=RTRTTRTTRTT

I should have advised against the use of the equal sign here, as it implies something else, but I let it go.  (It was a source of confusion later, so I regret that decision not to nip it in the bud.)

I had students stand with the rope, and we talked about something one of the groups had discovered -- it never makes sense to do two rotates in a row.  I demonstrated with a couple of different tangles.

Next, I had them make a TTTRT tangle to see if they could undo it.  They struggled a bit until someone realized that it was the TTT pattern starting to untangle.  Then they could just read the solution of the board.

I made up a tangle that they couldn't solve the same way -- something like TTTTRTTTRT.  We'd need another method.

At some point, a student had referred to the starting position as "a nothing tangle."  I suggested that we follow this and give it a number.  The consensus was that 0 made sense for the starting state.  Then we needed to decide what a Twist does to the number -- several suggested that it should add 1 to the value of the tangle.

At this point, I paused to give them a moment to see that since we were defining both the starting point's value (zero) and the role of a twist (+1), there was no "wrong" way to choose a value or an operation.  We could choose the starting tangle to be a 7, and a twist to be a "multiply by 5."  Of course, this would likely make the overall solution of the problem unduly complicated, so it makes sense to keep things simple.  (I'm pretty sure that it would be solvable with these first conditions, but not certain.)

Now the tricky part:  What does a Rotation do?

We looked at tangles that we knew how to solve:

TRT would be 0, then 1, then x, then 0.  But the final zero gives us x+1=0, so x must be -1.

Which means that R gets us from 1 to -1.

Perhaps R is "subtract 2"?  But then TTR would be a zero tangle, and it clearly isn't.  Hmmm.

Perhaps R is "multiply by -1".  Then TTRTT would be a zero tangle.  And it isn't.
Double hmmm.

After some quiet struggle, I had the students start with a zero tangle, and do a rotate.  The question is: What number do we have here?

I tell them to do a twist.  What happens?  Nothing changes.  Which means the number must be a solution to the equation x + 1 = x.  Hmm.  At this point in the past, I've had a student recognize that x might be infinity.  No such luck this year.

What number plus 1 equals itself?  I gave students the hint:  "Think of a playground taunt."   Sometimes this is enough -- not this year.  I give them enough coaxing to get: "I hate you infinity," and the retort:  "I hate you infinity plus one."  They see what I'm getting at, and they also wonder aloud if I had a difficult childhood.  (Not so much, but I probably need a more positive prompt.)

Anyway, once they see that infinity fits the bill, the question becomes: What operation can we perform on the number 1 that will bring us from 0 to infinity and then back to 0?  They puzzle and guess some things, but eventually someone suggests dividing by 0.  This leads to someone else suggesting that taking the reciprocal will do it.  They piece together that it's the negative reciprocal, and we're off to the races.

As you can tell, I haven't figured out a slick way of having students discover the rotation operation.  They'll look at some tangles and makes some algebraic equations, but it's different when the operation is the variable instead of the number.  It seems like relying on insight and instinct (and hints from the teacher) is the only way to go.

We tested our hypotheses (T is "add one", R is "take the negative reciprocal") on some of the tangles we'd solved.  Then, for homework, I gave them a more difficult tangle to undo.  We calculated that the value of the knot was 14/8.

They returned the next day with some success, and enough agreement on a solution for us to try it out.  We made the original tangle.  Then I inserted each of the four ropes through a plastic bag.  We then followed the "untangle" steps which created a tangle of rope and plastic.

The big payoff:  as students stood in their spots holding the ropes, I tore away the plastic bag and - TADA! - an untangled rope.

My (continuing to develop) conclusion:  This is a good exercise for this type of situation, and a class of about 12.  My class of 16 meant I should have probably had a third group, but I was lacking both ropes and space to do so.  With 8 in a group, it was possible for students to be disengaged, or to have their ideas lost in the ruckus of a larger group. 

I also find that it's easy to think that the four students actually holding the ropes are engaged, but sometimes they are just following directions and not really thinking.

I continue to have trouble getting someone to discover the rule for rotations.  The project is always more satisfying (for me and for the students) when a student has the "negative reciprocal" insight.

Still:  I like this activity a lot.  It's different, it's accessible, and it has a nice concluding moment, worthy of cheering.  It also has some sophisticated extension questions (Can you make any rational number into a knot?  Can any rational number be untangled?), although I must admit I haven't had students pursue these in the past.

For sure, I'll do it again.

First Day of Calculus

In two days, I begin my 36th year teaching, and probably my 25th year teaching AP calculus.  So of course, I figured I'd try something different.

Usually I've tried to give the course some context -- we're going to find slope at a point and area under a curve:  here's why we care.  Last year, I built a  given-the-velocity-find-the-distance activity.  It was okay, and I'll probably use a revised version of it at some point in the year.  (I've already videoed the speedometer on my new e-bike in preparation.) 

This year, I'm going to jump right into limits.  Here's may plan for the first two days or so.

Day 1 (30 minutes):  First thing, a Which One Doesn't Belong (word doc link):

Since most (all?) of the students will not have done a WODB before, I suspect that (D) will be the overwhelming selection as the "right" answer.  I'll need to use my sneaky teacher skills to get them to start looking elsewhere as well.  No sweat.

Next, we'll do a Desmos Polygraph I made that will (I hope!) force them to start reaching for vocabulary they don't yet have.  ("Teach us the right words!" they'll demand, "Teach us!  Pleeeeeeease!")  

That will likely finish the 30 minutes.  It will feel half-done, but that's okay.  

For homework, inspired by CalcDave's 8-year-old* (!) blog post, discovered via Sam Shah's also 8-year-old blog post, I've made a questionnaire of my own.  I added a little more algebra to mine, and a little less get-to-know-you.  (I've taught the majority of my students before, and I know most of the rest from other things.)  I do need to test their mettle, so to speak, as I'd like to determine whether the AP pace and demands will work for them moving forward.  

Day 2 will be a couple of Speed Dating exercises.  We'll do the first one without hardly any guidance from me as to what a limit is.  I will tell them how to read the statement ("the limit as x approaches 2 of f(x)") and make that part of the exercise, so that they get used to the language.  Other than that, the'll need to figure out where the right answers come from.  This is what I've taken from the #MTBoS philosophy:  don't tell students anything.  

After that's run its course, we'll regroup as a class and I'll answer any questions they have.  Once that happens, we'll do the second speed dating exercise, this time (if all goes right) with a bit more comfort and success.  Homework will be practicing limits by graphing, and then we can get into algebraic limits on day three.

And we're off..

*As a side-note, I keep waiting for a time when I don't look back 8 or 10 years and think, "Holy cow -- I had no idea what I was doing back then."  So I'm assuming Sam and Dave have moved on to something that they feel is progress, while progress for me is moving into someone else's past.  My past is open for occupancy.  If only I'd blogged back then.

Twitter Before Breakfast

I'd like to say that Twitter has become a daily presence in my life.  But it's much worse than that.  Okay, that's society talking.  I think it's better.  Here's what Twitter had me thinking about this morning - a Sunday morning in the summer --  from 8:03 to 8:21.

8:03.  A tweet from Amie Albrecht, now the Australian who inserts her presence in my day more than any other, as she has now surpassed Courtney Barnett.  

I always gave this exact answer when I was a kid and people asked why I liked math.  As I've continued to grow and learn, I realize that math is all about the journey and less about the destination, but I do find it comforting that there is a "truth" at the end that we can all agree on, even if that truth is that the truth is unknowable.  I can prove that root 2 is irrational, I can demonstrate that infinite sets have different cardinalities, I can prove the Pythagoream 40 different ways.  (Don't challenge me on that last one.)  Proofs are sometimes sensible, sometimes highly counter-intuitive, and sometimes incomprehensible to me (see: Fermat).  Yet, without a "right" answer at the end, the whole beautiful exercise could end in murky conclusions like a debate in economics or even science.  Those debates are obviously important, but I enjoy the comfort of knowing that an absolute (albeit abstract) truth exists out there.

There's a good discussion in this tweet's thread.

8:06.  Another one from Amie.  I was planning to do the Quarter the Cross exercise on the first day of geometry, and here's Amie adding another dimension - a card sort!  Check it out.  Thanks, Amie!

8:08.  Again from Amie.  (Not really a surprise -- she's tweeting while we're all sleeping.)  It's really a link to Fawn Nguyen's blog post about using different colored highlighters to mark student work.  I've seen this before and liked it, so I added it to a Wakelet folder (probably for the third time.)

8:09.  Andre Sasser asks about getting permission from students to post their photos.  I've been wondering about this, and also how to blog honestly about the specifics of what's happening in a classroom without running the risk of embarrassing my students should they read what I've written. I'll always try to make sure I'm communicating fondness and positivity toward my students, but I can't control how they interpret what I've written.

8:11.  I don't follow this account, but it shows up via likes every so often and I like it.  I like how the post begins:  "I pet Sadie."  I picture the build-up to this like the lunar module landing on the moon:  "I am approaching Sadie.  All systems are go. 10 feet.  Tail is wagging. 5 feet.  Tongue is drooling.  Houston, we have contact."

8:12.  Math photo scale week.  I am reminded of my candy loving (now 23-year-old) daughter, thinking I'd be almost certain to take this same photo when she were younger.  I had also seen one of these crazy machines at the movie theater last night, so I was thinking about my daughter and candy.  

Also, I want to take some #MathPhoto18 pics, but I keep forgetting to check the theme.

8:14.  Somebody saw "GOD" in a sliced eggplant.  Good ol' Twitter.

8:15.  Back to Andre with a grocery bag suggestion.

This had more meaning to me that I expected.  For the past three years, we've had two girls from Ghana at Millbrook, one of whom has lived with us.  They were back in Ghana this summer for the first time in three years, and they return this very day.  I'll be glad to see them, in part because they are literally my supermarket adventure partners.   Here's what it's like to shop with them:

I guess I gotta buy that shopping bag.

8:17.  A reminder about EquatIO.  I haven't really figured out how this will work for me, but it's on my list to look into this summer.  Getting late, though.

Then, in yet another fortuitous coincidence, when I went to log on to my computer to write this, Windows had a message for me:

EquaIO wants to "Read and change all your data on the websites I visit."  Uh, no.  I don't even give myself permission to do that.  I'll check on it later...

[Note:  I checked it out, Accepted Permissions and I'm really excited to use EquatiO this year.]

8:21.  A retweet from Blank Panther Girl.

I've been thinking about this a lot this summer, without reaching any coherent chain of thought.  It began with our trip to Ghana, where we met wonderful people and saw a country that had robbed of its resources, both physical and human, for hundreds of years before this century, largely to the benefit of people like me.  We got to meet our two Ghanaian girls' families, which was emotional beyond explanation.  The students who were with us worked with a school on some math, some storytelling and some soccer.  I know the school appreciated what we did, but it's hard to know how to feel.  This photo below of the class playing SET sums it up -- it makes me proud, confused, sad and happy all at once.  

Marian Dingle's keynote at TMC18 (watch, read) gave me further reason to reflect and even less clarity (in a good way!).  I also thought about my experience with our two Ghana girls, and how it's offered me the opportunity (if that's the right word) to see the pervasive, yet often very subtle, nature of the racism that exists in my world.  

And that's a heavy note to end on, but that's about 20 minutes of my Twitter experience.  In the middle of the summer.  All before breakfast.