Monday, December 10, 2012

Hobbit Ratios

Nothing amazing with this little project, but it's something to get started and you guys can run with it.

In geometry we're talking about ratios of similar shapes.  Comparing a 2x2 square with a 6x6 square, we see that the side lengths triple, but the area increases 9-fold.  I had them discover this by doing a number of different shapes and finding their areas then comparing.  Similarly, for volume, the shape whose sides have ratio N has a volume that is N^3 times as much.

Anyhow, with The Hobbit movie coming out this week, I thought I'd tap into that excitement a bit and have them do a little map reading.  I found this map online:  It's huge, but printable.  There are other online maps, but not with a scale and with different resolution screens or whatever, I thought I'd give them all a print out so we can be a little consistent.

Then I typed up these instructions on what to do with the map.  I couldn't think of how to incorporate volume into the instructions at the time, but now I think I might include making a scale model to the same proportions of the map.  Maybe how wide and tall a mini-Isengard should be made to fit on the map or something.

Anyways, feel free to steal and/or enhance this lesson for yourself.

Thursday, November 15, 2012

A Day in the Life... Nov 15, 2012

Sam suggested that many may not know what the life of a math educator is like.  Others may know the generalities, but even other math teachers can find comfort in knowing the details of another person's day.  It can be nice to know that you don't have it as bad or that we're all in the same boat.  Anyhow, here's how my Thursday, November 15, 2012 went.

Last night, I went to bed at 10pm. I go to bed every night at 10pm. I keep a very strict bedtime on school nights. When the clock ticks to 10, I drop what I'm doing and head to bed.

I got up at 3:45 this morning.  I usually get up around 4, but I've been feeling a little under the weather lately, so maybe my internal clock is being weird.  I make a quick trip to the bathroom (normally I'd leave that detail out, but you'll see how it comes back later).  Breakfast is important to me, so I had some cereal and juice as well as a chewable vitamin and head to the living room.

I click on the TV for the early AM news where the banter almost makes me want to turn it back off, but I guess they don't care much this early in the morning.  I just want to see if traffic is ok (St Louis has lots of bridges closing and other work being done all the time) and what kind of weather we'll have today.  They just want to tell me about all the Christmas sales going on (it's the Ides of November for crying out loud!) and what in my house could be poisoning my children (I have none).

I open my computer and check my RSS feed for any actual news that is important.  I have about a dozen webcomics that update often, any math blogs that may be interesting, a quick scan of tech/science blogs, and a visit to youtube to see if anything interesting has come up that I can use in class or just for some morning laughs.  Overall, this takes about an hour to catch up.

So, now it's 4:45.  I pull out a blanket and take a morning nap on the couch.  My actual alarm (first of the day) wakes me up at 5:45 from a dream I was having where my students were all failing and my inbox was full from parents being upset with me--basically full of anxiety.

But, I'm awake now, so time to make sure that doesn't happen in reality.  Jump in the shower and shave, then to sneak back into the bedroom (so as not to wake my wife) and turn pull the dimmer switch up enough just enough to keep myself from making a horrible fashion choice for the day.  I think I made a bad choice with the stripes-on-stripes and the gray + silver, but whatever -- I teach at an all-boys school and I don't think anybody will notice.

Selfie Thursday

As I do every day, I grab my bag, feed the fish, and touch my wife on the shoulder to wish her a good day, she mumbles back the same wish for me, a quick "I love you" and I'm out the door around 6:25am.

It's a chilly day (my computer said 26, but the car said 34), but I have Wisconsin blood in me, so it doesn't bother me.  I had my phone ready because I knew I'd be doing this today, so I snapped this without looking while driving over a bridge:

Sunrise on the way to school

I'm the first one in the teacher's parking lot at 6:45, but I like to get there early to set up for class and be ready for any students that come by before school.  Whenever they ask, I tell them I get there around 7:00 and can help them with any questions they have then.  With a test in both courses tomorrow, it was a real possibility some would show up this morning.

Swing by the faculty lounge to check my mailbox on my way to class.  There are two tickets I requested for our fall play "Count Dracula" that my wife and I will attend this weekend.  For me, it's about 50% actually wanting to see the production and 50% supporting my students.

Fancy tickets may be a sign of good things to come?

I teach at a Catholic school, so I pass by this interesting piece every morning.  It's made by one of the brothers who lives on campus.  I guess he's somewhat famous since I'd heard of Brother Mel before I got here.

Here are some pictures of my classroom:

The box under the TV is supposed to let me wirelessly project from my computer, but something's messed up and it won't stay on for more than 10 minutes at a time, so I don't use it.  The posters on the left are "Parent Functions" and the unit circle is on the right next to my math clock.  Desks are in rows, but they won't stay that way throughout the day.

My desk faces the wall, because I never use it.  These posters are of our daily prayers.

From the back of the classroom.

The desk I do use.  The power outlet on the other side of the room doesn't work, so this is the only place my computer can go and be powered.  Whatever, I don't use it during class except to put in attendance.

A closeup of my desk.  Mostly just a flat surface to hold all the papers and books I let float around.  Kids will use that rolling chair sometimes, but as long as they don't fight over it, I don't care.

I have to check the schedule for the day, because I can never remember.  We're on a block schedule, so only 4 periods for the day.  Today is a "Regular Day," but every Wednesday is a Late Start (for the kids, we come in at standard time for meetings) and every other week has a mass when we'll follow the Liturgy schedule.  We also rotate which period starts the day and today is a "B Day" (meaning B block corresponds to first period).

One of the pieces of paper on my desk reminded me that I have to recommend math courses for my current students.  This is to be turned in by Friday, but I should take care of it now while I'm thinking of it. It's already been sitting on my desk for about a week.

I don't have control over the temperature in my room and the guy next door (who has control over his room and mine) doesn't get to school until 7:30.  Fortunately, my key works in his door, so I run over there to set the thermostat at 68, since it's pretty cold in my room already.

Maybe it overcompensates, though, because as soon as class starts, my geometry kids want to open windows (it's still in the 30s outside).  Then they fight over how many to open and how much.

I closed the two wide open ones and left the cracked one to settle the argument.

The intercom buzzes and we begin school with a prayer from an assistant principal and some morning announcements.  Most of the announcements have been sent to me in an email that I am to read to the students in a bit, but he highlights the important ones and adds anything that was not on the email.

October was a month the Church focused on right to life things and at one point we had students participate in standing at the Planned Parenthood building to talk with expectant mothers about their options.  The teacher in charge of that came on the announcements as well to discuss their successes and remind the students to continue collecting their change to help fund the purchase of an ultrasound machine to allow willing women to see their child before making their choice.  Some kids roll their eyes and this, but most politely listen.

Anyhow, I then read our other announcements which takes about 10 minutes to get through everything.  Only about 5 of my 25 students are listening to them, but you never know when something comes up that they need to hear, so I go through them all.

I am beginning today with a geometry class.  They have a test tomorrow over vectors, so today is a review day.  I put up problems on the main whiteboard and they get in groups with their whiteboards and collaborate to review the problems.  I'll bounce around from group to group to help with any issues they can't resolve.

A group that decided to use different colors and also split it up so that one person did #1, another #2, etc. 

A student using his phone for a calculator and another group that split up the problems.  I tried to make sure they discussed them so they all knew how to do each one, but who knows.

They did fairly well with the problems given them and seem prepared for the test tomorrow.  They finished a bit early, so I let them play hangman or tic-tac-toe on their boards for the rest of the period.

On "Regular Days," we have a period of time called "Encore" after the first period.  It's 30 minutes of free time for students.  They can grab a snack at the cafeteria, attend a short eucharist service, make up tests, meet with teachers for tutoring, play outside, etc.  If they have a D or an F in a class, though, they are assigned a room to go to so they can work on that class.  I have 13 students assigned to me currently and we worked on some practice problems for their test in Trigonometry tomorrow.

Their grading term was done last Friday, so that's when the new checklist came out.  We are asked to inform parents about their low grades and offer ideas about what we can do to help them climb out of the hole.  SBG is a lifesaver here, so I just reiterate that point to these parents.

I always dread emailing parents.  I am not a parent myself and in general am not a fan of telling adults what to do, so I always fear that they'll come back aggressively and I won't know how to respond.  While that happened sometimes back in TN, this doesn't seem to be the case here.  The families here are paying good money for their kids to attend this school and most are not rich, so they are sacrificing for their child's education.  I'll post more about this in a separate post, but they are very supportive of the teachers and trust us to help the kids.

Nevertheless, I took until Tuesday to email the parents about their child's grades and felt guilty every day for not doing it earlier (even losing sleep over it some nights--maybe that was the cause of my morning dream?).  So, some of those students actually took the initiative to come in during this free time for extra help and it made me happy.  So, in the 5 minute passing period, I emailed their parents back to share the good news that we were making things happen.  Most wrote back quickly, too, to say they were happy and to let them know if they could help in any way.  It feels so great to have helpful parents.

So, on to the second period of the day, Trigonometry.  They, too, have a test tomorrow over Trig Identities.  They have severely struggled with it.  Not only in the usual way of not having an algorithm to follow and needing all of their algebra skills to come together at once, but apparently their algebra 2 teacher was pretty horrible.  Most students had barely heard of factoring at all, so I have needed to teach (rather than just review) that alongside the added complications of sines and cosines.  They get frustrated with it, too, but we're working through it.

We worked through the three problems that I had given them for homework and I posted another 7 more for them to work through in class.  For whatever reason, these juniors and seniors are not as enamored with the whiteboards as the sophomores in geometry, so they work on their own paper when they get in groups.

D block (which is third of the day today), is my planning and also the lunch period, yay!  So, I should have an extra long time off to grade my stack of papers, grab some lunch, and generally take it easy.  SHOULD.

Our school is on a sprawling 40 acres of land, so the actual school is all on one floor which is kind of nice except that the halls really bottleneck.  So, I took an alternate route to the faculty lounge and ended up behind the assistant principal who needed the course recommendations, so I handed those off on my way.

At this private school, when a teacher doesn't come in because he is sick or takes a personal day, we have to cover for each other when possible.  They inform us of this with a pink piece of paper in our mailbox.  I guess I got there too early this morning to find the paper in my box earlier, so when I get to the lounge for my break, I see the paper and have to dash back off to another room to cover for a sick teacher.

The plans on his desk say to let the students discuss their answers from the previous day's question for about 15 minutes, then they should read in their books and answer the questions at the end of the chapter.

Fortunately, it's a good group of kids and they were more or less on task.  I'm never sure of other teachers' norms, though, so it's awkward to be in someone else's space like this.  The kids discuss the answers for the questions they are working on.  Should I let that happen?  Are they being graded on this stuff?  Is it just busywork?  Are they learning by talking about the answers or are they just copying?  This is too much stress to just be watching these other kids!

They know me by reputation, though, so some asked whether it is true that I taught at Vanderbilt.  They're impressed when I said that I did.  Then they asked what my ACT score was and that impressed them more.  I feel awkward talking about myself in a classroom, especially when they are meant to be focusing on work and even more so when they are not my own students.  I know too many students who try to derail learning by asking personal questions, so I try to answer quickly and point them back to their work.

Halfway through the period, another teacher came in to replace me.  I now have 15 minutes to grab lunch from the line and eat it myself.  Then my own lunch duty time comes.  Since it's supposed to be my free period, I have one of the four lunches to watch over to make sure no food fights happen, nobody uses a water cup to steal soda from the fountain, no more than 8 kids sit at a table, etc.  None of this ever happens (we do have good guys at the school), so it's mostly baby sitting, but I have to be their Jiminy Cricket standing in the corner anyways.

After lunch, we hand out buckets of soapy water with rags to one lucky table and they get to stay after to wipe down the tables.  I like that they have some ownership of the space and help clean the tables, but I feel weird choosing a table for the chore.

The center table in the back got it today

The internet went down at school.  Teachers were freaking out all over the place because they couldn't check attendance (or twitter) and even the printers are wireless, so we couldn't print.  Oh well, my lessons today did not use the internet at all, so I selfishly ignored the conversation.

The offending server?

The last class of the day is Trigonometry again and it went as smoothly as the previous class.  They kids really are good at working with each other and actually making sure they understand rather than just copying in my classes -- especially on review days.  The job is so much easier with cooperative students.

Thus, school ends at 2:40 with a prayer over the intercom and any afternoon announcements.  

My duties don't end, though.  I was chosen to be the sponsor of the chess club this year.  A retired teacher is the one who actually runs the show, but they need a current faculty member to be the sponsor.

The guy who actually runs the club wrote this three volume tome of lessons for the kids to practice with.
I guess the chess hall of fame is here in St Louis (not far from where I live) and this guy has a major role there.  All to say, the school takes chess seriously (won nationals two years ago and are consistently top 10).  We have about 16 members on our team and they have practice every day after school for 2 hours and tournaments on weekends every now and again.  The one last weekend lasted 12 hours (6am to 6pm) on Saturday.

Today, we split the freshmen into two groups. The other guy took half and I took the other half to go over the lesson of how to finish the game when you have two bishops (and your king) and he only has his king. Then we did some lessons about how to make "double attacks" (in the format of how you might see puzzles in the paper where you set up the pieces in a certain way and then play it out from there).

After that, the kids get to play on their own for the second hour.  I just supervise mostly.  In the picture below, kids are playing a chess variant called "bughouse" where you have teams of two and after you capture a piece, you can give it to your neighbor to use on his board.  It's fast paced (5 minute time control) and fun for them, so they like playing with it.

Meanwhile, the other guy plays 4 games at once against some kids.  If they can beat him and have correct notation for it, he'll give them a 2 dollar bill.  They think it's fun to have the challenge.  Only the guy in the foreground came close, but he moved the wrong pawn at the end and lost.

So, now it's 5:00 and time for me to finally head home.  Here's a picture of the cathedral I pass on my way home every day (took while at a stoplight).

I got home at 5:30 and the rush is on.  All at once, I feel the need to strip off the tie and formal clothes for more comfortable PJs, grab a snack, and go to the bathroom.  Again, I'd leave off this detail except that I have included all of my restroom stops for the day here already.  Once at 6am, once at 5:30pm; there's no time in the middle of the day for such trivialities.

I check over all the internets I missed during the day (which takes longer than usual because it was off for half of the day at school) while turning on the afternoon news and eating a snack.  This is my time to relax before my wife gets home from her work around 6:30.  I really should do all my schoolwork when she's not here so I can enjoy my time with her, but I've already been schooling for 12 hours and need the break.

I write the two tests for tomorrow.  I don't write them ahead of time because I like to get a feel from the review day about how much they can handle and how difficult to make the questions.  My early arrival tomorrow will be used in making copies of the tests for classes.  I use TextMate on my Mac to write in LaTeX so the math looks nice.  I use geogebra to make the figures and skitch to take the screenshots.  Put the files in my Google Drive folder so that I can download and print/copy them when I get to school tomorrow.  I guess I should use a flash drive tonight, too, just in case the internet is down when I get back tomorrow.  It took me about an hour to come up with the problems, make the figures, and put it all together.

I have a feeling the tests will be short and they'll finish early, but I decide to err on the side of too easy this time.  Thanksgiving/Fall break is right around the corner and I know they're looking forward to that.  I know they also need a bit of a grade push about now to keep them motivated.  Anyways, I'll test the important things and there's no need to make it unnecessarily difficult when I can tell how well they understand the material with easier problems.

Wife gets home and 6:45 and immediately gets to work making dinner.  I hate that she doesn't have her own down time, but she says she enjoys it.  I guess she has her down time in the morning before going to work when I'm not there.

I got an email from at student at 7:30 tonight saying that he had signed up to retake a section of a quiz a while ago and wanted to actually take it tomorrow.  He mostly was writing to apologize on waiting so long to take it because I originally stressed that I wanted them to take it the day after they sign up.  He's a nice kid and I told him the email wasn't necessary, but I appreciated the thought.

I checked my Google Drive at 8:30 after dinner.  The policy in my classes is that they have until 8pm the night before to sign up for retakes.  So, if there were any results from the google form, I could write those retakes quickly for tomorrow.  Tonight there were none and there have not been any all week.  Sometimes there will be as many as seven in an evening, but they have fallen off recently.  Maybe after break students will feel the fire more.  Plus, we have a test tomorrow, so many like to wait to only have one graded assessment per day.

There's grading to be done (especially since I didn't have my planning period today), but I can use the internet being down as an excuse for not putting the grades in.  Hopefully I'll have my plan tomorrow and can enter them then.  Some students mentioned today that their weekend might depend on their grades going up, so I should try to help them get ungrounded.  I'll see what I can do.

Just for reference, I don't feel like today was all that busy or easy of a day.  This is fairly typical of my weekdays.  It is now 9:30pm.  So, I think I'll take this last half hour of being awake to help my wife make cookies (we have company coming through town tomorrow on their way to Thanksgiving elsewhere) and think about what to do with the time after they finish early tomorrow.  Maybe tomorrow's youtube mining will turn up something interesting.

Wednesday, October 10, 2012

A Quick Mnemonic for Fractional Exponents

I forgot that I invented this mnemonic for fractional exponents, so I thought I'd share in a quick post between periods.

My students often have trouble remembering whether the cube root of x^4 is x^(4/3) or x^(3/4).  So I use the analogy that fractional exponents are like trees.

When the sun shines, the POWER for the tree comes from the TOP (photosynthesis) while the ROOTS are on the BOTTOM.

Saturday, August 4, 2012

New Job

I got a new job this year at a Catholic, all-boys school on the block system. So, in addition to teaching new subjects (geometry, statistics/probability, and algebra I), I'll have to adjust to all those things. If any of you have tips on any piece of it, feel free to share here. I'll try to post when more specific problems present themselves.

Monday, July 23, 2012

Matrix Multiplication

Another quick tip. This time how to multiply matrices.

Begin with the matrix multiplication problem:

Then move the first matrix down. [Note: Since matrix multiplication is not commutative, this is important. Although it should be noted that the same effect can be accomplished by moving the second matrix up. But under no circumstances should the reverse be tried.] The answer will go in the new space you have created in the bottom right corner. Immediately you can see (if the product is possible) the shape of the answer. In this example it is a 2x2 matrix.

Pick a position in the answer matrix and follow across from the left and vertically from above to figure out which numbers you will use. Multiply pairs beginning with the outermost numbers (the blue 1 and 7 in the example) and sum with the product of the next pair in until you run out of pairs. The answer will go in the position where the arrows meet.

Remember not to use numbers from your answer when computing other spaces. For example, the 58 was not used to find the 64 below.

Continue with each position until the answer matrix is complete!

But what if the matrices in question are not able to be multiplied?

Consider the following case. Although it initially looks like our answer will be a 2x2 matrix, we see that the 3 does not have a pair, so these matrices cannot be multiplied in this order.

Friday, July 20, 2012

Factoring cubic binomials

To be finished later, but here is a quick idea that came up:

Factoring binomials of cubes (x^3 + y^3) or (x^3 - y^3)

My mnemonic is "SQuiggy CHases Many Purple SQuirrels." It stands for:

  • SQuare (the first term)
  • CHange (the sign)
  • Multiply (the two terms)
  • Plus
  • SQuare (the second term)

So, (x^3 + y^3) = (x+y)(x^2 - xy + y^2) and (x^3 - y^3) = (x-y)(x^2 + xy + y^2)

Monday, March 5, 2012

Hunger Games Sequence

I have a 45 minute drive to school each way. So, I've been picking up audiobooks to keep me company on the drive. Many of you have suggested reading The Hunger Games, so that was my latest acquisition. The whole thing is about 11 hours, I think, so I should finish it in about a week.

Anyways, getting through chapter one this morning, I listened to this bit:

The reaping system is unfair, with the poor getting the worst of it. You become eligible for the reaping the day you turn twelve. That year, your name is entered once. At thirteen, twice. And so on and so on until you reach the age of eighteen, the final year of eligibility, when your name goes into the pool seven times. That’s true for every citizen in all twelve districts in the entire country of Panem.

But here’s the catch. Say you are poor and starving as we were. You can opt to add your name more times in exchange for tesserae. Each tessera is worth a meager year’s supply of grain and oil for one person. You may do this for each of your family members as well. So, at the age of twelve, I had my name entered four times. Once, because I had to, and three times for tesserae for grain and oil for myself, Prim, and my mother. In fact, every year I have needed to do this. And the entries are cumulative. So now, at the age of sixteen, my name will be in the reaping twenty times. Gale, who is eighteen and has been either helping or single-handedly feeding a family of five for seven years, will have his name in forty-two times.

The Hunger Games by Suzanne Collins.
Chapter 1 as found here

Being a math teacher about to begin a unit on sequences and series in my Algebra II class (as well as a deeper study in Precalculus), I thought this would be an interesting problem to use. They are also all reading this book and seem to like it.

Wednesday, February 8, 2012

Teaching Resolution: Make More Mistakes

I have a problem with how I work in my classroom: I don't often make mistakes. I don't mean for that to sound conceited, but I generally like for people with authority to show few weaknesses and be both precise and accurate in their communication of concepts. It irks me when the principal (a former English teacher and currently moonlighting as a college professor for English education courses) writes an e-mail to the school that says something like, "We are starting this business that will be ran by..." or when our superintendent sends an e-mail to the entire district full of spelling and grammar mistakes.

I thought about it harder this past weekend, though and have come to a conclusion that this may not be the best strategy in the classroom. By limiting my own mistakes it sends a few false messages to students.

  • It implies that it is bad to make mistakes.
  • It implies that the concept or problem is (or should be) easy.
  • It puts me on another plane than my students so that they think I am way above their level and they will never be able to attain that level of understanding of the subject.

So, I am resolving to make more mistakes in my classroom. Some will be intentional; some may not be. Just today in precalculus, for example, I was trying to number our examples as we went and I put up numbers 1, 2, 3, 6, and 7, then I asked them to work through those seven problems. Those students who were paying attention got confused and corrected me. Yay!

Vocabulary or Jargon?

While we're entering a new era or spelling in our culture, the debate reigns whether to join the revolution or fight for the original ways. It's a debate on which I can argue both sides and about one in which I have a hard time deciding my own position. While "kids these days" are writing things like "lol ur rong" does seem somewhat uneducated, it also gets the point across and isn't the point of language to communicate ideas?

While that example may fall more squarely into the English teachers' realm, I have an internal debate about it in my own classroom, too. How important is it really that I use the words "denominator" and "numerator," when it's so much quicker (and more easily understood) to say "bottom" and "top?" When a line with negative slope is changed to get more vertical, how wrong is it for me to say that the slope gets "more negative?"

I realize that there are some words that need to be defined and used correctly so as to avoid confusion. I just finished a section of Algebra 2 where we talked extensively about the difference between permutations and combinations. The words here are important to the understanding of the logic and the formulation of the mathematical expressions needed to solve these problems.

On the flip side, though, in precalculus we are discussing vectors. What is the real difference between calling two vectors orthogonal versus just the familiar "perpendicular" word they already know? Later we introduce the term "normal" for this same concept. Do we really need three words to express the same concept? (Or are they not the same concept and I'm thinking about it incorrectly?)

I realize that even the words "denominator" and "numerator" are important when we involve more complex expressions. But even they get confusing when we discuss things like "multiply both the numerator and denominator of the entire fraction by least common denominator of the terms in both the denominator and the numerator" to simplify an expression like

So, what say you? When teaching my students math, how important is it to indoctrinate them to the traditional language of mathematics and somewhat confusing vocabulary? Is it possible to communicate some concepts using more colloquial language?

Thursday, January 26, 2012

iTextbook Thought

I have been intrigued by Euclid's Elements ever since I saw this website: I'd always thought it would be a cool way to explore geometry. Begin with the basics and then see what you could do with those things.

So, my idea for an iTextbook is based around those ideas. You begin with some "simple" definitions and postulates as in Euclid's book. Maybe give some reasoning behind the postulates by asking questions like, "Why do we need 2 points to draw a line? Challenge: I am thinking of a line in this window. See if you can guess it with zero hints. (Student draws random line and it won't match what's in the computer.) Here. I'll give you one hint: This point is on the line (point appears, kid draws another line, it won't match). You missed because here are some example lines that it could have been (show a bunch of lines through the point). Here is a second hint: This point is also on the line (connect the dots = winner)."

Expand by asking students to create some tools needed. For example, show a picture of a pile of items that you might find in a junk drawer. These include, but are not limited to, some string, some push-pins, a pen, tape, scissors, a straightedge of some sort, etc. Ask students "How could you create a perfect circle (see definitions 15-16) from these items?" Once they arrive at a correct answer, they "level up." A screen appears with something like, "You have acquired the circle tool!" and the glittery circle button appears in their tool-bar at the top which works like it would in any drawing program (click for the center point, then click again for a point on the circle to fix the size).

Basically, you build geogebra tools by finding out how to construct each thing. You have the line-segment tool and circle tool, so Challenge 1 is available to you: construct an equilateral triangle using the tools you have.

Once you show you can bisect a given line segment, you get a new "midpoint" tool. This unlocks new challenges in which you need to use the midpoint for other constructions.

After you construct an object, you get follow-up questions like: In constructing the midpoint, do the two circles need to be the same size? Is the bisecting segment always perpendicular to the original segment? Does the original segment bisect the segment you created? etc.

It'd be dynamic as with geogebra or sketchpad or anything like that, but you can only use the tools when you "level up" by figuring out how to get build them from more basic tools.

There's a "sandbox mode" where you can play around with tools you have or preview what will become available later. Challenges available are based on the tools you have acquired so far, so they are somewhat ordered, but not necessarily linear. They could be labeled with a difficulty level. Unlocking new tools unlocks new challenges. Hints can be available for the tougher constructions/proofs, like in most games, too.

Anyways, what do you think? The idea just struck me this afternoon, so I haven't thought through all the angles (pun intended), but it seemed like an interesting way to introduce parts of geometry.