Monthly Archives: November 2012

But Why Is It Wrong?

I know on the last post I said that I only had a few good ideas last year, and I’ve just stumbled upon another one.  Last year I realized I was resorting to showing why a certain operation was wrong by using just integers.  For example, sometimes I would see something like:

\frac{\sin{x}}{\cos{x}}+\frac{\sin{x}}{\cos{x}} = \frac{\sin{x}+\sin{x}}{\cos {x}+\cos{x}}

So, after cringing, I would ask something like “Is this true?”

\frac { 1 }{ 2 } +\frac { 1 }{ 2 } = \frac{1+1}{2+2}

Edit: Just fixed the equation above to reflect what I meant.  I got lost in the latex of it all (and for some reason, the latex parser in wordpress doesn’t like when you copy and paste latex from another site–I’ve had the exact same latex work and not work, one right above the other!).  Thanks to Steve Grossman for the spot!

They (usually) would recognize their mistake and add the fraction the right way.  I found myself doing this so often, that I decided to create an activity where they corrected mistakes (these were real mistakes I found on tests and quizzes–though I didn’t tell that to last year’s group because they were the first class I taught Precalculus to, so they’d realize it was their own mistakes!!) and showed why they were wrong using small integers.  Here are some examples

\sec^2(x) + \tan^2(x) \sec^2(x) \Rightarrow \frac{\sec^2(x)}{\sec^2(x)}+\frac{\tan^2(x) \sec^2(x)}{\sec^2(x)}

Is wrong because:

1+2 \neq \frac{1}{3} + \frac{2}{3}


1 + 2 = 3 but \frac{1}{3}+\frac{2}{3}=\frac{3}{3}=1 which is not equal to 3.

Or a slightly trickier one (for students):

\sec^2(x) + \tan^2(x) \sec^2(x) \Rightarrow \sec^2(x) (\sec^2(x)+\tan^2(x))

Is wrong because:

2 + 2 \times 3 \neq 2(2+3)


2 + 2 3 = 2+6 = 8


2(2+3) = 2(5) = 10 \neq 8

As good as the exercise sounds to me, I believe I failed in it last year, mostly because I did not provide enough structure or examples.  This year I have plenty of examples, and I am going to have to figure out how to provide more structure for the students.  I’ve heard that “error correction” is great for students, and I really think this extra step of understanding the error correction is essential, so I really hope that it goes over well!

EDIT: So I’m posting this after I did the exercise, and it went awesome!  Students were presenting the mistakes and explaining thoroughly why they were incorrect, even going so far as to explaining what they thought the student was thinking when they made the mistake!  I now see why this kind of error correction is invaluable.  The highlight of my day, though, was watching as one group of students (we’ll call them Jack and Jill) was presenting, Jack was explained the problem quickly, and Jill was watching the other students in the class and looking for comprehension.  When she realized that they didn’t follow Jack’s thought because it went too quickly, Jill stepped in and asked “you didn’t get that, did you?” and proceeded to explain the problem more thoroughly.  My students really are becoming teachers.  And I’m just sitting back and watching them.  Awesome.


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Scaffolding the Trig Identities

Last year my students struggled so much with proving trig identities.  You know, this kind of thing:

Prove:    \csc{\theta}\cos{\theta}\tan{\theta}=1

Part of it is that my students never did proofs in Geometry, so they have no former experience with the idea of “proofs” (they don’t get it in any other classes either :/  ).  Another part of it is that they flip out when they see a bunch of letters and numbers and have no idea where to begin (mostly a confidence issue).  But another big part of it is that my Precalculus students have forgotten (1) how to manipulate fractions (any operations with them), (2) how to factor, and (3) how to distribute or multiply polynomials (I dislike using the word FOIL, but I find myself repeating it over and over).  And if they remember how to, they have no conception of checking whether their work is accurate or not and more often than not make mistakes which throw off their entire equation.

SO I had the idea of starting with what they know (or should know), e.g. \frac{1}{5} + \frac{2}{5} = \frac{1+2}{5} = \frac{3}{5} and moving slowly into progressively more and more complicated equations, moving from just numbers, to variables, then trig functions.  I then begin another thread where the least common denominator is less than multiplying both denominators together (i.e. relatively prime), and work through “numbers –> variables –> trig functions”.  I think this is what my education grad teacher meant when she kept repeating the word “scaffold”.

Well, here are the results of my efforts from last year, and when looking through my previous lesson plans, I re-discovered it and thought “hey, I actually had a decent idea last year!” so I thought I’d share that here.  I don’t remember this helping a whole lot–mostly because I had the larger issue of students simply not doing the work, but I’ve already had students exclaim “I needed this, because I was always bad at it!” which makes me feel good.  (Even if it’s not helping, they think it’s helping and that’s a step…)

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A Day in the Life: Dictated (Dragon-style)

Most of this post was dictated using “Dragon Dictation” for the iPad.  A super-handy app if you have an iPad or iPhone/iPod and haven’t installed it yet.  It is faster than typing and during the day it was easier to get to the iPad and speak into it, however I apologize for any words that have changed because I did not speak clearly enough. Sometimes this can be quite humorous.

0550: I wake up to my alarm clocks. Yes that is plural. I need two because sometimes I do not wake up to only one and other at other times I accidentally hit the turn off button instead of the snooze button as happened this morning.

0610: I finally get out of bed after the alarm clock goes off a few more times and I stagger towards shower, which I usually take longer than I really should. I recall my dream was kind of a nightmare about teaching middle school and help students weren’t listening to me. Note: this nightmare is based on the reality that two years ago I did teach sixth-grade. I still have nightmares about it sometimes, and I have a ton of respect for teachers who teach middle school.

0705: After eating breakfast and getting dressed and thinking about how I should have graded all of the chemistry quizzes, I walk towards school. I think of dictating my morning on my walk, but it is way too cold for me to take out my iPad. I make the decision to let students grade each others quizzes and one of my chemistry classes. I’ve come to recognize the importance of students getting feedback very quickly on assessments as I compare the different years I’ve taught and how quickly I’ve been able to do that. The concern, when students grade their own quizzes, is that privacy is not protected. However sometimes getting quick feedback overrides that factor as I believe the cases today. It also saves me a little bit of time even though I was grading and working the entire time after school yesterday.

0720: After taking 10 minutes to dictate my morning, I write the agenda for the day on the board, as well as the homework.

0725: I remember to take my lunch over to the fridge in the teachers lounge. I go in and remember that over the weekend since it rained and snowed is a big pool of smelly water in there. It would smell much worse if it wasn’t so cold. Because of that, nobody has used the teachers lounge this past week and we are continuing to hope for a new school. Here is a picture of how old our school is. I would say that we need a new high school, but my definition of need is very strict, so currently I just want a new high school building.

0730: I quickly type up the warm-ups I had in mind into PowerPoint so that my students can theoretically began working as soon as they enter the classroom.  I also upload the notes we took from last week in Chemistry and post what’s going to be on the quiz Wednesday and Thursday for Precalculus and Chemistry respectively onto my website as I promised the students.  However, Weebly is not working completely, and I struggle to attach all the documents I needed to. (I blame our network)

0745: I turn on some music to get ready for the day, and I decided against letting my chemistry students grade their own quizzes because there are so, so many empty answers and wrong answers that it would be difficult for me to watch them not cheat. Perhaps I could give them each others’ quizzes, but again it would be very difficult to explain what the right answer is. I’m still debating it as school gets closer.  As I get my notes organized for the day and look over them, I remember to put the worksheet that I have to hand back to a student in a specific place (to help me remember to give it to the student).  The student misunderstood almost completely the method of naming ionic compounds.  There were also about a total of 10 students who were absent yesterday, six in one class, all of which have to take the quiz that the other students took yesterday. So therefore I have to send them out of the room and trust that they do not cheat. I e-mailed another teacher and I think he will be able to look over and make sure they are cheating.  I finish grading the quizzes 10 minutes before the first bell rings and put all of them into the grade book so that at least during one of my chemistry classes I can simply hand them back the quizzes instead of having them grade them. I really want to give them a chance to retake the quiz today and Thursday, although they have another quiz Thursday so that would be very difficult for many of them.  I then used Dragon dictation to remind myself to e-mail a parent because her child is not working as we had discussed in the parent teacher conference, and I believe that she will make some difference and his work efforts.

807: I am a little disappointed by the new YouTube interface because there is no button to stop playlists from continuing on. I typically like to have music going when students enter, but I can’t if it will keep going throughout the period. I suppose I could walk over and manually turn off, but that’s one more thing to remember to do.  I end up leaving the music running at the beginning of class.

1st Period, Precalculus. There are several things that I want to accomplish in class today: (1) do a warm-up related to the quiz tomorrow, (2) go over the HW (also related to the quiz tomorrow), (3) review radians, which will also be on the quiz tomorrow, and (4) begin learning new materials so we don’t get too far behind my other Precalculus class.  I only get around to (1) and (2).  One frustrating thing is that I come across two students who still haven’t taken out their books 15 minutes into class and started the warm-up.  FIFTEEN MINUTES!!?!?!  What am I doing??  I went around and checked HW (while they’re supposed to be working on the warm-up) but I usually end up giving them a little extra time after I check HW to do the warm-up, and I have a bad habit of waiting til most people are done, even though many of the students don’t deserve it because they were spacing out or doing other things than doing the warm-up.  What can I do to help them focus the minute they enter my classroom?  I’ll have to reflect more later.  We end up only doing about 4 different problems–I’m worried that the quiz will take a long time for some students when it really shouldn’t.

2nd Period, Chemistry.  Here’s what I want to accomplish in this class: (1) do a warm-up related to the quiz on Thursday, (2) check the HW (also related to the quiz on Thursday–do you see a theme here?), (3) have students grade each other’s quizzes (from yesterday), (4) do some more practice problems for the quiz on Thursday, and (5) continue with the notes that we started yesterday and are not related the the quiz (on Thursday).  We only got through (1), (2), and (3), and sort-of (4) via (2) (I made up a few examples on the fly to see if they understood what I had done while going over the HW).  No, I don’t normally assign two quizzes a week, but we didn’t have school Friday because our soccer team made it to the semifinals (weird, right?), so it got pushed back to Monday.  Also, naming Ionic compounds can get confusing if we just talked about naming Covalent compounds, so I wanted to make sure we hadn’t talked about the latter before quizzing on the former.  There will be plenty of time on the test later for them to get confused about those two. I ran out of time at the end of class. Chemistry ended with me giving them a problem, which usually doesn’t happen because I have a timekeeper who tells me when there are five minutes left in class. She did this but I continue teaching and ran out of time because I lost track of “we only have 5 minutes left”. I also promised students that they could retake the quizzes at lunch so I have to print off 38 copies of quizzes but I need all alter slightly so that they can retake the quizzes. Not sure when I’ll do that. Maybe during physics (I actually forgot to during Physics, even though I had the thought “I know there’s something I told myself that I would do now…”)

0950: Break between 2nd and 3rd period. I input attendance since I haven’t done it yet into the computer system.  We really should be doing it during class, but I find I just don’t have time, and I keep track of it on a clip-board so I can do it later if I forget (which is 100% of the time these days)  I also have 10 minutes to prepare for the next class (physics) and change classrooms (I share the “science lab” room with the other math/science teacher).

10 00:I briefly stop by the teacher lounge to see if anyone is there. Most of the other teachers taker 10 minute break there but I usually do not have time to since it is only my second year of teaching these subjects. I arrive in physics and have a little time to set up and think about what problems I want to show the students. I should stop dictating now because I will run out of time if I don’t work now.  Side note: I didn’t notice that my problem student from second period was missing until I put in the attendance. It is usually a pretty busy day when that happens.

3rd Period, Physics.  Class runs smoothly.  I meant to create a problem ahead of time for them to practice 2-D projectile problems, but I forgot and had to make one up as we went along.  Fortunately I’m getting good at those.  For the last 1/2 of class we go to the computer lab and work on the handout I created yesterday.  Some students finish it completely just as the bell rings, while others have a ways to go.  I feel like it was time well spent and I hope they understand acceleration and velocity just a little bit more.

4th Period, Chemistry.  This class is fun because it is smaller than 2nd period, I’ve already taught Chemistry that day, and it has all of my brightest students from Precalculus (scheduling of a small school).  We have fun, even when taking notes, and I get through (1), (2), (4), and (5) from above.  (Remember, I graded their quizzes, so they didn’t have to grade them–I’m not sure why I started with 4th period).

1149: Class lets out and I am late to chapel (every Tuesday and Friday) because I have to make a quiz so students can retake it during lunch. I feel little bad because we are supposed to be setting examples of being in chapel and not doing other things, and I enjoy chapel with the students–it is one of the unique things that we get to do here at a Christian school. However, the students really need to retake the quizzes and I didn’t have another one set up yet, so I really needed to make it. I was also beginning to grow weak from talking too much in my classes and I felt a strain on my vocal chords towards the end of fourth period. We’ll see if I have time to eat lunch during lunch hour today because I have to hold tutoring during lunch. That is a whole other story:

So last year we had tutoring after school (we received funds from the state to hold a program) which helped the needy students tremendously.  This year the government changed the policy and ran a check to see “how well our students were doing”.  They passed (which means that the after-school program was working) and because of this, decided “you guys don’t need after-school funding any more!”  Derp de derp.  SO, we do longer have an after-school program, and instead I wanted to offer tutoring to my students (some who live an hour away from school) and the best time to do that is during lunch.  I’ve found Tuesday and Thursday to be the best days when fitting into our schedule here, so every Tuesday and Thursday, I sit in my classroom while anywhere from 10 to 30 students join me and work on things.  I have tutors who earn points because they come and help out other students which is helpful for me. However I still I’m usually stretched between trying to give students re-tests on quizzes, teaching them what they missed because they were absent, or just trying to eat my lunch. This time, I managed to eat most my lunch and will be will snack on the rest later during my planning period.

5th Period, Precalculus.  The class goes well as it almost always does because I have very few students and they are the top in their class. It is a fun class to teach and we get through it even though I am winging most of the problems and figuring out how to do them on-the-fly. I know, I know, not good teaching practice, but I didn’t have time because of the other quizzes I needed to grade, and the students get to see me solving the and learn from that as well, which I think is often helpful.  This stuff is so fun for me, I really enjoy working with them anyways.  Plus, they are way ahead of the other class and anything that I can do to “slow them down” (read: give them more practice) is good.

Sixth hour is my planning period, and is usually my time to rest and catch up mentally. I was partway through checking my e-mails when my TA shows up and lets me know that he can help me already. That is actually wonderful news because he usually only helps during seventh hour. Between the two of us, we input the rest of my grades for the week and for the first time in nearly a month I am caught up on grades. Yay!

Seventh hour I spend helping some students in my study hall work on their chemistry because they did not understand what we’ve done class twice now. I do get excited, though when they start to show that they understand the material. They’re good kids and I do hope that they do well on their quiz on Thursday. I also spent seventh hour typing up a letter recommendation for a student who gave this to me about a week and a half ago. Not a lot of time, but I should have done it before now. The maximum is 2000 words and although I’ve typed a letter of recommendation before for the student, it was only 360 words. I guess I have some more to type. Hopefully I’ll get the post this on my blog before too long.

1520: The bell rings and I am talking about Google and other technology with the other math and science teacher which often happens after school (he’s a Mac person and I prefer Linux, so we jab at each other a bit–all in good humor). Around 4 o’clock he leaves and I am left here typing up what I’ve done for the day.  I lied earlier, I still have a few more papers to grade, but I am much further than I was earlier today. I am very fortunate to have a TA at the school. I do not know what I would do if I did not have that help.  He helps me manage my participation points, and has saved me hours of busy-work that otherwise I don’t know when would get done.  I am worried about going to another school eventually where I won’t have that help!

As I leave for the day I realize that I forgot to tell my seventh period study hall to put up the chairs. I tried to be as helpful as possible for the janitor because he works so hard and is paid so little, but without my two special-needs children (they were in Albuquerque for a doctor’s appointment), I usually forget to do that.  They are so good at remembering things like that and are so helpful and friendly, even though they can be a handful at times.

When I get home, I noticed that 13 Dropbox files were changed. (It tell because I turned on my computer and it downloads the files I worked on throughout the day.)  That means I created or edited/changed 13 different files throughout the day.  Thirteen files?!?  I can probably only count 2 or 3 of them, but it’s often the case that I did a bunch of things and made a bunch of small edits that I didn’t even think about.  Thankfully Dropbox keeps all my papers synced so that I don’t have the additional headache of e-mailing myself all the changes throughout the day.  (No, I’m not advertising for Dropbox, but you should get it if you don’t yet!)

As I look back on the day, one of the toughest things I think that I have to do is switch so quickly between subjects and between types of problems.  Fortunately discipline is not quite so big a worry as the vast majority of my students behave.  But especially during tutoring when I am going back and forth between Chemistry and Precalculus, sometimes every 15 seconds, it can get muddling to look back and think about.  No wonder I’m so shell-shocked when I get home (which is when I have to create new assessments, grade previous ones, and think about contacting parents, etc.

Wow, that was a lot more than I intended to put (thanks to Dragon Dictation, it didn’t take too long) and now I should really go write this letter of recommendation.  Thanks if you made it this far!


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Around the World–QR Code Style!

I enjoy the around the world activity, even if it takes a while to setup, because it gets kids moving around and gives them feedback as to whether their answer was correct or not.  What I didn’t realize is how much this activity can be enhanced, even beyond just the “oh boy, we get to use iPads” motivator, and so I want to share some of those ways here.

First, what is around the world?  If you know what this is, you can skip down to where I describe what I did differently specially with the QR Codes.

Around the world is an activity where you have answers taped or placed all around your room.  Underneath the answers (or behind them) you have questions.  A student begins by choosing any of the questions around the room, and solves the problem underneath to get an answer.  That student then looks around for the answer (which is showing), and once they find it, try to solve the question which is beneath that answer.  So the questions and the answers are not placed on the same piece of paper, that way, students have to work their way around the room (usually randomly, so they don’t figure out an order) until they end up back where they began.

**New with the QR Codes**

So obviously I replaced the Questions with QR Codes because students needed to see the answers in order to know which code to scan.  I used this website, The QR Code Generator, which I found to be the best because I could paste text and URLs and get back QR codes, all of which worked wonderfully with the iPad app “Scan”.  The only drawback is I couldn’t put subscripts (this was for Chemistry class) in the questions, since QR codes only read plain text, although I suppose if I thought it through and spent much more time, I could turn those into images (I’ll explain how I did that in a second).  Regardless, students made the connection that numbers in chemical formulas were supposed to be subscripts in this situation, so that went fine.

One excellent thing I can do, that you can’t do as well without QR codes, is that I was able to put up many wrong answers in addition to the right right answers.  Students would run to a QR code and scan it, just to get a picture like the one below:

(Yeah, I’m a Wake Forest Fan, and I like jab at the few students who care about Duke and/or UNC at our school)

Unfortunately QR Codes cannot store full images, so instead what I did was used Dropbox and linked the images, using the Dropbox URL, so that when students scanned it, it took them right to the image in my Dropbox.  Fortunately you can make it so that students can see the image, but not manipulate your Dropbox (which is important!).  By the way, I searched for the internet for the above images (I hope I did not just violate some copyright laws), and I used Pixlr to edit the images very quickly.

Another incredible thing you can do slightly better with these QR Codes is that you can give feedback for wrong answers (and not just pictures).  If I had spent more time putting this together, I could have done a lot more specific feedback, but as it is, I would tell them things like “the number in parenthesis means the charge, not the number of atoms” which would help them see where they went wrong.  I think the students appreciated both the funny “NO YOU GOT IT WRONG” pictures and the positive feedback to help set them on the right track.

At the end of the activity, I had some trickier places to look.  The final “real” question had an answer that was taped low on a desk, so it was harder to find.  That QR Code sent students to look inside of a drawer of a lab table (it told them specifically where to go).  That QR Code sent them to find a specific picture of one of the wrong answers and look on the back of that image.  That QR Code told them to bring me their answer sheet along with the latest scan, and then I gave them a super-secret QR Code, which happened to be a “Hall of Fame” Google Form.  Students got to enter their names and leave a comment.  Here are some of my favorite comments:


I just like to thank my mom and everyone who developed my treasure hunting skills! Thank you Jesus!!!! 🙂 yee haw!

We love chemistry 🙂

Follow the light, and it will lead you to victory

I’d like to thank my mom! I am awesome!

Yo!!! That’s how it’s done!!! :))))

I believe that my students had a lot of fun and I’m very thankful to the online community who shared so many of these ideas with me.  Some of them include Technology Integration for Math Engagement and The Teacher Garden.  I’m sure there were other website which I read and learned about QR Codes, but I apologize for not remembering all of them.

One other thing that I forgot to mention was that I allowed students to start at any QR Code in one half of the classroom.  I put the first half of all the answers on that side and the second half on the other side.  To start the activity, they just went looking for a question in the first half of the classroom.  So no, every student did not get to finish every problem, but they all did at least half of them and there wasn’t a traffic jam due to all of the students looking for the same answer.  If you’re wondering about numbers, for my class of 24, I only created about a dozen problems for them to do, and there was no “traffic jamming”.  The activity lasted about 25 minutes for the dozen, and some students didn’t do all of the problems because they started later on in the chain of problems (if that makes sense), however, all the students that finished had to complete at least half of the problems!

I just realized that I should have the first “answer” (which didn’t actually have a question in my situation) be a wrong answer to one of the later questions, so that students would start over if they didn’t get it right!  Bwhaha, that would have been mean.  I think some of them would have picked up on that… but still, *evil grin*

Please let me know if you try this out–it really did not take very long to put this together, so I am definitely going to do it again.  Here’s a picture of one of my students enjoying the activity (I tried to take more with my iPad, but I forgot and the quality wasn’t superb.)


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iPad Frustrations

Our school recently got some iPads.  I say “some” because we did not get enough for the High School to go one-to-one, but I think that was the goal before getting the iPads.  So as it is, we have these “carts” of 25 iPads each, and I really didn’t think I was going to use them that much, but I decided to go ahead and get a jump on them before the other teachers had time to sign them out and use them every day.

The activity was to capture a screenshot of a sine or cosine wave.  Students used Scan and QR Codes to go to my website and watch one of the 3 options of videos I made which include waves.  They would pause the video on a particularly significant wave and take a screenshot.  Then they would use the “Doceri” app to sketch on the wave and find Amplitude, Sinusoidal Axis, Period, and Phase Shift.

This worked really well when I did it on my own, or when I had one student do it on his own with his own iPad during study hall, but issues quickly compounded when I tried to get the whole class to do it.

Issue #1: turning in the assignments.  I wanted something easier than e-mailing, so I used the neat website DropItTo.Me in addition to the app “Scan” for QR Codes, and thought “hey, I have a method for students turning in work in under a minute!”  Wrong.  The issue was that the iPads always named the photos “image.jpg” and so every student who uploaded overwrote the previous file.  Doh!  Since there is no way to rename photos (not that I could find, at least) then this was a dead end.

Next attempt: e-mail the file to me.  Many students do not have a clue how to attach a file (yikes!) and most e-mail clients do not work well for safari on iPad because the iPad assumes you are going to use the native e-mail client (which we can’t use because iPads are not assigned to specific students–they’re a class set).  That failed in a big way, and the bell rang as students were attempting to frantically e-mail me.

Another issue that I saw coming down the road is that the iPads are wiped clean when you connect them to the cart laptop (well, not exactly, but if I want to add any applications, I basically have to wipe them) and so students could not save files on the iPads themselves unless I didn’t touch the cart between classes (which is what I’m doing this time so that students can retrieve their photos).

Later on, I had trouble with the iPads all syncing to the laptop on the cart, and I couldn’t install applications.  Well, I figured out what was preventing me (with near-administrative priviledges) from installing apps… it was that I had turned “off” the ability for students (or any user of the iPad) to install apps.  DOH!  Now maybe if you know the specifics of installing apps you’re saying “but they wouldn’t have the apple ID, right?” but they CAN use their own apple ID and install any apps they want.  Of course hopefully all the teachers are keeping a close eye on the iPads so they don’t do that anyway, but do the users really need the ability to install apps in order for anyone to be able to install apps?!?  Just another way that Apple fails when it comes to adaptability: iPads were made for individuals to use, and it is really difficult to modify the system to allow for classroom-shared carts.

Okay, that’s enough of my rant against Apple.  Perhaps next blog will be a positive blog about the fun things we’re doing in Physics.

UPDATE EDIT: I found an okay solution using Google Drive–I created an account for all of my classes, and added the “Google Drive” app to the iPads, and now students can upload the images and then rename them (and organize them) as needed.  The good part of this solution is that students can retrieve their work for further editing if necessary, and I can download the work on the due date very easily.  The bad part of this solution is that students can rename and even delete each other’s work, which is why I encourage  students to write their names on the Doceri projects, so other students cannot “steal” it.  I doubt students actually would, but copying work because much easier to do (but also much easier to discover).


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