Beyond the Algebra of Composition Functions

Now that my kids are comfortable with writing composition function equations, I gave them a few word problems to illustrate:

1. why composition functions can combine functions that have different types of domains. (For example, function f takes in dollars as domain, and function g takes in time as domain. It's possible to get an equation that represents f(g(x)). See first problem in Part 2 of the worksheet.)

2. how composition functions combine step-wise dependencies to represent them all in one swoop!

(The examples are a bit silly. But, they are intuitive and easy enough for kids to grasp/follow. After this, I made them do a bit more heavy-duty problems in the textbook, that are less light-hearted and a bit more "real", but also less fun.)

Check them out! (Part 1 is adopted from a lesson from NCTM. I just re-formatted the questions and re-worded them quickly. I was a bit scared by how long it took my 11th-graders to get through that first exercise.) I think the worksheets were pretty effective. I'm sure if you did it with a more accelerated class, they'd breeze right through this, and it'd help solidify their conceptual view of compositions.




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I also have an idea for how to teach inverse functions this year using a sort of telephone game. I haven't tried it out yet (that's for tomorrow's class), but I'm thinking of starting the class with writing a table of first / last names on the board and asking kids to evaluate
f(Isabella) or f(Alvaro) for last names. Then, I'll introduce / review that the reverse lookup notation is
f^-1(Alvarez) or f^-1(Garcia) and that this is called the inverse function.

Then, I'll call a few kids to sit in front of class in a row, and they each will get to pick a secret basic operation +, -, x, with an operand. For example, "+ 3" or "- 8" could be what they secretly choose. I'll give the person on one end of the row a number, and he or she would do the operation in their head, and then tell the result to the next person. So on, until they get to the end. Say that the original number is 10 and the final number is 61, I'd write f(10) = 61 on the board. Then, I'll ask them to go backwards, starting with the last person with the number 61. Each step, they should "un-do" their own step by doing the opposite of what they did before. That way, by the time they get to the front again, we should see that f^-1(61) = 10 happened by reversing each operation AND reversing the order of each operation. And I'll use that to introduce how to write equations of inverse functions!

Example: First person secretly chooses "add 3", second person "times 2", third person "minus 9", fourth person chooses to square.
f(x) = (2(x + 3) - 9)^2

That means that on the way back, in order to use the output to find the original input, we would need to: First square root, then "add 9", then "divide by 2", then "subtract 3".
f^-1(x) = (sqrt(x) + 9)/2 - 3

I think the nice thing about this demo is that we can repeat this process quickly if kids have questions about any part of the classwork/homework. (I'd just choose kids to represent each step of the operation, and have them sit in a row to demonstrate how the operations reverse their nature as well as their order.)

...I'm excited about this!! I think it will work and make sense to the kids!!! (Of course, I'll also teach them the "short cut" of flipping x and y and solving for the other variable. But I think that comes later, once they have a foundation of what inverse operations are and why they work.)

The other nice thing about the chairs exercise is that we can use it down the road to illustrate why f^-1(f(x)) = x, by putting twice as many chairs in a row, with the middle two operations canceling each other out, and then the next pair canceling each other out, etc. Example:

x --> Add 3, Times 4, Minus 1, Plus 1, Divide by 4, Subtract 3 --> you get x back, obviously!

Thoughts??

Addendum: The telephone game worked fabulously! It also was a good anchor for me to come back to in order to explain to kids why inverse function has nothing to do with flipping the signs. All I had to do was say to kids, "When I gave them the reverse input while going backwards, I didn't flip its sign, did I?"

Using the first and last names as warmup also had the added benefit of a giggle factor, when I explained to the class that f(Cuellar) = undefined. (Cuellar is the last name of a very silly and likeable kid in the class, but since it's the wrong type of domain value, it cannot be evaluated. The class thought it was funny that his evaluation gives an error.)

Happiness!

Four very happy bits of math teaching news:

1. I have launched my new "Precalculus algebra skills" videos website! Check it out. Some of my students (not sure how many... mostly the strugglers, I guess...) are very excited about this extra resource. Or at least that's what they say. We'll see if they actually follow through and watch the videos. The videos are very detailed and so are longer than the 2- to 3-minute length that was recommended to me by a reader. That might change in the future, but I'm not sure yet. (It really depends on student feedback, I guess.)

For now, the videos are just there on a voluntary-access basis. That might change in the future as well, if I can be sure that every kid has access to them from home. (ie. 6 weeks before the final, I might start assigning 2 or 3 videos a week, or something, as spiraling review at home.)

2. I'm going to Park City Math Institute in July! woohoo! I'm super excited. (It's a summer program for math teachers and other math geeks. I've read really great things about it!) It's going to be really interesting / probably very hectic, because I'll be bringing my same 2 suitcases-worth of stuff that I'll be lugging to Berlin. Berlin is expecting me to report immediately after the PCMI summer program so that I can begin working on my immigration paperwork -- I actually had to negotiate a later starting date with their HR department, in order to make this happen. Which means that I'll have to completely move out of El Salvador before I head off to the program. Which means that things can get verrrry interesting in June. :)

3. I came up with a GREAT new way to teach bisectors as a locus of points satisfying a property!! I gave my regular Geometry kids a Do Now where they had to:

A.) Copy down the definition of a locus (with examples: a circle is a locus of points equidistant from a center, and a line is locus of points (x,y) that fit into a certain equation y=mx+b).
B.) Draw two points, M and N, and find 5 other points that are each equidistant from both M and N.
C.) Draw an angle YXZ. Then, find 5 points that are equidistant from ray XY and ray XZ.

After the kids tried parts B and C for a good few minutes (and most of them had figured at least part B out), I picked a volunteer to stand between a plant (representing M) and a chair (representing N), so that he/she's equidistant from both objects. Asked the kid to step forward away from both objects, but still remaining equidistant to both the tree (M) and the chair (N) at all times. After the kid took a few steps, we noted on the board how the kid had no choice but to walk perpendicularly each step away from the original segment MN.

We then repeated the same exercise but using the corner of the class as the original angle. I picked a different kid to start in the corner, to walk away from the corner, but always staying equidistant to both walls. We noted that they bisected the angle, and connected this to the Do Now problem.

After these demos, kids thought this concept was so straight forward! The idea that perpendicular bisectors (or angle bisectors) contained an infinite number of points equidistant to both endpoints (or rays) became obvious to them. We drew the diagrams on the board, connected the points using a line, and discussed why that line is the locus of all points that satisfy those equidistant requirements. I was really excited because I believe that this is a really abstract concept, but making kids act out the points really made a huge difference!!

We also used wax paper to explore how to find a perpendicular bisector by folding. I gave these regular Geometry kids pieces of wax paper, told them to draw points A and B, and told them to figure out how to get the perpendicular bisector by folding. Everyone figured out that A has to go on top of B. Silly me for actually teaching it to them last year!! (Apparently it's just common sense.)

Then we went into this pizzeria / bisector project from NCTM. It has a lot of good math in it! (I've used it once before; thought then that it was really great as well. I use the scenarios where there are 2 pizzerias in town, versus 3 pizzerias in town, versus 5 pizzerias in town. Each one really highlights different skills, and do not become monotonous for kids in that succession. The only very time-consuming part is that in the scenario with 3 pizzerias, you have to find the area of each region by counting the blocks and estimating when the fractional blocks make a whole block. For 2 pizzerias, you can still use a trapezoid area formula, so it's not so bad.)

On Day 2 of the pizzeria project, we illustrated the definition of circumcenter again using objects and people. This time, I had two volunteers, one starting off in the middle of a plant and chair #1, the other starting off in the middle of a plant and chair #2. As they each walked forward along the perpendicular bisectors, the class observed how at some point they collide. And I stopped them and said that that point of collision is called the circumcenter, and at that point they are equidistant to the plant and BOTH chairs! Later during the pizzeria project, when kids needed to explain the significance of a house that was located on all 3 perpendicular bisectors, they immediately recalled it being the circumcenter and recalled that it was equidistant from all 3 pizzerias! Brilliant!!

4. Even though I had been a bit skeptical of my own group activity of writing composition formulas and analyzing domains using a "playing deck" of function cards (see bottom of this post), my 11th-graders LOVED it!! And every child's understanding of composition functions improved visibly between Round #1 and Round #5. By Round #5, they were consistently getting the equations and the domains correct. I was SUPER happy!!!

Love my job. Love, LOVE!

Making my First Math Video

I've been thinking about the inverted classroom model. There seems to be a few key advantages like getting kids to be more proactive about their own learning and allowing them the ability to rewind or re-watch as many times as it takes for them to absorb all of the important info, but I don't like it as a way to introduce skills and topics. The non-interactivity of monologuing about math for 10 minutes cannot be very effective in activating prior knowledge. Even when I do a mini-lesson in class, I like to call on kids to provide feedback, so that I can make sure that I am pacing the class appropriately for the slower learners of the batch and that I'm clarifying and re-wording parts that may seem unclear to them in my initial description. The other issue is that my kids don't really have trouble recalling simple facts or concepts. It's when they all mingle together that my kids start to get confused. (And although we work out their issues in class, they don't always remember what we did in those n-step problems a few weeks/months later. --Shocking, I know!)

But, I am considering implementing something like this to help kids review the more difficult parts of a past topic or assignment that has already been introduced / worked on in class. That way, they can look through the archive and only tune in to the "episodes" that are giving them trouble. Also, this means that if they do not remember how to do something that's a bit complicated, they can always go back and find a relevant problem and watch the video to see how it was done. (Versus my current model, which is we would discuss their difficulties in class, and who knows how good their annotations are going to be in helping them work out the entire process later on??)

In fact, I am going to begin making a video archive of problems soon. (Why wait until next year to try this out?? March seems as good a time as any.) My one class where kids really need extra review is in Precalculus, and it seems like we're always running out of time to teach new material, that we can only do so much review in class for the material we have already learned/practiced. I am going to experiment with making some videos about current and past topics/problems, and dumping them onto a webpage so that these kids can look them up as they need to instead of always seeking me to re-explain the same things.

Addendum: So, since I'm one of those impulsive types, I abandoned this blog entry midway through and went ahead and tried to make a video last night! The video wasn't half-bad in quality actually, even though holding the camera in my left hand and writing while standing (easier to hold the camera steadily that way, while leaning my wrist on a box) and thinking about what to say at the same time (and trying not to mess up) was quite tricky. In fact, there was a noticeable pause in the video when I had to do a simple subtraction, because I was just so distracted by everything that was happening at once. I also got cut off half-way when the camera ran out of battery, so I had to finish it off in a separate video -- fortunately, it was in a natural break of the topic. But in the meanwhile, if you can take a peek and give me some feedback on the math explanation and/or the format, that would be great! (I have to still work out the technical aspects of the job. Right now, after conversion into Windows Media Player format it is still too big for my taste and makes uploading kind of a nightmare. I tried converting into FLV and it got really grainy and difficult to see the letters on the page. Suggestions??)

Here are the links: Part 1 and Part 2 of how to find domains for combined functions. Are you able to see them on a regular internet connection? (My hope is that they're more or less stream-able.)

On Open-Ended Non-Questions

I came across this wonderfulness today in my Google Reader, and it made me pretty excited! (I know, I'm a total geek.) I love the idea that you'd give the kids an open-ended situation and leave it up to them to show as much understanding as they can about the situation. Seems like something that works extremely well in physics -- a subject whose goal is to encourage kids to gain a multi-layered understanding of everyday situations using a combination of various "laws" and models. It's also really exciting to me because this process highlights (to the teacher as well as the kids) that conceptual understanding of the same situation will continue to build, as you accumulate deeper knowledge on how to drill down further into pertinent details. For example, take a situation where a kid is told that a hot air balloon flies through the air. At the beginning of the year, they might only be able to say that it's because the air inside the balloon is less dense than the surrounding air. A week later, they might be able to say that this is because the air is warm and warmer air is less dense. Another week later, they might be able to say that the warm air is less dense because the molecules have more kinetic energy and therefore create pressure and expand the volume of the balloon. Another week later they might be able to explain the heating mechanism and why hot air doesn't escape through the hole of the balloon. And even later, they might be able to predict what temperature the balloon would have to have in order to carry a certain amount of weight. (I'm not actually totally sure if they can do those calculations AND I'm pretty sure you'd teach those concepts at a faster pace. I'm just using those steps to roughly illustrate how an idea about the same situation progresses over time.)

It sounds like something I would like to try in my own classes, but two difficulties immediately come to mind:

1. Is it necessarily applicable to a math classroom? Although open-ended explorations in math are indeed possible and interesting, a vast majority of the math processes we practice in our classroom encourage kids to eliminate cluttering information that they don't need. Whereas physics encourages the learner to take something very simple and expand it into something complex and multi-layered and to consider all factors involved, math (in my mind) is more like a funnel that gets rid of all of the cluttering details and focuses in on only the most important details. Even much of the WCYDWT stuff is inherently begging a certain question to be asked, and then more or less inherently requiring some specific math strategy (with some real-world messiness, of course).

Or am I missing something here and my view of the goals of math teaching is really much too narrow? (I'm thinking out loud here. Feel free to jump in. My mind really isn't made up about this one way or the other. After all, it's clear that the early mathematicians never limited themselves to thinking about the fastest way to get from point A to point B.)

Anyway, personally, I think there is always value in "playing around" aimlessly with a problem, even if it doesn't immediately lead you to something productive. Sometimes questions arise that way and other times solutions arise when you least expect them to, just because you've meandered your way through most of the issues. Past a certain age, all the problems that are worth solving are not solvable in your head anyway, so some playing around is usually necessary and giving kids these non-questions encourages that line of open-ended thinking.

2. Let's assume that my first point was moot and that this technique is entirely applicable to the math classroom. The beauty of this process, as I have described above, is that you can give the kid the same situation at the beginning of the year, the middle of the year, and the end of the year, and their understanding should continue to build upon itself and to encompass all existing knowledge, plus brand-new knowledge. What types of problems would I be able to give that tie together all of the things they would learn in the course of a year, so that they could demonstrate understanding at all different levels? (Here I feel that physics has another natural advantage -- a bunch of the ideas you learn in physics are all interrelated.)

I have no easy answers. I'm going to have to think about this one, so that I could possibly implement something like it next year.

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PS. Speaking of good test problems, I recently put a question on my regular Geometry exam that nicely incorporated some old and new concepts all in one shot. The question gave the kids 3 different y=mx+b equations to graph, and asked them to find the angles inside the triangle that is formed from these 3 lines. In order to do that, they needed to know: 1. how to graph lines (old knowledge), 2. how to use pythagorean thorem/distance formula to find edge lengths of the triangle (old knowledge), 3. how to use inverse trig to find the angles within the triangle (new knowledge). Synthesis!

Rollercoaster

It's been a pretty emotional week for me in Precalculus. It's hard for me to explain, but I reached a breaking point this week when I passed back a test where 2/3 of the class had failed. They had failed even though both they and I had been working steadily on the material for a while now. There was A LOT of material; make no mistake. They had a test on everything we had learned from early January to the end of February, and it took them 2 days in class to complete. The test included at least 2 major types of word problems (2-d and 3-d optimizations and piecewise functions), and various not-easy algebra skills, such as finding domains of function equations, completing the square, and function transformations. (I gave them one half of the test the day before a long weekend, and the other half of it the day after.) But we had also spent 3 days reviewing, and I had given them practice problems very similar to those on the test, just to jog their minds along the lines of my expectations. A bunch of the problems they had already seen very similar stuff to, on the previous 3 or so quizzes.

So, I reached a point where I had to have a serious chat with the class. I told them that I thought that their grades were, in short, disappointing. I said that it showed me that they didn't go back to make sure that they understood the old quiz problems, and are just more or less copying quiz corrections without really making sure that they have understood the material. I also said that if they had really tried to understand the material after each quiz, they would have a much easier time than cramming 3 days before the test. I told them that from now on, I wasn't going to accept any test or quiz corrections from them without them verbally walking me through every step. If they couldn't explain something, I would send them away and tell them to wait a few days before coming back to me. One girl spoke up and said that there were a lot of topics on the exam, and it was difficult to study. But this girl in particular had done well on Day 1 (not having known what was going to be on the exam for that day), and very poorly on Day 2, after the weekend. So I didn't hesitate to point out that she wasn't being truthful about why she had failed the exam.

Besides that, there weren't really complaints; kids knew that they weren't doing everything they could to be where they needed to be. I was UPSET. I told the kids that it really bothers me to think that we are at the cusp of going into brand new math topics, and they're struggling so much with things that are more or less review of previous years' material. I also told them that I know that a lot of them think that this is the last year they "have" to take math, but that it would be extremely unwise for them to not take math next year, and to expect to go into college having skipped a year of math. I said, "You guys can barely remember what I taught you two months ago! How are you supposed to remember it two years from now??" The class was pensive, and quiet. I also told them that I know that they need to study more at home, because in class EVERY SINGLE ONE OF THEM can always do the work with some help, which means that the material is NOT out of reach and my explanations are not too obscure for them. The problem is that they only think about math during the 4 hours a week when I see them. I said, "Math is a lot like playing sports. If you want to get good at something under pressure, you need to practice the same thing more than a couple of times. You wouldn't want to go into a soccer game having only practiced once a few days before; why would you do that for math?? I stay here after school everyday; what are YOU doing to improve your grade?? If you're not willing to stay with me after school everyday -- that's OK -- get a tutor! Some of my freshmen have tutors; there is nothing wrong with that. Only 1 of them is failing out of all 4 of my freshmen classes. Do what you need to do and make positive choices! Get it together!!"

Anyway, as usual, I thought to myself afterwards that I had perhaps been too hard on them. A major reason for their poor understanding can be attributed to their recent ridiculous absence patterns. After all, how can you be expected to do well on a two-day test when you've recently missed an entire week (or more) of school?? Another reason for them doing pretty badly, as one parent pointed out in a conference with me recently, is that they have poor basic skills coming into my class (for reasons I won't discuss). The parent was concerned because she recently started helping her kid with math, and she realized that her 11th-grade daughter can only calculate rise/run correctly sometimes. (And, even though I review these "old" skills whenever applicable in class, we have to be realistic here -- how many slope practice problems could/would I assign to a class of 11th-graders?) But, the result is that they often make silly algebra mistakes in word problems, even though they have the big picture of what they're supposed to do.

I have very complicated feelings about all of this. But, after my chat with the kids, they started miraculously showing up for extra help. Yesterday, 3 of them came after school. Today, 6 of them came after school. --AT 3PM, ON A FRIDAY?!?! I told them, "You guys sure choose some interesting times to be motivated." One of them who had NEVER come to see me before said to his friend, "This stuff is so easy now!" I rolled my eyes at him and said, "That's why YOU should have come to see me BEFORE the test."

I don't know what to think about all of this -- not about their poor state of affairs, nor about this sudden surge of motivation. I can only hope that their motivation is not just some sign of short-lived guilt. Because this class really, really needs that extra umph. None of my 11th-graders has issues processing complex material. They just need a LOT, LOT of work to get fluent at it. (And I hope that they're all planning on taking some type of math next year, so that all of this effort I am putting into them isn't going to waste.)

Best Group Work Ever!

Continuing with my "This was a trig Do Now that I had used that worked well for me" thread, I used two Do Now exercises to illustrate to my Honors Geometry kids how to break down quadrilaterals without ever lecturing. First one looked something like this:

1. Draw an irregular quadrilateral ABCD with one right angle A; use the corner of your white paper to get an easy right angle.
2. Measure AB and AD only.
3. Measure all 4 angles of ABCD.
4. Draw a dashed line from B to D.

Then, I had the kids fold the quadrilateral back along the dashed lines, completely solve for everything inside triangle ABD (all angles and all sides), and then open it up and completely solve for all remaining angles / sides, and finally to find the quadrilateral area! (I had to remind them that earlier we had found the area of a scalene triangle to be A = (1/2)(a)(b)(sinC). Other than that they were uber-independent and got the whole thing quite brilliantly. The activity was again self-checking using protractors and rulers, so it helped to build their confidence in their own rather elaborate calculations.)

After this, we worked a little bit on Kristen's awesome trig project problems. Since my honors kids are sharp, I didn't need to elaborate further on how to divide up quadrilaterals besides emphasizing that we had seen how right triangles make up scalene triangles; now we were going to see how scalene triangles make up quadrilaterals as well, and to use that as our basis for analysis.

Day 1 of the project was a bit slow. I didn't need to help them too much, but they were working slowly in groups, trying still to grasp when to apply which formula. They were tentative about discussing the problems, and mostly worked individually. I saw this as a sign that no one was sure-footed enough to pipe up in a group, and it made me wonder whether my kids were really fully ready to venture into quadrilaterals on their own just yet. So, I decided that what they needed was another scaffolding activity, just to ease the transition a bit.

So, on Day 2, I devised another Do Now that furthered this idea of cutting quadrilaterals into smaller pieces and applying Laws of Sines / Cosine to analyze the individual pieces. This time, I wasn't so interested in them actually carrying out all of the calculations. I just wanted them to make a plan of attack, so that they can begin to see the pattern/big picture:

1. On a sheet of white paper, draw an irregular quadrilateral ABCD. This time, don't use right angles.
2. Measure all 4 sides of ABCD.
3. Measure only angle A.
4. Describe, step by step, how you would solve for all angles of ABCD.
5. Also describe how you would find the area of quadrilateral ABCD.

Seems innocuous enough, but the kids had to work hard for this one. Only a few kids were able to run with it completely on their own. Most figured out how to cut the quadrilateral into two appropriate triangles. After waiting a while, I started to give the class hints along the way, one hint every few minutes. (I picked kids from the class to share how to do the next step, every few minutes.) They struggled through this, but the process of thinking about it was very worthwhile. After this Do Now, the kids worked on the remainder of the project packet with much more fluidity and confidence.

Today, we more or less wrapped up the whole project. I had given them 5 problems from Kristen's packet, and asked them to solve 4 (and had given them time in class to work on 4 problems), the fifth one serving as extra credit. By the end of today, a bunch of kids were on -- or had finished -- their fifth problem. I wasn't checking their answers any time during the project; I asked them to compare with their partners, and then to compare with another group if they still weren't sure. I asked them not to round whatsoever, so that in the end I can verify that everyone's answers are absolutely accurate (down to 12 sig figs, or whatever TI-89 allows). And it was glorious. As a contrast to how timid they had been at the beginning of the project, kids were LOUD. I had to shush them because some kids were working on it during my Study Hall (like an advisory period) and their exuberance was getting in the way of me helping other kids with other mathy things. (I normally have 12 kids in my own Study Hall; today I had about 30. A good 20 of them were honors kids working together on their trig projects, and the rest were regular Geometry kids asking me for help with studying for their test.) Every kid devised their own way of arriving at the same answers, even within the same group. Even my weakest honors kids were able to call me over, walk me through (in shocking clarity) what they had done to solve the majority of the problem and to ask me very specific questions about the remaining steps. The whole groupwork thing was so glorious that I took a mental picture of them working in groups today and really, really wished that I had an actual camera. It was truly one of the greatest days we had had this year, in terms of complexity, how great kids felt about their work and each other, and everything else. Afterwards, when we started reviewing for the upcoming trig test, I heard various kids whisper, "Now it all seems so easy!"

In the end, I told them how proud I was of them. I told them that I had taken this project from an 11th-grade teacher, and that I hadn't been totally sure that they could handle it but that they had handled it fabulously. And all of it is so true!

Visualizing Operations on Functions

For my Precalc kids, I started toying with this idea of presenting function operations using diagrams to help kids visually organize domain changes and to see how equations relate to one another.

Here's stab #1 (we started doing this earlier this week. I plan on finishing this tomorrow and easing our way into composition of functions...). My reasoning for organizing it like this is to show kids that addition, subtraction, and multiplication are all very forgiving operations. As long as f(x) is a valid value and g(x) is also a valid value, you can add/subtract/multiply them with no problem. Only division might cause additional exceptions in the domain.




I'm sort of envisioning the kids to then go into something like this, where they can picture composition as one function machine feeding into another one and using that idea to write equations. I also hope that they'll figure out on their own by the end of this second worksheet that g(f(x)) will only be undefined if either f(x) is undefined or if f(x) is some value that will in turn "break" or cause an error in g.





Finally, I'll squeeze in a game/activity, where they get in groups of 3 and split a deck of "cards." Every round I'll call out some order of composition -- for example, g(f(h(x))) -- and the kids would need to write the resulting equation and find the domain restrictions on that formula. And then, with those same 3 cards I'd reverse the composition order, and they'd do it again. It's not terribly fun of a "game", I guess, but as far as activities go, I hope it will be a little better than doing straight problems on paper, because they'd hopefully notice that the domain restrictions only come into play for certain types of operations (ie. square root), regardless whether that operation happens first or last in the composition.


Thoughts or suggestions?

...By the way, I recently read on someone else's teaching blog that they have all these really wonderful things planned for their Precalc kids. It made me feel a little math-envy, but alas, my kids need all of the basic reinforcements that they can get. (They're coming along, and are actually understanding words that are written on paper with numbers inserted in between!!!! It seems like half a miracle considering the zombie-esque state that they were in when I got them back in August.) But, our sloooowness in progress is making me very worried about their future, and I'm going to try my best not to become a total stressball over this during the next few months. It's going to take me another couple of weeks to just get through all of the Algebra 2-ish review-ish stuff with them, and then whatever trig I can squeeze in to the rest of the year, I'll have to be happy with!! The end of the year is coming SO FAST.