We just got back from a week-long trip to beautiful Turkey! Even though we had not much time to look around, from what we could see, it was truly an amazing mix of East and West, of modern and ancient cultures. Geoff and I both agreed that it is incredible that you can hear the songs calling Muslims to prayer 6 times a day in the backdrop of Istanbul, yet the city is vibrant with modernly (and risquely) dressed women. The city, at least, seems to embrace people's choice of lifestyles, more so perhaps than most parts of the West. (I think that in the States, as much as we claim to embrace liberal ideas, if you choose to pray 6 times a day towards Mecca, people at your work place would definitely look at you funny.)

Istanbul itself is also an amazing mix of cultures -- there are bar-lined streets galore, such as in the Taksim Square area, terraces overlooking the hilly city, trams that run from tourist point to tourist point, and yet the guitar and drums music seeping out from the bars are traditional-sounding, somewhere between Arabic and Indian.

The Turkish food is incredible; there are elegant restaurants to match the best of Europe. Three restaurants in our hostel area that we can immediately recommend are De L'Artiste, Morro, and Solera. Of these, Solera was my favorite, because they serve up local Turkish wine, coupled with delicious cold appetizers that are of a local variety, but beautifully done with elements of savory surprise. The city also has countless bazaars -- the only one we visited was the Grand Bazaar, but the price and the quality of the goods there were fantastic. Between Geoff and me, we bought: a silver necklace, an exquisite mirror for my sister, a lamp set, a rug, a leather-made silhouette puppet, and a beautifully woven pillow case. :)

While in Turkey, we did the typical touristy thing. We flew into Istanbul, spent a few days there, took a flight out to Izmir, spent a couple of days there on an all-inclusive tour, and then flew back to Istanbul for a few more days. Istanbul was amazing because it was a party spot, but backdropped by the ancient buildings. It's incredible to think about the unique culture exported from Istanbul to the rest of the world over the centuries. It is of little wonder that it prides itself as the Cultural Capital of the World.

This is me fake-crying because of the weather. :) It was snowing our first day in Istanbul! Colder than in Berlin!!!

While we were in Izmir, we got to visit Ephesus, the third largest ancient city. A good amount of the stuff has been rebuilt from the excavated material, and the excavation is still on-going, but it was still impressive to see the ruins left after thousands of years.

Besides Ephesus, we also got to visit Pammukkale, which is a beautiful calcium bicarbonate deposit formed by centuries of active hot springs. Some of the springs were cooled greatly during the winter, but others were still bath-water warm! You can only walk through the labyrinth of springs with bare feet, but we braved the cold anyway....

As an added bonus of going to Turkey during the off-season, we got to stay at a five-star hotel for a night as part of our all-inclusive trip to Izmir. The food was delicious and our room had bubble jet stream bath, and a view of the ocean. It had been a long time since we had fallen asleep to the sound of ocean waves outside of our window, so it was a real treat. (Especially because we had anticipated staying at a hostel.)

Anyway, here were some other random things we did:

Geoff was fed some stuffed clams by a cute-looking Einstein man! Clams stuffed with rice is apparently a local specialty.

We visited a local (Geoff's favorite) shisha spot. Surrounded by colorful Turkish lamps and lots of locals, it was the perfect spot to enjoy some apple tea and some shisha.

Here is Geoff inside the Circumcision Room at the Topkapi Palace.

We couldn't take any pictures at the Whirling Dervish religious ceremony, but it was pretty cool and inside a fixed up bath house.

Speaking of bath houses, we had our first experience with a Turkish bath house. For the equivalent of about 30 to 35 Euros, you can strip down naked, lie inside a sauna, and then have a person scrub you clean. (It's not coed. Geoff and I were in separate parts of the bath house; he had an overweight man leaning over him asking him, "Is this GOOD? IS THIS VERY GOOD?" while scrubbing him down. I had an old lady and she also asked me if it was very good while I lay completely naked and she scrubbed me down. It was very unique -- definitely an experience I would recommend.) The bath house we went to was a traditional one -- it had been running since the 1400s.

And, as a last note (harhar, no pun intended), I wanted to share with you my excitement to see the Sigma notation on a Turkish bill. I am pretty sure that in Turkey, they put random accomplished people (not just politicians) on bills, so they also put down some visual representation of why that person's famous. You know I had to share this:

## Saturday, December 31, 2011

## Thursday, December 22, 2011

### Categorizing Student Mistakes

I am experimenting with a new method of grading exams, in which I look through the problems to determine what

So far, these are the labels I've come up with:

My hopes are that in this way, I can help kids to focus first on the bigger conceptual issues, and next on the other types of issues. I'd say that if a high level kid sees that they're consistently making the same types of careless or incomplete-operation mistake, it is valuable feedback for them to keep in mind for the future. Versus if a lower-performing kid sees that they're at least not missing the major concepts, then that is a good feedback for them as well, so that they know they would just have to focus on the procedural issues.

Thoughts? I'm grading semester exams as we speak. (sigh.) It's the last road block between me and a real vacation...

*type*of mistake the student made, and instead of writing a lengthy explanation of the mistake (which is what I typically used to do), I just circle the place where they messed up and then write down the category of the mistake.So far, these are the labels I've come up with:

*Wrong approach- ie. the student was completely not on the right track.

*Conceptual mistake- the student had an inkling of what to do, but they made some severe error in the initial setup of the problem.

*Procedural mistake- the student understands at a high level what the question is asking and what procedures are required, but made some fundamental mechanical error in the procedure.

*Arithmetic mistake- mistakes involving combining decimals, fractions, or integers.

*Careless mistake- mistakes involving miscarried signs or wrongly recorded results, when the student exhibits overall competence in the process.

*Incomplete operation mistake- the student failed to completely answer the question or completely simplify their answers. Or, they were off to a good start and then bailed halfway...

*Mistake in interpreting the instructions- the student did not carefully follow the written directions and therefore did irrelevant calculations.

My hopes are that in this way, I can help kids to focus first on the bigger conceptual issues, and next on the other types of issues. I'd say that if a high level kid sees that they're consistently making the same types of careless or incomplete-operation mistake, it is valuable feedback for them to keep in mind for the future. Versus if a lower-performing kid sees that they're at least not missing the major concepts, then that is a good feedback for them as well, so that they know they would just have to focus on the procedural issues.

Thoughts? I'm grading semester exams as we speak. (sigh.) It's the last road block between me and a real vacation...

## Tuesday, December 20, 2011

### Doodle #3 and Rotating Calipers

I got a sketchpad and am fooling around on it, practicing sketching motions and emotions. Motions are easy, emotions are hard. So far, this is my favorite from the motions sketches:

If you're looking for some mathy updates, perhaps you should consider reading Geoff's very technical blog. He recently put up an implementation of the Rotating Calipers algorithm for finding the minimum bounding rectangle around any polygon. (The algorithm finds the best rotated rectangle, not just the best right-side-up rectangle, which would have been too easy.) His tech blog is super dry like the stock market books that he reads in his leisure time, but he likes it that way. :) He says he just wants to dump information on the internet to facilitate other people who might come across the same issues, so reader-friendliness isn't one of his main concerns.

Anyway, if you read through his implementation, you'd see that there's a lot of vectors math in there. It's neat... He's a computer programmer who

If you're looking for some mathy updates, perhaps you should consider reading Geoff's very technical blog. He recently put up an implementation of the Rotating Calipers algorithm for finding the minimum bounding rectangle around any polygon. (The algorithm finds the best rotated rectangle, not just the best right-side-up rectangle, which would have been too easy.) His tech blog is super dry like the stock market books that he reads in his leisure time, but he likes it that way. :) He says he just wants to dump information on the internet to facilitate other people who might come across the same issues, so reader-friendliness isn't one of his main concerns.

Anyway, if you read through his implementation, you'd see that there's a lot of vectors math in there. It's neat... He's a computer programmer who

*actually*uses high-school math on a regular basis to solve problems!## Sunday, December 18, 2011

### Doodle #2

Here's a second doodle I did today (you can click on it to see the full-sized version). My pen ran out of ink near the end, so I think this'll be the last sketch in a while and I'll go back to reading during my spare time.

I threw out the previous one (of the dancer) after I took a photo, since I had made it on scrap paper and there were other things on the back side. I think this one I'll keep. I didn't do such a good job on their faces, but I still liked the overall feel of the piece.

I threw out the previous one (of the dancer) after I took a photo, since I had made it on scrap paper and there were other things on the back side. I think this one I'll keep. I didn't do such a good job on their faces, but I still liked the overall feel of the piece.

### Ah, winter break...

I miss drawing. I think this Christmas, I'm going to buy Geoff and myself some charcoal and drawing pad, so that we can be drawing hippies on Saturday mornings. I made this today because I was bored and the art store was closed. It's based loosely on this picture, but of course I messed up on the arms since I was drawing with an ink pen (one I normally use for grading) and I hadn't made anything in years.

Hello, winter break. :) It's only Day 2, and we have already: partied a good bit, finished hanging all kinds of things up in our apartment (Geoff measured/built three art frames from scratch and stapled the canvasses to them! It was awesome watching him sawing and banging things together while I laid back, ate chocolate, and watched TV... but I'm extra happy that finally all of our Salvadoran art is hung up), and we rode our bikes today to eat yummy breakfast in the park. In about 5 days we will be off to Turkey. I can't wait!!!

PS. Geoff's parents sent us chilled champagne in the mail. Two bottles. I managed to convince Geoff to immediately crack open one bottle upon receiving them, because hey -- how often in your life would you get chilled champagne in the mail?! That seems like as good an occasion as any to enjoy them. :)

PPS. Our Christmas tree/bush is coming along. It's crooked and small (that's what she said?), but it's filled with holiday spirit! :) I am so excited about our first jointly owned Christmas tree ever!! (Last two years we lived in the tropics, and before that we each lived separately in NYC.)

Anyway, I hope your holiday spirits are bright. Setting up the Christmas tree made me all sorts of sentimental. It was the first time I had actually set one up without my sister around (even though it has been 7 or so years since we've spent Christmas together). Made me miss her extra much.

PPPS. You know that Australian claymation movie

Hello, winter break. :) It's only Day 2, and we have already: partied a good bit, finished hanging all kinds of things up in our apartment (Geoff measured/built three art frames from scratch and stapled the canvasses to them! It was awesome watching him sawing and banging things together while I laid back, ate chocolate, and watched TV... but I'm extra happy that finally all of our Salvadoran art is hung up), and we rode our bikes today to eat yummy breakfast in the park. In about 5 days we will be off to Turkey. I can't wait!!!

PS. Geoff's parents sent us chilled champagne in the mail. Two bottles. I managed to convince Geoff to immediately crack open one bottle upon receiving them, because hey -- how often in your life would you get chilled champagne in the mail?! That seems like as good an occasion as any to enjoy them. :)

PPS. Our Christmas tree/bush is coming along. It's crooked and small (that's what she said?), but it's filled with holiday spirit! :) I am so excited about our first jointly owned Christmas tree ever!! (Last two years we lived in the tropics, and before that we each lived separately in NYC.)

Anyway, I hope your holiday spirits are bright. Setting up the Christmas tree made me all sorts of sentimental. It was the first time I had actually set one up without my sister around (even though it has been 7 or so years since we've spent Christmas together). Made me miss her extra much.

PPPS. You know that Australian claymation movie

*Mary and Max*? Please watch it. It's phenomenal (although not really suitable for children) and made me both laugh and cry.## Thursday, December 15, 2011

### Holiday Geometry Activities

Today was the last full day before Christmas break, and another teacher and I had talked about gathering up both of our classes to do some fun holiday geometry. In the end, the other teacher was absent this week and then very busy when they returned, so they trusted me to just plan the session by myself.

This is how I structured it.

First, we did this snowflake prediction activity in partners. Everybody folded the papers together for the first snowflake, drew out their predictions, and cut it out. Then, I monitored that each pair of partners finished predicting the next two before I gave them each one piece of paper to have them cut out a snowflake to verify their predictions. (They split up what they would cut up, both to save time and save paper.) Then they proceeded to make more predictions, followed by more testing by cutting out snowflakes. This took about 40 minutes. Meanwhile, both the other teacher and I circulated to make sure that kids were understanding how to apply the idea of symmetry to making appropriate predictions.

Then, with the remaining 40 minutes, the kids got to choose between either doing a tetrahedron origami (mostly unassisted; the exercise was in reading and deciphering diagrammed instructions... the hardest part is reading the earlier instructions on how to create the regular hexagons out of a rectangular sheet of paper),

This second activity, by the way, was one that I learned at PCMI. :) You keep cutting each segment into smaller thirds (or any fixed fraction), and folding up the middle part. In the end, you end up with a very intricate design. I'll post a photo when I get a chance!

It was awesome! We wrapped up the class by talking a bit about the rotational symmetry of the tetrahedron and also about why we could fold hexagons up into tetrahedrons (same base shape, the equilateral triangle). It was a lovely way to inject some last-minute holiday cheers after all the heavy-duty algebra we had been doing.

Happy holidays!

This is how I structured it.

First, we did this snowflake prediction activity in partners. Everybody folded the papers together for the first snowflake, drew out their predictions, and cut it out. Then, I monitored that each pair of partners finished predicting the next two before I gave them each one piece of paper to have them cut out a snowflake to verify their predictions. (They split up what they would cut up, both to save time and save paper.) Then they proceeded to make more predictions, followed by more testing by cutting out snowflakes. This took about 40 minutes. Meanwhile, both the other teacher and I circulated to make sure that kids were understanding how to apply the idea of symmetry to making appropriate predictions.

Then, with the remaining 40 minutes, the kids got to choose between either doing a tetrahedron origami (mostly unassisted; the exercise was in reading and deciphering diagrammed instructions... the hardest part is reading the earlier instructions on how to create the regular hexagons out of a rectangular sheet of paper),

*or*making a geometric sequence/recursive pattern (see below).This second activity, by the way, was one that I learned at PCMI. :) You keep cutting each segment into smaller thirds (or any fixed fraction), and folding up the middle part. In the end, you end up with a very intricate design. I'll post a photo when I get a chance!

It was awesome! We wrapped up the class by talking a bit about the rotational symmetry of the tetrahedron and also about why we could fold hexagons up into tetrahedrons (same base shape, the equilateral triangle). It was a lovely way to inject some last-minute holiday cheers after all the heavy-duty algebra we had been doing.

Happy holidays!

## Wednesday, December 14, 2011

### Being the New Kid On the Block

I have to admit that my professional transition from El Salvador to Berlin has not been a warm-and-fuzzy one. At my old school, I had built a reputation with my students and their families, and their parents would regularly come up to tell me how much they appreciated their kids being in my class and how they wished that I could have stayed to teach their younger sons and daughters. At the end of my first year, an entire group of kids went to talk to the administration to request for me to move up with them to the next grade. Former students would come back to visit me, and even now they are sending me emails to let me know where they are off to next year for college.

When you move to a new school, for better or worse, you start off anew. You leave whatever reputation you have built up behind you -- the respect that you have gained from your colleagues and the administration, the affection from kids whom you've known over multiple years. Most of all, you leave behind the trust of your students and their families. When you move schools, you start again at the bottom of the totem pole and have to prove yourself every step of the way, to everyone who might be watching.

In my case, I took on the slower-paced classes in two grades this year, because 1. I didn't really mind, I enjoy teaching things at a manageable speed for the kids, 2. I wanted to make sure the kids at the bottom would get the extra attention/support that they needed. Well, in those two grades, I have had various resistance from a few kids who feel that they should belong to a faster-paced group. Those kids care not for fun learning or meaningful tasks; they just want to move along

ARGH! It is frustrating to feel like the new kid on the block. ...

Christmas break cannot come soon enough. The last week has been fairly demoralizing, and I don't have a lot of umph left in me before the holidays. :(

When you move to a new school, for better or worse, you start off anew. You leave whatever reputation you have built up behind you -- the respect that you have gained from your colleagues and the administration, the affection from kids whom you've known over multiple years. Most of all, you leave behind the trust of your students and their families. When you move schools, you start again at the bottom of the totem pole and have to prove yourself every step of the way, to everyone who might be watching.

In my case, I took on the slower-paced classes in two grades this year, because 1. I didn't really mind, I enjoy teaching things at a manageable speed for the kids, 2. I wanted to make sure the kids at the bottom would get the extra attention/support that they needed. Well, in those two grades, I have had various resistance from a few kids who feel that they should belong to a faster-paced group. Those kids care not for fun learning or meaningful tasks; they just want to move along

*faster*through the topics. How do you convince these kids that conceptual development is worth taking the time to get right?? I refuse to short-change their conceptual foundation in order to speed through the topics, and I don't believe that it is good for their mathematical growth in the long run,*or*good for their problem-solving abilities. (Case in point, one of those kids moved up to a faster-paced group on a trial basis, and went from getting 100% to getting 25% on assessments.)ARGH! It is frustrating to feel like the new kid on the block. ...

*I know*that being new doesn't mean I am less qualified to teach these kids, or that I'm making bad decisions for their learning. But it does mean that what I value carries a lot less currency around here, as far as my kids are concerned. sigh.Christmas break cannot come soon enough. The last week has been fairly demoralizing, and I don't have a lot of umph left in me before the holidays. :(

## Saturday, December 10, 2011

### Updates from Deutschland

I thought I'd take a minute and do some life updates. Time is flying by!

Things are going smoothly over in Berlin. I had put in a lot of work (ie. over 10-hour work days daily) from August to November to learn the new curricula and to earn the trust of my colleagues and student parents, and finally I was ready to re-focus on what I needed for myself. So, in a non-trivial gesture, I'm back doing yoga on a weekly basis and am LOVING it more than ever. (My new yoga teacher is really tough, but I love him!) I've also arranged for a private German teacher to work with me starting in the new year, since I feel like I cannot commit to 6 hours of classes like Geoff does during the work week. Socially, things are good and we've met a lot of fun people. :) Overall, despite the weather getting colder and the days getting darker, we are enjoying our first winter in Germany!

Recently, we got some free tickets to check out a Christmas market outside of Berlin, so we took the local train there last Sunday. It turned out to be a fairly small market, but the town was very charming!

The town had a lot of old people on bikes. I did not see a single young person riding a bike that day. I told Geoff that some day, we're going to retire to towns with old people on bikes. :) The town also had statues that looked like they were from old Grimmes' tales.

There were gondolas ferrying people back and forth between the two sides of the Christmas market. The ride was long and a bit chilly, but we had blankets and people were drinking hot mulled wine ("gluhwein"), which is common at Christmas markets and really all around Germany at this time of year.

We also saw a for-rent sign next to this cute little barrel of a room. It's even smaller than our apartments in NYC!!

Geoff took a photo of some locals moving a tractor via two gondolas.

All in all, it was a lovely day away from the city, and a much needed break from all of the stress I had been feeling from nearing the big semester exams. :)

I am looking forward to checking out some of Berlin's very own Christmas markets this weekend. Gluhwein, here we come!

Things are going smoothly over in Berlin. I had put in a lot of work (ie. over 10-hour work days daily) from August to November to learn the new curricula and to earn the trust of my colleagues and student parents, and finally I was ready to re-focus on what I needed for myself. So, in a non-trivial gesture, I'm back doing yoga on a weekly basis and am LOVING it more than ever. (My new yoga teacher is really tough, but I love him!) I've also arranged for a private German teacher to work with me starting in the new year, since I feel like I cannot commit to 6 hours of classes like Geoff does during the work week. Socially, things are good and we've met a lot of fun people. :) Overall, despite the weather getting colder and the days getting darker, we are enjoying our first winter in Germany!

Recently, we got some free tickets to check out a Christmas market outside of Berlin, so we took the local train there last Sunday. It turned out to be a fairly small market, but the town was very charming!

The town had a lot of old people on bikes. I did not see a single young person riding a bike that day. I told Geoff that some day, we're going to retire to towns with old people on bikes. :) The town also had statues that looked like they were from old Grimmes' tales.

There were gondolas ferrying people back and forth between the two sides of the Christmas market. The ride was long and a bit chilly, but we had blankets and people were drinking hot mulled wine ("gluhwein"), which is common at Christmas markets and really all around Germany at this time of year.

We also saw a for-rent sign next to this cute little barrel of a room. It's even smaller than our apartments in NYC!!

Geoff took a photo of some locals moving a tractor via two gondolas.

All in all, it was a lovely day away from the city, and a much needed break from all of the stress I had been feeling from nearing the big semester exams. :)

I am looking forward to checking out some of Berlin's very own Christmas markets this weekend. Gluhwein, here we come!

## Saturday, December 3, 2011

### Math in Psychology

Since my Kindle got fixed mid last week, I've been reading a rather delightful book called

I am also surprised by how mathematical the author is and how easily he ties abstract math ideas into concrete experiments. For this, I highly recommend the book to math teachers. For example, the author talks about how if you get one person to look at a jar of pennies to guess at how much money is inside, that person might over- or under- estimate by a lot. But then, if you repeat the experiment with a large number of people, their average errors will actually approach zero (in the absence of a systematic bias), and therefore if you average all of their guesses, that average is actually going to be quite accurate. This is a logical idea that kids can grasp, and it's a nice extension of the absolute-value error concept. By the same token, he ties this in general to public opinions. If you survey a large enough sample population on a certain issue, in the absence of a systematic bias, the average of their answers will represent the truth.

Another issue that the author addresses is basic numeracy when reading current event reports or statistics in the media. He illustrates with a simple picking-colored-balls-out-of-a-box example why, with small sample groups, we end up with more extreme values more often. And then he extends this to why when you poll different counties for health information, it's easy to see rural counties with more extreme health statistics. Again, it's not impressive math, but the ease with which he ties math to something real is delightful.

And, as an aside, try to answer this question:

"How many animals of each type did Moses bring into the ark?"*

If you're like me and (the author so says) most people, you let your mental image of the ark prevent you from noticing that Moses is the wrong biblical character here in this context. Our mind has the tendency to smooth over the little inconsistencies using preconstructed expectations, in order to make its job easier. And that's both advantageous and troublesome, depending on the context.

Anyway, so far, I thoroughly enjoy the book! :)

Addendum 12/07/11: I've reached a part of the book where the author talks about how we let our stereotypes affect our judgment of the likelihood of certain combined events. For example, after being exposed to a description to a liberal woman, people -- even those who are mathematically inclined -- would rank the probability of her being a "feminist banker" to be more likely than her being a banker, even though any added details should diminish the overall probability! --What an interesting intersection of math and psychology!!

*Thinking, Fast and Slow*. I've read books like this one before, on the clever psychological experiments that people have managed to design over the years and what they reveal about the human mind. But this book, in particular, pulls together a lot of interesting bits that I've either read or heard over the years and organizes them into one cohesive and surprisingly elegant theory. (I won't spoil the book for you here, but it's worth looking into. The theory is elegant and seemingly simple, but the details are very interesting and not always so predictable or obvious.)I am also surprised by how mathematical the author is and how easily he ties abstract math ideas into concrete experiments. For this, I highly recommend the book to math teachers. For example, the author talks about how if you get one person to look at a jar of pennies to guess at how much money is inside, that person might over- or under- estimate by a lot. But then, if you repeat the experiment with a large number of people, their average errors will actually approach zero (in the absence of a systematic bias), and therefore if you average all of their guesses, that average is actually going to be quite accurate. This is a logical idea that kids can grasp, and it's a nice extension of the absolute-value error concept. By the same token, he ties this in general to public opinions. If you survey a large enough sample population on a certain issue, in the absence of a systematic bias, the average of their answers will represent the truth.

Another issue that the author addresses is basic numeracy when reading current event reports or statistics in the media. He illustrates with a simple picking-colored-balls-out-of-a-box example why, with small sample groups, we end up with more extreme values more often. And then he extends this to why when you poll different counties for health information, it's easy to see rural counties with more extreme health statistics. Again, it's not impressive math, but the ease with which he ties math to something real is delightful.

And, as an aside, try to answer this question:

"How many animals of each type did Moses bring into the ark?"*

If you're like me and (the author so says) most people, you let your mental image of the ark prevent you from noticing that Moses is the wrong biblical character here in this context. Our mind has the tendency to smooth over the little inconsistencies using preconstructed expectations, in order to make its job easier. And that's both advantageous and troublesome, depending on the context.

Anyway, so far, I thoroughly enjoy the book! :)

Addendum 12/07/11: I've reached a part of the book where the author talks about how we let our stereotypes affect our judgment of the likelihood of certain combined events. For example, after being exposed to a description to a liberal woman, people -- even those who are mathematically inclined -- would rank the probability of her being a "feminist banker" to be more likely than her being a banker, even though any added details should diminish the overall probability! --What an interesting intersection of math and psychology!!

## Tuesday, November 29, 2011

### My Favorite Student

My favorite student (this year) is someone who is not particularly quick at picking up math concepts. After she got the first test back, she cried. She came to see me everyday during lunch to go over a section of the test at a time. We did the corrections together, and then when she went home, she re-did the problems at home that we had worked on together earlier that day. It took us several days just to go through the entire test. And then, because she had worked so hard, I offered to give her a re-test not for grade, but just to see if she now knew the material. On the re-test, she made no conceptual mistakes, and only some arithmetic errors. When she went home and did corrections again (this time, unprompted), she thought the corrections were very easy.

This kid works so hard that when I announced that we were having a big exam in December, she came to me within a couple of days to start working on the topics that will be included. We sat down and made flashcards for strategies while approaching problems, and we did some practice problems and put them also on the flashcards. She has been reviewing the flashcards since at home, and she says that they are very helpful as a starting point to solving problems.

This favorite student of mine came to me yesterday after school to ask me to teach her long division, because all of her friends are able to divide 3 digit numbers by 2 digit numbers and she couldn't remember how to divide. I was running a fever when she came by after school yesterday and my body ached all over and I longed to go home, but hearing her question warmed my heart. Big time. She is my favorite kid because she isn't embarrassed to get help, and I know that some day she will master all of the skills that she lacks, if she can keep up that amazing spirit of hers.

That's my favorite type of student. Over the years, I have seen other students like her grow into excellent math students, who can connect the dots faster than anyone else and to work quickly through complex scenarios of problems. As a teacher, may I always remember to appreciate when a kid has the willingness for hard work, regardless of what they currently achieve.

This kid works so hard that when I announced that we were having a big exam in December, she came to me within a couple of days to start working on the topics that will be included. We sat down and made flashcards for strategies while approaching problems, and we did some practice problems and put them also on the flashcards. She has been reviewing the flashcards since at home, and she says that they are very helpful as a starting point to solving problems.

This favorite student of mine came to me yesterday after school to ask me to teach her long division, because all of her friends are able to divide 3 digit numbers by 2 digit numbers and she couldn't remember how to divide. I was running a fever when she came by after school yesterday and my body ached all over and I longed to go home, but hearing her question warmed my heart. Big time. She is my favorite kid because she isn't embarrassed to get help, and I know that some day she will master all of the skills that she lacks, if she can keep up that amazing spirit of hers.

That's my favorite type of student. Over the years, I have seen other students like her grow into excellent math students, who can connect the dots faster than anyone else and to work quickly through complex scenarios of problems. As a teacher, may I always remember to appreciate when a kid has the willingness for hard work, regardless of what they currently achieve.

## Saturday, November 26, 2011

### Introducing Limits and Derivatives

I know you non-IB people probably cannot fathom this, but Calculus is only one of many units in the IB curriculum. I am introducing it to my kids now, and it's the last big topic I plan on teaching before we start to do mixed review in the spring time.

The reason why Calculus is condensed into a single unit in the IB makes sense to me, actually. Although there are so many applications of Calculus, I would be just as happy if a kid can walk away from Calculus knowing the big ideas of limits, differentiation, and integration, and to be able to do basic differentiation/integration of polynomials by hand without a formula sheet. Everything else, I'm OK with them relying on a formula sheet in order to remember the mechanics. So, it's very do-able as a single unit and to still develop the relevant concepts as a class.

To that end, I have already introduced instantaneous rates vs. average rates. Then, after that I wanted to pull in the idea of limits, which the kids had already seen a little bit of in the context of geometric series. So, I gave the kids this worksheet, and as they worked through it, we set up a huge grid of comparison charts on the board to go over when a "forbidden x value" inside a rational function will be a vertical asymptote, versus a "hole".... and to highlight the idea that a limit is what happens to the theoretical output as you approach those "impossible" x values. We are not done with the worksheet yet, but my hope is that by the start of the next class, the kids will fully grasp the idea that a function with a "hole" somewhere (as opposed to a vertical asymptote) can still have a limit at that breaking point, and many functions also have limits at extreme values of x. That (along with the previous instantaneous rate intro) will prime us for going into talking about the mechanics of differentiation in this next intro to differentiation lesson. One of the things I want to immediately tie into differentiation is that you can check your sensibility of your answers using a graph. I will also right away tie differentiation to the algebraic meaning of turning points. This way, they are immediately exposed to the key concepts in an integrated manner before we do any further practice.

Thoughts?? It's my first time teaching Calculus, so I'm still muddling my way through it while doing my best to sequence the concepts clearly. Your feedback is welcome!

The reason why Calculus is condensed into a single unit in the IB makes sense to me, actually. Although there are so many applications of Calculus, I would be just as happy if a kid can walk away from Calculus knowing the big ideas of limits, differentiation, and integration, and to be able to do basic differentiation/integration of polynomials by hand without a formula sheet. Everything else, I'm OK with them relying on a formula sheet in order to remember the mechanics. So, it's very do-able as a single unit and to still develop the relevant concepts as a class.

To that end, I have already introduced instantaneous rates vs. average rates. Then, after that I wanted to pull in the idea of limits, which the kids had already seen a little bit of in the context of geometric series. So, I gave the kids this worksheet, and as they worked through it, we set up a huge grid of comparison charts on the board to go over when a "forbidden x value" inside a rational function will be a vertical asymptote, versus a "hole".... and to highlight the idea that a limit is what happens to the theoretical output as you approach those "impossible" x values. We are not done with the worksheet yet, but my hope is that by the start of the next class, the kids will fully grasp the idea that a function with a "hole" somewhere (as opposed to a vertical asymptote) can still have a limit at that breaking point, and many functions also have limits at extreme values of x. That (along with the previous instantaneous rate intro) will prime us for going into talking about the mechanics of differentiation in this next intro to differentiation lesson. One of the things I want to immediately tie into differentiation is that you can check your sensibility of your answers using a graph. I will also right away tie differentiation to the algebraic meaning of turning points. This way, they are immediately exposed to the key concepts in an integrated manner before we do any further practice.

Thoughts?? It's my first time teaching Calculus, so I'm still muddling my way through it while doing my best to sequence the concepts clearly. Your feedback is welcome!

## Friday, November 25, 2011

### IB: Flash Cards for Concept Retrieval

I admit, I've spent a fair amount of time thinking about what I can do to help my 11th and 12th graders excel on the IB exam. Maybe that makes me a bad math teacher to be thinking about the test so much, but I think my instruction can still be independent of the test, but I have to keep the test in mind as our collective end goal, in order not to veer too far in my attempt to bring good mathematics into the curriculum.

One of my latest conclusions is that doing well on a math test requires two skills: 1. that you can read a problem and retrieve the correct key concepts and/or procedures. 2. that you can apply the procedure without mechanical (algebraic) issues.

(Of course, being a good math student requires much more than that. I want my kids to know where procedures and formulas come from, so that when they apply a formula, it's not just blind application and they don't falter just because one small thing has been changed. I also want them to see the layering of the building blocks and be able to guess at the next logical step. I also want them to see where math is applied, to understand how to model real-world situations using math, and to ask insightful questions. But, ultimately, when they sit down in front of that big math test, I still want them to kick ass and thereby gain the confidence they need in order to do all those other things!!)

To help them with skill #2 (applying the procedure), I've decided that spiraling is very important. Every test I give in class is made up of old IB test problems, and include not just the current topic but also old topics we have already studied. The more we hold kids accountable for skills that we have already learned, the more they will retain them over time. To help them with skill #1 (recognition of cues and retrieval of relevant concept), I've recently introduced to my students the idea of making flashcards for math. It's very simple:

On the front of the card, write down some clues in the problem. (Such as, "This quadratic equation has only one distinct root.") On the back of the card, write down the concept or key math skill that the problem is looking for you to apply. (Such as the discriminant.)

As the kids work on practice problems, they should learn to recognize those major clues in the problems and make flashcards. Each time they sit down to do practice problems, they should first quickly run through their flashcards to re-inforce the cue-concept connections, and then try to approach all problems without looking back at the flashcards. They can also make a few clear example problems on flashcards. (Question on one side, work and answer on the other.) This way:

1. They can study quickly without re-doing the same problems a gazillion times.

2. It makes explicit the cue-procedure connections that "strong math students" implicitly rely on.

3. Over time, if they keep reviewing the flashcards, they will eventually not need them anymore and will hopefully internalize those connections.

4. Particularly anxious or slow test-takers will benefit from having clearly retrievable facts stored on their minds during test time.

5. It's a great way for them to accumulate invaluable study material/improve their study habits over time!

What do you think? Have you tried something like this? I am recommending it to my strugglers in lower grades as well, as a way to reinforce their test-preparation skills. I am helping my entire Grade 7 class build these flashcards this semester, as we review for the December "big exam." I'll survey them afterwards to see how many of them actually used them before doing the practice problems (like I recommended), and how many of them think it actually helped.

PS. Happy Thanksgiving!

One of my latest conclusions is that doing well on a math test requires two skills: 1. that you can read a problem and retrieve the correct key concepts and/or procedures. 2. that you can apply the procedure without mechanical (algebraic) issues.

(Of course, being a good math student requires much more than that. I want my kids to know where procedures and formulas come from, so that when they apply a formula, it's not just blind application and they don't falter just because one small thing has been changed. I also want them to see the layering of the building blocks and be able to guess at the next logical step. I also want them to see where math is applied, to understand how to model real-world situations using math, and to ask insightful questions. But, ultimately, when they sit down in front of that big math test, I still want them to kick ass and thereby gain the confidence they need in order to do all those other things!!)

To help them with skill #2 (applying the procedure), I've decided that spiraling is very important. Every test I give in class is made up of old IB test problems, and include not just the current topic but also old topics we have already studied. The more we hold kids accountable for skills that we have already learned, the more they will retain them over time. To help them with skill #1 (recognition of cues and retrieval of relevant concept), I've recently introduced to my students the idea of making flashcards for math. It's very simple:

On the front of the card, write down some clues in the problem. (Such as, "This quadratic equation has only one distinct root.") On the back of the card, write down the concept or key math skill that the problem is looking for you to apply. (Such as the discriminant.)

As the kids work on practice problems, they should learn to recognize those major clues in the problems and make flashcards. Each time they sit down to do practice problems, they should first quickly run through their flashcards to re-inforce the cue-concept connections, and then try to approach all problems without looking back at the flashcards. They can also make a few clear example problems on flashcards. (Question on one side, work and answer on the other.) This way:

1. They can study quickly without re-doing the same problems a gazillion times.

2. It makes explicit the cue-procedure connections that "strong math students" implicitly rely on.

3. Over time, if they keep reviewing the flashcards, they will eventually not need them anymore and will hopefully internalize those connections.

4. Particularly anxious or slow test-takers will benefit from having clearly retrievable facts stored on their minds during test time.

5. It's a great way for them to accumulate invaluable study material/improve their study habits over time!

What do you think? Have you tried something like this? I am recommending it to my strugglers in lower grades as well, as a way to reinforce their test-preparation skills. I am helping my entire Grade 7 class build these flashcards this semester, as we review for the December "big exam." I'll survey them afterwards to see how many of them actually used them before doing the practice problems (like I recommended), and how many of them think it actually helped.

PS. Happy Thanksgiving!

## Thursday, November 17, 2011

### (Semi-Recent) Trip to Poland

I have been so bad about doing updates about our journeying! Part of it is because on our last trip, we didn't really take any pictures. Geoff had imagined that our camera would charge itself via the USB connector, and when that turned out to be not true, we didn't really want to fork out 50 Euros to buy a charger in Krakow. So, instead, Geoff took some pictures on his dinosauric iPhone 3GS, which made everything seem gray and romantic and more or less unrecognizable afterwards. :) (We even tried to take photos in the dark while "catching" someone else's flash. Sad times.)

Anyway, this particular trip happened in early October, during our school's fall break. We were too last-minute to book good airfare deals for going away, so we decided to rent a car to drive to Poland. FYI: Renting a car in Germany as a foreigner is not the easiest. Your non-European license is only valid within the first 6 months of moving here, and even then you have to rent from big companies that carry the international insurance -- which naturally means that you pay a bit more. Also FYI: Parking in Krakow is expensive and inconvenient. Supposedly it's not safe to just leave your German-plated cars on the street, because Polish people like to break in to German cars. So, if ever we intend on repeating this trip, we will be taking a local German train to hop over the border to Poland, and then we will take another local Polish train to Krakow.

Anyway, back to Krakow. You know, Krakow is poetic and beautiful, full of sad history. I didn't really know much about Poland before this trip, but I was intrigued to learn that it's really a country that has been continuously dominated by other empires and countries over the years, from Russian Empire to Prussian Empire to dominance by Austria, followed by a brief period of independence before it got dominated again by Nazi Germany. Also, we don't really think about this, but the Polish people were the second biggest group of victims, I think, in the Holocaust. Auschwitz was built originally to house the political prisoners from Poland, whom the Germans wanted to keep quiet and away from sight. Lots of them were murdered after swift "trials" at Auschwitz. The Polish people were not encouraged to attend schools during the war (and various strategies were used to prevent the intellectuals from gathering and studying), because Nazi Germany wanted to maintain a stronghold over the land and its people.

Anyway, when we went to Krakow, we visited Auschwitz and it was a very intense experience. The tour guides were superb and took their jobs very seriously, and it was very informative but obviously also very sad. When the Nazis evacuated the camp at the end of the war, they destroyed most of the evidence that it had ever been used as an extermination camp. There is only one small gas chamber that still remains, and we were able to go inside. It was very intense. The electrical barbed wire fences are still up and you can walk around and see the gutters that surrounded the camp and see the entrance to the camp and the train tracks that lead up to the entrance to drop prisoners off for sorting. Some of the sleeping quarters can still be seen, and even when we went in October, it was cold. One can only imagine what it was like in the dead of winter, when you are wearing next to nothing. Moreover, there are displays of all of the personal belongings, shoes, and even hair of the victims. The hair was haunting, because they weaved them into everyday products like rugs and resold them. It was deeply sad, and even though there are people who would argue that that place should not allow so many tourists to go per day, I cannot help but think that it is absolutely necessary for them to educate as many people as possible about the horrors that occurred there.

Besides Auschwitz, we also went by foot on our own and visited the old Jewish ghetto in Krakow, like featured in Schindler's movie. Much of the area still teems with old buildings, like they could have been around during the war. We visited an amazing and extensive museum on Poland's role and perspective in WWII, located inside Schindler's old factory. We also went into the small pharmacy that is famous for its Polish owner who refused to move out of the Jewish ghetto during the war, and who helped to hide Jews and to sneak in free medicine for people inside the ghetto.

We also saw the medieval Center of the city. It was beautiful, and filled with cathedrals, happening bars, and yummy restaurants. Our favorite was Marmolada, which is one of a mini chain of 4 restaurants that are all supposed to be different and delicious. This one was fantastic, from the wine to the food to the service to the price. It's also a 1-minute walk from the city center, right off of one of the little alleys.

Lastly, we visited a cheesy tourist trap of a salt mine in Krakow. You'd never think it, but there is a beautiful chapel that is carved entirely out of salt in the depths of a salt mine, made by 3 professional miners who did this as a hobby over the course of

Last note is that the Polish roads are supposed to be horrific -- even my coworker's wife, who is herself Polish and native from Warsaw, said so. The funny thing is that on our way to Poland, while we thought we were still on the German side of the border, the highway started to get really bumpy. You couldn't really see any difference, except the car was going up and down and up and down as though the road was filled with potholes. We were joking that perhaps we were already in Poland, and sure enough, in two minutes we started to see road signs marked in Polish letters...

The next big thing I'm looking forward to: German CHRISTMAS MARKETS!!!!

Anyway, this particular trip happened in early October, during our school's fall break. We were too last-minute to book good airfare deals for going away, so we decided to rent a car to drive to Poland. FYI: Renting a car in Germany as a foreigner is not the easiest. Your non-European license is only valid within the first 6 months of moving here, and even then you have to rent from big companies that carry the international insurance -- which naturally means that you pay a bit more. Also FYI: Parking in Krakow is expensive and inconvenient. Supposedly it's not safe to just leave your German-plated cars on the street, because Polish people like to break in to German cars. So, if ever we intend on repeating this trip, we will be taking a local German train to hop over the border to Poland, and then we will take another local Polish train to Krakow.

Anyway, back to Krakow. You know, Krakow is poetic and beautiful, full of sad history. I didn't really know much about Poland before this trip, but I was intrigued to learn that it's really a country that has been continuously dominated by other empires and countries over the years, from Russian Empire to Prussian Empire to dominance by Austria, followed by a brief period of independence before it got dominated again by Nazi Germany. Also, we don't really think about this, but the Polish people were the second biggest group of victims, I think, in the Holocaust. Auschwitz was built originally to house the political prisoners from Poland, whom the Germans wanted to keep quiet and away from sight. Lots of them were murdered after swift "trials" at Auschwitz. The Polish people were not encouraged to attend schools during the war (and various strategies were used to prevent the intellectuals from gathering and studying), because Nazi Germany wanted to maintain a stronghold over the land and its people.

Anyway, when we went to Krakow, we visited Auschwitz and it was a very intense experience. The tour guides were superb and took their jobs very seriously, and it was very informative but obviously also very sad. When the Nazis evacuated the camp at the end of the war, they destroyed most of the evidence that it had ever been used as an extermination camp. There is only one small gas chamber that still remains, and we were able to go inside. It was very intense. The electrical barbed wire fences are still up and you can walk around and see the gutters that surrounded the camp and see the entrance to the camp and the train tracks that lead up to the entrance to drop prisoners off for sorting. Some of the sleeping quarters can still be seen, and even when we went in October, it was cold. One can only imagine what it was like in the dead of winter, when you are wearing next to nothing. Moreover, there are displays of all of the personal belongings, shoes, and even hair of the victims. The hair was haunting, because they weaved them into everyday products like rugs and resold them. It was deeply sad, and even though there are people who would argue that that place should not allow so many tourists to go per day, I cannot help but think that it is absolutely necessary for them to educate as many people as possible about the horrors that occurred there.

Besides Auschwitz, we also went by foot on our own and visited the old Jewish ghetto in Krakow, like featured in Schindler's movie. Much of the area still teems with old buildings, like they could have been around during the war. We visited an amazing and extensive museum on Poland's role and perspective in WWII, located inside Schindler's old factory. We also went into the small pharmacy that is famous for its Polish owner who refused to move out of the Jewish ghetto during the war, and who helped to hide Jews and to sneak in free medicine for people inside the ghetto.

We also saw the medieval Center of the city. It was beautiful, and filled with cathedrals, happening bars, and yummy restaurants. Our favorite was Marmolada, which is one of a mini chain of 4 restaurants that are all supposed to be different and delicious. This one was fantastic, from the wine to the food to the service to the price. It's also a 1-minute walk from the city center, right off of one of the little alleys.

Lastly, we visited a cheesy tourist trap of a salt mine in Krakow. You'd never think it, but there is a beautiful chapel that is carved entirely out of salt in the depths of a salt mine, made by 3 professional miners who did this as a hobby over the course of

*seventy*years. They made it complete with statues and chandeliers and -- you would never believe this -- a breath-taking statue of The Last Supper, carved into the walls. Talk about a personal hobby project!Last note is that the Polish roads are supposed to be horrific -- even my coworker's wife, who is herself Polish and native from Warsaw, said so. The funny thing is that on our way to Poland, while we thought we were still on the German side of the border, the highway started to get really bumpy. You couldn't really see any difference, except the car was going up and down and up and down as though the road was filled with potholes. We were joking that perhaps we were already in Poland, and sure enough, in two minutes we started to see road signs marked in Polish letters...

The next big thing I'm looking forward to: German CHRISTMAS MARKETS!!!!

### Scatterplots and Patterns

I had a pretty great experience with my Grade 12's today using these made-up data sets to discuss with kids how to find multiple modeling functions and then how to narrow them down to one using either asymptotic behavior or the meaning of that type of function. I used it in conjunction with the function types handout that I had given them before, and they were able to ask me some good questions about the differences between functions and how do you know which one is the most appropriate, when a few look very similar?

Even though I wish that they could have come up with the handout information themselves about various function types, it was very valuable to hear them ask those relevant questions, and some of them (the ones who had brought laptops from home) were able to play around with plotting points, setting up sliders for parameter values, and then estimating best fit equations all in GeoGebra. Pretty great, for one day! (The rest of the kids plotted points in their graphing calcs and practiced looking through the functions to find possibilities.)

In the end, I passed out our official IB task and they looked over it and nobody panicked. I think that is a good sign.

Even though I wish that they could have come up with the handout information themselves about various function types, it was very valuable to hear them ask those relevant questions, and some of them (the ones who had brought laptops from home) were able to play around with plotting points, setting up sliders for parameter values, and then estimating best fit equations all in GeoGebra. Pretty great, for one day! (The rest of the kids plotted points in their graphing calcs and practiced looking through the functions to find possibilities.)

In the end, I passed out our official IB task and they looked over it and nobody panicked. I think that is a good sign.

## Wednesday, November 16, 2011

### An Unexpected Math Encounter

I forgot to write about this:

I went to a cabaret / dance thing last weekend. (It was more of a dance thing than a cabaret thing, which was totally cool except that I had thought it'd be more of a cabaret thing and wore heeled boots and danced with my boots on a slippery wooden floor all night.) At the entrance, there was a sign:

"Pay 9,50 Euros OR toss a die to try your luck. If you roll 1, you pay 7 Euros to get in. If you roll 2, you pay 8 Euros to get in. If you roll 3, you pay 9 Euros... etc. Up to a possible 12 Euros for entrance."

That made the math teacher in me happy to know that somebody paid attention in their math class. :)

I went to a cabaret / dance thing last weekend. (It was more of a dance thing than a cabaret thing, which was totally cool except that I had thought it'd be more of a cabaret thing and wore heeled boots and danced with my boots on a slippery wooden floor all night.) At the entrance, there was a sign:

"Pay 9,50 Euros OR toss a die to try your luck. If you roll 1, you pay 7 Euros to get in. If you roll 2, you pay 8 Euros to get in. If you roll 3, you pay 9 Euros... etc. Up to a possible 12 Euros for entrance."

That made the math teacher in me happy to know that somebody paid attention in their math class. :)

## Tuesday, November 15, 2011

### Fun with Ratios and Thinking About Functions

My middle-schoolers are coming along nicely in their conceptual understanding. I would like to share one project that my 7th-graders are working on currently. It's an old ratios assignment from when I taught middle-schoolers way back when, but still one that gets kids all riled up and excited, apparently.

First, I want to share a neat little trick with teaching ratios. My first and second years of teaching, I had a lot of trouble getting kids to "see" how to convert part-to-part ratios (such as girls:boys = 3:2) to part-to-whole ratios (such as girls:total = 3:5). I used all these hands-on manipulatives and guided exercises, but to no avail! My third year, I finally figured it out. All this time I had been trying to

So, nowadays I teach ratios problems using tables of values. The tables always start with zeros, because I want to continuously reinforce the connection between direct variations and ratios problems.

I've noticed that this way, no matter if the kids are starting with a part-to-part or part-to-whole ratio as given in the problem, they can always manage to find the other missing parts. And also, they can

Anyway, following some practice of the table method, I finally introduced cross multiplication. The kids were

In any case, back to the ratios project. The kids are given the specs of an original shape and partial specs of a new shape. They need to use their understanding of part-to-part and part-to-whole ratios in order to find the scale factor and to scale all sides appropriately. In the end, they measure and cut out both versions (initial and final), and in groups of 3 write out an explanation of the entire process -- which everyone has to agree upon -- and create an explanatory poster.

I like this project, because it allows me to differentiate easily. Some groups have scale factors that are whole numbers, and others have scale factors that are unit fractions (ie. 1/n where n is an integer), while others have scale factors that are non-unit fractions. Since they're not told what the scale factors are, finding it can be a bit tricky when it is a non-unit fraction.

Anyway, so far, it's going swimmingly! The kids have almost all finished measuring and cutting out their two shapes and are working on the final written explanations. If that sounds like fun to you, here are the project prompts I used this year (excluding the instructions for the posters) and the warmup exercise I used to introduce the idea of geometric scaling.

Love projects!

---------------------

For Grade 8, we are wrapping up two application assignments of the functions we have been learning. I did Dan's cup-stacking activity with them and they enjoyed it immensely -- 3 or so groups came within 1 cup of the final result! I also did a traditional profit and revenue assignment with them. It was the first time I felt like we were getting into

Today, I taught them to graph linear functions. But I did so by tying it firmly into the meaning of slope and y-intercept inside a word problem, so that every time they look at an equation like y=3x/5 + 2 they will think to themselves that 2 is the starting value (such as # vacation days at the beginning of the year) and 3/5 is a rate -- 5 describing "how often" it happens and 3 describing "what actually happens." (For example, every 5 months you gain 3 vacation days.) It was the first time I did graphing by connecting it to my Mad Libbs worksheet on interpreting linear functions, and it worked really beautifully!!! Not a single kid was lost in the graphing of equations, when they were just thinking about the meanings of the rate and the starting value and putting that into graphical form. Kids were even figuring out for themselves the idea of rise and run in a fractional slope, even though they

:) Yay to experimenting. Small changes, big impact!

Addendum 11/16/11: Today I gave my kids a bunch of graphs for them to write equations for. Using just their understanding of the meanings of parameters inside the slope-intercept equation as related to word problems, they were able to

First, I want to share a neat little trick with teaching ratios. My first and second years of teaching, I had a lot of trouble getting kids to "see" how to convert part-to-part ratios (such as girls:boys = 3:2) to part-to-whole ratios (such as girls:total = 3:5). I used all these hands-on manipulatives and guided exercises, but to no avail! My third year, I finally figured it out. All this time I had been trying to

*explain*ratios, when instead I should have been letting kids*observe*ratios, because ratios are an intuitive concept!So, nowadays I teach ratios problems using tables of values. The tables always start with zeros, because I want to continuously reinforce the connection between direct variations and ratios problems.

I've noticed that this way, no matter if the kids are starting with a part-to-part or part-to-whole ratio as given in the problem, they can always manage to find the other missing parts. And also, they can

*see*how every part (or column) scales equally, thereby leading naturally to the idea of a scale factor. When the problem then asks, "How many boys would there be, if there are 966 girls?" they can figure out easily which column 966 goes into, and then extend (fairly easily, with a bit of practice) the idea of scale factors to find the other column values. --And this is with my Grade 7s, who had never before seen any ratios problems! In the past, when I used the same method with my 8th-graders, I didn't have to teach them anything at all; they could just observe the pattern and figure out the scale factor shortcut when they felt tired of extending the table tediously.Anyway, following some practice of the table method, I finally introduced cross multiplication. The kids were

*mad*at me; they thought that setting up and solving proportions was way more difficult than making the tables! I made them practice both methods so that in the future, should a teacher require cross multiplication and/or the problem involves decimals, it'd still be in their arsenal. But, I am happy that their conceptual understanding is strong enough to want to replace the cross multiplication.In any case, back to the ratios project. The kids are given the specs of an original shape and partial specs of a new shape. They need to use their understanding of part-to-part and part-to-whole ratios in order to find the scale factor and to scale all sides appropriately. In the end, they measure and cut out both versions (initial and final), and in groups of 3 write out an explanation of the entire process -- which everyone has to agree upon -- and create an explanatory poster.

I like this project, because it allows me to differentiate easily. Some groups have scale factors that are whole numbers, and others have scale factors that are unit fractions (ie. 1/n where n is an integer), while others have scale factors that are non-unit fractions. Since they're not told what the scale factors are, finding it can be a bit tricky when it is a non-unit fraction.

Anyway, so far, it's going swimmingly! The kids have almost all finished measuring and cutting out their two shapes and are working on the final written explanations. If that sounds like fun to you, here are the project prompts I used this year (excluding the instructions for the posters) and the warmup exercise I used to introduce the idea of geometric scaling.

Love projects!

---------------------

For Grade 8, we are wrapping up two application assignments of the functions we have been learning. I did Dan's cup-stacking activity with them and they enjoyed it immensely -- 3 or so groups came within 1 cup of the final result! I also did a traditional profit and revenue assignment with them. It was the first time I felt like we were getting into

*complicated*application, and although it surely challenged them conceptually, I could see that some of them, at least, enjoyed the discussions about why it makes sense that profit and revenue are both shaped like a parabola.Today, I taught them to graph linear functions. But I did so by tying it firmly into the meaning of slope and y-intercept inside a word problem, so that every time they look at an equation like y=3x/5 + 2 they will think to themselves that 2 is the starting value (such as # vacation days at the beginning of the year) and 3/5 is a rate -- 5 describing "how often" it happens and 3 describing "what actually happens." (For example, every 5 months you gain 3 vacation days.) It was the first time I did graphing by connecting it to my Mad Libbs worksheet on interpreting linear functions, and it worked really beautifully!!! Not a single kid was lost in the graphing of equations, when they were just thinking about the meanings of the rate and the starting value and putting that into graphical form. Kids were even figuring out for themselves the idea of rise and run in a fractional slope, even though they

__had never learned this before__!!:) Yay to experimenting. Small changes, big impact!

Addendum 11/16/11: Today I gave my kids a bunch of graphs for them to write equations for. Using just their understanding of the meanings of parameters inside the slope-intercept equation as related to word problems, they were able to

*easily*look at graphs and to write those equations without messing up! This has been the smoothest teaching of writing equations EVER.## Monday, November 7, 2011

### On Being Truthful

Today, when I was grabbing something out of my closet, my 7th-graders came into my room to get ready to start our class. One of the last kids to come in did not see me behind the open closet door, and she cheered loudly when she thought that I was absent. She was immediately embarrassed when she realized that I was present (and standing quite close to her), and her friends all laughed.

Innocent mistake. I would have just let it go, except when class started a few moments later and a few kids were still giggling about it, I steadied the class and I asked her if she had something to tell me about how she felt about my class. She said no, that [they] like [me]. And I added softly, "Because I would never cheer even when a kid who's really difficult misses my class. What you said... hurt my feelings."

I wasn't mean about it, even though the kid had embarrassed me (intentionally or not). I think that's something that I've learned to do with grace in the last few years. During the class, the kid tried to sneak me a couple of quick apologies, and then after class she hung around and explained that she was sorry that she had been offensive, that she hadn't meant for it to be hurtful but that she was hoping for some free time today to work on something during class.

I appreciated her for her apology. As a teacher of middle-schoolers, I understand that I cannot take these things personally. But, I think it was the first time I embraced the truth and frankly told a kid that what they said was hurtful to me. And hopefully, the next time it will make her think twice about being inadvertently hurtful to someone else.

I am trying to model for my kids the kind of adult I hope they would grow up to be. That's the most difficult task of all, because I'm not sure I am that person all the time. I think this year I am closer than the years before, because I am much more aware of it. But I've still got ways to go.

Innocent mistake. I would have just let it go, except when class started a few moments later and a few kids were still giggling about it, I steadied the class and I asked her if she had something to tell me about how she felt about my class. She said no, that [they] like [me]. And I added softly, "Because I would never cheer even when a kid who's really difficult misses my class. What you said... hurt my feelings."

I wasn't mean about it, even though the kid had embarrassed me (intentionally or not). I think that's something that I've learned to do with grace in the last few years. During the class, the kid tried to sneak me a couple of quick apologies, and then after class she hung around and explained that she was sorry that she had been offensive, that she hadn't meant for it to be hurtful but that she was hoping for some free time today to work on something during class.

I appreciated her for her apology. As a teacher of middle-schoolers, I understand that I cannot take these things personally. But, I think it was the first time I embraced the truth and frankly told a kid that what they said was hurtful to me. And hopefully, the next time it will make her think twice about being inadvertently hurtful to someone else.

I am trying to model for my kids the kind of adult I hope they would grow up to be. That's the most difficult task of all, because I'm not sure I am that person all the time. I think this year I am closer than the years before, because I am much more aware of it. But I've still got ways to go.

### Intro to IB Type 2 Task

This entry is for you IB math(s) teachers out there. I pulled together an introduction slideshow for the Type II task to make it super clear for kids what the mathematical modeling process needs to look like.

Here is the Prezi, and if you wish to make it fullscreen, just click on the "More" button in the lower right corner of the Prezi presentation to see that option. The plan is to walk the kids through the various steps of modeling, and then to jump right into helping them install Autograph (30-day trial version) on their laptops, and then to play around with the program a bit in class, and then to go over the function basics again, and then to take them through an easy sample task.

This is, of course, meant to be used as a follow-up discussion assuming that your kids have already seen these awesome demo videos Part 1 and Part 2 that I had found on the web.

PS. Also, to help kids keep track of all the function types and features, I made this rather comprehensive handout that you are welcome to take. I know, it's not perfect. I wish kids could just all look at data sets and figure out on their own what functions make sense based on their own awesome mathematical understanding, but I don't think I have enough time with my 12th-graders to make that happen overnight, so I'm giving them this as a bit of help, and I will go over it with them as a class to make sure that they are taking notes and trying to understand the important features of each graph. Better than them going on the internet and copying and pasting things they don't understand, no??

Here is the Prezi, and if you wish to make it fullscreen, just click on the "More" button in the lower right corner of the Prezi presentation to see that option. The plan is to walk the kids through the various steps of modeling, and then to jump right into helping them install Autograph (30-day trial version) on their laptops, and then to play around with the program a bit in class, and then to go over the function basics again, and then to take them through an easy sample task.

This is, of course, meant to be used as a follow-up discussion assuming that your kids have already seen these awesome demo videos Part 1 and Part 2 that I had found on the web.

PS. Also, to help kids keep track of all the function types and features, I made this rather comprehensive handout that you are welcome to take. I know, it's not perfect. I wish kids could just all look at data sets and figure out on their own what functions make sense based on their own awesome mathematical understanding, but I don't think I have enough time with my 12th-graders to make that happen overnight, so I'm giving them this as a bit of help, and I will go over it with them as a class to make sure that they are taking notes and trying to understand the important features of each graph. Better than them going on the internet and copying and pasting things they don't understand, no??

## Wednesday, November 2, 2011

### Responding to Student Needs

I've been slowly reading

Here is a quote that I liked that helps me view my current experience through a different lens:

I think this really nicely outlines all of the things that are on my mind constantly, as I struggle to grow into being a "good" teacher for my kids.

* My Grade 7's have a few lagging performers who haven't yet figured out that math is important and they need to come see me outside of class for help, so (in response) I weave all of their review of past topics that they're still weak with into our normal class. Every test is cumulative and hits every past topic, and we review accordingly beforehand. I also send daily emails home to let parents know when their kids miss an assignment. And, once every few weeks we play a game that reviews an old skill that I want them to fully master.

* My Grade 8's are a bimodal group, so I need to balance between keeping the really advanced kids challenged and giving the strugglers time to work on their factoring skills when the leading coefficient isn't 1. I do so by introducing every factoring tool possible (looking at graphs & using quadratic formula), and also giving the top-top kids extra graphing calculator assignments to work on independently.

* My Grade 9's are the lowest group in the grade, so there are kids in the group with serious language issues, others with no mathematical memory past the current day, and a few kids who are working very hard to bridge the gap of their learning. I need to provide tasks that are accessible to all kids and allow them to work at their individual paces, and I give them free reign to correcting/re-doing all old assignments as many times as they want. Frequent assessment with clear skills expectations is key to making sure all kids are given regular feedback, and I've met half of their parents already to discuss my concerns about the kids.

* My Grade 11's are a mixture of returning kids and new kids to the school. The returning kids are much more experienced with the first topics of the year, but to ensure that all kids have a fair shot at the IB exam, I'm teaching them all from scratch to make sure the new kids get proper training. Again, it's a balance act of approaching the topic from many angles to ensure that the returning kids are challenged and enriched and pushing the new kids along with some urgency to make sure they do a bit of extra homework to keep moving at the same pace as the others. In the longer run, I've worked out with a colleague that she can transfer the kids who need a bit more nurturing into my group, in exchange for moving kids who wouldn't mind/could handle moving at a much faster pace into hers. We're both happy with that arrangement and think it will maximize the benefit to all kids.

* My Grade 12's have a lot of gaps in their knowledge. They basically don't know/remember anything they are supposed to know from Year 11, and so I've been interleaving as much of old material as possible into current topics to help them review. I know that in the spring, I will have to do some very heavy-duty concept-mapping and explicit learning strategies like algorithm flashcards to get them familiar with the basic concepts of each topic, before we can start doing integrated review and test prep. I am prepared to make that happen and I have a plan for how to help them be successful. (It'll involve topic-based worksheets / individual tests when they're ready / moving on to individually review the next topic when both they and I agree they've mastered the basic ones, or repeating the cycle until they do improve.)

It's stretching me professionally to consider the various academic and emotional needs of my many groups and to attempt to address individual needs within each group. But, I am loving the challenge! :)

(...Now, if only someone could tell me if it's having any actual effect on the kids...)

*Lost at School*as part of a professional book club at work. Honestly, I don't think much of the book is ground-breaking stuff, but it's a nice common-sense teaching book to spark some common-sense teaching discussions. It provides a good focal point at work for discussions about what is important to all of us, rather than discussions about our individual concerns.Here is a quote that I liked that helps me view my current experience through a different lens:

Good teaching means being responsive to the hand you've been dealt.

It goes without saying that each group of kids is different. The task with each group is to get a handle on its collective strengths and limitations and work toward building a community where each member feels safe, respected, and valued. But that takes time and concentrated effort. It doesn't happen by itself. And it looks different every year. That's what it means to be responsive.

It also goes without saying that every individual in a classroom is different. [...] The ultimate challenge is to be responsive in both ways -- to the group and to the individuals in it -- simultaneously.

I think this really nicely outlines all of the things that are on my mind constantly, as I struggle to grow into being a "good" teacher for my kids.

* My Grade 7's have a few lagging performers who haven't yet figured out that math is important and they need to come see me outside of class for help, so (in response) I weave all of their review of past topics that they're still weak with into our normal class. Every test is cumulative and hits every past topic, and we review accordingly beforehand. I also send daily emails home to let parents know when their kids miss an assignment. And, once every few weeks we play a game that reviews an old skill that I want them to fully master.

* My Grade 8's are a bimodal group, so I need to balance between keeping the really advanced kids challenged and giving the strugglers time to work on their factoring skills when the leading coefficient isn't 1. I do so by introducing every factoring tool possible (looking at graphs & using quadratic formula), and also giving the top-top kids extra graphing calculator assignments to work on independently.

* My Grade 9's are the lowest group in the grade, so there are kids in the group with serious language issues, others with no mathematical memory past the current day, and a few kids who are working very hard to bridge the gap of their learning. I need to provide tasks that are accessible to all kids and allow them to work at their individual paces, and I give them free reign to correcting/re-doing all old assignments as many times as they want. Frequent assessment with clear skills expectations is key to making sure all kids are given regular feedback, and I've met half of their parents already to discuss my concerns about the kids.

* My Grade 11's are a mixture of returning kids and new kids to the school. The returning kids are much more experienced with the first topics of the year, but to ensure that all kids have a fair shot at the IB exam, I'm teaching them all from scratch to make sure the new kids get proper training. Again, it's a balance act of approaching the topic from many angles to ensure that the returning kids are challenged and enriched and pushing the new kids along with some urgency to make sure they do a bit of extra homework to keep moving at the same pace as the others. In the longer run, I've worked out with a colleague that she can transfer the kids who need a bit more nurturing into my group, in exchange for moving kids who wouldn't mind/could handle moving at a much faster pace into hers. We're both happy with that arrangement and think it will maximize the benefit to all kids.

* My Grade 12's have a lot of gaps in their knowledge. They basically don't know/remember anything they are supposed to know from Year 11, and so I've been interleaving as much of old material as possible into current topics to help them review. I know that in the spring, I will have to do some very heavy-duty concept-mapping and explicit learning strategies like algorithm flashcards to get them familiar with the basic concepts of each topic, before we can start doing integrated review and test prep. I am prepared to make that happen and I have a plan for how to help them be successful. (It'll involve topic-based worksheets / individual tests when they're ready / moving on to individually review the next topic when both they and I agree they've mastered the basic ones, or repeating the cycle until they do improve.)

It's stretching me professionally to consider the various academic and emotional needs of my many groups and to attempt to address individual needs within each group. But, I am loving the challenge! :)

(...Now, if only someone could tell me if it's having any actual effect on the kids...)

## Friday, October 21, 2011

### A Move-Around Friday Algebra Activity

I just had a really fun (and mostly productive) Friday afternoon with my Grade 7's! :) We had been working on drawing and solving some simple algebra balances, and I wanted to give them a bit more practice without it being too boring on a Friday afternoon.

This is how I structured it:

* Each kid gets a strip of one equation. They take a sheet of scrap paper and draw the balance scale that corresponds to that equation (They can ask me for help, of course, but NOT for me to see if it's correct).

* They go tape their algebra scales up around the room. (The kids were very silly about this; they taped them on all kinds of surfaces like the fan, the clock, and our lights.) The equation is written above the picture in nice, big letters with markers.

* They go stand in front of a scale that's not theirs and double-check the balance scale that has been drawn, fixing it if necessary. (I should have had them rotate once more to get a second look, but all of the mistakes except for 1 were caught.) When/if in doubt, I went over and took a look myself before they made any changes to the already-drawn scales.

* I handed out an answer sheet with the equations written on it. They are to roam around the room and find the balances that have already been drawn, and then solve them. Some of them are easier to solve than others; some of them required the kids to copy down the balance in order to work it out. (Each equation had a number/index, to make it easy for us to go over the answers in the end.)

* Since some of the kids were absent today, we needed to do two rounds of this in order to finish all the equations. The second time they sat down to draw the balances, they worked in pairs to make sure their drawings were definitely correct.

In the end, I went over the answers and asked the kids if they had questions, and everyone said that they understand the concept (but made some arithmetic mistakes)!! :) :) SCORE. It is a shame that some of my kids were absent today because of a soccer tournament, so next week I'll still have to think of a way to help those kids bridge their understanding gap. But, I think today was a very smooth run!!

Here is the material, if you wish to do this. Notice that I intentionally included parentheses into every problem; I really wanted kids to be able to see 5(x + 2) as drawing x and +2 each five times. I also threw in some "no solutions" and "all solutions" ones in there; the kids particularly liked those. :)

This is how I structured it:

* Each kid gets a strip of one equation. They take a sheet of scrap paper and draw the balance scale that corresponds to that equation (They can ask me for help, of course, but NOT for me to see if it's correct).

* They go tape their algebra scales up around the room. (The kids were very silly about this; they taped them on all kinds of surfaces like the fan, the clock, and our lights.) The equation is written above the picture in nice, big letters with markers.

* They go stand in front of a scale that's not theirs and double-check the balance scale that has been drawn, fixing it if necessary. (I should have had them rotate once more to get a second look, but all of the mistakes except for 1 were caught.) When/if in doubt, I went over and took a look myself before they made any changes to the already-drawn scales.

* I handed out an answer sheet with the equations written on it. They are to roam around the room and find the balances that have already been drawn, and then solve them. Some of them are easier to solve than others; some of them required the kids to copy down the balance in order to work it out. (Each equation had a number/index, to make it easy for us to go over the answers in the end.)

* Since some of the kids were absent today, we needed to do two rounds of this in order to finish all the equations. The second time they sat down to draw the balances, they worked in pairs to make sure their drawings were definitely correct.

In the end, I went over the answers and asked the kids if they had questions, and everyone said that they understand the concept (but made some arithmetic mistakes)!! :) :) SCORE. It is a shame that some of my kids were absent today because of a soccer tournament, so next week I'll still have to think of a way to help those kids bridge their understanding gap. But, I think today was a very smooth run!!

Here is the material, if you wish to do this. Notice that I intentionally included parentheses into every problem; I really wanted kids to be able to see 5(x + 2) as drawing x and +2 each five times. I also threw in some "no solutions" and "all solutions" ones in there; the kids particularly liked those. :)

## Wednesday, October 19, 2011

### An Excellent Resource Book

I wrote about this resource a while ago, but I wanted to come back to it since I've just had a chance to scratch the surface of looking at this resource book and doing its problems. The book I bought called

I sat down and made the answer key for the first 7 pages (involving linear functions only), and noticed that even the equations aren't totally easy to write (a good challenge for those higher level kids). Once they put the functions into the calculator, in order to navigate to the answers, they will still need to figure out how to zoom, trace, adjust table settings, and scroll through the table of values on the graphing calculator.

It's really wonderful! The topics in the book actually go hand-in-hand with what we're learning in class. First the book goes through some analysis of linear functions and related linear functions, and then immediately after, the second topic in the book is going to be linear regression.

I AM SO HAPPY!! It is actually excellent (and well-scaffolded*) material that I don't have to modify before giving to the kids. What a gem!

*Previously, when I was skimming through the book, I had thought the scaffolding was a bit too much. But now that I actually sat down to do the problems, I don't think so. That opinion could change when I move further into the book, but so far I like what I see.

*Graphic Algebra*is absolutely fantastic! I plan to use it with my most advanced 8th-graders as a tool to encourage them to think critically about algebra and to gain familiarity with the graphing calculator at their own pace. From now on, any time it does not make sense for me to start a new skill or topic with them, while the others in the class still need some more time to wrap up their practice, I am going to point them at pages xeroxed from this book. I think it's better this way, because this book is systematic and intentional in the way that it sequences its tasks, which my onesie, twosie on-the-spot differentiation tasks cannot accomplish.I sat down and made the answer key for the first 7 pages (involving linear functions only), and noticed that even the equations aren't totally easy to write (a good challenge for those higher level kids). Once they put the functions into the calculator, in order to navigate to the answers, they will still need to figure out how to zoom, trace, adjust table settings, and scroll through the table of values on the graphing calculator.

It's really wonderful! The topics in the book actually go hand-in-hand with what we're learning in class. First the book goes through some analysis of linear functions and related linear functions, and then immediately after, the second topic in the book is going to be linear regression.

I AM SO HAPPY!! It is actually excellent (and well-scaffolded*) material that I don't have to modify before giving to the kids. What a gem!

*Previously, when I was skimming through the book, I had thought the scaffolding was a bit too much. But now that I actually sat down to do the problems, I don't think so. That opinion could change when I move further into the book, but so far I like what I see.

## Tuesday, October 18, 2011

### Challenge of Pacing a Bimodal Group

I've been thinking about ways to address the bimodal performance that my Grade 8 kids exhibited on the last exam.

Good news: I am getting a lot of motivated kids to come see me during lunch for extra reinforcement to fill in the gaps of their learning.

Bad news: I think the bimodal performance is a result of my not very effective attempt to keep slower and faster learners paced similarly in the same class.

Imagine a typical class: I give kids handouts to work through at their own paces. The problems are scaffolded, so they get harder as you go. Kids are helping each other through the assignment, although there is a healthy amount of struggling happening per kid, and that means that each kid is moving at a different pace. The class works steadily for the entire period, and at the end of class, some of the faster kids finish the worksheet and grab a new one from me, while the slower kids are about two thirds of the way through with the worksheet. I consider assigning the last third as homework, but I also consider the fact that it's the hardest part of the worksheet, so I change my mind and tell the class that they can continue working on it the next day.

Fast-forward to the next class, or two classes from now. The faster kids are now a couple of worksheets ahead of the slowest kids. Instead of giving them pure algebra practice, we now are attacking the problems from all angles -- word problems with non-trivial context. My hopes are that this way, all kids are benefiting from the new assignments (whether it is additional algebra practice OR additional exposure to new contexts). I give the most advanced kids an extra challenging assignment, and I decide that it's definitely time for all of us to sync back up on our pacing, so I assign close to an entire worksheet as homework for the slower kids. The slower kids feel punished for working more slowly than their peers, even though I explain that we need to get everyone more or less on the same page now.

In the end, my slower-working kids are the ones who should probably be doing

How do I differentiate my instruction enough to fix this??

Good news: I am getting a lot of motivated kids to come see me during lunch for extra reinforcement to fill in the gaps of their learning.

Bad news: I think the bimodal performance is a result of my not very effective attempt to keep slower and faster learners paced similarly in the same class.

Imagine a typical class: I give kids handouts to work through at their own paces. The problems are scaffolded, so they get harder as you go. Kids are helping each other through the assignment, although there is a healthy amount of struggling happening per kid, and that means that each kid is moving at a different pace. The class works steadily for the entire period, and at the end of class, some of the faster kids finish the worksheet and grab a new one from me, while the slower kids are about two thirds of the way through with the worksheet. I consider assigning the last third as homework, but I also consider the fact that it's the hardest part of the worksheet, so I change my mind and tell the class that they can continue working on it the next day.

Fast-forward to the next class, or two classes from now. The faster kids are now a couple of worksheets ahead of the slowest kids. Instead of giving them pure algebra practice, we now are attacking the problems from all angles -- word problems with non-trivial context. My hopes are that this way, all kids are benefiting from the new assignments (whether it is additional algebra practice OR additional exposure to new contexts). I give the most advanced kids an extra challenging assignment, and I decide that it's definitely time for all of us to sync back up on our pacing, so I assign close to an entire worksheet as homework for the slower kids. The slower kids feel punished for working more slowly than their peers, even though I explain that we need to get everyone more or less on the same page now.

In the end, my slower-working kids are the ones who should probably be doing

*extra*problems just to keep up with the material, but the reality is that they need to spend extra time at home just to finish the*regular*assignments that everyone else can finish in class. Naturally, their learning results are not where I want them to be, because my pacing is at least somewhat dictated by the fact that half of the class is absolutely mastering quadratic operations forwards and backwards.How do I differentiate my instruction enough to fix this??

## Sunday, October 16, 2011

### IB Portfolio Task Preparation

I just spent an entire afternoon trying to wrap my mind around the IB Portfolio Type II Tasks. I found a fantastic podcast on Youtube that I am going to share with my kids: Part 1 and Part 2. What I really liked about this podcast is that the teachers walked through the technical aspect of the graphing program, as well as reviewed the rubric in parts relating to their thinking-out-loud about the sample task. I am going to assign as homework these two videos for my kids to watch at home, and then together I will go over it with them again in class, pausing every so often to go over the mathematical content and to allow them to install/run Autograph with my assistance. (There are some things that the video glosses over that the kids might find tricky to navigate on their own.)

It seems to me like many Type II tasks want the kids to "analytically" come up with their own equation, which can mean plugging in multiple points and solving the system of equations in order to find the coefficients, OR using what they know about the meanings of the coefficients (ie. amplitude or frequency) in order to construct the equation algebraically from the graph. Then, the tasks call for the students to either modify their equation to match additional data or ask them to generate (using technology) a

In order for the kids to be successful, we'll need: 1 day of going through the basics of regression and rehearsing the technical aspects of the graphing software; at least 1 day of looking at different types of non-perfect numeric patterns, in order to figure out what type of regression is necessary (Some types are far messier than others; the IB tasks aren't messing around with simple quadratic or even cubic regression. It looks like I'll have to go into some fairly involving functions review with my batch of Grade 12's...); another 0.5 day of looking at asymptotic behavior.

Even though the task is going to surely be challenging, I think it's going to be really good for their mathematical minds to tackle this task. As a teacher, I really like how the IB Program has high expectations for all kids, because those are some very noble and worthwhile goals for us to shoot for in the math class!

(More to come about Type I tasks a few months from now...)

It seems to me like many Type II tasks want the kids to "analytically" come up with their own equation, which can mean plugging in multiple points and solving the system of equations in order to find the coefficients, OR using what they know about the meanings of the coefficients (ie. amplitude or frequency) in order to construct the equation algebraically from the graph. Then, the tasks call for the students to either modify their equation to match additional data or ask them to generate (using technology) a

*different*type of regression equation. The latter isn't always trivial to do, especially since sometimes the task gives them a fairly hairy form of equation to play with. And then, assuming that they can successfully do the mathematical modeling, they'll need to firmly link the asymptotic behavior of their regressed equation to the context of the problem, in order to examine whether that asymptotic behavior makes sense or if the modeling domain should be restricted only to the given set of data. In some cases, the kids may even need to have some background knowledge of the topic in order to properly discuss the asymptotic behavior.In order for the kids to be successful, we'll need: 1 day of going through the basics of regression and rehearsing the technical aspects of the graphing software; at least 1 day of looking at different types of non-perfect numeric patterns, in order to figure out what type of regression is necessary (Some types are far messier than others; the IB tasks aren't messing around with simple quadratic or even cubic regression. It looks like I'll have to go into some fairly involving functions review with my batch of Grade 12's...); another 0.5 day of looking at asymptotic behavior.

Even though the task is going to surely be challenging, I think it's going to be really good for their mathematical minds to tackle this task. As a teacher, I really like how the IB Program has high expectations for all kids, because those are some very noble and worthwhile goals for us to shoot for in the math class!

(More to come about Type I tasks a few months from now...)

## Saturday, October 15, 2011

### Algebra Scales Worksheets and Longer-Term Grade 7 Vision

I've been taking a pretty holistic, integrated approach to my Grade 7 curriculum. Our last test looked something like this and it covered some mental percentage arithmetic, some patterns and writing of equations, and then some basic word problems. The class did fair on the exam; most kids got most things correct, except for the setting up of the (rather complicated) word problem. It showed me that we're going to have to come back to practice that skill before the next go-around on an exam, but that as a class we're ready to move on to a new topic -- proportions, but in an integrated fashion still, while looping in all the things we already know.

So far, in terms of actually solving equations, the kids have not progressed to formal symbols and algebra yet. Since I promised them that every Friday is going to be something "fun", I made up yesterday a sheet of algebra scales problems for them to do. (Normally, I'd do this kind of thing on the computer, but for reasons that are not easy to explain, I couldn't get easy access to the computers at the school. I figured doing it on paper is almost as good.) Here was the file I used, and the kids were a bit nuts about it while working loudly but enthusiastically in groups! The problems are scaffolded up to letting the kids see when there are no solutions or there are infinite solutions, and then on the back side they needed to draw their own scales from a given equation, in order to solve for x. I used different levels of the scale to show why we might have something like 5x + 3x + 2 = ... and why that's equivalent to 8x + 2 = ... In a few weeks, once the kids have internalized this visualization method (including negative coefficients and negative integers), we'll go over how it translates formally to algebraic symbols.

It's probably nothing new, but I think if you don't already have a worksheet like this, you may find my scaffolding helpful, so here it is: Positive things on scales and Negative things on scales.

Isn't algebra fun?? :) My hope is that by the end of the first semester, my kids will: be comfortable with the idea of predicting linear patterns forwards/backwards and reading/making graphs; understand intuitively what proportions are, when to use them, and how they are tied to linear patterns; have great number sense and be able to quickly compare quantities that involve calculating fractions or percents; be able to set up and solve linear equations given a word problem. I think the list is ambitious, but definitely doable. That would leave us time in the second semester to do some probability/basic geometry and to begin tackling a "harder" algebra topic such as quadratics.

So far, in terms of actually solving equations, the kids have not progressed to formal symbols and algebra yet. Since I promised them that every Friday is going to be something "fun", I made up yesterday a sheet of algebra scales problems for them to do. (Normally, I'd do this kind of thing on the computer, but for reasons that are not easy to explain, I couldn't get easy access to the computers at the school. I figured doing it on paper is almost as good.) Here was the file I used, and the kids were a bit nuts about it while working loudly but enthusiastically in groups! The problems are scaffolded up to letting the kids see when there are no solutions or there are infinite solutions, and then on the back side they needed to draw their own scales from a given equation, in order to solve for x. I used different levels of the scale to show why we might have something like 5x + 3x + 2 = ... and why that's equivalent to 8x + 2 = ... In a few weeks, once the kids have internalized this visualization method (including negative coefficients and negative integers), we'll go over how it translates formally to algebraic symbols.

*As always, I am a firm believer that symbols need to be introduced only after the visualization of the operations becomes second-nature to the kids.*It's probably nothing new, but I think if you don't already have a worksheet like this, you may find my scaffolding helpful, so here it is: Positive things on scales and Negative things on scales.

Isn't algebra fun?? :) My hope is that by the end of the first semester, my kids will: be comfortable with the idea of predicting linear patterns forwards/backwards and reading/making graphs; understand intuitively what proportions are, when to use them, and how they are tied to linear patterns; have great number sense and be able to quickly compare quantities that involve calculating fractions or percents; be able to set up and solve linear equations given a word problem. I think the list is ambitious, but definitely doable. That would leave us time in the second semester to do some probability/basic geometry and to begin tackling a "harder" algebra topic such as quadratics.

## Thursday, October 13, 2011

### Integrity Brain Dump

I've been reflecting extensively about integrity, because this year I am really trying to do a better job with the character education program I am teaching as part of the Grade 8 homeroom. Already I've talked to the homeroom kids about: appreciation as a tool to smoothing over difficult confrontations; manners and cultures; why we come to school (and why that's a privilege); the idea that our intelligence is not fixed, and that our knowledge can improve our brain's processes and in effect we can "get smarter" over time; setting goals and brainstorming study skills. I have found that by opening my own life experiences up to the kids, I am making the textbook lessons more meaningful for me (and thereby, hopefully for them as well). Every week, I try to tie the lesson to something a little bit deeper, and the kids have reciprocated by opening up with their own thoughts and experiences as well, allowing me a sneak peek at their thoughtful and insightful side, which I don't always see in the context of a math class.

Something that has been on my mind (although not necessarily in the Grade 8 character-development curriculum) has been the issue of integrity. The more I think about it, the more integrity seems to weave itself into everything that we do, which means both that it's necessary for me to talk about integrity issues with the kids, and also that it is very difficult to get a clear, convincing, effective message across in a single discussion. I am going to do a brain dump over here; feel free to add your own thoughts.

------

When we think about integrity, we mostly think of

The first stumbling block I have is how to talk to kids in a convincing way about the

* Being worried about consequences (ie. punishment) that may occur to us personally

* Upholding our reputation (another form of personal consequence)

* Being afraid to hurt or damage others who trust us

* Believing that our individual needs should come after our commitment or obligation to a group/policy/community (Relativistic morals?)

* Believing that the action is wrong on an absolute scale

As you probably agree, our aim to talk to kids about integrity is roughly equivalent to moving them along the spectrum of reasons towards the intrinsic motivators rather than the extrinsic motivators. But, doing so is difficult. The best that I can come up with is talking to kids about my own view of my own personal integrity, in order to shed light on what it means to me, personally. (I grew up in a family that raised us on stories that carried inherent values, but I'm not sure if my kids have the same relationship with their parents.)

So, here are some examples I've come up with for where I think I exhibit a personal integrity, in a way that is perhaps subtler than not cheating and not lying:

When we put ourselves in the shoes of a hiring manager, it's clear why integrity is important. Whom would you rather hire -- a conniving employee who might lie about their results, or an employee who would own up to their mistakes and reflect upon them with others? For the same reasons, we prefer our politicians to have integrity, because so much goes on behind closed doors in politics that we have to be able to trust them (at least a little). When the situation is gray legally but not ethically so, we can only hope collectively that those in charge (such as the bankers giving out mortagages) are doing their part to ensure that the interest of the larger community is protected. Integrity, therefore, is clearly something that we value as a community.

When does a kid encounter integrity issues? All the time, I bet. Inside and outside of school, I bet. Imagine a kid whose parents want him to go home early, while the temptation is to hang out late with his friends. He can either: 1. stay out and then make up an excuse afterwards, 2. go home and then sneak out, 3. try to reason with his parents to get a later curfew, but going home at the promised time, even if it's early. Which is the least pleasant but the most honest option for a kid? Probably #3. In situations big and small, whether or not they even give it a second thought, they are constantly being confronted with choices and reinforcing their own integrity, or the lack thereof.

So far, these are all just my thoughts. I am giving an anonymous survey next week in my homeroom and I'll be tallying the opinions to get a feel for what kids consider to be cheating behavior, whether that definition is tied to the outcome (ie. if they successfully cheated or not), and what they think are the most important factors that discourage them from certain types of behavior. From those surveyed results I'll plan some discussion points and go about it, leaving much of it open-ended to hear what the kids have to say about the issue.

The point is that I want to open the door for more proactive conversation about integrity and less reactive conversation. Thoughts? Ideas??

Addendum 10/14/2011: In case you are interested, here is the survey I am going to pass out.

Something that has been on my mind (although not necessarily in the Grade 8 character-development curriculum) has been the issue of integrity. The more I think about it, the more integrity seems to weave itself into everything that we do, which means both that it's necessary for me to talk about integrity issues with the kids, and also that it is very difficult to get a clear, convincing, effective message across in a single discussion. I am going to do a brain dump over here; feel free to add your own thoughts.

------

When we think about integrity, we mostly think of

*not lying and not cheating*. In school, we only talk about integrity as a reactionary mechanism -- usually after something bad has already occurred, such as group cheating or plagiarism. What we need to do is to actively address integrity issues as a school as a prevention mechanism -- and that can be done in our individual curricula (ie. history and English, where morality is explored in the context of literature), or in our character-education classes.The first stumbling block I have is how to talk to kids in a convincing way about the

*importance*of integrity. Why should they bother not doing certain things, if they know that they cannot or probably will not be caught?? I think we are dissuaded from negative behavior for any number of these reasons below (depending on our own morality developmental stage):* Being worried about consequences (ie. punishment) that may occur to us personally

* Upholding our reputation (another form of personal consequence)

* Being afraid to hurt or damage others who trust us

* Believing that our individual needs should come after our commitment or obligation to a group/policy/community (Relativistic morals?)

* Believing that the action is wrong on an absolute scale

As you probably agree, our aim to talk to kids about integrity is roughly equivalent to moving them along the spectrum of reasons towards the intrinsic motivators rather than the extrinsic motivators. But, doing so is difficult. The best that I can come up with is talking to kids about my own view of my own personal integrity, in order to shed light on what it means to me, personally. (I grew up in a family that raised us on stories that carried inherent values, but I'm not sure if my kids have the same relationship with their parents.)

So, here are some examples I've come up with for where I think I exhibit a personal integrity, in a way that is perhaps subtler than not cheating and not lying:

My integrity is reflected in the way I work at my jobs. When I was 17, I had my first job evaluation by my Starbucks store manager. I was nervous, and I had asked my friend if he was nervous as well. This is what he said: "I always do my best on a job, whether or not someone is watching me. And in the end, that's all I can do. There is no reason to worry." I still carry that work ethic with me today. I don't ever compare myself with other teachers in my department; I need to do my best within my own frame of possibilities. Over time, that frame will expand. So, it doesn't matter if someone is years more experienced than me or if they choose to leave at 3pm; what matters to me is that I do my very best every single day for every kid in my class, within my own realm of possibilities. Furthermore, I don't ever worry about whether I've earned my place at work, because I've never lied or cheated on any test, project, resume, or interview to get there.

My personal integrity also means to me that if there is a legitimate way to get something done, that's what I'll do even if that means my life will probably be made a bit harder. I remembered today an incident where I went to a supervisor asking for a day off after months of not taking a single day off -- there had been days when I was so sick that I could not stand up, and still I had shown up to work. The supervisor told me no, that I couldn't have this day off to go to my friend's wedding, even though I was requesting it far in advance. I was really upset, but even then I refused to call in sick that day. My personal integrity means that I need to represent what is true, even if that truth would inconvenience me.

When we put ourselves in the shoes of a hiring manager, it's clear why integrity is important. Whom would you rather hire -- a conniving employee who might lie about their results, or an employee who would own up to their mistakes and reflect upon them with others? For the same reasons, we prefer our politicians to have integrity, because so much goes on behind closed doors in politics that we have to be able to trust them (at least a little). When the situation is gray legally but not ethically so, we can only hope collectively that those in charge (such as the bankers giving out mortagages) are doing their part to ensure that the interest of the larger community is protected. Integrity, therefore, is clearly something that we value as a community.

When does a kid encounter integrity issues? All the time, I bet. Inside and outside of school, I bet. Imagine a kid whose parents want him to go home early, while the temptation is to hang out late with his friends. He can either: 1. stay out and then make up an excuse afterwards, 2. go home and then sneak out, 3. try to reason with his parents to get a later curfew, but going home at the promised time, even if it's early. Which is the least pleasant but the most honest option for a kid? Probably #3. In situations big and small, whether or not they even give it a second thought, they are constantly being confronted with choices and reinforcing their own integrity, or the lack thereof.

So far, these are all just my thoughts. I am giving an anonymous survey next week in my homeroom and I'll be tallying the opinions to get a feel for what kids consider to be cheating behavior, whether that definition is tied to the outcome (ie. if they successfully cheated or not), and what they think are the most important factors that discourage them from certain types of behavior. From those surveyed results I'll plan some discussion points and go about it, leaving much of it open-ended to hear what the kids have to say about the issue.

The point is that I want to open the door for more proactive conversation about integrity and less reactive conversation. Thoughts? Ideas??

Addendum 10/14/2011: In case you are interested, here is the survey I am going to pass out.

## Wednesday, October 12, 2011

### Bimodal Performance

One of my classes just took an exam and exhibited bimodal performance. Half of the kids did very well and did not miss any conceptual understanding, only a very few (1 or 2) arithmetic errors throughout the entire exam. The other half of the class is missing large chunks of the conceptual setting-up-of-equations and making-predictions type of work. The split is about even.

So, what to do now? My intuition is to complete a concept map/graphic organizer as a class and then set them up in expert-novice pairs to work on remediation, and then the kids who need the extra practice are going to go home with a fresh sheet of problems and be asked to work through it to get additional reinforcement.

What would you do?

PS. I think one of my mistakes was to wait too long to have a large assessment. That's obviously easy to fix in the future, but I'm curious what you think I should do now, given the current situation.

So, what to do now? My intuition is to complete a concept map/graphic organizer as a class and then set them up in expert-novice pairs to work on remediation, and then the kids who need the extra practice are going to go home with a fresh sheet of problems and be asked to work through it to get additional reinforcement.

What would you do?

PS. I think one of my mistakes was to wait too long to have a large assessment. That's obviously easy to fix in the future, but I'm curious what you think I should do now, given the current situation.

## Monday, October 3, 2011

### Shanghai Exchange

I just spent a weekend hosting four teachers from a traditional public middle school in Shanghai. Our school has an exchange program with them, where they take a small group of their students here to visit in the fall, and we take a small group of our students over there to visit in the spring. The kids stay with host families and go with their host students to classes, and the adults are hosted by local adults in order to get an authentic feel of the place.

It turned out to be very useful that I can speak Chinese. Two of the teachers spoke English very well, as they are English teachers at the school. A third one understood a decent amount, but a fourth one did not speak any English at all. It was not

We started off in Alexanderplatz on Saturday, took some pictures at the Rotes Rathaus (Red City Hall), and then they declined paying to go up the TV Tower, which is the highest point in Europe. I then took them shopping (as per their request), and after lunch we went over to the beautiful Reichstag, which is where the German laws are made. We had made a reservation beforehand, so we got to go up to the beautiful glass dome of the Reichstag, and to take a self-guided audio tour that introduced us to the features of the buildings surrounding the Reichstag and the features of the dome itself.

On Sunday, we first went to Schloss Charlottenberg, which is a beautiful palace built hundreds of years ago during the Prussian dynasty, and still today reflects the luxury of those times. During WWII, much of the palace was bombed and destroyed, so much of what you can see today is the result of reconstruction in the 80s. Still, you can get a sense of the grandeur that once dominated this palace. After some hours at the palace, we headed over to the famous Checkpoint Charlie to take some photos, and then we walked along the Eastside Gallery, which is a stretch of the remnants of the Berlin Wall that has since been turned into a symbol for hope and inspiration as artists have made the wall into an elaborate art display.

During the course of the day, I got a chance to ask about the school in Shanghai. One of the girls told me that in Shanghai, there are four tiers of schools: city-level magnet schools, district-level magnet schools, "normal" schools, and private or independent schools. At the end of every level of schooling (ie. elementary school, or middle school, or high school), kids need to take a city-wide test and apply for the next schools. The system is very competitive, because in order to get into a good college (or perhaps any college at all), you need to be from a top high school, which means you needed to be from a top middle school. There are some exceptions to this system, however, such as the fact that a kid who lives within a certain close proximity to a school has the right to attend that school, even if the kid is not academically qualified. And, on top of that, there is a lot of pressure from the government to make sure that ALL kids pass every class by the end of middle school, regardless of whether the child was qualified to attend this school in the first place. So, that creates a lot of pressure on the teachers AND on the kids who need to struggle to pass just ONE class, let alone all classes.

In Shanghai, this teacher tells me that they teach roughly only half of the time that I teach, but that in every class they have 40 kids. I asked her if she thinks she has enough time to reach every kid and to take care of them, and she said no. Most of her prep time is spent on correcting the daily homework that was assigned. The kids go home and most of them do their homework through midnight each night. So, it is an eye-opening experience for their kids to come to our school and see that our middle-schoolers have barely any homework and enjoy so much freedom at home and in school.

I look forward to visiting their school in April to see for myself what it's like!

It turned out to be very useful that I can speak Chinese. Two of the teachers spoke English very well, as they are English teachers at the school. A third one understood a decent amount, but a fourth one did not speak any English at all. It was not

*necessary*, therefore, for me to speak Chinese, but the fact that I could helped all of them feel comfortable. Geoff tagged along to also be a tour guide, and we took the teachers to see all the usual touristy places.We started off in Alexanderplatz on Saturday, took some pictures at the Rotes Rathaus (Red City Hall), and then they declined paying to go up the TV Tower, which is the highest point in Europe. I then took them shopping (as per their request), and after lunch we went over to the beautiful Reichstag, which is where the German laws are made. We had made a reservation beforehand, so we got to go up to the beautiful glass dome of the Reichstag, and to take a self-guided audio tour that introduced us to the features of the buildings surrounding the Reichstag and the features of the dome itself.

On Sunday, we first went to Schloss Charlottenberg, which is a beautiful palace built hundreds of years ago during the Prussian dynasty, and still today reflects the luxury of those times. During WWII, much of the palace was bombed and destroyed, so much of what you can see today is the result of reconstruction in the 80s. Still, you can get a sense of the grandeur that once dominated this palace. After some hours at the palace, we headed over to the famous Checkpoint Charlie to take some photos, and then we walked along the Eastside Gallery, which is a stretch of the remnants of the Berlin Wall that has since been turned into a symbol for hope and inspiration as artists have made the wall into an elaborate art display.

During the course of the day, I got a chance to ask about the school in Shanghai. One of the girls told me that in Shanghai, there are four tiers of schools: city-level magnet schools, district-level magnet schools, "normal" schools, and private or independent schools. At the end of every level of schooling (ie. elementary school, or middle school, or high school), kids need to take a city-wide test and apply for the next schools. The system is very competitive, because in order to get into a good college (or perhaps any college at all), you need to be from a top high school, which means you needed to be from a top middle school. There are some exceptions to this system, however, such as the fact that a kid who lives within a certain close proximity to a school has the right to attend that school, even if the kid is not academically qualified. And, on top of that, there is a lot of pressure from the government to make sure that ALL kids pass every class by the end of middle school, regardless of whether the child was qualified to attend this school in the first place. So, that creates a lot of pressure on the teachers AND on the kids who need to struggle to pass just ONE class, let alone all classes.

In Shanghai, this teacher tells me that they teach roughly only half of the time that I teach, but that in every class they have 40 kids. I asked her if she thinks she has enough time to reach every kid and to take care of them, and she said no. Most of her prep time is spent on correcting the daily homework that was assigned. The kids go home and most of them do their homework through midnight each night. So, it is an eye-opening experience for their kids to come to our school and see that our middle-schoolers have barely any homework and enjoy so much freedom at home and in school.

I look forward to visiting their school in April to see for myself what it's like!

## Friday, September 30, 2011

### Restructured Speed Game

I love my 7th-graders! Today was the second time I've played games with them on a Friday afternoon, and they absolutely go nuts for it every time.

One concern I always have during a group game is that only the kids who are up for their turn are engaged. Today I improved that by letting TWO kids from each team go up together, and they can only collaborate AFTER they have both tried the problem independently. When they ring the bell, both people should already have written down the answer on their paper, and I'll randomly call on one of them to EXPLAIN their answer. (So, they had to confer before ringing the bell, and the stronger student was always eagerly explaining to the weaker student how to do it.)

The problems built up in difficulty, so I'd always make sure all students from the class understood the explanations to the previous question before starting another round. Students at their desks were also practicing because they didn't want to be the person to hold their partner up when it's their turn. Overall, the rotations went twice as fast, and everyone was constantly engaged.

By the end of the 80-minute period, even my weakest students were getting through the problems by themselves, and EVERYONE WAS SO EXCITED TO EXPLAIN THEIR ANSWERS! It was great!!

Grab the file here. This one was on filling in missing numbers into linear patterns, and some of it involved integers and decimals. The kids were brilliant! (We also had two exchange students from China playing with us. It was COOL; one of them explained to me in Chinese, and I translated for her in real time for the class, and high-fived her after that round! It was sweet.)

One concern I always have during a group game is that only the kids who are up for their turn are engaged. Today I improved that by letting TWO kids from each team go up together, and they can only collaborate AFTER they have both tried the problem independently. When they ring the bell, both people should already have written down the answer on their paper, and I'll randomly call on one of them to EXPLAIN their answer. (So, they had to confer before ringing the bell, and the stronger student was always eagerly explaining to the weaker student how to do it.)

The problems built up in difficulty, so I'd always make sure all students from the class understood the explanations to the previous question before starting another round. Students at their desks were also practicing because they didn't want to be the person to hold their partner up when it's their turn. Overall, the rotations went twice as fast, and everyone was constantly engaged.

By the end of the 80-minute period, even my weakest students were getting through the problems by themselves, and EVERYONE WAS SO EXCITED TO EXPLAIN THEIR ANSWERS! It was great!!

Grab the file here. This one was on filling in missing numbers into linear patterns, and some of it involved integers and decimals. The kids were brilliant! (We also had two exchange students from China playing with us. It was COOL; one of them explained to me in Chinese, and I translated for her in real time for the class, and high-fived her after that round! It was sweet.)

## Sunday, September 25, 2011

### Numerous Connections to Quadratic Functions

I think that as secondary math teachers, we ought to be aware of as many ways as possible to model quadratic patterns, so we can seamlessly integrate those techniques into our curricula. It's the first non-straight forward pattern that our kids encounter, and it seems to trouble them throughout their secondary careers. Thankfully, there are ties to quadratic modeling from MANY different topics!

To discuss this concretely, let's first have a look at a quadratic pattern:

(2, 7), (3, 9), (4, 12), (5, 16)

In this discussion, I am going to assume that you've already taught kids how to recognize that this is a quadratic pattern, since that discussion should absolutely precede any quadratic skills, methinks!

Neat, eh? I think it's pretty amazing how you can approach a simple quadratic pattern from so many different algebraic methods, and I am a true believer that the more of these integrated approaches we take, the better kids will understand each individual method and be able to transfer their knowledge.

By the way, an 8th-grader came up to me and made an insightful observation, the day following our quick class discussion about why we need 3 points in order to find a path of trajectory (quadratic function) using Dan Meyer's basketball pictures. She asked me, "We need 3 points to find a quadratic function, but we need at least 4 points to verify that the pattern IS quadratic, right?" and she drew me a table on the board with some first and second differences. Brilliant! Yes - 3 points is enough to find an equation, but that's like saying two points form a line. That's the minimal amount of info needed to get the equation, but as soon as you have redundant data points, you will know if your regression form is actually correct! How astute of her to make that observation!

To discuss this concretely, let's first have a look at a quadratic pattern:

(2, 7), (3, 9), (4, 12), (5, 16)

In this discussion, I am going to assume that you've already taught kids how to recognize that this is a quadratic pattern, since that discussion should absolutely precede any quadratic skills, methinks!

Method #1:Helping kids understand at a middle-school level that we find missing coefficients inside equations by plugging in given x's and y's.

For example, given our points above, y = ax^2 + bx + c becomes:

7 = 4a + 2b + c

9 = 9a + 3b + c

12 = 16a + 4b + c

This we can solve in MIDDLE SCHOOL using systems of equations/elimination method.Method #2:When kids get into high school and they start learning about matrices, we can set the same system up as a problem of 3-by-3 matrix, and help them use the calculator to solve it:

(It's a good time to make sure everyone understands why we're using inverse matrices, and what identity means in math, etc.)Method #3:Of course, now that they're in the realm of graphing calculators, we should also teach them how to use quadratic regression to get the values of the coefficients.Method #4:There is another sneakier way of finding the equation using pure quadratics algebra. It's called the Legrange Method, and I learned it this summer at PCMI.

You split the function into smaller, easier quadratic functions by splitting the table and hiding two points at a time:

With a bit of simple algebra, you can figure out that

y1 = 3.5(x - 3)(x - 4)

y2 = -9(x - 2)(x - 4)

y3 = 6(x - 2)(x - 3)

Then, when you simplify each and combine them, you'd get your equation for y! This way is not the easiest, but it does reinforce the idea of roots, plugging in numbers to solve for parts of equation, and expansion/simplification of binomials.Method #5:Individual elements within a quadratic sequence are created as a summation of arithmetic/linear steps! For example, if you look at the example we have here: (2, 7), (3, 9), (4, 12), (5, 16)

f(2) = 7

f(3) = 7 + 2

f(4) = 7 + 2 + 3

f(5) = 7 + 2 + 3 + 4

Inductively,

f(x) = 7 + { 2 + 3 + 4 + ... +(x-1) }

Or, if we use an arithmetic series formula to represent the sum-of-steps:

f(x) = 7 + [2 + (x-1)][(x-2)/2]

Since my arithmetic series has first term = 2, last term = (x-1), and altogether (x-2)/2 pairs of elements.Method #6:There is a name for this trick, but I forget what it is. If we draw out the quadratic sequence as dots, we get:

Here I used the gray dots to represent the initial value.

If you double the added dots at each stage (see blue/black dots below), it now becomes instantly easier to find the pattern, because the length and the width of the resulting rectangle is a simple linear progression in terms of x:

f(x) = initial value + half of rectangular area

f(x) = 7 + 0.5(length times width of rectangle)

f(x) = 7 + 0.5(x + 1)(x - 2)

Neat, eh? I think it's pretty amazing how you can approach a simple quadratic pattern from so many different algebraic methods, and I am a true believer that the more of these integrated approaches we take, the better kids will understand each individual method and be able to transfer their knowledge.

By the way, an 8th-grader came up to me and made an insightful observation, the day following our quick class discussion about why we need 3 points in order to find a path of trajectory (quadratic function) using Dan Meyer's basketball pictures. She asked me, "We need 3 points to find a quadratic function, but we need at least 4 points to verify that the pattern IS quadratic, right?" and she drew me a table on the board with some first and second differences. Brilliant! Yes - 3 points is enough to find an equation, but that's like saying two points form a line. That's the minimal amount of info needed to get the equation, but as soon as you have redundant data points, you will know if your regression form is actually correct! How astute of her to make that observation!

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