Do you do any PSAT prep in your classes? And if you do, how do you do it in a way that helps all kids stay invested?
I decided recently to give kids time in class to go over their PSAT score reports, which have all of the questions and item-level analysis (including a check if they had the question correct, an "o" if they had skipped the question, or a comparison of their answer versus the correct answer). They were supposed to go over them in groups, and with a partner compare the questions that they had either omitted or gotten wrong, to see if they can now figure out why the correct answer is as indicated. (Since I figured that it was unrealistic to expect my freshmen to be going over these on their own at home, honors kids or otherwise.)
Anyway, long story short, kids hated it. Many of them weren't invested because they had missed so many questions on the original test that they felt very discouraged. Others felt like it was irrelevant, since their understanding of the PSATs wouldn't impact their grade in my class. What should I do?? I feel like this stuff should be super important, but I have trouble getting my regular (non-honors) kids to take them seriously. :(
Tuesday, February 1, 2011
Friday, January 28, 2011
What Learning Looks Like in My Classroom
I do very little talking-in-front-of-class. It's a habit I have retained from my first year of teaching, when I couldn't manage to get a class to be quiet for me. Nowadays, obviously that's not an issue anymore, but I still prefer to give kids worksheets that are scaffolded enough to be accessible to every kid without following a long lecture. The only times when I do "lecture" are when I emphasize different approaches to a problem, go over definitions, or model thinking-out-loud. But, those are rare -- like, 10 or 15 minutes per class -- or on a really bad day, maybe 25 minutes. Truly "mini" lessons.
Instead of lecturing, I put kids in groups to work on problems that are just one-step harder than what they're able to do individually (and I let them choose their own groups daily -- they are well-behaved more than 90% of the time, and the other times I have to be very on top of them and to speak sternly and look them in the eye and -- if the group is really incorrigible -- move kids into corners, you know, the "yuge"). And they argue over the problems, trying to figure them out on their own. And I go around and facilitate. Some days/years the kids rely on my questioning more than others. Other days I poke my head into groups just to make sure everyone did understand everything, and no "weak link" is being left behind conceptually and copying answers. In groups, I also re-explain concepts in a way that links to the bigger picture. Kids all know that they work in groups basically daily, and that my 5-to-10-minute instructions in the beginning of class (after the Do Now) are just setting them up for successfully attacking the assignment du jour. And so when I say, "Heads up!" just about everyone is actually listening.
Doing so has very real rewards, at least from a process perspective. Some days I look around the room and I am very pleased to see kids arguing over math. Like, LOUDLY and VEHEMENTLY! And LAUGHING at their own silly mistakes! (They're actually so into the math, sometimes, that I have to tell them to keep their voices down.) Other days I see kids explaining math to other kids in the hallway while working on a late homework assignment. Or helping them study in the library for a make-up quiz. Or some days kids come and see me before they are going home, because they are feeling sick but still want to take the quiz on the day it's supposed to be, so that they can get it back at the same time as everyone else. There is a real sense of community, like "We are learning math together," and a responsibility for their own understanding. Most of the time, they prefer to explain/ask each other for help before they will turn to me as sort of the "final verdict."
Those things make me happy, because my kids are being (in lack of a better word) proactive participants in the learning community. I don't do anything extraordinary to instill it, but I certainly expect it from all of them, and amazingly, they rise to the occasion. :)
PS. I do read out loud all the numeric answers and go over the big ideas again at the end of every class, to make sure that kids have absolutely no doubts remaining and they can self-monitor their understanding/accuracy. That takes about 5 minutes, but if I have done a good job circulating during the period and poking my head into every group, there is really not much content I still have to go over.
Instead of lecturing, I put kids in groups to work on problems that are just one-step harder than what they're able to do individually (and I let them choose their own groups daily -- they are well-behaved more than 90% of the time, and the other times I have to be very on top of them and to speak sternly and look them in the eye and -- if the group is really incorrigible -- move kids into corners, you know, the "yuge"). And they argue over the problems, trying to figure them out on their own. And I go around and facilitate. Some days/years the kids rely on my questioning more than others. Other days I poke my head into groups just to make sure everyone did understand everything, and no "weak link" is being left behind conceptually and copying answers. In groups, I also re-explain concepts in a way that links to the bigger picture. Kids all know that they work in groups basically daily, and that my 5-to-10-minute instructions in the beginning of class (after the Do Now) are just setting them up for successfully attacking the assignment du jour. And so when I say, "Heads up!" just about everyone is actually listening.
Doing so has very real rewards, at least from a process perspective. Some days I look around the room and I am very pleased to see kids arguing over math. Like, LOUDLY and VEHEMENTLY! And LAUGHING at their own silly mistakes! (They're actually so into the math, sometimes, that I have to tell them to keep their voices down.) Other days I see kids explaining math to other kids in the hallway while working on a late homework assignment. Or helping them study in the library for a make-up quiz. Or some days kids come and see me before they are going home, because they are feeling sick but still want to take the quiz on the day it's supposed to be, so that they can get it back at the same time as everyone else. There is a real sense of community, like "We are learning math together," and a responsibility for their own understanding. Most of the time, they prefer to explain/ask each other for help before they will turn to me as sort of the "final verdict."
Those things make me happy, because my kids are being (in lack of a better word) proactive participants in the learning community. I don't do anything extraordinary to instill it, but I certainly expect it from all of them, and amazingly, they rise to the occasion. :)
PS. I do read out loud all the numeric answers and go over the big ideas again at the end of every class, to make sure that kids have absolutely no doubts remaining and they can self-monitor their understanding/accuracy. That takes about 5 minutes, but if I have done a good job circulating during the period and poking my head into every group, there is really not much content I still have to go over.
Thursday, January 27, 2011
Algebraic Prism
Here is a worksheet I gave to my H. Geometry kiddies today. It was something I had made for my H. Algebra 2 kids last year, but now slightly modified to remove the function notation, specific questions about the degrees of functions, etc. The L-shaped prism looks that way entirely on purpose: Can your kids still properly identify the base shape and the height when it's an "irregular" shape, lying on its side?
I introduced to them the "box method" of multiplying polynomials, and they loved it. (You algebra teachers know what I'm talking about. Way better than drawing distributive arcs any day!)
(#4 of Pg. 2 is a little ambiguously phrased, because I want kids to remember on their own that the easiest way to find volume, once you have the base area, is to multiply by the height. They then later show that there is a different way to come up with the volume, and that it gives you the same formula in the end. See #6 on Pg. 3.)
They absolutely loved it (...It's always funny what gets kids going...), and we got into some pretty good geometric discussions about how surfaces on opposite parts of the prism correspond in area (ie. Area of DKMF = Area of CJIB + Area of AHNG), and how a little geometric intuition can save us a lot of work when calculating algebraic surface area!
I introduced to them the "box method" of multiplying polynomials, and they loved it. (You algebra teachers know what I'm talking about. Way better than drawing distributive arcs any day!)
(#4 of Pg. 2 is a little ambiguously phrased, because I want kids to remember on their own that the easiest way to find volume, once you have the base area, is to multiply by the height. They then later show that there is a different way to come up with the volume, and that it gives you the same formula in the end. See #6 on Pg. 3.)
They absolutely loved it (...It's always funny what gets kids going...), and we got into some pretty good geometric discussions about how surfaces on opposite parts of the prism correspond in area (ie. Area of DKMF = Area of CJIB + Area of AHNG), and how a little geometric intuition can save us a lot of work when calculating algebraic surface area!
Wednesday, January 26, 2011
Measurement Unit: Episodes 7, 8 (Liquid Density; Other Measurement Methods)
We're done! With the measurement unit!! All there is left to do is the test. (Or, for my regular kiddies: review, quiz, and then a test after we have remediated the quiz material.)
Lesson 7 in the unit was on liquid density. This one was my favorite, because the kids had to first discuss as a class how to measure the mass of a liquid. After they came up with the general idea, I had them take notes on the definition and procedure for finding "net weight," and we related it to the labels we see on packaged food containers. Kids were excited that they now knew what "net weight" on the corner of their cereal boxes meant!
I then had kids split up into groups of 2 or 3. They were instructed to work on a rather tricky practice sheet of conversions and volume problems, and I pulled out a few groups at a time to rotate around to do parts of the liquid density lab. They needed to measure the density of oil, water, and maple syrup using graduated cylinders and triple-beam balances. (To make this manageable, I gave each group a "clean" cylinder that they would use to find the weight of the container, and whenever they needed to pour the liquid into the container, they would use a "dirty" container that the groups before had used for the same liquid. This way, we didn't have to keep cleaning the graduated cylinders in between every group.) In the end, once ALL of the groups had finished gathering data, we discussed as a class what would happen if we were to pour all 3 liquids into the same graduated cylinder. Then, we tested it! I showed them that even if you flipped the container and straightened it back up, the liquids would still separate themselves. (--To a degree, anyway. The syrup and the water begin to mix gradually, since syrup is water-based and it gets diluted over time.)
It was super fun!! :) Now after this, the kids really have a good grasp of how to measure mass, net weight (net mass), volume, and what all of it means. Lovely!
For my honors kids (who are truly done-done with the whole unit, including the review... the other kids are still a couple of days behind), we followed it up with this: a reading on how to measure an elephant, and another on how to measure the oceans. We discussed the articles after they had read them individually, to make sure that they understood everything in the readings. (I try to insert some relevant reading into every unit to contribute to their literacy.*)
And they got a chance to try their hands at putting together the most complicated concepts in the unit -- making predictions about a 3-D container. (The second page of this was my favorite. It's tricky, unless you really have a good geometric understanding of how the whole container fits together!) Surprisingly, the kiddies didn't really need any help with most of this stuff...
After class, a kid came up to me and said, "These (last problems) are not hard. But, you really need to know your stuff!" It made me feel a little extra proud of them for recognizing their own growth.
I am now looking forward to trigonometry goodness. :)
*What do you do in your classroom to support literacy? I do a lot of making-kids-write-about-stuff, but not enough reading!!!
---------------
PS. On the other-things front, I have accepted a job offer from an international school in Berlin! Geoff and I are SUPER excited. It's official news, and all of my bosses know and are happy for me. :) :)
Lesson 7 in the unit was on liquid density. This one was my favorite, because the kids had to first discuss as a class how to measure the mass of a liquid. After they came up with the general idea, I had them take notes on the definition and procedure for finding "net weight," and we related it to the labels we see on packaged food containers. Kids were excited that they now knew what "net weight" on the corner of their cereal boxes meant!
I then had kids split up into groups of 2 or 3. They were instructed to work on a rather tricky practice sheet of conversions and volume problems, and I pulled out a few groups at a time to rotate around to do parts of the liquid density lab. They needed to measure the density of oil, water, and maple syrup using graduated cylinders and triple-beam balances. (To make this manageable, I gave each group a "clean" cylinder that they would use to find the weight of the container, and whenever they needed to pour the liquid into the container, they would use a "dirty" container that the groups before had used for the same liquid. This way, we didn't have to keep cleaning the graduated cylinders in between every group.) In the end, once ALL of the groups had finished gathering data, we discussed as a class what would happen if we were to pour all 3 liquids into the same graduated cylinder. Then, we tested it! I showed them that even if you flipped the container and straightened it back up, the liquids would still separate themselves. (--To a degree, anyway. The syrup and the water begin to mix gradually, since syrup is water-based and it gets diluted over time.)
It was super fun!! :) Now after this, the kids really have a good grasp of how to measure mass, net weight (net mass), volume, and what all of it means. Lovely!
For my honors kids (who are truly done-done with the whole unit, including the review... the other kids are still a couple of days behind), we followed it up with this: a reading on how to measure an elephant, and another on how to measure the oceans. We discussed the articles after they had read them individually, to make sure that they understood everything in the readings. (I try to insert some relevant reading into every unit to contribute to their literacy.*)
And they got a chance to try their hands at putting together the most complicated concepts in the unit -- making predictions about a 3-D container. (The second page of this was my favorite. It's tricky, unless you really have a good geometric understanding of how the whole container fits together!) Surprisingly, the kiddies didn't really need any help with most of this stuff...
After class, a kid came up to me and said, "These (last problems) are not hard. But, you really need to know your stuff!" It made me feel a little extra proud of them for recognizing their own growth.
I am now looking forward to trigonometry goodness. :)
*What do you do in your classroom to support literacy? I do a lot of making-kids-write-about-stuff, but not enough reading!!!
---------------
PS. On the other-things front, I have accepted a job offer from an international school in Berlin! Geoff and I are SUPER excited. It's official news, and all of my bosses know and are happy for me. :) :)
Thursday, January 20, 2011
Measurement Unit: Episodes 5, 6 (Density!)
Following the introduction to volume formulas, my kiddies spent a day or so doing various conversions in class. The wildest is when they found out that it would take 200,000 liter-bottles of water to fill up our classroom! (And that 1 cubic meter = 1,000 liters = 1,000,000 cubic centimeters!! They were amazed by how huge that number sounded, even though they were absolutely convinced that 100 x 100 x 100 = 1 million little cubes inside the "huge" cubic meter.)
After that, I did with my kids today my second favorite lesson out of the entire measurement unit -- density lab! It's actually a two-labs-rolled-into-one sort of thing. I had to set up 6 lab stations. At 3 of the stations, there are irregular containers (two spray bottles of different sizes, and one curvy baby bottle) whose volumes need to be measured using transfer-of-water idea. (I provide them with some extra empty beakers and some bottles of water.) At 3 other stations, there are triple-beam balances, beakers, water, and some object (rock or cube) whose density needs to be measured. They had to practice using the various instruments to gather the mass, volume (either using rulers or displacement method) and to predict which objects would float/sink in water, and then to test their predictions. (They were so excited when their cube floated! It's pretty funny.)
The lesson was kind of hairy to set up (since it had involved borrowing a lot of stuff from the Math & Science Center, and digging up my old supplies of irregular containers and objects), but super fun and easy to run! The labs pretty much ran themselves. I just had to go around and make sure kids were cleaning up after themselves in between rotations and were resetting their scales. In the end, we talked about why metal ships float and why the Titanic sunk. (I had to ask, "Do you guys know about the Titanic??" You know, these kids are babies!! One of them told me that Leonardo DiCaprio is old.) And we talked about whether the object's density would change if we tagged on more cubes. (I had made the cubes out of lego-like manipulatives.) My kids were so smart! Some of them realized that g/cm^3 isn't going to change even if you add more cm^3. I also talked them through what happens if you double the volume of a cube -- what would happen to its mass? (After that, the kids were convinced that the overall density, or ratio of mass and volume, would still remain the same. --You like how I threw in a little math word there?) :)
Anyway, it was really fun! :) I am sad that we're nearing the end of our unit. We only have one more measurement lesson left for the honors kids (regular kids are just a couple of days behind) -- net weight and density of liquids! I'll be sad when it ends, but I am already thinking ahead about shooting Pringles cannons for the next unit on triangles and trigonometry.
(Sadly, I can't seem to eat all of the Pringles fast enough. I'll have to email my kids this weekend and ask them to eat some Pringles over the weekend and to donate some cans, so that we can have spare cannons in case one blows up during class!)
After that, I did with my kids today my second favorite lesson out of the entire measurement unit -- density lab! It's actually a two-labs-rolled-into-one sort of thing. I had to set up 6 lab stations. At 3 of the stations, there are irregular containers (two spray bottles of different sizes, and one curvy baby bottle) whose volumes need to be measured using transfer-of-water idea. (I provide them with some extra empty beakers and some bottles of water.) At 3 other stations, there are triple-beam balances, beakers, water, and some object (rock or cube) whose density needs to be measured. They had to practice using the various instruments to gather the mass, volume (either using rulers or displacement method) and to predict which objects would float/sink in water, and then to test their predictions. (They were so excited when their cube floated! It's pretty funny.)
The lesson was kind of hairy to set up (since it had involved borrowing a lot of stuff from the Math & Science Center, and digging up my old supplies of irregular containers and objects), but super fun and easy to run! The labs pretty much ran themselves. I just had to go around and make sure kids were cleaning up after themselves in between rotations and were resetting their scales. In the end, we talked about why metal ships float and why the Titanic sunk. (I had to ask, "Do you guys know about the Titanic??" You know, these kids are babies!! One of them told me that Leonardo DiCaprio is old.) And we talked about whether the object's density would change if we tagged on more cubes. (I had made the cubes out of lego-like manipulatives.) My kids were so smart! Some of them realized that g/cm^3 isn't going to change even if you add more cm^3. I also talked them through what happens if you double the volume of a cube -- what would happen to its mass? (After that, the kids were convinced that the overall density, or ratio of mass and volume, would still remain the same. --You like how I threw in a little math word there?) :)
Anyway, it was really fun! :) I am sad that we're nearing the end of our unit. We only have one more measurement lesson left for the honors kids (regular kids are just a couple of days behind) -- net weight and density of liquids! I'll be sad when it ends, but I am already thinking ahead about shooting Pringles cannons for the next unit on triangles and trigonometry.
(Sadly, I can't seem to eat all of the Pringles fast enough. I'll have to email my kids this weekend and ask them to eat some Pringles over the weekend and to donate some cans, so that we can have spare cannons in case one blows up during class!)
Sunday, January 16, 2011
Letting Kids Develop Their Own Graphical Vocabulary
...Ha! Lots of math posts today/this weekend. I guess you can tell that math is on my mind and that my boyfriend (who is usually the poor victim of my math blabbings) is busy with his own pet projects. :) :)
Anyway, I was planning a lesson to teach my precalculus kids some basic graph-analysis vocabulary -- things like concaving upwards/downwards, local minimum/maximum, inflection points. And I wondered: how much of this can the kids come up with on their own? When they are looking at a graph (or forced to look at it closely by describing it to their "blind-folded" friend), how many different features can they pick out without any assistance from me?
I guess we'll find out! This is modeled after a Geometry activity I had done in the past, where kids had to describe transformations in a coordinate plane in their own words before I taught them the proper ways of numerically specifying transformations. Except here, all my juniors will have to do is to get their friends to re-draw the graph without gesturing and without peeking at the original graph! --Easy?
I am interested in whether any group will develop ideas similar to concavity. At the minimum, they should figure out that they're going to need to specify slopes, "highest"/"lowest" points, some type of discussion of curvature, and (hopefully) roots! If they can already pick out all of these distinguishing features on their own, I have every reason to hope that the formal vocabulary will stick without too much trouble.
(Speaking of which, I don't teach middle-schoolers anymore, but it seems like this type of method of developing graphical vocabulary can be extended to middle school, when kids first learn to identify slope and y-intercept. If you give a kid a line and they have to get their friend to re-draw it based on verbal directions only, and they're not allowed to name (x, y) coordinates, what types of things would a kid pick out of the graph to describe??)
Anyway, I was planning a lesson to teach my precalculus kids some basic graph-analysis vocabulary -- things like concaving upwards/downwards, local minimum/maximum, inflection points. And I wondered: how much of this can the kids come up with on their own? When they are looking at a graph (or forced to look at it closely by describing it to their "blind-folded" friend), how many different features can they pick out without any assistance from me?
I guess we'll find out! This is modeled after a Geometry activity I had done in the past, where kids had to describe transformations in a coordinate plane in their own words before I taught them the proper ways of numerically specifying transformations. Except here, all my juniors will have to do is to get their friends to re-draw the graph without gesturing and without peeking at the original graph! --Easy?
I am interested in whether any group will develop ideas similar to concavity. At the minimum, they should figure out that they're going to need to specify slopes, "highest"/"lowest" points, some type of discussion of curvature, and (hopefully) roots! If they can already pick out all of these distinguishing features on their own, I have every reason to hope that the formal vocabulary will stick without too much trouble.
(Speaking of which, I don't teach middle-schoolers anymore, but it seems like this type of method of developing graphical vocabulary can be extended to middle school, when kids first learn to identify slope and y-intercept. If you give a kid a line and they have to get their friend to re-draw it based on verbal directions only, and they're not allowed to name (x, y) coordinates, what types of things would a kid pick out of the graph to describe??)
Measurement Unit: Episode 4 (Volume Formulas)
It's funny how a little bit of a change to a lesson can make a humongous difference.
Last year (and a couple of years prior), I reviewed volume formulas with my students by putting them into groups and rotating a bunch of prism- or cylinder-shaped containers around. (Largest container was a plastic bucket. Most were tupperware.) They had to measure the objects and to calculate the approximate volume, in cubic centimeters.
This year, I added a new piece. I wanted the kids to be able to visualize that their volume measurements are correct (or incorrect), without me giving them my answers. So, I collected a bunch of liter water bottles and filled them up in advance with water. At the end of the kids measuring/calculating all of the volumes, I said to them that we could certainly verify the volumes with cubic centimeter blocks, but it would require thousands of them per container, and it just doesn't scale. So, instead, we were going to use water. I held up a liter bottle and held up a plastic 10cm-by-10cm-by-10cm container and asked them to vote on which one they thought looked bigger. Once we got outside (Ooh! So warm and sunny!), a kid volunteered to do the demo where they poured the water carefully from the bottle into the cube. Gee whiz! They're exactly the same! To emphasize that this means that a bottle of 1 liter water is equivalent in volume to 1000 cubic centimeters, I showed them using a math manipulative item how you can neatly fit 1000 cm^3 cubes snugly inside the cube container, the same way that 1 liter of water had filled the same container to the brim.
Now that we knew that 1 liter = 1000 cm^3, we started to fill up some of the containers they had measured with liters of water. I kept asking kids what values they had gotten for each container, and volunteers kept pouring in more water to see if it would overflow. (We had some measurement instruments, obviously, to obtain increments smaller than 1 liter.) It was super neat. Every step of the way I kept exclaiming to the kids, "Remember that this means that you're adding in another ______ of those little yellow cubic centimeters!" (Kids were getting secretly competitive, obviously, when other groups' answers were getting eliminated and when the containers turned out to hold about as much as their own answers.)
It was super cool! We collectively marveled in the end at how even a relatively small container can hold a couple of liters of water -- proving that they can fit thousands of those little yellow cm^3 cubes!!
Afterwards, we went back up to the classroom. Our next task was to figure out how we can predict the height of the water inside a new prism-shaped container, once you transfer it from an old container that was filled to the top. I let the groups struggle with this for a while on their own, and most of them figured out one way of doing it (with some guiding questions, mostly). At the board, I had the kids explain their ways of doing it, and we saw that both ways -- finding % of volume taken up in new container, then multiplying it by the total height of the new container; or setting up V = l*w*h with volume of water and solving for h -- arrived at the same answers using very different geometric understanding. I made the kids show me work both ways, using their own measurements/numbers, to verify that they hadn't made an arithmetic error somewhere and that they did indeed understand both methods, and I heard kids say, "Tsssssss..." when they were finished. It's a noise that Salvadoran children make to indicate that something is unexpectedly cool, and that noise made me smile.
And of course, we did the actual experimentation. I filled Container A with water to the top, transfered it over to Container B, and announced to the class that the height rose to about 6cm. Kids got to see for themselves that their calculations brought them to the right ballpark of predictions!
How fun! Now we're ready for big boy conversions!
Last year (and a couple of years prior), I reviewed volume formulas with my students by putting them into groups and rotating a bunch of prism- or cylinder-shaped containers around. (Largest container was a plastic bucket. Most were tupperware.) They had to measure the objects and to calculate the approximate volume, in cubic centimeters.
This year, I added a new piece. I wanted the kids to be able to visualize that their volume measurements are correct (or incorrect), without me giving them my answers. So, I collected a bunch of liter water bottles and filled them up in advance with water. At the end of the kids measuring/calculating all of the volumes, I said to them that we could certainly verify the volumes with cubic centimeter blocks, but it would require thousands of them per container, and it just doesn't scale. So, instead, we were going to use water. I held up a liter bottle and held up a plastic 10cm-by-10cm-by-10cm container and asked them to vote on which one they thought looked bigger. Once we got outside (Ooh! So warm and sunny!), a kid volunteered to do the demo where they poured the water carefully from the bottle into the cube. Gee whiz! They're exactly the same! To emphasize that this means that a bottle of 1 liter water is equivalent in volume to 1000 cubic centimeters, I showed them using a math manipulative item how you can neatly fit 1000 cm^3 cubes snugly inside the cube container, the same way that 1 liter of water had filled the same container to the brim.
Now that we knew that 1 liter = 1000 cm^3, we started to fill up some of the containers they had measured with liters of water. I kept asking kids what values they had gotten for each container, and volunteers kept pouring in more water to see if it would overflow. (We had some measurement instruments, obviously, to obtain increments smaller than 1 liter.) It was super neat. Every step of the way I kept exclaiming to the kids, "Remember that this means that you're adding in another ______ of those little yellow cubic centimeters!" (Kids were getting secretly competitive, obviously, when other groups' answers were getting eliminated and when the containers turned out to hold about as much as their own answers.)
It was super cool! We collectively marveled in the end at how even a relatively small container can hold a couple of liters of water -- proving that they can fit thousands of those little yellow cm^3 cubes!!
Afterwards, we went back up to the classroom. Our next task was to figure out how we can predict the height of the water inside a new prism-shaped container, once you transfer it from an old container that was filled to the top. I let the groups struggle with this for a while on their own, and most of them figured out one way of doing it (with some guiding questions, mostly). At the board, I had the kids explain their ways of doing it, and we saw that both ways -- finding % of volume taken up in new container, then multiplying it by the total height of the new container; or setting up V = l*w*h with volume of water and solving for h -- arrived at the same answers using very different geometric understanding. I made the kids show me work both ways, using their own measurements/numbers, to verify that they hadn't made an arithmetic error somewhere and that they did indeed understand both methods, and I heard kids say, "Tsssssss..." when they were finished. It's a noise that Salvadoran children make to indicate that something is unexpectedly cool, and that noise made me smile.
And of course, we did the actual experimentation. I filled Container A with water to the top, transfered it over to Container B, and announced to the class that the height rose to about 6cm. Kids got to see for themselves that their calculations brought them to the right ballpark of predictions!
How fun! Now we're ready for big boy conversions!
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