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Copyright © 2010-7 Joseph Mack
15 Mar 2017, released under GPL-v3.
A video of a lesson on the Platonic solids, given to 3rd graders, 29 Apr 2009.
Material/images from this webpage may be used, as long as credit is given to the author, and the url of this webpage is included as a reference.
Table of Contents
Since about 2002, the science teacher at my son's school, Lyn Streck, has let me teach a science class to her 3rd graders. There's four classes in the grade and I give all lessons in one day.
The students cut out and assemble the NOAA Surface of the Earth Icosahedron Globe (http://www.ngdc.noaa.gov/mgg/fliers/04mgg02.html).
The maps are low cost ($0.50 ea), and printed on cardstock making a nice finished globe (I have them on my desk at work and home).
|You can download a pdf of the maps and print them yourself, but you'll need a large format printer, lots of ink and your own cardstock. It's cheaper and more convenient to buy them from NOAA.|
A new version of the map (released in 2008, here called v2 globe) has press out tabs. In the video I'm using the old version, v1, which needs cutting out. I didn't start using v2 till 2013, when my stock of v1 maps ran out. With the v1 globe each student needs
The class is 40mins and can only just be done in the time.
You want every kid to walk out with a functional globe, no matter how badly the kid botches the job. Following a large mistake, there isn't time to start again, and an adult will have to handle patching the globe.
The instructions for assembling the globe are relatively complicated, and for v1, the cutting takes time. Adults need to be on hand to help the kids. Despite my best attempts to give clear directions, with the v1 maps, some student will cut off the flanges (tabs), while others turn their scissors around and cut from the inside out, gouging the map (see the video for directions).
To help form the map into a sphere, the carboard is folded over the edge of a desk, to make 60° bends in the carboard. Despite my best efforts and demonstrating what to do and what not to do, and holding up the partially folded map, showing that the map is not being folded into a cube, and is being folded into a sphere, there is always some kid who folds the cardboard over 180° and attempts to iron the fold flat with a hard object.
The last part of the taping (closing the globe) is tricky and usually has to be done by an adult.
Besides building the icosahedral globe, other material covered in the class is
trusses (since the icosahedron is made of triangles). (Trusses were first mathematically studied by James Clarke Maxwell. Trusses are a good place to start when studying mechanics.)
The tetrahedron, octahedron and icosahedron have triangular faces and are rigid (when you try to deform the sides). The cube (hexahedron, with square faces) and the dodecahedron (pentagonal faces) are floppy. Early in the class I ask the kids if they can figure out why some of the Platonic solids are rigid and some floppy. For the first 10 years, I encouraged them try out the models themselves during the class and, if they have an answer, to let me know quietly (so not to let the cat out of the bag to the rest of the class). About half the time, some student figures it out (or already knows, I can't tell which). I had expected that most students would figure it out. As I discovered accidentally in 2011 (see below), the reason the kids weren't figuring out the answer, was that I wasn't giving the kids enough time to play with the models.
The trusses and the earlier versions of the Platonic solids were made from toothpicks and hotglue. Making these models took several sunday afternoons. (Buy your toothpicks all at once; toothpicks bought a year later, at the same store, in the same type of box, were a different length.)
When I asked the kids to figure out why some of the toothpick and hotglue models were floppy and some were rigid, I first had to convince the students that some of the solids were intrinsically rigid and some were instrinsically floppy. With the glue providing some rigidity at the vertices, even for the floppy models, the usual answer was that the rigid solids had more glue. While the kids saw the glue providing rigidity, they wouldn't look further to structural elements as the answer. As well the kids often broke the relatively fragile toothpick and hotglue models while testing their rigidity.
My first approach was to tell the kids that the rigidity wasn't coming from the glue. Quite reasonably kids don't trust adults, who are in charge, make arbitary rules and can make a kid's answer wrong any time they want. As all the kids could plainly see, the glue was providing (some) rigidity. It took several years before I saw the futility of expecting them to believe that the glue wasn't providing rigidity. I needed models with demonstrably floppy vertices. This led to the models with rubber tubing vertices in the video. Since I've changed over to these models, students have stopped proposing glue as the agent of rigidity.
In 2015, after displaying the relatively fragile toothpick and glue models in the front row, I moved them behind the more resilent rubber tubing and drinking straw models, and let the kids play with the more resilient models.
With about 30 models arrayed on various tables, one student asked in amazement if I'd made all of them. I said "yes, and it was great fun".
By the wonders of digital editing, on combining videos from four different classes, the students give correct answers to all the questions.
Errata. A volunteer at the school kindly edited the videos of the four classes for me in their spare time and I wasn't able to get the following fixed.
There is some duplication of material; the same part of the talk from different classes appears more than once. A virus not mentioned in the talk is HPV.
(There is not enough time in class to refer to these notes; you have to know the material off by heart.) With only a few minutes at the end for the kids to explore trusses and their globe, an extra 5 mins, if you can find it, can make or break the class. In 2011 I made these changes to the class
v1 At the suggestion of the teacher's assistant, I precut the cardboard sheets (for the v1 maps) (paper cutter/guillotine; a sharp one helps) removing extraneous cardboard and separating the legend and base from the map. This eliminated the first cutting step for the kids.
v2 Initially I pre-popped out the maps and handed the kids the whole piece of carboard, with the popped out pieces resting in the original locations.
In 2017, I handed the kids only the popped out pieces. This saved a couple of minutes. It's too bad that education now is paced too fast for the kids to explore around what they're doing, and be able to pop out their own maps (the class was originally 50min, with 16 students; now it's 40mins with 19 students. The 10 mins was taken from a couple of classes to allow the students to take an extra class in being well rounded.)
In the early years, I had held off putting out the maps until the kids were ready to start building the globes. I had expected the kids would be distracted by the map and not pay attention to my explanation of the platonic solids. However stopping the class to hand out the maps adds time to the class. In 2015 I put out the maps before the class. To head off the kids being distracted by the maps in front of them, in the introductory phase of the class, I first acquainted the kids with the maps by having them put their name on some blue spot (ocean) and on the stand, with marker pen. This seemed to give them time to satisfy their curiosity about the maps and they were happy to pay attention to my discussion about the platonic solids.
The first thing the kids need to understand is that the tabs are pushed in from the outside and that the tab is taped over on the inside, where it can't be seen. This is obvious to an adult, but not to the kids. If you start the kids taping the globe, an adult will have to cut out the tape holding the tabs facing the outside, instead of the inside.
Instead, first I have the kids make the stand. This has only one tab and a mistake can be fixed relatively easily. Closing the loop to make the stand requires the kid to put the tab in from the outside, and align the colours, so that the white is just not showing. I tell them to show me their stand before putting on the tape.
In 2015, in the first class, despite holding up a sample and showing them how to put the tab in from the outside, half of them got it wrong (==random), which means that none of them were listening (or my explanation was unintelligible).
For the next class I emphasised that they had to show me the tab before taping telling them that otherwise the next part of the globe would not work. By the last class of the day, I'd emphasised it enough that they'd all showed showed me the correct insertion before taping.
I found that none of the kids knew how to tear off a piece of tape. This must have been the first time they've ever used a tape dispenser. They pulled the tape down straight, rather than tearing from one corner. This required enormous force for success. I wasn't prepared for this and didn't know how to handle it. The next year, in the preparation part, I showed them how to tear off tape.
I told the kids to only tear off a piece of tape long enough to just cover the tab (1cm). Instead they tore off a piece about 4cm long. I let this go.
Several stages of checking are required (cut out, fold along all the black lines) so that the final map folds to a globe.
At the time I did the video, as soon as the first student was ready for the next stage, I stopped the whole class and demonstrated the next stage to everyone. This didn't work. The kids are busy with their efforts and quite reasonably don't have attention for much else and they can't listen. This is particularly so when the first student reaches the last stage - closing the globe - at which time some students are still back two stages - cutting out (for v1). I would have to repeat the taping instructions individually to confused students a few minutes later as they reached the taping stage.
Instead, in 2011, as student(s) were ready, I showed them individually how to apply the tape. The tape and the to-be-taped globe were right infront of the students, who could see exactly how I did it. Initially only the first student was ready, then a few minutes later, a group of four more students were ready... Although this method required me to be in a constant frenzy, repeatedly explaining the next step to whichever student was in front of me, few students had to return for clarification, reducing the time on this step.
With these changes, after assembling their globes, the students had time to gather around the rubber and drinking straw platonic solids, to figure out why some were rigid. Talking and poking, they collectively figured it out and when I asked for the answer, they all knew it was triangles. Over the previous 10yrs of this class, only a couple of times has a student figured out the answer. I'd concluded that the problem was too difficult for them, when all that was missing was a few minutes poking the models.
I then had about 5 minutes at the end of the class to talk about the trusses and explore oceanic trenches and the location of mountain ranges on their globe. You still have about 1/4 of the class with uncompleted globes; these are handled by the adults, while you're talking.
In 2016 following a disasterous first class, when the kids barely completed the globes, and we didn't get to talk about trusses, I made the following changes.
In 2017, I accepted starting the first class at 8:30am, because that day fitted my schedule better. Usually I start at 10:30am. Starting at 8:30am doesn't leave enough time to train the assistants to assemble a map and to put out the maps, and pieces of tape. I arrived at 7:30am (when the school opened) and we got it all done. Still it was a rush.
We popped out all the map pieces and just gave them the pieces. We threw away the outer cardboard.
We put out 15 short pieces of single sided sticky for each student, as suggested by Miss Connie in 2016. We were trying to avoid the problem that the kids can't dispense tape and they spend a lot of time picking up and putting down the dispensers. In 2016 we put the tape on the same side of the table as the kid, with the tape sticking out into mid-air. The kids wound up folding the tape down onto the vertical side of the table, making the tape unusable. This was through leaning on the tape, or for something to do while they were listening. In 2017 we put the tape on the opposite side of the table to the kid. Now they had to reach for the tape, and not quite being able to grasp the tape, because of the distance, often wound up folding the tape down onto the table. We told the kid to move around the other side of the table, once they started assembing the map.
In 2017 we did not fold the maps over the edge of the desk. The maps assembled just as easily as when the maps were pre-folded along the creases. Not doing the folding saved time
In 2008 a new version of the map (here called v2) was released. (I had plenty of the v1 maps and didn't start using the v2 maps till 2013). With the v2 map, no cutting is required. The map presses out by hand from the cardboard sheet. The tabs fit into precut slots in the cardboard. Like the v1 map, an adult can make the v2 into a perfect globe in relatively short time. With the v2 map (like the v1 map) the kids, particularly under the gun of time, need adult help.
After a little folding of the edges, the map can almost be assembled into a recognisable globe. Unfortunately it springs apart in your hands. I tried a rubber band around a diameter, but the globe collapses. I expect you need 3 not-too-tight rubber bands placed symmetrically to make this work. Thus sticky tape is still required.
With the v2 globe, the tab in the middle of each flange is required (for v1, I cut it off). If you use the v1 method, of double sided sticky tape to seal the flange to the inside of the globe, then each flange requires the positioning of two half sized pieces of double sided sticky tape. The dexterity required is OK for an adult with plenty of time, but is beyond 3rd graders, under the gun to finish on time.
For the v2 globe:
There are two rows of tabs to tape down; top and bottom. Pick one row and then start taping from one end to the other interlocking the pair of edges with the tab Do not start taping in the middle; it's hard to get your hand in to insert the tape.
Insert the tab (from the outside), carefully line up the colours on the maps, so the white cardboard underneath just disappears. Alignment of the edges (using the colours on the outside) is critical, or you can't close the globe. (You won't find this out till the end. If the globe doesn't close well, an adult has to cut pieces out of the globe.) Once the edges are aligned, put a piece of tape lightly on the tab (inside of the globe). Check that the map join is placed correctly. If you want to change it, an adult can lift the tape off the inside of the globe with a boxcutter, and the kid can do it again. If the alignement looks right, press the tape hard between your fingers.
As you work along the top/bottom, you'll find that accessing the inside of the globe needs longer fingers or more dexterity than the kids may have.
While the alignement of the colours at the edges of the join is critical, the placement of the tape in the inside is not; almost anything that holds the globe together will do. You can tape along or across the join of the edges.
Once you've done the top, invert the globe and start on the bottom, taping from the end.
At the end you are left with 3 pairs of edges to join As well the three pairs of edges each have a tab in the middle. For almost everyone in the class, an adult will have to join these edges. We just used single sided tape on the outside of all 3 joins. It doesn't look great, but the kids get the class done in time.
If you want no tape showing, you need 6 pieces of double sided sticky tape. Each of these is inserted, after first slightly prying open each of the 6 gaps, to insert a piece of double sided sticky tape on the blade of a small screwdriver. This is OK for adults, but doesn't fly for kids in a time limited class.
In the v1 globe, I had the students cut off the square tabs. Each pair of edges was joined by a single piece of double sided sticky tape. At the last step, an adult handled the final pair of edges. You squeezed the globe (or pried open the final pair of edges) so that the edges opened slightly like a mouth. A piece of double sided sticky tape was inserted into the gap on a not-too-sticky flat object, like a small screwdriver, letter opener, or one blade of a pair of scissors. The tape was pushed hard against the cardboard on the inside of one of the edges, where it would stick, allowing the flat blade to be carefully withdrawn, leaving the double sided tape stuck to the cardboard. You then allow the globe to unpucker, when the two edges meet and stick. A bit of massaging of the join from the outside, with your fingers, forms a reasonable seal.
With the v1 map, to save time, I precut the maps, removing the main extraneous carboard. With the v2 map, popping out the map is simple for an adult, once you've done is a couple of dozen times, but it's new to the kids and naturally they're interested in how everything works. I found the v2 maps went slower than the v1 maps. For the first year, I pre-popped-out the base and legend, leaving them attached at one or two points, so they didn't separate from the main piece of cardboard. I also pre-popped half of the map (leaving some for the kids to pop out, since it's fun.) For 2014, the class was 5 mins shorter (now 35mins) and I had to pop everything out before the class, just giving them the relevant pieces of cardboard, and throwing away the trim.
About half the joins have to be done with adult help or the class won't finish in time. It helps to have a couple of adults on hand.
In the v2 map, the slots are 1-D cuts rather than a real slot. You can't get the tabs in easily, unless you expand/open the slot a little bit with scissors or a screwdriver. One of the problems was aligning and joining the edges. The kids should push the tab in till the colours match. If you push the tab all the way in, rather than stopping just a bit short, i.e. when the colours match up at the edges, then at the end, there isn't enough room to close the icosahedron and you'll have to cut bits of the icosahedron to close it. The resulting icosahedron isn't symmetrical. With all the instructions being foisted on them, the kids had a hard time adding colour matching to their tasks. In 2014 I found that telling the kids to leave the thinnest white line between the adjoining colours worked.
It's obvious to an adult that the flanges and tabs go inside the globe, but it wasn't to the kids. It took me a few classes to notice that many of the kids were joining the maps with the tabs facing outwards. For the last class in the set of four classes for the day, I had the kids do the base (which has one tab-slot pair) first, rather than last, so I could check that the tabs were folded inwards and the colours matched. Despite my excruciatingly clear explanation, holding the finished circularised base up to show the kids, and showing that the tab was inside and putting the tape on the inside, and telling the kids to check with me if they weren't sure what they were doing, almost half the kids had the tabs pointed outwards and taped down so I couldn't easily undo them without tearing. The success rate for my explanation is no better than random and indicates that my explanation was no better than no explanation at all. For 2014, I had them do the base first and made sure they had the tabs facing inward before letting them put the sticky tape on the inside.
Is the v2 map better? Most of the students don't do a great job of cutting, so the resulting v1 globe isn't particularly symmetrical and the colors don't match up well across joins. The cutting problems are gone v2, but the colour matching at the edges is still a problem. At the last step in the assembly of v2, there's a lot for the adults to do. Even so, the class is finished a few minutes earlier, which helps. I'd say it was a step sideways, but what you've exchanged is different for adults and kids. Some things are easier and some things are harder. The cutting is gone, making that easier for the kids, but closing the globe is impossible for kids and has to be done by adults. (Note: For 2013 we had the kids close the globe by putting single sided sticky tape on the outside, rather than the adult method of double sided sticky tape between the flanges and the inside of the globe. Many of them do this on their own, without waiting to be told how to close the globe.)
Suggestions: As part of the design phase, the map needs to be tested on kids (adults can stand in for kids in the early rounds of testing). The students have a lot of trouble getting the tab into the slot, particularly when closing the sphere, when their sphere isn't symmetrical and the tab and the slot no longer line up. A pointed triangular tab, rather than a square tab, would help get miss-aligned tabs into the slot. It would be nice to have a sphere that can be assembled without sticky tape. The tabs could be non reversible - e.g. have an "L" shape that catches on the inside of the slot, using the natural spring of the cardboard.
The 240 piece icosahedral jigsaw globe of the earth, seen in the video, comes in 3 versions, with 240, 560 and 960 pieces (search in the internet with "3-D puzzle of the world" or "icosahedral puzzle globe of the earth"). Presumably they're all icosahedrons. An icosahedral globe will have multiples of 20 pieces (4-fold as for my 80 faced toothpick icosahedron as seen in the video, or 3-fold as seen for the 60 subunit capsids of icosahedral viruses).
I have no vested interest in the use of these maps. I like maps and bought a few of these to decorate home and work, several years before I met Lyn. I think everyone should get a chance to build one of these.
There is a large spread in skills in any class. Usually there will be one kid for whom the whole process is obvious, and who won't need any direction, while there will be another kid who has no idea what's going on and has to be walked through the whole process and then thrown over the finish line. There must be at least a log difference between the top and the bottom kid in each class. I had not expected this, but now after being a soccer coach and referee for 10yrs, teaching several different classes at the school, and teaching astronomy to boy scout troops, I find the same spread no matter what you're teaching. Presumably professional educators know about this spread, but I didn't. Previously all my teaching had been at university level where the difference between the top and the bottom kids wasn't so marked.
I remember in one of these 3rd grade science classes, when I was showing the kids how an eclipse occurs, having one of the whizzes from my soccer team, looking at me with exasperation - he had no idea what was going on. So as the kids move between environments, the kid on top and the kid on the bottom changes every 40 minutes.
Goal is to make the NOAA globe, starting from a flat sheet of carboard (hold both up). Everyone will go home with a completed globe.
You kids probably haven't heard of many of these viruses. That's because you don't have to worry about these diseases now. Vaccinations, modern medicine and public health measures have handled most of the problems. When I was growing up, every kid was off for a week from school, sick and home in bed, with some viral disease every semester. Vaccinations have made a huge difference to the health and lifestyle of children. You get your shots before you start school or daycare and you go about your daily life without worrying about when you'll next be sick.
Anyone know the most significant advance in modern medicine in the last 100yrs, as far as preventing disease, saving lives and increasing our lifespan? (washing your hands and having safe drinking water - it turns out the best safety measures are also the cheapest.)
The video shows cutting out the v1 globe. The v2 globe, which I'm using now, instead pops out and is done a little differently. With the v1 globe, once the kid's folds were checked out, the kids could tape the globe up in one go, only requiring adults to close the globe. Here with v2, I'm doing it more stepwise till I know what the kids can do.
As I was watching the kids I noticed their lack of dexterity compared to adults. The other thing I hadn't noticed previously is that their fingers are shorter than adults. An adult's fingers are long enough that a dexterous adult can put a piece of sticky tape on the back of a tab in one move. A kid just can't do this.
When 1/2 to 3/4 of the kids have taped up their globes, move on to the next section. Have adults finish the globes of the last kids (the slow kids are not going to finish in the time of the class). If not, you won't finish the class and the kids who have finished, with nothing to do, will go crazy.
Tell the kids who have finished their globes, to look at the rubber and drinking straws platonic solids to figure out why some are rigid and some are floppy. They can also roll the polyhedral dice.
In one of my earlier sessions, I had them all try to figure it out together; once the first kid figured it out, they all knew and they could check the answer on the spot. However that way only one kid gets to figure it out. That didn't work.
The next year, when they had an answer, I had them not tell each other and instead asked them to sit down. After a few kids had sat down, I asked the class for answers. To my dismay, I found that only some of them had the correct answer. The kids who were wrong, and now back in their seats, couldn't check the right answer. This didn't work either.
Being stumped, I asked Lyn (the professional educator). The way you do it is
- let the kids come up and try to figure it out all together, but to keep quiet if they think they know.
- when they have an answer, come check with me, but tell me quietly enough that the other kids don't hear their answer.
- if they're right they sit down, or play with the polyhedral dice, but keep quiet about their answer.
- if they're wrong, try again. Most kids are wrong (but are convinced they're right) and need multiple attempts.
- when about half the class is up at the table (i.e. half the kids have finished their globes), get everyone to sit down. Ask the kids who had the right answer to identify themselves by raising their hands, then to call out all together, the single word that is the correct answer.
Only about half the kids got up to the table; the rest were working on their globes. In 2014 I had four classes of 18-19 kids, in which 1,0,2 and then 3 kids found the right answer (i.e. a total of 6 kids out of 74, only half of who made it to the table).
I didn't pay much attention to the wrong answers. Once I realised that the kid hadn't got it, rather than listening to a long winded wrong answer, I just said to try again. One kid was talking about squares and triangles, so I asked which ones were rigid and which were floppy and then they got it. All the other kids who got it just said "triangles" and that was that. Some kids didn't know what "rigid" meant. Afterwards, recalling the wrong answers, it was clear that the kids had no idea about falsifiability. They said it was due to the structure (well, what is it about the structure?), that the rubber in some of the models was more bendy...
Falsifiability is a modern (20th century) concept, and was only arrived at after several centuries of thought. I've had a lifetime of reading examples of falsifiability and the consequences of mistakes. You can't expect kids to rediscover falsifiability on demand, in a matter of seconds, at a table. I was told about the requirement for triangles in rigid structures long before I started thinking about why structures stood up, so I'm asking the kids to do something I hadn't done at their age.
Perhaps I should ask them "if you think you have the answer, before you come to me, check that your answer is true for all the rigid structures and false for all the floppy structures" .
I have an 80 piece and a 240 piece icosahedron (in the video). These show how you can make smoother spheres, still using triangles. The piece in the video was shot between classes. I've never had time to show them to the kids.
The class is sequenced to follow the section on plate tectonics and mountain building. The questions are designed to reinforce this material. Only obvious (high and low) places on earth shown on the globes, are discussed in the class e.g. Greenland, Antarctica, Himalays, Andes, Colorado Plateau, Hawaiian Is, deepest part of the ocean (it's not in the middle, it's at the edge) - Marianna's trench (deeper than Mt Everest is high), Atacama trench to the west of the Andes (26kfeet deep).
Posters on the wall
With Lyn, at school in the icosahedron class kit:
With Lyn at school, materials not specific to this class, that have to be located:
Pick a day 4, i.e. first class at 10:30am, two classes in morning, two classes in the afternoon. In 2017, the first class was at 8:30am. I had to put out 18 maps, and with the help of the assistant - 300 pieces of sticky tape, all between 7:30am and 8:30am. We did it, but it wasn't easy.
2017: I brough my boxes, checked Lyn's boxes (nothing was missing or broken). This didn't take long. I popped out maps for 4 classes; this took a while, maybe an hour.
In 2017, all the posters were rolled up from last year. They had to be multiply taped down to the board to keep them from curling up. It took a bit of work to get them off the board, rather than just lifting them up from the untaped bottoms. In 2017, I rolled them up again. In 2018, I should unroll them a week ahead.
2014: this took about 5hrs.
2015: after getting organised last year, this took 2hrs. Nothing was broken.
2017: after finding everything a week earlier, I setup: posters on wall and models on table. This took about an hour.
v1: for each pair of kids, a single tray containing: sticky tape (two single sided and a single double sided reel), two kid grade pairs of scissors, marker pen.
v2: marker pen and single sided sticky tape.
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