tag:blogger.com,1999:blog-14356076023562356152024-03-13T04:30:39.682-07:00Courage To CoreAnonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-1435607602356235615.post-25042908628830746242016-03-05T14:06:00.001-08:002016-03-05T22:13:20.111-08:00Is Pi Always 3.14...?<div class="MsoNormal">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgt-mpK0Nu4j26qehCEaBOc61vyx1-AdeN32CCzpBd2LMFZlWJVMaM6tFZe1UTXkmC9VxWjetyG6xlPXNZSmXbunApcJVmE05po_aynPVpX2T5wq0svhl2meZkVcgSFyb53Sqwo6pePLC2A/s1600/12745894_1573944596264059_693231664512900006_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgt-mpK0Nu4j26qehCEaBOc61vyx1-AdeN32CCzpBd2LMFZlWJVMaM6tFZe1UTXkmC9VxWjetyG6xlPXNZSmXbunApcJVmE05po_aynPVpX2T5wq0svhl2meZkVcgSFyb53Sqwo6pePLC2A/s200/12745894_1573944596264059_693231664512900006_n.jpg" width="200" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">One of the greatest experiments students can do to celebrate
pi day is measuring the circumference of a circle and dividing by the measured diameter
to approximate pi. This is a cool exercise not only because it is an engaging
experiment, but also because it gets students to think about and observe an
essential characteristic of circles, namely that every circle is similar to
every other, and C/d is a constant. They might even start to wonder,
why is it that pi is approximately 3.14 and not some other value? This is a
wonderful mystery that made me think: Is there a way to make pi something else?</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Well, let’s first review where pi comes from. Imagine you
get a puppy and name him Pie-Dough. Because Fido was already taken. You’ve got
a nice big yard for him to play in, but he’s always getting into things he
shouldn’t, so sometimes you put him on a 20 meter leash that is tied to a tree in
the middle of the yard. Of course, he is always running around at the end of
the leash. Here’s what that looks like:</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Of course, Pie-Dough is happily running in a circle. The
circumference of that circle divided by the radius (the leash) is 2pi. No
surprises there. But things are about to get pretty fantastic. Imagine that you
move to the North Pole, and you put Pie-Dough on a super lightweight leash
that is, get this, 500 kilometers long! And Pie-Dough has crazy puppy energy,
and still runs in a circle at the end of the leash. But I have a question for
you: Is the leash pulled straight? Check out the image below, with the leash in
blue and the path of Pie-Dough as a black circle:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyvhhFYrHccrSNCJJ014LqU3GpqLN-QAMNbakb8WuSGdD7qhCNLAZ9IMlz2cGLLwllJV_zwt-BVCkBB90rBCy4JlxZz1OvASlzuij4gTADwXCZxq24oWMHbRdVWiCnhM_yPJcwUYO3bEFr/s1600/small+circle+sphere.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="209" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyvhhFYrHccrSNCJJ014LqU3GpqLN-QAMNbakb8WuSGdD7qhCNLAZ9IMlz2cGLLwllJV_zwt-BVCkBB90rBCy4JlxZz1OvASlzuij4gTADwXCZxq24oWMHbRdVWiCnhM_yPJcwUYO3bEFr/s320/small+circle+sphere.gif" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Because the leash is so long, Pie-Dough’s leash is actually
curved along the surface of the earth! He actually doesn’t get to run in as big
of a circle as he could if the leash were straight. Can you see why? So now, if
you divide the circumference of his circle by the leash length, will you still
get 2pi?</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Nope, you won’t. Because his “radius” is curved. The
Circumference divided by the “radius” will be less than 2pi. This is even
clearer in the diagram below, where Pie-Dough’s leash extends all the way to
the equator (he can swim oceans, no problem). Can you figure out C/r in this
case?</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaVuAFG39GtzVhrEPP1kAQWL3rqd-QH9oLb__ieTRiRhquc8mmh1d-5FAXw8G-7Xpp0BBjkD6d5h4VJ5JAlDcngY9WU2HLpmIqMSQ4FbcjcLmYDsw65hx0v4NX-gM6pwrqy21Th-YF2xuZ/s1600/big+circle+sphere.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="207" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaVuAFG39GtzVhrEPP1kAQWL3rqd-QH9oLb__ieTRiRhquc8mmh1d-5FAXw8G-7Xpp0BBjkD6d5h4VJ5JAlDcngY9WU2HLpmIqMSQ4FbcjcLmYDsw65hx0v4NX-gM6pwrqy21Th-YF2xuZ/s320/big+circle+sphere.gif" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">So you might be thinking, well yeah, the ratio is different
because I’ve distorted the “radius.” That’s true. But another way to think
about it is that I’ve changed the space we are working in from a flat surface
to a curved surface. We stopped looking at the world from Euclid’s perspective,
where the ratio C/r = 2pi, to a non-Euclidean perspective, where C/r < 2pi. Our
universe changed from a flat plane to the surface of a sphere! In fact, the actual universe on large scales is curved, so there is good reason to look at it from a non-Euclidean perspective. The universe can be curved like the spheres we saw above, or it can be
curved in such a way that C/r > 2pi. What does that look
like?</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs88Y_czAX3kuDc0IRL_rSTKwa9sZNORLKfEsUvz9Jhk7FdmPQOJSnNQ2EGzTTJhL1ePhiMxyKN12ywENzJ9gxWrQvalANoio4xdsYk_4qUGX9h81h9ZlvvT-tavww-Bkx5q-r52tJWzpq/s1600/saddle.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="209" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs88Y_czAX3kuDc0IRL_rSTKwa9sZNORLKfEsUvz9Jhk7FdmPQOJSnNQ2EGzTTJhL1ePhiMxyKN12ywENzJ9gxWrQvalANoio4xdsYk_4qUGX9h81h9ZlvvT-tavww-Bkx5q-r52tJWzpq/s320/saddle.gif" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">So in the diagram above, imagine Pie-Dough’s yard is the
funky red saddle shape. The sphere there is just to roughly show you Pie-Dough’s
possible paths, if he could fly off the red saddle. But let’s say he can’t fly
off the surface and he just runs around on the curved path where the blue
sphere and the red saddle intersect. The points A and B show some examples of
where Pie-Dough could be, with the leash shown as a dotted line from the
center. Actually the leash should also be curving, stuck on the saddle, but this was the best approximation I could make using Geogebra. If you know of a better visualization let me know!</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV-9viYPTrpQd1O3keNvXtE-VZakemI9URmd75Tpl7XQ70U6Ozs0v4JB_M3oZuMRSIgLAlZMAo1uts73vpueMEQyOH_1Ei-ME-J_gjBEubfzbfFgSUFbmceQcYDefUCuZ11qZYWCe6fVRo/s1600/saddle+pic.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="212" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV-9viYPTrpQd1O3keNvXtE-VZakemI9URmd75Tpl7XQ70U6Ozs0v4JB_M3oZuMRSIgLAlZMAo1uts73vpueMEQyOH_1Ei-ME-J_gjBEubfzbfFgSUFbmceQcYDefUCuZ11qZYWCe6fVRo/s320/saddle+pic.png" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">As you can see, Pie-Dough can run quite a distance along
that topsy-turvy path, a lot more than he could when he was stuck on a flat
surface. In this case, because the universe he’s running in is curved like this
saddle, the “circumference” is huge compared to the “radius,” giving us a ratio
of C/r > 2pi.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">So there are two things I think are cool about this. First
off, at large scales, the actual universe can be curved as shown above.
Secondly, if you have a baseball, you can explain to someone how the
circumference divided by the radius can be less than or greater than 2pi! It's all about thinking of it from the perspective of Pie-Dough--when he runs on his path in his curved universe, for him, the yard is flat, the radius is straight, and the path is a circle. But we know that because his universe is actually curved, the geometry get's weird, specifically, it gets non-Euclidean. Pie-Dough, we're not in Kansas anymore!</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Oh, by the way, here’s another way to approximate pi quite
accurately. It’s a freebie so check it out!</span></div>
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<a href="https://www.teacherspayteachers.com/Product/Pi-From-A-Pendulum-2428949" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWsEMMa5C_XS1M_VNClB8HewQ9zUnK9RDoykQkQSa0-P24LyVXL-W0wyojFBUbI2zFPizqk47a01NbbpVquFuYwCETVKUeiAzy2FqRG9my-NoaxFsV_vSEdlmtbOkAwHAoqdFBxUBMBcPe/s200/Pi+CP-01.png" width="200" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Finally, check out these other great Pi-Day posts by
inspiring teachers! Happy 3.14! </span></div>
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<!-- end InLinkz script -->Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com2tag:blogger.com,1999:blog-1435607602356235615.post-55883849276182516542016-02-28T23:28:00.001-08:002016-02-28T23:49:13.671-08:00Leap Day Sale!<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDxJzpSfpqF6p8P1G4-6XK9dy849B0JkrqVPCnYYeWq7GVZyo3-d3SjSybFvnhrQlFr-OPt0-yZIPnfTmkZIHD0acjw5UjPesibpKhs4yac-2sRruZmFvp5deAyK4C9-PYISA47NDWWkc9/s320/leap+day+sale.jpg" width="320" /></div>
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Visit the blog post at <a href="http://scaffoldedmath.blogspot.com.es/2016/02/secondary-math-leap-day-sale.html" target="_blank">Scaffolded Math And Science</a> to get a list of links to great secondary math stores having a sale today!</div>
<br />Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-53634149789067853632016-02-20T09:24:00.000-08:002016-09-16T18:02:09.062-07:00It's Your Boat<div class="separator" style="clear: both; text-align: center;">
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Captain, here’s what you need. You need a magic potion. Could be a little brandy to take the edge off, or coffee to put the edge back on, but you need to walk into your classroom and grab the wheel and right the ship. Or maybe the ship is maddeningly on course, like some robot ship from the future. Or there’s no ship, everybody’s in a life preserver, including you, smiling weakly because hey you’re still floating. Or everything’s going swimmingly, which makes you feel like the water’s probably a bit sharky. Maybe you’re the captain and first mate and crew and wondering if anyone else is going to do any work around here. You think you should just go back to farming. You really need a magic potion.
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<a name='more'></a><span style="font-family: "arial" , "helvetica" , sans-serif;">So what can you do? Be yourself, or your alter ego. Answer questions with a certainty that conveys authority and motivates buy-in, or admit you don’t know an answer to demonstrate that you’re all in this together. Command the room with captivating faux-thoritarian theatrics, or cede the stage to student directors. Never work harder than your students—give them ownership. Always work harder than your students—show them you care. No wonder you're seasick! It’s like a conference where the keynote is about the great new Common Child Succeeds Behind Pearson system but in the next room over there’s a guy evangelizing for a universal Pre-School Semester-At-Sea program (all grades are sink/swim). Meanwhile, you know, class starts in like 5 minutes and you still have to pee from all that, um, coffee. </span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Let me level with you. The deck never will be. That’s also true for the playing field, budgets across departments, the answer blanks on the photocopies you just made, and so on. Kids love fair and as a teacher you will spend a lot of time modeling how to deal with life’s not. Accept the things you cannot change and all that. Maybe I shouldn’t have had that nip of brandy at the top, but we can’t change the past.</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">But Captain, there’s a lot of decisions you still get to make, and despite the conflicting salvos of advice you’ll hear, it doesn’t have to be chaos on the stacked deck of the USS "Hello, my name is ________."<hello ____________="" is="" my="" name=""> Because even though there’s not one right way to navigate the passage ahead, there’s a right way for you and your charges today. I know that sounds squishy as a jellyfish, but there’s a big difference between anarchic relativism and customizing your classroom to create a unique environment that plays to the strengths of you and your students. And these days, dear internet reader of internet writer, you are a pirate. Steal ideas, improvise solutions with the tools at hand, experiment and risk the Cliffs of Insanity for the sake of engaging the students in the adventure. Even though treasure would be nice (actually, really nice), you didn’t sign up for the doubloons, and your kid crew isn’t paying attention to you because they are stoked on job training per se. They want you to be a pirate, and they want to set sail. They are on board. (Even if sometimes you’d love for them to walk the plank.)</hello></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><hello ____________="" is="" my="" name=""><br /></hello></span>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><hello ____________="" is="" my="" name="">And I know that might sound a little smug, as if I’ve always been the King of the World, but some days I’ve ridden the stern of the Titanic, maybe been the ‘berg itself. Jack Sparrow’ed to keep them on their toes. Played the fearsome Blackbeard to keep them in line. Lurked secretively as Dread Pirate Roberts, hardly there except to save the day occasionally. And maybe this blog post using pirates has totally jumped the shark—ye went from pirates to zombies a long time ago, and in 2016 I should totally be doing super-heroes to keep up. Pirates was a risk, but taking risks is something I learned from a teacher once. She was pretty new, and she asked me for my advice. I told her what I thought she should do, and she took it on board. Next day she tossed it overboard like so much chum, did something else. Aye, I said, it’s your boat.
</hello></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><hello ____________="" is="" my="" name=""><br /></hello></span>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><hello ____________="" is="" my="" name=""><br /></hello></span>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><hello ____________="" is="" my="" name="">Check out more great teaching advice at the links below!</hello></span><br />
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com22tag:blogger.com,1999:blog-1435607602356235615.post-3495105110410002362016-01-25T01:29:00.000-08:002016-02-21T13:49:04.837-08:00Beating the Math Game: Helping Students Embrace the Struggle<div class="MsoNormal">
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Doc Running at <a href="http://everyed.blogspot.com.es/" target="_blank">Everything Ed</a> has organized another great blog hop, this time on strategies to help students who struggle in math:</span></div>
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<a href="http://everyed.blogspot.com.es/2016/01/math-monday-strategies-for-struggling.html" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyr5eCVKBgY4-CjXx5k5BlXX-hLk0loCcHEjB0nQAYuq2NFiA645LahmspMSGekT-sb6aw9DiQxpWCA7zRmFQP-LcSk58adB1moliHhl4Q6g6LErKEBqPL3B_ifAdhgsFLO97V0_uHoBmg/s200/huh.jpg" width="200" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Gaming is such an integral part of today’s culture that it
can be hard to remember that in the early 80’s, video games were still a
novelty one step away from pinball machines. Today, the graphics and audio as
well as the complexity of the narrative have advanced to cinematic proportions,
and today’s online multi-player games allow extensive social interaction and
coordination. But my 20<sup>th</sup> century adventures with the pixelated
worlds of Atari and Nintendo 64 were nearly as compelling. How is that possible?</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR3a4R-btTITVHAsGuX_2pA5Z450ByizIDvmMhSNeYxf7MSCiPqt0QSGtrin3nNk9VESmtx9VUdLZWkF_kJJbewsV2f_AdjVM-6IPFAvE8YUutskDKD_s6_fAbIeGui52A0eYHJHA0zm6J/s1600/pacman.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR3a4R-btTITVHAsGuX_2pA5Z450ByizIDvmMhSNeYxf7MSCiPqt0QSGtrin3nNk9VESmtx9VUdLZWkF_kJJbewsV2f_AdjVM-6IPFAvE8YUutskDKD_s6_fAbIeGui52A0eYHJHA0zm6J/s320/pacman.gif" width="320" /></a></div>
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<a name='more'></a><span style="font-family: "arial" , "helvetica" , sans-serif; text-align: center;">It’s not just that I was younger then. It’s that one big
part of gaming is learning and creatively applying algorithms precisely to get
a desired result. It’s worth remembering that despite the mind blowing
sophistication of gaming today, the player’s actual interaction with many games
has hardly changed since the early days—you are often trying to solve spatial puzzles
by pressing buttons in the correct order at the correct moment. It’s deeply
satisfying to master algorithms and make progress with increasingly complex
puzzles. But it takes resilience in the face of repeated failure. We love that
process.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsrDR9UuE-pq6gV9N3VFITKpya4XzhtZf-9dFk_OffchS9Ti29Ao6o9TpRa6YVw-3bOUcpBkDLcH6aV9NkvKVcrzrX-1OoYH8v6q0vu2YMNDh7C_d1rPKq3PNMlOaI0bTsX8CfEZ59wqvu/s1600/xgzcv.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsrDR9UuE-pq6gV9N3VFITKpya4XzhtZf-9dFk_OffchS9Ti29Ao6o9TpRa6YVw-3bOUcpBkDLcH6aV9NkvKVcrzrX-1OoYH8v6q0vu2YMNDh7C_d1rPKq3PNMlOaI0bTsX8CfEZ59wqvu/s1600/xgzcv.gif" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; text-align: center;">But why is it that many students shy away from complexity
when its presented in a high school math class? In literature students wade
into Shakespeare to understand the best of the English language. In science, students
must do experiments, and we don’t dumb down the periodic table or the Kreb’s
cycle to avoid complexity. It would be obscene to treat every historical event as a simplistic, perfectly resolved narrative of good and evil. And yet, math has a
reputation for being utterly algorithmic, solvable like the simplest video game
from the 80’s. In fact, many students expect that most math problems actually
have a cheat code, and beg the teacher to release them so everyone wins the
game.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhImcd7_OGxsvq4OLc8PMNkzlgiLICXVEG_AMmHGxXiB_QYaZ0SJi1kxF5slfAVVBxvkny49oRWZ674bO3pTrXd6XLENUNSdF5ftypQPOb4jZnymA8uOoo_9WNKRL5JD7Swvm2_KWISOnGa/s1600/cheat+code.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhImcd7_OGxsvq4OLc8PMNkzlgiLICXVEG_AMmHGxXiB_QYaZ0SJi1kxF5slfAVVBxvkny49oRWZ674bO3pTrXd6XLENUNSdF5ftypQPOb4jZnymA8uOoo_9WNKRL5JD7Swvm2_KWISOnGa/s1600/cheat+code.gif" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Why is it relatively easy to embrace the struggle in other
disciplines but frequently hard for students to do so in math? The short answer
is that the game of math looks like Pac Man when actually it’s more like
Minecraft. The objective of the game of math isn’t just to crack levels, but to
build worlds. Sometimes you’ll be a construction worker following a set of
instructions, but often you’ll be an engineer, coming up with the procedure the
construction worker will use. Sometimes you’ll be the architect, designing
something that is both beautiful and functional. The more your world mirrors
the real one, the closer you are to winning the game.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">So how do we make a real game of math for students?</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">1) Give students a
steady diet of open-ended but do-able problems grounded in the real world for
which they don’t yet have the cheat codes. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">The first time you played any
game, you died pretty quickly. Also the second time. You knew this would
happen, and you responded well to the experience, because you knew that the
game was designed for you to learn by experimentation, failure and restarts.
Students may come to you expecting a cheat code and hoping to play perfectly
from the start, but they need to learn that the game doesn’t work that way. Many
students are unaccustomed to working without a rehearsed algorithm pre-placed
in their hands and will need time to engage with problems creatively.</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">2) Give students guided
experiments which help them develop the tools to solve the real world problems you’ve
given them. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">Traditionally we have given students hammers and nails and told
them to practice incessantly, with the promise that one day they would build a
house. Instead I suggest students try and build a dog house with what they’ve
got on hand. Then, just as they are asking for better tools, we help students
develop the tools they need to build a better dog house. In this way students
have a deeper understanding of and greater ownership for the algorithms in
their toolbox. A great example is a linear function: an awesome tool whose
efficacy resonates really well when introduced after a real-world problem that
can be modeled with a line.</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">3)</b><span style="font-family: "arial" , "helvetica" , sans-serif;"> </span><b style="font-family: Arial, Helvetica, sans-serif;">Have students work together in pairs and
small groups. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">Students who lack basic skills or the confidence to engage
creatively with problems can fade into silence unless they are continually
challenged to interact. Small group work pushes weaker students to remain involved
and gives you regular feedback. Advanced students learn to articulate ideas
intelligibly for their peers.</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">4) Go back as far as
you can reasonably go. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">Students who are accustomed to using cheat codes
will often lack a real understanding of basic concepts and procedures. The
cumulative nature of our world-building means we can’t leave these skills
behind. Revisiting fractions, order of operations and so on pays dividends
later.</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">5) Create problems
which can be answered using a variety of tools. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">The diversity of abilities
in the classroom means that we have to create problems and experiments which
allow everyone to contribute at their level. There are many problems whose solution
can be achieved through approximate measurement, graphical estimation,
guess-and-check with a table, or algebra. Creating scenarios which can be
approached from multiple angles engages students at different levels, offers a
method of checking results and demonstrates the open-ended nature of the game of
math.</span></div>
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<b style="font-family: Arial, Helvetica, sans-serif;">6) Allow students
time and space to struggle. </b><span style="font-family: "arial" , "helvetica" , sans-serif;">Given the time and curriculum constraints of
the school year it can be challenging to allow students to grapple with
problems and experiments. Students need time to make mistakes and try again,
and if they are not accustomed to that experience in math they will need even
more time.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">The game of math is not won on points or time, but on the
complexity of the world constructed. If we want students to build real world
models and solve open-ended problems, they must reach beyond cheat codes to
become not just construction workers, but the engineers who design the
algorithms and the architects who design the models. The struggle will require
repeated failures and restarts, but that’s what makes gaming fun.</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG3o4EjOYCK0uUqmiaednesPBAC3wox98bBElOlzao3a-ixI_pRmO8WV3zQ92XY5xZ8_J0C3RJvDUB1qriwFJuxtGXz2kKtOa_kGz3C8xO2YwnFatTL6aC1jLDauiR3s8axmELWAWw6nwN/s1600/gameover.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="192" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG3o4EjOYCK0uUqmiaednesPBAC3wox98bBElOlzao3a-ixI_pRmO8WV3zQ92XY5xZ8_J0C3RJvDUB1qriwFJuxtGXz2kKtOa_kGz3C8xO2YwnFatTL6aC1jLDauiR3s8axmELWAWw6nwN/s320/gameover.gif" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Check out the other great post in this blog hop below!</span></div>
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<a href="http://www.inlinkz.com/new/view.php?id=601100" rel="nofollow" title="click to view in an external page.">An InLinkz Link-up</a></div>
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<!-- end InLinkz script -->Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com8tag:blogger.com,1999:blog-1435607602356235615.post-57648066556637612022016-01-19T01:31:00.003-08:002016-01-19T23:17:14.787-08:00Teachers Pay Teachers Sale!<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.teacherspayteachers.com/Store/Courage-To-Core" style="margin-left: 1em; margin-right: 1em;">
<img a="" alt="" border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiD8TMcr7kPtD_D67b4jIB_O5hWeHX0ZeA26nxCn4bX_k655ml2p6Ioj1RO2bh0sp6dAoqgdIZXU767G3cBZxm4EWdvxLxG4t1QHJWuXZKL6PVcvahYZhZlcPYTpiAnTGBhIWItc0ZxRwYa/s200/sale_160_200.jpg" width="160" />
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-27592958002988350002016-01-17T05:58:00.001-08:002016-01-24T02:14:59.782-08:00Small Group Math Facebook group!<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhPan0GwlXeN4F_Y3608Wru-Pa9r28OdPG5MdNIVI8AYcbxuK2BzTtvkK_Oo6sRdpImmopI4VfMYeF4VLtgQFba85XWHaqq5cv7_C1uiyQfi45hpKKpFCqqK_MfPtP5igEEHcjj4yI3EW/s1600/Picture1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="239" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHhPan0GwlXeN4F_Y3608Wru-Pa9r28OdPG5MdNIVI8AYcbxuK2BzTtvkK_Oo6sRdpImmopI4VfMYeF4VLtgQFba85XWHaqq5cv7_C1uiyQfi45hpKKpFCqqK_MfPtP5igEEHcjj4yI3EW/s320/Picture1.png" width="320" /></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Calling all middle and high school math teachers! As you know, I am a big exponent of small group work. Maybe you are equally animated by seeing kids collaborate, or you are intrigued but cautious, wondering how to make it work. Either way, you'd love to join a group where you can share ideas, ask questions, and find a resources from a variety of sources that makes it easier for you to weave small group work into the classroom environment. Lucky for all of us, such a group exists! It's obviously called Small Group Math. Go join it at the link below, and invite your fellow math teachers to do the same!</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://www.facebook.com/groups/961493860591651/" target="_blank">https://www.facebook.com/groups/961493860591651/</a></span></div>
Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com2tag:blogger.com,1999:blog-1435607602356235615.post-55759444633631216282015-12-08T04:27:00.003-08:002016-01-24T02:19:25.451-08:00Making Mastodons RealFor many years, in addition to teaching I ran an outdoor education program at a school in urban southern California. Most of my students lived far-removed from the natural world and outdoor education was literally a breath of fresh air for them. Usually we took students to Joshua Tree National Park, a land of giant granite boulders sculpted by the wind and glowing gold in the setting sun. One evening we drove into the park with a new group of middle schoolers, their faces pressed to the window in awe of the hulking stones looming like mastodons. One kid, who had studied the basics of geology in his science class, nevertheless innocently exclaimed, “Woah, who put these here?”, thinking that the fabricators of Disneyland had been hard at work in Joshua Tree. This climbing trip made those stones real for the students, bridging the gap between the abstraction of nature and the reality of rocks.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkIK-3zmdR_vg-p869a3po1RJeZGBagSyCv7O2yOUrb42sRTXi_lbn9i26e7sVhaXiTEmLedLeo6iZhssxDZRr_Xj8rUm8sjHGqCAZzDwDOVVZ7I61j8DBbRN9sobVv-Sr_5CRmLN5Wyt8/s1600/climb+jtree+for+letter.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkIK-3zmdR_vg-p869a3po1RJeZGBagSyCv7O2yOUrb42sRTXi_lbn9i26e7sVhaXiTEmLedLeo6iZhssxDZRr_Xj8rUm8sjHGqCAZzDwDOVVZ7I61j8DBbRN9sobVv-Sr_5CRmLN5Wyt8/s200/climb+jtree+for+letter.JPG" width="133" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Climbing in Joshua Tree</td></tr>
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<a name='more'></a>How do we bridge the gap between the abstraction of mathematics and the reality of the world it quantifies? We can’t always take a field trip, but we can challenge students to do small experiments, measuring and calculating in the context of some real-world problem. In this way mathematics becomes a tool for understanding something about the world, not only an abstraction. Don’t get me wrong, the formal abstractions of mathematics are awe-inspiring. But my students in southern California needed some help connecting their understanding of geology in the classroom with the actual world in their midst. And students come to our math classes yearning for the numbers to become real in some way.<br />
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The best way to get a student interested in rock climbing is not to climb with them. It’s to go on a scramble, a hike through uneven terrain, involving some boulder hopping and creative wiggling. Eventually a kid will want to go higher, and I tell them, “Ok, let’s get a rope!” In the same way, I love seeing students creatively engage with problems at the limits of what they know until finally they are asking for a tool to solve a problem, or even developing one. Building anchors and tying into ropes for its own sake is pretty boring, but when you really want to climb something it’s brilliant.<br />
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Courage To Core represents my fifteen year effort to develop materials that give students a certain amount of autonomy to grapple with real world problems and build the tools to solve them. I’m so glad that Teachers Pay Teachers has given me the chance to share these materials with you. My most recent resource is linked below—I hope you and your students enjoy its narrative and experimental nature, and that together we can bridge the gap between the abstract and the granite.</div>
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<a href="https://www.teacherspayteachers.com/Product/Trigonometry-and-Special-Right-Triangles-TR3-HSGSRTC6-HSGSRTC8-2246812"><span style="color: blue;">Introduction to Trigonometry and Special Right Triangles</span></a></span><br />
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com4tag:blogger.com,1999:blog-1435607602356235615.post-28709399520056215502015-11-16T12:26:00.000-08:002015-12-08T04:29:22.442-08:00High School and Beyond, Spanish Style (An Interview)<div class="MsoNormal">
Thanks to <a href="https://www.teacherspayteachers.com/Store/Darlene-Anne" target="_blank">Darlene Anne</a> for organizing this intriguing blog hop around the theme of Creating a Global Classroom!<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNhJywlytfeonOzHTr6vNS18pJxS1DllmbW0wr6SoQxCYUHlQPCj1Uu8L2uHsCUt6mvOhB38MyWvYiddR91lUiwba8o400EZ5wmziXvwueSKSw-MnGtYYGZRQRcZ3hEzBh2txd21H4ZYpw/s1600/global+classroom.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNhJywlytfeonOzHTr6vNS18pJxS1DllmbW0wr6SoQxCYUHlQPCj1Uu8L2uHsCUt6mvOhB38MyWvYiddR91lUiwba8o400EZ5wmziXvwueSKSw-MnGtYYGZRQRcZ3hEzBh2txd21H4ZYpw/s320/global+classroom.jpg" width="320" /></a></div>
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As I live in Spain this is a great opportunity for me to share a little about the educational system here. I've interviewed a health teacher at a vocational school in Alicante to bring you insights into the path of a typical Spanish student. Loida Moya Smith is our interview subject--she's a nurse who teaches at the Secondary Education Institute of Leonardo da Vinci in Alicante. She offers one of many perspectives on education in Spain. Enjoy!</div>
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<b style="mso-bidi-font-weight: normal;">We call the last four
years of schooling before university “high school.” I gather it’s somewhat
different in Spain?<o:p></o:p></b></div>
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<div class="MsoNormal">
Yes, we have primary school from age 6-12, then secondary
school from age 12-16, then students can stop studying if they want. At 16 they can either continue with vocational training for 2-4 years, or
the Bachillerato. Bachillerato is a two-year program that finishes with an exam
called Selectividad. Your grade on this exam along with your grade in the
Bachillerato program are averaged to determine university admission.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">So if a student wants
to go to medical school for example, do they take the same Bachillerato course as
every other student, or do they need to select a particular path?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
They would need to select the Technology and Sciences track
in Bachillerato then take the Selectividad for that. Students can select either
Technology and Sciences, Social Sciences or Art, with some sub-divisions.
Everyone has to take Spanish language, Spanish literature, a foreign language,
philosophy, physical education, ethics, contemporary science and Spanish
history. Then if students choose the Technology and Science, for example, they
take courses like physics, technical drawing and math.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">In the US, university
admission is based on your grades for four years, the difficulty of the classes
you take, your performance on AP exams, standardized tests like the SAT, your
application, teacher recommendations, extra-curriculars and more. Are their
other factors at play for admission in Spain besides Bachillerato and
Selectividad?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
No. It’s always been based on just those two things. But it
used to be that you had 4 years that were included in the Bachillerato and now
it’s just 2. And that means that students only have pressure for the 2 years and
they don’t try hard enough because it doesn’t count and students aren’t that
motivated.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Don’t kids or parents
think, if I don't do well now I won’t be able to succeed in
Bachillerato later?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Many times we call parents when we see low grades, and they
listen and accept what we say, but they don’t generally put a lot of pressure
on them. <o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Why is that?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
I think actually this goes all the way back to the Spanish
Civil War (which ended in 1939), when the priorities were protecting and
preserving family in the face of adversity. My grandparents were more concerned
with keeping us safe and together. Like many people of that time, they had a
farm which provided food, and even I grew up in that same environment. All they
wanted to do was keep the family together, so much so that when I left my home
to study in Ceuta it was initially difficult for my family to accept that I was
leaving our town. So even though success in school and career is important, the
first priority is always a happy kid that is close to the family.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Do you think the
economic crisis of 2009 has changed anything?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
I think so, young people are more motivated to find a job
outside of the country. <o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">How much does it cost
to go to university in Spain?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Around €2000 for a public university. It could be €10,000 to €20,000 for a year at a private university. Usually students can find the money
for a public university if they don’t have it—I attended university on a government
grant.</div>
<div class="MsoNormal">
<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Are you much more
likely to get a job with a university degree?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Any health related degree is good. It’s hard to find a job
with a law degree.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">What about vocational
training?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Students who are studying in health or science related
technical fields, like nursing assistants and laboratory technicians, they are
finding work pretty quickly. <o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Is there an effort to
continually update vocational training to fit the evolving needs of the
economy?<o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In the past five years there are lots of new health related
degrees, which is my area of expertise, and I know there are new programs related
to tourism, restaurant work, graphic design and computer programming.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">Would you recommend
vocational training or university for a student? <o:p></o:p></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Vocational training can really motivate a student. Students
can get a mid-level degree which enables them to work, but then they can take
two more years and get a superior degree which provides better work
opportunities and they can go to university without jumping through the
Bachillerato and Selectividad. I think unless a student is very motivated
academically, going through vocational school and working first gives students
life experience, and then they can go to university with a clearer vision of
their future.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">What advice would you
give to an American student considering attending a university in Spain?</b></div>
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Be ready to socialize. It can be tough to study in Spain—you
have to have clear goals and be prepared to try hard and stay focused. We have
great universities and in particular our medical research institutions and
nursing programs are very strong. There are clubs and teams but it’s not
heavily promoted—the main extra-curricular is the night-life.<o:p></o:p></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_pfImp1aE0iPKJEkrRqY-VQNqRkeu3lL4-wMh3y_okhaJm5jfmhtkXDy8Danun7XH5DepG4RF4RM_Qx_ktAbuK04RxlXAB9IhG1X_qkIyJ7kdIyBCuom6PKL9h2XT7EZ5CCLk1wdgsj2-/s1600/8loida+hike.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_pfImp1aE0iPKJEkrRqY-VQNqRkeu3lL4-wMh3y_okhaJm5jfmhtkXDy8Danun7XH5DepG4RF4RM_Qx_ktAbuK04RxlXAB9IhG1X_qkIyJ7kdIyBCuom6PKL9h2XT7EZ5CCLk1wdgsj2-/s320/8loida+hike.JPG" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Loida Moya Smith<br />
Health Education<br />
IES Leonardo da Vinci, Alicante</td></tr>
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<div class="InLinkzContainer" id="578894">
<a href="http://www.inlinkz.com/new/view.php?id=578894" rel="nofollow" title="click to view in an external page.">An InLinkz Link-up</a></div>
<script src="//static.inlinkz.com/cs2.js?v=116" type="text/javascript"></script>
<!-- end InLinkz script -->Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com5tag:blogger.com,1999:blog-1435607602356235615.post-38377693326360168232015-10-26T11:13:00.001-07:002016-04-25T16:14:09.098-07:00Predicting The Great Pumpkin<div class="MsoNormal">
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Thanks to <a href="http://everyed.blogspot.fr/2015/10/math-mondays-math-treats-blog-hop.html" target="_blank">Doc Running</a> who has organized a Halloween blog hop, you are in for a treat from several great math teachers! I've got a big story for you that I hope you'll share with students, and links to more great resources. Read on!</span></span><br />
<br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">I always smile when I think about
the scene in Big where giant Tom Hanks is coincidentally eating a tiny ear of
corn. Partly I smile because it’s great comedy—he approaches the task
with a seriousness reminiscent of Chaplin eating his shoe.</span><br />
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<div class="separator" style="clear: both; text-align: left;">
<span style="font-family: "arial" , "helvetica" , sans-serif;">But I also like this scene because science! Corn (more precisely, maize) was domesticated </span></div>
<a name='more'></a><span style="font-family: "arial" , "helvetica" , sans-serif;">around 9000 years ago. Like all domesticated plants, maize has a wild ancestor, in this
case a Central American grass called teosinte. The seeds have hard shells and
look like rocks when dry to be camouflaged against the ground. They didn’t want
to be eaten!</span><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><o:p> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://teosinte.wisc.edu/questions.html" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaiFYKba5xeJWNVe8Jn4mCWWPeDNvZpl-n_4klckxBzAwmcbq6vP_Z2TxqzP-dKFyk_goB9TtMxRBWKyiXQtPTtHp240npn5b6i2-KfZtYNs1n56C4_Jhr8H9aVFNWSAdjuiwbqSXzZPD0/s200/Z_diplo_ear_w.jpg" width="128" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: xx-small;"><span class="My_Banner" style="background-color: white; text-align: start;">University of Wisconsin-Madison</span></span></td></tr>
</tbody></table>
</o:p></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">But early farmers selected the
most edible and planted them, continuing the process until they created
miniature versions of our modern maize. Below are the ancient cobs that archeologists
found, measuring just 6 cm ( 2.4 inches). Was Tom Hanks eating a variety of domesticated maize from 4200 BC?<o:p></o:p></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><br /></span></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.pnas.org/content/98/4/2104.full" imageanchor="1" style="margin-left: auto; margin-right: auto;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" src="http://www.pnas.org/content/98/4/2104/F1.medium.gif" height="200" width="193" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><h1 id="article-title-1" itemprop="headline" style="background-color: white; border: 0px; color: #202020; font-weight: normal; line-height: inherit; margin: 0px; outline-style: none; padding: 0px; text-align: left; vertical-align: baseline;">
<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: xx-small; line-height: inherit;">Proceedings on the National Academy of Sciences of the United States of America</span></h1>
</td></tr>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><a href="https://www.blogger.com/blogger.g?blogID=1435607602356235615" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></span></div>
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">Well no, he was actually eating
baby corn—modern maize harvested tiny. But still, it’s amaizing
how big domesticated corn is compared to its wild ancestor! Impressed? Well then, consider the giant pumpkin. This record holder from 2014 weighed 2323 pounds!<o:p></o:p></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><br /></span></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://greatpumpkincommonwealth.com/index.php/gpc-champions/overall" imageanchor="1" style="margin-left: auto; margin-right: auto;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" src="http://greatpumpkincommonwealth.com/images/Champions/Overall/Screen_Shot_2015-09-02_at_9.51.41_PM.png" height="164" width="200" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Great Pumpkin Commonwealth</span></td></tr>
</tbody></table>
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<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">Is this just a normal pumpkin
that’s been overfed, or is it some genetic freak? A little of both. The seeds
of a monster can sell for thousands of dollars as growers select the genes for
giants. But they also tend to these plants with a care usually reserved for
royalty—fertilizing, watering, removing pests and turning the pumpkin to avoid
cracking or rot over 4 months of growing. And these farmers are doing a great
job—nearly every year there has been a new record for heaviest pumpkin ever. In
1976, the record pumpkin weighed only 451 pounds.<o:p></o:p></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">In interviews, growers and
botanists agree that the records will continue to fall, with some speculating
that a 4000 pound pumpkin is close at hand. That’s about the weight of a Ford Taurus!
Does the math support this prediction? Let’s look at a graph. The x-axis
represents the year and the y-axis represents the weight of the world-record
pumpkin for that year. <o:p></o:p></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "calibri light"; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><br /></span>
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<a href="https://s3.amazonaws.com/grapher/exports/gvii0gldvk.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="275" src="https://s3.amazonaws.com/grapher/exports/gvii0gldvk.png" width="400" /></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">Performing a linear regression is the
statistical process of finding the line of best fit for the data:<o:p></o:p></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><br /></span></div>
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<a href="https://s3.amazonaws.com/grapher/exports/w9xtr1vute.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="275" src="https://s3.amazonaws.com/grapher/exports/w9xtr1vute.png" width="400" /></span></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTjXHjlUbIM-dsvPFfTxLKTI8zXGXNJgSqH_IWmZZSqoU26TPzKw_vUxUIgCZHE6Z0igdawwZJqymdwlB5tL7bx3goeVk0URiNnnT8Z5pOIWbjoufu7w2PoFg_4qgAddIlkkAbBIGniY8C/s1600/Screen+Shot+2015-10-26+at+7.29.54+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="37" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTjXHjlUbIM-dsvPFfTxLKTI8zXGXNJgSqH_IWmZZSqoU26TPzKw_vUxUIgCZHE6Z0igdawwZJqymdwlB5tL7bx3goeVk0URiNnnT8Z5pOIWbjoufu7w2PoFg_4qgAddIlkkAbBIGniY8C/s200/Screen+Shot+2015-10-26+at+7.29.54+PM.png" width="200" /></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "calibri light"; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"><br /></span>
<span style="font-family: "calibri light"; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"></span>Wow, fits pretty well,
doesn’t it? Hefty slope, right? And yet we can see that starting in 2007, the mass
of the pumpkins has often increased at a faster rate than 46.7 pounds per year. Does
this mean that a quadratic function would do better? Or how about an exponential function?</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><br /></span>
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://s3.amazonaws.com/grapher/exports/mkjsrbg55c.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="271" src="https://s3.amazonaws.com/grapher/exports/mkjsrbg55c.png" width="400" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Quadratic</span></td></tr>
</tbody></table>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://s3.amazonaws.com/grapher/exports/qikivnjbmf.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="255" src="https://s3.amazonaws.com/grapher/exports/qikivnjbmf.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Exponential</td></tr>
</tbody></table>
<span style="font-family: "arial" , "helvetica" , sans-serif;">The quadratic does better than the linear function, and the exponential model seems even better. So let's take the risk and use the exponential to make a prediction. The exponential model predicts the world record in 2015 will be….2252 pounds! Wait, that’s not a world record! The world record is 2323 pounds set
last year, and look on the graph—it was pretty far above the exponential curve—a really special
pumpkin in the march of the greats. There’s still time for a record breaker,
but the biggest pumpkin so far this year is 2230 pounds. I don’t think the
record will fall this year, but what do you think?</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "calibri light"; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"></span><br /></span>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Either way, pumpkins have come a
long way from their humble, wild ancestral origins:<span style="font-family: "calibri light"; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"></span></span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;"><br /></span>
<span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;"></span></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cucurbita_texana_2.jpg/1920px-Cucurbita_texana_2.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img border="0" height="213" src="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cucurbita_texana_2.jpg/1920px-Cucurbita_texana_2.jpg" width="320" /></span></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-family: "arial" , "helvetica" , sans-serif;">"Cucurbita texana 2" by John D. Byrd, Mississippi State University</span></td></tr>
</tbody></table>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Congratulations pumpkin growers.
Tom Hanks and maize both grew, but you have truly redefined what it means to grow big!</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><br /></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif; mso-ascii-theme-font: major-latin; mso-hansi-theme-font: major-latin;">Now for a couple treats! First off, I made the above graphs using <a href="https://www.desmos.com/" target="_blank">Desmos</a>. I still like and use Geogebra, but I have to praise the uncluttered, intuitive structure of Desmos. Sliders and regressions which take a little work in Geogebra are quick, easy, and they look good right away in Desmos. And of course it's free and you store stuff in your account on their servers. Go try it! Secondly, if you like this post and want to
share this adventure with your kids, get the</span> mission for <a href="https://www.teacherspayteachers.com/Product/Predicting-the-Great-Pumpkin-2174042" target="_blank">Predicting the Great Pumpkin</a>! Here's another fun one: my Halloween freebie, <a href="https://www.teacherspayteachers.com/Product/Halloween-Candy-Jar-Estimates-2149640" target="_blank">Candy Jar Estimate</a>. Oh, and check out these other great treats from excellent math teachers!</span></div>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com9tag:blogger.com,1999:blog-1435607602356235615.post-1586167872018149082015-10-15T07:12:00.000-07:002016-04-25T16:13:20.978-07:00Tricks for Counting Treats<!--[if gte mso 9]><xml>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="List Continue 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Message Header"/>
<w:LsdException Locked="false" Priority="11" QFormat="true" Name="Subtitle"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Salutation"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Date"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text First Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Heading"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Body Text Indent 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Block Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Hyperlink"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="FollowedHyperlink"/>
<w:LsdException Locked="false" Priority="22" QFormat="true" Name="Strong"/>
<w:LsdException Locked="false" Priority="20" QFormat="true" Name="Emphasis"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Document Map"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Plain Text"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="E-mail Signature"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Top of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Bottom of Form"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal (Web)"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Address"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Cite"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Code"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Definition"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Keyboard"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Preformatted"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Sample"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Typewriter"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Variable"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Normal Table"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="No List"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Outline List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Simple 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Classic 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Colorful 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Columns 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Grid 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table List 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table 3D effects 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Contemporary"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Elegant"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Professional"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Subtle 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Web 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Balloon Text"/>
<w:LsdException Locked="false" Priority="59" Name="Table Grid"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Table Theme"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 1"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 2"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 3"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 4"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 5"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 6"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 7"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 8"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="Note Level 9"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Placeholder Text"/>
<w:LsdException Locked="false" Priority="1" QFormat="true" Name="No Spacing"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading"/>
<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" SemiHidden="true" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" QFormat="true"
Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 5"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 5"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" Name="Medium Grid 1 Accent 6"/>
<w:LsdException Locked="false" Priority="68" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" QFormat="true"
Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" QFormat="true"
Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" QFormat="true"
Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" QFormat="true"
Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" SemiHidden="true"
UnhideWhenUsed="true" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" SemiHidden="true"
UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/>
<w:LsdException Locked="false" Priority="41" Name="Plain Table 1"/>
<w:LsdException Locked="false" Priority="42" Name="Plain Table 2"/>
<w:LsdException Locked="false" Priority="43" Name="Plain Table 3"/>
<w:LsdException Locked="false" Priority="44" Name="Plain Table 4"/>
<w:LsdException Locked="false" Priority="45" Name="Plain Table 5"/>
<w:LsdException Locked="false" Priority="40" Name="Grid Table Light"/>
<w:LsdException Locked="false" Priority="46" Name="Grid Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="Grid Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="Grid Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 3"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 3"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 3"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 3"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 3"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 3"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 4"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 4"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 4"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 4"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 4"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 4"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 5"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 5"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 5"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 5"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 5"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 5"/>
<w:LsdException Locked="false" Priority="46"
Name="Grid Table 1 Light Accent 6"/>
<w:LsdException Locked="false" Priority="47" Name="Grid Table 2 Accent 6"/>
<w:LsdException Locked="false" Priority="48" Name="Grid Table 3 Accent 6"/>
<w:LsdException Locked="false" Priority="49" Name="Grid Table 4 Accent 6"/>
<w:LsdException Locked="false" Priority="50" Name="Grid Table 5 Dark Accent 6"/>
<w:LsdException Locked="false" Priority="51"
Name="Grid Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
Name="Grid Table 7 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="46" Name="List Table 1 Light"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark"/>
<w:LsdException Locked="false" Priority="51" Name="List Table 6 Colorful"/>
<w:LsdException Locked="false" Priority="52" Name="List Table 7 Colorful"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 1"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 1"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 1"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 1"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 1"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 1"/>
<w:LsdException Locked="false" Priority="46"
Name="List Table 1 Light Accent 2"/>
<w:LsdException Locked="false" Priority="47" Name="List Table 2 Accent 2"/>
<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 2"/>
<w:LsdException Locked="false" Priority="49" Name="List Table 4 Accent 2"/>
<w:LsdException Locked="false" Priority="50" Name="List Table 5 Dark Accent 2"/>
<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 2"/>
<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 2"/>
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There are lots of treats (including freebies) in this post! Read on...</div>
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<br /></div>
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<br /></div>
The leaves are changing color in the Blue Ridge mountains of
North Carolina. The surf is picking up at Pacific Beach in San Diego. The elk are tangling in the forests near Jackson Hole but soon the first snow will urge them to
warmer plains. Great flocks of warblers are touching down and stirring the
glassy waters of the salt marshes of Louisiana. A welcome reef of cloud dulls
the Mediterranean sun in Alicante. Autumn.<br />
<a name='more'></a></div>
<div class="separator" style="clear: both; text-align: center;">
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Autumn. At high schools everywhere, the clubs and student
governments and teams have gathered themselves and now they raise their voices
in a full-throated Autumnal cry: Let the fundraising begin! <o:p></o:p>One of my favorite fundraising efforts is the ever-enticing
challenge to guess the number of candies in a jar. Of course, there is always
the possibility that the candy conceals a couple tennis balls, but I always
assume the jar is authentically full and attempt to calculate the number. It’s
tough to do without access to the jar, but if you’ve got a ruler and can get
the dimensions you’ve got something to go on.</div>
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But then what? Then you go home and fill a much smaller jar with
the same candies!<br />
<br />
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Then calculate the volume of the small jar and the large jar. (I leave that to you!) and set up a
proportion:<br />
<br />
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But there was clearly some<span style="mso-fareast-font-family: "MS 明朝"; mso-fareast-theme-font: minor-fareast;"> imprecision in measuring in the
interior dimensions of a not exactly cylindrical cylinder!<span style="mso-spacerun: yes;"> </span>Also, my larger jar in this case was actually
narrower, which might actually reduce the efficiency of the packing of the
candies, which are pretty large.<o:p></o:p></span></div>
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Without trying to solve a packing problem, and assuming we couldn’t just read the
volumes from the label or use a measuring cup, what other methods could we use?
Perhaps we can add some tools. If we had an empty jar equal to the target jar and
some water, we could find out how many small jars of liquid are required to
fill the large one!<o:p></o:p></span></div>
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="HTML Acronym"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
Name="annotation subject"/>
<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed="true"
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<w:LsdException Locked="false" Priority="61" Name="Light List"/>
<w:LsdException Locked="false" Priority="62" Name="Light Grid"/>
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<w:LsdException Locked="false" Priority="72" Name="Colorful List"/>
<w:LsdException Locked="false" Priority="73" Name="Colorful Grid"/>
<w:LsdException Locked="false" Priority="60" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" Name="Light List Accent 1"/>
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<w:LsdException Locked="false" Priority="29" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" QFormat="true"
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<w:LsdException Locked="false" Priority="48" Name="List Table 3 Accent 5"/>
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<w:LsdException Locked="false" Priority="52"
Name="List Table 7 Colorful Accent 5"/>
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<w:LsdException Locked="false" Priority="51"
Name="List Table 6 Colorful Accent 6"/>
<w:LsdException Locked="false" Priority="52"
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</w:LatentStyles>
</xml><![endif]-->
<!--[if gte mso 10]>
<style>
/* Style Definitions */
table.MsoNormalTable
{mso-style-name:"Table Normal";
mso-tstyle-rowband-size:0;
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Of course, we are still relying on an imprecise measurement
of the fraction of a small jar we used at the last pour. Can we do better?
Let’s add a scale to the toolkit. Now, if we find the mass of water which fills
each jar, we’ve got a perfect proxy for volume.<o:p></o:p></div>
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It turns out the the true number of
candies was 95!
And it comes as no surprise that the best solution came from a great proxy for volume.
Not only is the mass a great proxy for volume, a gram of water has exactly a
volume of 1 cubic centimeter.</div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig4z46-R9CMsDSdlIyY7ZO_JSKIHexDQdu3BI55uO2vPa5Hi2exjUGhzmV-DSqI9OUUPeSzhD18TTLsHR-WhskmpCsu7rZxWJzLIwsZQQUJoh0MUYGscaLXOIsaVUc-shecxis0xJkz7SS/s1600/IMG_6099a.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEig4z46-R9CMsDSdlIyY7ZO_JSKIHexDQdu3BI55uO2vPa5Hi2exjUGhzmV-DSqI9OUUPeSzhD18TTLsHR-WhskmpCsu7rZxWJzLIwsZQQUJoh0MUYGscaLXOIsaVUc-shecxis0xJkz7SS/s320/IMG_6099a.JPG" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="MsoNormal">
I’ve created a freebie which you
can use to guide your student through the same series of experiments. Check it
out <a href="https://www.teacherspayteachers.com/Product/Halloween-Candy-Jar-Estimates-2149640" target="_blank">HERE</a>, as well as all the abundant treats for you at the intriguing blogs of the innovative
Teacher-Authors of Teachers Pay Teachers below! Happy Halloween!<o:p></o:p></div>
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<br /></div>
<!-- start InLinkz script -->
<br />
<div class="InLinkzContainer" id="572289">
<a href="http://www.inlinkz.com/new/view.php?id=572289" rel="nofollow" title="click to view in an external page.">An InLinkz Link-up</a></div>
<script src="//static.inlinkz.com/cs2.js?v=116" type="text/javascript"></script>
<!-- end InLinkz script -->Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com23tag:blogger.com,1999:blog-1435607602356235615.post-65671504411190532662015-10-11T01:25:00.000-07:002015-10-11T03:44:49.151-07:00Free is good. Free stuff from great teachers is awesome!<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial, Helvetica, sans-serif;">Is it too soon for candy? Nah. Treats for you!</span></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://freetodiscoverthecc.blogspot.com.es/2015/10/my-teachers-pay-teachers-journeybegin.html" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqAANd47CRxJJXlJ8yzy-qM81t2G9BOGgc1CyU_X1Q6wtAzz8IQK2QVUrETrYD86YDyKxBnTvOHoayY1LXkpTv3ChYC5_sk-I0CIT_TWZg_Go9w1Q_8J-v9GjzdeggjUJD1aBLUzxiDL9u/s640/Slide1.jpg" width="400" /></a></div>
<br />Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com2tag:blogger.com,1999:blog-1435607602356235615.post-71991348983791440062015-10-10T13:31:00.003-07:002015-10-19T05:17:37.432-07:00The Shepherd's Calculation<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">From the balcony of my new apartment I can see the Mediterranean
Sea. This is what it looks like:<o:p></o:p></span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial, Helvetica, sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPj4fhw1ftcIO-ByWRKi9ejIJ2dwPt6hIOFVlc0UKnj7cbWIAoqSMEJcLPKfjBxT-UlEzkZU-4LdA_inhOkO1iVn9QtH0rKUtyxhM3wfLp4OfdonKZN4Dgys5WIGpynGIkhbTD29VBW8am/s1600/IMG_5931A.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPj4fhw1ftcIO-ByWRKi9ejIJ2dwPt6hIOFVlc0UKnj7cbWIAoqSMEJcLPKfjBxT-UlEzkZU-4LdA_inhOkO1iVn9QtH0rKUtyxhM3wfLp4OfdonKZN4Dgys5WIGpynGIkhbTD29VBW8am/s320/IMG_5931A.JPG" width="320" /></a></span></div>
<br />
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">I live in Spain, where I’ve taught math at international
schools for the past four years. I taught and surfed in southern California for
many sunny seasons, and last week I discovered that you can surf in the
Mediterranean too! Can you tell which photo below depicts surfing in California
and which surfing on the Mediterranean? <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Arial, Helvetica, sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyVLfTC3eOIxuRmJE_gAn9beSUkHR7jR3rNfdBW99CFfBR71z0TKxIHnTh5fNqAEdECAsADYzx9TBpjWvQEUV2hRcf_D1mCcOsEcUSQEo8XcPggedGG7hisY-5qv0QCDT4BYIV_7uzOF1l/s1600/IMG_5962a.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyVLfTC3eOIxuRmJE_gAn9beSUkHR7jR3rNfdBW99CFfBR71z0TKxIHnTh5fNqAEdECAsADYzx9TBpjWvQEUV2hRcf_D1mCcOsEcUSQEo8XcPggedGG7hisY-5qv0QCDT4BYIV_7uzOF1l/s200/IMG_5962a.JPG" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG-ObFLJm5lA6HfKl4R4UI4av1WwvOOBwB-E4w1ex7T2Z8MUmp-SXlwAWFNI_7AaIY_vlrH_NUtnhLKKOTIiGLG0WDIXUotxpKV_zvpb8v9GMxWjGL9IPjAwKLmvs1sm7xX7iYXzIT032j/s1600/IMG_6039a.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="111" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgG-ObFLJm5lA6HfKl4R4UI4av1WwvOOBwB-E4w1ex7T2Z8MUmp-SXlwAWFNI_7AaIY_vlrH_NUtnhLKKOTIiGLG0WDIXUotxpKV_zvpb8v9GMxWjGL9IPjAwKLmvs1sm7xX7iYXzIT032j/s200/IMG_6039a.JPG" width="200" /></a></span></div>
<span style="font-family: Arial, Helvetica, sans-serif;"></span><br />
<a name='more'></a><span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">That was a trick question actually—I promise no more trick
questions in this post, but I couldn’t resist: They are both pictures of
surfing in the Mediterranean! I still can’t believe there are waves here. And
here’s another surprising revelation, in photo form, from my balcony:</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmgM9NvZ9vB4_v6MSTw5vEJ2lN6nfQlekcVt0v4KiukFB26zf9eN9Urs_-N-xtgJEPpVcO0H5FspBmR6-PLNnuyUCkWbxHNIU2f6FfYtGIlDnhJ3zR7rMzpmSM03Y1_yaWIY7LRctr7_hH/s1600/IMG_5971a.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmgM9NvZ9vB4_v6MSTw5vEJ2lN6nfQlekcVt0v4KiukFB26zf9eN9Urs_-N-xtgJEPpVcO0H5FspBmR6-PLNnuyUCkWbxHNIU2f6FfYtGIlDnhJ3zR7rMzpmSM03Y1_yaWIY7LRctr7_hH/s320/IMG_5971a.JPG" width="320" /></a></span></div>
<br />
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<span style="font-family: Arial, Helvetica, sans-serif;">So, there are shepherds here! Well, primarily he’s a
goatherd, to be honest. His name’s Claudio. Here he is with his assistant:<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzCrdbOBibIYcsseWF_umOzFMcqBiDDe7Ys8ah9K6qXfK_LgwkkaqVAdQmRmLpAuJt5vgVAVpUoyD-9pYfvY3qiOaUCbdZ0a-MgPMELELjLWGkm0vh-0OozwQBK8B38qnSOA3WAvlzCmt8/s1600/IMG_6089a.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzCrdbOBibIYcsseWF_umOzFMcqBiDDe7Ys8ah9K6qXfK_LgwkkaqVAdQmRmLpAuJt5vgVAVpUoyD-9pYfvY3qiOaUCbdZ0a-MgPMELELjLWGkm0vh-0OozwQBK8B38qnSOA3WAvlzCmt8/s320/IMG_6089a.JPG" width="211" /></a></span></div>
<br />
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<span style="font-family: Arial, Helvetica, sans-serif;">I really wanted to talk to this guy, because I couldn’t imagine
there was sufficient remuneration to be gained from these ruminants! How much
could he possibly make? Well, I won’t reveal Claudio’s current economic situation,
but from talking with him, here’s a super-simplified version of how he milks
this gig for all it’s worth:<o:p></o:p></span></div>
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<br /></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><i>He gets about 3 liters of milk per day per goat, and 2 liters
per sheep. That’s is a lot of milk! Cows in modern dairies produce about 25 liters
a day now, which is so mind boggling that I dairyn’t consider it further!
Claudio and his big family can only milk about 400 liters a day, absolute maximum.<o:p></o:p></i></span></div>
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<br /></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><i>The sheep are a bit more difficult to manage, and they are
bigger. Goats take up one “stall” in the barn, but each big sheep takes up two!
He’s got 300 stalls. (It’s more complicated than this, but go with it!)<o:p></o:p></i></span></div>
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<br /></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><i>He used to make about <span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span>1 for each liter
of goat’s milk, and <span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span>2 for each liter of sheep’s milk,
because the sheep’s milk is richer. Not punning there, it’s apparently
creamier!<o:p></o:p></i></span></div>
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<br /></div>
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<!--EndFragment--></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">So, naturally, let’s make some inequalities and equations!</span><o:p></o:p></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsOh2nmZRh4QPRMjnXozHeBiSpYCn3SwXcach1yRqaqZAjG9F3zcwBsNWzv9ZNds05yej43jBBVUpkn1UISpTjVWi5Hduem9rVDG-vlzJiM5s9hw9TNO6Negc9JYvzKc1OmqOhq9kaffaH/s1600/Screen+Shot+2015-10-10+at+10.23.46+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="210" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsOh2nmZRh4QPRMjnXozHeBiSpYCn3SwXcach1yRqaqZAjG9F3zcwBsNWzv9ZNds05yej43jBBVUpkn1UISpTjVWi5Hduem9rVDG-vlzJiM5s9hw9TNO6Negc9JYvzKc1OmqOhq9kaffaH/s640/Screen+Shot+2015-10-10+at+10.23.46+PM.png" width="540" /></a></div>
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Of course, we can graph the inequalities. Perhaps you
recognize this as a classic linear programming problem!</span></div>
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<br /></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Here’s the graph, which shows us all the possible
combinations of goats (x) and sheep (y) that meet Claudio’s constraints, in the region
that is double-shaded. For example, clearly 40 goats and 50 sheep is possible
for Claudio, because (40,50) is in the double-shaded region. (Note: Even though goats
can be stubborn, their numbers are never negative!)<o:p></o:p></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBRRois52zlYVUrarjcf2tok2sg3HhKp70D2qxezq5zJ1n1DLvDOTUL1do7w_jjZPRQKT6k9SwchgDGlgCxh3nY6ZqyersTXLROQneXiyEhIe5bIzGXmhXNeGplZwyRjEJbUTR5o_hlo4/s1600/Goat+Inequalities.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="294" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBRRois52zlYVUrarjcf2tok2sg3HhKp70D2qxezq5zJ1n1DLvDOTUL1do7w_jjZPRQKT6k9SwchgDGlgCxh3nY6ZqyersTXLROQneXiyEhIe5bIzGXmhXNeGplZwyRjEJbUTR5o_hlo4/s320/Goat+Inequalities.png" width="320" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">Finding the maximum revenue requires finding the corner
points of this region and plugging them into the Revenue equation to see which
combination returns the most money. Why the corner points? Below is what
happens when you graph the revenue equation for different revenues. Notice that
the revenue line doesn’t cross into Claudio’s world until we hit a revenue of
about </span><span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span><span style="font-family: Arial, Helvetica, sans-serif;">650.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
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</div>
<div class="separator" style="clear: both; text-align: left;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnY5opUfz2oXlf4Fd9RDxh5-p7ol5cvbKC49RRCpUdp6JyaVAdAG34_U_0XTvP4lqBdPgnL8jdw5aACahB5tcK1zhegUkz8Tb2pezeaOibIUbnVQc7ZQmBbo0J78nh-VFo9uni3oVNLPnN/s1600/goat+ship.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnY5opUfz2oXlf4Fd9RDxh5-p7ol5cvbKC49RRCpUdp6JyaVAdAG34_U_0XTvP4lqBdPgnL8jdw5aACahB5tcK1zhegUkz8Tb2pezeaOibIUbnVQc7ZQmBbo0J78nh-VFo9uni3oVNLPnN/s640/goat+ship.gif" width="520" /></a></div>
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">The combination of goats and sheep where the revenue line
enters Claudio’s world is about 50 goats and 125 sheep. I leave it to you to
determine the exact answers algebraically!</span></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-family: Arial, Helvetica, sans-serif;">You might have noticed that above I said he used to make <span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span>1
per liter of goat milk and <span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span>2 per liter of sheep milk.
Nowadays, the milk market has dried up, and he makes only <span style="mso-ascii-font-family: "Calibri Light"; mso-hansi-font-family: "Calibri Light";">€</span>0.5
per liter of each kind of milk. If you run the numbers again you’ll see the
reason why he seems to have more goats than sheep now in the picture. He and I didn’t
talk about this, but it seems like the only reason he still has sheep is that
he had them before, and you never know when the market might shift. Among the
many realities complicating all this is that his local milk market isn’t as
liquid as you might think—he only sells to one buyer and doesn’t always sell
all his product.<o:p></o:p></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Nevertheless, Claudio and his family have been doing this
difficult work for generations and Claudio seems relatively content. For my
part, I feel lucky to have visited with someone living a truly pastoral life
right next door, and I hope that you enjoyed getting some insights into the
economics which drive the animal drivers! I’ll eventually put this into a
Mission on my Teachers Pay Teachers store (sooner rather than later upon
request). Look for more real-world math at my TpT store, Courage To Core, and give the kids something to graze on! (Click the kid below!)</span><o:p></o:p></div>
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<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.teacherspayteachers.com/Store/Courage-To-Core" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="234" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijV_XRR4C0mlff14D5smHveNH33C5T3p5-FuRXf9pxWTh_aCgIJlU8ybQKqfhGJvEepEQBzUyhyphenhyphenElubwqyFUb_0enjSPoCV_qublSBscwpH10Pz1buEI1Q7oAW9tZ0JJQZ0TbmeZL-aFMW/s320/IMG_6073a.JPG" width="320" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">And for more intriguing posts about math in real-life, click below! Dart games, fish tanks, tasty tarts and much more!</span><br />
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<a href="http://www.missmathdork.com/2015/10/math-is-real-life-october-2015-edition-family-competition/" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://www.missmathdork.com/wp-content/uploads/2015/08/mathisreallife-revised-500x500.png" height="320" width="320" /></a></div>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com2tag:blogger.com,1999:blog-1435607602356235615.post-9495060295927957642015-10-08T11:07:00.000-07:002015-10-08T11:09:29.057-07:00The Good Dinosaur Likes the Grand Tetons......and so do I! A photo of the Tetons is part of my logo above, and I've climbed there many times. A beautiful place, and Pixar knows it. They built a world for The Good Dinosaur using USGS topographical data for the Teton area and then rendered it digitally. The result is rather magical, not unlike the famous turn of the century paintings of Thomas Moran. Click the images below to learn more about both artists!<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Thomas_Moran_-_The_Three_Tetons.jpg/1599px-Thomas_Moran_-_The_Three_Tetons.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="212" src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Thomas_Moran_-_The_Three_Tetons.jpg/1599px-Thomas_Moran_-_The_Three_Tetons.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Thomas Moran. <i>The Three Tetons.</i> 1895</td></tr>
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<tr><td style="text-align: center;"><a href="http://io9.com/how-pixar-changed-all-the-rules-to-make-the-good-dinosa-1735364564?utm_campaign=socialflow_io9_twitter&utm_source=io9_twitter&utm_medium=socialflow" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://i.kinja-img.com/gawker-media/image/upload/s--D9V4X2me--/c_scale,fl_progressive,q_80,w_800/1465116888861896776.jpg" height="238" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Pixar. <i>The Good Dinosaur.</i></td></tr>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-51709940676634636042015-10-08T10:11:00.003-07:002015-10-08T11:08:59.947-07:00Make your own Halloween Mask. So cool.<div style="text-align: center;">
Click the image below to learn how to do this:</div>
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.slate.com/blogs/the_eye/2015/10/08/steve_wintercroft_s_wonderfully_spooky_designer_diy_halloween_masks_channel.html" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="314" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYNLT39REJXnHYgAf46foUSLeabiIIFSR8Pez20rup1-9Y-JZ_j-92ajs4hbosRYiB7JXmHvXWzXd4L2UuVnG0OD6keaJB3ugyWocm7BFfrbd91I3LPVSPh0C-a0hX6vUwpget1lxGtcf9/s320/Screen+Shot+2015-10-08+at+7.06.36+PM.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Steve Wintercroft</td></tr>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-89907381139448065832015-10-06T09:11:00.002-07:002015-10-19T05:18:50.160-07:00Alt-J's School of AthensThis is the first in a series of posts delighting in the variety of intersections between math and music. Often, those intersections will be lyrical, that is, the lyrics of a track will literally include math terminology. And that is the case for this song.<br />
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My students often like to share their musical interests with me, as I'm known to have a wide-ranging and contemporary aural palate. They sometimes seek to shock me, or to see if their choices rise to the level of my considered tastes. In fact I'm neither easily shocked nor particularly sophisticated, so music is always a great starting place for a conversation.<br />
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Students in my Math Topics class last year brought me this stormy and menacing track from Alt-J. Some of you will recognize the band's name as a keyboard shortcut on a Mac--try it! I love the way the drums drop out frequently and the voice holds you in. Sometimes everything falls away and the song is yours alone for a beat.<br />
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There is a bit of evocative geometry woven into the song. Equally compelling is the video, which re-imagines Raphael's famous 16th century fresco in the Vatican. His fresco was entitled The School of Athens and features Plato, Aristotle, and more than one mathematician. The video shows us the School re-imagined with contemporary youth who in their hardened stances seem pre-disposed to vices rather than virtues. And yet, while this vision of the School is hardly a Platonic ideal, there is a suspenseful defiance in the poses they strike. It does remind me of students I've seen standing in the courtyard, asserting selfhood, straining at leashes that are not just physical, <a href="https://www.youtube.com/watch?v=gi_QqX5ioWM" target="_blank">waiting for something to happen</a>, <a href="http://www.rollingstone.com/music/features/radioheads-the-bends-at-20-alt-js-thom-green-looks-back-20150313" target="_blank">wanting to be the something that happens</a>.<br />
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That's when Socrates would step into the scene, and ask the question: what are you here for? And maybe the dialogue would begin. The song is about the shape of relationships. The video takes the shape of Raphael's fresco to show us potential energy. Waiting for Socrates.<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://en.wikipedia.org/wiki/The_School_of_Athens" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="248" src="https://upload.wikimedia.org/wikipedia/commons/9/94/Sanzio_01.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Raphael [Public domain], via Wikimedia Commons</td></tr>
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<tr><td style="text-align: center;"><a href="https://www.youtube.com/watch?v=Qg6BwvDcANg" imageanchor="1" style="margin-left: auto; margin-right: auto;" target="_blank"><img border="0" height="129" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGLtHH4Xe_iKWbQVDFmhji-Tb7l1IpJ_HaPrFAsJqsHyitvmErSjdcrA-234DS_qhNt0m7ZcKw9gmCQhlVKIrP_ViAcG8M63JRnAmN2UaaIUZ2f7xbOIreBJHIuFuhfBuBQkjSuRgO1HEj/s320/Screen+Shot+2015-10-06+at+6.02.25+PM.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Tesselate, Alt-J [official video]</td></tr>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-11132713971163814202015-10-01T10:05:00.000-07:002015-10-19T05:19:45.167-07:007 Ways to Drop the Mic in Math Class<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRE77IOrAnkbmauSUMNmV502qDTeOmCFry2L-FLUzN4hXPkS68hoJNNlNJll1xFekglR891vm7AUQjHDrB1gducjdJNt8H1fDUEaUkvaioOcR7cTCyln2WeYVm1q6C0zRKDOjopm98DSzb/s1600/IMG_5935.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRE77IOrAnkbmauSUMNmV502qDTeOmCFry2L-FLUzN4hXPkS68hoJNNlNJll1xFekglR891vm7AUQjHDrB1gducjdJNt8H1fDUEaUkvaioOcR7cTCyln2WeYVm1q6C0zRKDOjopm98DSzb/s320/IMG_5935.JPG" width="320" /></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">I like to think I’m an entertaining story-teller. Even the
driest of mathematical procedures (quadratic formula derivation, anyone?) can
come to life when I’m in performance mode. But over the years I’ve lectured less
and less, giving students more autonomy to follow their own paths of inquiry.
Students work in small groups on real world scenarios, experimenting, drawing
conclusions and solving complex problems while I facilitate, motivate and
occasionally lecture. Here a few tips on how to drop the mic and let students
pick it up in math class.<o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">1) Ask Curious Questions<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">How much taller is a human compared to a carpenter ant? How
much faster can a sailboat go if you double its length? How many trees per
person are there in the world? <span style="mso-spacerun: yes;"> </span>If you
flip two coins, are you more likely to get two heads or one head and one tail?
If you double the radius of a pizza, how much more food do you get? What
function is the best model for a car accelerating from a stop light, and why?
Can you figure out the percentage of green m and m’s in the world from one bag?</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Every student has an innate curiosity about how the world is
put together. It can seem that the abstractions of algebra are outside daily
experience, and yet there are ample opportunities to draw numbers from the
real-world and spark excitement. The above questions can springboard into deep conversations
about exponential notation, square root functions, probability, area, quadratic
functions and sampling.<o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">2) Not IKEA. Not Home Depot<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Assembling a piece of furniture from IKEA is such a choreographed
experience that creativity is a dead end by design. Conversely, if I walk into
Home Depot to build a house without a blueprint the project will end before it
begins. It’s important to find materials which strike the right balance between
providing guidance and allowing students to experiment. Courage To Core math
materials are classroom tested tools for algebra and geometry, and many other
teachers are creating and selling great materials on Teachers Pay Teachers, Amazon and beyond which offer students the chance to actively engage in carefully guided experiments.<o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">3) A Contained but Spacious Playground<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">A classroom which gives students greater autonomy to collaborate
needs a structure. (See my prior post on how to create expectations and
effectively play the role of facilitator in classrooms centered around group
work.) Building a culture of self-directed students takes patient effort at the
outset, but once groups are humming along it can be an efficient and effective
learning structure and a great way for a teacher to observe each student in
action. <o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">4) Be the Lifeguard They Trust<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">In swim class when you were a little kid, you let that one
lifeguard throw you in the deep end of the pool. He’d let you struggle when you
were capable, but you knew he’d fish you out if you were in real trouble.
Students need to know that they can visibly struggle with mathematics and that
you’ll let them go at it as long as they need to. They also need to know that
you’ll throw them a lifeline if their group is lost at sea. Finesse that line
carefully.<o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">5) The Calm Center of the Storm<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Students are adept at following rules, but they are often
even more adept at blurring the lines. Play at school can be an act of mild
rebellion or it can be intrinsic to a learning environment that is designed to
engage the voracious appetites of young minds. Once you allow students to engage
more freely, the classroom can be a more productive yet more chaotic place. The
usual distractions still interrupt work flow, but when group work is working, students
take more responsibility for maintaining the work culture, and conversation and
invention are steered toward productive ends. <o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">6) Nothing Wrong Means Something’s Wrong<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Once we have fired a curiosity with a good question and
given them the basic rules of engagement, students need to experiment, fail,
and experiment some more. The path of least resistance is also the path of
least persistence. Mistakes are the necessary accidents on the path to deeper
understanding. Of course, this process can take time…<o:p></o:p></span></div>
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<u><span style="font-family: Arial, Helvetica, sans-serif;">7) Game Never Over<o:p></o:p></span></u></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">It can be tough to fit student-directed work into the rigid
schedule of the school day, and tempting to sweep kids towards wrapping up when
they are still deeply working in progress. As much as possible I don’t put time
limits on activities, so that students can self-pace and own their hard fought
success at the end of the proverbial day. In my experience, at the beginning of
the year students are less efficient as they adapt to the structure but by the
end of the year move through assignments efficiently and effectively.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">A student-centered, collaborative classroom environment
takes a bit of effort to set up, but the rewards are great. Students learn to
communicate, collaborate, persevere, bounce back from failure, think creatively
and problem solve more confidently. Check out my prior posts for more hints on
how to implement a collaborative model in high school math class. If you drop
the mic you may help more kids find their singing voice.</span></div>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com5tag:blogger.com,1999:blog-1435607602356235615.post-47218281630637596702015-09-30T08:11:00.000-07:002015-10-15T13:16:28.748-07:00A Generous Tide Lifts All Boats<span style="font-family: Arial, Helvetica, sans-serif;">Although I've only been in the Teachers Pay Teachers Community for about a month I've started to follow a few teachers closely. One of my favorite bloggers and authors on Teachers Pay Teachers is Doc Running, who teaches Math and Social Studies with a student-centered approach that allows students to be the active agents of their learning through workshops and activities. Even more, the blog feels like a TpT hub--she really goes the extra mile for his peers at Teachers Pay Teachers. Recently she posted a couple thoughtful introductions of new sellers on TpT. She's also got well articulated opinions about education policy, and these are also sprinkled throughout the blog. To me, Doc Running is emblematic of the value of Teachers Pay Teachers as a forum for sharing ideas and resources, and for helping elevate the profile of individual teachers and the profession in general. Check it out!</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" height="78" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs3X3ozITl9VX_iQ_XX6Y5NeGZM8Hm90rCVcXulD16ylX5Klgw9p5fBualiI2qsmMizbHPKpuxI2tFLWP08govogxvJaJ4zNhKXBPG0iUCW8NFXzEmfahYxBMJ4-4UZBJ6flZwFFaaN-dk/s320/Screen+Shot+2015-09-30+at+5.36.48+PM.png" width="320" /></span></div>
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<a href="http://everyed.blogspot/" target="_blank"><span style="font-family: Arial, Helvetica, sans-serif; font-size: large;">Everything Education</span></a></div>
Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-17305651467061814432015-09-28T17:56:00.002-07:002015-10-06T13:00:49.886-07:00Edutopia!<a href="http://www.edutopia.org/discussion/10-tips-making-group-work-work" style="font-family: Arial, Helvetica, sans-serif;" target="_blank">Edutopia</a><span style="font-family: Arial, Helvetica, sans-serif;"> posted my "Ten Tips For Making Group Work Work" on their site! Cool! I really like reading stuff on Edutopia for five reasons:</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">1) A lot of their content is produced by teachers.</span><br />
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<span style="font-family: Arial, Helvetica, sans-serif;">2) Often, the educational philosophy they espouse resonates with me.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">3) The tone is resolutely positive and debates are civil.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">4) They are called Edutopia.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">5) George Lucas.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Check them out!</span><br />
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<a href="http://www.edutopia.org/discussion/10-tips-making-group-work-work" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="233" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiCydtjmGvFqMgefcaHaYblzN4frFVuXamGuEikIMttPmcy93-wdqbop8O6abUgj5j58Qjb2qOdHBrE6OHAusTlXtTZNlwKzSp_q99El2MUANn6xqkyR66BJNYgerr5JCY9A34JhNDm_umX/s320/Screen+Shot+2015-10-06+at+9.39.43+PM.png" width="320" /></a></div>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-59702683880566344562015-09-27T10:20:00.001-07:002015-09-27T10:22:04.037-07:00GIFS are powerful!<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizDwS6o4ZsWf-ik0JLleIrMCH6Y39BvIwoiGFCM_7WIF_ZdAnvxng_hyphenhyphen6ybuJ8UZ9qoWMX7Rc870uHkfG6YkVnnrcqKYGaNPIUyzaMPHJgNAL25iGpjwjnNZIc6nFF7ueX3j6xBEC_O7_U/s1600/quad+power.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: Arial, Helvetica, sans-serif;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizDwS6o4ZsWf-ik0JLleIrMCH6Y39BvIwoiGFCM_7WIF_ZdAnvxng_hyphenhyphen6ybuJ8UZ9qoWMX7Rc870uHkfG6YkVnnrcqKYGaNPIUyzaMPHJgNAL25iGpjwjnNZIc6nFF7ueX3j6xBEC_O7_U/s1600/quad+power.gif" /></span></a></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">And pretty easy to make! I bought the cheap app Animation HD, for my ipad, then used imgflip free online to create the gif. Way easier than using photoshop in my opinion. It's cool because in less than 30 seconds you get a little introduction to quadratic functions, with automatic review since it loops!</span></div>
Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-4369197928469725642015-09-25T09:07:00.000-07:002015-09-26T02:49:49.049-07:00The CTC on the TpT!<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNqJbaG_Z6y24VJyORaFs_s2ibhmvm2OGvUFsFNKtHJAQ7yrgAo8t6uBgJAKTpW686hkG9F6mU00ACBxLDMMrxbKoFn-WxY7CpW554KY1tVp36oyb5DSSaG6Vc5d6E1JGeaYPTQsuIaXV5/s1600/Screen+Shot+2015-09-25+at+6.17.55+PM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="90" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNqJbaG_Z6y24VJyORaFs_s2ibhmvm2OGvUFsFNKtHJAQ7yrgAo8t6uBgJAKTpW686hkG9F6mU00ACBxLDMMrxbKoFn-WxY7CpW554KY1tVp36oyb5DSSaG6Vc5d6E1JGeaYPTQsuIaXV5/s400/Screen+Shot+2015-09-25+at+6.17.55+PM.png" width="400" /></a></div>
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<a href="http://blog.teacherspayteachers.com/tough-math-concepts-tpt-teacher-authors-to-the-rescue/" target="_blank">A quick take on the difference between graphing a line and understanding a linear function.</a></div>
Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-40173990739721120442015-09-24T14:23:00.001-07:002015-09-26T02:46:05.669-07:00Ten Tricks for Effective Group Work<div class="separator" style="clear: both; text-align: center;">
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When I first considered putting students in small groups to work
on missions the biggest fear was that students would run amuck, losing focus
and wasting time when they weren’t being told what to do. Even more, I was
worried that some elements of the activities not might not work, and imagined
the exhausting challenge of dealing with the pandemonium of various groups
variously confused. It would be so much easier to simply keep a tight reign on
a quiet class as I gave them a solid and entertaining lecture. And yet it
seemed that group work provided an opportunity for students to engage in the
risk-taking, creativity, conversation, failure and recovery which gives
students greater ownership over learning. Was I willing to take the personal
risk to give up some control and allow students to become the principle agents
of their own success?<o:p></o:p></div>
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Once I answered that question reluctantly in the affirmative,
the question became, what structures and expectations could I put in place to
facilitate effective group work? Below is my list of the most important
elements which make group work work. Note that these are biased towards my personal
experiences teaching high school math. I’m sure you’ve got ideas as well, so
put them in the comments section!<o:p></o:p></div>
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1) Determine groups yourself, but mix them every couple
weeks. As best as possible each group should represent diversity in current capacity,
work ethic and social confidence. <o:p></o:p></div>
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2) It can take some groups a while to establish a work
culture. Interventions can vary beyond talking to a student who is being a
distraction. After class I may ask a stronger student to help focus the group,
guide a peer more directly or encourage a peer to be more vocal. Or I may have
a conference with the whole group to talk about their challenges, reset the
guidelines for them and let them know I believe they can succeed.<o:p></o:p></div>
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3) Try to be a facilitator, and carefully observe and praise
the quality of student collaboration and the vigor of their effort in addition
to their problem solving. Occasionally pull up a chair with a group and
observe, help or ask questions.<o:p></o:p></div>
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4) If needed, give an impromptu mini-lecture to a small
group on a nearby whiteboard or a portable whiteboard. Sometimes a student will
say something intriguing and I’ll give a mini-lecture that extends on their
comment, even though it’s not in the activity they are working on.<o:p></o:p></div>
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5) Pause the whole class a couple times to share a short
visualization, praise a group or share anonymous observations, explain how to
avoid a common error or allow groups to share out in response to a question.<o:p></o:p></div>
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6) Give a mini-lecture near the beginning and end of class
to frame the day. Short.<o:p></o:p></div>
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7) Set the expectation that students begin work immediately
when the come in to class, picking up where they left off yesterday or starting
the next activity before you deliver any introduction. Students should also
work to the end of class if there is any time after your wrap up.<o:p></o:p></div>
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8) To whatever degree possible, allow groups to work at their
own pace, sometimes over more than one class period. Groups that finish early
can work on the practice problems towards the end of assignments, extra practice
you give them, correct their work with the answer key or move on to the next
assignment with their group.<o:p></o:p></div>
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9) Give group assessments. How to design and grade them is a
topic for another post!<o:p></o:p></div>
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10) Enjoy your rapport with students!<br />
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That’s how I get the most out of group work with my
students. Do any of these resonate with you? What do you do to maximize the
effectiveness of group work?</div>
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Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com4tag:blogger.com,1999:blog-1435607602356235615.post-42651540384977086872015-09-21T08:05:00.001-07:002015-09-21T08:12:31.188-07:00Better Together: 11 Tips for Great Group Work in Middle and High School - Teachers Pay Teachers<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZCcGwv03cLALqnjrglVTe4MECowIUh9lJ7nytQnhFz6JF_6L3OdSN_KdGEdUpvW9WwTJ4nvt8BJZ19LY009Ux328le3pIDmgAK-kbRNY7C__P7rh-VszygbGf7oFVbAtKJIsM7dJ6htKU/s1600/P1010023.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZCcGwv03cLALqnjrglVTe4MECowIUh9lJ7nytQnhFz6JF_6L3OdSN_KdGEdUpvW9WwTJ4nvt8BJZ19LY009Ux328le3pIDmgAK-kbRNY7C__P7rh-VszygbGf7oFVbAtKJIsM7dJ6htKU/s320/P1010023.JPG" width="320" /></a></div>
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Courage To Core materials are designed for students to work in small groups to explore math deeply. It can be a challenge to organize and manage groups successfully. Jodi Rigotti at Teachers Pay Teachers recently assembled a fine list of teacher tips on how to help students get the most out of their group work. My personal favorite is that we set clear expectations knowing that students will need to practice in order to work effectively in groups. What's your favorite, or do you have any tips to add? On an upcoming post I'll discuss in detail the structures and techniques I use to get the most out of group work.<br />
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<a href="http://blog.teacherspayteachers.com/11-tips-for-great-group-work-middle-high-school/#.VgAac0AxkYw.blogger">Better Together: 11 Tips for Great Group Work in Middle and High School - Teachers Pay Teachers</a>Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-70351155822977050542015-09-20T14:54:00.002-07:002015-09-20T14:55:49.852-07:00A Snapshot of Interactive, Hands-On Math<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvT3sShsf84xP_976Ax61CoyTRe9qnCS6Pc_WepRAA2r52FK0-6ZLjm1rTes9ddKbCMIEnZwC7pzxsPbK70cy4b1Lr7k_02fIfd1gOLlBMv25E83xAmY7wJQxrE6lnbescsMLbazFCESJG/s320/Hands+Working+Groups.png" width="208" /></div>
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<a href="https://www.teacherspayteachers.com/Product/Linear-Models-and-Problem-Solving-LF5-2049504" target="_blank">What they're working on.</a></div>
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<br />Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-54259097603489422322015-09-18T07:07:00.001-07:002015-09-18T07:21:17.533-07:00Flexible Learning Spaces<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB5nhGbKWR47bWWKEVGxzdBkej0k3l9-WYhhoZmN3ANKkM5lsXjLwQ_dmLRP-26uU0bXc9UIaCZ3PD7W22AVqc-9y3dyB9qoK50uoNkaOVzGzFXj76cEgkGz6Xx7TcRpDsmFi1nx16YUPg/s1600/Group+Work+Photo+small.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiB5nhGbKWR47bWWKEVGxzdBkej0k3l9-WYhhoZmN3ANKkM5lsXjLwQ_dmLRP-26uU0bXc9UIaCZ3PD7W22AVqc-9y3dyB9qoK50uoNkaOVzGzFXj76cEgkGz6Xx7TcRpDsmFi1nx16YUPg/s320/Group+Work+Photo+small.jpg" width="320" /></a></div>
In the collaborative space of my math class, students usually work in groups of 3 or 4 which I've assigned. And yet, sometimes it may better to allow students to work in pairs, or to shuffle the groups, or allow a particular student to work on their own. This sort of flexibility has to exist within a classroom where structures have already been established, but it's great to have that flexibility. I was moved to reflect on this after reading a great piece, with photos, about flexible learning spaces in libraries. Check it out! <a href="http://itsaboutinformation.blogspot.com/2015/09/flexible-learning-spaces.html?spref=bl">It's About Information: Flexible Learning Spaces</a>. How do you make your classroom a flexible learning space, what are the challenges of doing so, and how do you deal with them?<br />
<br />Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0tag:blogger.com,1999:blog-1435607602356235615.post-71831602285017984802015-09-18T05:48:00.002-07:002015-09-18T10:06:39.696-07:00An Introduction To Courage To Core<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiszApo_FUQz1zl2ZPr6ZEVY2FkaJFTlsC4qyk5Kr6o8kjfHUWH-T8AZSlKTI_k6TJpom8-s4mFScuTpxR86gbfRSEuUFMuZxRIFt-rCD8BwhayHxctydQzz5oZ0jNig3-zr0Wraf9vrbRa/s1600/little+flowers+teton.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiszApo_FUQz1zl2ZPr6ZEVY2FkaJFTlsC4qyk5Kr6o8kjfHUWH-T8AZSlKTI_k6TJpom8-s4mFScuTpxR86gbfRSEuUFMuZxRIFt-rCD8BwhayHxctydQzz5oZ0jNig3-zr0Wraf9vrbRa/s320/little+flowers+teton.JPG" width="320" /></a></div>
<span id="internal-source-marker_0.6955657196231186"><span style="font-family: inherit;"><span style="font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">Introduction</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">My first year of math teaching was in 1995 at an innovative private high school in San Francisco named <a href="http://www.lwhs.org/academics" target="_blank">Lick-Wilmerding</a>. Students worked in small groups to explore algebra and geometry, reporting their findings in documents written as if they were lab reports. Students would receive direct instruction when necessary, but frequently, questions were explored and answered by members of the groups themselves. I adapted hesitantly to this new structure, as I myself had been taught by teachers and professors using lecture combined with extensive homework practice. Over my subsequent years of teaching in the states I explored a range of methods tailored to the various cultures of the schools at which I taught. This included straight lecture and homework, a mixture of lecture and independent class work, and collaborative group work supported by a textbook. Over the last 8 years of teaching at <a href="http://www.ljcds.org/page" target="_blank">La Jolla Country Day</a> in San Diego and two international schools in Spain, the <a href="http://asvalencia.org/" target="_blank">American School of Valencia</a> and the <a href="http://www.asmadrid.org/neweb/" target="_blank">American School of Madrid</a>, I’ve developed <a href="http://www.couragetocore.com/" target="_blank">Courage To Core</a> which allows my ninth and tenth grade math students to explore mathematical concepts as if exploring scientific concepts in laboratories.</span></span></span><br />
<span style="font-family: inherit; font-weight: bold; white-space: pre-wrap;"></span><br />
<a name='more'></a><span style="font-family: inherit; font-weight: bold; white-space: pre-wrap;">The Objective</span><br />
<span id="internal-source-marker_0.6955657196231186"><span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;">As educators we hope to develop students who are inquiring, drawing upon their own innate <a href="http://news.nationalpost.com/news/fewer-tests-more-play-qatar-asked-finland-to-open-a-progressive-school-in-doha-and-its-a-huge-success" target="_blank">curiosity</a> to explore the world around them and develop and delight in an understanding which persists beyond high school. We aspire to help them become <a href="http://www.edutopia.org/blog/student-centered-learning-starts-with-teacher-john-mccarthy" target="_blank">communicators</a>, who work readily and effectively with others to enrich their own understanding and are capable of verbalizing and writing about their ideas. We hope they become deep <a href="https://www.youtube.com/watch?v=YIPO3W081Hw" target="_blank">thinkers</a> who can apply technical skills and conceptual understanding to complex new problems. We strive to help them take intellectual <a href="https://medium.com/bright/in-education-how-do-we-create-a-culture-of-experimentation-8c360f8ee32" target="_blank">risks</a>, to share novel ideas and pursue creative lines of thinking, and to respond to failure with resilience. My approach to helping students become <a href="http://www.theatlantic.com/education/archive/2015/07/what-grit-looks-like-in-the-classroom/399197/" target="_blank">resilient</a>, self-directed, collaborative problem solvers is to give them missions which require them to work together. In the missions they investigate mathematical ideas in real-world contexts, recognize patterns, share conclusions and practice applying learned skills. As a teacher I work to facilitate, advise, challenge, motivate and instruct, depending on the needs of the group.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">The Math Intuition</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Research shows that not only do we have an innate sense of mathematical relationships, but that sense is rather complex. <a href="http://www.scientificamerican.com/article.cfm?id=a-natural-log" target="_blank">Research</a> suggests that a child’s numeracy is likely logarithmic rather than linear, which means they compare the relative numbers of things by thinking exponentially. More broadly, a young child recognizes that a toy car is the “same thing” as a real car, and his ability to model things demonstrates more than just a sense of proportionality, but also a capacity to represent the real world symbolically. A graph is precisely such a symbolic representation if it is a graph of profits increasing over time, or the curvature of a rocket’s path, or the pace of a heart beat as a runner increases her speed. It is incumbent upon us to harness the natural capacity of students to understand the world through intuitive numerical relationships and models. For this reason CTC strives to connect numbers and functions to real world scenarios. </span><br /><br /><span style="font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">Collaborative Investigations with CTC</span><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">With Courage To Core, students work in small groups of three or four to explore mathematical concepts and procedures. Missions guide students with leading questions and a logical flow which directs their work to a learning objective. At the same time the investigations are replete with open-ended questions which require students to discuss and decide how best to represent the given situation. They work with each other at a pace determined by their group. Sometimes they will solicit help from another group, or from the teacher. The teacher walks around the class observing and providing hints and guiding questions as needed. The teacher's role is that of a consultant, a counselor, a tutor, a manager, a resource and a lecturer. The teacher is looking for determined collaboration and creative problem solving, and expecting students to make mistakes and then correct them. The student's ability to persist is more important on a daily basis than quick arrival at a correct solution. Correct solutions can be arrived at by many methods, and that they will arrive eventually through effort.</span><br /><span style="font-weight: bold; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="font-weight: bold; vertical-align: baseline; white-space: pre-wrap;">The Recommended Structure of the Class for CTC</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Students work in small groups for about two weeks before the group members are shuffled. Every couple weeks, they complete a set number of missions and correct any errors in their with the answers. Students take frequent group tests in advance of individual tests. For group assessments, all students must record the same answers, but ultimately one student’s assessment is graded at random and everyone in the group receives that grade. Group tests allow students to grow comfortable with formative assessment and get the most out of it. Although assessment is often associated with high-stakes multiple choice testing, group assessments provide a great alternative. <a href="http://www.nytimes.com/2011/01/21/science/21memory.html?pagewanted=all&_r=0" target="_blank">Research</a> shows the value of frequent assessment in learning, which is consistent with the idea that students need opportunities to demonstrate their understanding verbally and in writing.</span><br /><br /><span style="vertical-align: baseline; white-space: pre-wrap;"><b>CTC is Fun!</b></span></span></span><br />
<span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;">Students often enjoy working in groups. We want to see students connecting socially and intellectually with each other in class. <a href="http://www.nasponline.org/resources/factsheets/socialskills_fs.aspx" target="_blank">Research</a> also shows that social skills developed during school are among the most important factors in future success, and CTC provides students an opportunity to be social and productive members of the school community. With CTC, students are given a space to engage their curiosity, work with others, make mistakes and correct them, and develop a deeper connection to mathematical ideas through real world scenarios. It's fun!</span></span>Anonymoushttp://www.blogger.com/profile/06011451984359039574noreply@blogger.com0