Dan Meyer opened his site 101qs.com some time ago now, and I have to say it has been rather engaging to see what the results have been.  You enter the site thinking you have some pretty top notch photo or video prompts that are bound to produce wonder and amazement, and yet somehow you find your perplexity score slowly going the way of the buffalo. At first it was disheartening.  You think to yourself how could my amazing collection of gumballs stacked on dominoes in the shape of a sierpinski triangle at the burning man festival not provoke the question, “How can we derive the gravitational constant?” The question now becomes, "Is that the point of 101qs?"  What do we want out of these videos and pictures?  Do we want everyone to be on the same page or do we want multiples questions to stem from each video?  Do we want both?  Is 101qs giving us what we want?

Clash of the Titans

 Over at Dan’s site people have been discussing these last set of questions and we find, naturally, Dan promoting his brand of “Make the prompt scream the question you are looking for” and Karim Ani saying, “There are more interesting questions that go beyond a one minute clip or picture.”
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When I read debates from guys like these I imagine this, but with better vocabulary.
I think many of us, including Dan and Karim, find ourselves right in the middle of these two conflicting axioms. On one hand we desire that students seek for themselves.  We desire that they personally invest in interesting questions that provoke grand thoughts about life the universe and everything.  Yet we also recognize that they are young whippersnappers who have little experience beyond their hometown, school, and even neighbourhood; they may not have the capacity to think beyond the world of their home.  We then as teachers must make the hard decision of how to lead them to these wonderings.

How many “how many” questions can we put up with?
One of the biggest critiques of 101qs right now is that the questions are too simple.
  • How many dominoes?
  • How many combinations are possible?
  • How many tickets?
  • How many gumballs?
These questions have all been taken from the top of our list of most perplexing images.  Sure they represent different math (in this selection alone we have rates, combinations, volume, and area covered!), but the depth of questioning is shallow to say the least, and this is coming from teachers, who have experienced this world and are confronted with questions that affect our humanity and interactions with the world at large.  “How many dominoes?” just doesn’t cut it when we live in a world that is torn apart by famine, poverty, super-consumption, and an economy that seems to be hanging over us like the sword of Damocles.  We the teachers need to see how we can bridge this gap.  Students are certainly engaged by these problems, but how do we turn them, and mathematics as a whole, into more than a daily puzzler? If our subject, and this website, is turned into only a collection of neat little brain teasers, then we have missed the point of our roles as teachers.
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Sadly, the interactive iPad version comes out next year.
Finding the Middle Ground

Just looking at the top of my site you will see clearly that I have drunk Dan’s Kool-Aid.  I love anyqs, and the whole three act process.  I know Dan's framework has directly made me a better teacher, and helped me to focus on more engaging class discussion, but I am beginning to find myself wanting more out of it.  In my comment on Dan’s blog I express that my main answer for this question is that we must understand that  101qs.com, in its infancy, is only presenting the first act of a fuller narrative.  We must use these videos as starters to the greater questions.  Students want to experience mastery of concepts, they want to feel positive about their abilities, and we can enter these students much easier into this subject that we love so much with a question like “How many gumballs are in that dang machine?”  It is fun, it’s nice, and students say to themselves, “I can do this!”  Then they take to the math, and they realize, “I need to know a lot of information to solve this.  What shape is that thing?  Is it a prism?  No, it’s a ball shape!  What is the volume of a ball?”  We enter with our sequels to these stories to propel their thinking, “How many gumballs could fit into this room?  Into this whole school?”  Students start adapting their thought to new situations, and they begin to see that this one little problem can extend far beyond just this picture.

Going for the Home Run

Once we have given students these chances to use their knowledge in differing situations we unload the brain busters.  We ask the hard questions, we develop projects from here. After seeing so many of these “How many little things can fit inside one big thing (gummy bears, tickets, dominoes, gumballs, teacups...) the question that has been arriving in my mind is this, “Why do we measure things in g, mL, etc. and not gummy bears, or gumballs?”  Get students to debate, and then create their own measurement systems.  Have them create conversion charts, and think about how to create a system that measures mass, volume, as well as linear measurement.  Have them reflect on the advantages and disadvantages of having their own measurement system.  Have them present their measurement systems to the class, have them create lessons about the Billy-Standard-System.  Then show them why the metric system is so awesome! They started their journey in the “How many?” river, but they are now swimming in the “What if?” ocean.

Where to Now?
Have I pulled this stuff off? Not yet, I’m still young, but I know that this whole 101qs thing is Dan Meyer’s attempt to pull the stunt on us as I want to do with my students. “Take some pictures,”  he says, but underneath it all is the possibility to go wherever we want.  So yes, if we discount these “How many?” questions as paltry, then that is all they will ever be, but if we keep pushing, we can turn these low floor questions into high ceiling discussions and projects.  So grab your camera, grab your phone, and get some first acts loaded.  We’ll work on the rest together later.
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There has been plenty of talk about the increasingly popular Khan Academy. I am not here to say anything “new” but to demonstrate how it can be used or misused.  As you may know much in math education barely scratches the surface of the grandeur of mathematics (a fact that Vi Hart bemoans); this is clear to many of us, our students included.  Though they may not put as fine a point on it, their questions such as “When am I ever going to use this in the real world?” or even “This doesn’t make sense” shows that math, this wonder of logic and creativity, has been lost by many a struggling student.  I am not saying that math must be used in “the real world” (whatever that really means), but what I am saying is that if students have no grasp why they are doing any particular math, then it will continue to be an alien language incomprehensible by mere mortals. Khan Academy can help to alleviate this, but only if presented at the right moment.

The danger that I see in the KA approach is that he begins with the 'mathiest' part, right from the get-go! A good learning cycle generally looks like this:
  1. Hook: This grabs students' attention, and usually poses a question.
  2. Investigation: This is where students try out a concept on their own, assisted by teacher questioning.
  3. Direct Instruction: This can be to fill in the holes, or to formalize what they have learned.
  4. Practice: Try out the formalization.
  5. Create: Going further by application.
Now this is not set in stone.  Project Based Learning flips this around a bit (or rather makes the creation the hook), but in general we want to have students in a state of ‘messing around with concepts’ before we formalize them, otherwise they have no anchor for the formalisations.  This is the reason KA cannot be a teacher replacement simply because students remain in steps 3 and 4 with Khan.  This is why when I teach exponents I start with Vi’s videos.  I can hook students on doodling in math class, and having them make Sierpinski triangles and Hydra Head Binary trees.  If I start my lesson with, “Graph this equation!,” I lose them. This is how I would teach this particular lesson:
  1. Hook: Watch Vi Hart’s Video (full link at the bottom of the post), and pause right before she tries to get to the bottom of the page.  Challenge my class to do so.
  2. Investigation: Students get frustrated that they can’t reach the bottom.  I ask them, “Why can’t they? What are some strategies to get further?”  Watch more of the video, ask them if they could do this (with a really small pencil), how many hydra heads would be at the bottom of the page?

    Here you let students, draw a T-Chart, here you let them look at the pattern.  Here you get them to draw a graph, and ask them, “Does that look linear?” but only after they make the realisation that, “We get MORE AND MORE EVERYTIME! ZOMG!”

    Here you can also talk to some students about going into negative exponents, “What does that do?”
  3. Direct Instruction: Now we can talk about what they are graphing.  We term this as an exponential function (non-linear), we talk about our x-axes, and our y-axes. We have them try it with other functions (and maybe with a Sierpinski’s Triangle).
  4. Practice: If you are into textbook work, they can do this.  If you want them to look at some more interesting examples of exponents have them do that. If you want them on Khan Academy, get them here now.
  5. Create: Estimating Population, Bacteria growth, etc. etc. Have them apply what they have learned here!

That is a brief outline, but you can see the point.  KA just does not fit anywhere but in Direct Instruction and Practice, and this is not a bad thing.  It is only a bad thing when we try to force it where it does not belong, if we try to say it is engaging, when it really isn’t.  KA can fill a void, but it should fill the right void. It may be better than a textbook, and you know what, I am all right if it replaces my textbook, I can handle that, but if we expect KA to fill the gap of the same humdrum curriculum that poor Vi had to sit through when she was a kid, it just doesn’t measure up. I tried too many times last year to tell students “How to do math.”  Please just let them do math, and if Uncle Sal wants to let them in on some of the formalities, by all means let him, just make sure you are not making him Mr. Sal; he’s not the teacher in your classroom, you are.

Am I being unfair? Which video would you rather start off with? Here are the original videos in their entirety:
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Dear SBG,

We have been together for a few months now, and since we are coming up on our 2nd term report card, I just wanted to share some of my feelings with you.  We have had such a great time, and in our relationship I have learned so much from you; it is amazing.

I remember first hearing about you back in University, and I just always thought that you were out of my league. I was accustomed to my life with Traditional Grading and I never thought I could break free of that life. You saw how awful it was last year when Trad and I were together. It seemed that every time report cards came around we would get in a fight and there was nothing I could do but limp along in the dead end hole of a grade-book. I didn’t understand what was going on, and it felt like we were just making numbers for the sake of making numbers.

Then in the summer, after hanging out in a few blogs, you and I came to see each other again. I was no longer worried about finals and writing papers, like when we first met, and I was free to sit down and really get to know you.  First it was just reading, then we started getting more serious: making grade trackers for my students, preparing learning targets, and creating spreadsheets with conditional formatting. It was the stuff of fairy tales. You even helped when it came around to prepping for my current school year!  I couldn’t believe how much clarity you brought to my life.  No longer did I have to worry about what I was going to teach; we had worked that out together already.  I knew what I wanted my students to learn and, thanks to you, I was able to focus on how I should teach my students, what a beautiful prospect.

When I look at you now, I know that what we have is special. I see so much in you. I now know that Billy’s three in the Pythagorean theorem means he knows how to apply the formula to a given problem, but sometimes forgets to find his square root on the last step. I look at what we’ve created, and I can get a sense of the class’ ability just by our colour scheme! This is something I never had with Trad and what I look forward to in the future.

Not all is perfect though. It’s funny I see you with @jybuell and @thinkthankthunk and wonder if that will ever be us.  I know, I know, you guys have a history and I can’t expect that we would work as well together so early on, but darn-it, I cannot wait until that time.  We have so much still ahead of us.  We have to improve our student buy-in, we need to incoroporate reassessment more naturally in the day to day culture of the classroom, and we have to have more student input in the process.  I am deeply sorry that not one student has helped develop a rubric this year, little to no peer assessment has occurred and our self-assessment has been very shallow. We still have the hills to climb, but based on these first few months, I know that we are headed in the right direction.

SBG you have done so much for me to relieve my anxieties of teaching and understanding my students.  You have shown me so clearly what those cherubs can do and what they know. How can I ever repay you?  Thank you so much for being a part of my practice.

Sincerely,

Timon
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Like so many good things that I have created in life, this post comes entirely from someone else's idea.  Dan Meyer had the first speed of light task, which I am linking for my sequel ideas, because, well, his just rock.  I wanted to include this post, because I feel like I will be able to use this version a lot easier in my grade eight class, it allows talking about cool conspiracy, moon landing stuff, Mythbusters, and is practice for me taking something that I see and turning it into something I can use in class.

Act 1 - The "Laser"

I love this set up because it is what scientists actually did!  They have reflectors on the moon at which you can shoot your lasers and receive signals back to prove that people were on the rock that is orbiting our Earth! Well, now all we need to know is how long did it take for the laser to to hit the moon and come back?

Act 2 - The Measurements

This act two is simple, I would probably use this for a started (shouldn't take too long at all), but here they are...

Act 3 - Fire the "Laser!"

Well there it is; a simple fun intro to the speed of light.

Sequels

As promised, Dan has some awesome work with the same problem, which can be found here.  Check it out, you won't be disappointed.
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A few weeks ago I was talking to a number of students in the hall at lunch.  I cherish these moments, because when outside of my classroom students remove a certain number of proverbial layers that they bring into the class.  One layer they commonly wear in class, and therefore the one that I love to see vanish most, is their fear of saying something wrong.  Often in class I see students’ puppy dog eyes, fishing for the correct answer that will allow me to move along and bother them no longer.  I am making students work for the gratification of being correct more and more, but that is for a different post.

My point however is that when out of the class I am no longer the “all mighty holder of all truths Mathematical, Sciencentific and French;” the stakes are low in their interaction with me beyond my door and that layer disappears.  I have taught some of my favourite science, math and French lessons at lunch and afterschool simply because I have students that are a) not scared of being tested, and b) genuinely interested in the learning that is occurring. They have no reason to be there other than sheer curiosity.  The other benefit of being outside of the scheduled class time is that students become much more candid.  Since they are not worried about the correct answer, they therefore reveal much more of their true thinking, worldview, and understanding.

In the particular moment that prompted this post, I was chatting with a group of students made up of a number of boys who were winded, red in the face, and generally exhausted.  The matter was that they had walked during their run in PE, and as a result had to run the route again.  This was a BIG DEAL, because it meant that any time that they spent on this second run, was time spent away from lunch hour. The boys then exclaimed, “We got our fastest time of the whole year today!”

I replied, “Great!  Now you know how fast you can really run!” This remark was met with many a disgruntled face.  One girl in the group said, very sincerely, “Mr. Piccini, that’s not how it works.  You see, what you do is you run just hard enough to make it look like your trying, but that you know you can improve your time later on, so on the record it looks like we are improving.”  This amount of utter honesty almost shocked me.

I then asked her, “Why don’t you just try your hardest and actually improve your speed?” to which she promptly said, “There’s no point!  Put me in basketball or soccer and I will run because there is a point to that game, but regular running, there is no point to that.”

This is what really hit home for me.  These students were not lazy, nor were they rebellious; they saw no point in running for the sake of running, and therefore they labeled it as something that could be done with just enough effort to get the teacher off their back. I wanted to tell them about the physical and mental health benefits, the conditioning for sports, and the pleasantness  a good run can have, but I realised like so many other lessons a lecture was not the way to show them the point.

This got me thinking about my class.  In Science, Math, and French have I shown my students the point?  Do I make it clear to them everyday why we are doing this? More importantly do I create for them moments and opportunities where they realise ‘What the crap? This is something I need to know now,’ because it is not enough simply to tell them. Necessity is the mother of all invention, and when we put students in a state of necessity, they are no longer receivers, but they are seekers and inventors.  I want to nurture a generation of such students, because once they become true seekers “the point” becomes moot; when they become seekers they do not need to hear the point, because as seekers,  they now have a new responsibility: they must create their own point.
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Ever wonder why companies make the decisions that they do? My wife and I drink more pop than I am willing to admit, and one thing I noticed while at the store is that the twelve packs of Coke and Pepsi do not have the same design.  Let's look at them (warning I do not know if this works in the States). This is also a precursor to this lesson.

Act 1 - The Boxes

I asked my students which one uses the least amount of cardboard, and in relation, which company made the best choice for the environment?

This was a good starter.  I had kids talking about which looks bigger, and a trend over at least two classes (I'll see later on in the future), is that the majority of students say that Pepsi is the clearer waster, or they have equal amounts of carboard.

Act 2 - The Measurements

I ask students what do we need to know to solve this?  Students came up with volume, area, dimensions etc.  at which point I introduce them to our cm cubes.

We talk only a little bit about the difference between volume and surface area.  In fact I do not use the words in particular, we talk only about "How many cubes are necessary to build this prism?" and "How many squares can we count on the outside?" an idea I lifted from Christopher Danielson.  I gave students the following sheet and basically said, GO!
surface_area_investigation.docx
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Here I talked to students a lot about short cuts.  It was great when they said they had no short cuts, and I ask, "You just counted?"

"Yes," replies the student.  I then proceed to count one-by-one each outside square.  Before I can even get to three, the student interrupts and says, "No, no, no, you times the length by width for one side, then times it by two.  You do this for three of the sides, since they each have an equal side" (paraphrased).  Bwa ha ha! I laugh to myself, they have it, and I didn't have to write a single formula on the board!

Bringing it back to the pop box now only involves asking, what do we need to know to solve the pop box, problem.  Students tell me they need the dimensions of the box so I give them these...

Act 3 - The Reveal

This is also a good time to have students talk about anything we are missing.  My wife after a few days of seeing me make this asked about the overlap, and if that makes a difference, I haven't looked into it yet, let your students do it too!

Sequel

  • The boxes  are designed with the following can configuriations...
How many different configurations can you find, and which one of these uses the least amount of cardboard? Do any of them use less cardboard than Pepsi and Coke's configurations?
  • Write a letter to one of the companies that explains your mathematics.  Suggest for them why you think they should switch or keep their design.

Reflections and Moving Further

This was extremely engaging for one of my two classes.  I think for me the pay out of the reveal is not as great as most of these 3act stories, so it can lose some students.  I want to work on that out for next year.

What else I need to do is to prepare more guiding questions to start off the investigation, so students can be invested in the problem earlier.  They have not seen surface area before and so are not acquainted with it.  Students knee jerk reaction at this age is to find volume (well actually they just say lxwxh without knowing what that means), so I want to guide them away from that. If they focus on volume to early on they count the problem off to early.

What I really liked about this investigation was the next day when I wanted to teach them surface area of triangular prisms I handed them the nets, and asked what do we have to do now. Students quickly came up with the idea that we needed to find the area of each separate shape and them together.   They knew the only formula they needed was to find the area of the triangle. We talked about the relation between rectangles and triangles.  This was a fairly easy process, because students can see if you cut a rectangle down the diagonal it makes two triangles (therefore bh/2).  This was a fairly simple review of the area of a triangle and extension of their learning of Surface Area.  It was fantastic to see how quickly they adapted to the new information! Students were able to further adapt this to other nets that used basic shapes, and it was a very simple natural extension of the same logic (Thanks to Kate Nowak for the idea of nets and surface area).

My only hiccup came with cylinders.  I was ill prepared to lead students into a discovery of the area of a circle (technically they learned it last year... but they're only in grade 8), so I fell back onto lameness. I hope to alleviate the lameness, any suggestions? I have learned that great teaching comes from thinking through all aspects of the lesson and leading them through the inquiry.  My cylinders lesson bombed, because I was lazy and had little prepared!  I was humbly reminded how easily it has to slip into bad teaching, let's change that Timon!
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I am learning more and more about how this whole problem based learning shtick works, and I have to say I love it.  Today was an example of how much I love it, and how worthwhile it is.  I have been struggling to motivate the use of the Pythagorean Theorem in a natural way so that students can jump in.  Enter Mr. Piccini's Summer Purchase...

Act 1 - My Living Room

I started off the "story telling," by discussing how it was time for my wife and I to upgrade our T.V. I showed them the picture and asked them: "What problems do you think I may run into?"  I realised after that this was a little too much of the "Guess what is in the teacher's head" but the question was so natural they needed to ask "Will the TV fit under the shelf?"  We talked about what it meant for a TV to be 40".  Almost half the class knew that it meant the diagonal.  I asked what do we notice about the shape of a TV is when cut down the diagonal.  Clearly there were two triangles, and thus began our exploration.

Act 2 - Exploring the Pythagorean Theorem

For the next part I stole the great investigation from Dan Meyer found here.  This worked so well it was unbelievable. Students were very quick to see the relationship between the sum of the small areas and the large area.  Students enjoyed the manipulative nature of this exploration, and the result was clear and accessible to the students.

We did some practice and then came back to my T.V. the next day.

We started the next day with whiteboards, and I had students write what information we knew and what information we needed to know.  I reinforced to students that I did not want to renovate my wall because I am lazy, but I am not too lazy to move my speakers.  They said then that we needed to know the height to the shelf. So I gave them this...
Depending on how hard you want to make this problem you can do either cm or inches.  Inches is the easiest.  Since students recognized that we need information on the diagonals, we need to talk about what a good diagonal measurement would look like for this space.  For that I broke out geogebra, to show students how sizing TV's looks.  I used these two geogebra apps.
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We talked about what the different shapes of TVs look like, and what your average wide screen would look like.  Using the geogebra animations the students discussed how the proportions always stay the same.  At which point we could talk about ratios.
At this point students needed guidance on how to find the dimensions of the space, but it came quite naturally to talk about the largest possible TV that I could fit in my living room.  Students found that this was a 55" TV, and as it turns out...

Act 3 - The Reveal

Reflections

This was such a simple execution, but it gave the students a real challenge, and a real-world application to try from the get-go.  I had a debrief with my students in one class, and they said they loved how visual, real, and engaging it was.  I never expected to hear that.  The only complaint was from students that finished really quickly.  I was not ready with sequels, and in fact, I still can't think of any. I never thought that this specific problem would have as much engagement as it did, but once students got rolling they were hooked. All in all I was happy today, and I sure beat the socks off this version of the problem.
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So, I am kind of on a binge right now with my 3 act posts, but I have had this one for awhile and I have not posted it, but after reading this comment on Dan Meyer's blog I decided I should put this up as a possible elementary level 3 acts lesson.  Here it is....

Act 1 - Mmm Juice....

I have a goofy little set up for this, where I say I'm making drinks for my friends, do I have enough?
Ask them what they think?  If so how many glasses do you think I can fill?  Give me a number that is too high and a number that too is small.  Ask your students what information we need to know.  Then we are on to Act 2 (and here is where the differentiation comes in...

Act 2 - Measurements (Grades 4-6?)

Act 2 Measurements (Grades 8-10)

If you want students to find the volume of each of these containers you can use these pictures to do the trick (jump-high-five for differentiation!).

Act 3 The Reveal

Sequels

  • How many cans of juice do I need to give one cup of juice to the whole class? To the whole school?
  • How big of a container do I need to hold all the juice for the whole class? The whole school?
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When it comes to anyqs, I find that the ones that interest me the most are the images or videos that blow my mind by sheer scale.  A seven wonders of the world sort of deal rings deep within my soul, and so I offer you MEGACOIN!  

Act 1 - The Coin

Picture
HOLY CRAP THAT IS HUGE! Here is where I would ask "How much is this coin worth?" Let students ask questions.  I am debating if I want to let them know it is made of pure Gold or should I let them follow any rabbit trail they want.  They'll probably guess it's gold.

Act 2 - The measurements

There is a lot that goes on with this one.  We have volume, density, rates, ratios, it is all fantastic!

Act 3 - The Reveal

And the final answer is...

The Sequels

I have actually been able to think of a few cool sequels based on this (which makes me happy since this is the hardest part for me).

  • What could the dimensions for a true million dollar coin be? A billion dollar coin?  A trillion dollar coin?
  • Using the following chart, how much would this coin be worth today?  Did the people that bought these make a good investment?
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I have a penchant for dinosaurs and monsters that roamed the Earth millions of years ago.  I still remember sitting in the theatre when I was six years old to see Jurassic Park.  I remember the first moment that I saw the Brachiosaurus in that movie and ever since I was hooked!  I have never grown out of that.  Watching Jurassic Park still makes me giggle like a school girl. When I see things like quake circles in glasses of water, and the piercing cry of the velociraptors, my heart soars!.  Suffice to say when I first learned of something called the Megalodon I was hooked.  Here are my 3 acts.

Act 1- Say "Ahhh!"

Megalodon was a shark: a really really big shark.  What is different about finding remains of a shark than say that of a Tyrannosaurus is that a shark is mostly cartilage.  That means that scientists mostly only get to find jaws and teeth.
So I want to ask students: "If you found a tooth this size, how big would the shark be?"  *Side note for extra awesome go buy a replica on ebay!* Get guesses, draw pictures! Give them a grid to show scale between you and the size of the shark, all in the name of awesome!

Guiding Questions

  • If we know this is a shark what is the next step that we should take?
  • What sharks could we compare it to?
  • Which shark looks most like Megalodon's?  Check here maybe.
  • How can we compare the sizes of these teeth?
  • How does that affect the shark's size?

Act 2 - The Scientific Process

Now experts recognized that this is a shark tooth by comparing it with one of the fiercest sharks around.  The Great White shark actually shares much the same characteristics as the Megalodon.  Hopefully students will recognize that if we have identified it as a shark tooth that we can compare it to these sharks.  This is where some comparison pictures come into play.
Students now can make some educated guesses, and we can talk about what are some good methods for estimating Megalodon's size.

Act 3 - The Reveal

The first estimations were done very simple proportional reasoning, which gives us the overestimate. At this point I might have students measure out on a string how big the Megalodon actually is!

Sequels

There are a few sequels that come to mind.
  • Given a length of a shark, draw the size of its tooth.
  • Experts thought that assuming a directly proportional relation was not accurate measurement.  Some experts (Gottfried et al.) found the following information to create a formula that predicts the size of a shark.  Use the information in this table to find the equation, and use it to make a new estimation of our Megalodon's size. 

Tooth Size (cm)Shark Length (m)
2.5 2.2
3.8 3.4
5.1 4.7
6.4 5.9

Well tell me what you think!  I am so excited about this, but will it translate?  What do ya think?
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