Counting Trees

The "Counting Trees" Formative Assessment Lesson from MARS/Shell Centre is one of my favorites. I think it's open ended in an interesting way and I love that kids need to be okay with not knowing the exact right answer.
The other day I heard this story on NPR about how many trees there are in the entire world.


Of course my first thought was the FAL and how I would use this story in conjunction with that lesson. I wonder how I would structure it. Would it be a hook to the lesson or a "beyond"? What parts would I have kids listen to or read? I haven't read the Nature article yet, so I wonder what's in there.

I loved that the story went through a whole process of asking a question, making a conjecture, revising the conjecture, and so on--exactly the kind of thinking process that I want to highlight for kids.

It also raises some interesting questions about rate (how long it would take to plant 1 billion trees), density (if so much forest has been depleted, what did forests used to look like?), and large numbers (what does 3 trillion trees even mean?!)

Exponential Water Tank

Apparently some guy once said, "The greatest shortcoming of the human race is our inability to understand the exponential function." I'm pretty sure I'm not on board with that statement, but I do agree that exponentials are very challenging to make sense of. Linear growth is intuitive; exponential growth is not. 


This video shows a tank of water filling up at an exponential rate. I think it would be interesting to have kids watch to really think about how fast its growing, and talk about what it means for something to grow at the same rate, or a constant rate. I wish there was a side-by-side video or another version showing linear (or quadratic) growth. 

http://bl.ocks.org/hanbzu/9787042

Another thought with this video: use it in the introduction to exponentials and have kids make some predictions, do some graphing, etc. The clock in the corner is handy. Less handy is the fact that there's no marked height anywhere. If you projected onto a whiteboard could you mark height? 

Not-So-Easy Area

This seems to be called the Azulejos puzzle. I haven't actually gone through and figured it out yet (or cheated and watched the solution videos that pop up on YouTube), but it's super cool. I love that it takes a pretty straightforward area (a rectangle with obviously countable squares) and plays with it. I imagine that kids would have lots of non-mathematical theories about why this illusion happens. The YouTube commenters are thrilled to share their theories about how the person in the video is using slight of hand. But no magic, just math.

This task, "Doesn't Add Up" from NRICH Maths--my new favorite (or should I say favourite?) source of math tasks--feels similar to me. There's something going on that fools you, and it would be interesting to see what kids think is going on and how they solve it.

The NRICH Maths task also brings back memories of a long, painful evening I spent trying to write a shaded area problem for a test. I was trying to come up with a triangle that had parts shaded so I could ask kids to find the shaded area in at least two ways (getting at ideas of decomposition and recomposition). But the measurements I kept trying kept getting me different answers from different strategies! Even though I was trying to work systematically starting from one measurement and determining the others from there, it kept coming up weird. Finally I figured out that free-handing my triangles and adding values meant I wasn't actually accounting for angle measures and slopes of lines so the shape I was drawing was decidedly not drawn to scale. Finally I just got out some graph paper so I knew I was drawing to scale; I couldn't believe it took me so long to figure out what was wrong. On the plus side, at least I caught my error and didn't put an impossible problem on a test (wouldn't be the first time, won't be the last, but at least it wasn't another time).

Food Equivalencies

What does 2000 calories look like?

There are nice visuals around how many ____ are equal to how many ____ in terms of food. The bacon to cinnabon equivalency is my favorite, I think, because we think of bacon being so unhealthy. (Not that we think of cinnabons as being so unhealthy, but I, as a teenager, was definitely more likely to get a "snack" at the mall of a cinnabon, but would never have gotten a stack of bacon as a snack).

Questions to think about:
-Are all calories the same?
-What sets of 2000 calories can you imagine eating (I can definitely imagine eating 2.5 cinnabons, but not a whole pizza)? How does the mental association of these foods impact your eating?
-These equivalencies represent calories. Which would be the same if we looked at fats/protein/carbs/etc.
-Why is 2000 calories the recommended daily allowance?

Maybe it's not super mathematically interesting, but I do think it gets into units and the meaning of the equals sign in an interesting way.

Honey, I mis-calculated my math?

In this Honey, I Shrunk the Kids trailer (an amazing movie...) at 0:41 the littlest kid does some calculations. "We're a quarter of an inch tall and 64 feet from the house. That's the equivalent of 3.2 miles."



If his calculations are correct, how tall were the kids originally? Whose height was he using to make his calculations?

Other questions that might be relevant:
-How did he know "we're a quarter of an inch tall"?
-If all the kids are a quarter of an inch tall, did the shrink ray really work? Alternately, if all kids are not a quarter of an inch tall, did the shrink ray really work?
-What evidence is there in the video that the kid's calculations are off?

Lunchtime Clock

This clock speeds up and slows down such that you get an extra 12 minutes of lunch.

First, I have to say that it took me a long time to figure out just what was going on. It was confusing to me when the clock speeds up, when it slows down, when the extra 12 minutes is actually happening, and so on. It took me awhile to even figure out what time the lunch hour is supposed to be! Some googling of "Lunchtime Clock" helped me make some sense of the actual context of this. Definitely if I were to use this video there would need to be some built in sense-making time (no pun intended). See my scaffolding questions (below the interesting ones).

I'm not sure what mathematical category this fits in because I haven't done any math around it yet, but it's definitely some interesting math. Here are some questions off the top of my head.

Interesting questions
-Why is the "slow" interval longer than the "fast" interval? What is the relationship between these two intervals? What is the relationship between the intervals and the percent increase/decrease in clock speed?
-How do you know that speeding up and slowing down by 20% will add exactly 12 minutes to your lunch?
-How would you re-program if you had a 40 minute lunch instead of an hour lunch? A 90 minute lunch? (Consider differences between adding 12 minutes to a 40 minute or 90 minute lunch versus adding 20% to your lunch hour, regardless of the original lunch hour's length).
-If you wanted to start off just getting an extra 5 min of lunch (so your boss didn't notice), how would you have to re-program the clock? What percentage of the normal speed would you have to reduce/increase the clock to, and over what period of time? What if you wanted 15 extra minutes? n extra minutes?
-If you tried adding more time to your lunch hour, at what point do you think your coworkers/boss would notice the difference?
-You have an 11:30 meeting--what time will it say on the lunchtime clock? A 12:30 lunch meeting? Can you come up with a rule to tell what time it actually is by looking at the lunchtime clock?
-During that lunch hour, does the lunch time clock ever display the correct time?

Scaffolding questions
-What time does this clock assume that your lunch hour happens?
-Over what time interval is the clock moving slower than usual? Over what time interval is it moving slower than usual?
-What does it mean to "speed up by 20%" and "slow down by 20%"? 20% of what?

What other people on the internet seem to care about
Look, the internet made us a clock to play with! So many extension questions to play with!
http://www.lunchclock.com/ 

Also, check out the YouTube comments -- lots of math and interesting strategies! Might be a good hook for "evaluate the reasoning of others" to just go through and try to make sense of the comments. Of course, given that it's YouTube, probably best to screen the comments first.

--------------------
Update: I tried this task with a group of about 100 secondary teachers (6th-12th grade) and it was a HUGE hit. I gave pretty straightforward prompts:

• You planned an 11:30 phone call with someone who, strangely, hasn’t set up their own Lunchtime Clock yet. What time should you look for on the Lunchtime Clock to know when to make your phone call 
• Oops, you read the Lunchtime Clock wrong and missed the phone call! When your phone date calls back, the Lunchtime Clock reads 12:30. What time is it really?

Teachers worked for at least an hour straight and I'm not sure anyone came to an agreed upon answer. I didn't share the internet lunch clock because I just wanted people to use the video for data points. I was surprised at how few people actually found data points and tried to fit a function (piecewise or otherwise). 

I was thrilled by the different representations people worked from: lots of variations on tables, attempts at equations, some interesting graphs, and a whole array of non-standard representations and model that came out organically as people tried to explain their thinking. 

I was hoping that doing the task with teachers would give me a better idea of how to use this with students and where it might sit in a particular course or vertical progression. But the teachers left me even more unsure, actually. I really want to spend sometime working on this task myself... 

Joey Tribiani's Proportional Reasoning

Friends Season 3, Episode 18: "The One with the Hypnosis Tape."

1:44:10 on the DVD, Joey and Ross are trying to convince Frank Jr. not to marry his fiancee because of the age difference.

Joey: "You're 18, she's 44. When you're 36, she's gonna be 88."
Frank Jr.: "You don't think I know that?"

Double or Nothing

http://blog.mrmeyer.com/?p=5394

Parks & Recreation clip about betting on a pool game.

The questions:
How many games did they play?
How long were they playing for?

Will It Hit the Corner?

http://blog.mrmeyer.com/?p=5954

The Office clip!