Bongard Problems

https://en.wikipedia.org/wiki/Bongard_problem
http://www.foundalis.com/res/bps/bpidx.htm

So awesome for understanding what a property is.

It would be interesting to do these with numbers in addition to diagrams.

Thanks, CMC North 2015!

Where in the world?

I'm not sure whether this would be a fun problem or not. I definitely think it falls into the "doing math" category when it comes to that math task categorization taxonomy.

Lots of famous cities have those signs with lots of arrows pointing in different directions with distances to famous cities. My math problem idea: show a picture and ask "Where in the world is this?" I'd want kids to figure out what to do from there, and to use the power of the internet and Google maps to go crazy.

I also found this sign that gives times instead of distances. Definitely an added level of challenge.

To me, this is a problem about loci and intersections of loci, so I can imagine asking a follow up question about how much information from the sign is actually needed to figure out where you are. In theory, this could get pretty interesting because you're working on a sphere instead of a cartesian plane.

Here's why I have questions about this activity:
-How do you find a picture of a sign that's not too easy to google?
-Just by typing in "___ kilometers from Moscow" how quickly will kids find their answer?
-Does a sign with obscure cities (like a road sign on the freeway) make the task more interesting or less interesting? More challenging or less challenging?
-Which is really all to say: how much math is in this task? What math is it?

Also: in searching for a good picture to add to this post, I found a 3D Signpost App. This feels like it has more potential. I especially like this screen:

It feels like there's lots of potential for kids to make their own versions of the screens from this app. And if there's some crossover with a geography or world history class or something, even better. 

How big a TV should I buy?

This article made me think about some geometry:

http://www.cnet.com/news/how-big-a-tv-should-i-buy/ 

The graphic is interesting because it involves angles and lengths. It seems like there's a lot that kids could play with.

• If a couch is positioned at exactly 9 ft from the TV, what size TV should you buy? 
• If your couch is positioned at a different distance, what size TV should you buy to keep the same ratio? 
• If you own a TV of a given size, how far away should you place your couch? Does it matter if the given size refers to the diagonal or the width? 
• When the article says "the TV should fill 40 degrees of your field of vision" what percentage of your field of vision is this (you'd need to think about your peripheral vision)? 
• THX and SMPTE recommend 40 degrees and 30 degrees respectively. How does this change the size TV you should buy? Or where you should put your couch if you have a specific size TV? 
• This could easily be connected to costs
• Does it matter where a person sits on their couch? The diagram works from a person sitting in the middle of the couch. Will other people on the couch still have the TV fill 40 degrees (or 30 degrees) of their vision? How much more or less of your vision will the TV fill at different points along the couch? 
• What do these distances and angles mean about where you should (or shouldn't) sit in a movie theater? 

The article also includes a link to a chart that accounts for pixel resolution. I haven't looked at it closely, but I'd be interested in exploring the patterns in the table: 

http://referencehometheater.com/2013/commentary/4k-calculator/ 

Biggest Trig Warmup Ever

This looks so fun! What a way to get kids to think about the composition and decomposition of the "sweet 16" triangles on a unit circle. I haven't found many get-up-and-move-around activities that feel appropriate to high schoolers, so I love this. The materials prep isn't even that bad because you only need the two triangles.

Extra challenge for when you have more students in the class: throw in the angles an increments of 15degrees that aren't already part of the key unit circle angles. Really this is a physical variation on the question of how many angles you can build from just from a set of drafting triangles.

The Painted Cube, Revisited

I'm a big fan of the "Painted Cubes" problem (although to be honest, I don't think I have ever actually used it with kids...). I feel like most math-y people have seen the problem at some point: If you build an n x n x n cube out of unit cubes and paint all the faces, how many of the unit cubes will have one face painted? 2 faces? 3? 0?

This problem from NRICH Maths is a a cool extension. I love the use of the word "some"--it's so interesting how that word both increases access and makes the problem so much harder. I haven't actually done any work on the problem yet, but it just makes me want to play!

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).

More Grocery Shrinking Options

There has to be a volume problem/project in here. The setup: you work for some company that wants to cut costs by shrinking its product... but you don't want the consumer to know. Consider 3 options for shrinking the product without making it too different:

• Change the shape of the packaging (box 1 in the comic above). Needs to be a change that's subtle enough that someone won't notice. Could be totally dramatic (changing rectangular package to a cylinder or frustrum). Could be subtle (shave a little bit off here, a little bit off here, no one will notice). Could be sneaky (when bottle manufacturers make the bottom of the bottle not flat).
• Make the actual product smaller (upper right box in the comic above). Seriously, how much could you save by making the holes in cheerios or bagels bigger? 
• Filling the package with something else (the ice cream gnome above). Less ice cream, more toys!

I wonder what givens you'd need to give kids to start with and how open-ended it actually could/should be. I would love for kids to actually build their packages so that they could show why it's a sneaky change!

(Yes, this is teaching kids to be corporate scammers, but hey, preparation for the real world, right? Maybe it will also teach them to be suspicious of corporate scammers!)

Geometric Fruits & Vegetables

http://laughingsquid.com/geometric-fruits-veggies-photo-series/

Seems like there is a lesson on fractions here. How many watermelons are there in the first picture?

Another thought: This could totally be done as a number talk!

Another thought: I wonder how some of these pictures are analogous to the border problem? Is there something about making the pictures larger or smaller that could lead to a generalization? This one seems like it would work well for growing/shrinking/generalizing.

Another thought: This one is hecka cool for kids to think about visualizing how shapes get dissected and put back together in 3-D.


UPDATE:

My colleagues and I included some of these photos for Number Talks in our Math 6 curriculum. (For those who are unfamiliar with Number Talks, I highly recommend learning more!) We called them "Tasty Number Talks" and they have been a huge hit with teachers and students. Unfortunately I have not yet gotten to see a Tasty Number Talk in action, but I hear good reviews! I wonder what the artist Sakir Gokcebag would think about his art being used for math class...

Exploratorium Math: The Square Wheel

What happens when three math teachers go to the Exploratorium? We briefly pass through all the fancy science exhibits and then spend like 20 minutes staring at this seemingly simple display of a square wheel rolling on arched ground. 

The key is keeping the center of the wheel always at the same height above the ground. So how do you design the piece of the circle that makes up the "ground"? I still haven't figure it out, but we came up with some interesting stuff that was too difficult to continue without pen and paper. 

In summation: my kids are going to build square wheels. But I should probably try it first. 

Questions: 

• Is it better to start with the ground or the wheel? 
• If bottom of each circular ground piece is a chord, does the central angle intersecting the ends of that chord always have to be the same? 

Exploratorium Math: The real-life hyperbola

This still amazes me every time I see it:

How incredible is it that you can move a straight line and it makes a curve?! I guess that's what math is all about. But I like this exhibit because if you just showed me the tilted bar spinning around without the plexiglass, then asked me what kind of hole I should cut in the plexiglass that will allow the tilted bar to pass through... I would never guess the hyperbola. 

Questions I want to ask (myself or my students): 

• How does the angle/slope of the tilted bar relate to the shape/equation of the hyperbola?
• How does the angle/slope of the tilted bar relate to the slopes of the hyperbola's asymptotes? 
• What's up with the intersection of the hyperbola's asymptotes? Are they perpendicular (I can't tell at all from this picture)? What changes can you make to the tilted bar or the way its rotated that will either (1) keep the asymptotes perpendicular (if they already are)? (2) make them perpendicular if they're not already?
• Where are the foci of the hyperbola? How do they relate to the position of the tilted bar? 
• If you make the bar longer, but keep it at the same angle, what will need to change about the hyperbola and the way it's cut out of the plexiglass? 

I pretty much just want to make kids build this.

Culturally Situated Design Tools

http://csdt.rpi.edu/

There's a lot of stuff on here and I haven't had time to make sense of it, but I don't want to lose this link. I especially appreciate that so much of it is connected to higher-level math because authentic, interesting culturally relevant pedagogy feels nearly impossible to find for upper-level high school math. This feels like it's using the "gimmick" of the cultural connection in a useful, non-cheesy way.

The thing I'm most interested in right this second is how to adapt the concepts and ideas presented to activities that (1) do not require use of technology--or that build from the technology to push beyond guess and check, (2) are more groupworthy.

Thoughts?

XL Wine Glass

http://www.amazon.com/DCI-10040-XL-Wine-Glass/dp/B000VKOK6O

"Holds a full bottle of wine!"

Hmm... why does this look so much smaller than a real wine glass? Is there really that little wine in a bottle? Is this glass really that big? How could we find out?

How Big Really?

https://thetruesize.com

Enter in your zip code and find out how big something is (environmental disasters, historical events, famous landmarks, and so on) compared to where you live.

Twitter's New Design + Golden Ratio

http://mashable.com/2010/09/29/new-twitter-golden-ratio/

I wonder what else follows this.

Some questions to ask:
-Which design is more pleasing to the eye?
-How do differently-sized computer screens and browser windows effect this layout?

Buildings Shaped Like What They Sell

http://www.mentalfloss.com/blogs/archives/74157

Definitely something awesome for the proportional reasoning or 1-2-3-D growth units.

Will It Hit the Corner?

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

The Office clip!

Rolling glasses

http://blog.mrmeyer.com/?p=4018
Which glass will make the biggest circle?

Question: would my 9th grade geo students be able to handle this? What would they do with it?