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!)

Steep Streets of San Francisco

I kind of wish I taught in San Francisco because I feel like this could somehow turn into a fun field trip:
http://www.7x7.com/arts-culture/real-top-10-list-steepest-streets-san-francisco

Maybe the math problem isn't that interesting, at least not at high school, but talk about a real-life application of slope. I think it's interesting to think about slope as a percentage--you never see a 30% line in math class, but you might see things like 30% grade on street signs. I wonder if kids notice those signs (especially if they don't drive yet and especially if they don't live in a hilly area), and I wonder what they think that grade means. It might be interesting to connect trig to this somehow and think about the angle that the street is tilted at. There always seem to be trig problems about wheelchair ramps--are road grades another piece of this? Is it even interesting?

Sort of related, I have to say I'm always kind of amazing at how steep a road or hill actually feels when I think of the grade mathematically. In my math class mind, a slope of 3/10 is definitely not steep (really, anything below a slope of 1 doesn't seem that steep. I wonder if this is because I often think about slopes in terms of pile patterns, so a slope <1 means that you're not even adding one block per pile...). But driving on a 30% grade is terrifying and just walking up it is painful. Would a kinesthetic experience with slope actually support students' understanding?

Update: 

I finally decided to turn this into a student activity, but the article above doesn't really give enough information to make something interesting. With a little more research into Stephen Van Woorley (the guy who did the calculations the article was based on), I found some much more interesting stuff:

• The story of how he came up with these measurements. I think this will be my basis for the student activity. 
• More steepness calculations, including some good history. An interdisciplinary opportunity??
• A sweet map! I mean, really, who doesn't love a good map-based data visualization?

What is Wrong with these Charts?

http://flowingdata.com/2013/07/15/open-thread-what-is-wrong-with-these-charts/

Good lord.

Another interesting question: what is right with these charts? Poor Pete always does something right.

Dictionary of Numbers

A Chrome extension that gives context to numbers.

From playing around with it, it's not as easy as I'd like, but there are some cool things. Did you know that the International Space Station weights 1 million pounds? I like the idea of giving context to numbers so that kids/people can have a reference point, especially for big numbers.

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.