Is the Google Offer worth it?

This came from some real-life math I was doing today as I was trying to book at flight from SFO to Chicago. Google Offers had a deal on airfare (on Virgin America): $50 for 50% off your next flight (round-trip or one-way). 50% off--awesome! But how good of a deal is this? Is this Google Offer always worth buying? How can you tell when it's worth it and when it's not? Is there a situation where it could even end up costing you more to use this Google Offer? How can we create a model to tell us when this Google Offer is worth it, and when it's not.

To complicate matters, this offer came out at the same time that I also happened to have a coupon for Virgin America--a coupon that I got for free from an online promotion--for 20% off my next flight. How does this change when it's worth it and when it's not worth it to buy the Google offer? How does it change the model?

Use multiple representations to explain your solutions (t-table, graph, equation, etc.)

Russian Stacking Dolls

Proportional reasoning meets volume, maybe? There have gotta be some connections between growth factors in 1, 2, and 3-dimensions. 

Also, just some cool pattern finding and different methods for problem solving.

http://blog.mrmeyer.com/?p=10211&utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+dydan1+%28dy%2Fdan+posts+%2B+lessons%29

More Than $1800 in Savings

This month's Costco coupon book says on the front "More than $1800 in savings, plus more than $3500 in online savings from Costco.com." Wow, what a good deal?

I'm curious to know how this is calculated. I only counted 76 in-store coupons, most with a savings of under $5. I'd love to talk to kids about the marketing strategy behind Costco's assertion, as well as just how much money you'd have to actually spend in order to save $1800 (or $3500!).

100 times the bigger?

http://blog.mrmeyer.com/?p=9173

Orbeez (little toys) purport to grow to 100 times the volume. What does it mean for something to be 100 times bigger? What kinds of measurements would you predict for something to be 100 times bigger? Is it misleading to just talk about volume?

How Big is that Country?

This site lets you pick a country and compare a number of its quality-of-life stats to the US. But I think the most interesting part is the map that overlays the country on top of your geographic area. Given that I feel like I understand the state of California as a good scale, it's really interesting to see how the area of other countries compares.

http://www.ifitweremyhome.com/

Given how bad kids seem to be at geography, this might not be as interesting or surprising to them, but it raises questions about map scaling that at least I think are worth asking.

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... 

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?

"Large" and "Small" Beer Cups

http://deadspin.com/5728087/the-great-qwest-field-beer-scandal-of-2011

A large beer costs $1.25 more than a small beer, but they hold the same amount.

Qwest Field responds:
http://www.seahawks.com/news/articles/article-1/First--Goal-statement-regarding-beer-cup-size/c9b409ac-be93-4e80-8b7e-24dcc4cc49e3

The real question is: how is it that these cups look like they're different sizes but actually hold the same amount of liquid. There's gotta be a problem in there about what kinds of constraints on height and diameter are enough/too much to allow the cups to "look" different sizes. E.g. how small of a diameter would make the small cup look like the big cup?