Tiny Pop Stars

 

This is why we need to learn about accurate scaling of graphs. 

Wavy walls

 https://hasanjasim.online/15-illustrations-of-british-crinkle-crankle-walls-which-require-less-bricks-to-construct-than-straight-walls/

In England you might see a lot of "wavy walls" made of brick. According to the link above, the walls actually use fewer bricks than a straight wall because straight walls need at least two layers of bricks to be sturdy, but the wavy walls are sturdy enough with one layer because of the arch support provided by the waves. 

Lots of investigate in here. 

• Why is it sturdier? 
• How "straight" could a wall get before it starts to get unstable? 
• Is there a "too wavy" wall? 
• What connections are there (if any??) to sine and cosine functions? 
• Construct a wavy wall! Could be interesting to test strength somehow. 

Not sure how much engineering and physics is necessary to understand to be able to really make sense of this. But the walls do look cool :-) 

Making Cents

One of my favorite activities with students is asking them to calculate how many pennies it takes to cover the floor. Take ~20 pennies, a ruler, some string, and go crazy.

Well, this person really figured it out:

I would want to show kids this picture and have them generate questions that come up. Some questions that come up for me:

• How big is the floor? 
• How many pennies? 
• Are there more dark-side-down or light-side-down pennies? An equal number? 

Beatmaker

I got this idea from some amazing student teachers!

Using the Beatmaker from splice.com, they had us come up with beats using different instruments that hit at different times. What I thought was really cool was that we had to draw the pattern ourselves before we played it on the Beatmaker. There was a lot of thinking about how to mix and match patterns for different instruments, and what kind of visual patterns would result in what kinds of sounds.

I feel like there's a lot more to do with it, but I haven't investigated yet!

Number Bracelets

I went to Ruth Parker's session at CMC North 2015 and loved it. Of the many things I learned, I am perhaps most excited about Number Bracelets.

Here's how they work:

• Start with any two single digits. Say, 3 and 6. So the start of your bracelet is 6,3
• Add them together. You get 9. Now your bracelet looks like 6,3,9
• Add the last two digits in the bracelet. When you get a double-digit number, you only write down the 1's digit. So now our number bracelet looks like 3,6,9,2 (because 3+9=12, and 2 is the 1's digit)
• Keep going until it starts to repeat: 6,3,9,2,1,3,4,7,1,8,9,7,6,3,9 --> notice that it starts to repeat here. 
• The length of your number bracelet is the number of digits/terms before it starts to repeat. So in this example, the number bracelet starting 6,3 has a length of 12

I have so many questions I want to answer about these!!!

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!

Average book length

Some statistics are useful, and some are not. I don't know if this one is useful, but it's kind of fun:

The average book has 64,500 words.

I don't know exactly what I'd do with this in class, but it seems like an interesting opportunity to discuss why we use different statistical measures to describe data, and to get kids thinking about which measures feel useful in which situations. Is mean really the right measure of central tendency for this statistic? Is comparing a measure of central tendency even useful? Why do we care?

There also might be something (less exciting, more practice-y) about using all the stats for each book to work backwards to calculate the standard deviation. That also raises the question of whether mean and standard deviation are really the right descriptors. I am very curious whether word length is a normal distribution. It might depend on what genres of books you include (children's books seem like the have the potential to skew the data).

What other statistical questions might kids generate? How could they use info about the books they're reading in English class to do some further exploration?

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?!)

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. 

Could you be an Olympian?

http://www.theguardian.com/sport/datablog/2012/aug/07/olympics-2012-athletes-age-weight-height

The Guardian has stats on height, weight, and age across athletes in the 2012 Olympics. Some questions I would want kids to ask/investigate:

• For your height/weight/age what sport are you most likely to play. That is, for what sport are you most "normal"? 
• I don't know if I'd include age depending on how old my kids are. If I mostly had 14 year olds, it would be tough because most athletes are older. But it would probably be fine for 17 year olds
• I might also leave out weight because that's a touchy subject, but maybe I'd give kids the choice. 
• It would be interesting to see how kids combined all three variables
• If you played ___, what percentile would you be in for height/weight/age? (I think this would require the assumption that the variables are normally distributed)
• Some kind of comparison of shape, center, spread across sports. Which sport has the longest window of time where you can reasonably play (we could discuss whether this referred to range or standard deviation)? For which sports is mean a better measure of center and vice versa? 
• For which sport are athletes the most different from the general American population

What's best is that all the data is available in a spreadsheet, so you can do whatever you want with it

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/ 

The Math of Fast Food Sweepstakes

Michael Ross did an interesting analysis of McDonald's Monopoly game and Tim Horton's "Roll Up the Rim to Win" (not living in Michigan anymore, I had kind of forgotten that Tim Horton's existed...)

http://regressing.deadspin.com/the-math-behind-mcdonalds-monopoly-1642081131
http://www.mikerobe007.ca/2013/02/the-economics-of-roll-up-rim-to-win.html

I like the data displays he uses in his Monopoly analysis are nice (2-way table vs. pie chart), and I like the comparisons used in his analysis.

Not sure exactly what math I would have kids do with this, but it's fun. Maybe it would give kids another mathematical reason to not eat at McDonald's.

Estimation 180 and Confidence Intervals

I love Andrew Stadel's Estimation 180 collection and sequence, for all the reasons why lots of people have been praising it. Estimation is an undervalued skill! Kids are terrible with units of measurement! It's super-accessible across multiple grade-levels! It's quick! It's fun! Etc!

One of the features is that Stadel always first asks for a guess that is too low and a guess that is too high, before asking for a the final estimate. I initially liked this because of how it increases accessibility for students and also trains them, in the long-term, to get more specific and accurate with their estimates and reasoning. Now I found a new reason why I like this practice: preparing kids for confidence intervals. Confidence intervals are just a more systematic way of making estimations, and really what the confidence interval is saying is "here is my range of guesses that are neither too high or too low." I imagine that when having kids share to high/low guesses for Estimation 180, there will sometimes be guesses that are actually correct, especially as students get better at their estimation and try to get "just a little bit" off in their too high/low values. That, in a lot of ways, is like a confidence interval! A 95% CI is saying that it is possible--5% of the time--that our interval doesn't capture the true population statistic (mean, proportion, etc). The too high/low guesses for whatever population statistic fall in those outer 2.5% tails. They're possibly correct, but it's highly unlikely. You'd be really surprised a value from those tails turned out to be the true population statistic.

Somewhere in here there has to be a lesson/activity where we collect estimations from a bunch of people and find out that the mean estimation is actually pretty close to the actual value. This is true for people guessing about the number of jelly beans in a jar, and so on. Can we use estimations to set up a confidence interval for the real number of jelly beans in a jar? Is that legitimate statistics?

Guess My Weight

I've always been interested in those people at amusement parks and fairs who guess people's weight or age or birthday month or whatever. One interesting question from that situation is which variable you should have the person guess for the best chance of winning. On one hand, birthday month feels nearly impossible for someone to guess by just looking at you, but the guesser does have a 1/12 chance of being correct. I can't remember the usual ranges for age and weight that let the guesser win, but it would also be interesting to think about how the amusement park sets those and if they're fair. To complicate things even more, how do social factors change what the guesser guesses (e.g. does the guesser under-guess age and weight for older people and women respectively, because that's what our society says is better?)

This problem is super-interesting:
http://nrich.maths.org/6957

I like that there is a lot of open-endedness to the solution and "correct answer." Unfortunately I am not sure what unit it might fall in because it involves a ton of different possibilities. Just a fun math problem? That's okay with me too!


--------------------

Update: I tried this task with a group of approximately 80 secondary math teachers (6th-12th grade). My version was slightly modified to (1) give it a little bit more of a hook and make it look pretty; and (2) obscure the task name so no one could google it... Teachers are sneakier than students. They shouldn't get to do what I did and just go straight to the sample student solutions!

I haven't looked at feedback from the session yet, but I personally enjoyed listening to what people came up with. There as a great deal of disagreement in the room about who should "win" and lots of different takes on a scoring system. Unfortunately we didn't have as much time as I wanted to, so I didn't get as much of an opportunity as I would have liked to push on some of the justification aspects, especially about why a scoring system is fair or ideal.

A lot came out about mean and standard deviation that I also didn't get a chance to make sense of. I wanted to ask people mean and standard deviation of what? What's their sample? Is mean or median a better measure of center? Part of that last question might rely on an assumption about guessing whole number weights. What happens when this task changes from discrete to continuous mathematics? I doubt the answer changes, but the questions you ask will definitely change.

Baby Name Distributions

http://fivethirtyeight.com/features/how-to-tell-someones-age-when-all-you-know-is-her-name/

I am fascinated by baby name trends, but I don't know if students are. What I am most interested in with this is the intuition students will have about names, and how that helps set them up to understand distributions, especially bimodal distributions. I know that I have fairly set ideas about what names come from what eras (I hear "Agnes" and I picture an elderly woman; I hear "Kaylee" and I picture a young girl), so it helps with thinking about when median may or may not be the best measure of center. Also, there's a nice graph with interquartile ranges, which demonstrates why we care about the interquartile range. Finally, there's so much baby name data out there that kids could definitely research and construct their own graphs based on the questions that (I expect) will come up from looking at all this.

Graphing Stories: the Next Level

https://teacher.desmos.com/carnival/walkthrough#cannonman

This is so cool! It's like graphing stories, but the interactivity really helps hone in on kids' misconceptions around graphing. All those things about graphs just being pictures, about understanding what makes a function (beyond the vertical line test...), etc. are captured in the well-chosen scenarios.

I talk a lot of smack about blended learning or personalized learning or whatever they're calling it these days, but I am in no way opposed to technology use in the classroom. The Function Carnival is a great example of technology usage because it provides something that pencil and paper can't. Sure, it's probably engaging to a kid because it's on the computer and it has fun animation, but technology purely for engagement's sake is not enough. This technology also doesn't just feel like a way for teachers to measure some percentage of material learned. Those things are fine, but not really enough (at least for me). But this tool helps kids deepen their understanding of function from very different perspective. That is what technology should do--it should enhance teaching, not replace it.

Who's Lurking behind These?

42 strange things that correlate:
http://tylervigen.com/

Obviously, it's interesting fuel for the "correlation is not causation" discussion, particularly because it's interesting to think about what the lurking or confounding variables might be.

What I also think is interesting about these graphs is some of the graphs that seem to follow each other closely, but don't really have that high of a correlation coefficient. For example, Number people who drowned by falling into a swimming-pool vs. Number of films Niclas Cage appeared in. Most of the data has an r above .9, which is good, but I think it would be interesting for kids to talk about why the curves on that graph seem to rise and fall together, but the correlation coefficient is not really that convincing of there being a statistical correlation. 

Also cool: if you click on one of the variables, you can see how it correlates with a whole mess of other variables. This site could clearly could be a huge time suck for stats teacher trying to find interesting data to work from. 

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? 

Mario's a Baller

http://www.supercompressor.com/tech/13-things-you-probably-didn-t-know-about-nintendo

See fact #9: Mario has a 27’ vertical leap.

This seems like a fun addition to "How High Can Your Teacher Jump?" or any kind of proportional reasoning kind of thing.

What would we look like if we measured human heights in pixels? What would that mean for how tall Mario is compared to a human? How much bigger is Big Mario vs. Little (pre-mushroom) Mario? How big would YOU be if you ate a mushroom (or alternatively, if you're full size now, how tall would you be after running into a goomba)?

Do kids even recognize pixelated Mario anymore these days?

Lies, Damned Lies, Beautiful Lies

https://visualisingadvocacy.org/blog/disinformation-visualization-how-lie-datavis

I am in love with this article. Obviously data visualization can be just as persuasive as data provided in different ways (raw data vs. percentage vs. percent increase, etc.), but I like how this article specifically calls out the visual persuasion tactics.