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.
Wednesday, October 8, 2014
Tuesday, September 23, 2014
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?
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?
Wednesday, July 2, 2014
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.
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.
Saturday, May 31, 2014
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.
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.
Wednesday, May 28, 2014
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.
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.
Saturday, May 10, 2014
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.
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.
Monday, March 31, 2014
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?
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?
Friday, March 14, 2014
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.
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?
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?
Tuesday, February 11, 2014
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.
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.
Sunday, February 9, 2014
When are Babies Born?
Are births evenly distributed across time of day, day of the week, and time of year?
http://journals.lww.com/greenjournal/Fulltext/2004/04000/Timing_of_Birth_After_Spontaneous_Onset_of_Labor.8.aspx
The raw data is in the tables. I think kids would be interested in this question, and the results are kind of surprising.
http://journals.lww.com/greenjournal/Fulltext/2004/04000/Timing_of_Birth_After_Spontaneous_Onset_of_Labor.8.aspx
The raw data is in the tables. I think kids would be interested in this question, and the results are kind of surprising.
What Are the Odds of Twins Born in Different Years?
The best thing about teaching probability and statistics, in my opinion, is that it's so much easier (and fun!) to find interesting contexts for problems. I feel like I don't have a good sense of what data is actually interesting to kids, but this question seems like it would capture some imaginations:
http://freakonomics.com/2014/02/05/what-are-the-odds-of-twins-born-in-different-years/
This also feels like a Fermi problem in some ways.
Rather than having kids actually answer this question, I think it would be more interesting for them to try to make sense of the methodology described in the post, the comments, etc. Analyzing someone else's process is a big part of evaluating is reasonableness. Oh, hey SMP #3...
http://freakonomics.com/2014/02/05/what-are-the-odds-of-twins-born-in-different-years/
This also feels like a Fermi problem in some ways.
Rather than having kids actually answer this question, I think it would be more interesting for them to try to make sense of the methodology described in the post, the comments, etc. Analyzing someone else's process is a big part of evaluating is reasonableness. Oh, hey SMP #3...
Wednesday, February 5, 2014
Population Density Redistribution
http://www.washingtonpost.com/blogs/the-fix/wp/2014/01/30/what-the-u-s-map-would-look-like-if-population-matched-state-size/
I wonder what else you could have kids redistribute.
I wonder what else you could have kids redistribute.
Sunday, January 5, 2014
Graphing Stories
I love having kids play with the story that a graph tells, or creating a graph that describes a story. I like this take on it: video.
http://graphingstories.com/
I'm not sure I agree with all their categorizations, but that's fine. I like that there are a decent amount of video clips where the graph will not end up just looking like a picture of what happened (e.g. time vs. altitude when someone is climbing a mountain doesn't really help kids make sense of axes).
Just as important as having this library, it makes me think about other quick, easy activities that you could film and have kids graph. Extra bonus: have kids make their own 15-second films or find YouTube clips of something you could graph.
http://graphingstories.com/
I'm not sure I agree with all their categorizations, but that's fine. I like that there are a decent amount of video clips where the graph will not end up just looking like a picture of what happened (e.g. time vs. altitude when someone is climbing a mountain doesn't really help kids make sense of axes).
Just as important as having this library, it makes me think about other quick, easy activities that you could film and have kids graph. Extra bonus: have kids make their own 15-second films or find YouTube clips of something you could graph.
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