Headlines from a Mathematically Illiterate World

http://mathwithbaddrawings.com/2013/12/02/headlines-from-a-mathematically-literate-world/

The longer I teach, the more I think that the math that feels most important for students to take away from my class is about learning to read, interpret, and critically evaluate the logical/illogical statements that float around every day. I do actively enjoy pure math kinds of things, but if it came down to a choice of residue, I'd give up a robust understanding of derivatives for my kids leave being able to read a graph in the newspaper and evaluate the reasonableness of the latest study's claim.

I think it would be really fun for kids to find ridiculous statements in news articles and correct them like this. It would be interesting to develop that critical eye for poorly worded statements, both from a language and a mathematical perspective.

Love, and Love Lost

Visualizations of Love:
http://love.seebytouch.com/#LetMeShowYou

Clearly some are more mathy than others, but I like the variation in the types of visualizations.

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

And quantifications of love no more:
http://quantifiedbreakup.tumblr.com/

Sometimes when kids tell me that they're in a bad mood or not feeling well, I respond that doing math problems always makes you feel better. Looks like I wasn't making it up. Doing math as therapy is real!

Where's Waldo?

If this isn't real-world math, I don't know what is.

http://www.slate.com/articles/arts/culturebox/2013/11/where_s_waldo_a_new_strategy_for_locating_the_missing_man_in_martin_hanford.html

I like what you can do with this around probability and area. It also makes me think about what kids hear/understand when they use and read the word "random." Do they think that Waldo is placed randomly? Why or why not? What would the pages look like if he were to be placed randomly?

I also like that it takes something that looks like it has little order, and uses math to help you see something you wouldn't otherwise be able to notice.

Really Long Tweets

This "What If" from xkcd seems like a simple proportional reasoning problem--and could totally stay that way--but actually turns into something way more complicated, particularly when you consider growth over time. 

A Math Major Talks about Fear

Would kids believe it if it came from this girl instead of me? 

Poor Pete Tries Data Visualization

http://wtfviz.net/

The title may not be school-appropriate, but the awful data representations are a goldmine of "What's wrong with this?" problems.

What's Number Are You?

http://www.bbc.co.uk/news/world-15391515

According to this website, of all the people living on Earth, I was the 4,611,347,584th person to be born. Can you figure out how old I am???

There are lots of other interesting mathematical things about this. First, just the fact that the population of the Earth increased by about 2.5 billion people in my lifetime--more than half of what it was when I was born--is just staggering. I know that large numbers are hard for kids to conceptualize (adults too! me (non-kid, non-adult) too!), but there's something in here that makes you say WOAH. Even if you put in a kid's birthday (I picked a random 15 year old), there were still about 1 billion people born in their lifetime.

Of course there's also interesting stuff with exponential growth, how we would calculate this number, and so on. I'm also interested in the statistic of how many people have been born on Earth since the beginning of time. This website gives an interesting summary of that calculation, including this video:

The World Bank has a shorter video on the same topic: 

I like the stat in this video that 7% of all the people who have ever lived are alive today. Holy smokes!

Record Setting Stupidity

http://recordsetter.com/

This is a proportional reasoning teacher's dream. All of those problems I've used about competitive eaters, giant foods, and other world records, here are videos to match! Those weird facts are already a good hook for kids, but reading about someone cramming hot dogs down their throat will never top actually seeing it.

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.

Elevator Math

Eli Lansley of the Lansley Brothers Blog posted these pictures of interesting floor numbering systems in buildings.

The first he found in Israel, the second in Hackensack, New Jersey

 
I have definitely taught kids to think about negative numbers and number lines by thinking about elevators. I don't know much about teaching younger grades or teaching negative numbers for the first time, but I can totally imagine giving kids a picture like this and asking them to describe the building where this elevator lives. Or some kind of problem about a building that adds an underground garage with the question "How should they label the number on the elevator button?" Even with the standard US systems for labeling underground space (P1, P2, P3 and stuff like that), I think it's interesting to talk about what order those should go in. Maybe it's not that exciting, but it feels like there's a little something there.

Also thinking about elevator numbering, I feel like there might be something with the way that many European countries label their floors by having floor 1 as the floor above the lobby. Or with Americans skipping the 13th floor. Is there some kind of function rule you can write to figure out how many floors the building actually has? It would be a piecewise function because the rule for calculating how many floors for buildings above 13 would be different than for buildings with fewer than 13 floors.

Upgrading & Early Termination Fees

Here's an interesting story about a guy trying to negotiate upgrades for his family to the latest iPhone. All comments aside about the scam that constant upgrade "opportunities" offer, the story is an interesting mix of straight math and emotion. So basically, economics. Homo Economicus might have just skipped the upgrade altogether (or took his family back to flip phones).

The story in itself seems like more of a hook than the basis for a problem you'd actually spend the bulk of a class period on, but it opens up larger questions about early termination fees, upgrade costs, etc. Given deal X from company A that gives you free or cheaper phones with a new account, at what point is it "worth it" to leave your existing phone company and suck up the early termination fees?

There are lots of other interesting variables that could factor into the decision:

• How many phone lines you have
• Variations in cost between the actual monthly phone plans of the two companies
• How soon you'd be eligible for an upgrade with your current company
• How soon the other phone lines would be eligible for an upgrade
• What kind of quantifiable measure you put on being cooler than your friends and/or having the latest thing. Would it change your mind about waiting out your time until an upgrade if you knew that all your friends would have the new iPhone before that? How many friends would you care about? Can you quantify how much you care? 

Reminiscing about the Good Old Days of Gas Prices

http://consumerist.com/2013/09/17/you-will-probably-never-pay-less-than-3-for-a-gallon-of-gas-ever-again/

The Consumerist says that we will never again in our lifetimes pay less than $3/gallon for gas. When I first got my driver's license, I remember paying less than a dollar! I also remember when gas prices started to go up and I swore I would never pay more than $2/gallon. (Maybe the math problem in here is figuring out how old I am...). The Consumerists's logic seems to be that because today was the 1000th consecutive day that average gas prices are over $3, we can safely say that we've reached the point of no return. 

Here are my mathematical questions about this situation: 

• Are you convinced the 1000th day is "enough" of a pattern to say that we're never going back? 
• Would you be as convinced if the 1000 days were not consecutive? What kind of pattern of conductivity would be "enough" for you? 
• Does it matter that this about the national average of gas prices? 
• Does it matter what that average price is for you to be convinced? For example, if the national average price of gas is $3.89, I'll probably be convinced that <$3 is unlikely. But if the national average is $3.02, I feel confident in saying that I won't have too much trouble driving around and finding a gas station where I can pay $2.99. --> This seems like a perfect opportunity to talk about the importance of standard deviation!
• How could looking at a graph of gas prices help add to an argument? I feel like there's a very powerful visual argument of showing the $3 line and looking at when the graph stops dropping below that. 
• If you wanted to convince someone with a graph, what data would you show? 
• National averages only, or the range? 
• How often would you want to chart data points? Daily? Weekly? Monthly? 

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!)

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.

How many servings?

Can you figure out what's wrong with the serving size for this Trader Joe's deliciousness? Can you figure out what's right with the serving size?

I already hate it when the nutritional info tells me that the serving size is something 1/x of a package and there are x servings. Duh. But this takes it to an even more ridiculous level. I'm not sure what interesting mathematical questions you could ask, there's gotta be something.

One potentially less math-y, but still interesting thing for kids to think about is how manufacturers decide to label how much one serving is. I know I've looked at things like candy bars and the nutritional info says 2 servings, despite the fact that no one would ever eat half a candy bar and think, "I'm good. I'll save the other half for my next meal." But then I look at the calories and think, "Dang, at over 500 calories in this whole candy bar, maybe I should eat just one 'serving' and save the other half for later." But 260 calories in a serving doesn't look so bad. I wonder if there's something interesting around having kids think about how much a serving size actually is for, say, potato chips, and then rewrite the nutritional label to match their serving. Along the same lines, it would be interesting to compare "1 serving" of potato chips as measured in weight vs. number of chips. Did the manufacturer accurately represent the weight of a serving of 10 chips? How would it change all the other nutritional info? This is starting to sound like a middle school proportional reasoning exercise for sure.

What could you add for older kids to make it more challenging math? Package design and labeling has got to include a ton of math. Any packaging designers out there who want to help me out?

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?

Here is Today

http://hereistoday.com/

I always get a little annoyed/weirded out by these kinds of time comparisons because to me the underlying message is, "Today is not that big of a deal. Get over it." It's similar to the "There are starving kids in Africa" argument for why your problems aren't that important. Yes, I know that my problems (or a 15-year old's problems) are not that dramatic in the larger scheme of things and that many of the things I'm pre-occupied with today will be relatively insignificant by the end of the year, the decade, etc. But that doesn't mean those things aren't real and important to me in this moment. It diminishes and invalidates someone else's emotions to tell them, "So what? In geological time, you're invisible." Everyone else is invisible too, but that shouldn't make them any less important or worthy of our love and attention.

But on to the math. What's cool about this interactive graphic is the proportionality and evolution of the part versus the whole. "Today" remains the numerator, but the denominator changes and our concept of "today" changes as a result. Seems like an interesting way of thinking about and understanding fractions, proportions, percents, and relative size. I think it would be interesting for kids to create or think about their own life maps in this way. What does today look like in comparison to your entire 15-year old life? Thinking about an important time period in your life, what is its relative size compared to today? Compared to an different time period in your life? It also feels like there's something interesting in there around fractions greater than 1--your life thus far is 1, so what will your 20 year old life look like?

Food Equivalencies

What does 2000 calories look like?

There are nice visuals around how many ____ are equal to how many ____ in terms of food. The bacon to cinnabon equivalency is my favorite, I think, because we think of bacon being so unhealthy. (Not that we think of cinnabons as being so unhealthy, but I, as a teenager, was definitely more likely to get a "snack" at the mall of a cinnabon, but would never have gotten a stack of bacon as a snack).

Questions to think about:
-Are all calories the same?
-What sets of 2000 calories can you imagine eating (I can definitely imagine eating 2.5 cinnabons, but not a whole pizza)? How does the mental association of these foods impact your eating?
-These equivalencies represent calories. Which would be the same if we looked at fats/protein/carbs/etc.
-Why is 2000 calories the recommended daily allowance?

Maybe it's not super mathematically interesting, but I do think it gets into units and the meaning of the equals sign in an interesting way.

Making Babies

I saw this cartoon on someone's Facebook page. Yeah, yeah, imaginary numbers are great for puns. That's not the math I'm interested in here. The way I interpreted this column, the 8 and 4 are the parents of the 6. True, they could be two "adult" numbers of any sort (teacher and parent, two teachers, whatever), but I interpreted them as parents I think in part because 6 is the average of 8 and 4. It makes sense: we think of children as being the genetic combination of their parents ("He has mom's eyes and dad's nose") and in many ways the average. Thinking purely about skin color, mine is about the average of my dark-skinned father and white mother.

So what other questions can we ask or think about?

• Thinking about using the arithmetic mean, wouldn't all these numerical children hit a certain limit at some point? (Kind of like how people say that in 100 years everyone will have light brown skin?)
• What are other ways we could do genetic counseling for two numbers trying to procreate? Geometric mean? 
• What traits are dominant or recessive (or something else)? Two even numbers or two odd numbers will create an even child, but an even and odd will never pass on their even-ness/odd-ness to the next generation. What about multiples or 
• How could you do some eugenics to make sure you weeded out "undesirable" offspring (uh oh, I am taking this to a dark place...). 
• How far can I take this metaphor before it starts to break down? 

Maybe the most interesting question would just to give kids two numbers and ask what number their child will be. I wonder what kids would come up with. I especially wonder what elementary schoolers would come up with versus high schoolers. My guess: elementary schoolers would be more creative. 

Percents of Percents

Percent increase and percent decrease still confuse the heck out of me, no matter how many problems I do or how many times I teach the topic. The language gets 

• Has the high school graduation rate of black males increased by 6.6% or 5.1%? 
• Has their high school dropout rate decreased by 37.9% or 5.8%? 
• Has their college enrollment rate increased by 32.7% or 1.7%?
• Has their incarceration rate decreased by 25.3% or 2.1%?

Two bigger questions:

• Would the positive changes for black males highlighted by this list of statistics still be as powerful if they had cited the change in percentage rather than the percent change (of the percent)? Either way, they still show increases where we would want there to be increases (high school graduation rate, college enrollment, college "by the numbers") and decreases where we would want there to be decreases (dropout rate, "incarcerated"). 
• In what other ways could this information be presented (pure numbers, different types of graphs, etc.) that would make them more or less powerful? How do you think the author chose this table? 

An underlying question: 

What is the difference between "net increase" and "percent increase"? Is there a difference? How does this very, very subtle difference change how we present and interpret statistics about changes that are measured in and by percents? The Wikipedia article on percents has some interesting things to say, including how the use of the term "percentage points" can help clear up confusion. 

As teachers, especially teachers of English Language Learners, how do we support students in navigating this very tricky language. It seems particularly important/frustrating given that percent increase/decrease problems are a not insignificant part of the California High School Exit Exam. (Really, no concept is insignificant when one or two questions could make the difference in whether you earn a high school diploma.)