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!!!
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?!)
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:
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?)
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!
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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.
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:
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...
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!
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).
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 recommendlearningmore!) 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...
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?
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.
Cool pattern finding - not your usual pattern hunt. Plus I like the kinesthetic entry point--kids can think about the pattern in terms of how they're folding it, not just in terms of some of the "usual" pattern finding strategies.
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.)
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...
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?
What is the cheapest way to park for an hour?
What is the most expensive way to park for an hour?
You only have bills, so you ask the nearby store clerk for change. What is the best way for him to give you change for $1 if you want to maximize your parking time? What if the store clerk is running low and doesn't have enough change to give you all coins of the same value?
Is there a systems of inequalities somewhere in here? I can't tell.
