Until next year!

Hello friends! It's been quite a while since I last posted on this blog. I really apologize for not having a better and more timely conclusion, but this is my temporary break from this blog. It's the summer so not much tutoring has been going on. I have had a really amazing time writing for it and I hope just one person got something out of it.


I am wrapping up my year of service with the Massachusetts Promise Fellowship, Boston Partners in Education, and AmeriCorps. Next year I will be taking on another Mass. Promise Fellowship position at Tutoring Plus in Cambridge. I will be working on a different project to individually support students and help manage an afterschool program in the Cambridge Public Schools. I'd really like to thank Boston Partners in Education for making my first year amazing. I wouldn't be continuing on with AmeriCorps again if I hadn't gotten so much support and learned as much as I have learned.

Don't forget math is still everywhere, I'll keep an eye out for you all :)


I am hoping to return to Boston Partners to volunteer again with Math Rules!, in which case, I'll be updating this blog again. Thank you for visiting, thank you for commenting, thank you for reading along.

Until then, Math Rules!





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Wrapping up

There's less than one month left in the Boston Public School year (we're running a bit late because of "snow" days), but Math Rules! will be wrapping up in the next few weeks. I've been working on end of the year surveys for our teachers, volunteers, and students. I also helped put information together for a report. From both of these, I have some good news and good data to share with everyone.


Teacher at the Eliot School - Jennifer DiSarcina (center), winner of Boston Partners in Education's Educator of the Year

This year in Math Rules! -
In raw numbers, we finished the year with 47 volunteers in over 50 classrooms helping almost 250 students throughout the year. Math Rules! volunteers provided an estimated 2000+ hours of academic mentoring and math support for Boston Public students and teachers.

Compared to last year -
*We served 38% more students than last year
*Helped out in 27% more classrooms
*Recruited 25 new volunteers
*Retained 23 volunteers from previous years
*90% of nominated students have a volunteer (up from 70% matched students last year)


Me and my students at the Orchard Gardens

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I also took excerpts of volunteer responses from the volunteer surveys and compiled them, all of them are stunning reports of awesome math students!

“G- is an independent young man who knows his stuff. I worked with him one day and not only did he know how to do the problem, he explained it to his classmates. Then he asked if he could move and do work with a buddy. He's a kind boy and I hope he opens up to me more.” – Volunteer at Tobin

Sometimes interesting math poses will help with your math
“He has generally shown his ability of constant focus on the daily tasks. He usually encourages neighboring colleagues to concentrate on the work and is not easily distracted by others. He has a good attitude and a good helper to his friends.” – Volunteer at Tobin

“...the students know me a lot better now. They are always excited to see me, and they've shown more interest in maths as compared to before.” – Volunteer at Tobin

Mr. Marcus requested I take a picture, so studious!
“I can tell that M- tries to remember the details of what I teach him each time we meet. He is getting faster and more accurate when solving math problems. I think math is becoming more fun for him now that he is getting better at it.” – Volunteer at Tobin

“I feel that D- has a lot of potential. I feel he is very sharp young man. I feel that he improves in each class which is very exciting for me.” – Volunteer at Marshall


“M- is very smart. She needs the occasional reminder and follows right on cue. She thinks out the process of the math strategy and then she practices the strategy until she has mastered it.” – Volunteer at the Orchard Gardens

“I've seen B- transform over the months. He used to be really resistant to having a tutor, but he's one of my best behaved students now. He concentrates, separates himself from distractions and tries the hardest problems in the packet. I wish I had more time to devote to him during our tutoring sessions.” – Volunteer at the Orchard Gardens


“N- is a very confident, and beautiful young woman. She is a hard worker. She does struggle but after practicing, the obstacles become less and less of a challenge and she zooms through her math.” – Volunteer at the Orchard Gardens

“…in the last three weeks I've been there I've noticed them becoming more independent and focused.” – Volunteer at Orchard Gardens


“L- is a capable student who is always sure to ask questions about topics she doesnt understand, and keep me informed about her social life. She doesn't always grasp concepts from the start but can work through them with help and works well independently” – volunteer at Quincy


A big thank you to all volunteers and students for participating in Math Rules! this year. It looks like a lot of you all did really well this year. These survey stories are just a brief glimpse into the great work and quality time spent for both volunteers and students.


If you're a Boston Partners volunteer, we hope to see you all at the Volunteer Recognition Event on June 8th!


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Good luck 3rd and 4th graders!

Tomorrow and Wednesday, 3rd and 4th graders will be taking the math MCAS at many schools. Good luck!

This is what I gave my students as a good luck + encouragement:


Good luck on your MCAS,

Final tips:
1) TAKE YOUR TIME!!

2) Read instructions and problems carefully. Read one sentence at a time. Reread if you need to.

3) Cross off answers that you’re sure are not right.

4) Show ALL work, you can get partial credit!

5) Answer the question in full sentences.

6) Check over your work if you have time.


Believe in yourself and you’ll do fine!
YOU GOT THIS!


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Student tutors

Tutoring yesterday at the Tobin School involved more MCAS preparations and working in pairs. Students from the class next door came over to tutor the kids in my class. Again, this week, I concentrated on my girls and how they were doing. Some of the pairs worked well together but one of my girls made her partner a bit upset. It might've been my extra attention and my student got overexcited. So I backed off and let them work together again. The other pair worked on different topics.

It was really amazing to see students teaching other students. Our teacher then handed out a test they had previously taken and encouraged the tutor students to guide them through the test - "They should be able to take the same test again and do much better!"

I actually learned how to divide a second way yesterday also! One of the tutor-students despises long division and taught me the open array to divide. The open array works best with single digit numbers, but I used the open array to divide 792 by 17.


The strategy is a multi-step process, but I compressed it into one picture. So I'll explain.

1) The array is set up with an open array with the divisor on the side. For 487 divided by 9, you have the 9 on the side.
2) Start building to the answer by making arrays with easy numbers (10s, 20s, etc). I tried to explain using estimation to help speed up the process. If you know 50 x 9 is close but not quite, it's much easier than starting off with lots of 10s.
3) Continue building until you get as close as possible with a remainder (or not).
4) While you're adding more arrays, remind your students to keep a running total at the bottom. If you track it well, you won't go over and waste valuable standardized test time.
5) Finally add up your multiplicands/multipliers at the top to get your final answer. Don't forget your remainders!

The technique works well with single digit divisors, but like I said, it will work with multiple digit division problems as well. A strong sense of estimation and close tracking of sums will greatly help your students. I enjoy this strategy because it's using partial sums to get to a final answer.


Having students teach each other is great reinforcement for what the students have already learned. If you can teach it to someone else, it means you really know what you're doing. I also used this technique at MathSTARS with some of my 9th graders who were working on physics. Not only did tutor-student reinforce her knowledge of the physics concepts, but the student being helped could understand and relate to the extra help.

I also use the student-teaching method when there are too many students who need help and it's harder to help all your students at once. For example, when all four of your students are asking questions about completely different problems. I tend to ask the student who is done already, or finished the particular problem to help his/her classmates.


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My weekend math

Hi math fans! I'm just going to briefly talk about my experience with math over the weekend. I didn't get a chance to do any paper and pencil math, but I did watch a few basketball games and revisited one of my favorite awful-yet-amazing TV shows Xena: Warrior Princess.


I don't get a chance to watch a lot of professional basketball, but one of my roommates is a die-hard Spurs fan. So we watched the Spurs play the Suns on Friday night and again on Sunday night. Some players are much better mathematicians than others. Steve Nash is a genius at basketball math and while Tim Duncan is a good defender and guard, his math for making free throws needs work. Just listening to the sportscasters was interesting, they talked about basketball statistics a lot, how many shots they've taken and how many they've made. The percentages for whole team, how many points the bench has, etc. Lots of math, but no one went to the AT&T stadium to hear or think about math. It's sad that the Spurs lost, but I got to see lots of math in action.

I also found that Netflix has Xena online for members, and I took advantage of that by watching a few episodes of season 1. Keep in mind, Xena the Warrior Princess is a very old-school action show with low budget graphics, exotificiation of cultures, campy humor, and plotholes galore. But I grew up watching the show with my family on Saturday nights, and I'm obsessed with Lucy Lawless (who is more talented than most people give her credit for).


No matter how improbable or completely outrageous the show is, Xena is an excellent mathematician. One of her main weapons is the chakram, or a metal disc with sharp edges that she either 1) chucks at the bad guys or 2) uses to cause avalances. She takes her geometry and trigonometry into consideration before she throws her chakram and always hits her target. She uses angles to ricochet her chakram all over the place. It's some pretty accurate math if you ask me. Sure, the laws of physics make it difficult to tell if she can actually do such chakram acrobatics, but it makes for a good show.

A video of her chakram

And that's what a math nerd thinks about over the weekend. More soon!


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When I grow up...

I want to be older.

It's been a few days since MAM ended, so now we're back to regular tutoring entries with a few extra MAM bonus days thrown in.

For Math Rules! volunteers who are reading my blog, the following are my takeaway points:

• Check in with them and ask how things have been going before you start tutoring. It's a good gauge of how the session will go.
  
• Encourage your students in the following couple of weeks before MCAS, let them know they can do well no matter what happened in previous years.
  
• Remind them to take their time to read the questions carefully, check their answers, and answer in complete questions for word problems.
  
• If you can, take an extra five minutes at the end of your session to chat with them about college and careers. Talk about your college and/or career. For the most part, students are very interested in college but may or may not have someone to ask. Even if they're 3rd graders, if you get them thinking about the future it will help them in the years to come.
  
• Help them realize that you're there for them and want them to succeed in everything, not just math!

Yesterday I worked with my fourth graders. They're preparing for the MCAS in two weeks so I helped two students who needed a bit more practice. It's sad because it seemed like my other two students wanted to work with me too. In the end, I think it was better that I was only working with two because I could focus and give them more attention. Also, my teacher has changed her schedule - to prep for MCAS the students are doing 1.5 - 2 hours of math until the MCAS.

MCAS prep means working off of last year's test and trying to figure out best approaches to problems. We went over the standard algorithm for multiplication and division (long division) computations. We also talked about alternative approaches to multiplication problems such as using the array to help with multipication problems. Finally we focused on word problems, which was a good exercise for them to take their time and carefully read all the instructions.


I tried my best to give them test taking strategies like checking answers, rereading the word problems, not falling for the time-wasting tricks, showing all work on the test, and making sure they knew their multiplication and division algorithms.

I found out that both of my students didn't get very good MCAS scores last year and I think it's because they rushed through and didn't check their answers. When we went through the packet, they seemed dejected when I told them their final answers were wrong. It's interesting because most of the work is right, but the final answer is wrong and then they assume they failed.

One of my students kept repeating that she hadn't gotten anything higher than 2s on practice tests throughout the school year. I tried to encourage her by telling her "that was last year, this year you can do better!" I'll see how her self-confidence levels are next week.


We also got a good chance to chat in between problems. I found out that one of my students wants to be a teacher or singer when she grows up. My other student didn't know, and when they asked me, I said I don't know either. :) She retaliated and said "What do you mean you don't know?" That's one of life's secrets, adults sometimes don't actually know what they want to be when they grow up, it's a self-discovery process. We also discussed college briefly which was a good thing, they brought it up and asked me if I was still in college and if it's hard. I told them that it is tough, but if you work hard, you'll end up learning so so much.


We also got off topic and one of my students asked to interview me for a job, which is ironic because I'm in the process of interviewing right now. I also got a chance to interview her and found out some things I didn't know before. She took on a different persona and told me she was a 21 year old singer living in Florida who was interviewing for a teaching position. It was a great chance to bond and talk about jobs and our personal lives. I told her about my new pet hedgehog (shameless plug) and I found out that we both have four siblings.

In a program meeting, we started talking about how to improve Math Rules! for next year and I'll definitely take into consideration how many students are working in a group and the length of math time. These longer math sessions mean more time with the students, more personal attention, and it gives me some space to chat between math problems.


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MAM Day 30: 0100 0010 0110 1001 0110 1110 0110 0001 0111 0010 0111 1001

Today is the last day of Math Awareness Month! :( It's been an incredible month! I've learned so much about the importance of math in everyday life and how integral (hehe) it is in everyone's lives. From my first few giddy days of planning out the topics and drawing pandas, to the middle of the month where I struggled to stay on schedule, to some of my favorite posts: counting, math and language, relative size, taxes, music and math, math in nature, and finally optimization of space, I've really enjoyed this month-long blogging project. I personally have learned quite a lot about different topics and could do much more research on some of them. If you've checked in on my blog at all this month, I hope you learned something or were at least amused at my enthusiam of math. My goal for the month was to blog for every day and link it to how it affects most everyone in one way or another. To show just one person that there is no escaping math, math runs our lives and it doesn't get the credit it should. Because the bottom line is - Math Rules!


For my last post of the month, I figured why not go back to the basics? As the age of technology continues to reign, changing the way we communicate, educate, and live life, the human race has become incredibly dependent on our computers, networks, and electronics. However, the information age wouldn't be possible without math and the binary language. For those of you who have never heard of binary this is a crash course, 60 second video explaining binary. This website also explains "In a more general sense, binary systems can be anything which offers only two options, not necessarily limited to numerical systems. In the case of electronic switches, for example, the binary system consists of current-no current. A true-false exam is another example of a binary system. Yes-no questions are also binary in nature." If you'll remember back to Greg Tang's workshop, binary is also known as base two.


While binary runs our most advanced tools today, binary was the primary math language for Australian aboriginal people and cultures and groups of people who relied on smoke or drum messages. Morse code is a binary language. The utter simplicity of binary is what led to the efficiency of the computer. The digital world relies on the simplest of math languages! Anything that relies on a digital processor, computers, servers, digital watches, phones, video games, TVs, digital alarm clocks, computers that track car maintenence, even digital refrigerators and other appliances all are running on the binary language. In almost every computer built since the 1950s, the binary system has advanced digital computer capabilities to an incredible degree by simplifiying information processing.


A special message from me:
01010100011010000110000101101110011010110111001100100000011001100110111101110010001000000110001101101000011001010110001101101011011010010110111001100111001000000110100101101110001000000111010001101000011010010111001100100000011011010110111101101110011101000110100000100001001000000100100100100000011011000110010101100001011100100110111001100101011001000010000001100001001000000110110001101111011101000010000001100110011100100110111101101101001000000111011101110010011010010111010001101001011011100110011100100000011010010111010000101100001000000110000101101110011001000010000001001001001000000110100001101111011100000110010100100000010010010010000001100011011011110110111001110110011010010110111001100011011001010110010000100000011100110110111101101101011001010110111101101110011001010010000001101000011011110111011100100000011010010110110101110000011011110111001001110100011000010110111001110100001000000110110101100001011101000110100000100000011010010111001100101110

So there we have it! This is the last official Math Awareness Month blog post. If you have a second, check out the MAM survey so I can get feedback on the month. I'll continue updating about volunteering with Math Rules! and with MathSTARS with the occasional math related topic of interest. Thanks for checking in on my blog!

Click here to take the Math Awareness Month survey!



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MAM Day 29: Are we there yet?

We've covered math on the T, and math on foot, but math in a car is one of the obvious places where you'll find math. The speedometer is a great start, it's measuring speed in miles per hour. We can use the speedometer and road signs to know how much further we've got to go. If we've been traveling at roughly 70 miles/hour or MPH, and we're 200 miles away from our destination, we should get there in roughly 3 more hours. Easy right?


What I've always wondered is why speedometers go so high, typical highway speeds don't go much higher than 75 mph.
The other big math process we use for cars is determining gas mileage. Gas mileage is how many miles your car can run per gallon of gas. Calculating gas mileage requires that you know how many gallons your car's gas tank holds, and also keeping track of the odometer. The odometer tracks how many miles your car has traveled in total. Newer cars have multiple odometers that can also track the miles on a single trip, very handy for tracking your gas mileage.


The odometers are the counters in the middle.
So if you wrote down the miles off the odometer when you fill up, and then write down the miles at your next fill up (your car should be as empty as possible to track accurate mileage). Find the difference in miles and you have the total miles traveled on that tank of gas. Divide the total miles by how many gallons of gas your car can hold and you've got gas mileage in miles per gallon. Here's an exercise for kids to calculate gas mileage.

Obviously, the higher your gas mileage, the better. Cars typically get better gas mileage on the highway because they aren't wasting gas for starting and stopping. Heavy traffic will cause your gas mileage, your wallet, and your sanity to suffer. My aunt suggested that I keep track of gas mileage to make sure the car is running well. If your gas mileage changes significantly, you'll know something's up and you should take your car to the shop. If you track gas mileage for a few months, you'll get a sense of how long an average tank of gas will last.


I love license plates!
There's tons of numbers on a road trip, from license plates to road signs, billboards to just counting other cars. A great resource for car math ideas.

A video of a mental math game in the car

Although I don't quite understand the sport of car racing (it's incredibly wasteful of gas and I'm not sure how much athleticism is really required for drivers and the spectators) there is a lot of math involved with car racing from car speeds, laps, times in comparison to other cars, and the prep work that's required before a race, tire treads, weight restrictions, and finally the all famous speed of pit crews.

Here's more complex car math including horsepower and torque and tire diameters and gear ratios. Everything else is car lingo I don't understand.

Happy road tripping!



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MAM Day 28: Math on foot

To write this blog today, as a full time pedestrian and part time rider, I am continually interested in rates of walking behind other people, rates of running, juding speeds of cars, etc. So I searched rate of change, math problems that deal with rates of walking or running and found a few great examples.

I found myself longing for my calculus days, when I could understand these pretty complex word problems without a second thought. Working through these following problems, I struggled with the algebra, setting up the problem, and trying my best to remember how calculus makes these problems so much easier.

The rate of change is a foundation of calculus and derivatives. Since I've forgotten most of that and don't particularly want to reteach myself in a few hours, we'll skip over as much of the calculus as possible. Welsh corgis are excellent at doing calculus, maybe they can review calculus with me.


If you'll remember those math problems that state: a train leaves from City A at some time going at x miles per hour and another train leaving from City B leaves at a later time going at a different speed, the word problems tend to ask "When do they meet?" or "Which train gets there first?" These are the problems I'm talking about, distance = rate x time or d=rt. While they're fun to work through, they take up a lot of time and mental concentration to get all the parts right.


If train A and train B are traveling on the same track, shouldn't the question be "When do they crash?"
I worked through this problem using paper and pencil and found that I struggled with the alegbra. Despite my rusty algebra and calculus skills, I found working through the problem to be relaxing. Plus I couldn't stop once I started. The question asks if it's faster to run half a distance then walk the other half, or to run half the time and walk half the time.


This problem was also interesting to read through. I tried to maximize her sponsorship money, but after graphing the equation I realized that she would make the most money if she ran the entire way, but that's not realistic, or what the question was asking. Oh well.

Back to my initial interest in this topic. As a walker in Boston, we've all experienced getting stuck behind someone who is taking their time looking at buildings and the scenery. However, when you're in a rush or trying to catch a bus or T, I severely dislike it when a slowpoke is in front of me on the sidewalk or walking down the stairs. We all walk at different speeds, and I shouldn't blame anyone but myself for being late, but after doing some research, it seems that rates of walking are much more mathematically complex than it would appear.


Effectively navigating the streets of Boston takes a lot of skill.
Walking sometimes requires an optimization analysis, especially if you're running late. Let's say I've got somewhere to be at 6 but leave the office a later than I planned. Should I walk to the nearest T or should I just walk it to the station I would be transferring to later? It ultimately depends on how fast you walk versus how long a wait would be for the T. Other factors like weather also influence people's choices. People cut through grass fields, take smaller roads, use alleys for shortcuts, use the stairs instead of wait for the elevator, etc. Path optimization is anoter part of math on foot.

I also think about judging speeds as a walker. If you're crossing the road when the crosswalk lights haven't said it's ok (jaywalking if you will) you have to properly judge how fast a car is going. You might also have to judge the rate of change in speed if the driver chooses to slow down and let you pass or speed up to try and hit illegal street crossers. I'll try my best to stop jaywalking, but in the meantime, I'll use my rate judgement to stay alive. This goes to show survival skills depend on math.



So today's blog post was brought to you by: rates of change, distance, rate, time, and speed.



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MAM Bonus: Interview with my sister

I called my sister yesterday to talk about how much math she uses in her school work. She's a second year architecture student at the University of New Mexico. I originally wanted to use it in a blog post, but we started talking about other math related things and I just thought it was interesting enough to get its own blog post. Now, my sister is about 7 times smarter than I am so I hope you'll bear with me through our somewhat philosophical discussion of math.

So I started off asking how much math she uses every day to design. Turns out, because UNM is more focused on the aesthetics of design, they don't rely on math as much as I had hoped. The prerequisite for getting into the architecture program is Trigonometry, and a required Calculus 1 by the time architecture students graduate from the program. Not too much math, if you ask me. Which is a shame because I know my sister is a math rockstar.


She didn't build this one, but similar models.
So I asked her if she uses math in some or all of her projects, I would assume so because designing on paper or the computer and translating it into 3D models requires some sort of math. She told me one of her first projects involved calculating the angles of the sun in conjunction with building a model. She used her trig skills on this project, and many other projects as well.

Most of her models rely on a scale, one inch on the model represents 8 feet or 10 feet in real construction terms. She also pointed out that they calculate slopes from time to time when they're putting in handicap accessible ramps and making sure it's not too steep a slope.

It would seem that she doesn't use hard math as much as I thought architecture students would, but perhaps it's because of her program, or the fact that she's only a student and not a certified and experienced architect (though I hope she gets there soon!).

So I asked her if she uses math every day. She countered with "can we really live a day without math?" After 27 days of everyday math, I would have to say we really can't. If we are living in civilized society, there is no escaping math. Time and money are the two biggest resources people deal with every day, and both are math and number related.


Yesterday I noticed that the bus system relies on numbers and routes, if you mess up your numbers, you'll end up somewhere you didn't want to (this happened to me one day when I hopped on the 23 bus instead of the 22). It happens! And if you're not paying attention to your numbers, you could be stuck somewhere. A bus rider asked me what time it was. People who pay fares need to know how much is enough and how much is too much for a bus fare. Three examples of math while waiting for the bus. You can't get away from it.


Which bus is this? The 23 or the 22?
My sister and I concluded that math has permeated the organization of society so much that it's nearly impossible to not be dealing with math on a daily basis. Most of the time it's unconscious, we don't think about it too hard. After learning how to tell time, we can easily let other people know what time it is, after learning how money works, we can buy and sell things we need, etc. Earlier civilizations relied on barter and trading before money was invented, but there was still some sort of value and wealth system, a cow is worth more than a chicken, and the guy who has 100 cows is the "richest."

But what about societies and cultures that don't rely on time or money? My sister came up with the basic instincts of survival. If humans are living with others, we need to count and estimate in order to survive. To effectively pool our resources and make sure everyone we're interacting with has enough to eat and drink we have to measure and divide accordingly. Humans living outside our definitions of "civilized" society are still dealing with math in one way or another. And if you're stranded on an island, without anyone to share food and shelter with, I would be willing to bet that you're counting the days until you get saved or perish.

Finally, we then discussed if animals have some sort of understanding of math over survival. At first I'm wasn't sure that animals are aware of groups, sharing resources, and possess number sense. However, after doing some internet research, there is evidence that some species do understand concepts of more and less, and can compute simple math problems. An optimizing Corgi, an example of animal calculus.Monkeys can also perform simple addition.


Hans the math horse
I guess it goes to show that even if animals aren't solving complex mathematical enigmas or doing multivariable calculus that some animals posses math skills. Animals are far more complex than we give them credit for. For all we know, animals may be doing more math than my sister's architecture program. I sure hope not...



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MAM Day 27: Optimization of space

Hey math fans! Following yesterday's topic of packaging, today's topic is the optimization of space, something I've been looking forward to the entire month. I'm not sure why I'm so fascinated with the optimization in general, but the optimization of space is something I see it quite a bit in my day-to-day.

Optimization generally refers to maximizing or minimizing a mathematical function and finding the max or min value of the function. Merriam-Webster defines it as: an act, process, or methodology of making something (as a design, system, or decision) as fully perfect, functional, or effective as possible. The optimization of space then means being able to spatially fit as much stuff into a defined space as possible, the optimization of volume in a way.

I have quite a few real world examples, but yesterday, my friend Sonal over at Science Club for Girls had a peculiar problem. We were on a fast food adventure with another friend and she had lots of Science Club stuff in her back seat. To make enough room for both of us in her car, we had to reorganize the boxes. By giving up her rear view, all three of us fit in the car and we had a successful adventure.


Some of us love optimizing space
Another example that helped me think up this topic is the public transportation system in Boston, the T. I'm sure lots of us have experienced a crowded T before, after work during rush hour, rainy days on the Silver Line, Sox game nights, Bruins game nights, weekend nights on the B Line, after big city-wide events like the 4th or the Boston Marathon. There are lots of cases when this happens, and most of us dread a packed T car. I tend to feel really uncomfortable being squished in with so many other people, trying my best not to fall over or push and shove some stranger.


Look familiar?
But it's an optimization problem when you're on the outside, trying to push your way into the T so you can get home. How many people CAN fit in one of those cars? It all depends on how efficiently people can maneuver themselves into the space, how much personal space they're willing to give up, and how much or little stuff people have with them, etc. There are a number of factors, but if everyone on the T can give up just a little more space, more people can pack onto the T car. It's about minimizing our personal space and minimizing our discomfort, in order to maximize how many people can get on the T. It amuses me for about 5 minutes before I intensely dislike feeling like cattle :(


In Japan they have subway workers who push people into the subway.
Optimization of space happens when you're packing for a trip, packing a suitcase, backpack, duffel bag, whatever. How do you fit all your clothes into the bag? Sometimes, it comes down to an optimization problem. If you need all your stuff to fit, but don't have enough room, sometimes reorganizing everything helps things fit better. Other times you have to leave something behind, take stuff out so that everything else can fit. Either way, it's an unconscious optimization problem.


My roommates and I were discussing how parking garages manage to fit so many cars into one area. We also reveled in the design of parking structures that allows cars to both go up and down at the same time. We guessed that they're structured like DNA helix or rotini pasta, there are two rotating levels that let people go up or down and sections in between that let people choose which way they want to go. But parking structures are optimizing how to store cars. Instead of having a parking lot, we can store literally tons more cars in the same space.


Not quite the same as a helix parking structure, but you get the idea right?


Speaking of the DNA helix, talk about optimization of space! Two or three meters of DNA fit into one single human cell!
How about living situations? People who live in apartments (a large majority of Boston residents) are literally living on top of one another. The optimization of residential space comes down to how comfortable people are living in small spaces. The City of Boston's population of 620,000 is spatially living in an area of about 50 square miles. My hometown has about 100,000 less people living in an area three and a half times larger than Boston's area. The population density of Boston is 12,800 people/square mile, while my hometown's population density is only 2,800 people/square mile. Lots of people live in apartment buildings in Boston, while more people live in houses in New Mexico.


These pictures are population density maps that show how many people are living in certain areas of the world or the United States.

A few months back, I also came across a photo series that is incredibly revealing about how Hong Kong residents have adapted to this optimization of human living spaces problem. It seems less space is a small living movement in minimization of stuff and space.


One of the most interesting photo series I've seen yet.



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MAM Day 26: Real World Geometry Part 3

The last installment of real world geometry is packaging of food and other products. From Earth Day statistics: $1 out of every $11 Americans spend for food goes for packaging.

If you're buying food at the grocery store, chances are you've experienced packaging and the incredible industry that is dedicated to researching and designing new and updated packaging. Packaging influences what catches our eyes when we walk into a store, it influences what we buy or don't buy, and it influences what brands we are most loyal to. Packaging designs are an art really, an art that is based on the geometry of what goes inside.

A great article on The Power of The Box


Some packaging is fancy origami. But think of the spatial research and thousands of hours spent on the design of a package before it hits shelves.

I'm sure most of you have seen how sports drinks, sodas, and bottled water gets updated every once a year or so with a "new package." Gatorade is unveils new designs quite often. Pepsi (the parent company of Gatorade) also goes through brand and packaging redesigns often.

In addition to the increases in portion sizes, Coca Cola bottles have gone through significant redesigns. I'm highly intrigued by how much work actually goes into a new packaging for any product. It seems like a healthy intersection of art, design, and math that makes this such an exciting topic for me. New bottle designs are necessary for a company that wants to stay on top, with new designs, new marketing, new advertisements and a rebranding as a hip and contemporary company.

Coca Cola is also dedicated to reducing packaging and therefore waste.

Mustard and ketchup redesigns are also really exciting. Remember when the newfangled no drip/water squeeze bottles come out and everyone was really excited that mustard/ketchup juice wouldn't gross you out or ruin your hotdogs? All thanks to packaging research.

A science fair project on juice box packaging.


More fancy packaging designs

How to package and store spherical things



Packaging also includes other products that aren't food related. I recently bought tablet for my computer (and no it's not the same thing as an iPad) and was shocked at how much packaging accompanied my tablet. The box outside, an inner molded cardboard holder for my 12.8 oz tablet, lots of plastic bags, wrapping around the cords and bits and pieces of other various "packaging." It's hard to know what's recyclable and what isn't and I would wager that most people just toss everything that they didn't want into the trash instead of separating recyclables out. Despite the waste that packaging produces, experts have to design (using geometry!) and make sure products are packaged the way the company wants.





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