The Math Mom

Two New Math Apps

2012-01-20T08:07:00.001-05:00

Both of these apps will help your kids master addition, subtraction, multiplication and division. They do not teach but provide fun rigorous practice for elementary school kids.

Bubbling Math for iPad and Mac by Tappy Taps.

This type of graphics will probably be most attractive to the girls (age 5-9) . The interface is very intuitive for novel as well as experienced players. Parents or kids can select level of complexity for each kind of the operation. Parents can review the day-to-day progress and wrong answers. As child progresses new lands and places are unlocked. I wish there would be more to the story that would push your child keep going or alter the plot based on the child's responses. Still, it is a great beautiful alternative to simple math sheets. iPad app is $1.99, Mac version is $3.99 but now if FREE.

Operation Math Pocket by Spinlight.

Terrific comics-style graphics for older kids (9-12), especially boys. Kids need to be comfortable with reading and comprehension to understand the overall plot. I wish there would be a woman agent to attract girls as well. As you solve +,-,*,/ exercises you unlock some spy gear and accomplish spy missions all around the world. Nice, fast-paced, engaging. As with the Bubble Math app, it could be great to get to the next level of interactivity where the story is feasibly affected by child's responses. However for the price of $2.99 on iPad and $1.99 for iPhone/iPod (during the limited offer-launch sale) this beauty is a bargain.

Can you trust the mirrors?

2012-01-13T17:07:00.001-05:00

If one of your New Year's resolutions was to shape up, I have good news for you: You look much thinner than what you think you are. Don't believe me? Take an old lipstick or washable markers, stand in front of a home mirror and carefully draw your outline on a mirror. Now measure the size of the face you just drew. It would be about half of what you expected. Viva-la-mirror!

How can it be that you on a surface of a mirror is about half the size of the real you? Here is the math:

Take a look at the triangles ABC and DEC.
Your face and its virtual image in the mirror have a size AB. The projection of your face on the mirror surface is the red line DE. AB is twice the size of DE because CH is twice the size from you to the mirror.

Are you really half the size? No, I was just kidding.
Our brain focuses on what is reflected in the mirror, rather than on the surface of the mirror. AB and not DE.

This peculiar trick is from a great book by John D. Barrow "One Hundred Essential Things You Didn't Know You Didn't Know. Math Explains Your World."

Top image by Dru!, distributed under CCL.

Baby Feeding Math

2011-12-02T04:47:00.000-05:00

We have a new baby in the house and with her arrival came various logical riddles that appear very simple, yet the solutions are not trivial although can mean quiet, health and happiness of the whole household.

What time did I last feed her?
What breast did I last feed her from?
How much did she drink?

While I have been trying to remember the last feeding times, I have not always been able to keep track of the "last breast." In many cases you can feel the pressure of the milk, however not always - just recollect the sleepless zombie state new parents find themselves in. I mentioned this problem in my newsletter and many of you wrote suggesting placing a sticker on a bra, writing L or R on a notepad, moving a ring from one hand to another. Someone proposed this as a great idea for a startup.

Today another creative solution arrived by mail from the Similac baby formula company. A present or an advertising trick, depending on how you look at it. An elegant hand bracelet:

Instructions advise to put the bracelet on the hand corresponding to the side of the last feeding (when breastfeeding) and slide the heart-shaped window to the mark corresponding to the time of the last feeding. Two riddles are solved with one bracelet! Very neat - I am already using it.

Another puzzling and usually annoying question that is frequently asked by nurses, doctors or grandmothers is: "How much milk does your baby consume in one feeding?" When baby is on a formula - it is easy to answer. When breastfeeding - the only way to answer this would be to pump, measure, then feed. Too much work just to satisfy the curiosity. Feeding time may give a clue to the answer, but as experience shows milk sucking speed depends on the baby's age, temper and hunger state. What other optios are there? You may weigh the baby before and after the feeding or weigh the mom, but the 100-200mg consumed will likely get lost in the noise of the weighting scale or a burp.

Undoubtedly various creative solutions to these questions are in the works. Have your own ideas - rush to patent it; use someone else's idea - share it with us.

Multiplication Song

2011-11-17T15:21:00.000-05:00

Like many of you, I have memorized the order of the letters in the English alphabet thanks to the "ABC" song. In my case it happened at the age of 30, from my kids' Sesame street DVD (with Billy Joel). Yes, I learned the alphabet before in school but it never settled so naturally inside my brain until I heard it fused with music (and over and over again). No matter that I am practically tone deaf.

My daughter quickly mastered the Hebrew alphabet also from a song - "Alef, Beth" by Naomi Shemer.
My late grandma surprised us all starting to sing some childhood songs in Yiddish at the late stages of her Alzheimer, when she no longer recognized any of us. This all is not a coincidence.We are more likely to remember a poem that was put to music than "bare" rhymes. Scientists have shown that music facilitates quick memorization, long-term memory retention and efficient retrieval of information. So, shouldn't we be singing more at school and work?

In the past few months, a large amount of readers and friends consulted with me on how to help their 3rd graders memorize the multiplication table. Well, I have been struggling with this with my own daughter and nothing seems to do the magic. We tried memorizing the diagonal, or starting from the upper-left corner and slowly increasing the square size, we practiced on windows of buildings that we passed by, we tried online games and iPhone games. She got better and better at it but was easily forgetting complex parts after a few days without practice. Plus, it was not fun at all. Perhaps we should put the multiplication table to music.

I checked on YouTube and found a few attempts by various artists. If you are aware and like some other clips - please add the link below or email me.
Share this with your kids. Better yet, write your own! There are few professional musicians among our readers and I challenge you all to try - contribute a rhyme, or music or both.

The Multiplying Multiplication Song !!!

Mrs. D's Multiplication Rap Remix™

How to make a festive paper decoration

2011-10-14T08:26:00.001-04:00

My daughter convinced me to share with you the tips and tricks on making this wonderful paper decoration for Christmas, Sukkot, birthday or any other celebration.

She made it in school and I noticed how every math-curious person who came into our house in the past week (including my father and cleaning lady) carefully folded it and tried to reverse-engineer the creation process. We did it as well and after a few failed attempts finally managed to figure it out.

You will need: two pieces of regular (A4 size, white or better colorful) paper, scissors, ruler, pen, eraser and stapler.

Start with a full sheet of paper.

fold it in half and again in half:

Hold on from the folded corner and cut the open corners of the sheet diagonally:

You are left with a folded piece in the shape of a triangle. Use ruler and pen to make the following lines on this triangle. Note that lines are parallel to the triangle's base. They start at the opposite sides of the triangle and neither line goes all the way to the other side:

Cut along the lines:

Unfold the triangle:

and unfold again very carefully:

Gently tilt the middle part so that you can hang the decoration from it:

You got it!

Now you can make another one of those and staple them together:

Enjoy:

And now when you mastered this try reverse-engineering this Escher's drawing:

Online Math Games: Bad or Good for your kids?

2011-09-22T16:30:00.001-04:00

You get a link from your child's teacher. You load the page and easily interest your 8-year-old in the selection of colorful and innocent math games. Alien addition, sailboat subtraction, penguin multiplication, drag race division and tens of others. You point her to multiplication that you both have spent the entire August practicing. You patiently wait while she spends 5 minutes on choosing the color for her penguin and then leave her in front of the naive penguins jumping from iceberg to iceberg. You go to fill the dryer with laundry or check your email with a cup of coffee. When you return, you are most likely to find your darling in front of youtube, pbskids, frustrated and certain that she is bad at math. How did this good turned bad so quickly?

I have always been ambivalent about recommending online math games. CD-ROMs - yes. We have a few excellent ones in my household and both kids enjoy playing them for hours. They are grade-based, start you on the Easy level, and do not rush or humiliate you. With online games it is a different story. Many of them, such as recommended at my kids' school Academic Skill Builder Games take into account the speed of answering along with the correctness of results. Every game is a race where for starters you are guaranteed to come last unless you are choosing a game way below your skill level.
Ready, set, go.
The timer is rushing you.
You notice the red, blue and green penguins are getting ahead while your pink cutie is sinking in the water each time you confuse 6 x 1 and 6 + 1. No time to recover. You press the wrong button again, you squeak in frustration, you lick your tear. Game is over. Your name is last in the results list alongside the naive little pink penguin. Math is fun, isn't it?

A perfect five minute scenario for a quick bloom of serious math-phobia. Beware of online math games. Never ever let your kids start them alone. Instead - demonstrate and test the game on yourself first. Let kids see you struggle with the keyboard, the rules of the game, the penguin sinking disaster, let them see you lose and not give up. It may take a few trials. When you finally manage to finish first, play it together with your kid with one of you answering and another pressing the buttons. Finished in the first three? This is fun. Now you both are ready for the full transfer of the game power to the kid. The level is right, the rules are clear, kid has warmed up and ready for an exciting math practice.

If you are away at work and have to instruct your kid to play such games remotely - recommend to them to start on a very easy level way below their grade skills, carefully reading the instructions and slowly raising higher as they get more comfortable and confident with the speed and rules of the game.

These are my thoughts. Teachers and parents - what is your experience, strategies, trials, errors and successes in the online math gaming world?

It depends on how you measure it

2011-09-16T05:11:00.003-04:00

I always thought that cooking is simple as long as you have a reliable recipe. Over the years I learned to trust some of the newspaper food columnists, while taking with a grain of salt my kids’ class cookbooks because some words fade during the kids’ mail transit and Xeroxing.

Once you have a good recipe there is usually no one to blame for a bad outcome other than yourself. Well, occasionally you can tell that the eggs were too big or the stove is new. My eccentric grandaunt used to prohibit us from entering the kitchen during baking and blame us for disturbing the cake’s gestation period if it didn’t rise. But there is really not much more. One cup of flour is one cup of flour anywhere in the world, one tablespoon of butter is one tablespoon of butter. Or is it?

This week’s article in the New York Times surprises us by suggesting that it is not. It refers to an experiment when ten different people were asked to scoop 1 cup of flour and pour it into a bowl. The weight of flour in the individual bowls varied between 4 and 6 ounces depending on the strength and technique of scooping used by each participant. This meant that some of these people may be making a cake with 1.5 times as much flour as others.

What else can we use instead of the traditional and universal cup and spoon volume measures? The weight (mass), says the article advocating for simple kitchen scales. Note that weight is equal mass as long as we cook on earth.

Image by jamieanne, distributed under CCL.

Let’s recollect some math and physics:

Mass = Volume x Density

If the mass of 1 cup of flour in the bowls varied from 4 to 6 ounces, it means that the density is to blame. Faster scooping, scooping up vs down techniques, different storage, type of flour, shape of the cup – all of them can influence the density of the flour. To get the same amount in your recipe rely on mass.

The difference may be even more drastic when dealing with grated cheese. According to the article “the heavier shavings of a box grater can fill a cup with twice as much cheese as” “billowy ribbons of machine-shaved cheese.”

So, get yourself a kitchen scale for the next holidays. Use mass-based recipe source and you will:

Get consistent recipe-matching results every time.

Easily double or halve the recipe.

Have less stuff to clean. You can use only one mixing bowl by slowly adding ingredients into it directly from the containers and zeroing weight on the scale after each addition.

What about your old favorite volume-based recipes? One cup of oil in mom's sweet corn bread, one cup of honey in the Rosh haShana cake. Should you just convert them to mass? This Pyrex measuring cup clearly marks 1 cup volume as 8 oz mass.

Remember the formula:

Mass = Volume x Density

For water measurements: 8 oz = 1 cup x Water Density
Oil, melted chocolate and honey are obviously denser than water. Higher density gives higher mass: around 10 oz for one cup of oil, and 12 oz for one cup of honey. So, beware of the Pyrex' cup.

It is just you now in the kitchen with your scale and math.

Amazon links to buy kitchen scale:

Presenting Math as an Art

2011-08-19T15:54:00.051-04:00

I was disconnected from the internet for most of the August and got a luxury of time to read a few deep-hidden non-fiction books from our library. One of these books was a shocker - a light, eloquent, captivating and constructive critique of the ways mathematics is percieved and presented at schools, universities and most of our households. It is a very refreshing and convincing analysis from a former scientist and experienced math teacher and I am going to recommend it to every teacher, parent and high school student. The book is: "A Mathematical Lament" by Paul Lockhart. It is subtitled: "How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form."

Dr Lockhart suggests that math is an art and should be treated exactly like music and painting. We should not be learning math because it is applicable to our daily lives or essential in our future occupations, but rather because the process and the result bring joy to our soul like listening to or producing music does. Forget all the bad math associations you have nurtured and think of a pleasure that discovery of any pattern brings to us. And the sweet mental game of attempting to explain this pattern.

Take for example this curiosity:

1 = 1

1+3 = 4

1+3+5=9

1+3+5+7=16

1+3+5+7+9=25

Fascinating! Sum of the subsequent odd numbers produce results that are all square numbers.

But why?

Can any square number be represented as a sum of subsequent odd numbers?

If we let our mind wonder about this for a while we may be able to come with an explanation, perhaps even as thrilling as this one:

Any square number can be drawn as a square, such as this big and colorful 5x5. Any square can be split into such colorful parts.

Count the number of little pieces in these parts. They are 1 (purple), 3(green), 5(yellow), 7(blue) etc. These are our subsequent odd numbers. Looks like any square can be split into such pattern.

Lockhart writes that in school "the rich and fascinating adventure of the imagination has been reduced to a sterile set of facts to be memorized and procedures to be followed." Being math expert, math lover and experienced math teacher he suggests a number of directions to improve such math education:

"Mathematics is an art of explanation." This means that students should be allowed to pose their own problems, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble togather their own explanations and proofs."
Math history is fascinating and provides a great base for captivating storytelling. Instead of handing kids formulas to memorize retell them stories of ancient and contemporary mathematicians and their search for solutions to the fascinating problems that are wrapped in these formulas today.
Like other arts math should be subject to debate and critisize in schools. "Is this argument sound? Does it make sense? Is it simple and elegant?"
Forget proper notations that are frequently a source of frustration. "Math is not a language, it is an adventure." DaVinci, Pollock,and Warhol each created art in their own way.

The author also cleverly warns that although math is and should be presented as an "art" subject, such shelving may be dangerous as "useless" art disciplines are frequently the first to be cut off from the curriculum in time of budget cuts.

This book will definitely change the way you think about math.

It may boil your blood from anger with the author or on opposite, the current math curriculum against which he convincingly stands.

It may explain you why you always hated math classes but liked riddles and puzzles.

It surely will inspire you and teach you a few tricks that you will rush to share with your kids as I did.

Daily Roadside Math

2011-07-16T21:42:00.018-04:00

Walking around Vancouver today I noticed this stretched bicycle sign:

Why is it so distorted? Was it painted by a machine that accidentally accelerated shortening the bicycle at the top?

There must be a reason for this distortion. Probably something facilitating our perception. Similarly to the Ambulance sign painted with a horizontal flip on the ambulance truck to allow the sign be correctly read in a rear view mirror.

Let's see. Who is it for? A warning for approaching cars and bicycles that this road is bicycle only. They should be able to see it from 20-50 feet away. Let's check how the sign looks from such distance:

Just perfect!
What exactly is happening with this visual illusion?
The drawing of the bicycle is cleverly made to account for the perspective projection distortion in our eyes (and brain) and allows for the bicycle to be perceived clearly from the average height and far distance. Note how the bicycle wheels appear perfectly round on the second picture.

This is simplified demonstration of our eye perceiving bicycle wheel as perfectly round from a distance and as an oval from close up.

Now, how about an explanation to this signage?

From Magic Squares to Sudoku

2011-05-26T14:33:00.018-04:00

In 1415, the German renaissance painter and mathematician Albrecht Dürer created this busy copperplate engraving of a woman melancholically contemplating amongst a clutter that includes an hour glass, a balance scale, a sphere, a polyhedron (aka philosopher’s stone), a purse, and keys. Like his contemporary’s Mona Lisa, this mysterious painting has been extensively studied, speculated upon and written about. It is believed that it was planned to be the first in a series of paintings describing various temperaments: melancholic, phlegmatic, choleric, and sanguine. It is titled: “Melencolia I”

Among the peculiar geometrical tool in this painting, in the upper right corner, one can find … a magic square!

Wait; weren’t magic squares invented by Ben Franklin three centuries after?

Magic squares are tables of numbers where sums of every row, every column and both diagonals produce the same number. In Drurer’s square the sum is 34:

Magic squares have been around for thousands of years. Magic squares of size 3x3 are mentioned in the Chinese literature dating 650 BC. Magic squares of size 5x5 and 6x6 are found in an encyclopedia from Baghdad written in the ten’s century. They were engraved in stone and metal, worn as talismans, their usage believed to ensure longevity and prevention of diseases (from Wikipedia). Magic squares can be spotted in art all around the globe. The Indian Parshva temple still contains a 4x4 magic square carved in the 10th century. The Passion facade of Gaudi’s Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square with a sum of 33 that is the age of Jesus at the time of Passion.

What has fascinated people about Dürer’s square is that in addition to the regular magic properties, it contained more mathematical patterns and messages:

1) Numbers in each of the following five quadrants add up to 34.

2) Any pair of numbers symmetrically placed about the center of the square (such as these) sums to 17:

3) The two numbers in the middle of the bottom row give the date of the engraving: 1514

4) The numbers 1 and 4 at either side of the date correspond to, in English, the letters 'A' and 'D' which are the initials of the artist.

What about Ben Franklin? As a child and an adult he amused himself with creating large size magic squares such as this 8x8:

While his squares are not truly magic squares because sum of each of the diagonals is not equal to the sum of numbers in every row or column, they are still pretty cool. To compensate for a lack of the diagonal property Franklin defined a broken diagonal or “bended rows” properties where the sum of numbers in each highlighted bended rows is the same.

Magic squares could be considered the ancient relatives of the modern Sudoku game. In the late 19th century, French newspapers started publishing matrix puzzles – 9x9 magic squares with some numbers removed. Then the puzzle slowly evolved taking away the sum requirements and instead requiring that each row, column and diagonal have all the numbers from 1 to 9. And finally, the 3x3 sub-division was added producing the modern day Sudoku (you can click to play):

Thousands of Sudoku are published in the world daily. You have likely seen someone doing Sudoku on his way to work, at the playground, laundromat or in the coffee shop. A good story attesting to Sudoku’s popularity was published in the Sydney Morning Herald. In June 2008 an Australian drug-related jury trial costing over Australian $1,000,000 was aborted when it was discovered that five of the twelve jurors had been playing Sudoku instead of listening to the evidence.

When to expect when you are expecting

2011-05-19T22:50:00.007-04:00

They were expecting their third child – a daughter – a week after Labor Day. Perfect timing: their older kids will be out of the house and back to school, and the doctors back in town. All was well except that a few months before he realized that the September 12th due date fell inconveniently close to a professional conference in Europe, which he organized. “To go or not to go?” was pulsating daily in his brain. Will she manage here with 2.5 kids on her 9th month of pregnancy without me? What are her chances of delivering 3 weeks earlier? Shall I bring my mother to help here instead of me? If I stay – something will surely be messed up at the conference. Who will do my part? What would my colleagues say about me abandoning them? It is an honorable responsibility and I want to see it through.

***

Due date is calculated based on the empirical observations that it takes an average of 38 weeks from the moment of conception to the moment of delivery. But no one knows when exactly conception took place and how fast the sperm moved and fertilized the egg, plus it may be embarrassing to admit that you did it while at your in-laws or on a friend’s couch. So we look around for the closest related landmark – the menstrual period. Its first day usually is 10-18 days before the conception. So, your due date is approximately:
38 weeks + [10-18] before conception days =roughly 40 weeks from the first day of your last menstrual period
And ironically enough, when you conceive you are already considered 2 weeks pregnant.

The actual birth dates are spread around this 40 week due date in approximately normal distribution (bell curve).

(Graph from Ref 1.)
Graph mark of 24% at the 40th week mean that your chances of delivering on any day of this week are approximately 24%/7=3.4%. You can see that around 90% of babies are born within three weeks of their due date, and 21% are born within 3 days of it. In many geographical places, the right side of this distribution is sharply raised and cut at 42 weeks because the majority of the women that have not delivered by the 42nd week are being induced.

Why does the gestational age vary? Because not everyone ovulates and conceives exactly 14 days after the beginning of the menstrual period; some babies mature faster while some are too comfy inside; second and further babies seems to be more in a rush to come out; mother’s age and weather may influence the dates; and women of some ethnicities tend to give birth a bit earlier than others. (Ref 2, 3, 4)

Back to our expecting couple. He took from Wikipedia the data about probability distribution of actual births with respect to their due date.

He centered the graph on their due date, September 12th. Total area under this bell curve is equal to 1 = accumulated probability of giving birth. The chances of delivery during any interval are equal to the area under this interval. He marked in red the probability of her delivering prior to August 27th while he is away. He added up all the red area under the curve obtaining 0.043. He proudly showed it to her, explaining that her chances of giving birth before he returns are only 4.3%.
She told that this may be true but he is a 100% asshole. She also reminded him that both of their older kids were born two weeks earlier.
He still went abroad and run the conference.
She gave birth a week and a half after his return (on September 8th) to a healthy and beautiful baby girl.
They lived happily ever after, but he was never forgiven.

References:

iPhone and iPad apps of TheMathMom Stories and Puzzles

2011-05-04T22:51:00.006-04:00

New: iPhone and iPad apps of TheMathMom Stories and Puzzles. Get them while they are free. Hundreds have been downloaded in the last week since these apps were released. Read, answer or share from anywhere on the go. Click on the icons to download apps from iTunes. Please share your feedback on iTunes.

Spring IQ Test

2011-04-21T14:30:00.016-04:00

Ready to shake up your brain after the long and cold winter?
This IQ Test is easy, fun and playful. No math skills above Kindergarten level are required. Answer quickly, without hesitation.

Puzzle #1: You participate in a race and you are advancing ahead of a racer who is second. What is your place in the race right now?

Answer #1: If you answered that you are on the first place - you are absolutely wrong. If you went ahead of the runner who was second, you now took his second place.

Puzzle #2: Two fireman entered a school and one of them claimed to be a father of the other's 2nd grader. Can this be true?

Answer #2: Hopefully, you have answered yes. No complex family relationships are necessary, the other fireman must have been the 2nd grader's mother.

Puzzle #3: A girl entering a library was surprised to see her friend walking toward her. The friend was accompanied by her mother, two grandparents, younger brother and a little puppy. The friend was also holding a doll. How many people in total just came to the library?

Answer #3: Don't even think of answering 5 or 6. They are all leaving the library. Only one girl just came in.

Puzzle #4: How many pairs of animals did Moses take on the Arc?

Answer #4: It was Noah, not Moses.

Puzzle #5: Mary's father has five daughters: 1-Arlene, 2-Darlene, 3-Marlene, 4-Sharlene. What is the name of his fifth daughter?

Answer #5: If you answered anything ending with "lene" - wrong again. Run this puzzle by your toddler kid. She will help you figure out that Mary's father should have at least one daughter named Mary.

Better luck next time. Enjoy sharing this.

Top image of the brain-resembling coral is by boogieswithfish, distributed under CCL.

How to beat Chinese 2nd graders in the frog game

2011-04-12T21:24:00.013-04:00

This is a very cool game that is believed to be a 2nd grader computer test in China. Three green frogs, three brown frogs and all you need to do is switch their places over. But wait before you challenge with this your second grader. Click on the image below to try it yourself. Once, twice, ... ten times. Game site allows only small number of simultaneous players and occasionally is unavailable. Bookmark it and reload after a few minutes. It is well worth the wait.

If you start thinking that this is a prank and it is impossible, trust me it is possible. It may take a while, it may be non-intuitive but you can do it. Only if you become absolutely desperate check the solution steps below.

Can you do it now? Challenge your friend or your kid.

Birthday Treasure Hunt

2011-04-06T13:03:00.042-04:00

It became a tradition in our family to create birthday treasure hunts for kids. This started back in my childhood when my parents probably wanted to make sure that my brother and I remember to brush out teeth, make beds and dress up among the excitement of opening our presents. So, for each of our birthdays, parents left us a chain of notes that led the kid from his bed to the bathroom, where the greeting poem would be hidden somewhere under the toothbrush holder and a note to search further in his closet. After dressing up and scavenging for the first present in the closet birthday kid would be led to the kitchen for further greetings, presents, riddles and assignments.

These days I am recreating these treasure hunts with my kids and occasionally their friends at the birthday party, adding pirate or mystery flavour and more intellectual challenges (including math). Kids love it, remember these treasure hunts for years and always request it ahead of their birthdays.

I decided to share here one of the recent treasure hunts as it is easily adaptable to any kids' age and any household. To create your own treasure hunt you need a few (I used 6) treasure boxes filled with various presents or candy or goody bags and a chain of notes to search for one such treasure after another.

The first note that I handed my kids said:

Along with this note was a folded piece of paper sealed with a candle wax. On it: After a few minutes of running around the house kids figured out that the note refers to the painting we have that features women picking nuts and the old wooden deer statue on a shelf across from it. The first treasure box was hidden behind the deer. In it among the prizes was a second note leading to the next treasure: Note that the little wavy "x" is a multiplication sign. We have two coded multiplication sentences. Kids needed a bit of guidance, but after some hints quickly realized that: X=3, O=7, Z=9, S=6 Picking up letters 3,6,7 and 9 from the KWGOTARQAV gave them GARA. From this they figured out that it must be "garage" and found the second treasure there. Together with the treasure in the box was another note: Third treasure was hid in the warmer drawer of the STOVE that could be unscrambled from "ETOSV." In it was the next note: After running around the rooms for a while kids discovered the fourth box hidden in my son's closet. In it with the bouncy balls and candy was the next note: Almost there! The fifth box hidden in the bookshelf behind the world atlas contained the final note: I played hangman with the kids and they quickly figured out that the words are: UNDER PRINTER. This is where the last and the best treasure box has been waiting. Exhausted and satisfied after 15-20 minutes of searching around and bending their minds, kids proceeded to start opening the candy and all agreed that this was fun.

I am sure you can use these ideas and come with various versions of such adventure that will be more creative, fun or complex. Please share them! This is quite time consuming but it is joy to make, even more so to participate and it leaves the long lasting childhood memories. Plus it strengthens the association between math, word puzzles and fun.

France, Bicycle and Eclair.

2011-03-24T13:34:00.006-04:00

A treat math story today shared here with the permission of the author: Lawrence J. Krakauer. You can find more of his blog posts on his Memoirs and Musings blog.

During my French study summer in Pau, France, in 1961, I found myself one afternoon in the company of one of the girls in our group, Diane. Diane and I wanted to get to some event that was scheduled to take place in Pau's Place Clemenceau, and we were a bit late. I had a bicycle, which I was renting from the entrepreneurial son of the family I was staying with.

Unfortunately, the bicycle was not one that two people could comfortably ride. We started out walking together, with me pushing the bicycle, but then I had a better idea - a way we could get there faster, while still arriving at the same time. We could do that if we each bicycled half way, and walked half way.

To do that, I consulted my map of the city of Pau, and picked a point along our route that seemed to be half way to the Place Clemenceau. We agreed that Diane would ride the bicycle to that point, lock it up at an agreed-upon corner, and walk the rest of the way. I would walk to that point, pick up the bicycle, and ride the second half. We should both arrive at our destination at about the same time.

So Diane set off, carrying the bicycle lock, but leaving me the key. I proceeded to walk to the agreed upon corner, and as I arrived there, I was pleased to see the bicycle, chained to a pole. My brilliant idea was working! But just as I unlocked the bicycle, who should step out of a nearby doorway but Diane. I looked at the sign above the door - it was a patisserie (a pastry shop). In one hand, Diane held an éclair.

It seemed that upon arriving at the designated corner, Diane had noticed the patisserie, and had found it irresistible. So there we were, now half way to the Place Clemenceau, together again. Since the past is past, I guess the logical thing to do would have been to have picked another location halfway from our new starting point. But I recall being so annoyed at Diane for ruining my wonderful idea that I got on the bicycle and rode off, making her walk the rest of the way (as we had originally planned).

Let me add that Diane was not alone in being magnetically drawn to Pau's patisseries. All the girls in our group seemed to spend a lot of their time in these French gardens of Eden (for some reason, the girls much more than the boys). Once, during a morning French lesson, the professor decided to expand our vocabularies by asking the students to name a few French pastries. Overwhelmed by the response, he called a halt to the exercise after the students, the girls mostly, had named about fifty or sixty, some of which the professor had never heard of. Thus I discovered where many of the girls had been spending their free afternoons.

--- --- ---

Of course we all expected some romance to appear in this story of American students spending a summer in France. But assuming no romantic involvement between Lawrence and Diane, the half bike/half walk idea looks brilliant. How much time do you think could be saved by such an arrangement?

Assume that the distance they need to cover is S km. Both walk at 5 km/hour and both bike at 15 km/hour. We use km as we are in France.
In a half bike and half walk arrangement suggested by Lawrence, it will take:
0.5 x S /5 + 0.5 x S / 15 = S/10 + S/30 = 4S/30 = 2S/15

Should they politely continue walking and pushing the bicycle, it will take them S/5 hours. Or 3S/15.

It is easy to see that the half-and-half arrangement is S/15 faster than all walk or takes 2/3 of the walking time.
For a distance of 1km, it will be 1/15 hour = 4 min faster. The walk will take 12min and half-and-half only 8min.

For a distance of 5km, it will be 5/15 hour = 20 min faster. The walk will take 60 min and half-and-half arrangement 40 min.

I love this story, the clever trick and the unexpected patisserie surprise. Life is much more than a set of equations, however these equations do come handy from time to time.

Top image by mhall209, distributed under CCL.

Math in Special Effects: Motion Tracking

2011-02-21T16:48:00.062-05:00

While reading an article on architecture this week, I stumbled upon a statement saying that in the future 80% of jobs will require math and physics skills. I mentioned this to my kids to motivate their occasionally fading math homework spirits and it seemed to energize them a bit. But the best trick that really lights up their minds is an illustration of how math is relevant to the occupations of their dreams. In our home this may mean a story about the collapse of a roof in the new Paris airport terminal for my son, who (currently) wants to be an architect, or discussion of NFL players' statistical models, or a girl movie mentioning the math of ice skating (Ice Princess).

But while math is omnipresent in most professions, some of the best playgrounds for math skills have always been: special effects, national defense and medicine. Math majors always felt like kids in a candy shop in these fields, liberating hours of manual labor and allowing for projects to be done cheaper, faster, safer and more realistically. As a special effects veteran, I am happy to share with you a math trick that I have used in the past in the movies "Air Force One," "Multiplicity," "Desperate Measures" and "Starship Troopers." It is called Motion Tracking.

Take a look at this breathtaking commercial and don't forget to invite your kids. Note the boy who is holding a giant gorilla's hand in the beginning of the movie, the man who is bouncing the soccer balls and the race car drivers.

And now, check out the clip of how this was made. No time? Play at least till you see the boy walking around with a box.

Why is he holding this box and why it is marked with a tape?

The box is obviously being replaced with the giant gorilla's hand (computer generated). The tape on the box allows to reconstruct the exact motion of the box relative to the camera and then make the gorilla move it's hand with exactly the same motion.

Why are there four tape marks on the box? Wouldn't one be enough?
No!
Imagine the box right in front of you, moving in your direction. One piece of tape attached right in the middle of the box. The tape may not be moving at all as the box approaches you. When you have two tapes, you will notice that the distance between them is increasing as the box is moving closer.
The math behind this requires you to use at least three tape marks to solve a system of equations and find the exact motion of the box with respect to the camera. Use of more than three marks allows to do it more precisely, minimizing any possible errors.

In order to render the gorilla as holding the boy's hand, we need to know how far away the gorilla should stand from the camera and how it should be rotated. This is defined through two parameters: R that is the 3x3 rotation matrix and T that is 3x1 translation vector. Gorilla and its computer generated hand are represented as a collection of 3D points. Each such point P has (X, Y, Z) values in the computer graphics library. To place gorilla next to the boy animators tell their rendering programs to use rotation R, translation T and some camera parameters. How do they know what R and T are? From the box!

Special effects artists are tracking the tape on the box to recover its R and T. Imagine this box in your hands. Every point P on this box (including the tape point) has 3 coordinates (X, Y, Z). Next to the boy, these coordinates are P1:

P1 = R x P + T

We don't know P1 but we know its projection on the image in each frame of the commercial. Let's call this projected point p.

P1/Z = p/f where f is known camera's focal distance

from this:

p = (f/Z ) P1 = R x P + T

We know f, p and P and want to find R and T. P is a coordinate of a point on the box where tape is attached, when this point is in your hands. p is coordinate of the same point in the image of the commercial. As p has two coordinates on the image (x, y) you get two equations for every point you use.

If you use some special rotation matrix (small angle rotational matrix), you will get a system of linear equations with 6 unknowns: 3 rotational components and 3 translational components. To solve a linear system with 6 unknowns you need at least 6 equations.

Remember we have 2 equations per point? Three points minimum but better more to account for possible imprecision in point tracking: some points being blurred on the image or box is tilted so that some points become invisible.

If you got it right then the gorilla's hand will be tightly attached to the boy's hand. If your math is wrong, then the gorilla and it's hand will be unrealistically plastered next to the boys as in some cheap old commercials. By the way, in the second video you can see similar tape marks on the wall that the soccer balls are bouncing off, and on the race car driver's helmets. In both cases, the position of these objects with respect to the video camera is being tracked in order to add more computer generated objects into the scene, map textures or render reflections.

Interestingly, exactly the same motion tracking strategies are used to reconstruct battle scenes from video for analysis and training purposes in the field of national defense, or to track patients' motion during a surgery in medicine. Now you have something to tell anyone who mentions that linear equations and matrices are boring.

There are many tricks that allow special effects artists seamlessly combine real and computer generated characters in one sequence. Matching character motion is one of them. Others include realistic texture and fur rendering and light matching as described in the following stories: The Silly, Wacky, Revealing and Useful Shadows
and The trouble with facial hair.

All About Kim

2011-02-21T12:54:00.011-05:00

As you probably know: those who solve five puzzles in our Family Puzzle Marathon are receiving pink-and-black TheMathMom mug; those who solve ten get a honor of a puzzle being written about them. Now, meet Kim - a master puzzle solver with 50 puzzles on her list. To celebrate her achievements we decided to run an interview with her. This is tell-all: from the childhood math experiments to girl scout cookies and NFL players. Kim reveals all her influences and passions. Below is a portrait of Kim made by her children.

Q: You are obviously a master of logic and numbers. Whom do you attribute your love of math to?

A: I have always loved math. I remember creating my own books of word problems for the other kids to do back in first and second grade. My father went to MIT and so maybe this math thing is genetic for me. Some credit then to my Dad. One of my brothers tutors math (he is really, really good at it), and he was always fiddling around with ways to teach math. So back when I was in elementary school and he was in high school, I was his first guinea pig -- I mean, student. It was fun for me always knowing stuff way ahead of everybody.

Q: What do you like more: odd numbers or even numbers?

A: This is a toughie. They both have their pros and cons. Odd numbers are so interesting. There are lots of great ones, like Lucky 7, and all those really cool numbers that end in 5. And all those prime numbers -- you just don't see a lot of even prime numbers... On the other hand, since I have two kids, evens are good, too. They are always divisible by two which often comes in handy. And since there are four of us in our household, half of those evens are divisible by four, too! Really handy. So it's hard to pick.

Q: Do you find irrational numbers rational?

A: Rational, not so much, but exciting, yes! I mean, look at π -- you just don't get numbers that cool everyday. And √2. Yeah, I admit it they aren't exactly rational, but a lot of my best friends aren't rational either. We can't be so judgmental about our numbers. We have to look at what they do for us. Those irrational numbers can be pretty useful, and I'm willing to forgive some of their shortcomings.

Q: Do you know any math song?

A: You bet I do!! Did you ever used to watch Schoolhouse Rock? There were a bunch of great math songs. My husband (sssh, he probably wouldn't want me to write about this) knows the lyrics to pretty much all of them. I don't, but I do know a few. I always really liked the one about the number three (Three is a Magic Number). I have to admit, though, my favorite Schoolhouse Rock song wasn't about math, it was the one about the little bill who wanted to be a law.

Q: What was the coolest life situation you applied math to?

A: One morning my son was wondering how much tape there was in a 90 minute cassette tape. There are a bunch of different ways you might figure something like that out, like marking the tape, running it in a tape player for a few seconds, and measuring how much it moved. Or doing some measurements and calculations based on the diameter of the tape when it's all rolled up. But we decided to take a different approach. We broke the tape, held one end of it still and sent my son running the perimeter of our property until he had unrolled the whole thing. It's longer than you might expect.

Q: What is the most interesting professional math problem you ever had to solve?

A: One of my favorite projects I've worked on was several years back when I was working with colleagues building a simulation model for an NFL football team. They wanted to understand the dynamics of building dynasties (back then other teams weren't trying to be like the Patriots, they wanted to be like the 49ers). The question was: how did they build such a great team and keep it that way? What were they doing right that nobody else was? We built a model for them, and even had it with us when we joined them on Draft Day to help them assess some of the picks they were considering.

Q: What is your favorite angle? Why?

A: It would be easy to say 90 degree angles, but they are so square. I have to say, I find those bigger angles to be sort of obtuse, and I prefer those little, cuter, ones. 45 degree is nice because how evenly it splits a 90 degree angle. 60 degree angles are also nice, and I like how nicely they share they when they come together to form a triangle.

Q: Some personal questions, if you do not mind: what is ((your age x your weight) / your height)?

A: Ah! The Age-Mass Index. Yes, this is a very important measure, and I expect doctors will start using this as soon as they can make sense of it. I come in at just under 75 pound-years/inch.

Q: What is your favorite girl scout cookie?

A: This is very important topic in our household because I have a Girl Scout, and we have seen (too) many boxes come through our home. I can tell you that our Troop sold far more Thin Mints than anything else (with Caramel deLites a distant second -- these are called Samoas in other parts of the country). My favorite changes from time to time. I do love a good Thin Mint, but it's hard to argue with the Shortbread or the Thanks-A-Lots either.

Q: Do you do your own taxes?

A: Of course, I do. And I will make a confession here: I kind of enjoy it. I'm not sure I'd like it so much without the nice folks at Intuit who make TurboTax though.

Q: How do you share your love of math with your kids?

A: When he was really little, I remember my son asking about if there were any numbers bigger than infinity. That was really exciting for me. My kids will tell you that I share my love for math a little too much. For me, everything is a math problem! How much flour will I need if I triple this recipe? How should I divide this lasagna into 7 pieces? How long will it take to finish this book if I read 25 pages per day? That sort of thing. My favorite thing in mathematics is probabilities, and these fit into "real life" all the time. Just a few days ago we were talking about the importance of having a low false-positive rate if you want a good test for a rare disease, and they totally got it.

I love to proudly wave around my Math Mom coffee mug. And I share the Math Mom puzzles with them, too.

My favorite math games with the kids are Set, Mastermind, and 24.

Q: OK, so you love math. Do you think math loves you back?

A: Yes, no doubt about it!

Valentine's Math Quiz: All The Math You Will Do For Love

2011-02-14T14:40:00.003-05:00

Dating and love - what do they have to do with math? This Valentine's quiz created by TheMathMom for the Boston Globe demonstrates the applicability of math to love and dating affairs. Ready for the challenge? Click on the image below to start:

Would you like a power-point version of this quiz to share with your students, teachers or your kids? Download it here.

Read the Math Mom's column The Math of Dating -- Mark Yourself Excelled for more about the math as the guiding hand in some of your romantic adventures.

2011 Is One Sexy Prime

2011-02-04T23:23:00.011-05:00

This story was published by Boston Globe on January 25, 2010 (online edition) and February 10 (paper edition). Link.

2011 is a sexy prime. And not only in a Playboy calendar.

Like prime ministers and prime ribs, prime numbers are the elite of their kind. They are unbreakable.

Unlike other numbers that can be split into smaller factors (e.g. 6=2 x 3), prime numbers can only be divided by 1 and themselves. Three is prime, so are 5, 7, and 11. The only even prime is 2.

A lot of special numbers around us are primes. Emergency phone numbers 911 and (in some European countries) 101 are primes. The fearful 13 is a prime.

The world of primes is very colorful, especially when you look at their relationship with their neighboring primes. Those that differ by 2, like 3 and 5, are called twin primes. Those that differ by 4 are cousin primes. Two subsequent primes that differ by 6 are called sexy primes — because ‘‘sex’’ is Latin for ‘‘six.’’

2011 and 2017 are two sexy primes. There are also lucky primes, happy primes, Cuban primes, Titanic primes, and many more.

Primes have always been the most talked about numbers; even Pythagoras believed that these unbreakable numbers have some spiritual properties. It was discovered that you can create any number by multiplying or adding the primes together. Check it out: Every even number larger than 6 can be represented as a product of primes and as a sum of two primes. For example: 8 = 2 x 2 x 2 = 5 + 3

But about 40 years ago, primes became celebrities. Math paparazzi all around the world are now hunting for primes. Awards are offered for those who discover hundred-million-digit prime numbers and every new prime spotting is noted and celebrated.
Why? Because any bank transaction, any online purchase, any medical or military data exchange requires data encryption.

Cryptographers found a great encryption mechanism using the observation that when a number is created as a product of two large primes, guessing or computing those primes from the total number is almost impossible. Those primes are used to encrypt the message and serve as a secret digital key known only to the encrypting and decrypting parties.

The more unbreakable you want your code to be, the larger the primes you should use for encryption. Thus, the search for the ultra-large prime numbers.

Want to help search? The good news is that there are infinitely many prime numbers. Euclid proved that. But the bad news is that higher you climb in digits, the more rare these prime numbers are.

The probability of any random number being prime is inversely proportional to the number of its digits (roughly). So, for a hundred-million-digit number, the probability of it being prime is 1/100,000,000, or 0.00000001.

Also, for such large numbers it takes about a month — even for a computer — to check whether any given number is a prime or not.

In addition to helping scramble secret messages, primes can be used to help hide the messages inside an image or audio signal.

As appropriate for celebrities, primes are surrounded by some mysticism. They are not regularly positioned among other numbers, nor are they randomly spread.

Take the recent and future prime years: 2003, 2011, 2017, 2027, 2029. Some are 10 years apart and some only 2. Like fortune tellers, number theory scientists are trying to detect and predict patterns in prime number appearances. Turns out that if you arrange all the numbers in a spiral instead of a number line, and then connect all the primes, you will find some surprising diagonal patterns.

Back to our special year. The upside-down reflection of 2011 on a digital calculator or clock is 5011, which is also a prime. Parents who gave birth on the second Tuesday in January could just remember a bunch of 1s for their child’s birth date: 1/11/11. You can still make it for 11/1/11 or 11/11/11 if you plan ahead. There is also 9/10/11 for grabs.

Since prime is so hip and relevant this year, I suggest we should start using it more outside the context of mathematics, prime ministers, and ribs.

This year’s Oscar ceremony hosts — James Franco and Anne Hathaway — are undoubtedly two sexy primes, while the Oscars (and Superbowl) winners could be referred to as lucky primes. And who would argue that Leonardo DiCaprio and Kate Winslet are Titanic primes?

Tom Brady, what can we say about him? With a #12 and the recent devastating Patriots’ loss to the Jets, he is ‘‘almost prime’’— that, believe it or not, also happens to be a math term.

With 11 (also a prime number) out of the 12 months ahead of us, there is still plenty of time to make this year a very happy and important prime.

Top cartoon by Peter Mesnik.

Know your primes? Take TheMathMom's prime numbers quiz created specially for Boston Globe to test your skills. Over 20,000 people took this quiz!

The game of my childhood: Bulls and Cows or Mastermind.

2011-02-01T09:54:00.048-05:00

Post-Cold War Russia of 1980th. We still believed that US is a threat and a serious competitor in the life style, science, sports and space. In school we trained to assemble and shoot from Kalashnikov and run to the bomb shelter (each school was equipped with one since World War II). I recollect perceiving US as a country of chewing gum, handsom guys like Tom Cruise, brown paper supermarket bags and homeless people sleeping under the bridge. The rest of my childhood was pretty much like yours - cartoons; after school sports, music, ballet; weekend ski, hikes, or friends and family gatherings.

One of the games I remember enjoying playing was a code breaking game named Bulls and Cows. All you need is a pencil and paper. Pick a 4 digit number with all the digits being different. This is your code. Ask your friend to break your code - guess this number.
Let's try it!
You pick 5817
My first guess is: 0123

You write my number under your number and see how many digits match. If I guess a digit correctly and it is in the same position as in your number, it is a "bull." Same digit but in a different position is called a "cow."
5817
0123 (0 bulls, 1 cow)

You tell me 0 bulls and 1 cow.

I give my second guess: 3456

You write again this number under your secret number in a column and reply how many bulls and cows are there in my guess:

5817
-----
0123 (0 bulls, 1 cow)
3456 (0 bulls, 1 cow)
6789 (0 bulls, 2 cow)

Now, I am using some logic to figure out that digits 3 and 6 are not in your number. Note that these digits present in two of my guesses and if either of them would be in your secret number the total number of bulls + cows would be higher than 4. I use these digits to guess the rest:

3698 (0 bulls, 1 cow) => only one of 9,8 is in your number
7836 (1 bull, 1 cow) => 7 and 8 are in your number
7048 (0 bulls, 2 cows) => 0, 4 are not in it, we know the place for 8 now
5872 (2 bulls, 1 cow) => 2 is not in it
1857 (2 bulls, 2 cows)
5817 Yes!

Perhaps you played this game as well - in US, Brazil, Egypt or Israel. Math is not only a universal language but also a toy shared across continents, cultures and regimes. According to Wikipedia, Bulls and Cows was known around the globe for hundred of years.

Nowadays, most of us here in US know and love this game as Mastermind - with color pegs instead of the numbers. The colorful version - Mastermind - was invented in 1970 by Mordecai Meirowitz, an Israeli postmaster and telecommunications expert. Apparently, his idea was at first turned down by many of the leading toy companies, but he persisted, and took it to the International Toy Fair at Nuremberg where a small English plastic company bought the IP rights and made it the most successful game of the 1970th. If you think something was sexist in the name "Bull and Cows", check out the peculiar original Mastermind box cover that called it a "Game of cunning and logic for two players:"

Since then over 55 million copies were sold in 80 countries. In US this game is made and distributed by Pressman Toys and is available on Amazon:

**********************************************************************************

Practice your logic skills with this Bulls and Cows, also known as Hit & Match game. Recommended for ages 10-100, parent assistant is advised for younger kids.
Can you guess a four digit number? It may be a secret military code or a treasure box key. All the digits in this secret number are different. For each of your guesses computer will tell you how many digits are perfectly matching the code (Match) and how many are in the code but in a different place than you guessed (Hit). Can you do it in less than 6 guesses? The game is created by Gilad Oved for TheMathMom. Find more of Gilad's games and MCAS math practice tests on his website http://www.gv-8.com/

Click to play:

What do Nobel Prize winner and Donald Duck have in common?

2011-01-20T13:59:00.015-05:00

Bertrand Russell who was a British renaissance man - philosopher, historian, social critic, mathematician, and the winner of the Nobel Prize in Literature - wrote in his books:

"In adolescence, I hated life and was continually on the verge of suicide, from which, however, I was restrained by the desire to know more mathematics." (Conquest of Happiness)

"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." (The Study of Mathematics)

Here are few amazing videos where the old Donald Duck and the young math whiz share the obsessive, captivating, mysterious, applicative and beautiful math.

Share this 7 min clip from the old Disney Donald in Mathmagic Land with your elementary school kids:

And this Math Doodling will fascinate and inspire even your teenagers:

These links have been added to The Math Mom's Math Resources Page. Please remember to check this page anytime you are looking for interactive math CD-Roms, iPhone apps, books, TV programs, toys etc.

Reversing Math Stereotypes

2011-01-06T08:44:00.017-05:00

What can you do with kids that use "nerd" and "geek" as offensive words? Tell them about Bill Gates - a self-made billionaire who lives in a house with a swimming pool equipped with an underwater music system; describe to them Google founders who go surfing in-between business meetings around the world and turned their own Boeing 767 jet into a party plane; point out those geeks polishing their statistical presentations in the roomy Business Class section of your next flight while enjoying an enviably large dessert selection.

A new TV show and a book may further help you demolish down the stereotypes.
Head Rush - an amazingly cool Science Channel show for ages 9-15. Click for program schedule in your area.

For younger kids, an old favorite - PBS's Cyberchase - is guaranteed to captivate, encourage scientific curiosity and associate "geek" with "hip." Link to the website with show schedule and online games.

When most of us visualize mathematicians, we image emotionless men in plaid shirt with heavy prescription glasses. A book that is fresh out of the publishing oven is challenging such stereotypes by depicting mathematicians' passions, collaborations, love affairs and even murder crimes.
Loving and Hating Mathematics: Challenging the Myths of Mathematical Life
Reuben Hersh and Vera John-Steiner Princeton University Press 428 pp. $29.95 (2011)

You can find more Math and Science resources (books, TV&Radio show, iPhone apps, CD Roms) split down by ages on TheMatMom's Resources Page.

The Math of Parallel Parking

2010-12-23T21:22:00.018-05:00

This is how I was taught to park: you pull parallel to a parked car with an empty spot behind, like in line dancing, placing your front wheels next to the other car's front wheels. You start backing up while rotating your steering wheel all the way to the right. When your front wheels are in line with that other car's rear wheels, you rotate your steering wheel all the way to the left. It works half of the times for me. Quarter of my attempts result in my car being at least 2 feet away from the curb, and another quarter in my back wheels hitting the curb too early that usually requires a lot of hocky-pocky to fix. I know that I am not alone when I observe all these randomly parked cars near my kids' school. A few of my friends suggested that something should be written about the art and math of parallel parking. Turns out, there is!

Image by TriciaWang, distributed under CCL

A year ago a British scientist, Simon Blackburn, published a formula for perfect parallel parking.

A US Math teacher, Jerome White, later created a simulation applet based on this algorithm and noticed a flaw. You can play with this safe simulation applet here. The Blackburn's formula was far from optimal, requiring much more parking space along the curb than necessary. Throughout this last year White and his colleagues tried to revise the perfect parking challenge. The formulas got more complex but the simulations look a bit more like my hocky-pocky parking (Jerome White perfect parking page). The best part is that they proved what all of us stubborn and angry car drivers have always been hoping for: you can squeeze into any parking space if it is at least as long as your car, if you are willing to maneuver an unlimited number of times.

The Tests, The "Sputnik Moment" And What We Can Do About It.

2010-12-09T08:22:00.019-05:00

This week the results of the respectful PISA test (Program for International Student Assessment) have been officially released. It is shocking, it is visual, it is convincing and it can be viewed here: Test Result Table

In this test 15 year old students from 65 countries were tested in science, reading comprehension and math. We are not at the top. In science we are 23nd, very close to the average. In reading we are above the average - on a 17th place, but in math we are... Where are we? This gray line at the very bottom. It is us. Actually, we are not at the tail. Only the first half of the countries are listed here. But we are surprisingly below average - on the 31st place.

Shanghai, China - is leading in all the columns, with other Asian countries and areas at the top. Finland is doing great, but even our neighbor - Canada - is way ahead of us all around. Educators and government officials have been calling these results "The Sputnik moment" and "The wake-up call", with papers full of suggestions for reasons on China's success and our failures.

It may be the curriculum, the teacher status and salaries, the length of the school year and the amount of the homework. But I truly believe that the most important factor in all this is our cultural attitude toward math and sciences. There is a big difference between presenting them as cool, necessary and applicable as opposite of boring, optional and narrowly useful. Not at school, but rather in our homes, in the media, and everywhere in our culture. Here is a very relevant post I wrote a few months ago comparing our math and sports attitudes and offering ideas on taking example from our sport excellence to improve our math...

Are we sure it's OK to be Smart?

Imagine Olympic athletes apologizing on TV for being muscular and agile. Or an engraving on a lipstick tube that reaffirms “It is OK to be beautiful.” Alternatively, try visualizing a clothing store ad luring us in with the words “Nothing is wrong with being well-dressed.” Sounds rather unreal. But how about the disclaimer on the back of a set of Brain Quest flash cards for kids that states “It's O.K. to be Smart!” Does it seem similarly ridiculous? This one is actually real. A great toy company feels that it should warn us not to worry about being a nerd. Are we sure it is OK to be smart? Are we serious that it is just OK???? Why don't we say: “It's cool to be smart!” or “It's awesome to be smart!” And why do we need this affirmation at all? Shouldn't it be obvious?continue reading.

6 Ways Math Helped Me Gain More Home Space

2010-11-30T21:42:00.042-05:00

I started writing this blog because I wanted to share my outlook on math as a friendly, exciting and very applicable part of our daily lives and a cool tool to be equipped with. I continue writing because with your help I am constantly discovering the surprising guidance math could provide in situations one has never imagined possible.

A few months ago, I wrote a post on how math could help us get more out of the home space we have. Now, looking around my home, I suddenly realize that without conscious intention, I put many of my advices to work. Observing the results, I believe that we now use our not-so-large home more effectively and enjoy it more than before. Here are the 6 simple things that I tackled:

1)Shoe Closet
My husband has size 14 shoes. They are almost 12" or 30cm in length each. It took me years to realize that he needs much more shoe closet space than I or any of the kids. But even when we divide it 1-to-3, with him taking a whole half of the closet and me sharing the other half with the kids, his shoes outflow his space. I do have hard time believing him that he does need separate sneakers for running, basketball, squash and hiking. But it is too late - all this (and much more) is already in the closet. One simple approach would be to divide the closet floor proportionally to the shoe sole area. But a truly ingenious geometrical solution is to consider the other dimensions. On this Ikea rack, small or large size shoes take the same space when arranged vertically.

2)Living Room
Writing about the statistics of our use of each of the rooms, I created a home usability spreadsheet that can be downloaded here. Studying it, made me realize that our living room was vastly underused. We were barely spending 6 hours per week in it, that is 2.5% of our total home-spent awake time. Why didn't we use the new reading couches and bookshelves we recently purchased? Mainly because the light in the room was never strong enough for the evening reading. Last month we installed four cute little spotlights that are strong enough for late night winter reading or a family game. We put a few games, books and magazines on the table and encourage kids to cuddle with a soft warm blanket on the sofas to read, chat or play.

3)Kitchen
A discussion with my friend who recently moved into a new home and from all the rooms that needed renovation, decided to concentrate time and money first of all on the kitchen because "we all spend so much time there" made me review my family's room usability chart. Indeed, after the office (I work from home), our open kitchen-dining-homework area is the busiest in the house and perhaps it is about time we invest to redo the antiquated 70th style kitchen to something more fitting our taste.
Feel free to upload and modify this room usability spreadsheet to investigate what is being under and over used in your home.

4)Towel closet
Learning from a flight attendant's suitcase packing tips, I started rolling my towels instead of folding and immediately gained extra space in our closet. Plus, kids find it much more fun to help me now. However, their favorite rolling place is the floor.

5)Bathroom
Ever since we moved to the East Coast from LA, we were wondering why toilets and vanities seem to be much lower here. The standard 30" height of the bathroom cabinets apparently came from the old days when water bowls were just put atop a dresser. Such a low height may work well for kids of certain age and height, but little kids still need a stool and teenagers are the height of their parents. Therefore, unless you are running a home-school, you can be much more comfortable with a higher vanity that does not require so much bending. Especially, if you have a bad back. To test this, we tried hand washing and teeth brushing in a kitchen sink, as kitchen counter height is usually 36". We liked it and when redoing our bathroom after an occidental flooding, we knew that we wanted to fit our home to our "tall" needs.

6)Mirrors
When turning our utilities room into my husbands' office and contemplating the style of the bi-fold doors that should cover the washing and drying machines, I recollected my own advice: "Mirrors make room appear larger and brighter by reflecting the light instead of absorbing it as walls do." Now, we can't believe that we got so much visible space out of the junkyard of utilities and storage.

Please feel free to post in the comments your own advices and stories on creating more space or making home more efficient with the use of trivial and not so trivial mathematical thinking. You can also share your own Home Usability Chart with your kids discussing how to make heavily used rooms more fun and underused rooms more practical.

In Search of a Perfect Drinking Glass

2010-11-24T22:52:00.008-05:00

When our kids were little, hardly a day went by without someone spilling a glass of milk, juice or water. We were prepared – with sippy-cups, plastic cups, paper cups and abundance of paper towels. We knew that while the kids’ motor skills are still developing, ours are too sleepy to function as designed, and the kitchen is too small to allow the juggling circus we were trying to perform there. Still, there were days when I thought there is a spilling spell on our family. Then, I started blaming the cups. Plastic or paper – they were unbreakable but so light and unstable that just a light push would flip them out of balance.

Ever since, I have been preoccupied with the search for a perfect drinking cup. Take a simple, clear water glass. Every houseware store has a wide selection, every season brings new designs. But have you found the one that works for you? What qualities does it possess?

Learning from the kids’ plastic cups lesson, I defined my first requirement as stability on impact – the perfect glass should be difficult to take out of balance. This means it needs to have heavy and/or large bottom and/or thick walls. The retro-style glass pictured below was proudly purchased by my husband, and it seemed to satisfy my first demand. Was is perfect? Nope - too heavy to lift even for adults, and too bulky to store. Plus, while it may work well in a diner, I did’t like the look of it on my dining table.

So, stability was clearly not enough, and the list of properties of the perfect water glass started growing:

stable

not heavy

beautiful

I bought more and more: Ikea, Crate and Barrel, Cost Plus, Pier Import, Marshalls. Some were too short to hold the amount of water accompanying a meal. Some were too tall to fit in my kitchen shelves. Some broke on the first week, some I couldn't break as much as I tried to. A few years ago I thought I finally found it. The perfect glass: stable, light and elegant. The dips on the sides made for a sure and easy grip. I bought 12 only to realize that I can’t stack them, plus the cave-like space inside the glass doesn’t allow it to be cleaned well in the dishwasher.

Back to the list. The perfect glass should be:

stable

not heavy

beautiful

stackable and not too tall for efficient storage

large enough to hold 1 cup of water

cleaned well in the dishwasher

I read about a design exhibition in Copper Hewitt Museum in NYC that showcased elegant Swedish drinking glasses “which swell gently outward just below midpoint.” The swell was a special design feature to stabilize the grip so that the glasses could be comfortably held by people afflicted with neural or muscular disorders that produce numbness or tremors.

Credit: Karin Eriksson

These glasses would look nice on my dining table, they appear stable and easy to hold even for a child. But they seem too gentle for a daily hectic kitchen life.

I decided to go back to the simple solution and finally settled on something that looked OK, was moderately heavy, rather sturdy and looked promising to become the one and only.

At the store I went over my list:

stable – I sneaked a light push that didn’t affect the glass

not too heavy - yes

beautiful - OK

stackable and not too tall for efficient storage - yes

large enough to hold 1 cup of water - yes

cleaned well in a dishwasher – should be because it is smooth

not too gentle for a daily handling – thick enough yet not heavy

But guess what, some sandy grains are always accumulating at the bottom of the glass in our dishwasher. I suspect it may be too tall and narrow to be cleaned well automatically.

I am still searching for the one and only. But does it exist? Have you found one?

Great math games for iPhone, iPod Touch and iPad

2010-11-10T11:43:00.001-05:00

iApplications are cheap, easy and handy. Chances are your child has already downloaded a few to your or her device. It may shoot, scream (or worse), play, read or teach. Given the diversity of flavors, you may want to be involved, and indeed you can guide your kids quickly and efficiently. Suggested below are some interesting educational apps that can help you present math to your kids as a toy and a friend.

As much as you may want to download the app and entrust the phone to your kids instructing them to figure app out on their own, please don't do it. Learn from my mistakes. Make sure you spend a few minutes to set up your child with any of these games. While most children are very comfortable with new technologies and can intuit the game rules faster than we do, the default complexity level and speed may not be the one appropriate for your child. Test it yourself, then enjoy doing it together, then set them up on their own and pat yourself on the shoulder while listening to their excited squeaks.

iHangMath by Ocean Blue Apps
A creative adaptation of the old favorite Hangman game. Only instead of the words you have a random multiplication sentence to decode. Great for kids who know the rules of long number multiplication as well as adults. $0.99 iTunes link

MotionMath by MotionMath
Learn fractions the Wii-style. MotionMath utilizes the iPhone's ability to sense motion with its gyroscope and gives players a physical connection to fractions and the number line. Users need to tilt the phone to guide falling balls with ratio values into proper positions along the number line. Cool and simple in its design, the game is fun but not at all trivial as balls are falling very fast. Level can be adjusted after the first game. MotionMath won an Editor's Choice Award for Excellence in Design from Children's Technology Review. $0.99 iTunes link

PopMath, PopMath Lite, PopMath Plus by Pop Soft
This is a great app for kids at elementary school age. Practice addition, subtraction, multiplication and division by popping corresponding bubbles on the screen. There is a free version and two $0.99 versions. iTunes link

MathGirl Number Garden and Math Girls Addition by Icanology LLC.
While numerous studies have shown that there is no difference between boys and girls in math capabilities, girls are still occasionally being stirred away from math in our culture. Presenting math (and sciences in general) in a relaxing girl-themed context has been shown to improve girls' math attitude. Preschoolers and young elementary school ladies will be charmed by these Apps' names and pink flowery graphics. They will practice grouping and counting objects in the Number Garden and adding them in the Addition app. The game follows a popular among kids pattern of solving puzzles, earning points and then spending them to buy something. In Number Garden kids populate their virtual garden by buying flowers, animals and even lakes. Each app is $1.99 iTunes link

Along with the new creative apps, there are app versions of old popular games that kids and adults will be thrilled and comfortable to play on i-devices.

Tangrams
Little kids will enjoy this centuries-old game while practicing some basic geometry skills. Free. A pro version is $0.99 iTunes link

Set by Pockent, LLC
Great card matching game for all ages. Develops logical thinking. Set Lite is free. A full app is available for $2.99. iTunes link

Rush Hour by ThinkFun, Inc.
Start with the Free app, which offers 10 puzzles on each complexity level. This is a fun early introduction to programming. iTunes link

Treat Lite versions of the apps as product samples. They just provide a taste but are never enough for a delicious meal. If you like what you play, get a full app version that offers many more puzzles and challenges.

What do you and your kids like? Please share your favorite math-related apps in the comments of this post.

Vodka, Jesus and a new Millennium

2010-10-20T10:11:00.017-04:00

Last week an interesting email story titled "Scientific Resolution" has landed in my mailbox. Here it is, translated from Russian:

Any scientist who can't explain to an eight-year old what he is doing is a charlatan.
Kurt Vonnegut

At the end of the 1990s, hot discussions have been puffing all around the world on whether the new millennium should be celebrated on January 1st 2000 or January 1st 2001. Everyone argued everywhere, occasionally with screams and fights. So did we, in the editorial department of our news channel. Arguments varied. One lady even pencilled 2000 check marks on the paper to support her position.

When we ran out of arguments we continued with insults. At that moment one of the camera operators stuck his nose inside our room and told us that there is a distinguished professor being interviewed in the next studio over for some scientific program. Perhaps he can serve as a judge in our by now boiling discussion. Let's grab him, and force him to resolve this troubling millennium question. And let's agree that whatever he says we'll have to accept and get back to work.

The scientists looked very authoritative:
Around 100 years old, member of various world science academies.
Physicists, mathematician etc. First class!

He spent the last 40 years in the US, so he spoke Russian with a American accent.

We grabbed him and asked! In a second he smiled and pronounced:
"My friends, imagine you are having a picnic and drinking vodka. You have
3 crates. Each crate has 10 bottles. Now, you answer me: "twentieths bottle will be in the 2nd crate or the 3rd?"

This is a real scientist! I have hard time imagining a fool that THIS professor can't teach.

End of the story.
***

I loved the story but was doubting that this professor is indeed right. He may be Professor, Lord, Duke, or a President - I thought annoyingly - but while the 20th bottle belongs to the 2nd crate and the 40th bottle belongs to the 4th crate, I knew very well that on my 40th birthday I will be starting my 5th decade. Does Jesus's life and birth, defining our AD, have more in common with my life count or with the vodka arrangement?

Millennium window at St Peter & St Paul, Shoreham, Kent. Image from Flickr, distributed under CCL

By the time we are 1 year old, we already lived 1 full year.
So, the first decade of our life is from birth until we turn 10.
From the day after our 10th birthday until our 20th birthday is the second decade of your life. Therefore, the year when we are 10 belongs to the 2nd decade of our life. While 10th vodka bottle obviously belongs to the 1st crate.

We all know that our AD count is associated with the birth of Jesus Christ. It must have started with zero at the time of the Jesus's birth and is therefore similar to count of our age. Then, similarly to your every tenth birthday opening a new decade, January 1st of 2000 will be opening a new Millennium.

Turns out I was wrong. According to Wikipedia, "Most historians agree that Dionysius nominated Christ's birth as December 25 of the year before AD 1. Therefore, there is no year zero in this scheme, so the year AD 1 immediately follows the year 1 BC." This means that our AD year count is more similar to the vodka bottles count, starting with 1. And like the 10th bottle is the last one we drink from the first crate, year 2000 is the last year of the old Millennium and the new Millennium is starting on January 1st 2001.

The authoritative professor was absolutely right and brilliant in the simplicity of his explanation!

What is wrong in Ma & Pa Kettle Math?

2010-10-14T21:36:00.004-04:00

This fun video can easily convince you that 25 / 5 = 14.

Still doubting? Good!
We are being cheated right under our eyes. Ma & Pa use correct multiplication and division rules but freely mix and match digits in singles and tens places.

The last trick is the easiest to catch.
14
14
14
14
14
--
25

He adds all five 4th obtaining 20 correctly.
Then adds all five 1th, but places this 5 into the singles digit instead of the tens one.

This is what he should have got:
20
5
---
70

Mom does the same multiplying (I use 0s to kep digits in place):
14
05
--
20
05
--
25

when 5 should go into tens producing 70.

And in the very first argument Pa again puts 1 and 4 next to each other producing 14 instead of adding them to get 5.

Watch it a few time stopping after each math machination and you will see that 25 / 5 is still 5 and Ma and Pa are amazing actors.

Waka-Waka. Your turn to play.

2010-10-06T21:42:00.029-04:00

It is a special treat for me when you share your own life-math puzzles, and a pleasurable challenge if you ask me to help you solve them.
Life is a Puzzle. Let's solve it with numbers.
We have a wonderful community of readers, with diverse backgrounds and with various experiences, and together we can crack almost any problem.
Case in point: Last week, a girls' soccer coach and a reader, Lauren, asked me to help creating a play/rest timetable for her team. This is a fun real-life math challenge to share with your family and friends.

Image from Flckr distributed under the CCL

Lauren wrote:
We have 11 girls on our soccer team.
The games are split into 2 halves, each one is 30 minutes long for a total of 60 minutes in a game.
3 of our girls play goalie
There are 5 girls on the field and 1 goalie per shift for of a total of 6 girls on the field at one time.
The goalies should play equal amounts of the game and play consecutive shifts without subbing until their shift is over. After playing goalie or before playing goalie, the girl should be on the field as one of the 5 playing.
Each of the other 8 girls on the team who are not playing goalie, should play equal shifts on the field, alternating being in the game and being on the sidelines.
How many shifts do the 8 girls get and how long should the shifts be?
Also, how many shifts do the 3 girls get in goal and how long should their goalie shifts be?
Is there a solution to this problem?

As Shakira is singing to the World Cup players:
The pressure is on
You feel it
But you've got it all
Believe it!

Let's see. We have 60 minutes in a game overall.
At any given moment there are 6 players on a field. So, we have 60 x 6 = 360 playminutes total to divide among players.
60 of these minutes are goalie minutes and the rest 300 are regular playing minutes.

Start with dividing goalie time, as it is easier:
60 mins, 3 goalies, to be fair we need 60/3 = 20 mins for each goalie at the goal.

Now, we have 300 regular playing minutes. 8 players that should get equal amounts of time on the field and plus it seems that we should leave some play time for the goalies to run around with a ball. Lauren doesn't say whether each of 3 goalies should play the same time as regular players, so we assume that this is not a requirement.
Assume that regular players each play time T1 and goalies each spend T2 min running in the field.
So we have:
8 x T1 + 3 x T2 = 300

There are actually many solutions to this.
Pick any reasonable number of minutes as T1 and get a matching value for T2 from this equation.
The easiest will be to let each of the 8 regular players play a full half = 30min.
If T1=30min
8 x 30 + 3 x T2 = 300
240 + 3 x T2 = 300
3 x T2 = 60
T2 = 20

Each of the 3 goalies will then spend 20 min at the goal, 20 min playing in the field and 20 min resting on the sidelines.
Each of the 8 regular players will then spend 30 min playing. 4 will play first half and 4 second half.
Easy and fair.

But what will happen if somebody gets sick? How should coach Lauren re-adjust the calculations on a day of the game based on the show up? Assume we have G goalies and P regular players. How can we divide the game time using coach Lauren's description of fair play? Submit your thoughts in the blog post comment.

Image from Flckr distributed under the CCL

Aside from sport, similar scheduling tactics is applicable to many other domains such as dividing a time share property, traffic control, shift management at the restaurant and even chores at home.

Math Support for the Old Testament

2010-09-17T10:45:00.009-04:00

This story has been paraphrased from a wonderful book “Blood Matters” by Masha Gessen.

In Judaism, Cohanim are a priestly caste endowed with a set of particular privileges in synagogue and burdened with a set of restrictions in ordinary life. The Cohen lineage is paternal (Cohen is the singular and the Cohanim the plural form in Hebrew). All Cohanim are believed to be the sons of Aaron, brother of Moses. Moses was supposed to receive the priesthood along with the leadership of the Jewish people, but when he argued with God that he should not be the leader, it was given to his brother Aaron. Aaron received the priesthood along with his children and any descendants that would be subsequently born. See the illustration below of Moses ordaining Aaron. Keeping in mind that the Old Testament's chronology for the exodus occurring approximately 1500 BC, this all would date back around 2010 + 1500 = 3610 years.

Only recently, the belief that the Cohanim are all descendants of a single predecessor has been confirmed with the help of genetic testing and… math. Here is the story of how it happened.

Fifteen years ago, an Israeli doctor, Karl Skorecki, who is an Ashkenazi Jew of Cohen lineage, noticed a very differently looking Cohen of North African origin, a Sephardic Jew, in the synagogue and wondered whether all Cohanim’s DNA carries traces of their common predecessor. As Cohen lineage goes from father to a son, and males receive the X chromosome from their mother and Y chromosome from their father. Y chromosome’s DNA would be the one to have the presumed “stamp” of Aaron.

Skoreski contacted an Arizona geneticist by the name of Michael Hammer, who had distinguished himself in Y-chromosome research. It was a perfect research hypothesis: simple and verifiable, and Hammer loved it. Together, Skoreski and Hammer analyzed the DNA of a large set of Jewish men who have been told by their fathers (and usually by their last name) that they are Cohanim. Since then this research has been repeated a number of times and extended following the significant progress in our ability to decode DNA.

And indeed, all these studies proved that about 46% of the men claiming to be Cohanim are descendants of a single male – presumably Aaron. They also were able to determine when this common ancestor lived. To do this, they looked at the level of relatedness – or, rather, the degree of difference – among the Y-chromosome DNA of his various descendants. The difference is measured in mutations, because the rate that mutations accumulate on the Y chromosome is relatively constant. Using an estimate of the average time it takes a mutation to occur, the researchers calculated that the common ancestor had indeed lived somewhere around 3,000 +/- 1,000 years ago. In other words, the man, presumably Aaron, the first Cohen, may have lived during the exodus from Egypt. More than this, statistical analysis revealed that this common ancestor likely have lived in the Near East, as Cohanim DNA patterns contain similarity to those of Beduins, Yemenis and Jordanians. Just like that - science meets the Old Testament.

To read more about this, see Wikipedia and the latest article by M.Hammer and his colleagues.

The Refinancing Dilemma

2010-08-26T11:46:00.024-04:00

It used to be that a small talk would start with the topic of weather. This month, however, it is: “How is your refinancing going?” You will bond in an instance and also find what today’s rates are. Like a circus animal that is eyeing a treat it could snatch after hard work, I motivate myself to think about refinancing by visualizing all the wonderful things I could potentially do with the money saved by it. $100 per month – an extra shopping trip, or let it accumulate to a $1,200 per year – you got a small family vacation. Wait a few years and use this for a new kitchen. Definitely worth investing a few hours to play a math game with your lender and then copy and sign the pile of paperwork.

Here are the rules of the math game:

you prefer to pay less than you already do on a monthly basis

you do not want to increase the overall sum of your payments from now till the end of your mortgage or sale of the house

with every year you own more and more of your home, you do not want to significantly slow this process (named equity buildup), and in fact you probably prefer to speed it up

With the rates falling so low throughout the last year, we all probably got some sense to obtain fixed mortgages. If you plan on staying in your house for longer than the fixed part of your adjustable mortgage is, you may want to grab a fixed one.

So, you have likely refinanced in the last year to a fixed rate or just a better rate mortgage. Now, rates fell again. Should you refinance? Refinance now or wait? This has been the main topic of discussions in my neighborhood, while we walked the dogs, compared whose grass is greener, or watched the kids play outside. We read some articles; we talked to our mortgage lenders; we bounced around some numbers. Here are the refinancing basics and some tricks we learned along the way. Neither of this should be taken as a direct advice but rather some food for thought.

What you would need:

Call your lender to find out your current rate, how much you borrowed, how much you currently owe, and how many years you still need to pay.
For example, say you borrowed a $417,000, for 30 years, with a fixed rate of 4.875%
1.5 years later you still owe $407,000.

If you think you may owe more to your bank than the current price of your home – beware of appraisal, the bank may not allow the refinancing. Check this before going any further.

Ask your lender or shop around for the best options for your refinancing.
For a $407,000 amount the following options may be considered this week:
Option 1: 4.25%, 30 year fixed with closing costs of $2,600
Option 2: 4.375%, 30 year fixed, no closing costs
Option 3: 4.375%, 25 year fixed
Option 4: 4.25%, 20 year fixed

Start comparing your current mortgage and each of the alternative options. See my colorful table below. I used Bloomberg mortgage calculator to fill in details for each of the mortgage options.

Click on the table to open full size.

All the dollar amounts marked blue in reality are smaller than they are. Part of them is tax-deductible. See more details below.
Refinancing costs unfortunately are NOT tax deductable.
The key question in refinancing is how long you think you are likely to stay in the house. Rows 6, 7 and 8 describe a few possible scenarios. In many cases, one refinancing option is better in the short term but another on the long term.
The numbers in lines 6, 7, 8 are obtained by pressing View Report button on the Bloomberg mortgage calculator. Look at the table in this report. What you owe is your Ending Principal Balance, last column of the report.
What you paid is: (monthly payments) x 12 x (number of years).
Number in line 9 is your total payments. It is listed at the top of the report as well as inside the Bloomberg’s calculator GUI, below the monthly payments.

Now, let the math game begin:

Shall you consider refinancing? What option to choose

In general, the more you owe and the longer you still need to pay, the more it makes sense to refinance. Smaller interest rates shrink your interest. And in the first years of your mortgage term, 70-50% of your payments are part of the interest. Once you get to the tail of your mortgage, you are mostly paying principal and shrinking the small interest fraction by extending your payout time doesn’t make sense.

Shorten your term:

If you still have a large chunk of your mortgage to pay, you are very likely to find a profitable refinancing option in the current environment. Compare each of the candy-shop colored columns with the yellow one (your current mortgage). Note row #4 (monthly payments amounts), row#5 (number of years to pay) and #9 (total accumulated payments). If you are doing well financially right now and can afford paying a bit more monthly, you can get advantage of the low rates and shorten your mortgage term. In our table these are last Pink and Grey options. Look how significantly it saves us in the example on a long term: $150,000 before tax! Monthly payments are higher but many people get part of them back in taxes. The actual payment difference is: (Payment difference) x (1 – tax rate)
In the example, if we assume 28% tax bracket, the actual payment difference between yellow and grey options it is:
($2,520 - $2,206) x (1 – 0.28) = $314 x 0.72 = $226.08

Save monthly and in the accumulated payments:

If you are struggling financially, you can use advantage of the low interest rates and refinance with the same or slightly longer term, reducing your monthly payments. In many cases, it will reduce your total accumulated payments as well. In the example, these are options green and orange, 30 year fixed rate with and without closing costs.

No closing costs vs. closing costs options:

The no closing cost option is almost always your best bet if you are not certain you would stay in the house for the whole term of your mortgage. Interest payments are tax deductible. Closing costs are not. Closing costs are like a Costco membership fee – worth paying only if you know you will get the money back in savings throughout the term of use. What if you need to move? Life is unpredictable: job relocations, family dynamics, kids’ needs or new hobbies. You loose if you move soon after the refinance. How soon is this soon? Calculate how many years will it take for you to recover these up-front closing by comparing closing costs option to a no closing costs:

Months to recover = Closing costs / monthly savings

Monthly savings are = (Difference between mortgage payments) x (1 – tax rate)

In the example, the difference between green and orange mortgage payments is $30.
Monthly savings are = $30 x (1- 0.28) = $21.6
Months to recover from the upfront $2,600 closing fee =
$2,600 / $21.6 = 120.37 months (that is over 10 years)

Only after these years you will start seeing the benefit of closing cost as compared with the no closing cost option. So, consider this option only if you certain that you will stay in your house for at least as long and if you believe interest rates won’t be falling anymore tempting you to refinance again.

Wrapped closing costs trick:

If you really like your closing cost option but don’t have any open cash to pay for the closing costs, check this interesting trick: instead of paying closing fee up-front, borrow more equity and use these extra for closing costs. This Option is described in the blue column of our colorful table. If you are still able to reduce your monthly payments as well as overall accumulated payment and plan to stay long in the house – consider it. Why stay long? Because if you sell or refinance in a year or few – you will still owe to your bank these extra closing costs you wrapped into your mortgage. Therefore, this option is quite an opposite of pink and grey – instead of speeding up, you slow your equity buildup on your home.

Invest then pay off:

With the interest rates so low and interests being tax deductible, you are paying very small rates on your loan. For example, on a 4.25% loan with a 28% tax rate, you would pay only 4.25% x (1 -0.28) = 3.3% interest. So, for every $1,000 bank offers you today, you would have to return only $1,033 in a year. There are those of us who are confident that they could generate much higher profits in a stock market or other investments. For them, an extra $1,000 today with a 10% profit and 20% tax rate on a long-term investment will result in an 8% profit or $1,080 in a year. So, instead of using any extra money they may have to pre-pay or shorten the mortgage term, they can profit more by investing: $1,080 - $1,033 = $47

Do your math, consult with your lender, chat with your neighbors, and pick what works best for you. All the above is not a specific advice but only points to consider. Interest rates do deviate a lot, so save your comparison tables for the future. Whether you refinance or not, remember: you just cleverly saved by doing something or deciding not to. And as soon as you start receiving those monthly savings, celebrate and treat yourself to something memorable.

Acknowledgments: I appreciate the help of our lender, Connor Shortsleeve (leaderbank.com) and my neighbors in preparation for this article. Top image from Flckr, distributed under the Creative Commons License

Italian Notes

2010-08-12T07:13:00.019-04:00

One of the perks of being professor's wife is the opportunity to travel. This summer my husband was invited to teach in the North-Italian city of Trento and our family spent a few weeks immersing ourselves in the local Italian life. Here are my brief notes of this experience, which, as you might expect, are spiced with some simple math.

Evening strolls to the Piazza for gelato or sorbet. Pick apricot, fig or granny-smith-apple sorbet made with 50% fruit pulp. Or contemplate between creamy Stracciatella vs. Sicilian Almond gelato. Great news is that a fantastic Italian gelato chain called Grom is starting to conquer the US - first shops opened in New York. Their all-natural gelatos are scooped with an elaborate whipping technique and served at slightly higher temperatures than ice cream. Therefore, they melt fast - cone lovers beware.

I devoted my Master’s Thesis to teaching computers to recognize human faces, believing it to be one of the most complex problems of object classification. Little did I know about the varieties of Italian pasta. Penne, pennette, rotini, conchiglie, tagliatelle, mezzelune. The endless American supermarket shelves do not hold even 1/100 of the actual selection. To make things even more complex, some of the names vary by geographical region. But if you are in the mood for a salad, life is easier. No need to decide between Italian, Blue Cheese, Thousand Islands or Balsamic Vinaigrette dressing. Most of the places we visited only offer Insalata Mista (mixed salad) served with a tray of olive oil, vinegar, and salt-and-pepper.

Beware when ordering randomly from the untranslated restaurant menu. There is approximately 5% chance that you will get a horse or donkey meat.

The evolution of Pesto knowledge:
Stage 1, Ignorance. You don’t know what pesto is.
Stage 2, Naive. You buy off-the-shelf pesto in you local supermarket and proudly dump it on your pasta.
Stage 3, Smarty Pants. You figured out the pesto ingredients and now make your own home pesto in a blender or food processor.
Stage 4, Expert. You understand that the blender heats the olive oil and basil, which, for the trained, absolutely spoils the taste. You meticulously hand chop the ingredients for an hour or/and use mortar to slowly pound them into a flavorful mix.

How many types of coffee drinks can you name? Italian coffee machines offer at least 15:

But you wouldn't find cafe Americano on it. It is served only out of a pity in some tourist spots as a tiny Espresso with a bucket of hot water to dilute.

You can tell time of the day by looking at the café tables: coffee with milk is a breakfast drink, no milk accompanies coffee in the afternoon or post-dinner.

Espresso cups are cleverly positioned off-center on the saucers – leaving space for a treat or at least spoon and sugar.

Bambino Menu is translated as a Baby Menu but a 14" diameter pizza it offers could easily satisfy a teenager or 3-4 real babies. Pizza is consumed with fork and knife.

Beachwear seems to be twice more expensive than in US despite being half the size. Men wear Speedo. Women wear bikini. My American friend, wearing tank-and-skirt bathing suit, was constantly bypassed in the changing line because people assumed that she is already dressed.

It was very hot and we slept with the balcony doors wide open, which did not allow closing the shades. Every night I would wake up because of a strong light from the neighbor's balcony in our L-shaped building. I waited until the jet lag subsided, but the light was still bothering me. I tried eye covers, but the light rays found their way through. I blamed the neighbors from whose balcony the light was coming, and was waiting for them to return from vacation and turn it off. They did come back, but the light was still shining away my dreams. It turned out to be an externally attached street-balcony lamp. I tried visualizing my enemy - sharp arrows of light thrown directly into my eyes. Wait a minute, can I create an obstacle? Can I hide behind a shield? A simple geometrical breakthrough! What if I can move the bed a bit (those light European beds from the 70th are so easy to move) so that the new straight line between the pillow and the light will pass through something else like balcony door or a wall that will block the light. Eureka!

The trip to Trento took over 24 hours - on two planes, two trains, two cabs - it is a small university town not yet discovered by hoards of tourists. One day we open our apartment door and bump into an English-speaking couple from across the floor. Where do you think they are from? Turns out that our homes in MA are just 1 mile apart! What are the chances? If we assume that everyone in the US has equal probability of traveling to Italy, and that anyone living in our assigned apartment complex is from US, then the chances would be:
Our town population/total US population
30,000 / 307,000,000 = 0.0000977 or 0.00977%
Doesn't look realistic. Aha, let's take into account that all those American visitors living in our apartment complex actually came to work in the university. And that Boston has the highest concentration of academics in the country, and that many academics live in our suburb. Then chances would look very real.

"Mommy, Daddy - are we there yet?
How much longer is the ride?
When are we supposed to arrive?"
If this sounds familiar, here is an answer that works magic:
"Look at this road sign. It says that our destination is 8 miles away. Now, let's check the speed: 65 mph. Can you figure out how much longer?"
Then, make sure to wait 5 minutes and tease your kids. "Are we there yet?"

And once you prime your kids’ brains with simple daily math, they start noticing or even inventing it themselves. Our daughter posed a math puzzle after visiting a real Medieval Castle. She was also probably cued by our "what she needs to do for her allowance" discussion.
"These guards that lower and raise the castle doors every morning and evening to prevent enemy entry at night. If they earn 1 euro for each of these operations. How much would they earn in a year?"

When I showed her how to add 365 to 365 in a column, she thought this to be the coolest trick and to my sincere amazement demanded more of such additions to play with. It is all in a context!

Some of Italian frescoes, such as this one by Andrea Mantegna (1430-1506), are fascinating studies of perspective. Imagine painting this, without a slide projector, directly on the ceiling.

Italy is lavishing with pleasures to offer: emerald lakes, warm seas, cultural events for any taste, human history, celebrated food, sophisticated fashion. But eventually it is always great to be home. I am still coping with the jet lag that makes me drowsy and sleepy. Is the magnitude of suffering proportional to the distance traveled? Do you think it is a linear or an exponential relation?

You - in a Puzzle!

2010-07-22T09:40:00.003-04:00

Take a look at these five interesting puzzles about some people you may recognize.Then - put yourself in a puzzle! Just enter your name and the names of a few friends below and enjoy solving your own mysteries.

Click on the image below to play:

Facebook Birthdays: Virtual is the New Real

2010-06-16T22:33:00.005-04:00

Image from Flickr

Virtual is the new real. How else would we know what really happens to our neighbors if not for Facebook? Karen, who just had her third baby and is chronically sleep deprived, entangled in her sheets and fell out of bed at night. But no worry - bruises only. Irina's husband is snoring and she is wondering what she can get him for Father's Day to ease her nights. Rich and Linda just found a snake in the back yard and killed it. Stephanie is thrilled that she is managing to breastfeed and poop at the same time, or maybe it was her older toddler that was pooping while she was breastfeeding in the same bathroom. You see, these are things that my neighbors would never discuss with me in person. They may mention the other events - renovations, vacations, work, but not these humble but honest and meaningful life moments.

Most of my neighbors would not announce their anniversaries or birthdays when meeting each other in person, but thanks to Facebook, we know who is celebrating what, in what outfit, what they are eating and whether they will be having a hangover the next day. The angels of Facebook created a gentle birthday reminder app that shows a little present icon and a name of those from your Facebook friends who are having a birthday today or tomorrow, or the day after tomorrow. If you want to show that you care, you can politely note "Happy Birthday!" and skip the rest of the wishes.

My neighbor (same person who fell out of bed) and who has 532 friends on Facebook sent me a message (on Facebook, of course). She asked whether with so many friends it is guaranteed that on any given day she may open Facebook, she'll see such friends' birthday notifications.

What do you think?
The Math Mom title comes with responsibilities...
Sometimes it is easier to solve questions that start with "What are the chances that something happens" by looking into the opposite scenario: "How likely is it that something does NOT happen." Especially if there are a lot of people that are involved in this "something". Like 532.

So, the chances that on a given day you will have a birthday notification are:

1 - (chance that NO ONE has a birthday on this day or next day or day after)

this will be:

1 - (Laura does not have birthday today-and-next-two-days AND Daniel does not have birthday today-and-next-two-days AND Caroline does not have birthday today-and-next-two-days AND continue with all your 532 friends)

(We assume here the default setting when Facebook starts notifying about a birthday two days in advance, but you can configure this and then adjust the formulas correspondingly.)

For any of your friends (Laura, Daniel, Caroline etc) the chances of not having birthday on any one specific day are (365-1)/365=364/365 as there are 365 days in a non-leap year.
The chances of not having a birthday notification today, tomorrow or day after tomorrow are (365-3)/365=362/365.
We assume that birthdays are uniformly distributed throughout the year, that is not exactly true but good enough for a large sample of Facebook users from all around the globe.

So, we will have:

1 - (362/365 AND 362/365 AND .... ), with 362/365 repeated 532 times for all the friends

when you compute the chances of two independent events happening together (A AND B) you multiply the probability of one event by the probability of the other: A x B

Therefore, we will get: 1 - (362/365 )⁵³²

this will be:

1- 0.012 = 0.9876= 98.76%

This means that there is almost 99% probability or 99 out of 100 chance that my neighbor will see a birthday icon on any given day she opens Facebook. Sounds like she may need another application to send those daily "Happy Birthday" messages.

But if you are not so popular, and have only 50 real Facebook friends, you may skip the birthday corner. The chances of seeing the icon are only 34%:

1 - (362/365)⁵⁰=
1 - 0.66 = 0.34 = 34%

So, let's not be afraid to get relaxed, open, and honest in our real virtual worlds. We'll see ourselves surrounded with friends and likely to be celebrating birthdays daily.

Cartoon by Peter Mesnik from Beyond The Gate

The Math of Your Home Size. Part 2: Your Guide to Measuring and Expanding Your Home.

2010-06-02T12:16:00.021-04:00

You may want to start with Part 1: Make the Best of the Space You Have.

Ready to follow your expansion dreams? Here is your guide on what you can do. The guidelines vary from town to town, and could be adjusted at any time, but you need to know the basics in order to adapt to these changes and know what you can ask for. Sit together with an architect, and play with some math to see what space you need, how much you could spend, and whether your goal is to increase the official home living area. Then, you might consider consulting with an experienced real estate agent or appraiser to see what your home’s increased value would be after the expansion.

In order to check your expansion options, you need to start by figuring out your current home size. It’s like the starting point of a dieting journey, except that here we hope to gain rather than lose. How do you measure it? Pretty intuitively: take your home plan if you happen to have it (such as the one displayed below) or create one by drawing a bird's eye view of your home. Go around the wall perimeter, measuring and marking every wall. For homes you usually measure around exterior walls, for condos – around the interior. Adjust the shape if necessary, then split it into rectangles, compute the area of every rectangle (length of one side times the length of the other), add it all up, and voilà. But, wait! Do you count the bathrooms? What about the staircase? And the back porch that you just turned into a sunroom? Is it counted as part of your square footage?

These mystical GLA initials next to the square footage in your mortgage papers or appraisal report, as it turns out, mean Gross Living Area. The GLA is defined as the total area of finished, above-ground, residential space, excluding unheated areas such as porches and balconies. See the sample home (Dover design) and the floor plans from the CadSmith Studio LLC

All the areas highlighted in pink and blue are counted as part of the GLA.

1st floor: 28' x 38' + 16' x 24' = 1064 sq ft + 384 sq ft = 1448 sq ft
2nd floor: 28' x 38' = 1064 sq ft

Total GLA: 1448 sq ft + 1064 sq ft = 2512 sq ft

Here are home areas in question that are counted as part of the GLA:

bathrooms are in

halls are counted

closets are in

even the dead space inside stud walls is calculated as living space (except for the eaves in cape style and some other styles of homes)

staircase area is counted twice – once for each floor

porch turned into a sunroom is counted if heated and used year around

high-ceiling finished and heated attic is counted

Here is a list of areas NOT counted in the GLA:

open foyer -- essentially a two-story hole that is not counted twice, but just once

porches and balconies (unless heated and used year around)

finished basement – surprise, surprise - is out of the GLA unless it is completely above ground level and used year round; however, an appraiser will usually include them in a separate category of “basements and finished areas below grade.”

low-ceiling attic (see details below)

Ok, we know what we've got now, the current GLA. Ready to expand?

First, beware: increasing your GLA will likely lead to greater home value but also will increase your taxes.

According to broker and appraiser Sam Schneiderman, President of Greater Boston Home Team, the least expensive way to create an extra room or two of living space is usually to finish your attic.

Will a finished attic increase your GLA? Yes, if you can reach it from inside your home, without sneaking through the garage or romantically ascending exterior stairs, and if it fits your town's legal room height requirements. In some neighborhoods, for instance, the portion of the attic that has at least 5' height and an average ceiling of 7' is counted as legal GLA. Check your numbers carefully with your local building inspector and appraiser, keep it legal, especially if you plan on placing your mother-in-law or live-in nanny there.

If your attic is too small or non-existent, consider insulating and heating your garage or basement. Remember that the basement most probably won't add to the official GLA numbers, unless it is completely above ground. But then do you call it a basement? :)

If you do decide to expand, then you have to know a bit of expansion math. Luckily, it is quite logical. Unluckily, each town has its own rules and may have their own methodology when it comes to calculations. The most common rules according to Mr. Schneiderman are:

Setback Rules: To keep peace with neighbors, guard privacy, and avoid fire hazards, town building codes require some minimum space between your house and your neighbor’s house. They are called setbacks and vary at the front, side and back of your lot. Setback space is off limits for construction. You can find setback requirements in your town's building code. Then, take your lot plan and mark these “buffer” areas off-limits for any new construction. (If you can only expand by encroaching into the setback areas, ask the town building inspector about the likelihood of applying for and receiving a variance.)

Max Lot Coverage Rules: Still have some space left on the lot plan to expand sideways or up? Check your Max Lot Coverage % in your zoning district. There could be a limit on how much of your lot can be covered by your house. If applicable, maximum coverage rules are calculated as follows:

House Footprint / Lot Size <= Max Lot Coverage Therefore, you can expand as long as: (House Footprint + addition) / Lot Size <= Max Lot Coverage Your house footprint is just what it sounds like – square footage of the print your house makes on the lot.

Floor Area Ratio (FAR) rules: Most towns usually limit how much living area is allowed on lot of a given size. The limits can vary depending on the neighborhood or zoning district. Check the Floor Area Ratio (FAR) to see if you can expand your GLA. If applicable, the FAR rule is typically:

GLA / Lot Size = FAR

Therefore, as long as:
(current GLA + addition) / Lot Size <= Maximum FAR you can expand without a variance

Maximum Building Height Rules: As you may have suspected, you can't build an Empire State Building on your lot (because town fire equipment may not have Superman on staff to reach the top if it catches fire!) Most areas have a maximum height limit on buildings. Again, your town's building code is your bible. In my neighborhood it is 2.5 stories and 35'.

So, learn your expansion math dance with your architect. Ask your town's building inspector to approve your steps (calculations). Only after that, hire your contractor partner and get started with measured and riveting twirls. It all requires team work. Don't skip your steps.

While it may seem like a lot to consider, the math of it all is elementary. Beware that rules vary from town to town, and are updated from time to time. Your bank appraiser's calculations may disagree with the town building inspector's, but now you know what they are talking about and what you should be asking for. Enjoy playing with these numbers to plan, dream and negotiate your home's expansion to meet your needs and realize your dreams.

Meredith and Derek dream up the plan for their house on Grey's Anatomy

P.S. I would like to thank Sam Schneiderman of Greater Boston Home Team, Bonny Smith of CadSmith Studio LLC, Needham (MA) building inspectors as well as Anat Yosfan of AnatYDesign for their help in preparing this article.

The Math of Your Home Size. Part 1: Make the Best of the Space You Have.

2010-05-27T21:13:00.014-04:00

Our homes are museums of math. In every corner, there is a fascinating math puzzle awaiting you. Math can help you measure the real square footage of the space you rent or own; shock you with a revelation that your floors are not level; or advise on how to make your living room more livable. You can play with math to negotiate a larger closet, a taller toilet, or a homework table in your kitchen. For those dreaming of expanding their riches, math is a great adviser on whether to finish your attic, garage or basement and if it would actually increase the official square footage. Math can greenlight a large addition or pronounce a sad verdict on your expansion dreams.

In the following two columns, let's explore how math can help you better utilize the space you already have, and if desired, also guide your expansion plans.

Part 1: Make the Best of the Space You Have

Ikea Image from Flickr

Whether you occupy 500 square feet or 5,000 square feet, you are likely wishing you had more space for storage, work or entertaining. The best home advice for those running out of space: instead of figuring out what you can add to make things work, think of what you can re-arrange to make things work better. By merely shuffling your furniture and decorations around your house, you can create the refreshed and practical environment that you are looking for.

Think statistics: is your living room empty 25 out of 30 days a month, while you work out of a stuffed tiny home office? Of course you have hopes to populate this living room with book reading or chess playing kids on a daily basis, even if past experience shows they are more likely to be found in the family room watching TV or doing homework in the kitchen. Don't wait for your life style to fit your home, make your home fit your needs today!

To get a clear picture of these needs you may want to create an occupancy chart of your home. List all of your rooms and family members, and then add up the hours that each person spends in each room every day. Analyze the results with your family. Use this 95% idle living room to create a glamorous CEO office for yourself, make dreams come true with a pool or sewing table, or expand the busy dining/homework area.

Once you start thinking about how to use your space more efficiently and fairly, it is guaranteed to get addictive. Let's do some basic geometry. If your shoe size is 10 and up, the total square footage covered by a pair of your shoes is at least two times the square footage covered by the shoes of your 1st grader. Feel free to use this argument to snap up at least twice the space in the family shoe closet.

Not everyone is created equal. If your height is above average, you may have noticed that you need to bend 90 degrees over most bathroom sinks. It is, however, a nice stretching exercise, after sitting on that dwarf-ish toilet seat (14” height is the most common). But how can you check that your hands are clean when you bend so low over the sink you are in danger of summersaulting into your mirror image? On the East Coast, such low toilet seats and vanity varieties are surprisingly very popular, either due to a low average height of the original settlers, or because we put our kids' needs way ahead of our own. Consider treating yourself to height-appropriate bathroom furniture during the next round of renovations.

Have you always assumed that your floors are straight until your kids showed their toy cars sliding in one direction? Have you ever hung a picture using a level and then discovered that it is tilted with respect to the walls? This may be a hint that your floors are not level and/or your walls are not perfectly vertical. Not to worry: align the picture with the closest wall, forgoing level and gravity. Similarly, use pads to realign your furniture. And feel free to blame such floor tilts for the natural accumulation of body fat midway up front or at your bottom – its your body's attempt to balance your center of gravity.

Are you looking for creative ideas to materialize more storage space? Zoom in on your usability chart. Look for some idle corners or niches. Our ingenuity sparks when facing a real challenge. Do you happen to have both, shower and bathtub, in your bathroom? An unused bathtub and everything above it could be turned into a nice size closet of around 5' x 2.5' x 7' that could hold your linens, towels and travel luggage filled with winter sweaters.

Image from Flickr

Be generous with mirrors as they make rooms appear brighter by reflecting the light instead of absorbing it as walls do. They can fool our perception of space, making any room look larger. But remember the laws of reflection – if you can see something, this something can see you – mirrors can limit your privacy. To increase space not only perceptually but also physically – you'll have to learn the intricate steps of expansion math dance with your architect, town building inspector and your contractor. But this is another story - for next week.

To read more about mirror reflections click here.

Farewell Martin Gardner - a guest blog

2010-05-24T08:42:00.008-04:00

By Nicolas (Paris, France) - a friend and puzzle-solver.

Martin Gardner just passed away. If anybody influenced me into science and math, that should be him.

Martin Gardner was a journalist who wrote columns in Scientific American, short witty pages on fun math topics. He is the one who introduced things like Conway's game of life to the general public, raising overall interest in mathematics as a fun thing to do.

Gardner entered our family life in the 70s when my father brought back the latest issue of Scientific American. I remember folding my first hexaflexagon when I was maybe 6 or 7 and bringing it to school to wow the teacher. I did not understand the concept of Moebius stips then but I certainly got the fun part of it. My parents subscribed to the French
edition (Pour la Science) later on and we would get Gardner's columns every month with great impatience. From there on we bought all of his books, did his problems, read the solutions and discovered we had sometimes invented better ones. We played the game of life with pen and paper until we got our first programmable pocket calculators and later our own kid computer (a Sinclair ZX81). His columns were always bringing something new, teaching us tricks in algebra, geometry, paradoxes and logic, you name it.

I recently discovered that Gardner has also helped MC. Escher achieving the incredible fame he has today among scientists, and probably inspired Douglas Hofstadter into writing "Goedel, Escher and Bach", another inspirational book to which I probably owe my scientific-oriented career.

30 years later, I still own a copy of Martin Gardner's "Aha!" (dated 1980) and read it to my kids every now and then, enjoying the mathematical seeds he planted into my mind as a kid and propagating them one generation further.

Some of the gems found in Aha:

The Loch Ness monster's length is 20 meters plus half of its length. What is his total length? Click to submit or read the solution.

Try to guess my phone number. You can ask me 20 questions and I will answer by Yes or No. How can you do that? Click to submit or read the solution.

These two miners come out from a coal mine at the end of their working day. One of them has his face covered in soot and the other one still has a perfectly clean face. They both glance at each other in the elevator bringing them back to the surface. When they get home, the one with the clean face cleanses himself thoroughly and the other one does not. Why? Click to submit or read the solution.

[This latter story is much more profound than it looks. It took me years to actually see its relevance in the way we perceive our own selves compared to other people.]

Farewell Martin. Your achievements will live on forever.

References:
Interesting video about Martin Gardner
Martin Gardner's page on Wikipedia
Conway's game of life
Sinclair ZX81
Goedel, Escher and Bach: an eternal golden braid
Aha! Insight

You can find some of the Martin Gardner's books on our Math Resources for Adults page.

Folding and Packing Experiements

2010-05-13T14:39:00.009-04:00

How do you define optimal packing? For your luggage or a closet shelf – optimal packing is, first of all, packing that keeps all items safe (not broken or wrinkled). Second, it is packing that takes the least amount of space. And third, it is packing that allows one to easily find any single item and take it out without a major disturbance to other items. Packing problems extend well beyond clothes – just think of all the large boxes you’ve received, ordered through online commerce, that contained small volume items and lots of Styrofoam peanuts or bubble wrap or air bags. Dissertations have been written about optimal packing. Scientific wars fought and large prizes awarded. People have been inventing square tomatoes, space bags, nested dolls (perhaps), and new polymer materials - just to minimize the packing headache.

No wonder that last week many of New York Times readers were captivated by an on-line slide show demonstrating tips for efficient luggage packing from an experienced flight attendant. Could you manage flying with only one carry-on bag, even if your trip is ten days long? Heather Poole shows that it is not only possible but also easy: 10 Days in a Carry-On

She rolls her clothes instead of folding them! What an ingenious idea. It does work well for sleeping bags, why didn't we think of using it for clothes? Tight rolling pushes the air out of the clothes, they wrinkle less, and it is easier to take out one piece without messing up the others. I decided to check how this would work in my towel closet and recruited my daughter to help :

Our twenty towels folded:

Same twenty towels rolled up:

It looks like we have gotten around 15-20% of empty space in the rolled version. It is also much easier to see each towel. My daughter could easily pull her favorite one from underneath without turning the whole closet into a battlefield. In a folded scenario some tiny towels are obscured – I could spot only eighteen out of the twenty. Combining small and large towels on one shelf is much easier when they are rolled. No collapsing towers of folded towels of various sizes. And most importantly, my daughter and I agreed, rolling was much more fun! One small change in our habits - one large optimization step for our closet.
So, let’s Roll n’ Rock.

Try another story from The Math Mom: Why mirrors flip left-right and not up-down.

Fun with Statistics

2010-05-06T12:06:00.035-04:00

Statistics is a science that makes order out of a mess. It studies data, detects and analyzes patterns, and utilizes them to make predictions. We use it daily, packing or ordering school lunches for our kids and analyzing past experiences to predict what they are likely to eat or not. We rely on it when making a credit card payment over the phone and trying to convince the voice recognition system that we would like to use the same bank account on file. However, I always find myself in the outliers: "I said Yes, I didn't say ten dollars!"

Statistics is also a touchy topic. A slightly different analysis of the same data or even a slightly different presentation of the same data could lead to vastly different results. Famous British intellectual and politician, Benjamin Disraeli, once said "There are three kinds of lies: lies, damned lies, and statistics." So, make sure to have your lawyer by your side when making any kind of statistical statement. And, be scrupulous when trusting any statistical results.

This week TED posted an entertaining short talk that utilized statistics to tell people how to give the best TED talk. What is TED? It is an online community of "riveting talks by remarkable people, free to the world." Every invited talk is video recorded and available online to enjoy, learn and rate. Sebastian Wernicke analyzed how seemingly superficial talk parameters. such as the color of power-point slides, the color of a presenter's clothes, and more logical parameters, like talk length and key sentences, affect TED talk rating. And he has a recipe for you: when invited to give a TED talk, wear blue or green, stick to the same color for your slides, talk long and talk about French coffee, happiness and the brain, and do not mention "The New York Times." Well worth watching (8 min):
Sebastian Wernicke: Lies, damned lies and statistics (about TEDTalks)

Top image by kgradinger, distributed under Creative Commons license

Why are we behind in math? Here are some answers.

2010-04-22T21:32:00.010-04:00

My kids (1st and 4th grade) do at least one and two hours of math at school correspondingly and then at least 15 min as part of their homework. Probably the same as I did in their age. We go to science museums (and these museums are always delightfully packed, by the way). Kids enjoy math computer games that I buy for them. Some of the children's shows they watch on TV have math infused in them in very creative and engaging way. I keep hearing from neighbors and friends how their older kids spend hours on their math homework. Our kids' class sizes in US are, on average, smaller than in other countries my relatives or friends live. 40 students per teacher is not an unusual ratio there. In addition, we, as a country, have enormous knowledge capacity and amazing new ways of sharing it. Given all this, it sounds absolutely unbelievable that we are far from leading in something as basic as math.

Are we certain there is no mistake? Trends in International Mathematics and Science Study from 2007 compared 4th graders in 36 countries and 8th graders in 48 countries. The average score of US 4th graders was somewhere between 9th and 12th place, the average score of 8th graders in-between 6th and 11th place. See the tables. Not at the tail, but not at the head either. Other studies are unfortunately placing us further down. The Broad Education Foundation states that American students rank 25th in math compared to students in 30 industrialized countries. Our top math students rank 25th out of 30 countries when compared with top students elsewhere in the world. By the end of 8th grade, U.S. students are two years behind in the math being studied by peers in other countries.

Most of us are genuinely surprised and troubled to hear this statistics and of course we all have been wondering why is that happening and what we can do to lead in math. Just in time, last Friday's Science Magazine editorial written by math professors and education experts provides some answers. All the authors are members of the Council of Chief State School Officers (CCSS) Mathematics working group.

They write that US math performance is “compromised by a lack of focus and coherence in the curricula. Higher-performing countries teach central topics more coherently and in greater depth.” Therefore, they recommend selecting fewer grade-appropriate topics but devoting more time to study each of them in depth. The authors also mention a lack of national standards and curricula for mathematics. Each of the US states is currently free to select what should be taught and how to assess whether students have learned it. As a result, most of math workbooks are created as restaurant menus. Each containing five times more material than international school books, because states are frequently choosing different topics to cover. Editorial authors also mention that state math standards currently tell teachers what to cover, but do not define and measure how well kids lean it. A common assessment system may be of help. As the problem seems to be well-defined by the educators and scientists now, it should be possible to address it. Correct! Using the curricula of the high-performing countries, and taking into account research findings, the committee consisting of teachers, mathematicians, statisticians, math educators and cognitive scientists devised a set of new core standards. These new Common Core State Standards have been released last month with the hopes that they will be adapted by all states making the knowledge and skills expectations uniform across the country. In addition to this, the editorial mentions that consortia of states are already organizing a common assessment system for student achievement. And they helpfully finish with a statement that “the promise of an improved education system is thrilling.”

Top image from Flickr

Try something different from TheMathMom:
While educators are working on adopting new standards and curricula, you can work home front and share joy of math with your kids through these great card tricks.

TheMathMom.com in Redbook Magazine: time to put math and moms in the same equation

2010-04-19T09:34:00.006-04:00

TheMathMom.com has been featured as a Hot Blog in the May 2010 issue of Redbook Magazine - now available at the newsstands. Page 158, a big picture and one small paragraph... Redbook says that it is about time we put moms and math in the same equation!

Here is a full version of my Q&A with Redbook:

Redbook question: What inspired you to start a blog about the intersection of motherhood and math?

TheMathMom answer: I started writing my stories when I realized that most of my girlfriends do not share my passion for this subject, my perception of math as a reliable adviser in almost any life situation, a weapon of defense, a playful toy and a key to the future of our kids. One of my girlfriends, a Harvard graduate, encouraged me to share my stories, saying that she does not want to transfer her own math phobia to her daughters. TheMathMom's website illuminates and demystifies mathematics, showing that we all use it daily, we are much better at it than we think and we can have even more fun by applying it consciously. Understanding the universality and usefulness of math, we can present math to our kids as a candy through fun puzzles, encouragement, our own excitement and positive stereotyping. There are soccer moms and hockey mom, very similarly every mom can be a Math mom.

Redbook question: Can you give one or two of your favorite examples of how moms can use math to make their daily lives easier?

TheMathMom answer:
Example 1:
A year ago I was in a Manhattan store that advertised a 30% discount on top of 50% off. At the register I contemplated whether I should insist on the specific order of the discounts: is 30% after 50% be the same as 50% after 30%? And will these discounts together be same as 80% off? When you start figuring out the math, you realize that the order of the discounts does NOT matter.
Here is why:
30% off means that you pay 70% or 0.7 of the price
50% off means you pay 0.5 of the price
30% off after 50% is 0.7 x 0.5
50% after 30% off is 0.5 x 0.7
0.7 x 0.5 = 0.5 x 0.7 = 0.35 or 35% of the price, getting a 65% off
As you see these two discounts together will give you only 65% off, and not 80%. Still, 65% looks like a great deal.

Example 2:
Many of us have to figure out whether the financial aspect and logistics of working with two or three school age kids makes sense. Here is a modest estimation of the approximate amount of time kids will be off during a calendar year that should be taken into account:
3 kids x 6 sick days per year per child on average +
10 early closing days at school +
3 snow days +
4 holidays not observed by your company +
70 vacation days (summer + mid year)
= 105 days when children supervision is required

It is great to be prepared for all these days emotionally, financially and strategically. To know in advance what we are getting ourselves into before the school year starts. Is your work flexible enough to allow it? Can you afford hiring someone to take care of the kids for you? Finding, scheduling and paying for vacation camps and babysitters. Realistic planning can make our lives much less stressful.

Check out this useful kitchen advice from TheMathMom: You put a dirty plate in, you take a clean plate out. And you repeat it all around.

Don't transfer your math phobia to your kids! Five simple steps to follow.

2010-04-12T13:40:00.005-04:00

This short essay has been written by The Math Mom for CafeMom online community in March 2010.

So, you think you are bad at math. But what If I told you that you excel at it daily, usually without even realizing it. You do math when packing the dishwasher; when carefully carving the weekend schedule to fit around the kids' activities; when convincing yourself to buy expensive new boots because in a cold climate their cost-per-wear is reasonably low. Math is for everyone to enjoy, play and use. If you'll feel comfortable with math, you won't be transferring our math fears to your kids.
There are Soccer Moms, Hockey Moms, Baseball Moms. Could you be an enthusiastic and supportive Math Mom? No worries - you do not need to be a math expert to be a Math Mom. When was the last time you saw a Soccer Mom play soccer? Here are a few simple steps that can help you perceive math and present it to your kids as a cool tool, a toy, and a friend:

Encourage and expect high achievements. Success in any kind of learning is enhanced by encouragement and challenge, and inhibited by threats or shame. Think of how we challenge and support our kids on the football field or ski slopes. Remember those trophies that everyone gets just for trying? The pride we take when our child scores a homerun. Recall how we share the excitement of our kids’ successes in sports and work on their failures. Just adapt a similar mindset for math learning right there at your homework table.

Stimulate curiosity and search for answers together. Kids are naturally fascinated with numbers, patterns and logic. All we need to do is nurture this curiosity and expose our kids to the beautiful math of this world. Encourage questions and search for answers together on the web, in books, or ask around. What day of the week your birthday is going to be in five years? How does a 50 SPF cream different from a 15 SPF? What are the chances of getting a hot pink gum ball out of all the available colors in the Gumball machine?

Explain your reasoning. Have you been contemplating whether to order school lunch or pack one from home? Or how long will it take to heat five hot dogs in a microwave, if it takes 30 seconds for one? Our daily lives are full of math puzzles and we all became proficient at solving them. Pass this expertise on to your kids. Explain to them why it makes sense to wait until this toy goes on sale next week. Count together how many 20-cent lollipops will they be able to buy with what they just saved?

Avoid stereotypes. Beware of the movies that depict smart kids as unpopular geeks. Instead, point out those geeks polishing their statistical presentations in the roomy Business Class section of your next flight. Another old nonsense - boys are better at math than girls. A multitude of recent scientific studies in US and around the world proved that math skills are equal across genders, that culture is a factor in female math achievement and that girls’ confidence in math is dampened by parents' and teachers' gender stereotypes. Math is for everyone to enjoy, play and use. Don't let your child miss on this important one.

Positive Attitude. Math resources are everywhere around you: in the kitchen, at the supermarket, in your closet. However, remember that the most important resource is your attitude. Children are experts at deciphering our passions and priorities even when they appear to reject them. Sending your daughter to ask dad for help with her math homework implies that dads do math but moms do not. Making time to sit and patiently try to figure out together, teaches a great deal. No one knows all the answers, but it is crucial to give the confidence and to show how to look for them. This is your role.

You are the best math guru your kids can get. And they really need one, because proficiency in the universal language of math is a key to the global competitive world of their future.

Are we sure it's OK to be smart?

2010-03-24T11:40:00.018-04:00

Imagine Olympic athletes apologizing on TV for being muscular and agile. Or an engraving on a lipstick tube that reaffirms “It is OK to be beautiful.” Alternatively, try visualizing a clothing store ad luring us in with the words “Nothing is wrong with being well-dressed.” Sounds rather unreal. But how about the disclaimer on the back of a set of Brain Quest flash cards for kids that states “It's O.K. to be Smart!” Does it seem similarly ridiculous? This one is actually real. A great toy company feels that it should warn us not to worry about being a nerd. Are we sure it is OK to be smart? Are we serious that it is just OK???? Why don't we say: “It's cool to be smart!” or “It's awesome to be smart!” And why do we need this affirmation at all? Shouldn't it be obvious?

Growing up on the other side of the world, I was brought up to believe that it is as admirable to be smart as it is to be strong, and that exercising our brain is as important as exercising the muscles of our body. Smart kids were respected in schools and colleges, and admired as much as beautiful and athletic kids. Therefore I was especially surprised to learn how vastly different the attitudes are towards science and sports, here in the US.

It is fascinating to observe our infatuation with famous athletes, the unmatched popularity of school sports, the pressure we frequently put on our kids to succeed in sports, and the lengths we go to help them achieve it. Who among us has not risen early on a Sunday for 7am hockey practices, swim meets, or baseball clinics? Or maybe you’ve driven or flown to out-of-state competitions in gymnastics; or changed family summer vacation plans to match a 9-year-old’s football training schedule? Why don't we treat science and, math in particular, in a similar manner? Why has pampering our bodies become much more acceptable and admirable than pampering our minds?

Perhaps it is due to the materialistic reward that high achievement in sports brings to the famous few (usually by means of advertising endorsements). But among the self-made millionaires and billionaires, there are significantly more people with math and economic acumen than athletic prowess. And this is not only true on the level of super-achievers. On average, mathematicians earn $94,960 annually according to the most recent data from the Bureau of Labor Statistics. A January 2009 study of the best and worst professions by JobsRated.com, rated mathematician as the best career, taking into account work environment, income, employment outlook, physical demands, and stress (Ref 1).

Ok, the financial reward may be comparable, but it seems that becoming the next Shaun White, Michael Phelps or Lindsey Vonn is more achievable than making a fortune with math. We share our living rooms with these athletes daily via TV, we read their inspiring biographies, follow their advice on what to eat, how to dress or behave, and fantasize being just like them. Well, with only around 17,000 professional athletes in the US and with a total census population of around 308,788,000, chances of becoming one are only around 1 out of 18,000 or 0.0055%. Are those statistics pointing to the rarity of athletic success true? Take a look around. Chances are you are not familiar personally with a famous athlete. But you probably know someone on your street or in your extended family who made a million from a start-up company, very likely with the help of math. Check out the business class section on your next flight: do these people look like professional athletes or Internet entrepreneurs polishing the statistical graphs of their presentations? Sounds like math may be a ball worth catching.

Sports and math are universal languages anyone can speak and work in, anywhere on Earth. Similar to athletes who can easily train anywhere and hop from one country's team to another, those fluent in the language of math have the whole world open to them as well. While we speak different languages, pray to different gods, and celebrate different holidays, most people on our planet use Hindu-Arabic numerals and study the same laws of logic. A math degree expands the geography of a job search way beyond national borders. Just compare this to a law or medical degree that frequently requires re-certification even when moving from one US state to another.

It does look convincing that math could offer an impressive combination of financial stability and job security along with comfort and low stress. And perhaps you happen to have a neighbor with a Yale math degree who leisurely works from home, travels around the world, and drives a Lexus. You may also recall Olympic stories of horrific sports injuries while comparing them to the paper cut math could inflict on you. Considering this all, you are becoming more convinced about the advantages of a math career. But what about the road to these achievements? The hours, days or years spent solving abstract equations, the tears and frustration that frequently accompany pages of exercises? Turns out that math's learning and achievement processes are intriguingly similar to that of sports. You have seen those tears before, on the high school grid irons, ski slopes and skating rinks, but you probably reacted to them differently.

Take for example, the following statement about sports:

Rigorous consistent practice strengthens your muscles, enables your brain to memorize specific movements, perfects and automates the whole pallet of possible movement patterns. Your body gains strength. You get better at sports and enjoy it more.

Now, substitute math lingo for sports lingo:

Rigorous consistent practice strengthens neural connections, enables your brain to memorize specific tricks, perfects and automates the whole pallet of possible solution approaches. Your expertise grows. You get better at math and enjoy it more.

Sounds quite similar and logical, right? Then why are we so stressed when our children bring home one or two extra pages of math homework? Why are we at a loss for words when kids complain of being bored when asked to do long lists of abstract exercises rather than a few applicable word problems? We would not ask a football coach to explain why our kid needs to do 20 repetitions of monotonous exercises during practice four times a week just to play 15 minutes at the local town competition. And most of us accept that it takes a year of weekly practices and daily stretching to learn to do a gymnastic cartwheel properly.

Whenever my own kids complain about math homework being difficult, I tell them that this is great. Their brain “muscles” (neural connections) strengthen and grow when being pushed over the comfort zone. Easy homework is like passing the ball back and forth with your little brother. Fun and relaxing, but it won't improve your athletic achievements. No muscle pain, no gain.

Many of us intuitively discover these math-sports similarities. One middle-school teacher wrote to me about her idea for a “math warm-up:”

I did not have research to prove this, but I used to tell my students on the day of a test to get to math class early. I suggested that they use the few minutes before class began to do "warm up" problems, so their brain was ready for the test. I felt that without the warm up it would take longer for their brain to accommodate to the new information it had to process, and the warm up got their mind ready for the material they would now have to think about. I thought about this with your analogy, because it is the same for athletes warming up their body before they add intensity to their workout, but in this case, it is a warm up for a math test.

Both math and sports require a lot of monotonous, deliberate practice to succeed. Frequently we find our progress at a plateau and this is exactly when parents' encouragement and support are crucial. Perfecting one approach or taking a shot at an alternative strategy takes a lot of patience and determination but eventually bumps you up a level.

What about those who believe that math capabilities are inherited: one either has or doesn't have math aptitude? My quick survey of some soccer dads (who brought their five-year old daughters to the Kindergarten soccer practice) revealed that most of them believe that girls are not as good at math as boys are. Some explained their answer was a result of observing substantially more men than women in the engineering and scientific fields. All were surprisingly unaware of a multitude of recent scientific studies, financed by our taxes, aimed at improving the future of our little soccer princesses. Studies demonstrating that math skills are equal across genders; that culture is a factor in female math achievement and that girls’ confidence in math is dampened by parents' and teachers' gender stereotypes (Ref 2, 3, 4, 5, 6, 7).

So, we have entered an infinite loop here: by expecting less and demanding less from our girls, and kids in general, we encourage low performance and interest. This feeds our stereotypes and continues the vicious cycle. To break it, let's carefully look at the messages we send to our kids. Would you believe that these scientific studies showed that by merely suggesting to kids that they may not have racial or gender predisposition to math you can cause them to perform worse than they would have otherwise (Ref 8)? Or that by simply changing the class room decor from a Star Wars style to more feminine, you can make girls feel they belong and foster stronger interest in computer science (Ref 9)? It sounds like our math and science confidence is unbelievably fragile.

But then I recollect observing my six-year-old daughter on a ski slope recently. Frustrated with the cold air, wet snow and bulky clothes, she was trying to go up a tiny hill. Every other step she was sliding back or falling down with her skis crossed up in the air. We were both at the end of our patience and ready to give up. Until a fairytale ski instructor passed by, stopped for a few minutes and sprayed my daughter with an abundance of encouraging epithets: “You are doing amazing! Look at you, you are a natural skier. Great job.” It changed everything. My daughter beamed a victory smile, relaxed, and her patience was recharged for climbing the hill; she had fun and now is eager to go back.

So, how can we, as parents, help our kids become more comfortable, interested and successful in math, similar to the ways we support them in sports?

It is actually much easier than it sounds. We call ourselves soccer moms and dads even if we don't play soccer. One does not need to be an expert in math to become a Math Mom or Math Dad. Success in any kind of learning is enhanced by encouragement and challenge, and inhibited by threats or shame. Think of how we challenge and support our kids on the football field or ski slopes. Think of those trophies that everyone gets just for trying. The “Go, Alex, go!” you scream till you lose your voice. The pride we take when our child scores a homerun Recall how we share the excitement of our kids’ successes in sports and work on their failures. Why not adapt a similar mindset for math learning.

Think of the aura of adoration around successful athletes. We point out athlete role models on TV, in books and magazines. We buy those great Matt Christopher sports series books, we read and share Sports Illustrated stories. Those tear-jerking hero-making Superbowl or Olympic biographies that we re-tell to our kids to encourage them to practice and build perseverance. Remember how you are carefully avoiding negative stereotyping of athletes cursing, drinking or showcasing other misbehavior. When was the last time you switched off a TV program that featured smart kids as losers and unpopular geeks? Probably never. Can we then blame the kids for not wanting to be such unpopular geeks and crazy “beautiful minds”?

How about concentrating on the cool Stanford graduate, ocean-surfing Google founders that created something that comes as close as possible to saving the world by uniting it? Or showcasing mathematicians that apply their skills to create Mars mission or household robots, Avatar-style special effects, data analysis solutions that suggest the next movie you may enjoy watching or beer you would like based on your past ratings. It is not hard to find such role models.

Like sports, math is everywhere and it is not so hard to present it to our kids as a toy, a tool, and a friend. No one will stay indifferent when you reveal how prime numbers are used as weapons in secret message coding in war and peace, describe why 50% off on top of a 30% discount won’t give you 80% off, or share your financial insight on whether to buy or make school lunches and what to do with the money saved. Just imagine, science Olympiads and robotics competitions presented like the X-Games on TV, with groomed Rodarte-dressed geeks who tell your kids that nothing is impossible.

To expand the world of wonderful math possibilities for our children, we need to apply the same steps to math as we have been so masterfully using to encourage our kids in sports. Motivate, challenge, promote concentration, praise, share the excitement of successes and work together on failures, showcase inspiring role models, and avoid negative stereotypes.

And most importantly, never apologize for being smart.

Matt Damon as math whiz in the movie Good Will Hunting

P.S. Brain Quest has recently changed their logo to “It's fun to be smart!”

For more information on math see the NSF report:
Math: What's the Problem?

References:

CNN & CareerBuilder: 10 jobs for math whizzes

Boston Globe: Study finds culture a factor in female math achievement

Girls' confidence in math dampened by parents' gender stereotypes

Highbeam: Math skills are equal among genders in primary and secondary school

ScienceDaily: Few Gender Differences in Math Abilities, Worldwide Study Finds

Proceedings of National Academy of Science: Female teachers' math anxiety affects girls' math achievement. Feb 2, 2010, vol. 107, no. 5, 1860-1863

ScienceDaily: How Dads Influence Their Daughters' Interest In Math

CommunicationCurrents: Stereotype Threat and Female Student's Math performance

Science Notes: of Geeks and Girls

The Focus on the Numbers - a guest blog.

2010-03-11T14:36:00.023-05:00

Today I would like to introduce a guest blogger, Boston high school student named Haley. Thanks to her teacher, I came across Haley's erudite and stimulating essay that raises interesting questions about our math education and certainly will be of interest to parents, teachers and students.

The Focus on the Numbers

Sitting next to each other in a ninth grade classroom are a boy and a girl. The girl rapidly writes down every word her teacher says, not actually listening, only copying down his exact words, flustered when she cannot remember the last sentence he said. Meanwhile, the boy listens without writing, soon realizing that nothing the teacher is saying interests him whatsoever. So instead, he takes out his workout plan, determining when he should take his next protein shake in order to be prepared for the weight room after school.
The next day tests are handed back. He receives a 59, while she receives a 99. When the bell rings, he throws out the test and makes his way to the weight room for the fourth time that week. Instead of doing homework that night, he researches nutrition information while she memorizes trigonometric identities which she will soon know by heart without understanding what they actually do or mean. He can now explain exactly what a trans fat is and how it affects your body and health, while she can list ten different ways to write the same equation.

But which of the two will have the chance to be accepted into an Ivy League college? Who will have a better chance of making more money as an adult? The one who has no passion, no true comprehension of what she is learning, and is memorizing information which she has no idea how to utilize in her own life. She can remember dates, equations, names, and facts for a few days, maybe weeks, and once the test is over, the process begins anew. After all, as Goethe said, "it's easy to be brilliant when you don't believe in anything" (Edmundson, Why Read?).

We judge these two very differently, very unfairly. The boy has no interest in the curriculum set by the school. Neither does the girl. However, she does it to maintain the reputation as being smart. He does not care whether or not other people know that he is smart. He does what he enjoys and works hard at it. As Abraham Joshua Heschel, one of the leading Jewish theologians and philosophers of the 20th century, says in The Insecurity of Freedom, we evaluate the student by his ability to answer questions rather than to understand problems. This boy dedicates himself to becoming stronger, more fit, healthier. He has a passion for it. He loves it. He strives to get better week after week, calculating how he can reach 225 pounds by Friday. But few people respect him for this struggle, few people even acknowledge his dedication.

We have lost sight of what is important. Our values need to be reevaluated. Children are living in a place where passion is not praised unless it will benefit education or their future. The present is lost in this focus on the future. The constant pressure to get good grades leaves children with little time to cultivate themselves, to find things that they enjoy, to dedicate time to their development. Of course, it is important for people to be well-rounded, educated and informed in mathematics, language, history. However, there are ways to teach these things without bluntly forcing it upon children. Simone Weil, a 20th century French philosopher, wrote in her Anthology that slavery is the work without any light from eternity, without poetry, without religion the only incentives are fear and gain. In order to be successful, students must want to do well. It is difficult for them to do things for a future which seems distant. They need support, not fear. They need options, opportunities to learn things that they want to learn, things that interest them. The set curriculums of school systems throughout the country are the major downfall of our education system. How can students be expected to have an interest in subjects that their own teachers show little interest in?

The mission of our education system is in need of reevaluation. Our children need to be given the opportunities and encouragement to find and develop their talents and interests. They are told that it is important to be unique, and that everybody is different. But then why is it that we try to force them all to fit one mold and to focus on the same thing? It is time to stop overlooking the importance of the individual. Remember the words of Socrates: "a man's first obligation is to care for his own inner self, his own soul."

Top image is sadly titled "High School Sucks", it is distributed under Creative Commons license from Flickr

Haley's essay is definitely thought provoking. Feel free to post your comments below.

Snowstorm Warning. No Shoveling is Required.

2010-02-17T14:36:00.024-05:00

In the America's Northeast, we take pride in surviving and enjoying the thick snowy winters. Many of you who live much closer to the belly of the globe do not get to experience the joy and frustration of real snowstorms. What if math could help me sprinkle snowflakes, to share with you online? Here is a fascinating Snowflake Maker application created by a friend and colleague, Henry Kaufman:

This beautiful holiday card from a few years ago looks like a snowman's needlework. Try pressing “Stop,” and “Start “ to observe the creation process at work. Press “Restart” and then “Start” to create your own snowflakes and notice how each is very different from another.

In nature, each snowflake is unique, but they are almost all symmetric and have six equal sides (aka regular hexagons). Snowflakes are ice crystals, and the water molecules in an ice crystal form a hexagonal symmetrical shape:

Wikipedia references studies that explain why no two snowflakes are alike. Each snowflake is composed of roughly 10¹⁸ water molecules that stick together, at different rates and in different patterns, depending on the changing temperature and humidity within the atmosphere when the snowflake falls through it on its way to the ground. A bit similar to us navigating through life and accumulating experiences, friends and objects that shape our views, behaviors and appearances.

The virtual snowflakes that you can interactively create above are designed by a process imitating a real evolution of a snowflake and are similarly unique in their patterns. These snowflakes grow gradually and symmetrically in each of its six directions. On each step, a circle representing a water molecule is dropped at some random distance from the existing snowflake cluster. This circle is a starting point from which a thin thread is elegantly woven until its end is within some small distance from the existing water droplets. At this point, a new droplet of water is added to the snowflake and the thread disappears. The same process is repeated over and over with new water droplets. The thin thread’s beautiful path is created by simulating a process called Brownian motion. The name of which comes from the Scottish botanist, Robert Brown, who first observed and recorded the interesting motion pattern of pollen particles floating in water. This motion is observed and simulated by letting one big particle be pushed from the sides by many smaller, chaotically moving particles. Imagine yourself at the center of a four-year-olds’ soccer practice. From the sky, your bumpy motion path will look similar to the thread weaves that appear and disappear on the card above. A more scientific demonstration can be found here.

Enjoy observing how snowflake is knitted from thousands of particles and use the panning and zooming controls to wrap yourself in this crafty blanket. Imagine the gentle tickle of snow on your face. Share the Snowflake Maker and fascinating snowflake geometry with your kids and reveal a secret: when in doubt whether the snow you are skiing on is artificial or real, look closer at the pattern. Only nature or a computer simulation like above can create elaborate crystal arrangements:

Snow machine produces just a blob of frozen water droplets:

More of Henry Kaufman's creative work can be enjoyed on his website: www.tumbao.net

Simple Math Advice on Remembering Birthdays and Making Shopping Lists

2010-02-12T11:15:00.022-05:00

You know those people who need to recollect the year of their parents', siblings' and even kids' birth, to tell how old these relatives are?
“She was born in April 1998, so she must be 11 years old.”
Well, they may be on to something...

Image by hunterji, distributed under the Creative Commons license

Take me as a bad example. Whenever I hear that one of my relatives' kids has a birthday, I call the parents to ask for the child's age and rush to buy a present for him. When we visit relatives for the holidays, I create long lists of names and terrorize my husband until he calls to find out the exact ages of all his nephews. A year passes and then we call again.
"She is already 13! We can't believe she is growing up so fast."
We should have known. She was 12 the year before...

My new best strategy and my advice: the first time you buy a present for your friend or relative, record the date and year of their birth in your phone book, contact list, or calendar. Next year, to recall their age, all you need to do, is simple subtraction. They will be really surprised you knew! Suggest this to your aunt as well – she may stop embarrassing your 10 year old by asking if he has a girlfriend yet.

Shopping lists.
We love them. We hate them. We live by them.
Except that sometimes we forget to write something obvious like milk or bread, and then as soon as we are back from one shopping trip, we know another is imminent. And as much as we love some of these stores, who wants to go food shopping more than once a week?

Image by sk8geek, distributed under the Creative Commons license

Try a new approach - instead of marking what you need to buy (and risking missing something), create a master list of all the things you ever buy, xerox it or print one copy each week. You can start with this Master Shopping List. Before heading to the store, cross out the things you see in your refrigerator. You will not forget milk or bread anymore and believe it or not, you will be playing with the Set Theory.

The products on your list that you need to buy will be a Complement Set of all the products you crossed out because you have them. Every time you discover a new product that you like – expand your original set. If you prefer to keep separate lists for different stores, such as wholesale and local supermarkets, fair enough. There is probably some overlap between those lists. To avoid buying twice, as soon as you are back from one store, cross out the duplicates from your second store list. These duplicates are intersection of your wholesale shopping set and your local supermarket set.

Set Theory applied to shopping lists

Once your universal list is in the computer, it is easy to arrange products to appear in the order you come upon them as you navigate the store. No more random dashes against store “traffic” with a full cart and two kids in tow. And who knows, you may start enjoying it so much that you would not mind food shopping twice a week. :)

Would you like to read more from The Math Mom? Here are a few stories for Valentines Day: The Math of Dating - Mark Yourself Excelled, The Math of Ex.

Card Tricks

2010-01-21T10:47:00.020-05:00

Snowstorms, earthquakes, terrorism attempts... What could be better this time of the year than cuddling up near the fireplace with your family and friends and enjoying some captivating card tricks. These tricks are fun, surprisingly easy and impressive. What is even more fascinating, however, is how you can use math to explain how they work.

Trick 1: I can read your mind
You need 21 playing cards (pick any cards from the pile), and one person to choose and remember one of the cards. You, of course, will guess which one was picked.

Put all 21 cards into 3 columns by placing one card in each column, then a second card in each column, etc. till each column has 7. Ask the person to tell you which column his/her card is in. Also, ask this person to think about their card while you look straight into her/his eyes, pretending to have mind-reading capabilities.

Collapse the columns back into a deck, one on top of another. But make sure that the column that the person just pointed to is in the middle. Do not mix up cards in the columns.

Repeat steps (1) and (2), saying that you are getting closer.

Repeat only step 1). The middle card in the column that person pointed to would be the card he/she originally picked.

You are doing magic, and all thanks to math. How does it work?

Let's rewind. You place the cards in three columns and your participant points to the column where his card is.

You then gather all the cards back into one deck, placing the column that the person pointed to in the middle of your pile. You know that the card he picked is somewhere between the 8th and 14th cards in your deck.

When you divide the cards into columns again, the 1st card goes in the 1st column, 2nd card in 2nd, 3rd in 3rd, 4th card in 1st column, 5th in 2nd, 6th in 3rd, etc. Then the column in which the selected card is will occupy places 8-14 in the figure below. The first two and the last two cards in each column definitely do not hold the picked card. The picked card is somewhere in the middle of one of the columns: one of the three middle cards in the central column or one of the two “middle” cards in the left or right columns. All these possible locations of the picked card are colored orange on the illustration below.

Then the person points to the column where his card is now:

You gather the cards into a pile again, making sure to put the column the person pointed to in the middle. The picked card is in position 10, 11, or 12. More specifically, if it was in the first column, it would be in the positions 11 or 12, central column – positions 10, 11 or 12, right column – positions 10 or 11. But you don't need to keep track of all these details or remember any specific cards. All you do is spread the cards into three columns, gather them, spread, gather and spread again. Math does the magic for you.

When you place the cards into columns for the third and last time, cards 10, 11 and 12 are now each middle cards in a different pile. The picked card is one of them.

As soon as the person points to one of the columns, you know that the picked card is the middle card in this column.

Trick 2: Magic Words Card Trick
In this super-trick you can guess not one, but a pair of cards! And you can do it simultaneously for many different players. Choose 20 cards and lay them out in pairs.

Ask every player to remember one pair. Gather all the cards, without mixing up the pairs. Then spread out all 20 cards in 4 rows of 5 cards each, using special magic words, ask every player to tell you which rows his/her cards are in, and … guess them!

Did I leave anything out? Of course – the magic words.

You use them to put the cards into 4 rows of 5 cards each. These magic words are your key to this puzzle. You need to learn them. But do not reveal them to your audience.

Here are the magic words in English:

Russian version of magic words:
НАУКА
УМЕЕТ
МНОГО
ГИТИК

Notice that every letter is present twice in this magic word collection, and there are 20 letters total. So, we have 10 pairs of same letters and 10 pairs of cards. Imagine these magic words laid out in front of you on a table just like on the image above. Take your cards and place the first pair of cards where the two letter Bs are, in the top row. Next pair goes where the letter I is, in the first and second rows. Third pair where the letter Ls are. Continue with the rest of the letters and card pairs. It does not matter which pair corresponds to which letter. The only important thing is to make both cards in each pair fill the spot of the same letter.

Notice that any two rows have only one letter in common. Therefore, when your opponent points to the rows his cards are in, you recollect the magic words that you used to spread the cards and think which letter is in common for these two rows (or one row if both picked cards are in one row). Once you know what letter it is, you just point to the cards that are placed in this letter's spots.

Enjoy! These precious fun moments of surprise and discovery stay in our memories. One day your kids will entertain their friends and kids with these puzzles. Just as I did, learning them from my father.

A little math at every corner

2010-01-07T13:49:00.008-05:00

Drive straight until you reach Main street, then count three traffic lights, take a right at the fourth light that should be Highland street, then second light left into Summer street. But be careful as the street sign is hidden behind a tall tree.
What? You missed the Highland sign?
I told you – fourth light. Now, you have to wait two lights, then you can do a U-turn. Remember: going South it will be a third light left into Highland.

We all are too familiar with such navigation without a GPS in an unfamiliar area. Everything takes twice as long and requires twice the amount of concentration. On our recent trip to Israel, we were thrilled to discover that this driving adventure is ingeniously simplified by assigning numbers to traffic lights.

No need to count (by the way, do you count before, while or after you cross the light) to look for obscured street names. Just drive to the light numbered 7, take a right, drive to the light numbered 2 on the other street, take a left and you are there. You may not always be able to read a long name like “Yerushalayim ave” but you will definitely notice the “15” above the traffic light.
A little bit of simple math at every corner.

Guess Your Age Trick - Revealed

2009-12-22T11:57:00.008-05:00

My mom asked me to explain to her this "miraculous" trick she received by email and was shocked to discover is capable to expose her age.

Try it - it will guess your age as well:

What is your favorite day of the week? For Monday - pick number 1, for Tuesday - 2, Wednesday - 3, Thursday - 4, ..., Sunday - 7.

Multiply the number you got above by 50.

Add to the result 44.

Multiply it by 200.

If you already had your birthday this year, add to your number 109. If you did not have your birthday this year, add 108.

Deduct the year you were born from the resulting number.

Look at the long result. Here is the trick:
First digit of this number is your favorite day of the week.
Last two (or one) digits of this number is ... your age!

This trick expires at the end of 2009. And it is much easier than is sounds.
Here is explanation how it works:

You pick number N, where 1 <= N <= 7

50 x N

50 x N + 44

200 x (50 x N + 44) = 10,000 x N + 8,800

10,000 x N = N0,000 (where N is the digit corresponding to your favorite day of the week)

If it is 2009, most probably your birthday this year already passed. Add 109 to the number above. You got: N0,000 + 8,800 + 109 = N8,909

Look at the resulting number: N8,909
See how the first digit is exactly the digit your picked at the beginning. Anything you pick N to be (from 1 to 7), it will be the first digit of the resulting number. Here is the first part of the trick.

Now, remember that if you take 2009 and deduct the year you were born, you get your age. If you take 3009 or 8009 and deduct the year you were born, the last digits will still show you age. When you deduct your age from the number you got above - N8,909 - the last two digits are still showing your age. This is the second part of the trick for you.

Now you are ready to play the trick on anyone around you.

But beware: if your grandma is lucky to be older than 99, and you want to play this trick on her, you will need to change it a bit. You will need 3 last digits, and not two. Right now for a 100 year old person who was born in 1909, you take N8,909 and deduct 1,909, you get N7,000. Last three digits will of course compliment your grandma, but won't do the trick. You need this 000 be 100. So, add one more step: (7) and ask her to add 100 to the resulting number that she got.

Why the trick expires in 2009? Because in 2010 you would need to add 110/109 instead of 109/108 in the step before last. You need your "magic" resulting number (N8,909) to end with the current calendar year digits, be it 09 or 10.

But for the next two weeks you don't need to worry about it. Just copy the steps above and spook your friends and relatives at the Holiday party.

When Does Purple Become Brown and The Case of a Mixed-Up Jacket

2009-12-16T04:03:00.026-05:00

Two colorful things happened on one recent week: I painted my nails deep purple for a party and bought new dark brown carpets for our bathroom.
My husband, on his return from a business trip, puzzled me when he remarked that the “purple” carpets seem too dark. “They were very brown when I bought them,” I said and rushed to double-check. After spending two frustrating months on choosing our home wall colors, I knew very well his strong dislike of anything purplish. Surprisingly, the carpets did look dark: purple or brown, one could not say. My deep purple nails appeared to be identical to the carpets.

So, I went on a quest, trying to figure out how two very different colors, brown and purple, can appear exactly the same. And when does purple become brown?

Playing with an online Color Tool, that decomposes any color into Red, Green and Blue components, I realized that both, brown and purple, have very little green in them, and much more red and blue. By taking brown and adding some blue to it we get purple.

So, the answer to the mystery should be in blue. Aha! Perhaps we cannot distinguish between brown and purple in some lighting? Turns out, this is true! “White lights” such as strong skylights and white fluorescent lights as in shopping centers have a lot of blue in them. See the hill on the left graph above the blue. “Yellow lights” such as sunlight and the incandescent light of my bathroom have much more yellow and red than blue in them – see the curve on the right picture.

Light is energy. Here is how it is distributed for “White” Skylight and “Yellow” direct Sunlight.

The color of the object that we perceive is a combination of its color and surrounding illumination (light coming from the light source such as lamp, sun or reflected from other objects).

A simplified color equation:

Object color that we perceive = Object’s “true” emitted color X Illumination

Therefore, any blue in the object is amplified by a skylight or fluorescent light, making it appear even stronger blue to our eyes. The yellow light of the incandescent bulb or direct sunlight on the other hand, highlights the warmer tones, suppressing the blues. That explains why my brown carpet and my purple nails appear exactly the same in the yellow-ish bathroom light and look very different in the fluorescent light of the store.

What is the use of it all, you may say. It tells us that instead of exclusively considering the color of our purchases at the store, we should consider the color as it appears in the lighting conditions this purchase is intended to impress. Buy, try, and perhaps return. Otherwise, your Holiday purple dress may look too brownish for your taste, and those shiny light purple eye shadows may not be a good match at all.

If you are carefully choosing a fancy blue shade of color for your room – make sure you have plenty of white light in it, otherwise your visitors will just call it plain blue.

Great example from the Benjamin Moore page: the same Greyhound color appears drastically different under various light situations.

And if your kid comes home with the wrong color jacket, don’t worry about color blindness and don’t argue – everything will look different in the morning light.

Why Should You Care About Math?

2009-11-23T23:58:00.007-05:00

President Obama announced a new Math and Science Initiative, alarming us that we are 21st in math among the 25 ranked industrial nations. Surprising and pathetic, is not it? Why should we care? Why is math important? First of all, it is the foundation for our technological progress. Everything, from medicine and space program to sports relies on math to analyze, simulate and improve. We are desperately fighting to be ahead in all these fields. Second, in the global, competitive world we live in, being 21st out of 25 means not getting the jobs we want. And nowadays almost every job utilizes math in some extent. Dentists play with computer models to simulate imperfections of our bite and new dumpster trucks come equipped with robotic arms.

Another reason is a pretty good lifestyle that math expertise may allow for. A recent CareerBuilder survey reported by CNN rated mathematician as the best career, taking into account income and stress, with mathematicians earning $94,960 on average in 2008. Fluency in the language of math may give your children not only the attractive financial comfort, it will help open the whole world for them. Math is a universal language you can speak and work with anywhere on Earth and in Space.

What can we, parents, do in order to help our kids get ahead of our poor 21st ranking? You can start by equipping them with the right attitude towards math. By presenting math to them as a toy, a tool and a friend. Something as hip and respectable as sports. Something exciting and important, worth working hard for.

Surfing Our Waves of Productivity and the Taxi Driver Paradox

2009-11-19T14:00:00.010-05:00

You can't run a family or any business without discipline. To accomplish it all, we create the to-do list; we set clear goals, and try to stick to them:
Two pages of homework per day.
Three loads of laundry per week.
Eight hours of work each day. Try to not leave before your manager does.
Sound familiar? But what about those work days when you are present but your mind is absent, exhausted from lack of sleep or worrying about a sick child at home? What about the days when your cutie is jumping under and over the table and hardly gets to write a single character of her homework in one hour? How do you keep your productivity up without losing your sanity? There is an interesting taxi driver paradox that can help us apply some wisdom of math to our productivity:

A while ago, economists conducted a survey of New York cab drivers. Most of the drivers were renting their cars for a flat fee for up to 12 hours. Therefore, they were free to work as much as they wanted or needed in those 12 hour intervals. The majority of drivers chose to work a pre-set number of hours per day, or until they earned some fixed daily amount of money. But the nature of the taxi business (as well as most other businesses) is non stable. Some days it seems that everyone is traveling somewhere and needs a taxi, and some days are dull and one has to spend hours looking for a customer. What if the drivers gave themselves the freedom of calling it quits early on days business was slow, and worked late hours when demand was high? Economists have shown that following this demand rule, instead of the pre-set income or hours rules, would actually be more beneficial: an average NY taxi driver in this study could have earned at least 15% more by adopting this approach of productivity and freedom. Interestingly, a small fraction of the drivers that have been on this job for years and presumably are more experienced, did appear to follow this demand strategy.

So giving up when things do not go according to plan is well compensated by overworking when things go well later on. Sounds intuitive enough, right? But when it comes to loss in productivity or income we hate the loss much more than we enjoy the gain. Therefore, like taxi drivers searching for one more fare on the empty streets instead of going home, we frequently make ourselves (or our kids) suffer while trying to reach pre-set goals when the energy for productivity is gone.

To fix my strict inner disciplinarian, reduce my guilt and stress, I like to imagine surfing the waves of productivity. Here comes a giant wave: you received recognition at work and are energized to put in longer hours. So, go for it, enjoy the smooth productive ride, prolong it and get the most out of it. Feel like moving mountains this weekend? These three loads of laundry will come in handy, then perhaps a Children’s museum, and a homemade cake. On such days, never leave something for tomorrow that you can do today.

The wave will eventually break. You may find yourself unmotivated, exhausted or sleep deprived - allow yourself time off. Don’t let guilt get to you. Remember – we will gain in productivity in the long run. When your child can’t concentrate on homework – try switching plans and doing what feels right. Tomorrow, she will get her wave of productivity and you will show her how to enjoy the top of the wave glide. Skip the useless nagging and guilt. Let’s take every day as a surprise.

Images courtesy of Kennimatic (top) and Holladay Photo (bottom), distributed under the Creative Commons license

Math Kabbalah

2009-11-05T23:50:00.008-05:00

Kabbalah is a study of mystical aspects of Judaism, which deals with the inner, hidden meaning of the traditions and Tanakh (Hebrew Bible) stories. A meaning that will show the elegance and purpose of this universe, some larger goal of our existence. Looking at the mysterious and elegant patterns below, which I received via email from a friend, I wonder whether it is Math Kabbalah?

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

More of these fascinating patterns and surprising stories about numbers could be found in Book of Numbers, by Tim Glynne-Jones. Here are few interesting facts from this book:

0 : Ancient greeks did not recognize 0 as a number. There is no year 0 in our Gregorian calendar.
1: One is the most frequently used number, so if you're going to fiddle your tax return, add a few more 1s to slip undetected.
2: It is believed that Mark Twain borrowed his pen name “twain” from this number. In the beginning of 19th century “twain” used to be a synonym for “two.”
3: This one is storytellers' favorite: three little pigs, three bears, three musketeers.
4: Over 500 films have been made with “four” in the title.
5: The number of oceans and circles on Olympic emblem (symbolizing five continents that accepted Olympic rules back in 1912: Americas, Europe, Asia, Africa and Oceania)
6: People who concoct fraudulent data tend to start their made-up numbers with 6 most commonly.
7: Some believe in even years of bad luck when you break a mirror. Seven digits is the most the average person can remember.
8: A very popular number in the Imperial system of weights and measures.
9: Is the trickiest number. The digits of all multiples of 9 add up to 9 or a multiple of 9:
9x9=81, sum of digits is (8+1 = 9)
9 x 137 = 1,233 where sum of digits is (1+2+3+3=9)

I remember the pre-feature presentation at the Landmark Cinemas: “The language of cinema is universal,” echoed in many different languages. Perhaps the biggest power, mystery and magic of the language of math (like language of cinema or sport) is it's ability to transcend cultures, generations, religions and political regimes. Whenever you are right now, chances are you are using the Hindu-Arabic numeral brought to Europe by Arabs of North Africa. Whatever you do, you are applying the laws of logic defined by famous Greeks, Persian, French, Germans, Russians, or English mathematicians. And you can communicate and contribute using this ancient language of math anywhere on this planet.

The Peculiar Math of Tennis Court Use

2009-10-22T20:52:00.014-04:00

It sounds very simple: you ring a friend, you motivate each other, put on some inspiring tennis clothes, take a racket, balls, water and head off to a court.

If one is free – you go ahead and play. If all are busy – you chat and wait for one to open up. But you may want to know how long you can expect to wait. You contemplate sneaking on to each court, maneuvering among 50mph balls, and bugging angry players about when they’ll be done. How else do you know who has been on the court for 2 hours and who has been playing for only 5mins? Shall you put this cup of coffee aside and start warming up or you still have plenty of time to call and line up all the playdates for the upcoming weekend?

How do you make court use fair and simple? Hire a court manager? Great, but expensive. A sign-up board? We all know that the chalk gets wet, lost or stolen. Well, here is my town's novel solution to this old waiting problem. Listen carefully: if you find it feasible, you could apply it to any shared resource, such as your home's bathrooms. Or, perhaps, you could even suggest and patent a better one.

The following rule is posted at the entrance to our tennis courts:

and the boards on each of the four adjacent tennis courts:

So, this means that if you are playing on a court marked “:00” and someone is waiting, then at the next round hour you need to clear the court. Same for the three other courts, only you have to clear them at 15, 30, 45 minutes past each hour.

How exactly does this rule work?
If you come at 2pm, play for an hour, undisturbed on the :00 marked court, and then at 3pm have to free the court if anyone is waiting - it all makes great sense. But what if you come at 2.55pm, see the empty court marked ":15" and start playing. Twenty mins. later new players come and reading the rules, demand that you free the court right then, at 3:15pm. According to this rule, you should do it. You, and not the couple at the ":00" marked court who have been playing for over an hour. How is this supposed to be fair and logical?

It appears that this tennis court rule is a simple strategy to insure everyone can play for at least an hour and wait no longer than 14 minutes before getting to play. But this “60 min. play / 14 min. wait” outcome only works if everyone understands the math behind the rules and agrees to follow them for the long-term benefit, despite some short-term absurdity. For this “60 min. play /14 min. wait” result to be achieved, newcomers must go to the court corresponding to the next 15 minute mark.

So, anyone arriving at 2.55pm should head to the ":00" marked court, even if this court is busy and the other court is empty. This will lead to some waiting time (only up to 14 minutes), but will also allow newcomers to enjoy the whole hour from 3 to 4pm, uninterrupted. It may also make current ":00" players move to any other empty court at 3pm, but assuming that they also followed the rules, they already enjoyed a whole hour of undisturbed playing before that.

What do you think of this arrangement? Can we be trusted to read and understand the rules, and choose the appropriate timing strategy for long term gain, despite the obvious occasional short-term absurdity? What strategy do you use on your courts? Catch the question and pass on the answer - the ball is back in your court.

Top image courtesy of Bruno Girin, distributed under the Creative Commons license

Three Thrilling Math Adventures For Your To-Do List

2009-10-15T23:11:00.013-04:00

Have you ever thought that Math could take your breath away? Try these thrilling math adventures:

“Sum of All Thrills” Epcot ride

Imagine designing your own roller coaster, bobsled or jet ride and then riding in it. This is what new Epcot ride “Sum of all Thrills” is all about. It is build by the defense contractor Raytheon that is trying to figure out some creative ways of interesting kids in math and science and growing new generation of local innovators. Raytheon also has an interesting MathMovesU website for middle school kids where they can play cool math games. When you do go to visit Epcot, check out the Marrakesh Restaurant. I visited 10 years ago but still remember the taste of the Beef Brewat Rolls that they served. And please, post a comment below about the rolls and the ride if you survive your creation.

A Puzzle Thriller: “Fermat's Room”

“Fermat's Room” is a new well-rated Spanish movie recently released in US on DVD. Here is what Netflix reviewers tell us about it: Psychological Mystery/Thriller that gets off to a very sure-footed start and seemed to hold out the promise of a very fast-paced film. Drawn together by a mysterious host, four mathematicians wind up locked in a shrinking room and bombarded by a series of tricky riddles. To avoid horrific deaths, the frantic scholars must quickly figure out the link that unites them all. A slow and steady ending ties everything up in a satisfying end.
I have this movie lurking to me from the Netflix envelope and promise to share my opinion below as soon as I'll watch it.

Math Tour of Manhattan

Mathematician Glen Whitney is occasionally giving math tours of Manhattan in an attempt to bring publicity to a Math Museum (MathFactory) he is planning to create in the city There are movie location tours, fashion tours, shopping tours, culinary tours. Why not a math tour? Even if you are an island native, you may enjoy discovering why fire hydrants have pentagonal lug nuts, what is the optimal queuing strategy for efficient shopping or how to compute the shortest distance to get anywhere in the city. Nick Paumgarten wrote more about the tour in this New Yorker article. But beware, you may not like Glen's opinion about lottery being a tax on mathematically illiterate. Whatever your interests are, this tour will definitely give you a fresh perspective on some familiar things. And this is exactly the stimulation we all need to start thinking outside of our comfort zone and come up with some fantastic ideas of our own. Perhaps we could finally figure out how to make a school pick-up routine much more efficient.

Have more of your own math thrills to ad? Post them in the comments below or email me.

A Kaleidoscope of Puzzles

2009-10-08T21:53:00.001-04:00

I selected 21 of TheMathMom's Puzzles to satisfy every taste, complexity and type. Enjoy reading, solving and discussing them with your kids, parents, co-workers and friends. If your kids will see you playing with math and treating it as a toy and a friend, they will feel much more comfortable with it. Vote to help me select the most popular puzzles. Read to the end - the last puzzle (Egypt vs Vegas) is the latest unsolved mystery from the Family Marathon site. Solution is finally revealed! Don't forget to click a Submit button when you are done voting. Voting is anonymous, no reader information is requested. Have fun!

Click to read and pick the best puzzles

Love, Life and Math: My Life Story

2009-10-01T21:22:00.005-04:00

It has been my drug, my meditation, my weapon and my best friend. It fed and dressed me, led to travel, men, lavish parties and even Cuban cigars. It has placed my name on a movie screen, put my work in museums, and allowed me to manipulate cells deep inside the human brain. It is called Math.

In Russia, where I was a child in the 80’s, math was respected and celebrated as a tool of progress and technological advance. It was presented to school children as a toy with tricky wrapping that one had to outwit to open. We were challenged and encouraged to tackle it. Indeed, math has become the toy of my life, my key to the world, leading me across continents, opening doors to exciting projects and people, and even assisting in the realm of romance.

I was in high school, when I applied math to my love life for the first time. I fell in love, and barricaded in the heavy, still vacuum of my room, was desperately counting clock ticks, waiting for “the call” from the only person whose existence mattered. My brain was justifying the silence by devising elaborate excuses for why he hadn’t called “yet.”

It suddenly occurred to me that insecure, self-pitying anticipation could be turned into a confidence-boosting calculation of the probability of his call. What are the chances of his call, given the rumors of another girlfriend he may have had? I played with the concept of conditional probability in my mind. How does the likelihood of him being interested diminish with each passing day without a call? The results did not look promising. My attention, however, was diverted from the lost "love of my life" to a world of quiet concentration where I was queen, which significantly shrank his importance. Mathematically directing myself away from loveless depression, I tuned in to the world again and realized there would be many more love adventures to enjoy.

Immigrating to Israel, I discovered that math, more so than my religion, connected me to the young people in my new country. We spoke and read different languages, we lived through different histories, had very different worries, but we all studied math with Hindu-Arabic numerals and learned the same rules of logic. I met my future husband in a graduate math class. At that time we did not speak a common dialect but shared the language of math. Life was easy.

A rainbow of hip vocations presented themselves after I attained my math degree. The first came with delicious benefits. I was creating a product database at a chocolate factory. They had an “all you can eat at work” policy...

The coolest job of all was in movie special effects. The setting was like a dream: on the ocean side of Los Angeles, in a quiet nightclub atmosphere, in an abandoned military hangar, lit by a web of Christmas lights and lava lamps, surrounded by a life-size Princess Leah statue and old Star Wars spaceships, accompanied by a pet parrot. We, computer graphic programmers and animators, were making Hollywood history and immortalizing characters on the big screen.

With math I have helped move Godzilla through the cables of the Brooklyn Bridge, and created a Monet-style animation of rap singer Puff Daddy, while experimenting with painting-by-numbers. Inspired by a brilliant talk at a SIGGRAPH conference, I tried to digitally erase the cat's whiskers for the movie “Stuart Little.” An affair with another novel technology led me to use a reflective ball to capture the direction of the sun in a vast South Carolina field when shooting the movie “Patriot.” Some experiments worked and some did not, but the math behind them was thrilling, adventurous and playful.

I earned my Green Card by meticulously planning a hi-jack attempt on Air Force One. It was returning from a summit in Moscow and carrying Harrison Ford as President of the United States. Our team of animators and engineers helped all the explosions look realistic, simulated Air Force One refueling in the air, and made everyone believe that the President had escaped from the plane in a computer-generated pod we created for him, which was dropped through computer-generated doors at the bottom of the plane. I even had a chance to meet the President, I mean Harrison Ford, at the Sony stage, along with other actors who were practicing jumping from a plane in front of a giant air blower. We celebrated the movie opening with an extravagant Hollywood party that included Cuban cigars.

After 9/11 I thought I could help defend my third homeland, the U.S., from terrorism by teaching computers to recognize suspicious behaviors. I abandoned the idea when I realized that my system would have detected me as one of the false-positives, when after a 15 hour trans-atlantic flight with two little kids, sleep-deprived and afraid of admitting to smuggling an apple in my bag, I would be nervously avoiding a gaze of security men.

Instead, I joined forces against an even broader insidious enemy – cancer. In radio surgery, we use nifty math algorithms to target the cancer tumor with the precision of a single hair, radiating and killing the cancer cells while minimizing the damage to surrounding tissue. Math also offered a horrifying thrill when I assisted with brain surgery procedures on a death row inmate known to have killed his cell-mate back in 2005.

Now, ensconced in the mature days of parenthood, I am titillated by entertaining mathematics in our daily family routines. When figuring out the meaning of a double-negation note from school: “Please mark yes or no below if your child will not attend school on Friday.” When convincing myself to buy $250 winter boots because their cost-per-wear appears reasonably small: $250 / (100 cold days x 3 years ) = less than $1 per wear. When advising my son when to jump from the swing in order to enjoy the longest flight into the sand. Or, when finding an optimal home location that will minimize our family's combined commute time.

I am trying to pass on my math infatuation in the same manner that one passes traditions and language through generations. My 5-year old daughter runs around playing super girl and sings, “I can be anything I want to be.” I believe that the love of math may be the real super power that I can share with her, that will bring magic to her life journey.

An Ode to a Spoon

2009-09-24T21:46:00.013-04:00

I never thought a spoon would deserve a column. Until I fell in love with one. From the very first glance I knew this was the one: brilliantly designed, tastefully elegant, modestly shiny, and not mine...

Possessing it could brighten anyone's existence. No more awkward spoon maneuvering inside a narrow sugar or yogurt container. No more grains sifting along the sides of stylish yet slim, cheek-less spoons. Imagine, finishing all the milk with your cereal without needing a tablespoon or straw. Remember the unpleasant screeching sound when scratching the bottom or sides of a plastic or glass container with a regular spoon? This beauty's rounded contour would make it silent.

With a simple twist of your wrist, a soft little mountain of cottage or whipped cream majestically rests upon this spoon. Lift your hand a bit, and the spoon is right there, whispering to your lips. Think about the awkward elbow rotation, the longer travel paths to your mouth and the ridiculous diagonal fit of a regular spoon in it.

Left column: regular spoon. Right column: the perfect spoon. Note how both testers arms are extended in the left column as compared with the right and how regular spoon goes diagonally into mouth, while the perfect spoon fits naturally straight.

I’ll grant you that the lack of symmetry is unsettling, though thought-provoking and memorable. But isn’t that one of the reasons we like Picasso's paintings?

Afraid that asymmetry will lead to a lack of balance, I borrowed the spoon from the lucky owner (thanks, Rachel). In a Consumer Report-like testing environment we built at home, two testers used it for a week and raved of the spoon's superior usability. These two testers, who are notorious spillers, showed absolutely no issues with balance.

What are you saying? You are a lefty? Yes, you deserve a reversed copy.
Feeding your child? Sit to his right and just extend your hand, the spoon is already in his mouth. No need to play airplane, risking a spill of all contents before landing.
A fan of sporks? They are just undecided peasants compared with her majesty.

Do I deserve it? Me, who rarely notices how and what I am eating, distracted by the kids’ requests. Me, who could eat with a plastic spoon, crazy glued spoon or a spoon vandalized by a garbage disposal? Probably, not. Still, I will be searching for you, the perfect one. To own, to hold in my hands, to sip from it. My modest puzzling queen, I promise to adore you forever.

What to do When Facing a Dinosaur

2009-09-19T22:19:00.017-04:00

Sooner or later, most kids get fascinated by dinosaurs. Why did they die? Were they brushing their teeth? Could this cement glitch in our front steps be a T-rex fossil? My favorite question is: “What color were dinosaurs: green, brown or pink?” And the answer is that, for all we know, dinosaurs may have been pink with golden polka dots. Not practical for survival, but definitely attractive to young females. Without preserved flesh, dinosaurs’ colors are left to our imagination (*).

Want to impress your kids with some knowledge during their dinosaur fascination phase? Explain the math of “binocular (two-eyed) vision” to them and show why T-rex was one of the most successful predators.

Here is example of an image that one eye (human or dinosaur eye) could see:

Do you think this is a 30-foot high movie prop or just a 5 inch high plastic model? From one picture of a single object without a context, you can't necessarily tell how far away it is or how large it is.

If the object happens to be next to another object of known size, such as a real car, then we can approximate the size by comparison, as you can see here:

Without any objects of reference around, we only know that somewhere in this direction there is a dinosaur. The picture that one eye receives (such as the first dinosaur image on the top) is the same, whether this dinosaur is a tiny plastic model or a giant monster standing further away. See how on the illustration below the blue projection lines (from real size monster) are aligned with magenta lines (from the tiny model):

Our second eye comes to the rescue and solves the mystery of what is in front of us, how large it is, and how far away.

Magenta lines from the tiny model produce a very different image on the right eye than blue lines from a full size monster. Depending on what exactly the right eye sees, we know which dinosaur we are facing. This ability to perceive and fuse information from both eyes is called binocular vision.

The visual area in front that both eyes can see simultaneously, is called the binocular field of view. See light blue triangular area on the illustration below. For any object that happens to stand in this area we can define its size and how far away it is. Even if it is a pink dinosaur in your daughter's pink room.

Up until recently, people believed that with their large nose and deep positioned eyes, dinosaurs did not have binocular vision. In fact, in the Jurassic Park movie from 1993, one human character tells another that a T-rex can't see them if they don't move, even though the dinosaur is right in front of them. But recently, scientists designed some clever experiments with T-rex skulls and agreed that T-rex probably had a large binocular area of about 55 degrees (see brown illustration below left). This ability to perceive depth in these large areas in front, and to easily distinguish its prey from the surrounding background, helped T-rex to be an aggressive predator. So, the T-rex did see those kids in the movie!

In comparison, most other dinosaurs, such as the Allosaurus, and in fact our contemporary crocodiles, have deep and wide-positioned eyes and a narrow binocular field of around 20 degrees (see green illustration below right).

Interestingly, this nature math defined their behavior, turning them into patient, sneaky creatures that sit quietly and attack only when prey is right in front of their nose - inside the binocular field. Or they rely on their prey's motion to separate it from the background outside of the binocular field.

So, next time you see a dinosaur, make sure to stand outside of its 55 degree binocular field, and do not move, or just run as fast as you can in the opposite direction.

Here is a link to a movie showing how a dinosaur’s field of view was mapped using their skulls.

Note (*): "What color were dinosaurs? Nobody knows..." This fun dinosaur fact has been considered true for thousands of years. But two new scientific papers published in Science and Nature in 2010 suggest that it may sometimes be possible to find out the color of some of the dinosaurs' feathers from fossils.

Personalized Puzzles for Your Birthday, Rosh-ha-shana or Thanksgiving Party

2009-09-11T08:21:00.005-04:00

What could be better than fun logical puzzles to break an awkward shyness, unite strangers and create fun atmosphere at any party? Print these puzzles on a piece of paper, glue them to cardboard and use as table centerpieces or place them directly on the plates. Birthdays, Rosh-ha-Shana, Thanksgiving or Christmas. Young or old - everyone loves puzzles. Better yet, use your friends' names and their real life events in these puzzles. Who would not be thrilled to see a puzzle dedicated to him or her?

Here is the link to a fun "Make Puzzles About Him" Application. See 10 fun puzzles I used for my husband's birthday party. Then generate puzzles for any friend of yours. Just enter his data and press "Generate New Puzzles" button. Enjoy!

Desperate Housewives' Money Saving Tips

2009-08-28T21:35:00.008-04:00

Only a few of us enjoy keeping close track of our spending and hold pride in following a strict financial discipline. Most of us fancy ourselves as smart yet reckless characters of the network TV series lavishly spending on designer shoes, exquisite restaurants and the latest gadgets. This pretend may have worked for some of us for years. The recent economy has sterilized our dreams a bit, forcing us to pay attention to where our money is flowing. Having been commissioned to write about the ways math could help us save in the current economy has made me seriously wonder where my own family's money is draining, unnoticed and often un-enjoyed. As I described before in my presentation, I have nothing against a $300 pair of winter boots as long as they are beautiful, comfortable, sturdy and you will get a great price-per-wear from them. But the $10-20 we leave daily on tchatchkes in the local pharmacy, various gift shops, snack machines or coffee shops - they do alarm me. Just a daily coffee and a treat that run $5 per day, would add up to $35 per week and $1,820 per year! It may be well-spent money if you savor and enjoy every sip of this precious coffee. Just be aware that making it at home may allow you to afford the Caribbean trip you had to skip this year. So, let's pay attention to such little daily spending and leave only those that are definitely worth it.

Photo by zazie, distributed under Creative Commons license

Turning the spending awareness on and imagining myself as a financially savvy Desperate Housewife, I realized the advantages of a liquid soap refills. First advantage is minimizing the vicious cycle of plastic manufacturing and recycling that we are paying for. Those small liquid soap bottles that were produced, packed, shipped, stored and sold, you use them for a month or so and then recycle, with many more vehicles driving it from your home to the dump, to the recycling facility, washing, melting and starting their life cycle all over again only to return to your home a few months later. Sounds like too much hassle for such a short life-span of a little bottle. And obviously, we are paying for every step of this process.
The second advantage is your direct saving with a surprisingly minimal amount of work. A 500 mL 16.9 oz plastic bottle of liquid soap that I bought in Marshall’s was $5, with its original price being $9. Costco is selling small size bottles of soft soap containing 7.5 oz for $2.23 as well as a gigantic size 1 Gallon (128 oz) bottles for $13.99. As usual, to compare we need to check the price for the same amounts - say 1oz. The liquid soap from Marshall’s as well as a small packaged soap from Costco were both $.30 per 1 oz. Surprise! Wholesale prices are not always cheaper. However, the gigantic soap from Costco was $0.10 per oz – 1/3 of the price I am currently paying. We could save 2/3 of the price and simplify our life by buying the soap only twice a year instead of monthly! So, consider refills. They are easy, green and hip nowadays, especially among desperate housewives.

Look for TheMathMom's article in September 2009 issue of the Boston Parents Paper on many creative ways math could come to the rescue of your family budget.
Click here to read this article online.

Hmm... Summer Camp, or a Trip Around the World?

2009-08-18T18:06:00.021-04:00

Summer camps are understandably yet absurdly expensive. In fact, sending your child to a four-week summer camp session costs the same as buying your child an airline ticket for a trip around the world, between $1,500 to $2,000 per vacationing soul.

Before rushing to call our travel agents, let's zoom in on this comparison a bit closer. While many summer camps require kids to bring their own lunch (to avoid dealing with obscure dietary preferences), the around-the-world option includes plenty of meals tailored to any diet: vegetarian, low-sodium, diabetic, Kosher, Hindu, Muslim and even offers a kids’ menu. Occasionally, you can get seconds by charming a flight attendant, or by gathering all your chutzpah to ask the person in the next seat whether she is planning on eating her dessert. But don't even dream of snapping up your fellow camper's food; sharing is prohibited.

Summer camps mean sunburns, tick panic, mosquito bites, and lice checks. Airplane travel is free of these dangers. With the occasional outbreaks of flu followed by a mandatory 7-day quarantine, many kids have to miss entire weeks of camp without refunds. But consider this, if your child gets sick on the around-the-world trip, not only the child, but all passengers and crew will be rushed to the nearest hospital for extensive checkups, and all will happily continue the journey when recovered.

Another advantage of the around-the-world deal is a clear guarantee that your child will not return in the evening to tell you that he does not like it. True, there are camps that follow this model: sleep-away camps. Your child is provided lodging, food and is not allowed to call and complain. Only snail mail is permitted and by the time you get it, the camp session is almost over. But for such an all-inclusive package, you have to pay the equivalent of a business class around-the-world ticket.

The airline option sounds like a better deal overall. You just need to choose a route. This depends on what you want your child to learn during summer vacation. When sending kids to camp, we like to think that they acquire real life survival skills like canoeing, swimming, archery, knitting. However, studies show that, in the summer, kids' brains take a reverse evolutionary zip and the real survival skills of the 21st century, such as math and reading, significantly deteriorate.

Perhaps, airlines can offer a special around-the-world package that would combine all of these learning needs. Start the flight with two-hour mandatory reading sessions, and follow with some math to show why what took 80 days in the time of Jules Verne could be accomplished in four weeks now. Then a flight attendant can demonstrate how to use gimp to decorate your headphones. Next on the list is a Captain's special: a short meteorology quiz. Then it’s time for lunch: a selection of mac-and-cheese or mac-without-cheese. Rest time could include a movie in the language of the country over which the plane is flying. Sport selectives would have to be yoga, aisle minigolf or the Endless current pool. The latter could also serve for timeless timeouts. Finally, a daily treat of milk and cookies for snack.

To promote hygiene and insure that kids change clothes daily, enforce a different-color-for-each-day-of-the-week camp uniform.

At the end of the trip, award each passenger an “I made it around the world” certificate, upon release to his or her family. Parents will be thrilled with the accompanying list of things taught along the journey, complete with strategies to reinforce this new knowledge throughout the coming school year. Surprisingly, this whole package would not cost so much. Prices run as low as four weeks of summer camp!

Image courtesy of FunKa-Lerele, distributed under the Creative Commons license

P.S. Humor always helps to forget about financial difficulties but if you are interested in a real money-saving advice on your summer camp spendings, here are a few tips:

Town-run summer camps are significantly cheaper than private camps. In my town they are $130 per week, at least 4 times cheaper than private camps. They obviously do not have the facilities and resources that a private camp could afford, but if you convince you kid's friends to come along, they all will have a blast.
When considering sleep-away camps, take into account that you are buying yourself a few weeks of freedom. No need to pay for babysitters or run early from work to pick someone up at 4pm from a day camp. You are free to work, exercise, shop and go out as much as you like and when you like. Plus, your are getting back a more mature and independent kid than the one you sent.
Summer is about freedom and playing with the friends. Babysitter sharing or mom cooperative where each one supervises all the kids once a week are good money-saving alternatives to summer camps for a limited time intervals.
Many of the camps offer financial assistance. It never hurts to ask.
Children of the camp staff often get significant camp discounts. If you do not have any work obligation for next summer, perhaps you could consider helping at the camp.

Feel free to add your camp cost saving advice in the comments of this story on TheMathMom's website.

Who is your role model - explained by math.

2009-08-11T14:42:00.007-04:00

This fun trick came in the email from my friend. Try it and see how math could explain this mystery.

WHO IS YOUR ROLE MODEL?

Do this without looking at the answers

1. Pick your favorite number between 1 - 9
2. Multiply your favorite number by 3
3. Then add 3
4. Then multiply by 3 (I will wait while you get the calculator)
5. You should have a 2 digit number
6. Add the digits together

Now scroll down.........

Now match your number with the number below to see who is your ROLE MODEL on the list below:

1. Albert Einstein
2. Nelson Mandela
3. Margaret Thatcher
4. Tom Cruise
5. Bill Gates
6. Gandhi
7. Brad Pitt
8. Queen Elizabeth II
9. The Math Mom
10. George Washington

I know... I just have that effect on people... one day you too can be like me.
Do not aspire to anything less...
PS... Stop picking different numbers. I am your idol, just deal with it...

Curious to know how this "magic" works?

Let's say you picked number N.
Following the instructions you multiply it by 3, getting 3N.
Then you add 3: 3N+3
And multiply by 3: 3 x (3N+3) = 9N + 9
Now you are instructed in (6) to add two digits that compound your number.
How could this be that the sum of two digits in your number is exactly 9?

Well, here is secret: Number 9 has a magical property. If you multiply any number by it, in the resulting number the sum of digits is divisible by 9.
Your number is 9N+9, for sure is divisible by 9.
This means that in your number, the sum of its digits is divisible by 9.
This sum could be 9, 18, 27, 36 etc.

But no matter what N you choose you always get a sum of 9 in this trick. Why?
Here is the second secret: Your number N was between 1 and 9, so your result is between 9x1+9=18 and 9x9+9=90
For a number between 18 and 90, the largest sum of digits is 8+9=17.
So, from the sequence of 9, 18, 27, 36 etc. you are stuck with 9 only!

Now, feel free to substitute your name under '9' and forward this further.
Enjoy the magic of math!

Snakes, Cockroaches and Math.

2009-08-06T13:53:00.012-04:00

A very smart girlfriend with a high management post in a large company has recently mentioned to me that she is afraid of three things in this world: snakes, cockroaches and... math. While I can not help with the first two, shivering from the word "snake" alone, I think I could warm up your heart to the last phobia: math.
By showing that you are already using math daily while cooking, shopping, parenting, dating or traveling. By proving that you are actually better at it than you think. And by inviting you to observe daily math and present it to your kids as a toy, a tool and a friend. Here is a link to the slides from my presentation: "Don't transfer your math phobia to your kids."

Math Fun with Summer Produce

2009-07-31T09:02:00.009-04:00

Summer brings abundance of math toys. Who needs cheerios or marbles when you have fragrant, smooth, sweet, colorful delights to count, compare, measure and then gobble up. How many peas are there inside each green pea pod? Do all the pods from the same plant host the same number of peas?

Check it out yourself and make a bar plot to describe the distribution of peas in your snap pea pods.

Heirloom tomatoes: yellow, green, purple, red. What color is the sweetest? What is the sourest? Heirlooms are expensive, but their rare availability makes me splurge on them once or twice a year. This year I decided that our kids are big enough to be able to enjoy the tomatoes’ exquisite taste. We did heirloom tomato sampling and ran a contest.

We cut baby heirlooms into four little pieces. Each of us tried a few pieces of each color and ranked the colors by their sweetness (5 was the sweetest, 2 the most sour). Surprisingly, our opinions varied vastly. We created a chart and added all the ranks. Purple tomatoes were voted the sweetest, yellow – grimacingly sour. This chart is still hanging on our refrigerator to reinforce the memories. Kids loved the game and a week later suggested that we do the same with the canollis that a guest brought.

Plums: seductively purple, innocently yellow, confidently bright magenta. Their variety and colors can compete with a lipstick selection. Can you tell a plum type by its taste? Cut them into pieces, close your eyes and see if you can do better than chance (guessing correctly over 50% of the time for two types tasted in equal amounts, 33% for three etc 100%/n types)? Can you distinguish between plums by shape, with your eyes still closed?

How many large plums are there in a pound? How many little plums? How many little plums can you trade for one large plum?

Do larger plums always have a larger seed?
You can eat your fruits and veggies and do math in the same time.

Carrots:
What can you do, what can you do, what can you do with the carrots?
You can measure your height with them. 10 carrots tall?
You can count your bites while eating them. 9 bites?
You can imagine you are a Musketeer and fence with them.
Wait, you can even scratch your back or head with them.
And, for the finale, be adventurous and swallow them!

(inspired by “What can you do with a shoe?” by Beatrice de Regniers and Maurice Sendak)

P.S. Scientific studies show that kids' brains take a reverse evolutionary zip during summer month and by September they are two months behind in math and reading. Check out this Math Resources page for more pointers to math-infused books and games.

Here is a great fruit puzzle to brainstorm as a family: Mess at the Marrakesh Market.

The Math of Lawn Mowing

2009-07-16T20:09:00.017-04:00

Google and Yahoo have apparently solved their lawn mowing needs by hiring a herder with 200 goats to spend a week, eating the grass and fertilizing at the same time. If you are fortunate enough to own a lawn, but cannot find any goats in the neighborhood, and local rabbits cannot handle this amount of work, you may need to take care of the lawn yourself. Probably, by means of mowing rather than eating the grass. Now, a natural question comes to mind: where to start and in what pattern should you mow to be most efficient?

Consider these three potential mowing patterns.

Which one should you pick and why? Obviously, you want to walk less. More importantly, you want to consume less gas and make fewer sharp turns, as it is rather inconvenient with a bulky mower.

Assume your lawn is 50' by 100', and the blade of your mower is 2' wide.

In the mowing pattern of vertical parallel lines (left), you will walk 25 stripes of 100' each, 2500' total. You will turn half a circle (180 degrees) 24 times.

In the second scenario, horizontal lines (center), you will walk 50 stripes of 50' each, 2500' total. You will turn half a circle 49 times, twice more than in the first case! (You should try turning 180 degrees with a heavy mower to understand how un-fun this is going to be.)

In the third case (right), you will be moving in concentric motions, and you will walk:
100' + 48' + 98' + 46' + 96' + ... + 56' + 4' + 54' + 2' + 52 '=
(100' + 98' + ... + 56' + 54' + 52') + (48' + 46' + 4' + 2') =
152' x 12.5 + 50' x 12' =
1900' + 600' =
2500'.
Same distance! But you will make 48 90-degree turns, which are much friendlier than 180-degree turns.

Overall, the concentric pattern is your best choice. It does not save you any distance, but it allows you to avoid pirouetting with your mower. Only sharp parade turns. Better yet, forget about lawn corners, smooth your mowing angles and turn concentric rectangles into elegant ovals. No sharp turns at all! Just like Google's goats.

Read more stories about math of cooking, dating, parenting, home management and travel from The Math Mom: www.TheMathMom.com
Planning any air travel? See why airplane flight route looks suspiciously curved while the shortest distance between two points should be a straight line.

If I can see You, You can see Me.

2009-06-25T14:43:00.016-04:00

I went to the gym the other day and was changing in the women's locker room when I noticed two little girls observing me in a mirror and giggling. Lockers were obscuring the direct view between us, and the girls seemed to assume that they are the only ones spying. They were shocked when I waved back at them. They hid, then came back to check whether I am still in the mirror, whispering to each other, and excitedly retreated again.

The mathematics of mirrors is one of the most elegant daily math puzzles we encounter. Flat mirrors do a perfect reflection: every single ray of light is bouncing back at exactly the same angle it came in. If I can glance at you in a mirror and see you, you can look back and see me. Our glance goes back-and-force along the same ray, symmetrically reflected by the mirror, making it a non-ideal spying tool. This all is also applicable to your car's rear view mirror. You can see your kids and they can see you. So, no nose picking!

Mirrors a full of magic. Here is a previous story from The Math Mom that sheds light on other mirror mysteries: Mirror Magic Revealed- why mirrors flip Left-Right and not Up-Down, and why ambulance sign is confusingly flipped.

Sarkozy's And My Height Trouble

2009-06-10T16:44:00.032-04:00

French President Sarkozy was caught on camera using a little step stool to appear tall and authoritative alongside Barak Obama and Carla Bruni. How high should the step stool be for Nicolas Sarkozy (5'5") to look manly near his wife, Carla Bruni (5'9"), to not be dwarfed by Barak Obama (6'2")? Simple subtraction tells us that he would need 4" and 9" correspondingly...

By no means am I making fun of Nicolas Sarkozy. In fact, I had my own share of height problems, all the way through middle and high school. This was long before the supermodel age, at a time when fragile teenage confidence distorted my self-image into a caricature of a tall and awkward monster; and when all who mattered were many inches shorter. For the prom, I had to either sacrifice the elegant 2" heel shoes or calculate how far away to stand from my date so he wouldn't perceive me as a T-rex.

Unfortunately, illustrators (unlike paparazzi) do not favor Sarkozy's and my type of “troubled” couple. They never draw women towering above men. So, please pretend the genders in the drawing below are reversed.

My calculations demonstrated that our apparent height difference would be inversely proportional to the distance between us.

Apparent height difference = actual height difference / distance between partners

Our actual height difference was 1”, with 2” extra potentially added by my elegant shoes, bought specially for the prom. 3” total, not in my favor.
Standing right next to me, about 7" apart, my date would have to tilt his head up to look into my eyes. Our percieved height difference would be 3”/7” and the angle that he needs to tilt his head - 23 degrees. However, from further away (30") our apparent height difference would be 3”/30” and he would just need to slightly raise his eyelashes. Around 6 degrees up. From this far, my monstrosity would be almost unnoticeable. Such a distance, unfortunately, would rule out any slow dancing...

Flats were my final choice, but the reward: my long prom night ended with memorable romantic midnight kayaking.

In a mood for more? Read another, older story about Math of Dating from The Math Mom.

Your summer vacation as a math puzzle

2009-06-10T13:42:00.020-04:00

School is almost over and we are facing the whole 2.5 months with our own kids. The bravest of us are meeting the challenge on their own, the more resourceful have prepared a full chain of support starting from the grandparents and ending with a sleep away camps. We all are either low on or hesitant to spend the money, yet we are eager to enjoy the beautiful and short summer months. Here comes the Math - our friendly adviser, to help us save while not missing the fun:

Babysitter Sharing
With 10 kids of various ages on my suburban street, we have often mused that instead of paying thousands for summer camps, we should hire two babysitters and rely on our kids' imagination to figure out the program. Babysitter sharing is usually a lot of fun for kids, and could be a great help financially because the babysitter fee does not grow proportionally to the number of kids they supervise. For example, you may pay $12 per hour for one child, while splitting a $18 per hour fee for two kids would cost $9. The best yet, is to ask one of the kids' favorite school teachers who may dare to supervise all kids from your neighborhood. It may run you $2-$5 per child per hour, while kids will be in trusted hands enjoying their friends' company.

Frequent Flier Tickets
Airline financial difficulties surprisingly bring some good news to our vacation dreams. Tickets are cheaper and frequent flier reservations are more realistic than ever. My resent attempt to reserve a last minute international frequent flier ticket demonstrated great flexibility in the way how miles could be gathered, but demanded a lot of math creativity. Everything seems to be possible:

transferring miles between different accounts
buying missing miles
buying a ticket for one passenger from another passenger's frequent flier account
combining miles from frequent flier accounts of two merged airlines (such as Delta and North West)
using one company's frequent flier miles to buy tickets on all partner airlines.

See below my real life frequent flier math puzzle and the solution that allowed me to get an international round trip ticket for just $240.

Never Stop Partying
It is summer; enjoying friends' company makes us happier and is a great way to network if you are looking for a job. To save money, forget about impressing your friends with a fancy take-out or sophisticated home-made meal. Consider a potluck get together. Everyone loves making their signature dish and you can splurge on drinks, chips and salsa. Add an intellectual twist to your party by asking everyone to bring a puzzle and solve all the puzzles together. My Family Puzzle Marathon site hosts a wealth of entertaining, easy and not so easy puzzles.

Send me your Math-to-the-rescue summer saving ideas.

A real life Frequent Flier puzzle:
Find the least expensive way to gather 85,000 miles for a reward-based international round trip on Continental.
Your family's frequent flier account status:
you have 12,000 miles;
your spouse 26,000 miles,
your kid has 9,000 miles.
You can transfer 43,000 miles from points earned on your credit card.
It costs $15 to transfer 1,000 miles between two frequent flier accounts,
and $32 to buy 1,000 miles from the airline.

Now, remember that behind all these numbers there is a very affordable flight to romantic Europe, exotic Asia or the lush Caribbean. What is the wisest transaction of these miles that can get you there?

Solution:
Transfer 43,000 credit card earned miles to your spouse's account.
26,000 + 43,000 = 69,000.
You still need 16,000 to get to 85,000.
Then transfer 12,000 from your account and 4,000 miles from your kid's account to your husband's account.
You would still pay for mile transfers between the accounts, but it will be: 16 x $15 = $240, not bad at all for an international round trip ticket during the peak summer season.

Great puzzles for your warm summer weekend

2009-06-05T12:20:00.013-04:00

We have a new serious puzzle solver - Alin. Make sure you check the puzzles early in the morning or late at night before mysterious Alin gets to them. Rachel T, Amy and Alin solved at least 3 puzzles each and qualified for a special prize. All the winners, please email me your addresses to ship the prizes to.

Three friends
Three friends want to buy a house that costs $520,000 together. All friends have the same salaries. First one promised to contribute one half of his salary to the house purchase, second friend promised one third of his salary, and third friend one fourth of his salary. How much was given by each friend?
Click to submit or read the answer.

Two rich friends
One friend is saying to another: "If you give me $100, I will be twice richer than you." To this, second friend replied: "If you give me $10, I will be six times richer than you." How much each of the friends has? (from the old Russian puzzle book)
Click to submit or read the answer.

Half confusion
Half of the half of this number is equal to half. What is the number?
Click to submit or read the answer.

Popsicle Sticks
You have gathered 22 Popsicle sticks. How can you use them to mark a rectangle of the largest area, without breaking the sticks?
Click to submit or read the answer.

Kids and Oranges
Can you divide 5 oranges between 6 kids so that you would not need to cut any orange into more than 3 parts?
Click to submit or read the answer.

Next Number
What number comes next in the sequence 1 2 6 42 1806?
Why?
Puzzle submitted by Ola M.
Click to submit or read the answer.

Math is everywhere. We encounter it when organizing a closet, at the preschool drop-off, preparing for a prom and swinging at the playground, while watching TV, shopping, relocating, or simply joking. Just look around. Share with us your own Math Adventure Story. The author of the most unusual math adventure story will get a prize! See all the currently submitted stories and share your story here.

Can you wrap a bus with a newspaper?

2009-05-29T14:19:00.022-04:00

There is an old Russian joke:
- Can you wrap a bus with a newspaper?
- Yes, you can. If newspaper has Khrushchev's speech in it.

Nikita Khrushchev was a Russian president (during the Cuban missile crisis) with little formal education but with a passionate and long public appearances. He is famous for taking off his right shoe and banging it on a desk during one of the international debates.

In 1956 Khrushchev gave a record speech that lasted 6 hours. However, the title for the longest presidential speech lasting 8 hours goes to Venezuelan president, Hugo Chavez. Cuban ex-president, Fidel Castro, is on second place with 7 hours and 10 minutes.

What about our American politicians?
The longest inaugural speech, containing 8445 words and lasting 1 hour 45mins, was given by William Henry Harrison in 1841, who happened to be the shortest serving president as he died from pneumonia after a month in the office. The rest of the presidential speeches were rather unremarkable in length, with US Senators capturing the light spot instead. One record of 15 hour speech goes to Louisiana senator Huey P. Long in 1935. He frustrated his colleagues while entertaining spectators with his recitations of Shakespeare and his reading of recipes for "pot-likkers." Another record for the longest individual speech goes to South Carolina's J. Strom Thurmond who filibustered for 24 hours and 18 minutes against the Civil Rights Act of 1957.

Can you really wrap a bus with a newspaper containing a political speech of such length?

Let's check this joke.

Senator Long's 15 hour long speech is known to have 150,000 words.
One page of NYTimes may contain up to 3,000 words.
150,000 / 3,000 = 50 pages
assuming double-sided printing we get 25 pages to use for wrapping

What is the size of each page?
Current NYTimes page is 21" x 12" (it used to be larger).
The square area covered by one page is 21" x 12" = 252 inch²

What is the size of a bus?

This bus-for-sale from China has a length of 389", width of 98" and height of 134".
Total area to cover when wrapping this bus is:
two sides + top + front + back, that is:
2 x 389 x 134 + 389 x 98 + 2 x 134 x 98 = 142,374 + 26,264 = 168,638 inch²

How many pages will we need to wrap this bus?
area of the bus / area of one page
168,638 / 252 = 669 pages

And we have only 25 pages from Long's 15 hour speech.
It may take 669/25 = 27 of such newspapers or a single speech of 27 x 15 = 401 hours to wrap a bus in.
Fortunately for us, our politicians are not as talkative.
Have a great week and use your Sunday paper wisely.

Read another story by The Math Mom: Sarkozi's and My Height Trouble.

Barbies and Bicycles

2009-05-15T08:39:00.007-04:00

In 1992 Mattel Toys got in trouble for releasing a talking Barbie doll that was randomly programmed to speak four lines from a 270 lines repertoire. Among these 270 lines were “I love shopping!”, “Wanna have a pizza party?” and ... “Math is tough.” Many years have passed since then and a lot has changed in our culture. But occasionally we still hear from our kids that math is tough and math is boring. And math departments of our universities still have more foreign-born students than local crop. (As you probably already discovered, I am one of those foreign weirdo math lovers.) Changing our culture's attitude toward math would take many more years. But if we want to prepare our kids for an interesting life in a global competitive world, let's start by presenting math for them as a toy and showing our excitement while playing with it.

A Try-It-at-Home-With-the-Kids Math Tip: Measure your street with a bicycle

You have probably seen some very serious people rolling a single wheel across the street. They surely could not be suspected as clowning around on a unicycle. They are measuring the length of a road in this very cool way, and you can try it too. Even if you do not have a unicycle. And even if you can’t ride a bicycle, as some of the people I know (sorry, mom). Just roll something with wheels next to you.

Get on your bicycle and count how many pedal rounds does it take to move from one side of the road to the next. Let’s say that it takes 12 pedal rotations to reach the end of your street.
Next, we need to find how much our bicycle travels with each pedal rotation. The easiest is to just mark with a chalk on a road the start and end point of a pedal rotation. Say, your bicycle moves 3 feet while you are doing one full pedal rotation. So, the length of your road is: 12 pedal rotations x 3 feet each = 36 feet.

Notice how many energetic pedal rotations your kid needs to make on his little bicycle to move these 36 feet. For your 12 rotations, he is probably making 36. So, he works 3 times harder than you, which explains why she is so tired after going around the block. Or, perhaps he was just bored and now that you can give him a real assignment of measuring the perimeter of the block, it will get surprisingly more interesting.

Latest Family Puzzle:
You are going for a business lunch with representatives of a company where many employees are known to be liars. Hungry and desperate to get an advice on what to order, you bluntly ask all three of them if they are liars.

The first one says, "All three of us are always telling the truth."
The second says, "No, only one of us is truthful."
The third answers, "The second man is telling the truth."

Well, whom can you trust from these three people?
Click to submit or read the answer.

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Read more cool, fun and non-intimidating math stories, tips and puzzles at www.TheMathMom.com

To fight or not to fight (with the police....)

2009-05-08T08:12:00.015-04:00

You know how I have been saying that math can be our friend, our tool and our weapon? Well, here is an interesting math role as a police persecution tool.

We spent our spring school vacation in Washington DC. Enjoying the spring blossom, museums and monuments. On the way back we got “flashed” by a traffic camera at what seemed to be a rather slow speed on a very wide road. Two weeks later we received an interesting traffic ticket by mail. It claimed that we were driving at a speed of 46mph in a 35mph zone. Were we really? How does the camera knows? Prove it! Carefully reading this high tech traffic ticket, I realized how police are using some cool technology blended with math to create a convincing case... against us.

Speeding was detected by police radar that instructed a police camera to take two photographs of the speeding TheMathMom vehicle. The snapshots were taken with a time interval of 0.2 seconds, as the little stamp on the bottom shows. How do these pictures prove that we drove at 46 mph? Let's see...

If we drove at a speed of 46 mph, let's calculate how far we would move in these 0.2 seconds captured on the pictures.
speed = distance / time

from this:
distance = speed x time
distance = 46 mph x 0.2 sec

converting seconds into hours:
distance = 46 mph x 0.000055556 hours

multiplying, we get:
distance = 0.002555576 miles.

A very tiny number. Let's convert it to feet. I Googled that there are 5,280 feet in a mile. So, 0.002555576 miles would be 13.5 feet.

Ready for the final verdict? Did our car really move 13.5 feet in-between these two pictures?

Our traffic ticket explained that white marks on the road are painted with 5 foot intervals. I marked our car's back wheel positions on both snapshots with the red lines, observing that our car moved somewhere between 12 and 14 feet in-between these two pictures. 13.5 feet does, unfortunately, fit here very well, proving that we are guilty as charged. Oooof, I sometimes hate math!

Try it at Home:
Check out the new Try It at Home page from The Math Mom with a collection of tips and tricks to try at home.