Raise your hand if you’ve heard of M.C. Escher. Now raise your hand if you know what tessellations are.
Surprise! If you know of M.S. Escher’s work, you are also familiar with tessellations — even if you don’t recognize the term. In fact, if you have or have seen a tiled floor, tessellations are familiar to you.
A tessellation is a pattern of identical, interlocking shapes. There can be no space between the shapes and none of the shapes can overlap. Escher created complex tessellations of birds, lizards and fish. But even simple, square tiles are tessellations.
This video shows how they are made. Don’t watch it expecting a tutorial. Just look at how the shapes are formed and then replicated and rotated to form the tessellation. A design like this one is pretty complex, but it’s interesting to see it in motion.
(There is no sound with this video, so there’s no need to crank up your speakers.)
Bonus! I found this really great video that shows how to make a tessellation. Check it out.
Where have you seen tessellations? When do you think they’re useful or interesting to see? Leave your comment!
So, I’ve seen an abacus (or the plural, abaci), but until this video, I’d never seen one in action.
Now, I’m not a big fan of speed in math, but I have to admit this is pretty darned cool — especially when the kids imagine the abacus to solve problems without it.
But how does an abacus work?
There’s a great (though complex) explanation here. But here’s the really cool thing: to use an abacus, you have to have a really strong, knee-jerk sense of numeracy. So, it’s not so much a shortcut as a demonstration of a great understanding of math basics.
Even if you weren’t in the predicted path of Hurricane Irene, you likely heard about pretty much nothing but last weekend. We Americans love our big storms. There’s been lots of grumbling lately about over-hyped media coverage, especially since the hurricane was downgraded to a Category 1. But this was a bad storm — big, slow moving and full of rain.
Remarkably, the storm followed the path that meteorologists predicted: hitting landfall in North Carolina and moving up the east coast, hugging the shore. How did they know?
This video from The Weather Channel gives a really good explanation.
By the way, I apologize for being absent online for much of this week. I didn’t have power until Monday, and then it took another three days for my internet connection to be restored. Hope everyone else fared well!
It’s the perennial question from students of all ages: “When will I use this stuff?” So when tutor, Ryan faced this query (probably for the upteenth time), he took to the streets to find the answer. What he found is in the video below:
And of course I have some thoughts — for teachers and students.
It is absolutely true that series (that’s what the funny looking E — an uppercase sigma — means in this problem) are not the stuff of ordinary folks in non-science fields. But they’re not as difficult as they seem. It’s the notation that’s confusing.
Skip this part, if you don’t really want to do any algebra today.
A series is just the sum of a sequence (or list) of numbers. That’s it. Nothing more, nothing less. So when you have
sum{n=1}{7}{3n-1}
you’re simply saying, “Find the sum of the first 7 values of 3n-1, where the first value of n is 1.” In other words: 2 + 5 + 8 + 11 + 14 + 17 + 20 = 77.
Now back to my opinions.
Okay, so I don’t need to know what a series is in order to visit the grocery store or get a good deal on a car or even figure out how much I earned this year over last year. But here’s what I wish some of those folks who were interviewed for this video had been able to say:
“That funny-looking E is a Greek letter, right?”
“Doesn’t this have to do with adding things together?”
“Hey, I dated a girl from {Sigma}{Sigma}{Sigma} once!”
And second, this tutor did pick a humdinger of a problem to focus on. Series (and their brothers, sequences) are not the main focus of any mathematics course. But honestly, they wouldn’t be taught if they weren’t useful somewhere. And boy-howdy are they useful!
1. Architecture: “An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. How many seats are in the theatre?”
2. Business: “A company is offering a job with a salary of $30,000 for the first year and a 5% raise each year after that. If that 5% raise continues every year, find the amount of money you would earn in a 40-year career.”
3. Investment Analysis: “A person invests $800 at the beginning of each year in a superannuation fund. Compound interest is paid at 10% per annum on the investment. The first $800 was invested at the beginning of 1988 and the last is to be invested at the beginning 2017. Calculate the total amount at the beginning of 2018.”
4. Physics: “The nucleii of a radioactive isotope decay randomly. What is the total number of nucleii after a given period of time?”
And this brings me to some additional news of the week. Sol Garfunkel (Consortium for Mathematics and Its Applications) and David Mumford (emeritus professor of mathematics at Brown) made a bit of a splash on Wednesday, with an editorial in the New York Times: How to Fix Our Math Education.
Their proposal is that we teach tons of math that applies to everyday life — and focus on those applications. (Yay!) And we ditch “highly conceptual” math for folks who won’t need it for their jobs. (Boo!)
Hopefully, you’ve already identified the problem: How do we know if a kid won’t decide to go into physics or engineering or high school math education? Hell, how do we even attempt to lure them into these fields, if they don’t see the math at all? (And by the way, physics, engineering and applied mathematics were recently identified as the top-paying degrees in the U.S.)
Look, I empathize with the student who isn’t interested in what any of the Greek letters mean in math class. And I think it’s true that most folks won’t use these skills at all after high school. (It is worth mentioning that everyone depends on series in their daily lives–they just don’t see the math.) But my response to the kid who asks, “What’s this good for?” is to tell him where it can be applied.
And if he says he won’t be going into any of those fields, I would say, “Suck it up, cupcake, because you’re too darned young to know for sure.”
Please share your thoughts in the comments section. Do you agree that these concepts should be taught in high school, even though most kids won’t use them in their everyday lives? How do you think we should encourage more students to go into science, technology, engineering and math (STEM) fields?
Earlier this week, I provided a guest post about math anxiety and kids for Imp3rfect Mom. I wasn’t surprised to get a comment from a reader asking about how to deal with her math anxiety.
My son is an adult so my question concerns me. I’m almost 60 and I’ve been mathphobic (big time) since I was in 6th grade. At that point math just crashed and burned for me and I struggled for the rest of school. Now I am self studying for a designation related to my job (the job itself doesn’t require math ability) but I have to learn some equations for the Time Value of Money for the last exam. I look at that chapter and just freeze. I actually am telling myself “well, if I just skip that part and study real hard, I’ll still pass the test.” This is ridiculous! How do I conquer 50 years of Fear of Math?
I’m sure you can hear the frustration in her writing. (Do you ever feel the same way?) I anxious about certain things–making difficult phone calls, traveling to places where English is not the predominant language, or asking someone for help when I’m lost. (That last one is so silly, isn’t it?)
I’ve talked about the roots of math anxiety–the insistance that the goal is the right answer, timed calculations and an expectation of perfection–but now it’s time to share some ways to cope.
Allow yourself to fail. This is not so easy when you’re dealing with your finances or preparing to take a test. But when you’re learning (or relearning) something, you will make mistakes. Heck, even when you have something down cold, you can screw up. If you’re feeling anxious about math, set up low-stakes scenarios when failure isn’t a big deal. Try things on your own, for example, and allow someone you trust to check your work.
Ask yourself, “How hard can it be?” I’ve said this before, if I can do this stuff, so can you. I don’t have the typical “math brain.” I can’t do mental calculations, and sometimes I forget really basic facts like 6 x 7. And believe me, if a fourth grader can do these tasks, so can you.
Make it fun. I swear, I’m not violating math secret #3 (You Can Skip the Love). You don’t have to have fun or love math to be good at it. Still, if you’ve read my book, you know what I mean. Too often, math is cut-and-dry, boring numbers. When it’s presented or explored using real-world stories with funny characters, it’s a lot more tolerable. So, whether you’re studying for a test or trying to explain a concept to your kid, try making up problems using Sesame Street characters or your crazy Aunt Miriam who has 76 cats and wears a fedora. The sillier the better.
Find resources that work for you. I’m a big DIYer. And everything I know about sewing, painting, renovations and carpentry, I learned from Google. I promise. Besides my book, there are amazing resources out there for folks who need a little refresher. You can even find videos on YouTube or Flickr tutorials. But be careful: sometimes mathematicians think they’re being really helpful, when they’re not. Don’t let yourself be overwhelmed by minute details or unrelated tangents. Click through these resources quickly until you find what you need.
Trust your gut. Just because a textbook or a friend has the information you need, doesn’t mean you need to follow that advice or process. This is the beauty of being a grownup–we don’t have to follow the rules that a teacher sets out for us. Think about when you feel comfortable with math. Is it in the kitchen? When you’re gardening? When you’re doing your budget? What is it about that process that is less threatening? Use what you know about yourself–and your learning style–to step into these other, scary places.
So I’d love to hear from you now. What tricks have you used to conquer your anxiety or fear–about anything? If you have dealt with math anxiety in the past, what has helped? Share your ideas in the comments section.
I don’t usually post on Sundays, but with Geithner’s debt-ceiling deadline looming on Tuesday, I wanted to share this really great video. Using some math and graphs, the narrator explains the debt, deficit and debt ceiling in ways that even your 4th grader can understand.
It’s a little long — almost 10 minutes — but trust me, it’s not full of the gobbledy-gook that economists are sometimes famous for. You will be smarter after you watch it. Promise.
Questions? Ask them in the comment section. (But please skip the political comments. Math is neither Democrat nor Republican.)
Also, be sure to come back tomorrow for an exciting August announcement!
Okay, so most parents really do understand how to encourage literacy. We read signs, the backs of cereal boxes, the comic section and of course classics like Harry Potter and the Deathly Hallows. But injecting a little everyday math into long summer days can be a bit of a challenge.
Good Morning America to the rescue!
In a regular feature, the morning show brings in a “sneaky teacher” to show parents how to continue learning through July and August. And my good friend and fellow freelance writer, Debbie Abrams Kaplan was featured last week.
It’s a cool video, but unfortunately, I can’t figure out how to embed it. So just click on the picture below to view it. It’s worth the extra step! (Debbie’s kids — and she! — are adorable.)
Ron S. Doyle is both a web designer and a freelance writer. In fact, he’s found a particular niche in developing web sites for other freelance writers. He’s also got a wicked sense of humor and uses math in his work.
Can you explain what you do for a living?
The highfaluting answer: I help clients build or restructure their online presence through web development and design, business analysis, project management, strategic brand management, consultation and training.
The mundane answer: I make websites!
When do you use basic math in your job?
I use proportions, algebra and basic geometric concepts at work every day. Most of what I’m doing involves simple addition, counting pixels. For example, if a website’s main container is 960 pixels wide, I have to make sure that all the margins, padding, borders and boxes inside add up.
This basic addition turns into algebra when a client comes to me and says “I have this 300 pixel wide advertisement that must go here” or “I want to embed this YouTube video there.” Then all the other elements become variables—and I change them to make everything balance with the ad or video.
It gets even more complicated when I start adding things like drop shadows or glowing edges to an object, which have a specific radius from the edge. A 3px drop shadow spreads 1.5 px past the edge of the object, etc.
Certain objects, like videos, also must appear in specific proportions, e.g., 16:9. For example, if I know I must fit a high-definition video into a space that’s 500 pixels wide, I know that the video will be a little more than 281 pixels tall.
16/9 = 500/x
16x = 4,500
x = 281.25
I also use proportions for my favorite design element: The Golden Ratio, 1: 1.618. It’s a proportion that naturally occurs in nature and is used widely in design and architecture. I agree with the ancient Greeks that it’s a beautiful shape and I try, whenever possible, to use it in my designs. Sometimes, it’s a fun little secret for me. For example, Ann Logue’s website doesn’t seem to have many boxes or rectangles at all:
But there are actually seven golden rectangles coded into the layout:
Do you use any technology (like calculators or computers) to help with this math? Why or why not?
I use paper to draw initial designs, a calculator to figure out proportions and design software like Adobe Creative Suite to help with measurements and placement of objects before I write any code. I suppose I could do it all while I’m writing the code, but I like to keep costs low for my clients—and I like going outside from time to time.
How do you think math helps you do your job better?
Math doesn’t just help me do my job better, it makes it possible.
How comfortable with math do you feel?
None of this math feels uncomfortable to me. All web designers use math, whether they realize it or not, but some have a natural ability to see things like the golden proportion without picking up a calculator. I don’t know if I have that innate aesthetic skill—so the numbers make me feel more confident in my design decisions.
What kind of math did you take in high school? Did you like it/feel like you were good at it?
I didn’t develop a relationship with math until the seventh grade. That year, I had a great algebra teacher; things just clicked and I’ve loved mathematics ever since. I took Geometry, Algebra II, Trigonometry, PreCalculus and AP Calculus in high school. I always felt confident in math class, except Calculus; my teacher struggled teaching the subject and I had a bad case of graduation fever.
As a psychology major in college, I didn’t love research but I enjoyed the statistical part of the work (and I took Calculus for Engineers even though it wasn’t required). Before I started my current business, I was a high school teacher. Trigonometry was one of my favorite subjects to teach.
Did you have to learn new skills in order to do this math? Or was it something that you could pick up using the skills you learned in school?
School definitely helped me feel confident with math, but I learned the skills I use today from building things with my father when I was younger. I spent a lot of my childhood with a tape measure with my father rattling off fractions at me—I understood 5/8 and 3/4 and 9/16 on a visual level long before I learned them in school.
Everything else I learned from Donald Duck in Mathemagic Land:
Do you have questions for Ron? If so, ask them in the comments section below.
Math for Grownups blog readers tend to fall into two camps: grownups who are not parents and really hate math (or think they’re not good at it), and parents who are worried that they’re going to pass along their math anxiety to their kids. And so I thought I’d spend a little bit of time addressing some of the concerns of these parents.
Earlier this week, my friend and fellow freelancer, Debbie Abrams Kaplan forwarded the summary of a new bit of research on kids and math. Debbie is the author of two great blogs: Jersey Kids and Frisco Kids, and she figured that I might find some blog fodder from this study.
Boy did I! A couple of things jumped out at me:
No one has ever studied how the basic math skills of first graders affect their later understanding of math throughout elementary school. (Compare that with the many studies of early reading skills, and this fact will blow your mind, too.)
There are three basic skills that will help first graders become good fifth-grade math students.
I’m going to tell you those skills a little later, but first I want to introduce the concept of numeracy. Quite simply, numeracy is the ability to work with and understand numbers. When children are young, numeracy includes the ability to count, recognize the symbols that we use for numbers (which is akin to learning the alphabet), and even do some very simple operations (like 1 + 1 = 2). For high school students, numeracy includes more complex problem solving skills and properties of real numbers.Among math educators, there are big debates about how we can better teach numeracy. I guess this is like the debates about phonics vs. context support methods in reading education. But now that this study is out, it’s clear parents can help lay a firm foundation for our kids’ later success in math. According to this study, published by a team of University of Missouri psychologists, rising first graders should understand:
Numbers — I’m going to take this to mean whole numbers, since most first graders aren’t very familiar with fractions or decimals.
The quantities that these numbers represent — In other words, kids should be able to match a number with that same number of objects (five fingers, two cats, etc.)
Low-level arithmetic — And I’m guessing researchers mean things like adding and subtracting numbers that are smaller than 10 (excepting problems with negative answers).
If you’re like most parents, this is probably a duh moment. What’s so hard about recognizing whole numbers or understanding what five objects are? But I don’t think many parents spend much time emphasizing these ideas — at least not in the way that we commit to reading to our children every night.So here are a few ways that you can help instill numeracy in your pre- or elementary-school aged children.
Count things. Count everything — like the stairs that your climbing or the cars that pass your house or blocks as you take them out of the box or those adorable little toes!
Have your child count things. You can do this in really simple ways. Ask him to get you five spoons so you can set the table. When she wants some goldfish, tell her she can have 10 (and watch her count them). When you’re planning his birthday party, have him tell you which 10 friends he wants to invite. (Write them down for him, so he has something visual to count.)
Notice numbers. When she’s really tiny, ask her to say the numbers that are on your mailbox or on a license plate. Older kids can name multi-digit numbers, like 157 or 81. (And if you want to really be precise and prep your kid for school, don’t say things like “one hundred and fifty-seven. In math, “and” represents a decimal point, which is something most elementary school teachers will really drive home.)
Teach your child to count backwards. This can be a great way for kids to start understanding subtraction. If you know you have 10 steps in your staircase, count backwards as you go down the stairs. Then count frontwards as you go up!
Start adding and subtracting. Give your child 5 raisins and show her how to “count up” to 7 by adding 2 raisins to the pile. Then as your child eats the raisins one by one, “count down” to find out how many are left.
You don’t need to make a big deal about math. And for goodness sakes, skip the worksheets, flashcards and even video games — unless your kid really loves them. Integrate these basic skills into your daily life, and you’ll see your child’s understanding grow. (And you probably won’t feel so stressed out about it all!)What kinds of things do you do with your young elementary-age kids? Any teachers out there want to share their thoughts with the class? Post in the comments section.
You really don’t have to know or care what “binary trees” are to appreciate Vi Hart’s genius. And I’m so excited to finally introduce you all to her.
Vi calls herself a “recreational mathematician.” In other words, she plays with math, and it’s really amazing stuff. Just a couple of years ago, she graduated from Stony Brook University, with a degree in music. (Her senior project was a seven-movement piece about Harry Potter.) Before that, she got hooked on math when her father took her to a computational geometry conference. (George W. Hartis now chief of content for the soon-to-open Museum of Mathematics in Manhattan.)
In short, she’s not a trained math geek. She just loves math.
She’s also funny and infectious. I dare you to watch this video and not laugh. And nope, you don’t have to know what binary trees are to get the jokes. (Psst, you don’t even have to love math to love Vi.)
I’ll post more of Vi’s awesome videos in weeks to come. Let me know what you think in the comments section!
As I was planning my posts for the week, I came across this fantastic video about the U.S. deficit and debt. At first I had it scheduled for Friday, but with Geithner’s debt ceiling deadline looming on Tuesday, I decided you would probably benefit from seeing it sooner. Who knows what will happen this week, right?
No matter what happens, you may have been wondering about the math involved with the deficit, debt and debt ceiling. This handy video will lead the way. It’s a bit long — almost 10 minutes — but very easy to follow and worth every second. You’ll end up being a much smarter person for watching it!
Questions? Ask them in the comments section! (But keep the political commentary for another site, please.)
Over the last year, I’ve come across lots of great math-related videos, and now that my blog is up and book is out, people are sending me links to many more. I thought Fridays would be a great time to share them. So, welcome to the first edition of Film Fridays!
Today’s little clip comes courtesy of my mother-in-law, who majored in math and then went on to have a seriously incredible career as a sales representative for American Greetings. She uses math like it’s a second language — no big deal, thankyouverymuch. (She also makes the most amazing pies ever.)
Still, this clip is a bit geeky — as many math videos are. What I encourage you to do, though, is find the artistry and magic. There will be no quiz. This is just for fun. (Details are below the clip.)
So while this looks absolutely magical, it really does boil down to some very simple math. The length of the pendulum determines how far it swings, and that in turn determines how many swings (or oscillations) it can complete in a given period of time. In plain English: a short pendulum swings faster than a long one. So the smarty-pants at Harvard built this pendulum based on the design of University of Maryland physics professor, Richard Berg. Here’s the nitty gritty, if you’re interested:
The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations.
In other words: Pretty.
I am so excited to show you more videos! I especially can’t wait to introduce you to Vi Hart, who does the most captivating math doodles you can imagine. (Wait a minute, who else does math doodles?) So check in next week. And if you have a video that you want to share, please send me the link: llaing-at-comcast-dot-net.
