The Odds were at odds with the Evens. It never seemed fair to them that two Odds made an Even but two Evens didn't make an Odd. Fifteen fired the first salvo by stepping into the order twice. Sixteen managed to jump in, but then Eighteen disappeared. Seventeen and Nineteen were prime suspects. The Numerologist stepped in and told everyone to get back in the right order or ELSE. Unfortunately Eighteen was still missing. The authorities launched an investigation but there were so many factors involved that they never could get to the root of the problem.

Riddle: What do 10011, 23, and 13 have in common?

Gorgeous! The photo above, of an anglerfish ovary magnified 4 times, was taken by James E. Hayden, and won 4th place in the Nikon Small World photomicrography competition. You can see lots more winning photos here. More about Hayden at the end.

Getting down to business, allow me to welcome you to a math buffet. I think there will be something here to tickle everyone's palate. (Photo by skenmy.)

- Maria Andersen is at it again! She's redesigned her math for elementary teachers course, and shows us some of the results in Transforming Math for Elementary Ed. This post is full of links to work her students did, some of it exciting stuff.
- John Golden offers us a game called Area Block modeled on Blokus (one of my current favorites to play with kids). I haven't had a chance to play it yet, but I am definitely looking forward to it. (Here's a bit of math teachers at play trivia. John and Maria work within 30 miles of each other in my home state. What state is it?)
- Denise walks us through an algebra word problem, translating from English to "mathish". Nice! And Jason wants our help with teaching students how to read a few different types of problems, in "When vocabulary isn't the issue" and "A reading experiment". The puzzle given in that second post looks fun!
- Pat Ballew gets us thinking about geometry with his "Notes on Cyclic Quadrilaterals".
- Do we discover math or invent it? How we answer that question affects how we talk about the math of the ancients. Dan MacKinnon reviews a book and discusses this intriguing issue.
- A short review of The Little Book of Mathematical Principles is at We Overstep.
- Pumpkin Patch offers us another game, Sum Math.
- The descriptions don't always look accurate to me, but you may find some goodies among the "100 Incredible Open Lectures for Math Geeks". Some are audio only, some are videos.

If you haven't stuffed yourself yet, here are a few more tidbits I noticed over the past two weeks.

- Two mysteries solved, one a murder, and one the opening chord of Hard Day's Night.
- Sometimes there's a hard way to solve a problem, and an easy way...
- Does a bullet that's fired from a gun land at the same time it would if it were dropped? Built on Facts gives us some equations and Myths Busters weighs in on YouTube.
- On thinking before you compute.
- Like comics? Indexed on confusion as a function of information, and a graphic novel about Bertrand Russell.

Here's the interview Nikon did with James E. Hayden. I wrote and asked him how he uses math in his work. He said:

As for the math - it is, of course a large part of doing any kind of research. In a lot of my regular work, we work with analysis programs that quantify different aspects of the images we capture. Everything from simple counts (how many cells in the field of view?) to time lapse analysis - how fast are the cells moving? In what direction? At what angle to the original movement? Does the rate of movement change over time? Do we need to quantify the interaction of the cells in some way? Is there a change in the fluorescence intensity values? And other interesting things like that. My assistant, Fred Keeney (who won an Image of Distinction in this year's competition) has been taking computer programing classes to help us automate these kind of analyses.

**Submit**your blog article to the next edition of

**math teachers at play**using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page. (Our schedule is changing to once a month. Denise at Let's Play Math! will host the next carnival on November 20.)

My AT&T internet at home has been down for 3 full days. If anyone wants a quick edit on this, call me on my cell phone: 510-367- eight oh eight and last number is one plus four. (There! Try to get me now, you spambots!)

ReplyDeletemagnified four times?

ReplyDeleteThanks for correcting the numbering. Now let's see if it stays corrected.

ReplyDelete@Barry: If not, we'll just have to explain it with a good story about our funky-freaky halloweeny number system. ;^)

ReplyDelete@vlorbik: Well, that's what the slide said. Maybe I should have asked the photographer if they had it right, huh? Thanks for catching that... (Still offline at home. If lucky, I'll have your answer by tomorrow.)

I'm still not 100% sure, but I think it's magnified just 4 times on the original slide sent in, and not too much more by our screens.

ReplyDeleteHere's Jamie's full reply:

Total magnification is often misunderstood and incorrectly calculated. When I submitted my entry, I stated that I used a 4X objective to take the image, so that is where the figure published with the image came from. There is additional system magnification in the microscope that would increase that amount, and then additional print magnification to get to the size it is printed in the calendar. Not knowing what physical size it is on each computer screen, I would have no way of knowing actual magnification everywhere it is seen. Total magnification ultimately depends on the physical size it is displayed compared to the original size the specimen actually measures. In a publication figure, I would include a scale bar in the image to use as an internal reference, but that is not used in these images. Looking at the image through the 10X eyepieces on my microscope would provide a virtual magnification on the retina of X40 (4X objective x 10X eyepiece. And some people try to use that for total magnification as well.

So the short and long answer to your question is "It depends." I use the 4X value to indicate the objective used to capture the image, but your reader is correct to question what the total magnification actually is. In truth, I take every value listed with each of the winning images with a grain of salt because I do not know what formula they used to arrive at the value they indicated. In a situation like this, I think the objective value alone has the most relevence, because any microscopist knows about how big something is when they look at it through a particualar objective. That does not make it accurate, just relative.

For precise measurement, the only truly accurate approach is to take an image of a known value (we use a stage micrometer - a laser-etched ruler that can be read under the microscope). This image is captured with the same optical configuration as the image in question. It is then used to determine the size of a scale bar that can be added to the original image, providing the aforementioned internal reference.

I am not in the lab at the moment, so I cannot tell you the value off hand. I'll look it up at work tomorrow.

I hope this answers your question.

I wrote back:

Well, I guess I knew that "it depends" must be part of the answer, since it's smaller on my blog than it is on the original site. But I was guessing those yellow and red blobs were cells or eggs, and I'd expect the real things to be hundreds of times smaller. (My screen is about the size of a sheet of paper.) Are the structures in this photo visible without magnification?

And he replied:

I've attached the reference image used for my micrograph. This image is the same size and magnification as the one in the contest. The entire scale is 2.2mm left to right. The larger increments are 100 microns (0.1 mm) apart and the very close ones over on the left are 10 micron increments. The entire field of view in the image is 3mm across, so that gives you an idea of how big the structure is. Yes, you can see the structure by eye, this is just the very inside part of the spiral. Eggs are pretty big. The yellow-orange in the image is yolk, So the larger spheres are eggs that are fairly developed. The largest circles are about 300 microns (0.3 mm) across (a little smaller than a period at the end of a sentence). The solid red cells at the base, near the greeen spiral, are primary oocytes that have not developed enough yet.

I suppose you can now see that math is very important when dealing with microscopy!

Nope. I didn't have it quite right. Here's Jamie's latest reply:

ReplyDelete"Without going into all the detail, the specimen magnification is about 4 times original size on the camera imaging chip, then many times more to the size you see it on the screen. The chip is only about 8.8mm across. If you see the image on your screen and measure it, and it is about 8.8cm across, that is 10X more magnification than the captured image, so the total magnification would be (x4) x (x10) = x40 If the image is about 17.6cm (about 7 inches) across on your screen, it would be 20 times larger than the captured image, so the total magnification would be x80."

On my screen, it measured 6.5 cm, so that would be about 30x magnification for me.

This comment has been removed by the author.

ReplyDeletemuch clearer now.

ReplyDeletebetter than i hoped for

actually. play on,

o math teacher.

I notice that there is no answer to your riddle (what do 10011, 23 and 13 have in common?)

ReplyDeleteI am curious why 103 is not on that list (I'm not challenging the riddle maker, I'm just wondering why the real world skipped over 103)

Jonathan

Thanks for the question, JD! I was hoping comments would come around to the riddle. I like your addition. Put in its proper place, we have a series, instead of the random collection I started with. So, part 2 of the riddle: Where does JD's number go?

ReplyDelete(Sequence, not series, sorry!)

ReplyDelete