Monday, April 4, 2016

Square Root Day (Day 132)

Today marks the first day of school after spring break. What this does mean is that the longest stretch of my school year without a holiday is actually from Easter to Memorial Day. This year, there is an eight-week stretch from the long Easter vacation to the long Memorial Day weekend. But "Long April/May" doesn't have the same ring to it as "Long March" does.

But today also marks a special day on the mathematical calendar. The date, April 4th, 2016, can also be written as 4/4/16, and we note that four times four is sixteen -- that is, 4 is the square root of 16. I write this in ASCII as sqrt(16) = 4.

Therefore today is known as Square Root Day. Here is a link to the website of Ron Gordon, a fellow California high school teacher and the creator of Square Root Day:

Gordon writes:

Square Root Day is Monday, 4/4/16. This marks an historical convergence with the opening of the Major League Baseball season. Don’t miss our only Square Root Day in the 16-year span from 3/3/9 to 5/5/25.

So according to Gordon, the last Square Root Day was March 3rd, 2009, and the next Square Root Day will be May 5th, 2025. Square Root Day. Indeed, he writes:

There are only 9 of these days in a century, so let’s all enjoy a little math fun on Monday. We're hopeful of generating a lot of interest and enthusiasm, and giving lots of folks a smile.

Therefore Square Root Days are rarer than Leap Days (of which there are 24 or 25 in a century), but more common than Pi Days of the Century (of which there are only two, counting both the truncation of Pi Day on 3/14/15 and Round It Up to Pi Day on 3/14/16).

Gordon first created Square Root Day on September 9th, 1981, or 9/9/81. As it turns out, I was only nine months old on the first Square Root Day -- yes, my age in months was a perfect square! Also, notice that believe it or not, Square Root Day actually predates Pi Day, as the latter holiday wasn't celebrated until six and a half years after the first Square Root Day! Yet Pi Day is much better known, probably because Pi Day occurs every year, unlike Square Root Day.

The second Square Root Day was on January 1st, 2001, or 1/1/01. This must count as the first of the nine Square Root Days that exist in the century, but notice that Gordon, at his link above, writes very little about this Square Root Day. This is probably because people tend to forget that 1 (as well as 0) is a perfect square (sqrt(1) = 1, sqrt(0) = 0), and also it is overshadowed by New Year's Day, part of the winter break rather than a school day. Gordon himself writes, "It just seemed goofy" to celebrate on the first of January.

The next Square Root Day was on February 2nd, 2004. This was the same as Groundhog Day, and so Gordon celebrated both holidays that day. I remember reading about the Square Root Day on March 3rd, 2009 -- I was a student teacher at the time, but I did not mention it in class.

Some readers may notice that the next Square Root Day in 2025 will be special, because 2025 is itself a perfect square -- sqrt(2025) = 45. Because of this, perhaps 4/5/2025, incorporating the digits 4 and 5, will become a sort of "Square Root Day of the Century," and be celebrated in addition to 5/5/25, the usual Square Root Day. But actually, I suspect May 5th will be celebrated rather than April 5th, only because the latter day will fall on Saturday, a non-school day, while the former day will fall on Monday. a school day. And besides, Gordon will probably make it a double celebration with Cinco de Mayo, just as he combined Square Root Day with Groundhog Day in 2004.

The last Square Root Day of the Century was back on April 4th, 1936 (as sqrt(1936) = 44), and the next one will be on April 6th, 2116 (as sqrt(2116) = 46). As many people alive back in 1936 are no longer with us (my late grandmother was one year old), and someone born today would have to live to 100 to make it to 2116, 2025 will be the only perfect square year for most of us.

Notice that 1936, 2025, and 2116 all contain ordinary Square Root Days (on 6/6/36, 5/5/25, and 4/4/16) in addition to Square Root Days of the Century (on 4/4/1936, 4/5/2025, and 4/6/2116). This is not a coincidence -- every perfect square year from 1681 (which is 41^2) to 3581 (which is 59^2) contains both types of Square Root Day! One of these is doubly special -- 55^2 is 3025, so May 5th, 3025 -- around the year that Futurama is set -- will be both types of Square Root Day.

As it turns out, we can use Algebra to see why years from 1681 to 3581 contain both types of Square Root Day. We know that (a + b)^2 = a^2 + 2ab + b^2, and here we wish to consider the case where a = 50 and b is an integer ranging from -9 to +9, so that a + b ranges from 41 to 59. We then see that a^2 = 2500 and 2ab = 100b. Both of these give numbers ending in -00, so the last two digits of the square (a + b)^2 are completely determined by b^2, which is a square ranging from 1 to 81. So the perfect square years from 41^2 to 59^2 contain both types of Square Root Days. QED (Be careful that when b is negative, b^2 is still positive. This is why for 2116 = 46^2, a = 50 and b = -4, so even though 46 ends in 6, the ordinary Square Root Day that year is 4/4/2116.)

By the way, the name "Square Root Day of the Century" for 4/4/1936, 4/5/2025, or 4/6/2116 is actually a misnomer. It would be just like calling ordinary Square Root Days like today, 5/5/25, and 6/6/36 "Square Root Day of the Decade," even though they seem to occur about once per decade. After all, the first decade of this century had three Square Root Days (1/1/01, 2/2/04, 3/3/09), while the decades of the 50's, 70's, and 90's contain no Square Root Days.

We can use Calculus to see what's going on here -- the xth Square Root Day (of either type) occurs in the year number y = x^2. So we expect Square Root Days near the xth Square Root Day to occur about dy/dx = 2x years apart. It just so happens that we're looking at the cases where x is near 5 (for ordinary Square Root Days) and 50 (for Square Root Days of the Century), so 2x just happens to be near 10 or 100. So I mistakenly called them Square Root Days of the "Decade" or "Century."

How exactly do you celebrate Square Root Day, anyway? We see how Gordon often has a party tying the Square Root Day to another celebration like Groundhog Day or Cinco de Mayo. This year, Gordon is tying the celebration to one of the biggest sports days of the year -- not just Opening Day in baseball, but also the last day of college basketball:

March Math Madness has a glorious final date: The Sweet 4 Squared, which has winnowed to the Final Two Squared, will culminate in the NCAA Championship which shares the Baseball spotlight on Square Root Day---see, college math isn't that hard!
Gordon writes that Square Root Day also has its own rituals in itself:

Things for folks to do on Square Root Day:It's a good day to.......get things squared away, try to fit a square peg into a round hole, go square dancing, tie a square knot, travel on Route 66, drink rootbeer from a square glass, root for the underdog, eat a square meal, or watch the pigs root around.

We know that on Pi Day, we can serve our students pie -- fruit or pizza. Gordon writes that on 2/2/04, he sent radishes cut into squares -- in other words, square roots -- to the groundhog. But unlike groundhogs, humans rarely eat raw root vegetables, so this might not be work in our classes. Perhaps we can serve our students anything that's shaped like a square -- square cookies or cakes, for example.

Finally, Gordon has a contest for Square Root Day. No, we don't have to answer math problems like J. H. Conway's toughies back on Pi Day. To win the share of $441.60, you must simply throw the largest Square Root Day party!

Gordon also adds a few links to other special dates. Later this year is 12/13/14, which he refers to as "Tic Tac Toe Day" as there are three numbers in a row. And next year is 11/13/15, or "Odd Day," as there are three consecutive odd numbers. To me, these are just a way to extend the magical repeating dates from 1/1/01 to 12/12/12. After 11/13/15, there won't be any more special dates like this until 2101, after most of us will be gone, unless you count repdigit dates like 2/2/22 (which Gordon does and calls "Trumpet Day") up to 9/9/99. By the way, the first time I've ever heard of anyone playing around with digits and dates was back in the first grade, on 10/9/87.

Now today's still a school day, so I should be posting something about Geometry today -- but I still want to tie it to Square Root Day. Unfortunately, today is Day 132 according to the blog calendar -- and not only is 132 not a perfect square, it's about halfway between 11^2 = 121 and 12^2 = 144.

The term square root is defined in Lesson 8-7 of the U of Chicago text -- the lesson on the Pythagorean Theorem:

From the area of a square you can determine the length of any of its sides. If the area is A, the length of the side is sqrt(A). That is why sqrt(A) is called the square root of A. Just as a plant rests on its roots, a square can rest on its (positive) square root.

As this statement implies, the square root is used in Geometry class when we are given the area of a square and we need to find its side length. But in practice, we don't really deal with square roots in Geometry until we reach the Pythagorean Theorem -- if we are given two sides of a right triangle, we can't find the third side without extracting a square root. And most subsequent appearances of square roots (such as triangles with sides of length sqrt(2) or sqrt(3) or the geometric mean sqrt(ab)) are either directly or indirectly related to the Pythagorean Theorem. So on this Square Root Day, I should give a lesson to review the Pythagorean Theorem.

Indeed, on her Mathematical Calendar 2016, Theoni Pappas gives the following question today:

x^2 / 9 + y^2 / 25 = 1 is an ellipse with foci located at (0, +/- ?).

To solve this problem, we let a be the major axis and b the minor axis. Then a^2 and b^2 are the denominators in the equation, with a^2 the larger of the two. So a^2 = 25 and b^2 = 9. The equation that we must solve is:

b^2 + c^2 = a^2
9 + c^2 = 25
c^2 = 16
c = +/- sqrt(16) = +/- 4

And not only is today's date the 4th, but it actually involves sqrt(16) on this Square Root Day. This question is an Algebra II conic section problem. I claimed that almost all square roots in Geometry are ultimately related to the Pythagorean Theorem, but what about this Algebra II question?

Well, the resemblance of the equation b^2 + c^2 = a^2 to the Pythagorean Theorem is too strong for it to be a mere coincidence. Let's see whether we can indeed derive it from Pythagoras. (Note: Here I'm using a for the major semiaxis and b for the minor semiaxis. Some authors use a for the horizontal semiaxis, regardless of whether it's major or minor.)

Recall the definition of an ellipse as the locus of all points such that the sum of the distances from any point to each of the two foci is constant. So let's take the point (b, 0) where the minor axis intersects the ellipse, and draw line segments to each of the foci, (0, c) and (0, -c). We're starting to see two right triangles form (which is what we want since we're trying to get to Pythagoras). One leg of each
triangle is the minor semiaxis b, while each triangle has another leg of length c, the distance from the center to each focus. But what is the hypotenuse of these triangles?

Let's take another point on the ellipse -- the vertex (0, a). It's easy to find the distance from this point to each of the foci -- the distance from (0, a) to (0, c) is a - c and the distance from (0, a) to (0, -c) is a + c, so the sum is 2a. By the definition of ellipse, this is the sum that must remain constant -- the total distance from any point to the foci must be 2a.

So in particular, the total distance from (b, 0) to the foci must be 2a. As the ellipse is clearly symmetrical, this implies that the distance from (b, 0) to each focus must be a -- that is, a is the hypotenuse of the triangle.

So our right triangle has legs b and c and hypotenuse a. Thus by Pythagoras, b^2 + c^2 = a^2. QED

Returning to Geometry, we are currently in a unit on circles, spheres, surface area, and volume. Last year, I actually did give a lesson on the Pythagorean Theorem between the lessons on the surface area of a pyramid/cone (Lesson 10-2) and the volume of a sphere (Lesson 10-8). In some ways this makes sense, as the Pythagorean Theorem is often used to find the radius, height, and slant height of a pyramid/cone. It also gives the students another chance to review the theorem, since my first Pythagoras lesson focused more on the Distance Formula -- yes, that's yet another formula with a square root in it that ultimately goes back to the Pythagorean Theorem.

This is what I wrote last year about today's worksheet. Notice that this worksheet actually came from a class I had subbed that day, so there's some commentary about subbing:

Only in third period did I actually reach the triangle worksheet that I posted here. Many of the students were confused when they were asked to solve for a side of the triangle. Part of this is because the first two problems involve integers, and then everything jumps into decimals. Students were allowed to use a calculator, but many of them had only a simple online calculator that lacked a square root function.

Perhaps if I were the one preparing this lesson, I would have included more simple integer questions before jumping into decimals with irrational square roots so quickly, but this was a lesson that was already prepared by the regular teacher. Then again, we can't shield the students from decimals and irrational square roots forever. A few of the problems that appear in Lessons 10-2 and 10-7 of the U of Chicago text on pyramids and cones require decimal square roots, and of course we might expect such questions on PARCC or SBAC as well.

There are a few more things I wish to say about this lesson. A few days after I posted it last year, I wondered whether I could have made it into a game -- and not just the usual "Who Am I?"

Instead, we can take today's lesson and play the game The Triangle Is Right -- a parody of the game show The Price Is Right. We divide the class into four groups and call out one student from each group to "Come on down!" to the front of the class. We then present a right triangle, such as #1 from today's worksheet, and ask students to find the missing value of x. Whoever comes closest to the correct value -- without going over, of course -- gains a point for the group. So for #1, the triangle has leg 3 and hypotenuse 5, so the student calculates:

a^2 + b^2 = c^2
3^2 + b^2 = 5^2
9 + b^2 = 25
b^2 = 16
b = sqrt(16) = 4

Say, just how many times is sqrt(16) = 4 going to come up on this Square Root Day, anyway! In this case, a student who says the exact answer 4 can earn two points, just like the $500 perfect bid bonus on the actual show The Price Is Right.

This game also solves the problem with decimals. Question #3, for example, has as its exact answer the value of sqrt(208.68). Suppose one student gives 14 as an answer. The next student can't bid 15, since this value is over, so he or she can say something like 14.01 or 14.1 -- this would be just like one-upping a $1,400 bid by saying $1,401 on the actual game show. The answer 14.4 is the closest a student can get without going over with just one decimal -- this is the closest realistic answer a student will give without a calculator with a square root button. So this is a nice game teachers can give today, since it is both Square Root Day and the first day after spring break.

(By the way, today's actual airing of The Price Is Right was interesting. The show actually referenced Square Root Day -- the game "Cover Up" is a game where players cover up the wrong price of the car with the right price. But instead of the wrong price, today's player had to cover up Square Root Day, along with the four such days so far this century, from 1/1/01 to today. And on this 4/4/16, during the 44th season of the show, we ended up with a perfect show, with all games won! The show also acknowledged Pi Day for the first time ever three weeks ago!)

Indeed, I like the idea of playing any sort of game with square roots today -- and this can be modified whether the level is anywhere from fourth grade up to Calculus. We can just ask the students to find the square roots of whole numbers -- we might as well start with sqrt(16). Then we can move on to sqrt(25), and then perhaps some numbers in between like sqrt(17) or sqrt(18). Provided that the students don't have a calculator (or at least one without a square root key), they can begin to appreciate a little more the importance of estimation -- since sqrt(16) = 4 and sqrt(25) = 5, we can estimate that sqrt(17) is about 4.1, sqrt(18) is about 4.2, and so on.

I remember back when I was a second grader. A friend knew that I was good at math and wanted me to teach him something that would make him sound smart, so I told him about square roots. I was able to teach him three square roots -- the two "goofy" roots sqrt(0) = 0 and sqrt(1) = 1 (which should be easy to remember). For the third square root, he wanted to know what was the biggest number I knew the square root of, so I said sqrt(144) = 12. And so for the rest of the day, he kept repeating to everyone how he now knew the square roots of 0, 1, and 144.

That day, back in the second grade (on 6/7/89, maybe? It couldn't have been 8/8/88 as that was still summer vacation), was perhaps the day I first realized that I wanted to become a teacher.

And so this concludes my first post after spring break.

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