Friday, January 12, 2018

Lesson 8-8: Arc Measure and Arc Length (Day 88)

Lesson 8-8 of the U of Chicago text is called "Arc Measure and Arc Length." This and the next section are the same in the old Second and modern Third Editions -- except that arc measure appears much earlier in the new version (in Chapter 3).

This is, of course, the lesson when students learn about the number pi. Two of my favorite lessons to teach each year are the Pythagorean Theorem and pi. Both two and three years ago, I rearranged the lessons so that Pythagoras appears near the Distance Formula and pi is taught on Pi Day. But since we're following the order of the text this year, Pythagoras and pi are taught here in the same chapter!

Indeed, since following the digit pattern means that our pi lesson isn't on Pi Day, you might ask, what lesson will I post on Pi Day instead. According to the blog calendar, March 14th works out to be the 128th day of school. Lesson 12-8 of the U of Chicago text is on SSS Similarity -- unfortunately a lesson about triangles, not circles or pi. Luckily, I have two months to figure out how I'm going to celebrate Pi Day on the blog this year.

The seventh grade U of Chicago text, called Transition Mathematics, is much more convenient for setting up the pi lesson near Pi Day. Today's lesson on the circumference of a circle is Lesson 12-4, and Lesson 12-8 is on spheres -- whose surface area and volume formulas definitely use pi. Keep in mind that I'm referring to my old Second Edition, not the new Third Edition -- the Third Edition of Transition Math teaches pi in Chapter 7 and stats in Chapter 12.

Much of my chapter rearrangement in past years was driven by my desire to celebrate Pi Day by teaching the famous constant. Thus I began the second semester with Chapter 12, so that we would be in Chapter 8 on measurement. The chapters following 12 are also related to similarity (such as trig) while the chapters following 8 are also related to measurement (such as volume), and so the net result was that we covered Chapters 12 through 14, and then back to Chapters 8 through 10. This year I wanted to follow the book order, at the cost of severing the link between pi lessons and Pi Day.

Two years ago I tried to combine Lessons 8-8 (circumference) and 8-9 (area), but the worksheet I posted on Pi Day leaned more towards area. A month later, I subbed in a seventh grade classroom where students were learning about circumference. And so the April worksheet I found in that classroom is what I'm actually going to post again today.

This is what I wrote two years ago about today's lesson:

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

And we all know what this means -- today was the day the students begin learning about pi!

Of course I am posting today's worksheet on the blog. For this activity, the students are given four round objects and a tape measure, and they are to measure the circumference and diameter of each of the objects. For example, one of the objects is a heart tin -- its circumference is about 47 cm and its diameter is about 15 cm.

You may notice that there's room to measure five objects, not just four. Well, the fifth object is the circle painted on the outdoor basketball court. This is convenient because its diameter is already marked (the free-throw line). But the students, instead of bringing the tape measure outside, use a nonstandard unit to measure the circle -- their own feet. With basketball on the mind of so many Californians today -- here in the south we celebrate Kobe Bryant's final game, while those in the north hope the Warriors win their 73rd game today -- it's great to incorporate the sport into today's lesson.

Notice that students are not to fill out the column "What relationship do you see?" yet. But some students try to come up with a relationship anyway. One student tries subtracting the diameter from the circumference, to write something like, "The circumference is 32 cm more than the diameter." I argue that this student is actually on the right track, if you think about it.

Meanwhile, a few students have already heard of pi, so they already know the relationship. One student cheats by measuring the diameters and simply multiplying each one by 3.14. The regular teacher will probably reveal the relationship between the circumference and diameter tomorrow.

Most of the students enjoy the lesson, but a few wonder why we are doing this activity. But most likely, these students are upset because they finish measuring the basketball court before any other group and is hoping for a reward. Instead, they are caught by another teacher for attempting to return to the classroom and fool around while I'm still out watching the other students.

Let's think about where this lesson fits in the seventh grade curriculum. Last week I wrote that if I were teaching the class, I'd try to reach Chapter 8 by Pi Day. As we see, this class came close -- certainly much closer than last week's Chapter 2 class.

But it can be argued that today is actually a "Pi Day" of sorts. You see, instead of 3/14, today is April 13th, which is 4/13. As the digits of pi appear in reverse, we can think of this as "Opposite Pi Day."

And now you're thinking -- here we go grasping at straws to come up with another math holiday. We already have Pi Day on March 14th, Pi Approximation Day on July 22nd, and Pumpkin Pi Day on the 314th day of the year in November. We had Square Root Day of the Decade on 4/4/16, Square Root Day of the Century on 4/5/2025, and several Square Root Days of the Month -- including yesterday, April 12th, which can serve as sqrt(17) day. And now I insist on adding yet another Pi Day on April 13th just because 3/14 reversed is 4/13! Do I really think that anyone is actually going to celebrate any of these extra so-called "Pi Days"?

Well, actually I didn't invent "Opposite Pi Day" -- the creator of the following video did:


It's yet another parody of Rebecca Black's "Friday" -- as I mentioned back on the original Pi Day, the fact that Friday and Pi Day rhyme is too irresistible for many math parodists to avoid. The poster of this song, who goes by the username "AsianGlow," probably just missed uploading the song on the original Pi Day, so rather than wait a whole year to post it, he just reversed the digits. He uploaded the video exactly five years ago today -- April 13th, 2011.

AsianGlow writes:

Of course by now you know of Rebecca Black making her infamous debut with Friday, but what about Pi Day?

How come Fridays get so much more love? Friday usually gets four times a month to party, but Pi Day only gets one... March 14th...

Does this mean I can only eat pies once a year?! OMG NO! Any date containing 1, 3, or 4 should be a piece of Pi Day! hahaha! See what I did there? You don't think it's funny? Well don't be so bitter and get some sweets in yo life! :D

This is the first video in a while that I did allll by myself! Well the filming anyways...
FilmRebelRoby - http://youtube.com/filmrebelroby - helped me with mastering my vocals.

April 13 is like opposite Pi Day! OMG it's still April fools! That's right, that's Pi!

Notice that AsianGlow's proposal that any date containing 1, 3, and 4 should be Pi Day only applies to March 14th and April 13th. We see that January 34th, April 31st, and 13/4 all just barely avoid being valid Gregorian dates, but they could exist in certain versions of Calendar Reform. In particular, all three date exist in a Leap Week Calendar where every third month (including January and April) have 35 days, 28 days in all the rest, and Leap Week labeled as Month 13. [2018 update -- I wrote about Calendar Reform on the blog two weeks ago, so you can read more about these calendars in those posts.]

I wouldn't have mentioned Opposite Pi Day here on the blog at all had I not subbed in a class learning about pi today. In some ways, I'm celebrating Opposite Pi Day today for the exact same reason as AsianGlow -- we both already missed the real Pi Day, yet we want to celebrate pi today. Both Pi Day and Square Root Day can be easily manipulated so that they fall during the unit on pi or square roots.

Though officially I covered pi on the blog on the original Pi Day -- and that includes both the circumference and area formulas -- the actual lesson focused more on area. This follows the lesson plans on Dr. Hung-Hsi Wu. Yes, I gave circumference first, but the area activity is Wu's. Today I finally post a lesson where students experiment with circumference. Of course, with these two lessons a week apart, the last part of that Common Core Standard:

CCSS.MATH.CONTENT.7.G.B.4
...give an informal derivation of the relationship between the circumference and area of a circle.

isn't fully emphasized here on the blog.

Before we return to 2018, let's think about last year, 2017. Yesterday I wrote about how I should taught the Pythagorean to my eighth graders last year, and so today I'll do the same regarding pi and my seventh graders last year.

Actually, I never reached the lesson on pi last year. That's because I was waiting, as usual, for Pi Day to teach the lesson, but I was out of the classroom before March 14th. In fact, I wrote in a post dated later that month what had actually happened on Pi Day. I decided to give my students one last surprise by delivering a pizza to my old classroom. But the bell schedule was mixed up that day, with school out early the entire week for second trimester Parent Conferences. It turned out that sixth grade was in the classroom at the time I delivered the pizza. Thus the sixth graders got to celebrate Pi Day with a pizza, even though seventh grade is the year that pi appears in the Common Core.

So had I made it to Pi Day, how would I have taught the lesson? Pi Day fell on a Tuesday last year, and at the time, Tuesdays were for projects. I assume that the Illinois State text had some sort of project where students had to measure the diameters and circumferences of various round objects -- in other words, an activity not much different from the one I'm posting today.

On the other hand, I posted that I should have made Tuesdays the traditional lesson day. Still, I see no problem with a brief measurement activity before the traditional lesson -- just as I'd given the eighth graders a brief Pythagorean Theorem activity before the traditional lesson two months earlier.

I had no control over Parent Conferences or the bell schedule. Again, I don't know when seventh grade had class that day -- only that sixth grade was the last class. If I were teaching, I wouldn't have been able to get the pizza -- but I could have sent my support staff aide to purchase it instead. After all, she'd bought a pizza for our eighth grade class four months earlier. (I mention this pizza in my Epiphany post from last weekend, in the "October 2016" section.) As a bonus, I could have had her get an extra pizza to share with my fellow teachers as they waited for Parent Conferences to begin.

So that seventh grade isn't left out of the party, I could bring some other round foods -- such as cookies -- for the students to measure. They only get to eat what they measure, so this is an incentive to do the activity correctly. Meanwhile, sixth grade gets a party but isn't learning about pi. Actually, I remember that there was a pi activity near the end of the Illinois State sixth grade STEM text page as a preview of seventh grade. The Pi Day pizza party would have been a great excuse to do this -- provided, of course, that I was given more than a day's notice as to what the bell schedule would be that day (which, as you may recall, wasn't always guaranteed on shortened days).

Suppose now that I had returned to the school and was planning a Pi Day lesson and party this year, then how would I do it? Again March 14th will be a shortened day -- not because of Parent Conferences (which I believe are a week later this year), but because it's Wednesday. The Common Planning meeting means that once again, kids will go home before 1:59. And so again, I'd be at the mercy of which class of students meets between the time that pizza parlors open and the time that school is dismissed.

If seventh grade gets the party -- which would make the most sense, since they're the ones learning about pi this year -- it would be the second straight Pi Day pizza party for this cohort, since last year's sixth graders are this year's seventh graders! But once again, I have no control over what the bell schedule is.

And as Pi Day is a Wednesday, this is a Learning Centers day. Of course, that day the centers are set up so that students are measuring circles and learning about pi.

For the second straight day, we're avoiding the elephant in the room -- classroom management. When I gave this activity as a sub, there weren't too many behavior issues. After all, if students misbehaved that day, I'd leave their names for the regular teacher. The problem is when I am the regular teacher and there's no one for me to leave names for. And my support aide -- the only adult my students respect in my classroom -- might not be present if she's out buying the pizza!

Notice that in the post from two years ago, I did mention one behavior issue. Let me repeat it:

Most of the students enjoy the lesson, but a few wonder why we are doing this activity. But most likely, these students are upset because they finish measuring the basketball court before any other group and is hoping for a reward. Instead, they are caught by another teacher for attempting to return to the classroom and fool around while I'm still out watching the other students.

Is there anything I could have done to handle this problem better? Perhaps I should have told the students to stay near the court until all students had finished measuring it. This is something I should have anticipated at the time, because it was a sign of things to come. Do the students know what they should do at each step of the activity? And do they know what to do when they are done?

I suspect that if I had given this worksheet to my seventh graders last year, they would have handled it better than most of the other projects they did that year. This is because there's a worksheet, so at least they would know what they are supposed to do. No students, for example, could claim that they had finished in ten minutes or less if most of the boxes on their sheet were empty!

Yes, I know that many of these recent posts have turned into spilled milk and discussion of my class from last year again. But it's important to reflect on my past failures in order to set myself up for future success if I ever return to the classroom. The sky is the limit!

Since today is Friday, this is my first official activity day for the second semester. But of course, Lessons 8-4 and 8-7 already have activity components. This is a tough week for many students as it marks both the return from winter break and the first heavy set of formulas to remember. And so it's nice to have many activities going on during the week.

And on these Friday activity days, I was going to create a worksheet containing the Exploration questions from the U of Chicago text. If you wish to include them in your class, here they are:

a. Measure the circumference of your neck with a tape measure to the nearest half inch or centimeter.
b. Assuming your neck is circular, use your measurement to estimate your neck's radius.
c. What would be another way to get its radius? (Cutting is not allowed, of course!)

These questions almost fit this worksheet anyway. Students can write "my neck" in the "Object" column and its circumference in the second column. The only difference is that now they must calculate its diameter, not its radius, to match the third column.

Monday is the holiday for Martin Luther King, Jr. Day. And so my next post will be Tuesday.



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