As you might recall from that day, the first three classes are generally noisy, and so the teacher has them complete a music theory worksheet (on whole, half, quarter, and eighth notes) while watching the movie of the next songs that they'll perform in the spring -- The Little Mermaid. Meanwhile, the two advanced classes have a performance coming up in two days, and so these students have a more serious practice session today.
Today is Sevenday on the Eleven Calendar, so let's check out the seventh resolution:
Decade Resolution #7: We sing to help us learn math.
And it's only fitting that my first singing resolution lands on a vocal music day. If you recall from my November 21st post, tutorial is an excellent time to start singing, and so I do. The songs I choose today are "One Billion Is Big," "Triangle Song," "Angle Dance," and "No Drens."
I also sing during the other classes when the audio connection during Little Mermaid fails -- some of these same songs. When we reach the proper point in the video, I sing one song from the movie. It's the only one for which I find sheet music in the classroom -- "Under the Sea." In all classes, we reach a point slightly beyond this song, around the time that the sea witch Ursula makes a bargain for Ariel's voice. (Apparently, in the classes when the audio doesn't work, the sea witch has apparently stolen everyone's voices.)
In the classes that will perform on Saturday, the songs that they practice today are, in addition to "Let It Go" (from my November visit), "Build Me Up Buttercup," "City of Man," and "Slow Dance." This last song mentions Valentine's Day in the lyrics, even though performance night is only January 11th.
I know that I just posted Mocha music last week, but today's visit to the music class means that another Mocha post is in order. I'll wait until early next week to post a Mocha version of the only song I have the sheet music for, "Under the Sea." Today I'd rather focus on Geometry, including the first Geometry problem from the new Rapoport calendar.
Today on her Daily Epsilon of Math 2020, Rebecca Rapoport writes:
What is the maximum area of a rectangle with perimeter 12?
We've done problems like this before. In an earlier post, I wrote that of all rectangles with a given perimeter, the square has the largest area. Even though perimeters and areas of rectangles are taught right here in Chapter 8 of the text, the fact that the square has the largest area is not mentioned.
So anyway, if the perimeter of a square is 12, then its side length must be the. And the area of a square of this size must be nine. This the desired area is 9 square units -- and of course, today's date is the ninth.
Of course, of all plane figures with a given perimeter, the circle has the largest area. But we must first figure out what the perimeter (circumference) of a circle is.
Lecture 4 of Michael Starbird's Change and Motion is called "The Fundamental Theorem of Calculus," and here is an outline of this lecture:
I. The two fundamental ideas of calculus, namely, the methods for (1) finding speed from position (the derivative) and (2) finding distance traveled from the speed (the integral), involve common themes.
II. We looked at the same situation -- a car moving on a straight road -- from two points of view.
III. The moving car scenario presents a situation to analyze the Fundamental Theorem. Let's do so where the position function is p(t) = t^2, and the velocity function is v(t) = 2t.
IV. The fundamental insight relates the derivative and integral.
V. Supporters of Newton and Leibniz had a lively and acrimonious controversy about who developed calculus first.
Starbird tells us that both co-developers of Calculus came up with the Fundamental Theorem that links the derivative to the integral. So he returns to the car scenario from Lectures 2-3:
p(t) = t^2, v(t) = 2t
Mile Marker 0 at time 0, speed = 0
Mile Marker 1 at time 1, speed = 2
Mile Marker 2 at time sqrt(2), speed = 2sqrt(2)
Mile Marker 2.25 at time 1.5, speed = 3
Mile Marker 4 at time 2, speed = 4
Mile Marker 9 at time 3, speed = 6
Derivative: position -> speed
Integral: speed -> position
The professor explains that in practice, it's easier to find derivatives than integrals, which is why we often integrate a function by first finding an antiderivative for the function. Using the above example, we conclude:
Distance from time 1 to 3 is 9 - 1 = 8 miles.
v(t) = 3t^2, t = 1 to t = 4
As Starbird explains, this is difficult unless we already know the antiderivative is p(t) = t^3:
p(t) = t^3, v(t) = 3t^2
Mile Marker 1 at time 1, speed = 3
Mile Marker 8 at time 2, speed = 12
Mile Marker 27 at time 3, speed = 27
Mile Marker 64 at time 4, speed = 48
Derivative: position -> speed
Integral: speed -> position
Distance from time 1 to 4 is 64 - 1 = 63 miles.
The professor returns to the some of the developers of calculus, including Isaac Newton (1642-1727) and Pierre de Fermat (1601-1665). He points out that Newton often made new discoveries about Calculus yet didn't publish them for decades, which led to controversies regarding whether he or Gottfried Wilhelm von Leibniz (1646-1716) actually developed it first. The book that Leibniz published was titled "A New Method for Maxima and Minima...."
Of Newton, Starbird quotes Alexander Pope: "Nature and nature's laws lay hid in night. God said, 'Let Newton be and all was light.'" And Leibniz said of his rival, "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is the much better part."
This is what I wrote last year about today's lesson:
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. In the first few years of this blog, 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 calendar, March 14th works out to be a Saturday -- which, unfortunately, isn't a school day. 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 (so at least the Chapter 12 Test that I'd give near Pi Day is about 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.
In the past, 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.
But I'll wait until tomorrow to reblog what I taught in the class I subbed for that day.
Before we return to 2020, let's think about three years ago, 2017. Yesterday I wrote about how I should taught the Pythagorean to my eighth graders that year, and so today I'll do the same regarding pi and my seventh graders that year.
Actually, I never reached the lesson on pi that 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 that 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. 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).
Last year at the old charter school, I found out that there was indeed a Pi Day party. It's possible that the pizza I'd brought the year I left encouraged the administrators to make Pi Day a regular celebration at the school. Unfortunately, it was soon after Pi Day when the renewal petition for our school was denied.
For the second straight day, we're avoiding the elephant in the room -- classroom management. 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!
Yes, I know that many of these recent posts have turned into spilled milk and discussion of my class from three years ago 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 Thursday, this is not an official activity day. And so I am posting a brand-new worksheet for pi that isn't activity-based.
I create a worksheet containing the Exploration questions from the U of Chicago text:
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!)
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