Notice that this teacher actually starts with a "first" (i.e., zero) period class -- in fact, her first and second period classes are both Statistics. But the regular teacher is present for both Stats classes -- she leaves during her third period conference, and then I take over for Geometry. Therefore "A Day in the Life" will start with:
10:00 -- This is tutorial. It's a bit odd for me to begin with tutorial, but that's just how it goes today.
10:30 -- Fourth period arrives.
Today's lesson is a review worksheet on Lesson 5-3 and 5-5 of (presumably) the Glencoe text. If I recall correctly, Chapter 5 of the Glencoe text is a hodgepodge of triangle-related topics -- for example, the concurrency theorems appear in this chapter. And these two lessons are on the various inequalities in triangles. As I've said before, these are spread throughout the U of Chicago text -- we begin with the Triangle Inequality in Lesson 1-9, the SAS Inequality in Lesson 7-8, and finally Unequal Sides and Angles in Lesson 13-7. Also, the SSS Inequality (a sort of converse to SAS Inequality) appear in the Glencoe text, but not the U of Chicago.
How do I present this lesson today? I might have considered this a perfect day to play the Conjectures or "Who Am I" game? Notice that even though I originally designed this game to be for introducing material, it's since turned into a quick review game. Today's lesson is clearly review, and the students do have an assessment coming up on Tuesday -- the Chapter 5 "Quest." Apparently, the teacher can't decide whether to label this as a "quiz" or a "test," so she just calls it a "quest" instead.
But lately, I haven't played "Who Am I?" as often. Ever since I started singing more often in class, my game has gone by the wayside. Thus perhaps it's better for me to use a song to motivate the students instead.
So which do I choose today -- "Who Am I?" or a song? Well, today is Eightday on the Eleven Calendar, so let's see what the resolutions have to say about today:
Decade Resolution #8: We sing to help us remember procedures.
Well, that settles it -- today's resolution mentions singing, and so I sing today. I choose "Angle Dance" from Square One TV, for two reasons. First, I sang this song at the old charter school around this time three years ago, and thus I think of this as a "January" song. And second, today's lesson does refer to the angles of a triangle (SAS, Unequal Angles, and so on).
I randomly choose students to do some of the review problems on the board. These are the students I ultimately place on my good list.
Overall, the class runs rather smoothly. This is the quietest class of the day, in part because it's much smaller than the other two classes.
11:30 -- Fourth period leaves and fifth period arrives.
By the way, you might notice that today's resolution refers to remembering "procedures" rather than learning "math." Of course, today's song is all about learning math (angles). So far, only a few of my songs refer to school procedures (such as "The Packet Rap"). But my goal for the new decade is for that to change -- and this entails writing new songs (especially if I become a regular teacher).
There are many posters decorated around the room. They have various labels such as "The Ten Commandments of Math" (the first is "Thou shalt not divide by zero!") and "do/don'ts" (also called "How to Pass" and "How to Flunk"). I suspect that many students don't bother to read, much less heed, the advice given on these posters. So one way to draw their attention to them is to turn the writing on them into song lyrics.
One poster that's particularly suitable for conversion to a song is:
"David Numberman's Top 11 Excuses for Not Having Your Math Homework."
Let me include some of these eleven excuses here:
1. I accidentally divided by zero and my paper burst into flames.
6. I was watching the World Series and couldn't determine whether it converged or not.
7. I dug all around the yard but never found the square roots I needed.
"David Numberman" is, of course, a parody of comedian David Letterman. The idea of changing Letterman to Numberman also occurred to the creators of Square One TV. Notice that there are no stand-alone clips of Numberman on YouTube -- I must link to a full episode instead:
Unlike "Angle Dance," David Numberman isn't really a Square One TV song. There is a theme song for the talk show (parody), but no song with lyrics.
Thus I can take the Numberman poster and create my own song. The Square One TV Numberman theme can be the refrain. During this part, I introduce myself as "David Numberman" (since my name really is David, and I am a "man" who's trying to teach "numbers") and that I'm about to tell them eleven excuses for not turning in homework.
Then the verses gives the excuses. I can use Mocha to compose a tune for the verse that sounds good with the refrain tune. Even though the poster presents the excuses in the order 1-11, I wish to be more like the real Letterman, who would give his list starting with 11 and ending with 1. The first verse can give excuses 11 to 8, the second verse from 7 to 4, and the top three reasons in a bridge. If I want, I can even add an "outro" telling the students that they should just turn in the homework rather than come up with all these crazy excuses.
I don't do any of this today. This is a future project -- something to do especially if I return as a regular teacher (and this Numberman poster is in the room). Still, this poster is what I'm looking at during fifth period, when the students are busy writing their answers on the board.
12:25 -- Fifth period leaves for lunch.
1:10 -- Sixth period arrives.
On one hand, this class is much louder than fourth period. But on the other, this much livelier has many more students volunteering to answer questions on the board than fourth period. This brings me to something that's been bugging me since Wednesday.
Recall that I subbed in a seventh grade special ed math class that day. I'm still upset that my arguing crossed the line into yelling that day. Reflecting back upon that day, I think I now know what I did that led to the argument.
I had randomly chosen a student to solve one of the inequalities -- this one only involved addition, not multiplication or division (much less by a negative). I was wanting to get through the early questions quickly so to reach the ones that needed to be flipped. But then the student I'd chosen was unable to solve the simpler inequality. At this point I panicked -- now I needed to explain to this student how to solve the addition inequality, which would have taken even more time away from the flipped inequalities.
This raises two issues. One is my struggle between wanting to answer all the questions myself just so we can get through them faster, and the other is to slow down to allow for student understanding. I had an old resolution that was about making sure that the students are doing some of the work instead of the teaching doing everything. But if I want the students to work, I have to slow down to their pace, and then we might not progress as far in the lesson as I hoped.
The other issue is that I've had problems with student volunteerism throughout the old decade -- going all the way to my student teaching. I know that if I ask for volunteers, the same five students would answer every question. Choosing students randomly ensures that all students are participating in the lesson. But then some students might complain that they know the answer, yet never seem to be chosen by the random number generator.
To avoid this argument, I allow the volunteers to answer the questions. Only once do I use the random number generator -- I feel that not enough girls are volunteering today, and so I randomly choose one girl to get up and answer. But this is certainly something that I wish to look at in the future -- whether reliance on random choosing leads to arguments and yelling.
2:00 -- Sixth period leaves. Teachers who have "first" (zero) period typically don't have a seventh period, including today's teacher. Thus my day ends here.
Lecture 5 of Michael Starbird's Change and Motion is called "Visualizing the Derivative." Here is an outline of this lecture:
I. Change through time is of fundamental interest in many settings.
II. Let's again analyze the speed of a car given its position.
III. Consider now motion with varying speeds.
IV. We can see acceleration in the graph of a moving car.
V. If we see a position graph, we can sketch the speed graph.
VI. Let's look at the whole trip.
VII. Let's summarize the relationship between a graph and the derivative.
VIII. When Newton and Leibniz defined the derivative in the seventeenth century, they used different words and different notation.
In much of this lecture, Starbird shows many graphs. His first graph shows position at a given time, and Starbird tells a story about how fast the car is going. (The car starts out fast, slows down as we approach work, then reverses as we go home.) We've seen similar piecewise linear graphs before in some Common Core problems, but, as he explains, in Calculus the graphs are no longer linear. The idea is that the steepness of the graph corresponds to speed.
The professor uses a rope to represent a function and a yardstick to represent first the secant lines through two points of the graph, and then the tangent line at a single point. He tells us that the derivative is the slope of the tangent line. He also shows us a graph of a function and then zooms in many times until the graph looks almost linear.
Starbird now returns to the graph of a car's position and adds on a new graph of its derivative, which is the car's velocity.
The professor mentions the Mean Value Theorem for Derivatives. The average velocity of an interval equals the instantaneous velocity at some point. He gives an example -- if two tollbooths are 220 miles apart and it takes you one hour to get from one to the other, you must have been going 220 mph at some point -- so now you must pay a traffic ticket.
Starbird concludes with notation for the derivative. Newton placed a dot above the function. We no longer use this notation, since a fly leaving its mark on a paper can accidentally take a derivative (and besides, it doesn't show up well in ASCII). Instead, we use Leibniz's notation -- d/dt(t^2) = 2t. We can also use a prime symbol -- if p(t) = t^2, then p'(t) = 2t.
Lesson 8-9 of the U of Chicago text is called "The Area of a Circle." We all know the famous formula that appears in this lesson.
Last year, my Pi Day activity was more geared towards the area. Therefore, I'm posting that Pi Day worksheet today for Lesson 8-9.
Meanwhile, many chapters in the second half of the book are longer than those in the first half -- and this causes a problem in setting up the chapter review and chapter test. Tuesday is Day 91, which is when Lesson 9-1 will be taught, and Monday is the Chapter 8 Test. This means that today needs to be the Chapter 8 Review as well as Lesson 8-9. Get used to this, since there are several more long chapters coming up in the text.
And that's not to mention the problem of having a test on Monday. In the past, I asked myself whether it's better to have a test on Monday (forcing the students to study over the weekend) or a test on Friday (forcing the teacher to grade over the weekend). The idea is that Friday tests are better, since teachers are more likely to grade than students are to study in reality on weekends. But the way the digit pattern is set up, there are two Monday tests but only one Friday test.
Perhaps this means that I should made yesterday the activity day after all. Oh well -- it means that in today's post, I must combine activities for pi with review for the Chapter 8 Test.
[2020 update: Earlier I wrote that I wasn't sure whether to have a full Chapter 8 Test or just make it a Chapter 8 "Quiz," considering the timing of this assessment. Hmm -- perhaps after what I see in Geometry today, I should make it a Chapter 8 "Quest" instead!]
This is what I wrote last year about Lesson 8-9:
I visited several other teacher blogs for ideas on lessons. One of these blogs has a lesson that's perfect for Pi Day:
https://theinfinitelee.wordpress.com/2016/02/08/lesson-area-of-a-circle-or-how-i-got-students-hungry-for-the-formula/
Laura Lee is a middle school math teacher from Minnesota. Here is how she teaches her seventh graders about pi:
I teach CMP. I love discovering pi in Investigation 3.1 of Filling and Wrapping. It’s hands-on and engaging. Students love discovering that pi is a real thing not just a random number and excuse to eat pie on 3/14. If the lesson goes really well, I’ll even get a kid to ask, “Isn’t this a proportional relationship? Isn’t pi a constant of proportionality?” To which I want to use every happy emoji ever created in large, bold, capitalized letters with lots of exclamation points!!!
But then Investigation 3.2 comes along, where students discover the formula for area of a circle as pi groups of radius squares. I’ve never been able to get students to make sense of this lesson. There seem to be lots of understandings (what a radius square is, how to calculate how many times one number goes into another, understanding multiplication as groups of…) needed in order to get valid data that would allow students to conclude pi groups fit inside the circle. It just gets messy and doesn’t seem to make any sense to students, so this year I decided to take a different approach.
I decided to go with composing and decomposing. I would show them how to decompose the circle into a rectangle that is πr by r. Here’s how the lesson went (spoiler alert- a Domino’s pizza makes an appearance):
Notice that last year, I posted a lesson that actually covered area before circumference. Lee's lesson restores the order from the U of Chicago text, with circumference (Lesson 8-8) before area (8-9).Let's just skip to the part where, as Lee writes, a pizza makes an appearance:
Method 2 (I had been talking about pizza all morning with my first 2 classes, so last hour I decided I should just order a pizza and use that to derive the area formula)
- Order a pizza (Domino’s large cheese worked great!)
- Reveal pizza to class, watch them go insane!
- Have students gather around your front table
- Slice pizza into 16 slices,
- talk about circumference of 8 of the slices or half of the pizza: πr, record this on the pizza box
- then start arranging pizza slices into a rectangle, listen to student “Ahas!” and “No ways!” when they see it is clearly starting to form a rectangle
- Talk about dimensions of rectangle and then the area
The U of Chicago text does something similar in its Lesson 8-9. The difference, of course, is that the text doesn't use an actual pizza.
Lee writes that for her, the key is proportionality. This fits perfectly with the Common Core:
CCSS.MATH.CONTENT.HSG.C.A.1
Prove that all circles are similar.
Then again, notice that Common Core seems to expect a proof here. How does Common Core expect students to prove the similarity of all circles without Calculus?
Unfortunately, none of our sources actually prove that all circles are similar. What I'm expecting is something like this -- to prove that two circles are similar, we prove that there exists a dilation mapping one to the other. For simplicity, let's assume the circles are concentric, and the radii of the two circles are r and s. So we let D be the dilation of scale factor s/r whose center is -- where else -- the common center O of the two circles. If R is a point on the circle of radius r, then OR = r, and so its image R' must be a point whose distance from O is r * s/r = s, and so it must lie on the other circle of radius s. Likewise, if R' is a point on the circle of radius s, its preimage must be a point whose distance from O is s / (s/r) = r, and so it must like on the circle of radius r. Therefore the image of the circle of radius r is exactly the circle of radius s.
Of course, this only works if the circles are concentric. If the circles aren't concentric, then it's probably easiest just to compose the dilation with an isometry -- here a translation is easiest -- mapping the center of one circle to that of the other. Therefore there exists a similarity transformation mapping any circle to any other circle. Therefore all circles are similar. QED
To get from the area of the unit disk (pi) to the area of any disk (pi * r^2), we are basically using the Fundamental Theorem of Similarity from Section 12-6 of the U of Chicago. This time, though, we are using part (b) of that theorem:
Fundamental Theorem of Similarity:
If G ~ G' and k is the ratio of similitude [the scale factor -- dw], then
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2.
Although Wu attempts to prove a special case of the Fundamental Theorem of Similarity using triangles, it's much easier to do it using squares, as the U of Chicago does. If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.
Next, I'll add some of what I wrote a few years ago (on a subbing day, just before I started at the old charter school) about the circumference activity:
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"?
And this is what I wrote last year about Review for Chapter 8 Test:
As I mentioned earlier, the Chapter 8 Test is on Monday, which means that the review for the Chapter 8 Test must be today.
In earlier posts, I mentioned the problems that occur when a teacher blindly assigns a worksheet that doesn't correspond to what the student just learned in the text. Since I'm posting a review worksheet today, we should ask ourselves whether the students really learned the material that is to be assessed in this worksheet.
For example, most students learn about area at some point in their geometry texts, but only the U of Chicago text includes tessellations in the area chapter. Yet the very first question on this area test is about -- tessellations. So a teacher who assigns this worksheet to the class will then have the students confused on the very first question!
Let's review the purpose of this blog and the reason why I post worksheets here. The purpose of this blog is to inform teachers about the transformations (isometries, similarity transformations) and other ideas that are unique to Common Core method of teaching geometry. The worksheets don't make up a complete course, but instead are intended to be used with a non-Common Core text -- the one that teachers already use in the classroom, in order to supplement the non-Common Core text with Common Core ideas. Another intent is for those teachers who do have Common Core texts, but are unfamiliar with Common Core, to understand what Common Core Geometry is all about. My worksheets are based mainly on the U of Chicago text because both this old text and the Common Core Standards were influenced by NCTM, National Council of Teachers of Mathematics.
So this means that a teacher interested in Common Core Geometry may read this blog, see this worksheet, decide to assign it to the class, and then have all the students complain after seeing the first question because their own text doesn't mention tessellations at all.
I decided to include the tessellation question because it appear in the U of Chicago text. But as of now, it's uncertain that tessellations even appear on the PARCC or SBAC exams. So it would be OK, and preferable, for a teacher to cross out the question or even change it. There's a quadrilateral, a kite, that's already given in the question, so the question could be changed to, say, find the area of the kite, especially if the school's text highlights, instead of tessellations, the formula for the area of a kite.
I admit that it's tricky to accommodate all the various texts on a single worksheet. I included tessellations since this is a drawing assignment that is fun, and I'd try to include them if I were teaching a class of my own. But I also want to include questions that may be similar to those that may appear on the PARCC or SBAC exams.
This year, I restored the tessellations lesson, but that was all the way before winter break began. Meanwhile, I couldn't really skip the triangle area questions, even though we only barely discussed Lesson 8-5 this year.
For example, Questions 2 and 9 are exactly the type of "explain how the..." questions that many people say will appear on those Common Core exams. And so it was an easy decision for me to include those questions.
Then there is a question where students derive the area of a parallelogram from that of a trapezoid. I point out that in other texts -- especially those where trapezoid is defined inclusively -- this isn't how one derives the area of a parallelogram. In the U of Chicago, the chain of area derivations is:
rectangle --> triangle --> trapezoid --> parallelogram
But in other texts, it may be different, such as:
square --> rectangle --> parallelogram --> triangle --> trapezoid
This year, of course, we've followed a different pattern:
trapezoid --> parallelogram --> rectangle --> triangle
Thus parallelogram still comes after trapezoid, hence this test question is still valid.
For example, most students learn about area at some point in their geometry texts, but only the U of Chicago text includes tessellations in the area chapter. Yet the very first question on this area test is about -- tessellations. So a teacher who assigns this worksheet to the class will then have the students confused on the very first question!
Let's review the purpose of this blog and the reason why I post worksheets here. The purpose of this blog is to inform teachers about the transformations (isometries, similarity transformations) and other ideas that are unique to Common Core method of teaching geometry. The worksheets don't make up a complete course, but instead are intended to be used with a non-Common Core text -- the one that teachers already use in the classroom, in order to supplement the non-Common Core text with Common Core ideas. Another intent is for those teachers who do have Common Core texts, but are unfamiliar with Common Core, to understand what Common Core Geometry is all about. My worksheets are based mainly on the U of Chicago text because both this old text and the Common Core Standards were influenced by NCTM, National Council of Teachers of Mathematics.
So this means that a teacher interested in Common Core Geometry may read this blog, see this worksheet, decide to assign it to the class, and then have all the students complain after seeing the first question because their own text doesn't mention tessellations at all.
I decided to include the tessellation question because it appear in the U of Chicago text. But as of now, it's uncertain that tessellations even appear on the PARCC or SBAC exams. So it would be OK, and preferable, for a teacher to cross out the question or even change it. There's a quadrilateral, a kite, that's already given in the question, so the question could be changed to, say, find the area of the kite, especially if the school's text highlights, instead of tessellations, the formula for the area of a kite.
I admit that it's tricky to accommodate all the various texts on a single worksheet. I included tessellations since this is a drawing assignment that is fun, and I'd try to include them if I were teaching a class of my own. But I also want to include questions that may be similar to those that may appear on the PARCC or SBAC exams.
This year, I restored the tessellations lesson, but that was all the way before winter break began. Meanwhile, I couldn't really skip the triangle area questions, even though we only barely discussed Lesson 8-5 this year.
For example, Questions 2 and 9 are exactly the type of "explain how the..." questions that many people say will appear on those Common Core exams. And so it was an easy decision for me to include those questions.
Then there is a question where students derive the area of a parallelogram from that of a trapezoid. I point out that in other texts -- especially those where trapezoid is defined inclusively -- this isn't how one derives the area of a parallelogram. In the U of Chicago, the chain of area derivations is:
rectangle --> triangle --> trapezoid --> parallelogram
But in other texts, it may be different, such as:
square --> rectangle --> parallelogram --> triangle --> trapezoid
This year, of course, we've followed a different pattern:
trapezoid --> parallelogram --> rectangle --> triangle
Thus parallelogram still comes after trapezoid, hence this test question is still valid.
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