8:15 -- This is the middle school where homeroom leads directly into first period. And first period is an eighth grade class -- the only regular English class of the day. These students are learning about how to write argumentative essays. They read an essay where authors make a particular point and counterpoint on the issue of whether violent movies lead to violent behavior.
9:20 -- This is the middle school where all classes after the first rotate based on day of the week. As today is Friday, the rotation goes 1-6-2-3-4-5.
Now the special English classes begin. These are double-blocked, with one of the classes 2nd-3rd periods and the other 5th-6th periods. But double-blocked classes can be awkward at a school that has a period rotation. On Fridays, sixth period isn't after fifth period, and so we have the situation where students come to my class in the morning and then come back at the end of the day. (Tuesdays are similarly awkward for the 2nd-3rd period class.)
And indeed, the lesson plan for this class is confusing. On her printed lesson plan, the regular teacher has the students working in their Real Books (yes, that's the real name of a textbook) for fifth period, but on the board, she indicates that they should do this for sixth period, not fifth. Well, as the students enter, they see the lesson plan on the board and pick up their Real Books on the way to their seats. I decide that it's easier just to follow the lesson on the board since they already have their texts.
The Real Books lesson is all about making inferences. Students read articles about how drought affects countries on five different continents and infer how the droughts affect them. Of course, the example for our continent is the American Southwest, including my home state of California.
10:15 -- Sixth period leaves for snack.
10:30 -- Second period arrives. While the 5th-6th block is just an intensive reading class, 2nd-3rd period is for English learners. Not all of them are Hispanic -- there are a few Asian students (one Filipino and one Korean), and not all of the Hispanics are Mexican (one is Colombian). One girl speaks very little English at all.
(Notice that this is not going to be a racial post. The emphasis here is on language, not race.)
The class begins with a "Do Now" assignment in their spiral notebooks. (Recall that "Do Now" is what some teachers call their Warm-Up, including my predecessor at the old charter school.) The students must list as many English words starting with "S" as they can. Then they are to continue working on a Powerpoint assignment (or the Google equivalent) with slides containing various vocabulary words. Today is the last day they can work on them.
11:25 -- The students remain for third period. Technically they should begin their third period assignment, but they insist on a few more minutes to finish the slides since they're due today.
For third period, the students read an article on Martin Luther King Jr., since his holiday is coming up next week. (And no, this still isn't a racial post.) There are five questions for them to answer in their spiral notebooks. Finally, the class wraps up with a traditional game of Bingo, with candy as a prize for the winner.
12:20 -- Third period leaves for lunch.
1:05 -- Fourth period is the teacher's conference period -- but it turns out that once again, I must cover another class. Fortunately, it's a math class -- actually, it's the same teacher I covered during the week before Thanksgiving. He must leave at lunch today for an appointment.
If you recall from my mid-November posts, this is an Algebra I teacher. But fourth period isn't an Algebra I class -- instead, it's the Math Support class. And by now, you should know what the assignment for this class is -- ALEKS on Chromebooks.
You might ask, are any of the students excited to see me again? The answer is no, of course not -- because Math Support is only a one-trimester course. Thus these second trimester Math Support students don't remember me at all from the week I spent here during the first trimester.
I mentioned this Math Support class in my Epiphany post last weekend. The regular teacher has this student write their work on lined paper and then turn it in to prove that they really are working on their ALEKS assignments today. This is probably what I should have done at the old charter school -- instead of a full IXL accountability form, just having all the students write their work on paper should be adequate.
Some students do their math homework instead of ALEKS. This includes lessons (which might be either Common Core 8 or Algebra I) on functions, especially linear functions. Of course, I help out those students who need it.
2:00 -- The reading students and I return to the English classroom for fifth period. So now I follow the teacher's printed lesson plan for sixth period (but written on the board for fifth period).
This is what the regular teacher calls "double rotation," which is like Learning Centers. The class of sixteen students is divided into two (predetermined) groups. One group does silent reading and fills out a reading log, while the other group completes various assignments on Chromebooks (either slides like the other class, or drill and practice software similar to ALEKS or IXL).
2:55 -- Fifth period leaves, and my school day ends.
Since I just renewed my New Year's Resolutions for 2019, let me get back in the habit of stating a focus resolution for the day. We might as well begin with the first resolution:
1. Implement a classroom management system based on how students actually think.
The main classroom management issue today is in the 2nd-3rd period class. There is obviously something brewing between two students -- a boy and a girl.
In second period, I've already written the boy onto my good list because in his spiral notebook, he names more words starting with "S" than any other student. But then during Chromebook time, he keeps tattling on another girl, saying that she's playing around on another nonacademic website instead of Google Classroom. He and another guy actually trick her into admitting it by claiming that I, the teacher, am already aware that she's on the wrong website! Since she admits to playing around, I have no choice but to write her name on my bad list.
Between second and third period, she and a group of three more girls leave during passing period -- the regular teacher prefers that they use the restroom during passing period, not class time. But they return a minute or two late. I have no choice but to mark all four girls tardy, since technically they're all late. The first girl informs me that one of her friends has been crying all day. As it happens, the computer for attendance isn't working, and so I submit attendance by paper. I mark the four girls late, but add a note to the office that three of them have been trying to console the fourth.
When the Bingo game begins, the original boy starts insulting the crying girl in Spanish. These insults are aimed at her looks -- one of the words starts with "G" ("F" in English) and another word starts with "F" ("U" in English). I don't speak Spanish, yet I'm familiar with these two words.
Meanwhile, the teasing boy comes close, but ultimately doesn't win Bingo. He is one number away, but another guy wins. After I give the winner his prize of two candies, the teasing boy takes one candy anyway and complains that I should have called "G-59." I let him keep the candy -- because technically, he really is the winner. He's so busy insulting the girl that he never realizes that I've already called "G-59"!
I write the winner on the good list, but of course I place the teasing boy on the bad list. I tell him that the only "G" he should have been thinking about is "G-59," not the "G" word in Spanish. If he'd left the girl alone, he would have won the Bingo game outright.
Notice that I place the teaser on the good list for second period (because of his "S" words in English) and on the bad list for third period (because of his "G" and "F" words in Spanish). This raises the issue once again of whether good and bad lists should be for academics or behavior. My rule of thumb is to write good lists for academics and bad lists for behavior (but technically, a student who completes little to no work is usually showing bad behavior by refusing to work).
When third period ends, the four girls all linger behind. The victim of the name-calling tells me that the boy has been insulting her and making her cry all day, even before my class. I tell her that perhaps she should inform a counselor or respected teacher of what's going on.
At the end of the day, I see the boy sitting in the office -- clearly some sort of punishment. I wonder whether the girls have taken my advice and informed the office, or perhaps my note on the attendance roster has spurred the administrators into action. It's also possible that the boy has continued to taunt the girl during their afternoon classes, and so another teacher has sent him to the office.
I'm not sure whether there's anything I could have done better in this situation. It's possible that I did the best a sub can do in this situation.
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.
2019 update: 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.
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.
We skipped this formula back when we covered Lesson 12-6 because at the time, we hadn't learned about area yet. [2018 update: Actually, this year we reverse the order -- so far we've covered area but not similarity at all.] 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 last year about the circumference activity, which I'm also adding to today's post:
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.
[2019 update: 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
2019 update: 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.
[2019 update: 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
2019 update: 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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