And believe it or not again, I'm actually subbing in a math class in this district! I was first hired in this district nearly five years ago -- right around the time I created this blog -- and yet today is the very first time that I've been in a math classroom. And indeed, I first applied to this district after seeing an ad that this district was especially seeking out holders of math and science credentials to become subs.
So who would have imagined that it would take five whole years before I sub for math here? Again, I'm not counting special ed or co-teaching a period of math (which I have done in this district), but covering for a real math teacher.
Meanwhile, most of my subbing calls this school year so far have been in my new district -- and yet my first actual math assignment is in my old district. Once again, the only math I taught in my new district since the first day of school was one period of special ed basic math.
Today's teacher has three sections of Pre-Calculus (which is actually called "Math Analysis" in both of my districts) and three sections of Algebra I. Of course, I definitely want to do "A Day in the Life" for today's subbing. In fact, this might be the first "A Day in the Life" that I've done for this district.
7:20 -- Yes, this teacher has a zero period (and it's actually called "zero" in this district). This is the first of three Math Analysis courses, and the only one that's designated as "honors."
The students in all three Math Analysis classes (honors or no) are taking a test day. This test is on Chapter 2 of the Pre-Calc text, "Analysis of Graphs and Functions."
There are 34 students listed on the roster -- 13 seniors, 16 juniors, and five sophomores. Our main traditionalists would be delighted to see so many younger students taking an advanced class. The 16 juniors are at "SteveH" level, bound for senior-year Calculus AB. The five sophomores, meanwhile, are at "Bruce William Smith" level, bound for junior Calculus AB and maybe senior Calculus BC. (In other words, there is no "Handicapper General" here trying to keep these kids out of advanced math.)
8:05 -- Zero period ends -- well, sort of. Actually, I need to explain the bell schedule here. Notice that this is the same school that I subbed at a few weeks ago, in my September 9th post.
Most of the schools in this district have a block schedule -- and as I explained in my September 9th post, it's odd periods on Mondays and Wednesdays, even periods on Tuesdays and Thursdays. Then Fridays are all-classes days.
A few years ago, I discussed block schedules on the blog. If we take a traditional six-period day and naively convert it to a block schedule with three blocks per day, each block would be two hours. As two-hour classes seem just to drag on and on, most block schools don't do this. Instead, they often take a few minutes from each block and use this to make a seventh period that meets everyday. (This seventh class might be labeled "advisory," "homeroom," "tutorial," or it might be a real class.) Others shorten the blocks even further by adding an eighth period, with four 90-minute blocks per day.
But this district keeps the three two-hour blocks. Instead, here the last 20 minutes of each block is designated as "embedded support," a sort of tutorial. Students who don't need any extra help are free to leave. Sometimes teachers would list conditions that students must meet in order to be released for embedded support -- indeed, still written are the board are the conditions that fourth period Algebra I met to be released early yesterday. These were to finish the test (that Algebra I took yesterday) and have no outstanding missing assignments.
Anyway, this "embedded support" system as described above leaves out zero period. Instead, zero period has its embedded support on Fridays (when all other classes meet). By 8:05, most zero period Pre-Calc students have finished their tests. Those who haven't must stay until they complete it.
8:50 -- By now zero period embedded support has ended, and first period arrives. This is the first of three Algebra I classes.
I already explained above that Algebra I already took its test yesterday (or Wednesday, depending on the block schedule). So instead, these students have an extra credit worksheet on solving linear equations and inequalities.
Just as I've never subbed for math in this district before today, I've never actually sung any songs in classes until today. (I believe I may have tried to play a Mocha-generated song on a guitar during a music class, but I've never sung a song with lyrics.) This is mainly because most of my songs were for trying to motivate middle school students, while most classes I've subbed here so far were either older kids (juniors/seniors) or special ed students (with an aide in charge of classroom management).
But there's a song I've written back at the old charter school that fits this lesson well -- and in fact, I made changes to it over the summer. I've been itching to sub in a middle school or Algebra I class where I can sing this song -- and the opportunity finally presents itself today.
Do you remember the song "Solving Equations?" It's one of the last songs I sang at the old charter school. Let me rewrite the original version of this song as posted on the blog:
SOLVING EQUATIONS
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
Alternate 7th grade verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Whatever you do to the left side,
The same done to the right side.
That's all you have to do,
To solve it!
Alternate 8th grade verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Move the variables to the left side,
Move the numbers to the right side.
That's all you have to do,
To solve it!
At the time I posted this song, the sixth grade Illinois State text contained only equations that are essentially already solved -- such as d = 3 -- as a very basic intro to equations. The seventh grade text contained simple one-step equations, while the eighth graders were studying equations with variables on both sides. Thus the lines in each verse correspond to the equations studied in each grade.
But notice that if we were really solving an equation with variables on both sides, we'd follow the advice given in the eighth grade verse ("move the variables to the left side," etc.) -- and then do the same operations to both sides, just as the seventh grade verse tells us.
Thus, I realized, this song might be better as a "partridge" -- one where we repeat the lines from previous verses. (Actually, the official name of this is "cumulative song," but I like to call it a "partridge" after the most well-known example, "The Twelve Days of Christmas.") Let me rewrite the song above as a partridge (and remove references to middle school grades):
SOLVING EQUATIONS
1st verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
2nd verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Whatever you do to the left side,
The same done to the right side.
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
3rd verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Move the variables to the left side,
Move the numbers to the right side.
Whatever you do to the left side,
The same done to the right side.
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
Now the third verse actually contains the steps to solving a multi-step equation. First we move the variable terms to one side and the constant terms to the other. Then we perform the same operations to both sides of the equation. Finally, we're left with an isolated variable on one side and a number -- the solution -- on the other.
Over the summer, I decided that the fourth verse would be about combining like terms, while the fifth verse would be on distributing. Once again, this follows the sequence of steps to solve equations but in reverse -- we must distribute before we can combine like terms, since like terms will often start in separate parentheses.
But in this class, the worksheet contains some problems with the distributive property, but none with combining like terms. Thus I don't sing a verse on combining like terms at all.
In fact, on the worksheet there are three types of problems. The first six are on absolute value, the middle eight are simple two-step equations (following the third verse above), and the last eight are actually inequalities (mostly with negative coefficients/distributive property).
And so here are the three versions of the song that I actually sing in class:
First six problems:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Absolute value on the left side,
Plus and minus on the right side,
Whatever you do to the left side,
The same done to the right side.
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
Middle eight problems:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Move the variables to the left side,
Move the numbers to the right side.
Whatever you do to the left side,
The same done to the right side.
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
Last eight problems:
When you see an inequality,
Or problems that involve it,
All you have to do,
Is solve it!
Distribute on the left side,
Distribute on the right side.
Move the variables to the left side,
Move the numbers to the right side.
Divide or times by negative,
Inequality to the flip side.
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!
During first period, one girl decides she wants to dance during my song. I tell her that her dance ought to follow the lyrics -- she should perform some dance move on her left side when I sing "left side" and on her right side when I sing "right side." In fact, she should perform the same move on both sides. (Get it?) Perhaps someday a full dance will be developed for this song -- this would make the second song I sing with its own dance move (along with "Angle Dance" from yesterday's post).
9:45 -- First period leaves and second period arrives. This is the second of three Math Analysis classes, a non-honors class.
This time, out of 32 students, there are 26 seniors, four juniors, and two sophomores. Notice that the younger students tend to be in the honors class, since juniors and especially sophomores in Pre-Calc are, in fact, honors students. (There's no need to mention the grade levels in Algebra I classes, since of course they're all freshmen.)
10:40 -- Second period leaves for nutrition. One student needs more time to finish his test, though -- he tries staying for a few minutes during nutrition, but then he must leave.
10:55 -- Third period arrives. This is the second of three Algebra I classes.
Even though this is just an extra credit assignment on the day after a test, I strictly enforce that students must show their work. I establish good and bad lists of names -- the good students are those who answer questions on the front board. In order to avoid the bad list, the students must copy the work from any problems done on the board onto their papers. I end up choosing six problems -- one of each type. (Ignore the weird numbering -- this is how they appear on the worksheet.)
1) |6m| = 42
2) |-6x| = 30
1) 6 = a/4 + 2
16) -9x - 13 = -103
5) -b - 2 > 8
6) -4(3 + n) > -32
But some students get lazy -- they just divide both sides of |6m| = 42 by six and then either ignore the second solution m = -7, or just add m = -7 without writing the intervening step 6m = -42.
I end up writing one name on the bad list. This guy simply graphs the solutions to the two inequalities without showing the work of solving them -- and this is despite the fact that these two problems (and all the work) are still written on the board.
In other words, if I'm nice enough to have all the problems worked out on the board, then I must be strict in demanding that the students copy all of the work onto their papers. There's no room for leniency regarding the work when I don't even have to help them by showing work on the board! In this case, I'm not teaching the students how to solve the equations or inequalities -- I'm teaching them how to show their work.
This was, in fact, the original intent of giving Pappas-like problems (whose answers are the date) during the Warm-Ups at the old charter school. (Once again, I should have given such problems as Exit Passes instead, since the Warm-Ups were for Illinois State problems.) Anyway, what happened was that one day early in the year, fewer than half the students showed their work. I feared that the students would think that I was unreasonably mean if I denied credit to the majority of the class. And so the "must show work!!!!" rule was neutered -- students could just write the date and nothing else and call me mean unless I gave them credit.
There's no way I'm letting that happen again. And so I had to place the guy who refused to show work on the bad list.
11:50 -- Third period leaves and fourth period arrives. This is the last of three Algebra I classes.
After having trouble with getting some kids to show their work on the absolute value problems, I replace the second problem to have done on the board with:
3) |k - 10| = 3
This underscores why just ignoring the bars doesn't work. Yet many students still refuse to show the extra step k - 10 = -3. They either write just one solution, naively add k = -13 to the solution k = 13, or just copy the correct solution k = 7 from the board without justification. And so I continually ask students, "Where did you get k = 7 from?" and insist that they show the step k - 10 = -3.
Once again, many students are simply too lazy to copy all of the work. They're more interested in minimizing the work needed to get me off their back rather than maximizing their knowledge.
The type of student who doesn't show all the work for |6x| = 42 (by ignoring the bars and then adding x = 7 at the end) is the type of kid who gets |k - 10| = 3 wrong on the test and doesn't know why. Of course, it doesn't help that the test was yesterday (so that "You need to show work now so you can succeed on the test!" doesn't work). And these students are freshmen who probably don't know what finals are (so "You need to show work now so you can succeed on the final!" doesn't work either).
Again, this is why a Pappas-style Exit Pass might be successful, as long as "must show work!!!!" is strictly -- and I mean strictly -- enforced. It's easier to answer students' complaints of "I already know the answer, so why must I show the work?" with "We all know the answer is the date. The task is to show the work, not to find the answer!"
12:45 -- Fourth period leaves for lunch.
1:20 -- Fifth period arrives. This is the last of three Math Analysis courses. There are no sophomores in this class, and so I don't count how many juniors and seniors there are in this class.
2:15 -- Fifth period leaves. As usual, teachers with zero period don't have sixth period, and so my day ends here -- except for one guy who needs more time to complete his test. Since he doesn't have a sixth period either, he stays about five extra minutes to finish the test.
There's much more I can say about today's math classes, as well as how math classes are taught in my old district. But this post is getting long enough, so we must move on.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The sphere and the cube are the same height. The sphere's volume is 4.5pi. What is the cube's volume?
We clearly need the volume formulas for the sphere -- V = (4/3)pi r^3 (Lesson 10-8 of the U of Chicago text) and cube -- V = s^3 (Lesson 10-3). But the "height" of a sphere is the diameter of the sphere, while the "height" of a cube is its side length s.
Even though we can solve for radius r and then double it to find diameter d, we can avoid annoying messy decimals by simply finding the volume of the sphere in terms of its diameter:
V = (4/3)pi r^3
V = (4/3)pi(d/2)^3
V = (4/3)(1/8)pi d^3
V = (1/6)pi d^3
Now we plug in the known volume and solve it:
4.5 = (1/6)pi d^3
4.5(6) = d^3
d^3 = 27
And notice that d^3 is already the volume of the cube, since its side length is equal to the diameter of the sphere. Therefore the desired volume is 27 cubic units -- and of course, today's date is the 27th.
Of course, it's now easy to find the diameter d = 3, but the radius contains a decimal r = 1.5. And this is why I wanted to avoid the radius and its messy cube r^3 when all we really need is the diameter.
Lesson 3-2 of the U of Chicago text is called "Types of Angles." In this chapter, students learn about zero, acute, right, obtuse, straight, complementary, supplementary, adjacent, and vertical angles.
In the new Third Edition of the text, this actually corresponds to Lesson 3-3. But the definitions of acute, right, and obtuse are actually combined with yesterday's Lesson 3-1. Only the last four definitions (mainly adjacent and vertical angles) remain in the new Lesson 3-3.
(By the way, Lesson 1-4 of the Glencoe text, which I covered in the special ed class that I subbed for last Friday, is most analogous to Lesson 3-1 of the U of Chicago Third Edition. Last week's students learned all about angles, including acute, right, and obtuse angles as well as bisectors, in that lesson.)
In between these, in the new Lesson 3-2, are rotations. I've mentioned before how strange is this that both the old and new editions define a rotation as the composite of two reflections in intersecting lines, yet the new edition has a section on rotations before defining reflections! The U of Chicago most likely placed this section here so that in introducing rotations, students become more familiar with angles. (Again, I point out that Hung-Hsi Wu of Berkeley, in his recommendations for Common Core Geometry, teaches rotations before reflections, but he defines rotations differently. His lessons have nothing to do with the new Lesson 3-2.)
In fact, Jackie Stone -- a Blaugust participant last month -- also introduces rotations when teaching her students about angles, just like the U of Chicago text:
https://mathedjax.wordpress.com/2017/08/23/what-is-that-how-do-you-use-it-blaugust/
What was intended to be a five minute “review” of these skills to launch into the real lesson activity of the day turned into a much more in depth “teaching” of how to use this tool. Although they might NEVER use a protractor outside of my class again I do find the task of measuring something using a tool useful. The task also spoke to the CCSS Math Practice Standards of attending to precision and using tools strategically. It is so challenging (especially at the beginning of the year) to determine what are appropriate scaffolds to help students work on a task. Moving forward, I plan to assume less which is actually a good thing because then we can talk about refined meanings of things. For instance, because of their lack of background we were able to really talk about that the measurement in degrees was actually a measurement of a rotation. I think next year my approach might be different.
This is what I wrote last year on this lesson:
Section 3-2 of the U of Chicago text discusses the various types of angles. It covers both the classification of angles by their measures -- acute, right, and obtuse -- as well as related angles such as vertical angles and those that form a linear pair. Complementary and supplementary angles also appear in this lesson.
As usual, I like to look for other online resources by other geometry teachers. And I found the following blog post from Lisa Bejarano, a high school geometry teacher from Colorado who calls herself the "crazy math teacher lady":
http://crazymathteacherlady.wordpress.com/2014/08/23/
Bejarano writes that in her geometry class, she "starts with students defining many key terms so that we can use this vocabulary as we work through the content." And this current lesson, Section 3-2, contains many vocabulary terms. So this lesson is the perfect time to follow Bejarano's suggestion.
Now two textbooks are mentioned at the above link. One is published by Kagan, and unfortunately I'm not familiar with this book. But I definitely know about the other one -- indeed, I actually mentioned it in my very last post -- Michael Serra's Discovering Geometry.
The cornerstone to Bejarano's lesson is the concept of a Frayer model. Named after the late 20th-century Wisconsin educator Dorothy Frayer, the model directs students to distinguish between the examples and the non-examples of a vocabulary word.
Strictly speaking, I included a Frayer-like model in last week's Section 2-7. This is because the U of Chicago text places a strong emphasis on what is and isn't a polygon. But Serra's text uses examples and non-examples for many terms in its Section 2-3, which corresponds roughly to Section 3-2 in the U of Chicago text.
Here are steps used in Bejarano's implementation of this lesson:
1. I used the widget example from Discovering Geometry (chapter 1). It shows strange blobs and says “these are widgets”, then there is another group of strange blobs and it says “these are not widgets”. I have students define widgets in their groups. Then they read their definition and we try to draw a counterexample. Then we discussed what makes a good definition and we were ready to go!
(Note: Bejarano writes that she found the "widget" example in Chapter 1 of Serra's text, but I found it in Section 2-3 of my copy. Of course, I don't know how old my text is compared to Bejarano's -- mine is the Second Edition of the text, dated 1997. Hers could be the Third Edition or later.)
Notice that "widget" here is a non-mathematical (and indeed, a hypothetical) example that Bejarano uses as an Anticipatory Set. As we've seen throughout Chapter 2, many texts use non-mathematical examples to motivate the students.
3. Students worked in small groups with their 3 terms copying the examples & non-examples, then writing good definitions for each term. I set a timer for 10 minutes. (Yes, I'm skipping her #2.)
Now I don't necessarily want to include this as a group activity the way Bejarano does here. After all, I included the Daffynition Game in my last post as a group project [...]
And let's stop right here, because today's an activity day, and I haven't posted that Daffynition game yet this year -- so let me post it today! This is what I wrote last year about the activity. (Oh, and if you thought we were done with Serra's text after finishing Chapter 0 last month, think again!):
It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.
A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:
3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)
The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.
Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.
Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.
My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text.
OK, so let me post the worksheets. I decided to post only the first page of Lesson 3-2 (Lisa Bejarano's lesson) and then go directly to the Daffynition Game.
In cutting Bejarano's second page, I'm dropping some terms that don't appear until later in Chapter 3, but I also dropped "vertical angles" and "angle bisector," which do appear in Lesson 3-2. Teachers can either make sure to write those two dropped terms on index cards in the Daffynition Game, or else go full Bejarano and use the Frayer models as a full group project, just as the Colorado teacher originally intended.
[2019 update: Jackie Stone still posts weekly on her blog. Lisa Bejarano hasn't posted since November.]
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