Introduction
In past years on the blog, I would label my first Thanksgiving break post as "Floyd Thursby Day," but what does this mean? Well, Floyd Thursby Day is the Tuesday before Thanksgiving. The name refers to a certain traditionalist commenter who would go on websites and complain that too many teachers take the day off and call for a sub on Tuesday -- the last day of school before the holiday in his district.
But, that name is outdated. A Google search for Floyd Thursby and "Tuesday before Thanksgiving" leads mainly to this blog -- it's been so long since he's posted that his old comments no longer appear in any search results. And it's obvious that "Floyd Thursby" is a pseudonym -- the name refers to a certain character in The Maltese Falcon. So maybe I should stop calling this day "Floyd Thursby Day."
Perhaps a better name for today is "Thanksgiving Adam" -- the day before "Thanksgiving Eve" (as we've seen before with "Christmas Adam"). In recent years, I've also seen the name "Friendsgiving," referring to a meal eaten with friends on or just before Turkey Day. Even one of my favorite game shows, The Price Is Right, is scheduled to have a Friendsgiving episode tomorrow.
It's often believed that "Friendsgiving" comes from the TV show Friends, just as "Festivus" (the same as Christmas Adam) comes from the contemporary show Seinfeld. But the true origin is unknown -- and besides, many TV shows have a group a friends meeting and eating in late November. Perhaps the first Friendsgiving meal on TV consists of the toast, popcorn, and jellybeans eaten on The Peanuts.
Anyway, in today's post I wish to preview Chapter 3 of the CPM text for Integrated Math I. That's because this chapter covers a topic that's near and dear to my heart, ever since I started to blog. Chapter 3 is a Geometry chapter, and it specifically covers transformations -- you know, as in translations, reflections, and rotations.
Math III Chapter 5 Test
As much as I really want to get to the upcoming Math I lessons, I promised that I'd write the Chapter 5 Test for the Math III class -- and that I'd write the test before doing any blogging this week. Well, I did finish writing and uploading the test, so let me describe a little about the test right here. This chapter, by the way, is on inverses and logarithms.
First, I was instructed to write a test with two parts -- a paper section and a DeltaMath section. You might recall that last year at the old magnet school, I attempted to write several Calculus tests with both written and online sections, but only once did I pull it off successfully. Anyway, this year, nearly all of our Math III tests will contain two sections.
And according to my instructions, the DeltaMath section should have ten questions, while the paper section should have three questions, with multiple parts each. Indeed, at least one of the questions should require a rubric to grade -- a continuation of the problem I discussed in my last post, where students must evaluate each other's answers and attempt to grade them via a similar rubric.
The DeltaMath section is the easiest to write -- just set up the topics, and then DeltaMath will assign each student a different problem on test day. Here are the ten topics that I chose:
- Multi-step Function Inverses (Level 1)
- Multi-step Function Inverses (Level 2)
- Multi-step Function Inverses (Level 3)
- Multi-step Inverses w/Logs (Level 4)
- Multi-step Inverses w/Logs (Level 4)
- Inverse of Various Functions (M.C.)
- Identifying Inverses Graphically
- Evaluate Logarithms (Level 1)
- Evaluate Logarithms (Level 3)
- Features of Exponential and Log Functions
The trickier task is for me, of course, is to write the paper exam. The key is that the paper exam should assess topics that aren't covered well (or at all) on DeltaMath.
For the first question, I have the students find the inverse of a certain function, not by interchanging x and y (as DeltaMath emphasizes) but by writing out and reversing the steps in the reverse order. I often think of this as the "socks and shoes" or "wrapping and unwrapping" method. Moreover, the function I chose is a quadratic (written in vertex form) which requires that the domain be restricted in order to find the inverse. The polynomials on DeltaMath are all of odd degree and are invertible without any need to restrict the domain.
The second question involves finding the inverse of exponential and logarithmic functions. It has two parts, one for each type of function. For this question, any method can be used to find their inverse, including x-y interchange.
The third question involves finding some simple logs without a calculator (based on their knowledge of powers of ten for common logs, and perfect powers up to 5^3 = 125). Since no calculator is needed, it's best to put this on a paper test (as opposed to DeltaMath, where anyone could just open up a second tab for Google Calculator).
I'm the one writing this test for my own students and the rest of the Math III cohort. So I hope that my own students are successful on this test.
Geometry in Math I Chapter 3 and Beyond
As I wrote earlier, I've been waiting for years to teach a high school Geometry course -- and then when I finally get to a Geometry chapter, it ends up getting squeezed. Chapter 3 is supposed to be one of the three chapters we cover this semester, but the other Math I teachers keep saying "let's spend an extra week in Chapter 1 (or 2)," which then takes time away from Chapter 3.
At times like these, sometimes I wish we still had traditional Algebra I/Geometry/Algebra II rather than Integrated Math. But then again, if we were still traditional, I'd most likely be teaching the freshman and junior classes -- that is, Algebra I and II (just as I did back when I was a student teacher) with no Geometry at all. Thus it's only because of Integrated Math that I'm teaching any Geometry this year.
Still, it's worth it to compare the material coming up in Chapter 3 of the CPM text to the Geometry in the text that I discussed during the earliest years of this blog -- the U of Chicago text. Since I already mentioned transformations, you may think that CPM Chapter 3 corresponds to Chapters 4 (reflections) and 6 (other transformations) of the U of Chicago text. Well, let's look at CPM Chapter 3 in detail:
Indeed, the leader of the Math I curriculum team tells us that it's OK to push back Chapter 3 to after Thanksgiving, since only Lesson 3.1 on transformations is important for Math I. So my goal is to cover 3.1.1 through 3.1.3 the week we get back and 3.1.4 through 3.1.6 the following week, and only skip to Lessons 3.2 or 3.3 if someone tells me that material from these lessons will appear on some sort of Chapter 3 Test or final exam.
Believe it or not, some material from this chapter appeared on the District Benchmark Test. The lead teacher's original pacing guide had the Chapter 2 Test ending the week of October 24th, then a single transformations lesson -- a Halloween Transformations worksheet -- before giving the Benchmarks during the November 1st-2nd block.
Instead, not only did we take longer to finish Chapter 2, but we were required to give the Benchmarks a week earlier than expected. I still gave the worksheet on Halloween -- thinking that we were just days away from starting Chapter 3 so that the worksheet could serve as a preview. (Halloween was a Monday and hence a non-blogging day.) Then we kept delaying the start of Chapter 3 by inserting a midterm and extra Chapter 2 project.
There were three transformations questions on the benchmarks -- one question each on translations, reflections, and rotations. My students did well on the translation and reflection questions but struggled on the rotations. They likely remember a little about transformations from their Math 8 classes.
Indeed, only once have I taught transformations before -- in Math 8 at the old charter school. Back then, rotations were the most difficult for my eighth graders as well. (At my long-term school, the regular teacher returned before Math 8 reached the Geometry chapters.)
There are two more Geometry chapters coming up in the Math I text. Chapter 7 is on Congruence and Coordinate Geometry, and Chapter 11 is on Constructions and Closure.
And of course, there's plenty of Geometry coming up for the students next year in Math II. Indeed, about a month ago I had to cover a Math II class during my conference period (since in our district, most subs must work at elementary schools). The students had a DeltaMath assignment to work on that day, but I could easily see triangles on their screens.
(As usual, most teachers are out on Mondays and Fridays, and so most days when I must cover classes will be non-blogging days. As for last Friday -- our Floyd Thursby Day -- there were indeed a number of teachers out, but somehow I managed to avoid being called upon to cover one of their classes.)
A Complete Lesson Plan
Since I've been looking forward to teaching Geometry for so long, I'm really hoping that Chapter 3 will be successful for my students. But as I mentioned last week, a number of Math I classes haven't gone well lately, culminating in a disastrous Linear Art project. There's no way I want my Chapter 3 lessons to go as poorly as my Chapter 2 lessons.
Back when I was student teaching -- or, more precisely, when I was working as part of the special BTSA program for unemployed teachers (as explained years ago on the blog) -- my master teacher mentioned the ten parts of any successful lesson:
- Anticipatory Set
- Objective
- Purpose
- Input
- Modeling
- Checking for Understanding
- Guided Practice
- Independent Practice
- Homework
- Closure
Of course, back when she was teaching, this was before Desmos activities, tests that are half paper and half DeltaMath, and other new uses of technology. My challenge is to figure out where exactly Desmos and DeltaMath fit above so that there's enough time for the important Checking for Understanding.
Actually, DeltaMath is easy to figure out -- it is Homework, maybe Independent Practice. What's tricky is to figure out where Desmos belongs in the lesson plan.
Some Desmos activities introduce students to a new topic. In this case, Desmos serves as Anticipatory Set, with Objective and Purpose thrown in there. The problem is that sometimes my Desmos activities end up spanning the entire period. We might get to one example of the new material (Input and Modeling), but not much Checking for Understanding (especially if Desmos is on a Monday rather than a block day).
In last week's project, the students had to find equations for lines given two points, and so it's dependent on the success of the two points lesson. But finding a equation given two points is itself a two-step process -- first students must find the slope, and then they must use that slope and either of the two points to find an equation. Thus the previous lessons on slope and point-slope must also have gone well in order for the project to be successful.
I still remember the two points lesson, given just before Halloween. In sixth period, I started the Desmos lesson for two points. But then a few students wanted to know the purpose of this lesson -- after all, Desmos showed the graph of the lines for them on the screen, so they could just count squares on the grid to find the slope (instead of using the slope formula) and see where they cross the y-axis (instead of plugging into y = mx + b), and so the lengthy algebraic process was a waste of time. This led to yet another one of my recent heated arguments. (It goes without saying that this was on a Thursday, the day when most of my arguments occur.)
The problem, of course, was that my students still didn't understand how to find slope, so of course they'd be frustrated with a more difficult problem involving slope. And looking further back at the slope lesson, it also began with a lengthy Desmos activity with very little Checking for Understanding.
The Problems With Tech 55
One possible reason that my students are struggling is that they might be distracted with phone use during my lessons. If the students were staring at phones during the slope lesson, then of course they won't learn anything.
And indeed, on the first day of the project, some groups didn't draw anything at all -- instead, they took out phones and spent 100% of the period on entertainment. They might have been thinking, "I don't know what it means to draw a line with a fractional slope, so I just won't do anything."
So that's why I declared it to be Tech 55 -- put all phones away. Recall that "Tech 55" here refers to "Technology 1955" -- in other words, don't use technology that didn't exist when the 1955 generation attended school. But, as I found out, this didn't help -- even the students who obeyed Tech 55 still had trouble calculating slopes and just sat there waiting for me to tell them what the slope is.
Of course, it's likely that they did have phones out during the slope lesson. But I couldn't have declared Tech 55 during the slope lesson, because it was on Desmos, which didn't exist in 1955. The students needed to have their Chromebooks out to access Desmos.
Of course, I could have chosen a different year other than 1955 in my tech declaration. In 1999, for example, cellphones existed but not smartphones, so I could try "Tech 99" and say that all phones must be put away unless they're '90s-style phones that can't fit in your pocket. But if I were to say that, a savvy student would argue that Chromebooks didn't exist back then either -- yes, laptops did, but a Chromebook isn't a '90s-style laptop. The iPhone came out a few years before the first Chromebook, and so no single year can encapsulate what I really want -- Chromebooks out, smartphones away.
When I first came up with "Tech 55," I didn't realize how dependent regular lessons would be on Desmos and DeltaMath. As I wrote before, the reason for so much technology is to benefit students who must stay home for two weeks after testing positive for COVID -- they still need a way to access their lessons when they're stuck at home. But what it means is that "Tech 55" won't work.
The official rules at most schools usually state that all phones must be put away unless they're being specifically used for class -- in other words, the default position is that phones must be away, except for those very few times when they are used for the lesson. But for too many students, the default position is that phones are allowed except for those very few times when they must be put away -- which is why a counterargument to "put your phones away" is "But I already finished my work!" (when according to the official rules, "because I'm done" is exactly why the phones should not be out).
I actually have no problem when phones being out after the students are done (except for district or state tests where they must stay away) -- it reminds students that the math comes first before there can be technology. But they really must have completed the work first.
Looking at the lesson plan outline above, perhaps phones should stay away through Modeling. Yes, Checking for Understanding is the most important step. But having no phones out until after Checking might result in one student telling everyone else the answer so that phones will be allowed faster -- and that's exactly what I don't want. So I should let them take their phones out at Checking time -- the ones who understand will do the work even with phones out, and the ones who don't will just use their phones, signaling to me that they don't understand.
More on Checking for Understanding
So what exactly should Checking for Understanding look like in a Desmos/DeltaMath world? Once again, it all depends on the lesson.
First of all, Desmos should only take place on block days. But this is tricky, because often the pacing guide will place a lesson with Desmos on a Monday -- and I can't delay it to the Tuesday/Wednesday block because there's another lesson to be taught that day. So I should look at the pacing guide in advance and figure out whether I can rearrange the lessons so that Desmos falls on a block day.
If the Desmos lesson takes the place of the Anticipatory Set and the next few steps, then Checking for Understanding will be separate from Desmos. So I should have the students take out their notebooks and start doing some practice problems.
I keep going back to the slope lesson, since that's where it all began to go wrong. The Desmos lesson I did that day started out with the four types of slope (positive, negative, zero, undefined), then gave two examples on the slope formula -- and then it jumped to Marbleslides.
In hindsight, that's clearly not going to cut it as far as Checking for Understanding is concerned. While the Desmos does Check for Understanding for identifying positive/negative slope, it doesn't Check for the slope formula.
At this point, I should have had the students take out their notebooks, given them a few random slope problems, and had them start working on them. Keep asking more and more slope problems until I can assert that at least two-thirds of the students know how to do two-thirds of the problem. They might make common errors such as putting x and y in the wrong place or confusing the minus signs -- but I'd much rather them make those errors than see a slope and ignore it with "I don't know the first step" when they get to the project.
Sometimes the dry-erase boards that I mentioned last week can be used for Checking. Some teachers also have quick Checks such as "thumbs up if you agree" -- but my students aren't accustomed to such Checks and will ignore me if I tell them to use their thumbs.
The Next Three Weeks in Math I
Now that I know what an effective lesson looks like, let's apply it to the upcoming Geometry lessons in Math III to figure out how I'll teach them. The first week will be Lessons 3.1.1, 3.1.2, 3.1.3, and then perhaps some sort of quiz on Friday.
Lesson 3.1.1 introduces reflections. There is no Desmos for this lesson (good, since it's a Monday) -- instead, there's a worksheet based on problems from the text. The students are asked to reflect polygons over a horizontal, vertical, and diagonal line (slope 1), without coordinates. Then there are T-tables such that the coordinates can be labeled, and the rules for reflecting over the axes and y = x discovered. The only concern here is whether there are enough examples on the worksheet to Check for Understanding.
One interesting thing about Lesson 3.1.2 is -- like Lessons 6.2.and 6.3 of the U of Chicago text, but unlike many other Geometry texts -- rotations and translations are originally introduced as composites of reflections. This lesson begins with a Desmos activity, "Transformation Golf" -- and this should immediately bring to mind Lesson 6.4 of the U of Chicago text, on miniature golf. But the two golf games aren't identical. The U of Chicago version has only reflections, but on Desmos, all three transformations are available to move the "ball" (actually a L tetromino) into the "hole." Since there is a Desmos here, the notebook should be used to Check for Understanding of transformations.
Lesson 3.1.3 is the lesson I'm worried the most about. It's on slope -- and as we've seen, my students are really struggling to understand slope. The slope formulas for parallel/perpendicular lines are derived on a worksheet -- recall that these derivations are now emphasized in the Common Core Standards. At least only point-slope (not two points) appear on the worksheet. With no Desmos on this block day, it might be a good day to get out the dry erase boards -- both to derive the perpendicular formula (where rotating the dry erase board 90 degrees will help) and to get more practice with slope. Once again, there will be some sort of quiz on Friday.
The other lessons are straightforward. Lesson 3.1.4 is a worksheet where coordinate planes are brought back for translations and rotations. Both 3.1.5 and 3.1.6 have Desmos lessons landing on block days -- 3.1.5 has a Polygraph and 3.1.6 has more tetrominos that are transformed in order to play Tetris. Once again, the end of the semester is dependent on what will be included on the final exam -- so it's possible that we might not make it all the way to 3.1.6.
Again, Lesson 3.1 of the CPM text corresponds to Chapter 4 and 6.1-6.4 of the U of Chicago text. We will pick up Lesson 6.5 of U of Chicago when we reach Chapter 7 in CPM.
Rapoport Question of the Day
It's been some time since we looked at the Rapoport calendar -- it's often difficult for me to discuss it on school days. Anyway, on her Daily Epsilon of Mathematics 2022, Rebecca Rapoport writes:
What is the maximum number of regions into which 6 lines can divide the plane?
I've seen this problem and so I know the answer, but not its complete proof. After checking out the relevant Twitter thread, I see the following strategy based on induction:
The initial case is n = 0 -- with no lines, there is only one region, the entire plane itself.
Now we consider the induction case from n - 1 to n -- that is, the nth line is added. To maximize the number of new regions, the nth line should intersect all of the previous n - 1 lines. The intersection of each of those lines is a point -- and these points can divide the line into n sections (two of which are rays -- one on each end -- and n - 2 segments between the n - 1 points).
Thus the nth line can pass through at most n previously formed regions (one for each line section), thus dividing them to form n new sections (that is, the n old sections become 2n new sections, but the net gain is only n).
So the maximum number of sections after the nth line must be 1 + 1 + 2 + ... + n -- that is, it's one more than the nth triangular number (assuming this maximum is achievable, which isn't exactly obvious when you try to draw them). The sixth triangular number is 21, so therefore the maximum number of regions with six lines is 22 -- and of course, today's date is the 22nd.
NOVA: Zero and Infinity
Earlier today, I caught a rerun of the NOVA episode that I missed last week. While I'm trying to get away from long descriptions of books I'm reading or shows that I'm watching, I still want to give a summary of this episode -- focusing on the mathematicians who appear on the show.
"Zero and Infinity" opens with an attempt to communicate with hypothetical Martians about how Earth Time works. Then our host introduces herself as Talithia Williams, a math professor and statistician from Harvey Mudd College right here in Southern California. I've heard of Williams before -- she also has a Great Courses DVD on statistics, similar to the Michael Starbird video I showed my Calculus class last year. As one of the few black female statisticians I know of, I was considering ordering her DVD and playing it for my Ethnostats class last year. (Even though her lectures focused on the software "R" which we weren't using in that class, just seeing a minority woman involved in Stats would have been great for those students).
Williams tells us that one of the first cultures to develop the number zero was in India, perhaps almost two thousand years ago. Here she interviews Manjul Bhargava, a Princeton mathematician of South Asian descent. He explains that the original zero -- a dot -- originated as a linguistic symbol similar to a modern apostrophe, and then it became a musical rest before turning into the number zero. The seventh century mathematician Brahmagupta discovered some of the properties of zero -- adding, multiplying, and even subtracting from zero to develop the forerunner of negative numbers.
The host proceeds to interview Waleed El-Ansary of the Hispanic Society of America. He discusses how Indian mathematics were expanded upon by the Muslims about a thousand years ago. The ninth century mathematician al-Khwarizmi -- who created Algebra, for whom algorithms are named -- used zero in his writings. Slowly, algorithms began to replace the abacus as the main tool for doing various types of calculations, and Hindu-Arabic numbers, including zero, would supplant Roman numerals (after Fibonacci introduced them to Europe in the early 13th century).
Williams now discusses Zeno's paradox of the arrow as a transition from the concept of zero to the concept of infinity. She interviews Eric Bennett, a Paralympian archer, in the Arizona desert. He shoots an arrow with his single arm and points out that at every instant, the arrow is not moving, and yet the arrow is moving. The paradox is caused by a division by zero error, and it's resolved by considering limits to infinity. (Michael Starbird mentions this paradox in his lectures, and I tried to demonstrate it in last year's Calculus class by using a pencil for an arrow and moving it closer and closer to one of my students -- without actually reaching him, of course.)
The host proceeds to interview Stephen Gogatz, a Cornell professor, at the Museum of Math (MoMath) in New York City. He talks about infinity and how the rigorous consideration of infinity leads to the development of Calculus. Also, he divides a pizza into infinitely many slices in order to derive the formula for the area of a circle (which we've discussed on the blog before).
Recall that a few years ago, I read "Beyond Infinity" -- the second book written by Eugenia Cheng. And who's a better choice to speak with Williams regarding the infinite than Cheng herself? Unlike the other guests who meet on a college campus, the two lady mathematicians meet at "Hilbert's Hotel" -- the infinite hotel where even if it's full, one, two, or even infinitely many guests can be accommodated, as Cheng explains in her book. She also talks about Cantor's diagonal argument and how there are many more real numbers than integers (even though there are just as many rational numbers as integers).
By the way, Cheng released her fifth adult book about a month ago. Titled "The Joy of Abstraction," it expands upon the category theory that she introduced in her first book, "Baking Pi." I haven't seen it at any local library yet -- as soon as I do, I'll mention it on the blog.
(Oh, and I just happened to find the old book where I first learned about infinity -- a math text by Joseph Breuer that my high school gave away for free when I was a young student there. I decided to reread the book in anticipation of watching this NOVA episode.)
And I look forward to seeing more from Talithia Williams as well. I still might get her Great Courses DVD one of these days, even though I'm no longer teaching Ethnostats (or any Stats for that matter).
Conclusion
In Chapter 5 of the CPM Math III text, there is an "Ancient Puzzle" given there. The text states that the puzzle originated in India in the second century BC (thus making it older than the number zero) and it was solved by Muslim mathematicians about 700 years ago (sorry, al-Khwarizmi -- you missed solving this one by a few centuries).
The text gives the puzzle in modern notation, using the symbol "log." That's right -- the puzzle is all about the discovery of the logarithm.
While Math III never had a project go awry the way the Linear Art project did in Math I, some of the same issues regarding my teaching have come up in this class too. Logs are a difficult topic at this level, and so it will be more important than ever for me to Check for Understanding throughout.
Reducing and eliminating arguments is my biggest communication issue right now, but Checking for Understanding is my biggest academic issue. And the two aren't unrelated -- when my students understand the material, there are less likely to be arguments about having to do the work.
I have one Thanksgiving break post forthcoming. I wish everyone a happy Floyd Thursby Day -- or Friendsgiving, or how about Fibonacci Eve, since tomorrow is Fibonacci Day, 11/23.
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