Desmos: Geo-Coordinate Geometry Day 1 by Jenny White (Day 117)

Today I subbed in a high school self-contained special ed class. It's in my new district -- so in particular, it's not the same as the class I subbed in on Tuesday in my first OC district.

Ordinally, I don't do "A Day in the Life" for classes like this -- it's highly unrepresentative of the class I'd like to teach in someday. Then again, it's a two-day assignment -- I'll be back in this class on Monday. Yes, as we saw on Tuesday, special ed students often attend classes in-person at times when gen ed students don't -- and that includes Mondays.

Since it's a multi-day assignment. I will do "A Day in the Life today." I most likely won't do "A Day in the Life" on Monday, though.

8:00 -- As students are dropped off on campus, they begin some basic skills worksheets. Unlike Tuesday though, I don't notice any math on these worksheets.

There are three guys who attend in-person today -- usually there are four, but one is absent. There are also three girls who log in on Zoom -- one of them is celebrating her seventeenth birthday this weekend.

The aides consider taking the guys out for a walk, but but to their chagrin, the track is closed. They plan on going on the walk later this morning.

9:00 -- Snacks are served to the students. One student has forgotten his snack this morning, and so his mother comes in to drop it off.

9:45 -- The speech teacher arrives. I don't post the specific disabilities, including speech disabilities,.on the blog, but in this case the speech teacher gives a lesson to everyone, rather than pull students out. (I point out that this makes him more like the coding teacher at the old charter school.) His lesson is on common and proper nouns.

10:15 -- By now the track is open, and so the class goes on a walk. One of the aides has brought her pet dog, and some of the students chase him on the track and up the bleachers.

11:00 -- Lunch is served to the students.

12:15 -- The students watch a movie -- Disney's Lady and the Tramp. (Back on Tuesday, that class watched a Pixar film, Finding Dory.)

1:30 -- As I usually do in classes like this, I pass out some pencils and candy, even though it's not particularly near a holiday.

One of the aides takes out her ukelele and begins to play a song. (What is it with all the ukes I've been seeing in the past year -- first the one I saw in last classroom I subbed in before the pandemic, then the one the online-only girl played during my long-term, and now this instrument?)

She plays the songs "I Can't Help Falling in Love with You" and "Somewhere Over the Rainbow" -- both of which are ukelele standards. She wants someone to sing along with her -- and of course, I jump right in. I have no excuse, since I've been singing songs in class most of the week!

The one day I didn't sing was Tuesday, since I don't typically perform in classes like this one. Then again, since songs are being played anyway, I can't resist singing "The Big March Song," which I perform in between the two duets.

2:30 -- The students leave one-by-one as their parents arrive. One guy gives me a high-five (or air-five), and one aide informs me that this is the biggest smile this student's given in some time. I credit the candy and song that I give him.

This ends my day. I return for the second day of this two-day assignment on Monday.

Since I see a ukelele in class today -- you guessed it! -- it's back to figuring out how to play chords using Arabic 18EDL fretting on a four-stringed uke-like instrument. Yes, it seems as if I'm finding every excuse to post the "music" label these days. (Teacher with his uke in the classroom means music tag, uke chords on wall in teacher's old room means music tag, teacher's new room with no uke in sight means music tag, online girl playing uke from home means music tag, aide with her uke even though I don't get to play it means music tag.)

So far, our hypothetical four-stringed instrument is tuned to standard uke tuning (GCEA), and we wish to play a chord for every note of the 18EDL scale on the string G.

I'm still struggling to complete the scale, though. The stumbling block for me is the green F (gu F), at the eighth fret (the last note before the octave G at the ninth fret). Most of the other chords that are reachable with my hand are dissonant. The original song that I'm trying to play here, "Another Ratio Song," has a D7 chord here -- but then again, it's based on the real G major scale that ends with F#.

Then again, I'm not likely to sing that song when I return to this classroom on Monday. I will perform some song, but I won't decide which one until Monday itself.

By the way, since this is the first time I've seen the uke played in person, I do play close attention to the position of the aide's hands. As I suspected earlier, she doesn't play any barre chords -- with only four strings, barres are unnecessary. I also notice her right hand position and how the strings are strummed.

Today is Elevenday on the Eleven Calendar. There is no eleventh resolution -- instead the focus is on the Millennium Resolutions and communication. The speech teacher is there today to enhance the students' communication skills. I, meanwhile, do try to interact as much as possible with the in-person students and aides.

I also follow the old version of the fifth resolution -- sort of. That one is about the heroes of 1955, and the movie we watch today, Lady and the Tramp, was first released in that year (although it takes place nearly half a century earlier).

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

What is the maximum area that can be enclosed by 18 feet of fencing? (Round to the nearest square foot.)

Yes, that's right -- after missing Geometry for most of the month, we suddenly get three Geometry problems in a row on the calendar.

This is a Lesson 15-8 problem -- the Isoperimetric Inequality. Of all plane figures with the same perimeter, the circle has the most area. Thus we use the formulas for circle perimeter and area:

2pi r = 18

r = 9/pi

A = pi(9/pi)^2

A = 81/pi

A = 25.78 square feet

Rounding off here, the desired area is 26 square feet -- and of course, today's date is the 26th.

Let's finally get to today's Geometry activity. Last year, I posted an activity based on a district Performance Task -- but that's no fun, and it isn't pandemic-friendly.

Oh, and after I visited my old long-term classroom on Wednesday, the regular teacher finally removed me from the Canvas email list. This is good since I've been getting too many emails lately as each teacher adds me to Canvas just to sub one day for them, so I need teachers to remove me. It does mean, however, that I won't be able to keep up with what my old class is now learning -- and I'll no longer be inspired by their current assignments as examples of activities to post to the blog. (Then again, it's not as if the material from this chapter -- outside of the Distance Formula -- is taught in Math 8 anyway.)

This means it's time to return to our old standby -- Desmos. A quick search reveals the following activity on coordinate geometry, created by the math teacher Jenny White:

https://teacher.desmos.com/activitybuilder/custom/5c8ebddd6cf5480c7c980318

Recall that last year's Performance Task took two days, and so I have two days to fill this year. Well, notice that this activity is listed as "Day 1" -- yes, it's indeed a two-day activity. Thus it fits perfectly into our curriculum.

There's no need for me to post any worksheets at all in this post. The complete activity for today is found at the Desmos link above. I must say thank you, Jenny White.

Lesson 11-6: Three-Dimensional Coordinates (Day 116)

Today I subbed in a high school math class. It's in my first OC district. It's definitely worth doing "A Day in the Life" for this math class.

9:00 -- Today is an odd period day. This is the district where "first period" really means zero period -- and while this teacher did have first period last year, this year she doesn't. Thus I begin with third period -- which just so happens to be our favorite class, Geometry!

The students have a Go Formative to prepare for the upcoming test on Tuesday. It's the Chapter 8 Test -- and here I assume "Chapter 8" refers to the Glencoe text, since that's the text used in this district last year (unless they switched to something like APEX without my knowing).

Chapter 8 of the Glencoe text corresponds to Chapter 14 of the U of Chicago text, and so this chapter is on triangles and trig. Indeed, Glencoe goes beyond the U of Chicago, as it introduces the Laws of Sines and Cosines, which aren't usually introduced until at least Algebra II. (I blogged about the presence of these laws in Glencoe six years ago -- back when I tutored students who used this text.)

I do help a few students with some of the Go Formative questions. I'm surprised when I reach a simple trig question and the students don't recognize SOH-CAH-TOA -- apparently, this regular teacher chooses not to use that mnemonic.

For music, I sing the "Triangle Song" from Square One TV, since this is the triangle chapter. Despite this being a Geometry blog, I don't really have any songs in my repertoire from the second half of high school Geometry (except for yesterday's parody about volume and surface area). At least "Triangle Song" fits since it mentions an equilateral triangle (students are asked to find the area of an equilateral triangle on their Go Formative). Also, the fact that the sum of the angles is 180 degrees might come in handy (for example, when the Law of Sines is used in the ASA case).

Unfortunately, I don't get to any Law of Sines or Cosines problems today. I hope the students understand enough to be successful on Tuesday's test.

9:55 -- Third period leaves for break.

10:10 -- Fifth period arrives. This is an Algebra I class. There are also two student teachers who log into the Zoom for this class, which is a bit odd.

These students also have a Go Formative review, but their test is today, on the Laws of Exponents. I do one problem on the product rule, and I tell them to multiply the "big numbers" and then add the "little numbers" to answer it. I set up breakout rooms in Zoom for the review.

As for this period's songs, I begin with "U-N-I-T Rate! Rate! Rate!" since I originally had a second verse on the Laws of Exponents (back at the old charter school as part of the eighth grade curriculum). I follow it with "Solve It," my usual go-to song for Algebra I.

11:05 -- Fifth period ends. As it turns out, this teacher doesn't have a seventh period earlier (as her five periods are 2-6), and so this is basically conference period that leads into lunch.

1:00 -- Academic support begins.

2:15 -- Academic support ends, thus completing my day.

Today is Tenday on the Eleven Calendar:

Resolution #10: We are not truly done until we have achieved excellence.

Some online students ask to leave as soon as the Algebra I quiz is over. I've seen teachers release online students early before -- I do confirm with the student teachers that this is allowed before I do so.

If the students have really achieved excellence on their test, then I don't mind releasing students early when they finish -- and indeed, it fits with this resolution to let them go early in that situation. What I don't want to see is students rushing through the quiz just to leave early. (It's something that I've thought about before the pandemic -- for example, if no cell phones are allowed during a test and I tell them that they can use phones right after the test, then will students rush the test just to get to their phones?)

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

If all sides of an equiangular parallelogram are 5, what is its area?

Well, another name for "equiangular parallelogram" is a rectangle -- and since all sides are 5, this rectangle is in fact a square. Thus this question is really asking:

If the side of a square is 5, what is its area?

The answer to this question is 25 square units -- and of course, today's date is the 25th.

Lesson 11-6 of the U of Chicago text is called "Three-Dimensional Coordinates." In the modern Third Edition of the text, three-dimensional coordinates appear in Lesson 11-9.

I've decided to keep that activity from a few years ago and add a new Lesson 11-6 worksheet. The "Luck O' the Irish" actually fits in Chapter 11 on coordinate geometry, albeit in only two rather than three dimensions. If you wish, you can pretend parts of the graphs are in different planes and make it into a 3D lesson.

Ordinarily I don't post copyrighted material. But Cartesian Cartoons are so easy to find online that I see no harm in posting yet another copy of it. And besides, I've posted some of them before during years past. (The next pandemic-friendly activity is scheduled for tomorrow.)

Lesson 11-5: The Midpoint Connector Theorem (Day 115)

Today I subbed in a middle school cooking class. It's in my new district. Since it's a middle school class, I will do "A Day in the Life" today.

There is something unusual about this assignment, though. It's split between two different schools, and I must travel from one school to the other. If this sounds familiar, it is -- that's right, one of the two schools is the school where I completed my assignment. Recall that upon the return from winter break, our school rearranged the entire schedule in order to accommodate a certain teacher. Well, this is that teacher.

8:10 -- The day begins with -- yes, second period. No, it's not because "first period" means zero period (since that's in the other district anyway). It's because this is one of the reordered classes that was made to accommodate today's teacher.

Here's what happened with the schedule -- cooking is an elective that most students don't take, so the cooking teacher wouldn't have a full class load if she only worked at one school. Thus a deal was made for her to work at two different schools.

When hybrid began in the fall, it was originally decided to have a block schedule, with odd periods on certain days and even periods on the others. This turned out to be convenient for the cooking teacher -- she could teach odd periods on one campus (my long-term school) and even periods on the other.

Then administrators realized that the block schedule was ineffective during hybrid -- the students weren't learning as much as they could be. Thus the schedule was changed to that all periods meet everyday -- but unfortunately, naively scheduling all periods in order would force the cooking teacher to go back and forth between her two schools (which are located about ten miles apart).

The solution was to reorder the classes so that all even periods occur before all periods -- the official order is now 2-4-1-3-5. (Sixth period is independent study P.E. at district middle schools.) This is why I start today with second period.

Second period is an advanced cooking class, although today "advanced" matters little, since all classes are baking the same thing -- chocolate muffins. This is another one of those classes where the regular teacher is running the classes from home, so all I need to do is supervise and make sure the kids don't burn themselves or the school. One girl gives me one of her muffins.

9:05 -- Second period leaves and fourth period arrives. It's a beginning cooking class.

This is the class with the most discipline issues (semi-expected since it's a lower-level class). Some kids start chasing each other, hitting each other with potholders, and just playing around in general. Of course, the regular teacher sees it all and knows exactly who is doing what.

10:00 -- Fourth period ends. Two girls haven't finished their muffins yet -- unfortunately, they don't realize that their oven is turned off until it's too late. The previous class is supposed to leave their ovens on so that they're preheated for the next class, but the first class is smaller -- this pair just happens to choose the unused oven.

As they wait out the last few minutes, I sing the only song that I perform at this school. Admittedly, songs are often awkward on days when the regular teacher is Zooming in from home. I choose "Solving Equations," a cumulative song for which I can keep adding verses until the oven timer goes off.

After the girls leave, it's time for me to travel to the other school -- my long-term school.

11:00 -- I arrive at the second school.

When the new schedule was announced early last month, I was wondering why we couldn't keep first period the same and change the other classes -- that is, 1-3-2-4-5 instead of 2-4-1-3-5. Then only two classes would have to change.

Now that I'm covering for the cooking teacher, I can appreciate why the even periods have to be taught before the odd periods. The school day starts over half an hour earlier at the even period school than the odd period school. This gives her 30 extra minutes to get to the second school and still have a full load of five teaching periods.

Indeed, tutorial has been arranged in order to maximize her travel time. When I arrive at the second school, it's still the last few minutes of tutorial. Students are supposed to attend a certain period each week for tutorial -- if this works out to be the cooking class, those students go to the library instead.

11:05 -- The snack break begins as usual.

11:20 -- Third period begins -- no, not first period, but third. That's because while both schools have rearranged their classes to 2-4-1-3-5, my long-term school decided to renumber those periods 1-2-3-4-5 while the first school retained the numbering 2-4-1-3-5. Thus this teacher officially has two fourth period classes (one at each school) and no first period -- and this can be a bit confusing for her (ironic, since the class reordering was done solely for her benefit).

This is also a beginning cooking class, and it's the only class with an aide. As it turns out, it's the same aide I had from September to January during the long-term! She'd told me that she covered three classes, with two of them being my math classes. I never realized until now that her third class was in fact a cooking class.

This class is much better behaved than the corresponding class at the first school. I strongly suspect it's due to the presence of the aide. One guy gives me another chocolate muffin.

This leaves me with some more time for performing a song. I don't want to sing "Solving Equations" again, since I've already performed it in November during the long-term and yes -- I do recognize a few students from that long-term.

Since I still receive Canvas emails from my long-term, I know that the eighth graders are studying volume while the seventh graders have a lesson on surface area this week. This means that a good song to play is "All About That Base and Height," my Meghan Trainor parody:

Chorus:
Because you know I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base base base base.

1st Verse:
Yeah, it's pretty clear, I really want to,
Calculate your volume, volume, like I'm suppose to do.
'Cause I got that formula that all the students chose.
Just plug in all the right values in all the right places.
See that base! That's the area of the top.
We know the height, come on now make it pop.
If you got your calculator, 'lator, just multiply 'em,
'Cause every cubic inch is perfect from the bottom to the top.

Pre-Chorus:
Hey prisms, cylinders, don't worry about your size,
'Cause students all know the formula to find it right.
You know for the whole volume, it's just V = bh,
And for lateral area, it's L.A. = Ph.
(to Chorus)

2nd Verse:
I'm bringing area back! Go ahead with lateral
Area! Naw I'm just playin'. I know you
Want surface area! Then I have to tell you
First find lateral area then add the bottom and the top.
(to Pre-Chorus)

12:10 -- Third period leaves and fourth period arrives. This is the second of two advanced classes. I also sing "All About That Base and Height" for them.

1:05 -- Fourth period leaves for lunch.

1:45 -- Fifth period begins -- but this isn't a cooking class. Instead, it's an ASB class. It provides a second reason why our schools reordered the classes to 2-4-1-3-5 instead of 1-3-2-4-5 -- her fifth period class is here at my long-term school, along with her other odd periods.

As is typical for ASB, students go in and out of class doing their own projects and posters. I still find a way to sing "All About That Base and Height" for this students -- indeed, all three periods at this school have kids from my former long-term.

2:35 -- Fifth period ends, thus completing my day of teaching.

3:00 -- Before I leave campus, I meet with the regular teacher from my former long-term. He tells me that his mother has completely recovered from her cancer -- and she's also received both doses of the coronavirus vaccine.

Today is, I believe, the first time I've ever subbed a full-fledged cooking class. I believe that I once covered a self-contained special ed class where one of the skills the students get is cooking. (This was before the year I worked at the old charter school.) It's definitely the first time I ever had to work two half-days on different campuses.

Of course, it's especially interesting to return to my long-term school and see my former students and colleagues once more. I hope I get to come back to this school soon -- particularly on a Tuesday or Thursday, since I don't see that cohort in-person on today's visit.

Today is Nineday on the Eleven Calendar:

Resolution #9: We pay attention to math as long as possible.

This doesn't really come up today. I will point out that one guy -- a student I recall from the long-term as someone who almost never does his math work -- is the first (along with his partner) to finish his chocolate muffins, and in the advanced class to boot. If he's not motivated to do math, at least he's hopefully found something that he is motivated to do, namely bake.

By the way, the official name of this class is "Culinary Arts" at one of the two schools (and something more complicated at the other). A more traditional name of the class is "Home Economics" -- and in the old days, mainly girls take Home Ec. Today's classes are more gender-balanced, but girls are definitely the majority in the advanced classes -- neither has more than two boys in-person today. (I did mention that I took a one-quarter Home Ec class as part of a seventh grade Exploratory Wheel.)

Today on her Daily Epsilon on Math 2021, Rebecca Rapoport writes:

What is the difference of the areas of the non-overlapping portion of the squares?

[Here is the given info from the diagram: the two squares have side lengths 7 and 5.]

It's been a long time since I've posted a Rapoport problem. There are several Geometry questions on the calendar in February, but nearly all of them are on weekends or other non-posting days (including the long Presidents' Day weekend).

Notice that I don't give the area of the overlapping region -- or whether the overlapping region is even a rectangle (which it appears to be from the diagram). Let's see why not -- if the area of the overlapping region is x, then the remaining portion of the larger square has area 49 - x and the remaining portion of the smaller square has area 25 - x. Thus the difference between them is:

(49 - x) - (25 - x) = 49 - x - 25 + x = 49 - 25 = 24

Therefore the desired difference is 24 square units -- and of course, today's date is the 24th.

Lesson 11-5 of the U of Chicago text is called "The Midpoint Connector Theorem." In the modern Third Edition of the text, the Midpoint Connector Theorem appears in Lesson 11-8.

Unlike the rest of Chapter 11, this is a lesson I covered well last year. And so this is what I wrote last year about today's topic:

Lesson 11-5 of the U of Chicago text is on the Midpoint Connector Theorem -- a result that is used to prove both the Glide Reflection Theorem and the Centroid Concurrency Theorem.

Midpoint Connector Theorem:
The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.

As I mentioned last week, our discussion of Lesson 11-5 varies greatly from the way that it's given in the U of Chicago text. The text places this in Chapter 11 -- the chapter on coordinate proof -- and so students are expected to prove Midpoint Connector using coordinates.

It also appears that one could use similar triangles to prove Midpoint Connector -- to that end, this theorem appears to be related to both the Side-Splitting Theorem and its converse. Yet we're going to prove it a third way -- using parallelograms instead. Why is that?

It's because in Dr. Hung-Hsi Wu's lessons, the Midpoint Connector Theorem is used to prove the Fundamental Theorem of Similarity and the properties of coordinates, so in order to avoid circularity, the Midpoint Connector Theorem must be proved first. In many ways, the Midpoint Connector Theorem is case of the Fundamental Theorem of Similarity when the scale factor k = 2. Induction -- just like the induction that we saw last week -- can be used to prove the case k = n for every natural number n, and then Dr. Wu uses other tricks to extend this first to rational k and ultimately to real k.

I've decided that I won't use Wu's Fundamental Theorem of Similarity this year because the proof that he gives is much too complex. Instead, we'll have an extra postulate -- either a Dilation Postulate that gives the properties of dilations, or just one of the main similarity postulates like SAS. I won't make a decision on that until the second semester.

Nonetheless, I still want to give this parallelogram-based proof of the Midpoint Connector Theorem, since this is a proof that students can understand, and we haven't taught them yet about coordinate proofs or similarity.

I preserve the worksheet with this version of the proof -- but once again, a coordinate proof is also given in the U of Chicago text.

Lesson 11-4: The Midpoint Formula (Day 114)

Today I subbed in a high school special ed class. It is in my first OC district. It's basically the same special self-contained class that I mentioned earlier this month, in my February 3rd post. There's no "A Day in the Life" today.

The students are given a worksheet with five three-digit addition problems. They are allowed to use a calculator, and most of them do -- but one guy tries three of them without the calculator. In these problems, one number is a multiple of 100, which is why he decides to take them on. He gets two of them correct, but makes a mistake on the third, writing 800 + 223 = 923. I tell him to try it on the calculator, after which he realizes his error. This is in fact an advantage to having a calculator in this situation -- it allows me to correct silly mistakes without arguments.

Another guy uses the calculator for all five problems, except that he presses the + key instead of =, so for 800 + 223 he presses 800 + 223 +. This gives him a running total of all the addends so far, and so since this is the fourth problem, his answer for this problem is over 2000. I direct him to start over and make sure that he presses the = key at the right time. After the worksheet, the students work on a familiar website, IXL.

Today is Eightday on the Eleven Calendar:

Resolution #8: We follow procedures in the classroom.

This is one of the few resolutions that these students need to follow -- they have many procedures not found in a gen ed classroom, including those for getting lunch and bringing it back to the classroom.

I don't sing any songs today -- I almost never do for this class. Also, just as I alluded to in my February 3rd post, these students stay on campus after lunch for "academic support" -- the gen ed students go home and log in to Zoom if they need tutorial help. Today, the students get to dance around to music in a special room during academic support. Thus it's not as if they needed me to sing any songs today -- they get their music from the computer during dance time anyway.

EDIT: The delayed 2020 Putnam exam was given on Saturday, February 20th. This post is dated the Tuesday after the exam -- traditionally the day when I discuss Putnam problems on the blog. I maintain this tradition by inserting this discussion into this post.

This year, I wish to discuss two of the Putnam problems. As usual, the first will be Problem A1, since at least in theory, this is the simplest problem:

A1. How many positive integers N satisfy all of the following three conditions?

(i) N is divisible by 2020.

(ii) N has at most 2020 decimal digits.

(iii) The decimal digits of N are a string of consecutive ones followed by a string of consecutive zeros.

Recall that this is technically the 2020 Putnam that was delayed due to the pandemic. It's a Putnam tradition to mention the year in at least one of the problems. This year, a whopping four of the twelve problems mention 2020 -- A1, A5, B1, and B4.

And indeed, I recommend that students factor the year number on the night before the exam, since the problem that mentions that number often requires its factorization. Indeed, Problem A1 involves multiples and divisors, and so the factors of 2020 are very relevant. (Before you ask, notice that for the next Putnam, 2021 = 43 * 47.)

We observe that 2020 = 20 * 101. Further factorization of 20 isn't relevant because we already know a divisibility rule for 20 -- the last two digits must be 00, 20, 40, 60, or 80. Since our numbers consist only of 0's and 1's, the only digit pair we need to consider here is 00. Now we know how many zeros our numbers have -- at least two. And as soon as we have the zeros, we don't even need to think about the factor 20 again -- the zeros take care of it.

Now we must consider divisibility by 101. (This is called square-alpha at Dozens Online -- that is, it's one more than the square of the base.) It's easy to see that 1111 = 101 * 11 is the first multiple of 101 that contains only 1's. Putting our 1's and 0's together, this gives us the first number on our list:

111100

And we can have any number of zeros after the four 1's -- from two up to 2016 (since the total number of digits is at most 2020):

111100...00 (2015 list entries starting with 4 ones)

It might take a while to figure out the next repunit (that is, all 1's) multiple of 101 -- it's 11111111, which is equal to 1111 * 100001:

1111111100...00 (2011 list entries starting with 8 ones)

One repdigit divides another if the number of 1's in the first divides the number of 1's in the second. So this tells us that the number of 1's must be a multiple of four. Combining this with the fact that there are at least two 0's, this gives us all the numbers in our list:

2015 list entries (starting with 4 ones)

2011 list entries (starting with 8 ones)

2007 list entries (starting with 12 ones)

2003 list entries (starting with 16 ones)

..

15 list entries (starting with 2004 ones)

11 list entries (starting with 2008 ones)

7 list entries (starting with 2012 ones)

3 list entries (starting with 2016 ones)

Thus our final answer is the sum 2015 + 2011 + 2007 + 2003 + .. + 15 + 11 + 7 + 3 -- and recall that no calculator is allowed on the Putnam. But notice that this is the sum of an arithmetic sequence, and so we can use our "Gauss 1-100" trick to find the sum:

2015 + 3 = 2018

2011 + 7 = 2018

2007 + 11 = 2018

2003 + 15 = 2018

..

How many copies of 2018 are there? The number of addends in the sum is 2016/4 = 504, and so the number of pairs is half of this. But let's take half of 2018 instead -- in other words, there are 504 addends and the average of them is 1009. This is easier to multiply without a calculator:

1009 * 504 = 504500 + 4036 = 508536

which is the correct answer. There are over half a million numbers in our list that satisfy the properties (i), (ii), and (iii) given above.

If I show this problem to high school students, they should at least grasp the concept of what we're doing, since it's just addition, subtraction, multiplication, and division. There are a few leaps of abstraction that might confuse students though -- for example, why is it that we can add extra 0's to the end of the number without affecting its divisibility by 2020? We'll have to remind them that adding an extra 0 multiplies the number by 10 -- and multiplying only produces more factors of the number without taking any of them away.

The second problem I wish to discuss is Problem B2. That's because it mentions a game that might be appealing to our students:

B2. Let k and n be integers with 1 < k < n. Alice and Bob play a game with k pegs in a line of n holes. At the beginning of the game, the pegs occupy the k leftmost holes. A legal move consists of moving a single peg to any vacant hole that is further to the right. The players alternate moves, with Alice moving first. The game ends when the pegs are in the k rightmost holes, so whoever is next to play cannot move and therefore loses. For what values of n and k does Alice have a winning strategy?

This is a a very nice game to set up in the classroom. We divide the students into pairs, and they can take turns playing Alice and Bob in order to discover the winning strategy. It's also good for the teacher to play Bob against a student to play the role of Alice. (The student should be Alice since, after all, the questions asks us to find her winning strategy, not Bob's.)

We might start with, say k = 3 and n = 8. A simple game might look like this:

1. 2-7 3-4

2. 7-8 1-2

3. 2-5 4-6

4. 5-7A

This means, on the first move, Alice moves a peg from hole 2 to hole 7 and Bob moves a peg from hole 3 to hole 4. On the fourth move, Alice moves a peg from hole 5 to hole 7 and wins (A for Alice).

An easy way to find a strategy is to consider simple cases first. For example, suppose k = 1 -- that is, that there's only one peg. It's now trivial to find a winning strategy for Alice:

1. 1-8A

And this obviously works for any value of n -- just move the peg to the last hole in one move. So we see that Alice has a winning strategy for k = 1 and any value of n.

Let's see what happens when k = 2. When n = 8, we can see that Alice has no way to win -- if Bob plays correctly, he can force a win. Let's try it:

1. 1-3 2-4

2. 3-5 4-6

3. 6-7 5-8B

Bob wins by mimicking Alice -- whenever she moves one peg, he moves the other. He has mentally divided the board into pairs of holes -- 1/2, 3/4, 5/6, 7/8. Whichever hole Alice moves her peg to, Bob moves the other peg to the paired hole. Sooner or later, Alice moves a peg into either hole 7 to 8, thus allowing Bob to move the other peg into the other hole and win.

Of course, Bob can only make these pairs if n is even. And k doesn't have to be 2 -- it can also be any even number. Then both the pegs and holes are paired -- whenever Alice makes a move, Bob moves the paired peg (so if Alice moves peg 3, Bob moves peg 4 and vice versa) into the paired hole.

This suggests that Alice has a winning strategy if either k or n is odd -- that would prevent Bob from making pairs from the start. For example, if k = 2 and n = 7, Alice should start with the following move:

1. 1-3

Now with pegs in holes 2 and 3, Alice can divide the rest of the board into pairs -- 4/5 and 6/7. Now she can do to Bob what he does to her in the k, n even case -- whenever Bob moves a peg, Alice makes the paired move and wins:

1. .. .. 3-5

2. 2-4 4-7

3. 5-6A

If k is a larger even number like 4 and n is odd, Alice should begin with the following move:

1. 1-5

which allows her to pair all the remaining holes starting with 6/7, 8/9, and so on.

If k is odd, it's actually helpful for Alice to imagine that there is an invisible (k+1)st peg in the (n+1)st hole, so that there are now an even number of pegs to be paired. Alice's first move is then to pair this invisible peg with one of the real pegs. If both k and n are odd -- for example, k = 3, n = 7 -- then Alice should begin with:

1. 3-7

Now when Bob moves, he breaks the 1/2 pair, allowing Alice to make a pair on her next move. (Notice that even though both k + 1 and n + 1 are even, this invisible peg doesn't produce a win for Bob. That's because this invisible peg begins unpaired, so Alice can pair it. On the other hand, in the k, n even case, all the pegs begin paired, so Alice can only break a pair that Bob can repair on his move.)

If k is odd and n is even -- for example, k = 3, n = 8 like the original problem I gave above -- then Alice should pair the first peg with the invisible peg:

1. 1-8

Then the hole pairs are 2/3, 4/5, up to n/invisible peg.

So Bob wins with k, n both even, and Alice wins with at least one of k, n odd. This is a fascinating problem that I hope our students can enjoy.

Lesson 11-4 of the U of Chicago text is called "The Midpoint Formula." In the modern Third Edition of the text, the midpoint formula appears in Lesson 11-7.

This is what I wrote last year about today's lesson:

Lesson 11-4 of the U of Chicago text covers the other important formula of coordinate geometry -- the Midpoint Formula. As the text states, this is one of the more difficult theorems to prove.

In fact, the way we prove the Midpoint Formula is to use the Distance Formula to prove that, if M is the proposed midpoint of PQ, then both PM and MQ are equal to half of PQ. The rest of the proof is just messy algebra to find the three distances. The U of Chicago proof uses slope to prove that M actually lies on PQ. Since we don't cover slope until next week, instead I just use the Distance Formula again, to show that PM + MQ = PQ, so that M is between P and Q. The algebraic manipulation here is one that's not usually used -- notice that instead of taking out the four in the square root of 4x^2 to get 2x (as is done in the last exercise, the review question), but instead we take the 2 backwards inside the radical to get 4, and then distribute that 4 so that it cancels the 2 squared in the denominator.

Lesson 11-3: Equations for Circles (Day 113)

This week the Big March starts in earnest. As I've written before, the first week of the Big March isn't terrible, since at least it's a four-day week. The second week starts the string of five-day weeks -- these are what actually make the Big March the Big March.

Let's get to today's lesson. I've written above that one of the most difficult units always seems to begin right around the start of the Big March. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the equation of a circle, which just a few years ago was part of Algebra II! (Meanwhile, English classes tend to read Shakespeare during the Big March, for example.)

Lesson 11-3 of the U of Chicago text is called "Equations for Circles." In the modern Third Edition of the text, equations for circles appear in Lesson 11-6.

This is what I wrote last year about today's lesson:

The first circle lesson is on Lesson 11-3 of the U of Chicago text, on Equations of Circles. I mentioned that I wanted to skip this because I considered equations of circles to be more like Algebra II than Geometry. Yet equations of circles appear on the PARCC EOY exam.

Furthermore, I see that there are some circle equations on the PARCC exam that actually require the student to complete the square! For example, in Example 1 of the U of Chicago text, we have the equation x^2 + (y + 4)^2 = 49 for a circle centered at (0, -4) of radius 7. But this equation could also be written as x^2 + y^2 + 8y = 33. We have to complete the square before we can identify the center and radius of this circle.

In theory, the students already learned how to complete the square to solve quadratic equations the previous year, in Algebra I. But among the three algebraic methods of solving quadratic equations -- factoring, completing the square, and using the quadratic formula -- I believe that completing the square is the one that students are least likely to remember. In fact, back when I was student teaching, my Algebra I class had fallen behind and we ended up skipping completing the square -- covering only factoring and the quadratic formula to solve equations. And yet PARCC expects the students to complete the square on the Geometry test!

I also wonder whether it's desirable, in Algebra I, to teach factoring and completing the square, but possibly save the Quadratic Formula for Algebra II. This way, the students would have at least seen completing the square in Algebra I before applying it to today's Geometry lesson.

Here are the worksheets for today. (Yes, today is a rare short post from me.)

Lesson 11-2: The Distance Formula (Day 112)

Today I subbed in an eighth grade U.S. History class. It's in my new district -- indeed, it's at the same school I mentioned last week in my February 10th post, albeit a different class. (Last week, I subbed in a Grades 6-7 history class.)

Since it's in a middle school and I do have one classroom management crisis today, I will do "A Day in the Life" this time:

8:15 -- First period arrives.

The assignment for today is to look up vocabulary and answer questions for the lesson, which is on westward expansion during the early 19th century. The students cut and paste web images to accompany their vocab terms.

The class begins with a Warm-Up, which is some sort of journal entry. Unfortunately, I'm not able to access it or assign it to the students. One thing that I've found annoying about Chromebooks is that it's impossible to access any email other than district email. Even though the regular teacher sent me a lesson plan last night, he sent it to my personal account, not my district account. Therefore, I'm unable to see or assign any journal entry.

Of course, this problem is completely preventable -- all I had to do last night was forward the teacher's email from my personal to my district account, when I still had the chance. I didn't do it because I thought I didn't need to -- I did print a copy of the plan, but I can't click on a link from a piece of paper. From now on, I should assume that I'll always need a digital copy of the lesson plan -- the next time a regular teacher sends a plan to my personal account.

I will mention what song I choose to sing today. It's Square One TV's "One Billion Is Big" -- and it's easy to explain why I chose it. Last night, Prince Markee Dee of the Fat Boys passed away. It was just shy of his 53rd birthday (which would have been today). Thus I honor him by performing one of his songs today.

9:10 -- First period leaves and second period arrives.

I begin by explaining why there's no journal entry, and then we go directly to the main assignment -- and this I do in all subsequent classes.

10:00 -- Second period leaves for break.

10:20 -- Third period arrives.

I've heard of this happening to teachers in the pandemic era, but it's never happened to me until now. In third period, I was "Zoombombed" -- that is, someone logged into Zoom (actually Google Meet) who isn't actually enrolled in the class.

The Zoombomber posed as a district administrator. Even though Google Meet warns me that this person isn't using a district account, I see the name of our district in the username and so the account looked official. Then the Zoombomber clicks on "raise hand" and starts speaking loudly, saying things that have nothing to do with the class.

I figure out how to kick the Zoombomber out of the meet. Then he attempts to enter the meet again, this time using the principal's name as his username. Once again, Google warns me that it's not a district account -- and of course, the real principal would be using a district account. This time, I refuse to let him in, and I call the office. The (actual) assistant principal enters the meet, and one of the other students and I explain what's going on.

After seeing the Zoombomber today, I can start to appreciate why the district Chromebooks won't let us access personal email -- it's for security reasons, to protect us from people like the Zoombomber. Still, it makes it more difficult to access our account when we have a legitimate reason to.

11:10 -- Third period leaves and fourth period arrives.

Fortunately, the Zoombomber is gone, and there are no major problems in this class.

12:00 -- It is now time for tutorial. The students from fourth period remain in the room.

As I often do during tutorial classes, I sing extra songs. Once again, to celebrate the life of Markie Dee, I perform the other two Square One TV Fat Boys songs -- "Burger Pattern" and "Working Backward." It is the first time I've sung either one in the classroom. Neither song is printed in my songbook -- instead, I get the lyrics to "Working Backward" from this blog, where I posted three years ago. I've never written the "Burger Pattern" lyrics until now -- I simply find the video on YouTube, quietly play and write a small portion of the song, and then perform it. It's the perfect song to sing just before lunch.

I sing one more song that isn't from Fat Boys during tutorial. One girl decides to work on her math assignment in tutorial -- she's in Algebra I, and the assignment is on mean, median, and mode. It goes without saying which song I sing to her -- "Measures of Center Song," a parody of "Row Row Row Your Boat." (I also performed this when I was at this same school last week.)

12:30 -- The students leave for lunch.

1:10 -- Fifth period arrives.

Once again, there are no major problems, and I sing "One Billion Is Big" one last time today.

2:00 -- Fifth period leaves. As is usual for middle schools in this district, sixth period is independent study P.E., and so the students and I leave after fifth period.

Today is Fourday on the Eleven Calendar:

Resolution #4: We need to inflate the wheels of our bike.

Once again, it's not a math class. I do tell them that it's easier to remember math concepts if they're given in a song. The Fat Boys' songs help us remember the difference between a million and a billion, the definition of triangular number, and the working backward strategy.

Of course, I do pay homage to someone today -- Markie Dee. In fact, let me add the "music" label to this post and include lyrics for all the Fat Boys' songs today, including "Burger Pattern."

ONE BILLION IS BIG -- by the Fat Boys

1st Verse:
Have you seen the headline? We did OK,
We sold a million records in just one day.
That's a thousand times a thousand sold,
That's plenty of vinyl, a million whole.
A million dollar bills reach for the sky,
Stack 'em about three hundred feet high.
A billion dollars is a thousand times more,
A lot more money than we bargained for.

Refrain:
One million is big,
One billion is bigger.
One thousand times one million,
That's one billion.

2nd Verse:
We're getting kinda hungry for our favorite food,
Hey, what do ya say? Are you in the mood?
Let's satisfy our special taste,
And get some lunch at the burger place.
See that sign? "One billion served!"
Beat box, that's a lot of hamburgers.
One thousand times, when ya order fries,
A million times one thousand apple pies.
(Repeat Refrain)

3rd Verse:
If we multiply one million by ten,
How close are we to one million then?
If we take a look, we will see,
We got a way to go, my friend Markie Dee.
If we multiply by one hundred this time,
Let's take a look, and we will find,
That we're not even halfway there,
We need a lot more to be a billionaire.
If we order one billion cheeseburgers,
And eat one million cheeseburgers,
It would be enough to knock us off our feet,
'Cause we'd still have almost one billion burgers to eat.
One million's not even one percent of one billion. Wow!
(Repeat Refrain)

You might wish to skip to 3:14 for "Working Backwards":

Notice that the beginning of the video has a scene from "One Billion Is Big" (the headline "Fat Boys Sell One Million Records.") Oh. and by the way, I like the part where the three backup girls start singing the refrain. They come in dancing -- backwards, of course!

As far as I know, neither Barry Carter nor anyone else has recorded the lyrics for this song. So let me write down the lyrics to the best of my ability:

WORKING BACKWARDS -- by the Fat Boys

1st Verse:
Tonight our show begins at ten,
There's a lot we got to do 'til then.
We gotta rehearse, we gotta eat,
Put our makeup on and find time to sleep.
What time do we need to start?
Let's take the backwards day apart.
Working backwards, we'll figure it out,
And show the world what we're all about!

Refrain:
Working backwards is the way,
To solve this problem, working backwards.
Working backwards is the way,
To solve this problem, working backwards!

2nd Verse:
We got part of the day to fix our hair,
We need a half hour in the makeup chair.
They start their job at nine-thirty,
What time do we eat, my friend Markie Dee?
Thirty minutes is all it takes,
To eat a salad, to share a steak.
So tell the chef dinner starts at nine,
Working backwards is working fine!
(Repeat Refrain)

3rd Verse:
We need to nap before we eat,
About forty-five minutes of catching Z's.
We gotta rest, I hear my drum machine,
The lights go out at eight-fifteen.
We can't forget rehearsal time,
An hour and a quarter for beats and rhymes.
So rehearsal starts at seven o'clock,
Beat box, tell me, what time have you got?
Let's see -- my watch says seven o'clock already,
Wow, we better get moving right now!
(Repeat Refrain)

BURGER PATTERN -- bv the Fat Boys

Well I'm going to my favorite burger place,

I'm gonna eat those burgers and stuff my face.

And a shake and fries, I'm gonna munch,

I say, Markee Dee, I love my lunch.

Hey guys, what will it be,

A burger today or two or three?

Monday I had a burger, shake, and fries.

Tuesday three burgers, shake, and fries.

Wednesday six burgers, shake, and fries.

Thursday ten burgers, shake, and fries.

Today I'll have, I'll have, I'll have a...(I'm hungry!)

0 + 1 = 1, 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15

Fifteen, that's what it'll be,

I bet 15 burgers will fill me.

Fifteen burgers and shake and fries,

I see the pattern before my eyes.

Yeah, you see the pattern before your eyes,

You order 15 burgers and shake and fries.

This pattern didn't get here out of the blue,

It's gotta vibe in the business that we'll tell you.

You see, I start with nothing and then add one,

And add the next highest number to that sum.

I just added one so then I add two,

And then I add three, and you can do this too.

I keep this up and add four more,

You see I have the business I came here for.

So now I have ten and add five to that,

Hey, 15, that's really that.

This can go on forever if you know what I mean,

What's next is added to 15.

Hey, you're really a winner!

So tell me, my man, what time is dinner?

This is what I wrote last year about today's lesson:

Lesson 11-2 of the U of Chicago text is called "The Distance Formula." In the modern Third Edition of the text, the Distance Formula appears in Lesson 11-5.

Let's get to today's lesson. Many students have trouble with graphing throughout Chapter 11, and furthermore, today we learn the Distance Formula, which of course will be difficult for some students.

In the past, I combined Lesson 11-2 with Lesson 8-7, on the Pythagorean Theorem (and indeed, this lesson in the Third Edition is titled "The Pythagorean Distance Formula").

David Joyce has more to say about the Distance Formula:

Also in chapter 1 there is an introduction to plane coordinate geometry. Unfortunately, there is no connection made with plane synthetic geometry. Here in chapter 1, a distance formula is asserted with neither logical nor intuitive justification. Of course, the justification is the Pythagorean theorem, and that's not discussed until chapter 5. In that chapter there is an exercise to prove the distance formula from the Pythagorean theorem. The Pythagorean theorem itself gets proved in yet a later chapter.

Fortunately, the U of Chicago text avoids this problem. Our text makes it clear that the Distance Formula is derived from the Pythagorean Theorem.

Today I post an old worksheet from a few years ago. It introduces the Distance Formula -- but of course, it teaches (or reviews) the Pythagorean Theorem as well  -- including its similarity proof, which is mentioned in the Common Core Standards.

Today is also an activity day, so let's add an activity to this old worksheet. Once again, in order to find a pandemic-friendly activity, I look to the class where I completed my long-term assignment. The Distance Formula is also included in the Common Core Math 8 Standards, and the eighth graders reached it on February 8th. That day, the regular teacher assigned an EdPuzzle on distance, followed by a DeltaMath activity.

I've heard of DeltaMath before, even though this class didn't use it until after I left. Unfortunately, it appears there's no easy way for me to reach the Distance Formula lesson without setting up an account, which I don't have.

Once again, this is what I'll do -- post last year's worksheet, and then declare Deltamath to be the activity for today. It's up to you to access Deltamath.

We still got a way to go, my friend Markie Dee. Rest in peace.