Wednesday, August 17, 2016

Activity: Number Patterns (Days 2-3)

Remember that now I'm a full-time teacher, I no longer have time to post every school day. My plan is to skip every third day. So I'm posting today, Day 2, but my next post won't be until Day 4. The result is that I should be posting approximately three times every week during the school year -- this week I'm posting Tuesday, Wednesday, and Friday.

Let's begin with the Blaugust prompt for today:

17. The best teacher I ever had was …. because ...

Hey, I just wrote about my favorite teachers at the end of the previous school year! I usually don't like reblogging to satisfy MTBoS challenge prompts, but then again, the following prompt encourages it:

26. Reblog an old post - reflect how you see/use it now?

So let me fulfill both Prompts #17 and #26 by reblogging my favorite teachers. From June 2016:

-- My favorite elementary teacher was my second grade teacher -- who later became my fifth grade teacher as well. She was one of the first to notice that I was good at math, and so she came up with the idea of having a Pre-Algebra teacher from the high school (which went from Grades 7-12 in my district) send me a textbook. As a second-grader I would work on the assignments independently, then my teacher would send my work to the high school before I worked on the next assignment. By the time I reached the fifth grade and was in her class again, she had convinced the high school teacher to send me the textbook for "APA," or Advanced Pre-Algebra.

-- Incidentally, my favorite math teacher was that teacher who sent me the advanced work. I finally met her when I was placed in her Algebra I class in the seventh grade. I was the only seventh grader in a class full of eighth graders, but she made me feel welcome in her class.

-- Just like Fawn Nguyen, I had my favorite history teacher when I was an eighth grader. He was also in charge of the Thespian Club at our school, and so he decided to teach history in a unique way -- he would dress up as a historical figure and lecture as if he were that character. Therefore his lectures were more memorable to the students. A few years ago, he retired from teaching, and many of my classmates held a big party for him.

-- My favorite science teacher was my junior-year teacher. I was an up-and-down student when it came to science -- the first two years of Integrated Science were more biology-leaning and I struggled a little, but the third year had more emphasis on physical science, which is more closely related to my strongest subject, math (as we spent over a month discussing with Kline's book). And so I did very well in this teacher's class -- indeed, she told me that I would finish the whole test in a few minutes and spend the rest of the time making my writing neat, and of course my answers were correct. She wondered why I wasn't enrolled in the magnet program, and I replied that I had moved to my new district as a freshman, while magnet students are recruited in the eighth grade. And so my science teacher convinced the school to admit me to the magnet program as a junior. Even though I was no longer in her class, she was still my most memorable science teacher for this reason.

-- My favorite English teacher was my senior-year teacher -- or to be precise, one of two English teachers I had that year. You see, the magnet program I'd entered a year earlier was a year ahead in English -- that is, junior-level English for neighborhood students was equivalent to sophomore English within the magnet. This meant that I would have to double up on English my senior year in order to graduate from the magnet -- and I didn't look forward to this, since my strongest subject was math, not English. So even though I was the only senior in a class full of juniors, I enjoyed this English teacher's class the most. This teacher allowed us to be creative in our writing -- I remember that for extra-credit, I wrote parodies of the literature we were reading, except with my friends and me as the characters. There was also an essay contest for seniors in which we were to write about a journey we had taken -- I wasn't going to participate, except that the junior English teacher whose class I had to take decided to assign the same topic for an in-class grade! I was in the unique position of writing an essay for class and submitting the same essay to the contest.  So I wrote about my journey through my education (much of which I just wrote about in this post) -- and won $200.

When I reflect upon my favorite teachers, I notice that they have some traits in common. Two of my teachers taught subjects I didn't enjoy, English and history -- and made them enjoyable by presenting them in a unique way. The other teachers taught my stronger subjects, math and science -- and they recognized that I was talented enough in those subjects to move me up to the next level.

Some traditionalists lament the fact that the Common Core accountability movement encourages teachers to focus on the weaker students at the expense of the stronger students. They say that some strong students want to move ahead in their classes, but the teachers, who claim their hands are tied by Common Core, won't let them.

I'm torn whether I should focus on my stronger or weaker students as I get ready to teach in the middle school classroom this year. On one hand, neglecting the weaker students is why many people spurn tracking, so I want to help my weaker students get ahead. But on the other hand, I myself am the beneficiary of certain teachers noticing my special talents and allowing me to succeed in more challenging classes. Therefore I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

This is so important that it bears repeating. I owe it to my stronger students to support them and celebrate their talents just as my own teachers celebrated my own talents.

Recall back on Square Root Day the story I told about teaching my second grade friend the square roots of 0, 1, and 144. I admit that this incident, along with my admiration of my second grade teacher, formed the foundation of my desire to become a teacher. At first I didn't know that Grades 7 and higher even existed -- I knew that my elementary school was K-6, and I'd always believed that students went directly from sixth grade to college. I remember that as a kindergartner, to me the sixth graders looked like grown-ups, and so I expected that they were nearly college students.

Naturally, it was the arrival of my Pre-Algebra text that alerted me to existence of 7th grade. I wasn't sure whether I wanted to be a teacher because I wasn't sure I'd be good enough at any subject other than math, but the benefactor who gave me the Pre-Algebra text was a single-subject teacher who taught math and nothing else. And so I knew at that moment that I wanted to become a single-subject math teacher -- which meant that I'd most likely teach in a high school.

Now I will be working in a K-8 school, just like Nguyen, But while I will have three preps, Nguyen has just two -- interestingly enough she teaches 6th and 8th grade, but not 7th. This means that my new K-8 school will actually be more like what I thought school was like when I was a little kid, with 6th grade as (one of) the highest grades.

The Blaugust prompt directs us to reflect on this old post, so let me do so. First, as it happens, I was checking the website of my old elementary school and -- believe it or not -- my second/fifth grade teacher still teaches there, nearly 30 years after I was a student in her classes! (At least, she taught there last year, when the site was last updated.) And according to the website, she still teaches both second and fifth grade! (The only other teacher I recognize is my kindergarten teacher.)

Now as I wrote in that old post, I want to focus on my stronger as well as my weaker students. I won't truly know who my strongest students are until after the first Benchmark Testing Week. But I am aware of the best way to protect the top students in the class -- with rules.

As I wrote in yesterday's post, the middle school English teacher came up with the idea of having the students write about the rules. And so this is most of what we're doing on Days 2-3 in the classroom.

I begin by giving each group of four students a sheet of poster paper. Students divide the paper into four quadrants, and each quadrant is labeled with the four rules mentioned earlier on the blog:

1. The Teacher Respects You
2. Respect Your Honesty
3. Respect Yourself and Each Other
4. Respect Your Class Equipment

As it happened, I only had the sixth and eighth graders create rules blogs -- this is due to a mix-up in the Wednesday Common Planning Day schedule. On Wednesdays the students take music, but music lessons don't actually begin until next week. So I gave the seventh graders an alternate assignment.

All three middle school teachers have the students create rules posters. Afterwards, the three of us discuss what the students write, and then the English teacher prints up a common list of rules for all three of us to use. The ultimate rules list will be given to the students tomorrow, and from it we create a Behavior Contract for the parents to sign. We haven't discussed the final rules list at the time of this post, since tomorrow is the day I don't post to the blog.

Because of this, let me post my lesson plans for tomorrow. After passing out the Behavior Contract and spending half the period discussing the rules, I move on to the Music Break. Tomorrow I will play the Square One TV song "Count on It," another song about the need to learn math:

Here are the lyrics, courtesy the following link:

Count On It

Lead vocals by Larry Cedar

Sooner or later, you’re gonna see some math
You can count on it
Sooner or later, those numbers cross your path
You can count on it
You may be hoping it will go away
But let me tell you, math is here to stay
You can count on it, hoo, yeah
You can count on it
Everywhere you look, they’re measuring the action
You can count on it
Everywhere you look, they’re even using fractions
You can count on it
They’re keeping time, and they’re keeping the score
They draw the line, and they’re running the store
You can count on it, hoo
Yeah, you can count on it
Look at the dial; look at how far
Look at how much; look where we are
Look at the gauge; look at the graph
Check out the numbers; you’ve got the last laugh
‘Cause it ain’t mystery; there’s nothing tough about it
You can count on it, that’s right
Soon you’re gonna see that you couldn’t live without it
You can count on it, hoo
Don’t take a genius or a great magician
To make a pretty good mathematician
You can count on it, hoo, yeah
Yeah, you can count on it, whoo
Oh, you can count on it, whoo
Baby, you can count on it
(fade out over Larry singing skat)

In the video the song is played on saxophones, but the only instruments I'll ever play in class are drums and a guitar. I like to play each song over two days, so I'll play the song on the drums on Thursday and the guitar on Friday.

Finally, the main lesson is another Opening Activity, on number patterns. It's time for me to reblog again, since I wrote about it last year:

During the year, I pointed out that many texts begin with a lesson on inductive reasoning, which often entails completing number patterns. I like this sort of lesson at the start of the school year.

In this lesson, students will use inductive reasoning to find patterns in sequences of numbers, letters, and names.I began with some simple sequences where students had to find the next two terms. Notice that Exercise 5 is the Fibonacci sequence -- a nod to Fawn Nguyen, who gives her Geometry students a worksheet on that famous sequence on the first day of school.

Speaking of Fawn Nguyen, yes, many of my opening activities are based on Nguyen's. As it turns out, one of her best known lessons is on patterns -- except these are visual, not numerical, patterns:

Exercises 7 and 8 on my worksheet don't come from Nguyen. Instead, they come from the texts used by the students I tutored. Those students enjoyed trying to figure out the patterns in these lists of names. I know that I've spent so recent posts about future presidents, but these lists are all about past presidents. Exercise 7 refers to George Washington, John Adams, Thomas Jefferson, and James Madison, so the correct answer is another James (Monroe), another John (Quincy Adams), and then Andrew (Jackson).

Exercise 8 looks similar, but this time it refers to money -- George Washington ($1), Thomas Jefferson ($2), Abe Lincoln ($5), and Alexander Hamilton ($10). So the correct answer is Andrew (Jackson again, $20), Ulysses (Grant, $50), and then it's all about Benjamin (Franklin, $100). As it turns out, one can actually extend this sequence. I was recently watching old 1960's episodes of the game show Let's Make a Deal on the new BUZZR channel, and often the host Monty Hall would offer contestants $500 bills (William McKinley) and $1000 bills (Grover Cleveland). There was even an episode when Monty showed a contestant an extremely rare $5000 bill that the bank had allowed him to show on that episode only -- had the contestant won it, she would have received a check for $5000 as the bill would have to be returned to the bank. James (Madison, not Monroe) was on the $5000 bill, so the sequence would continue Benjamin, William, Grover, James. It may be a good idea for teachers to give the related number sequence 1, 2, 5, 10, 20, 50, ..., as a hint.

The worksheet was getting long, so I stopped here. but notice that there are still many other types of problems that I could give:

Quatros: 4, 108, 60, 52, 36, 144
Not Quatros: 2, 29, 106, 18, 15, 22, 6
Which are Quatros? 86, 737, 42, 72

Semirps: 2, 13, 11, 23, 53, 97, 71, 47
Not Semirps: 15, 25, 209, 21, 190
Which are Semirps? 123. 67, 51, 27

Notice that "Quatros" are simply multiples of four. The word "Quatro" comes from the Latin word for four -- and we'll see that root later on in Geometry when we cover quadrilaterals. As it turns out, the modern Portuguese word for "four" is quatro [yes -- the Rio Games are still going on now -- dw] A few other Romance languages pronounce the word for "four" identically to the Portuguese, albeit with a slightly different spelling.

As for "Semirps," any nerd -- or even a dren -- can see that "Semirp" is "primes" spelled backwards. I do find it a bit awkward that the text pluralized "prime" to "primes," reversed it as "Semirp," then pluralized it again to "Semirps." Then again, one advantage to calling them "Semirps" rather than "emirps" is that the extra s- may trick readers into thinking about the prefix semi-, which is Latin for one-half -- especially right after seeing the Latin root for "four" in the previous question.

I was also considering including the first two sequences from the "Improving Reasoning Skills" section, which contains some bonus problems:

1. 18, 49, 94, 63, 52, 61, ...
2. O, T, T, F, F, S, S, E, N, ...
3. 4, 8, 61, 221, 244, 884, ...

I was able to figure out the first one, and I'd seen the second one before, but the third question stumped me -- and I suspect that it will stump our students as well. (The second one in the sequence is as easy as One, Two, Three!)

One of my favorite websites when considering number sequences is the On-Line Encyclopedia of Integer Sequences:

The OEIS is one of the oldest sites on the Internet. Notice that it was first created in 1964 -- long before the Internet existed as we know it! Back in the 1960's, users had to submit queries by sending it a primitive form of e-mail. Nowadays, of course, it is web-based like most other sites.

Many famous sequences are entries in the OEIS. Here they are:

180,360,540,720 (Notice the geometrical interpretation -- sum of the angles of an n-gon!)
1,3,6,10,15,21 (triangular numbers)
1,3,4,7,11,18 (Lucas numbers, similar to Fibonacci)
1,3,7,15,31,63 (sometimes called Mersenne numbers)
2,6,15,31,56,92 (given by the polynomial that generates these, (n+2)*(2*n^2-n+3)/6)
3,12,48,192,768 (there are some signed sequences in the database, but here the signs were ignored)

In fact, the only sequences I didn't enter were the ones containing letters or fractions, since this is in fact an integer sequence database.

Here's a sequence related to one of the lists of dead presidents that I entered:


Finally, here are the answers to the bonus questions:

18,46,94,63,52,61 (but I really did figure this one out before entering it into the OEIS)
4,8,61,221,244,884 (too hard for me, no problem for OEIS)
2,3,6,1,8,6,8 (too hard for me, no problem for OEIS)

The last sequence did stump the OEIS, however. Neither one of us figured out the sequence:

6,8,5,10,3,14,1, ...

Maybe you can figure it out, then try submitting it to the OEIS! Unfortunately, the OEIS has been swamped with submissions for months. Still, you can see why I enjoy the OEIS as a handy resource for integer sequences.

I hope the students will enjoy tomorrow's activity.

I am not posting tomorrow -- the title of the post includes the notation (Days 2-3) to remind the readers that the next post won't be until Day 4, which is Friday.

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