(2 + 3i)(8 - 5i)
When in standard complex form, the imaginary part of the above = _____i.
To answer this, we can use FOIL -- actually just OI as only these contribute to the imaginary part. So we have -10i + 24i = 14i. The imaginary part is 14 -- and of course, today's date is the fourteenth.
This is obviously not a middle school problem -- it's Algebra II at the earliest. This means that once again, I give Illinois State questions for Warm-Up, not Pappas questions.
Today is the special activity day that I have planned, at least for sixth and eighth grades. As I wrote before, I have to do Student Journals in seventh grade because the class doesn't meet tomorrow. This class is learning about integer operations. This isn't exactly the topic I want to rush -- that this topic is already getting an extra day, but it really needs much more time. As you would expect, the students are confused when I had to jump from addition/subtraction to multiplication and explain why 7(-2) isn't 14 since the positive number is bigger -- or even worse, 5.
The sixth grade activity comes from Denise Gaskins -- one of the other teachers I met during an MTBoS challenge last month. I promised her that I'd do one of her activities in class and tell her about the experience, so here it is:
The particular activity I choose for today is called "Tax Collector":
This is what I wrote in my comment to Gaskins:
Hello! I commented here during last month's MTBoS challenge. We agreed that if I used one of your games in my classroom, I tell you about it, so here goes!
Anyway, I played Tax Collector with my sixth graders today, and they seemed to enjoy it very much! I began with me as the tax collector and the class as the taxpayers. As it turns out, the first number chosen was 11 -- and the end, the class just narrowly lost to the tax collector, 110-100. If they had started with 19 (or even 17) and made all the same subsequent choices, they'd have won the game!
Afterwards, I divided the class into pairs, with one student as the tax payer and the other as the tax collector. No taxpayer wins, or comes quite as close as our initial game with the whole class, even though by now they knew that it was best to start with a large prime like 19. A few taxpayers believed they had won, but often it was because the tax collector didn't take all the factors correctly -- for example, a taxpayer started with 20, and the collector took only 10 instead of 1, 2, 4, 5 as well.
All in all, it was the most fun we had in class in a while. Thanks for the activity!
In fact, you can see what makes the game so difficult for the taxpayer. If the payer takes a prime, the collector can take only 1. But the payer can never take a prime on any move but the first, since no matter what number the payer takes on the first move, the collector takes 1. So the payer can only take at most one prime the whole game -- the collector is guaranteed all but one of the primes. This is why it behooves the payer to take the largest possible prime on the first move. I'm not sure that my students recognize 19 as a large prime number, but they do learn to start with a large odd number.
The tricky part of the game is when to take an abundant number like 12. If the taxpayer takes such a number too soon, the collector gets many of its factors and takes the lead. But if the payer waits too long to take the number, all of its factors are gone and the collector keeps the number itself. The girl who mistaken thinks that she beats the collector does learn that it's good to take 10 before 20 -- if done correctly, after the payer takes 20, the collector gets only 4.
As I wrote to Gaskins, this is a great activity to get the students thinking about factors -- which is great during the current lessons on fractions and percents.
I want to discuss the eighth grade project -- but first let's look at the song for music break. Since today is Valentine's Day, I just had to sing the Square One TV song "The Mathematics of Love."
By the way, speaking of Valentine's Day, last year I wrote that one group that receives many V-Day goodies is teachers. The students give me candy all day, and one seventh grader even offers me a special holiday doughnut. Meanwhile, the K-1 teacher gives me some red velvet cupcakes to thank me for moving a large toy oven out of her classroom. Along with the fact that I wat waffles with syrup for breakfast and the school cafeteria serves yogurt for lunch, it means that the first thing I eat today that isn't sweet is dinner!
Barry Carter provides us with the lyrics:
The Mathematics Of Love
Lead vocals by Larry Cedar
Backup vocals by Reg E. Cathey, Cris Franco, Luisa Leschin, and Beverly Mickins
Featured vocals by Arthur Howard
In this song, the gag is that the lead vocalist reads all the numbers in the song as Roman numerals, so he sings "eye night," "eye eye hearts," "eye eye eye words," and so on. I admit that of all the songs from Square One TV, this one is my favorite.
Now as it turns out, my eighth grade project is based on this song. Part of the project has the students convert from Arabic to Roman numerals and vice versa. This is actually inspired by a project mentioned in the Illinois State text. Recall that two weeks ago was "Input, Process, Output," where the students are introduced to functions. There's a follow-up to this project -- but it's printed only in the teacher's edition, not the student texts. Anyway, the students learn that conversion to Roman numerals is also a function, with an input, process, and output.
The Illinois State teacher's text actually has the students focus on attempting to add, subtract, multiply, and divide Roman numerals. Instead, I mainly have the students convert the numerals and provide only one example for each operation. Naturally, the students find it easier to convert back to Arabic numerals before doing any calculations. The most common mistakes are just as you would expect -- confusing IV = 4 with VI = 6, and IX = 9 with XI = 11, and so on.
The text also asks whether the set of Roman numerals is closed under addition (or any of the other three operations). Notice that as soon as an additively closed set contains 1 (or Roman numeral I), it automatically contains infinitely many natural numbers, so we must ask whether one can write arbitrarily large natural numbers in Roman. At first glance it would appear not, since M is the largest number that has its own letter, and it only represents 1000. But sometimes we place a bar (which is called vinculum in Latin) over a Roman numeral to multiply it by 1000. So V-bar is 5000, M-bar is a million, and so on. So we can write Roman numerals as large as we please, if we use arbitrarily many vincula when writing them. Only then can the set of Roman numerals be closed under addition.
This is only part of today's activity -- the "mathematics" part. Since the activity is called "The Mathematics of Love," I need to represent the "love" part of the lesson. To accomplish this, I take a page from one of my usual fallback blogs -- Sarah Carter:
Even back when I was the age my students are now, I often found myself thinking about the boyfriend and girlfriend functions. As we see from Carter's blog, boyfriend and girlfriend are supposed to be functions, but in practice, many people cheat and so they're no longer functions, as one input (boy) is paired with more than one output (girl, etc.).
This example fits perfectly today, since it fits both the holiday (V-Day) and the content (functions). I begin with some examples of functions (such as father) and non-functions (such as brother) before moving on to boyfriend. This is actually part one of my activity -- part two is Roman numerals.
As it turns out, this week I was supposed to sign my sixth graders up for a Playworks activity, but I don't find out until it's too late. There was an open spot this morning, but I missed it. In a way, I don't mind that much -- the sixth grade math activity goes well. In fact, the students behave reasonably well during the activities -- of course they want to misbehave during traditional lessons. I'd much rather give up the SBAC Prep time to Playworks -- with all the talking that's going on, precious little test prep is accomplished anyway.
This is a two-day post, so we look ahead to tomorrow's lesson, particularly the science lesson. The next project in the Illinois State science text is on physical and chemical changes. Notice that the seventh graders at my sister school already had this lesson back in November -- the day my car broke down and I subbed for my counterpart.
I don't know what tomorrow's Common Planning meeting is about ye. But I do remember hearing that there would be another meeting with the Illinois State developers on the third Wednesday in February -- which works out to be tomorrow, the 15th. Right now, the administrators aren't checking to make sure that we're implementing the Illinois State curriculum (hence today's non-Illinois State sixth grade activity). Instead, the concern is with the SBAC Prep time and making sure that the students are taking the practice SBAC test online. But our student laptops are old, and they can't be upgraded enough to download a browser powerful enough to access the website. (That's another reason why I could have given up "SBAC Prep" time to Playworks!)
Again, this is a two-day post, covering today and tomorrow. So the next school day post won't be until after the five-day weekend, on Tuesday the 21st. But recall that my monthly posting date, for "Day in the Life" is the 18th, a Saturday. So not only is it a weekend, but it's smack-dab in the middle of a long holiday weekend!
Oh well -- my next post will be "Day in the Life" on Saturday.