Mr. Ramuel Mendoza Raagasfor Mathematics 16
on Lesson Study(Graduate Credit)
handled by Thomas Judson, Ph.D.

Sets Shots

Venue: a Metro-Boston area school (ideal: private school with young teen girls, or else ethnically diverse public school--- big no: all-boys private school); a gymnasium area may be used (indeed, it would be preferable)

Duration: at least two hours and half with twenty-minute break (under three hours all in all); alternatively, four hours activity with forty-five minute break (field day fitting the span of six hours, i.e. 10 a.m. till four p.m.)

Preferred time slot: starting at 1:00 p.m., ending just before 3

Ideal Class-size: around 22 to 44 students

Materials: (on the instructor panel's part, video footage, Topps cards, compact disks


We would like to maximize attendance. I guess, the school's own machinery to be counted on to deliver the warm bodies, especially with a swing of the dictum, "Attendance is a must."


In the target school's bulletin boards and legitimate posting areas, we should have set up teasers for our event.

Muhammad Ali, Sarah Hughes, Kareem Abdul Jabar, Sasha Cohen, Michael Jordan, Larry Bird

Letter of Introduction

One Noon Lecture

Hi, we are your friends from Cambridge. Your teachers have invited us to introduce you to a very special topic in math. This coming [specified date] As part of our one-time activity, we will need you to bring whatever materials you may find convenient to bring. We'll also bring ours!

See you this [specified weekday].

Class would be advised to bring in magazines, Topps cards

Be original!


The Extension Team

What is a set?

What to play?

Right from the outset, the instructor must be able to elicit ideas from the students.

The instructor(s) for this lesson are not expected to be such grand authorities in sports. As such, they should be prepared for original ideas straight out of the students' initiatives.

Whenever the students display such voluminosity of knowledge as to sports trivia, the instructor may eventually revert to the essential framework of the lecture, punctuating whatever strings of insights the kids draw up with key points at the very core of this lesson (i.e., the buzz words: intersection).

Setting up the play

[Dress options for instructors, either three-piece suit or sports wear.]

Hi, class. We're here to teach

This is Jason, and this is Carlos...this is Miguel, Pedro and I'm Ram.

Today, we will discuss a key concept in mathematics, something quite apparent even on the level of common sense, and yet carrying such deep implications as to scour even the most arcane fields of science. This is concept is an organ which figures into particles too small to be observed experimentally even today in this age drenched in technology. This same concept also reaches farther beyond what our most powerful telescopes can even tickle.

Instructor: Our lesson for today is the set.

What is a set?

[Instructor(s) will ask students to define or at least give their prior notion of what a set is. This prior notion may have been conceived with any previous lessons the students may have encountered on sets, or perhaps the promotional period leading to our visiting day would have given time for some good students to seize the initiative by looking up on or asking their parents about our pre-announced topic for discussion.]

I: Have you ever discussed what a set is in any of your math classes?

I:Well, what do you remember from it?

[Hopefully, the instructor won't have to do the entire initiating, especially for elder students, i.e., ninth-grade upwards.]

A set is a collection of elements. We may name a set's elements one by one. Alternatively, we may let out a guideling, or governing rule, for identifying all which may be members of a given set.

How does a set differ from a group? Well, we'll zoom in the semantic difference of a mathematical set from that of a group (group, that is, even in non-mathematical respects). At the heart of the terms, group and set do have a lot in common, but further down our discourse on sets in the mathematical set, we will relinquish any luxuries as to the tempting interchangeability between the two terms.

The mathematical set differs from (the common notion of) a group in that a set's elements may be remotely located from each other, even fixated so, to such extent that they never need move or simply be geographically proximate with each other, in which instance the elements would not lose their status of membership.

When we normally talk using the word "group", we understand that the group's elements must band together, just as the chemical elements which belong to the "halogen" group in the Mendeleevian periodic table must fall in a straight line together, one on top of each other.

The togetherness of halogens, however, is better appreciated conceptually, as their characterizations are bound to fall every now and then into adjacent paragraphs within a given chemistry textbook.

A group of friends may spend a summer far from each other, but we will cherish their being a group or more so when they get together, even with the utmost of corporal proximities.

The set of lovers of Formula One car racing is quite a huge one all across the world, but its members don't homogenously cover all the land that is in our planet. Even in countries without qualified NASCAR competitors, there are big fans rooting for Michael Schumacher.

Thus, the millions of members of professional track car racing need never group into the same space (although they would need to train their eyes and attention convergently onto television broadcasts, which they would be bound to enjoy simultaneously, via live, transmissions, that is)

We will need to figure out ways both of naming sets and of identifying the members which would go right into them.

Lets get the names right off first. We'll grab the shopping carts before picking up our groceries and other items.

Fortunately for us, naming sets is not much of a big deal. Naming sets is less a colourful process than naming computer variables. In naming sets, we only need to use big, or capitalized, letters from our English [Well, historically Roman.] alphabet. Such a simple process seems to restrict the number of nameable sets to a mere...


That we may only name twenty-six sets using the traditional notation of set nomenclature poses no grave threat, as even as you get into college (introductory courses, that is, not so deep into advanced set theory), you will be asked (for any given problem) to deal with at most only a handful of sets (perhaps three, and hardly more than nine sets all in all).

Instructor gets a magnetic board loaded with chips representing basketball players.

OK, we'll call this set B.

And this will be set C. [Instructor flashes a gripful of encased audio compact disks. The instructor prolongs his grip, making a point to try go instill in the students a sense of bewilderment as to the appositeness of this latest set]

The set carried the titles Scorpion's Love at First bite, Elv1s, and a Tchaikovsky.

How do we tell what belongs to a set?

roster and rule methods

How do we describe set C? That'd good

How else may we sharply describe set C.

Set C is the set of CDs containing music which is often used in performances of Ice Skating. E.R.: Huh? or preferebly Oh, yeah

unit set

What is the set of basketball players who have been noted to score one hundred points in a single game?

What is the set of basketball players who have been noted to score seventy-four points or more in a single game? Wilt Chamberlain is still the only player.

empty set

What are the years in which Los Angeles beat the Boston Celtics to clinch the championship

Let it be... B!

equal sets

[play video]

I: OK class, let's go watch this.

What do we have in here?

Expected Response: A game!

That's right

What kind of game?

Expected Response:A hockey game.

Which one is the home team?

Expected Response: The Bruins!


Let's just focus now on the home team at the court.

Let B be the set of all hockey players on the ice

The Universal set

In the most abstract sense, the universal set contains everything. As such the universal set would be--- the Universe.

How could we de-abstractize the dimension of the universal set? Context is our key.

one-to-one correspondence

Any given team on the court will have only one player wear a particular jersey number. As such, it never happens that more than two people in the playing field (i.e., for soccer, baseball) or the hardcourt (as in basketball) carry the same jersey number on their backs. Generally


Let I be our universal set

By rule, we shall define I as being the set of all people who have ever lived practicing the religion of Islam.

Muslims, you mean?

My such a big set, that would be far too grand to enclose over here in our little Venn diagram.

Mr. Raagas, I'm a Muslim.

Let A be the set of world-class athletes reknown the world over.

Which of these two sets is bigger? Grander. Set I is indeed bigger than set A, because set A includes over a billion people. Set A is quite small. From the top of our heads, we might have a dozen or so names in mind. In the intersection of the two sets

On the spot

OK, class we're going to make a set!

First, we'll do a wave down the rows. Do you know how to make a wave.

Starting with [kid's name], one by one we'll raise up each just one of the exhibits you've brought and yell out the name of the sport it's on. Like so...

How many of you have brought something on tennis?

How many of you have brought something on bowling?

For those of you have brought materials pertaining to more than one sport, pick in your heart the one that you'ed prefer today, and each of you will now be classified according to the sport you've chosen.

Okay all of those who picked volleyball, stay in that corner over there.

Okay all of those who picked golf, stay in that corner over there.

Okay all of those who picked soccer, stay in that corner over there.

[Approach each group one by one] [per group inquiries] OK, what are your names. I know we can identify each of our groups by the name of its sport Where's the volleyabll group? Oh, there you are fellows Each group has a name for its sport, but we will choose another name for each group. This new name will comprise of just one big letter. So the tennis can call itself. Your group label letter need not be the first letter of the sporty you've chosen

OK, in this situation we have right here: groups are sets. Your groups are sets, but in the broader, mathematical sense, sets need not be mere groups.

Which of you picked sports that use balls?

Which of you picked sports which don't use balls?

[watch out for snide adolescent humour]

Hypothetical Question: "Mr. Instructor, Is a shot put a ball?

Why, yes [Charlie---] a shot put IS a ball--- and a very heavy one at that, like a bowling ball, I guess.

[humor ONLY for college students]

When Ben Johnson won his temporary gold medal for Olympic sprinting, he didn't have much of balls to use for it.

Stop fiddling with sets when all twenty-six letters of the ENglish alphabet have been used up as set names. Venn Diagrams

The conjunction of numbers and sets

I hope by this point in your education, you will all get to appreciate the fact that math is made up of more than numbers--- oh, much more, indeed [to be voiced out by the instructor with an air of Aeschylian somberness].

Math is made of much more elements than either numbers or sets or even both of them put together. In a nutshell, math is embodied by numbers, sets, operations, illustrations...

One fold of math is comprised of pure conceptualization (like the Euclidean lines, and the infinite number of infinitesimally-sized points they are comprised); a second, of practical representation (even of useful approximation which we may subscribe therein); a third, of transformation, change--- a set may lose or gain members through the course of time, although such adjustments have not been traditionally considered in the construing of set theory for the pedagogical purpose of K-12 education; a fourth, of resignation (We may not be able to "register" all the members of a set, we have defined by rule). There may be more.

Sets are a key organ of math (even of science, in general). Thanks to sets, we can appreciate the value of closure. Closure often relieves researchers from the burden of scouring . No matter how many thousands more of geneticists surface over the next couple of decades, we can fairly say that they will never encounter anything to sequence in DNA other than the two pyrimidines and two purine we are all familiar with,

What do numbers and sets both do (in each of their , overlapping ways)?

[Give two to five minutes to fish for ideas.]

Sets and numbers both represent entities. Sets and numbers are two chief modes of representation. Does anybody here want to explain what we mean by "[to] represent?"

[Check if grammar school students comprehend don't lose thread of the conceptual words we employ.]

Now we go into sets which contain ONLY numbers as their elements. How many of you brought Topps cards? [Instructors must furnish Topps cards for those without them]

Both sets and numbers are indispensible in math and science. Sets are even explicitly indispensible in other academic disciplines such as sociology and philosophy.

Sets of Numbers

There are major sets already present in . EVery human being who receives an ample benefit of general education is made aware of these major sets. They are the set of numbers. These are sets whose members are entirely numbers. The first set of number we will discuss is, in a basic sense, the smallest set among the fundamental set of numbers. Yes, there are sub-sets smaller than this fundamental set of numbers, but such sub-sets cannot be deemed as fundamental as this original set. The smallest fundamental set of numbers is the set of counting numbers. What are counting numbers? Uhhh... we count with them? Yes, we count with counting numbers, but can we count using numbers different from counting numbers? The counting numbers are also known as natural numbers, and they contain all the numbers even a toddler is taught about. The counting numbers are different from Later on in your science and education you will get to "count" using numbers which don't belong to the set of counting numbers, but such "counting" is better formally known as "calculation" rather than "counting". Can anyone tell me here what the difference is between calculating and counting? Calculating and counting are different, and yet they are very strongly related. Counting is different in that a lot of it can be done without getting into any calculating (or calculations--- yes, that would be the term for it).

Whole Numbers

Then there are the whole numbers. The whole numbers is almost of the same dimension as the counting numbers, except that we include zero. Zero was a number invented by the Indians, a number unbeknownst to Graeco-Roman Hellenic-Classical Western civilization. WHo invented zero The Indians (a few) Again, who invented zero Who invented the double zero? Robert Parish


Charled Pierce made twelve out of nineteen field goals (12/19) against the grizzies this week. Robert Parish devoted 14 out of his 21 seasons to playing for the Boston Celtics,


MLB. Ted Williams had a batting average of .344 what is


Questions and Problems to hang upon the students after departing the visited classroom.

Roster Method Name several members or elements (but not all!--- perhaps just five would do) of the following infinite sets. K-12 Let S be set of all the sports offered collegiate level:
  1. Let R be the set of words of any Romance language which are cognate (and not faux ami) to any other words

Educao Esportes Futebol Ensino Fundamental Assinatura