Helen asks a deep question…

After doing station work with rational and irrational numbers, Helen asked, “Which is more common rational or irrational numbers?”

The answer follows- It’s in the last paragraph.

Rational and Irrational Numbers

Date: 2/2/96 at 0:4:38
From: Anonymous
Subject: Rational and Irrational numbers

What is the difference between rational and irrational numbers?

Date: 2/2/96 at 14:34:37
From: Doctor Syd
Subject: Re: Rational and Irrational numbers

Hello!

Before we talk about rational and irrational numbers, let's make clear 
one other definition.  An INTEGER is in the set: 

{...-3, -2, -1, 0, 1, 2, 3, ...}

It is just a positive or negative whole number.  Thus 454564 is an 
integer, but 1/2 isn't.

Now, a rational number is any number that can be written as a ratio of 
two integers (hence the name!).  In other words, a number is rational if
we can write it as a fraction where the numerator and denominator are 
both integers.  Now then, every integer is a rational number, since 
each integer n can be written in the form n/1.  For example 5 = 5/1 - 
thus 5 is a rational number.  However, numbers like 1/2, 
45454737/2424242, and -3/7 are also rational since they are fractions 
where the numerator and denominator are integers.  

An irrational number is any real number that is not rational.  By "real" 
number I mean, loosely, a number that we can conceive of in this world,
one with no square roots of negative numbers (numbers where square roots 
of negative numbers are involved are called complex, and there is lots 
of neat stuff there, if you are curious).  A real number is a number 
that is somewhere on your number line.  So, any number on the number 
line that isn't a rational number is irrational.  For example, the 
square root of 2 is an irrational number because it can't be written as 
a ratio of two integers.  

How would you imagine we would show something like that?  The proof 
is a proof by contradiction.  We assume that the square root of 2 CAN 
be written as p/q for some integers, p and q, and we get a contradiction.  
The proof has a little trick to it, but if you're curious about it, write back 
and I can tell you more!  

Other irrational numbers include:
square root of 3, the square root of 5, pi, e, ....

I hope this answers your question.  There are lots of neat properties of 
rational numbers, irrational numbers and real numbers.  For instance, it
turns out that if you were to try to gauge how many rational numbers, 
irrational numbers, and real numbers there are between 0 and 1, you 
would find that while there are infinitely many of each kind of number, 
there are many, many more irrational numbers than rational numbers.  The 
sizes of the infinities involved are somehow a little different.  Another 
property is that between any two rational numbers on the number line 
there is an irrational number; also, between any two irrational numbers 
there is a rational number.  

So, these are some things for you to ponder!  Write back if you have any 
more questions!

Week on October 1st- Motion and Exponents

It has been a big week in MST.  One student said, it felt like everything was easy and now it feels hard.  We have entered the world of exponents and exponent rules, such as negative exponents.  We are also diving into combining like terms and the distributive property.  I have reminded students that they are all new at this and that it takes time and practice to feel confident and comfortable with these skills.  We will continue to practice and work in small groups to make sure everyone understands.  If you want to know more about these skills, see the links below:

We have also begun our exploration of motion.  We began by watching a few videos of moving objects and talking about how they move and what we noticed about their motion.  The videos were:

We then watched the bugs bunny version of the tortoise and the hare as an entry into instantaneous and average speed.  We will end the week with a discussion of velocity and acceleration.

Math Topics:

Laws of Exponents

The Distributive Property

Combining Like Terms Game

Motion Videos:

cheetah video: http://www.youtube.com/watch?v=3pn2SI4KGJc

b-girl terra: http://www.youtube.com/watch?v=VdjNMCmWZXA

 

 

Who invented order of operations?

This was an EXCELLENT question asked by one of our students, Gabrielle Welch.  Dr. Math of Math Forum gives his answer below:  Sal Khan talks about the Order of Operations.

 

The Order of Operations rules as we know them could not have existed
before algebraic notation existed; but I strongly suspect that they
existed in some form from the beginning – in the grammar of how people
talked about arithmetic when they had only words, and not symbols, to
describe operations. It would be interesting to study that grammar in
Greek and Latin writings and see how clearly it can be detected.

At the other end, I think that computers have influenced the subject,
so that it is taught more rigidly now than it used to be, since
programming languages have had to define how every expression is to be
interpreted. Before then, it was more acceptable to simply recognize
some forms, like x/yz, as ambiguous and ignore them – something I
think we should do more often today, considering some of the questions
we get on such issues.

I spent some time researching this question, because it is asked
frequently, but I have not found a definitive answer yet. We can’t say
any one person invented the rules, and in some respects they have
grown gradually over several centuries and are still evolving.

Here are my conclusions, perhaps in more detail than you want:

1. The basic rule (that multiplication has precedence over addition)
appears to have arisen naturally and without much disagreement as
algebraic notation was being developed in the 1600s and the need for
such conventions arose. Even though there were numerous competing
systems of symbols, forcing each author to state his conventions at
the start of a book, they seem not to have had to say much in this
area. This is probably because the distributive property implies a
natural hierarchy in which multiplication is more powerful than
addition, and makes it desirable to be able to write polynomials with
as few parentheses as possible; without our order of operations, we
would have to write

ax^2 + bx + c
as
(a(x^2)) + (bx) + c

It may also be that the concept existed before the symbolism, perhaps
just reflecting the natural structure of problems such as the
quadratic.

You can see an example of early notation in “Earliest Uses of Grouping
Symbols” at:

http://jeff560.tripod.com/grouping.html

where the use of a vinculum (an early version of parentheses) shows,
both in its presence (around an additive expression) and its absence
(around the multiplicative term “B in D”) that the rules were
implicitly followed:
________________
In Van Schooten’s 1646 edition of Vieta, B in D quad. + B in D
is used to represent B(D^2 + BD).

2. There were some exceptions early in this development; in
particular, math historian Florian Cajori quotes many writers for
whom, in the special case of a factorial-like expression such as

n(n-1)(n-2)

the multiplication sign seems to have had some of the effect of an
aggregation symbol; they would write

n * n – 1 * n – 2

(using a dot or cross where I have the asterisks) to express this. Yet
Cajori points out that this was an exception to a rule already
established, by which “nn-1n-2″ would be taken as the quadratic
“n^2 – n – 2.”

There was also an early notation in which a multiplication would be
replaced by a comma to indicate aggregation:

n, n – 1

would mean

n (n – 1)

whereas

nn-1

meant

n^2 – 1.

3. Some of the specific rules were not yet established in Cajori’s own
time (the 1920s). He points out that there was disagreement as to
whether multiplication should have precedence over division, or
whether they should be treated equally. The general rule was that
parentheses should be used to clarify one’s meaning – which is still
a very good rule. I have not yet found any twentieth-century
declarations that resolved these issues, so I do not know how they
were resolved. You can see this in “Earliest Uses of Symbols of
Operation” at:

http://jeff560.tripod.com/operation.html

4. I suspect that the concept, and especially the term “order of
operations” and the “PEMDAS/BEDMAS” mnemonics, was formalized only in
this century, or at least in the late 1800s, with the growth of the
textbook industry. I think it has been more important to text authors
than to mathematicians, who have just informally agreed without
needing to state anything officially.

5. There is still some development in this area, as we frequently hear
from students and teachers confused by texts that either teach or
imply that implicit multiplication (2x) takes precedence over
explicit multiplication and division (2*x, 2/x) in expressions
such as a/2b, which they would take as a/(2b), contrary to the
generally accepted rules. The idea of adding new rules like this
implies that the conventions are not yet completely stable; the
situation is not all that different from the 1600s.

In summary, I would say that the rules actually fall into two
categories: the natural rules (such as precedence of exponential over
multiplicative over additive operations, and the meaning of
parentheses), and the artificial rules (left-to-right evaluation,
equal precedence for multiplication and division, and so on). The
former were present from the beginning of the notation, and probably
existed already, though in a somewhat different form, in the geometric
and verbal modes of expression that preceded algebraic symbolism. The
latter, not having any absolute reason for their acceptance, have had
to be gradually agreed upon through usage, and continue to evolve.

You can see a briefer answer in our archives at:

Ordering the Operations
http://mathforum.org/library/drmath/view/58237.html

and some of the current debates here:

More on Order of Operations
http://mathforum.org/library/drmath/view/57021.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

Approximating Square Roots and the Final (almost) Flight

Today we held the Paper Airplane World Expo at the ANCS Airfield .  We are very pleased with the attendance at the event, the quality of the planes, and the countries represented.  The top three planes for overall distance were:

  • Dequavious Mosley representing Everlast with a distance of 16.5 meters.

  • Agasha Irving representing Jamaica with a distance of 18 meters.

  • James Chitika representing Murica with a distance of 14.7 meters.

These participants along with the plane with the top speed will be invited to a grand master event on Wednesday, October 2nd at 8:35 am.  Please arrive 15 minutes early to prepare your planes and remarks.

In other news, we began stations in our class today to practice our 8th grade math skills.  This week students will be practicing evaluating and solving equations, working with integers and order of operations, and distinguishing between rational and irrational numbers with a card sort.  Small groups also got the chance to work with Ms. Ransijn to practice approximating square roots.