Congratulations to Nile Kennedy on winning the school wide airplane design competition on Wednesday. His plane beat out 9 other planes on the recess field at morning meeting.
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.
cheetah video: http://www.youtube.com/watch?v=3pn2SI4KGJc
b-girl terra: http://www.youtube.com/watch?v=VdjNMCmWZXA
We are a week into our study of exponents and square roots. We are working in stations to differentiate the work for students.
Each student has been given a student overview for the unit.
This week we derived the rules for exponents. We will continue to practice them throughout the next two weeks.
Here is a summary of the Exponent Rules from Math is Fun
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
(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
You can see an example of early notation in “Earliest Uses of Grouping
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
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
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
n (n – 1)
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
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
and some of the current debates here:
More on Order of Operations
- Doctor Peterson, The Math Forum
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.
This week, we will begin an exploration of rational and irrational numbers. Our goal for the week is that all students can do the following:
_____ I can convert fluently between fractions, decimals, and percents.
_____ I can express a repeating decimal as a fraction.
_____ I can determine whether a number is rational or irrational.
We will be starting station teaching where students will rotate bewteen Ms. Ransijn, Ms. Cannon, computers, and independent practice. They will spend 20 minutes in each space in small groups over the course of two days.
Some of the Khan Skills, the students will be working on in class include:
Videos to support the skills are:
This week we are continuing our study of the design cycle. We will begin to use our knowledge of scatter plots to make design decisions based on quantitative data with our paper plane design project. For details on the project follow this link.
On Friday, students presented their Everyday Object Redesigns to “venture capitalists” (played by other students). Each group of venture capitalists had a set amount of money to spend and got to decide how much to give to each Redesign.
The student’s were amazing in both their portrayal of venture capitalists and in their presentation of their own ideas.
See some photos below and videos coming soon.