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SPECIAL
POINTS
OF INTEREST:
Editor: Gail Homer Berry
V O L U M E
I ,
I S S U E
V I
M A R C H
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2 0 1 5
(INTER) SECT Report
2
Sign up for
ARML!
Students from
California:
www.mathleag
ue. org/
armlsfba.php
Remember to
read the new
FAQ and Policies documents before
committing!
Students from
other regions:
www.mathleag
ue. org/
armlwww.php
INSIDE
THIS ISSUE:
INTERSECT
Report
1
Gear up for
ARML
1
Igor’s Magic
Sign Chart
2
Kudos
2
Picture Page
3
About Us
4
Although the weather didn’t
cooperate with us in March—
blizzards forced us to reschedule
and even to cancel some contests—mathleague.org held four
regional middle school contests
which served hundreds of students.
Experienced mathletes noticed
that the tests were much more
difficult than normal. That was
deliberate; we tried to gear them
to the state MATHCOUNTS
level. Given the looks of concentration we saw during the tests ,
and the looks of respect we saw
afterwards, it appears we succeeded. The tests will be available for sale in our online store
this summer.
California (1st), Quail Valley
Middle School in Texas (2nd),
and Basis Independent in California (3rd).
One wonderful thing about our
online results system is that students can compare scores,—not
just rankings—across schools,
chapters, states, and the nation.
We plan to make next year’s
INTERSECT contests even better, and we hope you’ll come.
(Of course, we also hope that,
next year, the weather will decide to cooperate.)
Congratulations to the top three
overall teams nationally. We will
present trophies to the teams
from Redwood Middle School in
[Note: mathleague.org is not
affiliated in any way with MATHCOUNTS.]
Gear Up For ARML
If you’re a high school student in
Northern California, the San
Francisco Bay Area, or one of
several other western states, and
if you’ve been having a nagging
feeling that you need to do
something, this is it.
It’s time to sign up for ARML.
Last year, our top San Francisco
Bay Area (SFBA) team placed
second nationally at the American Regional Mathematics League
contest (ARML). The year before
that, they won. We have a history of our elite teams doing extremely well, but we also help to
field many other teams because
we want everyone to have a
chance to participate in this great
event.
For students in Arizona, Idaho,
Montana, Nevada, New Mexico,
North Dakota, South Dakota,
and Wyoming, go here: mathleague.org/armlwww.php. Apply
by April 21st, pay $220 by May
1st, get your permission slip
signed by parents and notarized
(no exceptions), and monitor
your email for practice announcements.
For students in Northern California and the San Francisco Bay
Area, the sign up deadline is
April 11th. Go to
www.mathleague.org/
armlsfba.php and follow the instructions.
Students and their parents (from
California only) are also respon-
sible to read the FAQ and Policies documents and abide by
them. This is especially important
regarding withdrawing from the
team: ARML is a lot of fun, and it
is enormously educational, but it
is also a serious responsibility.
Before you sign up, make sure
you can honor your commitment. Is your schedule clear ?
Can you participate appropriately in practices? Do you understand the consequences, both to
yourself and to your team, of
quitting after teams are formed?
But enough sternness. ARML is
also awesome fun. The bus rides,
the casino night, the t-shirts,
watching the coaches perform
embarrassing forfeits if a mathleague team wins…. Join us!
PAGE
2
Backstory: Igor’s Magic Sign Chart
On test 11505, Sprint #8, the
solution included a reference
to Igor’s Magic Sign Chart with
a promise that the newsletter
would explain all.
“We never recycle
questions; we write
new problems
every month. Our
tests emphasize
creative and critical
thinking, applied
reasoning, and
problem solving.
Not only do our
tests provide a
worthwhile
challenge on their
own, but they are
also excellent
preparation for
MATHCOUNTS,
AMC, ARML, and
other demanding
competitions.”
—Tim Sanders
Igor Konfisakhar is a professional math tutor who specializes in helping gifted students
realize their highest potential
as mathletes. He owns and
operates the St. Louis Math
Help Center and uses interactive classrooms to teach online
as well. His website is
www.stlouismathtutor.com.
A former mathlete himself,
Igor has many methods for
maximizing efficiency. The sign
chart is one of them: it is useful for graphing rational polynomial functions.
Let us take, for example, the
function y = (x-4)(x+2). The
“zeros” are 4 and –2, meaning
that the graph intersects with
the x-axis when x equals those
values. The function is neither
positive nor negative at those
points.
be positive. For example, (5-4)
(5+2) = 7. Thus anything greater than 4 on our number line
will be positive.
When the x value “crosses
over” the x-axis, however, the
sign of the function will
change. Where –2 < x < 4, the
function will be negative. We
can prove this by plugging in an
integer (e.g.: 1) and getting (14)(1+2)= -6.
When the x value crosses
over the x-axis again, it will
change signs once more; if x is
less than –4, the function will
be positive.
We are left with a quick chart
which looks like this:
The question on 11505 Sprint
8 was “Find the sum of all
integers x for which (x-2)(x+3)
(x-5)(x+7) < 0.
By Igor’s method, rather than
performing several iterations
of plugging in various points
and graphing them, almost at
random, we draw a simple
number line, marking –2 and 4.
If we set the polynomial above
equal to 0, the “zeros” are –7,
-3, 2, and 5. If x > 5, we’ll have
(+)(+)(+)(+) terms, for a net
positive. If 2 < x < 5, we will
have (+)(+)(-)(+), for a net
negative. Thus the function is
negative between 2 and 5.
First, we consider the (x-4)
term. If x is greater than 4,
then both terms will be positive and their product will also
As a general rule, the sign
“flips” each time we cross a
zero. In this case, an automatic
sign change means that where
–3 < x < 2, the function is
positive, and indeed, if we plug
in an integer (such as –2, 0, or
1), we see that we get (-)(+)(-)
(+) terms for a net positive.
Continuing this pattern, we get
the following chart:
The integers for which the
function is negative are –6, -5,
-4, 3, and 4. Their sum, –8, is
the answer to 11505 Sprint
#8.
With a little practice, a mathlete can generate such a chart
within seconds, getting a general overview of the function
to use for problem-solving or
sanity checks. Thanks, Igor!
Note: In the case of a “double
zero”, the sign will “flip” twice,
For example, (x+3)2 yields
zeros (–3)(-3). Just as multiplying by a negative flips the sign
and multiplying by another
negative flips the sign back, so
too does the function behave.
Thus with an odd exponent,
the changed sign will remain
changed; if there is an even
exponent, the changed sign
will change again.
We encourage you to experiment with this and see if you
can prove it generalizes. For
homework, try this:
y = -(x+3)5(x-4)2/(x+2)7(x-11)4.
Kudos
On test 11516, 8th grader
Michael Zhang from Redwood
Middle School earned a perfect
Target score at the contest
held in Saratoga, CA. This was
a particularly difficult test,
since it was for INTERSECT.
MATHLEAGUE
MESSAGE
On test 11506, at a contest
held in Dubuque, IA, 9th graders Kevin Liu and Pranav
Krishnamurthy, and 10 grader
Casey McClenathan, all from
Iowa City West High School,
earned perfect Target scores.
Proud of a student? Send your
pictures and perfect scores—
and parental permission to
publish them—to:
Gail Berry
[email protected]
MATHLEAGUE
MESSAGE
PAGE
3
Picture Page
Top left: Ar yan Ar or a, 2nd place in 3r d gr ade at the San J acinto contest Mar ch 14th. Top right: Students fr om Commonwealth Elementary School in Houston, TX raked in the hardware at the March 14th contest. Bottom left: (left to right) Eric Berry,
Jay Leeds, Maeve Dever, and Melva Loock celebrate their team trophy at (INTER) 2SECT. Bottom right: Jake Gresh, of Canyon
Springs Elementary School, with his 2nd place, 5th grade trophy from the February 26th contest in Prescott Valley, AZ and his
coach, Stephanie Stephens, holding the Can Springs 1st place team trophy. Center: Teachers from Harmony School of Innovation Ft. Worth held an impromptu contest amongst themselves during a middle school contest at their school on March 21st. We applaud
their competitive instincts and curiosity! We only hope that their proctoring duties didn’t suffer unduly while they were taking the
same tests as the students.
Improving math education
worldwide through
disruptive innovation.
About mathleague.org:
Staff Spotlight
We are an organization dedicated to helping
students from grades 3 through 12 learn math
through hands-on problem solving at contests.
We also sell old tests for practice and offer
online tutoring and classes.
Our questions (fresh every month) go well
beyond traditional curricula and are excellent
practice for MATHCOUNTS, ARML, AMC,
SAT, and other math tests. Because we believe
in making each contest a learning experience,
we return student answer sheets—plus solutions—so that students can learn from their
mistakes.
Our online grading system is fast, fair, efficient,
and robust; people are amazed at how quickly
we can start an awards ceremony after the last
test ends.
Tests tier so that students of all ability levels
can find appropriate problems to work on,
though we’re particularly proud of how well
we serve very gifted students.
For more information, please visit:
mathleague.org
why.mathleague.org
Contact information:
mathleague.org
PO Box 622768
Oviedo, FL 32762
[email protected]
Kendra Brashear has
provided administrative
support t o mat hleague.org since 2008.
Homeschooled from kindergarten through 12th
grade, she then majored
in accounting in college.
Kendra loves math and
children, so mathleague.org is a natural fit
for her. Her favorite test
is Number Sense.
She is the person who answers most routine
questions, helps to manage Tim’s schedule, cohandles the finances and bookkeeping, files paperwork, provides proof of insurance, signs
contracts, sends out mailings, maintains our
database, emails tests, keeps up with the mathleague.org mailbox, and juggles ARML team
details.
Tim likes to say that Kendra knows more
about mathleague than he does. A few months
ago, a co-worker “flew” Kendra’s desk while
she was on vacation. Two days in, we realized
how doomed we would be without her.
In her spare time, Kendra enjoys serving as a
youth leader at her church, playing softball, and
reading by the pool while snacking on a snickers bar.