I’m Done. What Do I Do Now?:
Providing Extensions for High-Ability
Students in Mixed Ability Classrooms
Jason Major
Michele Kane
Beaubien School
MA Gifted/NEIU
Center for Talent
Development
iMATHination
January 21, 2012
Why are you here?
Is it because you have seen or heard the
following:
I am sooo bored in math class.
I get assigned more problems that are exactly
the same because I finish early.
I tutor other students in class.
The textbooks that we use in class are really not
interesting—when will we ever use this?
I already know more than 50% of what we learn
in class.
When the teacher explains something, I already
know it.
It takes too long in math class before we get to
do any math.
There are too many notes in math.
Characteristics of Gifted/Talented
INTELLECTUAL
CHARACTERISTICS
• Exceptional reasoning ability
• Capacity for reflection
• Intellectual curiosity
• Rapid learning rate
• Facility with abstraction
• Complex thought processes
• Vivid imagination
• Early moral concern
• Passion for learning
• Powers of concentration
• Analytical thinking
• Divergent thinking/creativity
• Keen sense of justice
PERSONALITY
CHARACTERISTICS
• Insightfulness
• Need to understand
• Need for mental stimulation
• Perfectionism
• Need for precision/logic
• Excellent sense of humor
• Sensitivity/empathy
• Intensity
• Perseverance
• Acute self-awareness
• Nonconformity
• Questioning of rules/authority
• Tendency toward introversion
Source: Silverman, L. K. (1993). A developmental model for
counseling the gifted. In L.K. Silverman (Ed.), Counseling
the Gifted and Talented (pp. 51-78). Denver, CO: Love
Publishing Co.
Common Core and Gifted?
• Make sense of problems and persevere in
solving them
• Reason abstractly and quantitatively
• Construct viable arguments and critique
reasoning of others
• Model with mathematics (apply to new)
• Use appropriate tools strategically
• Attend to precision
• Look for and make use of structure
• Look for and express regularity in repeated reasoning
Suggestions to extend learning
and ensure a year (or more) of
growth for each year in school:
Art of Problem Solving
www.artofproblemsolving.com
Alcumus
For the Win
Countdown
Community
Online classes
Textbooks
Competition Math for Middle School
Mathcounts
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School
Chapter
State
National
Mathcounts club materials
6th, 7th, 8th grades
Solution
Well, the quick answer starts with the calculation that the snail’s
net gain is 5 meters a day. In four days it will have risen 20
meters, so on the fifth day it gets out.
But why would a problem this easy be so well known? Here’s
why. After three days, the snail has risen 15 meters. On the
fourth day, it can climb seven meters more, and that seven
meter advance puts it at 21 and beyond, so it can get to the
top and over the rim safely on the fourth day.
Stella's Stunners - Stella's Problems
ohiorc.org/for/math/stella
Previous IMSA - Abbott Fund
CyberQuiz 4Kids Challenges
A New Kind Of Triangular Numbers (6-7)
The Problem
Let’s build some triangles! Well, there are lots of ways to make triangles, so we need to set down some rules. Let’s start
small — say all the sides of our triangle need to have lengths of 1, 2, 3, 4, or 5. But here’s the interesting rule: each
pair of sides must have lengths which are relatively prime.
To say that two positive integers are relatively prime means that the greatest common divisor of the numbers is 1. Said
another way, the two numbers have no common factor greater than 1. So, for example, 8 and 12 are not relatively
prime because they have a common factor of 4, while 5 and 18 are relatively prime since they have no common
factor other than 1. 6 and 1 are relatively prime, too. We often abbreviate “greatest common divisor” by “GCD” to
make writing a little simpler.
So let’s start counting! You might start by thinking that since the greatest common divisor of a number and itself is itself
(the greatest common divisor of 7 and 7, for example, is 7), then no triangles we’re looking for have two sides with
the same length. In general this is true; however, since the GCD of 1 and 1 is 1, we must admit the triangle with all
sides of length 1 (we'll call it (1,1,1) for short).
Now can any other triangle have a side of length 1? Remember that the sum of the lengths of any two sides of a triangle
must be more than the third side (or, said another way, the difference of any two sides of a triangle must be less
than the third side). Since any two other side lengths must be different (aside from the triangle we’ve already
counted), they differ by at least 1, and so they cannot form a triangle with a third side of length 1. Thus, the only
triangle with a side of length 1 satisfying out rules is (1,1,1).
The moral of the story is that by resorting to mathematical reasoning, you don’t have to look at every single possibility
(though this is certainly possible — but it might take you a while!). You should be able to convince yourself that the
only other triangle possible given the conditions is (3,4,5).
Now what if we allowed the integers 1,2,...,12 as side lengths? Well, that’s your problem! The answer to this month’s
quiz is the number of triangles whose sides have integer lengths between 1 and 12, with the lengths of each pair of
sides being relatively prime. Happy hunting!
Problem authored by Dr. Vince Matsko, IMSA Math teacher.
IMSA Challenge
The Answer
The answer is 25.
We can begin with the triangle (1,1,1) as before. As argued earlier, there is no other possible triangle with a
side of length 1. What about a side of length 2? In this case, the other two sides must be odd (so each pair
is relatively prime) — but odd numbers differ by at least two. So there is no triangle with a side length of 2.
The next observation is that there cannot be two sides with even lengths. Since the odd side lengths are among
3, 5, 7, 9, and 11 (noting that both 3 and 9 cannot both be chosen because they share a common factor
of 3), it is not difficult to enumerate all possible triangles with odd side lengths: (3,5,7), (5,7,9), (5,7,11),
(5,9,11) and (7,9,11). This brings our total up to 6.
It is simplest to look at the remaining even side lengths — 4, 6, 8, 10, and 12 — on a case-by-case basis. In each
case, the remaining two sides must be chosen from 3, 5, 7, 9, and 11. With a side length o 4, the triangles
(3,4,5), (4,5,7), (4,7,9), and (4,9,11) are possible. It is worth noting that it is only necessary to check
consecutive odd pairs, since otherwise, the odd numbers would differ by at least 4. Our total is now 10.
With a side length of 6, we need only check pairs of odd numbers at most 4 apart — excluding 3 and 9. With
only three odd numbers left, this is easy: the triangles (5,6,7), and (6,7,11), bringing our total to 12.
With a side length of 8, we have the triangles (3,7,8), (5,7,8), (5,8,9), (5,8,11), (7,8,9), (7,8,11), and (8,9,11). Our
running total is now 19. With a side length of 10 (we cannot use 5), we have the triangles (3,10,11),
(7,9,10), (7,10,11), and (9,10,11), bringing the total to 23.
Finally, with a side length of 12 (we cannot use 3 or 9), we have the triangles (5,11,12) and (7,11,12). Thus, the
final total, which is the answer to this month’s CyberQuiz, is 25.
The American Math Competitions
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AMC 8
AMC 10
AMC 12
AIME
USAMO (or USAJMO)
IMO
Chicago Area Only
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City of Chicago Math League (CCML)
Chicago Junior Mathematics League (CJML)
North Suburban Math League
Illinois Math League
Illinois Council of Teachers of Mathematics
(ICTM) contest
• Payton Math Circle
(www.paytonmathcircle.org)
ONLINE MATH EXTENSIONS
http://www.themathleague.com/
http://www.continentalmathematicsleague.com/
Annual competitions for math lovers
http://mathcontest.olemiss.edu/currentproblems.php
The original Problem of the Week
www.khanacademy.org
Popular YouTube math teacher; all levels of mathematics are covered
http://nlvm.usu.edu/en/nav/vlibrary.html
National Library of Virtual Manipulatives
http://www.setgame.com
For visual-spatial discrimination
http://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/index.html
Learning links for kids
ONLINE MATH EXTENSIONS
http://www.hoagiesgifted.org/math.htm#math
Escher, Einstein, Fractals and more…scores of websites for high-ability math students
http://www.mathrealm.com/
Great resources and math extensions
http://www.ixl.com/
Math practice aligned with state standards
http://nrich.maths.org/public/index.php
Math enrichment from the UK
http://illuminations.nctm.org/
http://www.thinkfinity.org/partners.aspx
Partners including –NCTM
http://www.printable-puzzles.com/
Free puzzles-be sure to preview to ensure appropriateness
Khan Academy
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FREE
Work at individual pace
Thousands of lessons/worksheets
Track your progress with the help of a coach
Consider the “flipped” classroom
HOW ABOUT MATH
GAMES AT STATIONS?
Popular Games Provide Interest
Popular Games Develop Mental Math
•SET-For visual-spatial learners
•24
•Mancala/Chess/Bridge
•Logic Puzzles
•Figural Analogies
ED ZACCARO
http://web.me.com/zaccarohandouts/Site/HOME_files/15%20perplexing%20
problems.pdf
Math Circle
• Simply put, a group of people who get
together and do math
– Afterschool clubs including games like Chess Club
– Math Club
– Online
Math Journals
– Emphasize the reading/writing/problem-solving
aspects of mathematics
– Communication of ideas is essential
Mentors
• Favorite teachers can make connections
• Center for Talent Development; Worlds of
Wisdom –scholarship $$ are available
• Jack Kent Cooke Foundation
• Other older math students WANT to help
younger students…find someone in your area
who has had success and reach out to them
-National Honor Society Students are great mentors
Source :
Rotigel, J., & Fello. (2004).
Mathematically gifted students:
How can we meet their needs?
Gifted Child Today, 27(4), 4651.
For a copy of this presentation email us:
Jason Major
[email protected]
I will help:
--Talk to your school about offering math
clubs/math competitions
OR
[email protected]
I will help with additional resources about
gifted and talented children (even if they
are not identified)