MathAmigos and SFPS Grades 4-6 Workshop Math Circle Session #1: The Game of SET
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MathAmigos and SFPS Grades 4-6 Workshop
Math Circle Session #1: The Game of SET
Circle Leader: James Taylor, Math Circles Collaborative of New Mexico & MathAmigos
January 26, 2019
Higher Ed Center, Santa Fe
Agenda & Packet Contents
1. Intro to Math Circles
2. Math Circle Pledge
3. The Math Salute
4. The Circle: SET, Mathematics, and Isomorphism
a. Learn & play
b. 4-D and dimensions
c. “Make it smaller”: dimension down to 2-D. How many possible sets?
d. Representations, models, and refinement
e. Isomorphism
f. Torus TTT
g. How many 2-D sets redux: combinatorics (ABC orders), the Fundamental
Theorem of SET
h. Brings us back to “triangles in a lattice from the October workshop”
5. Discussion, Q & A
a. Especially how would you implement?
6. SET Math Circle and Common Core
7. Handout: What is a Math Circle?
8. Handout: Torus TTT. The affine plane (non-Euclidean geometry)
9. Handout: Ultimate TTT, with rules
10. Handout: Nine Chiles (from October 2018 workshop math circle)
11. Set decks, one per participant
12. Hand out this packet!
Some Resource Links
• The Joy of SET: The Many Mathematical Dimensions of a Seemingly Simple Card Game. Liz
McMahon, Gary Gordon, Hannah Gordon & Rebecca Gordon.
https://press.princeton.edu/titles/10824.html
• Sets, Planets, and Comets. Mark Baker, Jane Beltran, Jason Buell, Brian Conrey, Tom
Davis, Brianna Donaldson, Jeanne Detorre-Ozeki, Leila Dibble, Tom Freeman, Robert
Hammie, Julie Montgomery, Avery Pickford, and Justine Wong.
https://www.maa.org/sites/default/files/pdf/pubs/SetsPlanetsAndComets.pdf
• Set Enterprises: https://www.setgame.com/ and inventor Marsha Falco
https://www.setgame.com/founder-inventor (and a New Mexico connection!)
• Set Enterprises Teachers’ Corner: https://www.setgame.com/teachers-corner
• More mathematics of SET: https://www.mathteacherscircle.org/assets/session-
materials/BConreyBDonaldsonSET.pdf
• Math with Bad Drawings: https://mathwithbaddrawings.com/
• Math Pickle: http://mathpickle.com/ (math activities and games by grade level)
• AMS/MSRI Mathematical Circles Library: http://bookstore.ams.org/MCL
• Math Teachers’ Circles Network: https://www.mathteacherscircle.org/
• MTCircular magazine: http://www.mathteacherscircle.org/news/mtc-newsletter/
28 January 2019• Julia Robinson Mathematics Festivals (source of festival problems): http://jrmf.org
• National Association of Math Circles: They also have run and will be running math circle
workshops at the University of Colorado at Denver with grant money
attached.: https://www.mathcircles.org/
• From Russia with Math (for Kids), Scientific American article1
• My email: jtaylor@mathcirclesnm.org
Great web resources from James Tanton (James has lots of very short videos suitable for
showing to students being introduced to problem solving):
• http://JamesTanton.com
• http://gdaymath.com
• http://www.maa.org/math-competitions/teachers/curriculum-inspirations
• Consider doing Global Math Week in
October! https://www.theglobalmathproject.org/gmw
1https://blogs.scientificamerican.com:budding-scientist:from-russia-with-math-for-
kids:%3FWT.mc_id=SA_syn_BusinessInsider
28 January 2019“Set, Mathematics, and Isomorphism” and Common Core Connections
MathAmigos & SFPS Math Teachers Workshop January 26, 2019
Leader: James Taylor, Math Circles Collaborative of New Mexico
• Standards for Mathematical Practice: 1, 2, 3, and 7
• MP1: “Make sense of problems and persevere in solving them”.
• MP2: “to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own”. Scaling down the 4-dimensional
problem to 2 dimensions simplifies the problem. Seeing that 2-D SET is structurally the
same as a variation on a common children’s game introduces the notion of isomorphism—
which is an essential sort of abstraction in mathematics.
• MP3: “construct arguments using concrete referents such as objects, drawings, diagrams, and
actions“. Making drawings here is essential to creating a persuasive visual model of the 2-D
game. At first it is a flawed or incomplete model, but a critical first step.
• MP7: “Mathematically proficient students look closely to discern a pattern or structure”.
Simply learning to play the game is an exercise in playing with patterns and structure.
Simplifying the game from 4-D to 2-D is a structural exercise. Further investigation can lead
to viewing the problem through modular arithmetic and combinatorics, and a way of
grasping isomorphism.
Standards for Mathematical Content
Measurement and Data
Kindergarten
Introduction: “Classify objects and count the number of objects in each category.”
CCSS.MATH.CONTENT.K.MD.B.3
Classify objects into given categories; count the numbers of objects in each category and sort the
categories by count.
Geometry
Kindergarten
Introduction: “Identify and describe shapes.”
CCSS.MATH.CONTENT.K.G.A.1
Describe objects in the environment using names of shapes, and describe the relative positions of
these objects using terms such as above, below, beside, in front of, behind, and next to.
Introduction: “Analyze, compare, create, and compose shapes.”
CCSS.MATH.CONTENT.K.G.B.4
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using
informal language to describe their similarities, differences, parts (e.g., number of sides and
vertices/"corners") and other attributes (e.g., having sides of equal length).
28 January 2019Grade 1
Introduction: “Reason with shapes and their attributes.”
CCSS.MATH.CONTENT.1.G.A.1
Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-
defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining
attributes.
Grade 2
Introduction: “Reason with shapes and their attributes.”
CCSS.MATH.CONTENT.2.G.A.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given
number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
Grade 3
Introduction: “Reason with shapes and their attributes.”
CCSS.MATH.CONTENT.3.G.A.1
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share
attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g.,
quadrilaterals).
Grade 4
Introduction: “Students describe, analyze, compare, and classify two-dimensional shapes.
Through building, drawing, and analyzing two-dimensional shapes, students deepen their
understanding of properties of two-dimensional objects and the use of them to solve problems
involving symmetry.”
CCSS.MATH.CONTENT.4.G.A.3
Recognize a line of symmetry for a two-dimensional figure as a line across the figure such
that the figure can be folded along the line into matching parts. Identify line-symmetric
figures and draw lines of symmetry.
Grade 5
Graph points on the coordinate plane to solve real-world and mathematical problems.
CCSS.MATH.CONTENT.5.G.A.1
Use a pair of perpendicular number lines, called axes, to define a coordinate system, with
the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a
given point in the plane located by using an ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to travel from the origin in the direction
of one axis, and the second number indicates how far to travel in the direction of the
second axis, with the convention that the names of the two axes and the coordinates
correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).
CCSS.MATH.CONTENT.5.G.A.2
Represent real world and mathematical problems by graphing points in the first quadrant
of the coordinate plane, and interpret coordinate values of points in the context of the
situation.
Introduction: “Classify two-dimensional figures into categories based on their properties.”
CCSS.MATH.CONTENT.5.G.B.3
Understand that attributes belonging to a category of two-dimensional figures also belong
to all subcategories of that category. For example, all rectangles have four right angles and
squares are rectangles, so all squares have four right angles.
28 January 2019CCSS.MATH.CONTENT.5.G.B.4
Classify two-dimensional figures in a hierarchy based on properties.
Operations and Algebraic Thinking
Grade 4
Introduction: “Use the four operations with whole numbers to solve problems.”
CCSS.MATH.CONTENT.4.OA.C.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of
the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and
the starting number 1, generate terms in the resulting sequence and observe that the terms
appear to alternate between odd and even numbers. Explain informally why the numbers
will continue to alternate in this way.
28 January 2019by James Taylor2 jtaylor@mathcirclesnm.org
Math Circles Collaborative of New Mexico (mathcirclesnm.org)
Math Teachers’ Circle of Santa Fe
To quote from the National Association of Math Circles site:
“Math Circles bring K-12 students or K-12 mathematics teachers together with mathematically
sophisticated leaders in an informal setting, after school or on weekends, to work on interesting problems or
topics in mathematics. The Circles combine significant content with a setting that encourages a sense of
discovery and excitement about mathematics through problem solving and interactive exploration. Ideal
problems are low-threshold, high-ceiling; they offer a variety of entry points and can be approached with
minimal mathematical background, but lead to deep mathematical concepts and can be connected to
advanced mathematics.”
Math circles can be run at any level—from K-12 to adults (teachers, mathematicians, parents,
etc.). This does not mean that the problems are easy, even for younger students. They are
certainly not. In general, problems will start from an easily grasped place—perhaps involving
some physical object or manipulative—and build to a more generalized understanding of the
area of mathematics involved.
For example, we might begin with a pile of candy, and distribute it in some way among five
students (conference participants) at a table. Then we would have them share the candy in some
regular pattern several different ways, with each new way helping them to discover something
about the significance of the number of candies and candy sharers across the system. Or perhaps
tables of students will be give strips of paper with numbers on them, then be told something as
vague as “organize them.” The circle leader will do his or her best to not be too helpful.
This math circle process is about discovery and invention—constructing mathematics from our
innate (and teachable) ability to perceive patterns.
Mathematical circles are deeply about persevering through difficult problems that are often
unlike any the student has seen before (Common Core Math Standard #1). Circle problems and
session seldom start in any familiar place, and may seem to make little sense at first—but are
highly attractive mysteries. Students must learn the standard “tools of the trade” of problem
solving, such as working backwards, finding invariants, drawing pictures, making tables, the
extreme principal, making a simpler problem/scale down, wishful thinking, and looking for
symmetry. Further, since the leader will almost never reveal the answers, the students must learn
to explain clearly their solutions and approaches—the beginnings of mathematical proof.
Computing some quantity or following some rote algorithm will never be enough.
2 James Taylor runs the Math Circles Collaborative of New Mexico and Math Teachers' Circle of Santa Fe, in Santa Fe, New
Mexico. He retired after 21 years at Santa Fe Preparatory School as Computer Department head, computer and mathematics
teacher, as well as Director of Technology. He has been working with math circles for students and teachers since an early 2005
tour of math circles in the San Francisco bay area, and has had an interest in provocative and subversive mathematics education
much longer. James has taught many years of courses in mathematical problem solving at both the middle and high school levels,
and has mentored teachers in circle approaches in the upper elementary school grades. His work is supported by the American
Institute of Mathematics, the National Association of Math Circles, the Mathematical Association of America, the Alliance of
Indigenous Math Circles, the LANL Foundation, MathAmigos, and Northern New Mexico College. He has also been involved
since the late 1990s in teaching computational science and computer modeling in the US and Mexico.
28 January 2019A math circle could not be more different than a traditional classroom, though they may follow
many different styles and approaches. All participants must struggle through the problems,
usually in small groups. The problems may range from open-ended explorations of a
mathematical idea such as infinity or the nature of number, to math competition preparation, to
the investigation of a game with important mathematical underpinnings (such as the card game
SET3).
A math teachers’ circle allows teachers to directly experience what students experience in a math
circle. Further, the teachers get to have circle leading and pedagogy modeled for them by an
experience leader—often a mathematician—and at some point get to try their hand at designing
an leading a teacher circle. This experience will ideally lead to their introducing math circles at
their schools to their students. The director of the Math Circles Collaborative of New Mexico
and the Math Teacher’s Circle of Santa Fe, James Taylor, is available to visit schools in the
region and model circle activities in a teacher’s classroom. There are math circles and math
circle projects as well at Santa Fe Indian School, in Las Vegas, the Española Valley, Farmington,
Los Lunas, White Rock, Dixon, Mesa Vista, and throughout the Navajo Nation.
There is a constellation of activities and events which includes student and teacher math circles,
and extends the reach of the math circle approach: Math Wrangles, Julia Robinson Math
Festivals, and Exploding Dots (http://gdaymath.com/courses/exploding-dots/) and the Global
Math Project (https://www.explodingdots.org/g/big-mayfly-75).
A Math Wrangle (www.sanjosemathcircle.org/math-wrangle, with the rules at
mathcircle.tamu.edu/wp-content/uploads/2014/09/math_wrangle_rules.pdf) is a mathematics
debate. This is not as strange as it sounds. Teams of students go off into a room to study a set of
eight problems, then return to the debate setting and the coin toss. The winning toss permits the
team to challenge the opponents to solve one of the problems. That team may accept, and a
member presents their solution, deals with any rebuttal to their solution, and judges award points.
There is more to it, but in my experience students love the experience. Mounting a team requires
training, as does any debate program.
A math festival is a terrific way to introduce large numbers of students, teachers, and community
members to how playful and fun mathematics can be, math circle-style. The first New Mexico
Julia Robinson Math Circle (jrmf.org) was held February 24, 2017 at Santa Fe Community
College (stemsantafe.org/news-events/julia-robinson-mathematics-festival-santa-fe/) with about
150 middle school students and about 50 adult activity-table leaders. Students get to freely
choose math activities and play with them for about an hour-and-a-half. A festival engages not
only students, but also schools, teachers, the adult table leaders (volunteers), along with alerting
the larger community to some positive buzz about mathematics.
There are sample math circle videos available on a YouTube channel at
youtube.com/user/MathCircles/videos. The NAMC also has some video interviews with math
circle leaders at http://www.mathcircles.org/what-is-a-math-circle/ and
https://www.youtube.com/watch?v=7ZvWqvqljlo.
3 For starters, SET can be used to introduce isomorphism, topology/torus vs. plane/finite unbounded vs. finite bounded, magic squares,
unordered sets, dimension, finite discrete dimensions, combinatorics, modular arithmetic, projective geometry, multi-dimensional
geometry, error-correcting codes, non-geometric dimension, probability, statistics, modeling, linear algebra/vectors, number theory,
caps sets, seeing patterns, and more.
28 January 2019torus tic-tac-toe
7 8 9 7 8 9 7 8 9
3 1 2 3 1 3 1 2 3 1 3 1 2 3 1
6 4 5 6 4 6 4 5 6 4 6 4 5 6 4
9 7 8 9 7 9 7 8 9 7 9 7 8 9 7
1 2 3 1 2 3 1 2 3
7 8 9 7 8 9 7 8 9
3 1 2 3 1 3 1 2 3 1 3 1 2 3 1
6 4 5 6 4 6 4 5 6 4 6 4 5 6 4
9 7 8 9 7 9 7 8 9 7 9 7 8 9 7
1 2 3 1 2 3 1 2 3
7 8 9 7 8 9 7 8 9
3 1 2 3 1 3 1 2 3 1 3 1 2 3 1
6 4 5 6 4 6 4 5 6 4 6 4 5 6 4
9 7 8 9 7 9 7 8 9 7 9 7 8 9 7
1 2 3 1 2 3 1 2 3
Torus Tic-Tac-Toe Rules
Whenever you mark an X or O on a numbered spot, place your mark everywhere that number
appears. The rules are the same as the standard game, otherwise.
28 January 2019Ultimate Tic-Tac-Toe
Copyright © 2018 The Game Gal • www.thegamegal.com
Basic Rules (from Math with Bad Drawings4):
Each turn, you mark one of the small squares.
1. When you get three in a row on a small board, you’ve won that board.
2. To win the game, you need to win three small boards in a row.
But it took a while for the most important rule in the game to dawn on me:
You don’t get to pick which of the nine boards to play on. That’s determined by your opponent’s previous
move. Whichever square he picks, that’s the board you must play in next. (And whichever square you pick will determine
which board he plays on next.) For example, if I go here…
This lends the game a strategic element. You can’t just focus on the little board. You’ve got to consider where your move will send
your opponent, and where his next move will send you, and so on.
The resulting scenarios look bizarre. Players seem to move randomly, missing easy two- and three-in-a-rows. But there’s a method to
the madness – they’re thinking ahead to future moves, wary of setting up their opponent on prime real estate. It is, in short, vastly
more interesting than regular tic-tac-toe.
A few clarifying rules are necessary:
1. What if my opponent sends me to a board that’s already been won? Tough luck. If there are open squares, you must pick
one. While you can’t really affect that board, you can at least determine where your opponent will go next. [see edit below]
2. What if my opponent sends me to a board that’s full? In that case, congratulations – you get to go anywhere you like, on any
of the other boards. (This means you should avoid sending your opponent to a full board!)
When I see my students playing tic-tac-toe, I resist the urge to roll my eyes, and I teach them this game instead. You could argue that
it builds mathematical skills (deductive reasoning, conditional thinking, the geometric concept of similarity), but who cares? It’s a
good game in any case.
4 Rules with pictures: https://mathwithbaddrawings.com/ultimate-tic-tac-toe-original-post/
28 January 2019How many triangles can you make with the 9 chiles as
corners/vertices?
28 January 2019You can also read
