# SET Math Game lesson plan worksheet.doc

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```					SET Math Game
Kerry Weinberg and John Williams

Introduction: SET is a challenging card game involving visual perception and pattern
recognition. In the original game, players work with a randomnly dealt group of 12 cards and
try to find “sets” within this larger group. A “set” consists of three cards whose four attributes
(color, shape, number, and shading) must individually either all be different or all the same.
Once a set has been identified by a player, it is first checked for veracity by other players and
then given to the player who identified the set. Then, these removed cards are replaced with
new ones from the remaining deck. The game ends when no player can find any more sets and
the winner is the player who has identified the most number of sets (i.e. has the most number
of cards in their hands). The set which has the highest probability of occurring is 3 different/1
same and the winning strategy would be using this knowledge to quickly locate sets. In this
lesson, students will first be introduced to the concept of what a “set” is and then move onto
understanding how probability and combinations can be used to understand the likelihood of
certain sets arising and further how this can be used for a winning strategy of the game. Last,
students will play the original SET game using the knowledge they have gained throughout
the lesson.

Materials: SET cards, 1 worksheet per student

Learning Goals: First, students should be able to grasp the rules of the game and what it
means for three cards to be in a set. Next, they should be able to follow the probability
example initially given explaining the likelihood of finding a set in the deck. After being
shown the probability of a 4 different/0 same set, they should be able to extend this and
determine the probability of different types of sets occurring. They should then be able to
identify and discuss what they think it the most effective way to locate sets. Lastly, they
should be able to apply their newfound knowledge to a real SET game.

NCTM Standards Correlation: Algebra (pattern recognition) Data Analysis and
Probability (using laws of probability to support winning strategy) Problem Solving
(most effective method for identifying sets) Communication (group discussions for each
question) Connections (wrap up questions)

Preparation: Students should be previously acquainted with the formulas of
probability and combinations. If this material has not been covered in a long time, a
short review beforehand would be helpful.
Directions: Game rules are described in the student activity sheet and in above
Introduction.

Plan: First, introduce the game briefly and then separate the class into groups of 4-6 students
per group. Give each of the students an activity worksheet. Direct them to start off with the
“Get familiar with SET!” section. Help groups of students who are having difficulty with this
first part and make announcements to the class if necessary. You may want to give students
time to play the game and develop some intuitions about which sets are more or less
likely. Once the majority of the class is adequately familiar with SET, direct the class to
move onto discovering the probability of randomly drawing a set of any type from the deck.
Once students have moved onto examining the probability of different types of sets occurring,
visit the groups and ask them how they got their answers. When most of the class has finished
answering these questions, encourage students to move onto playing the game using
strategies based on the probability concepts they have just learned. At the end of class have
each group briefly describe their responses to the probability questions and then explain they
thought was the most effective way to identify sets.

Wrap Up Discussion Questions:

What is the probability of choosing three cards that are not a set? (Hint: The probability of
an event not occurring is 1 minus probability of an event occurring.)

What would happen if you eliminated the attribute of color (i.e. the cards were all the
same color)? How would this change the probabilities and therefore your winning
strategy?

What do you think is a good systematic method for determining probability and
combinations?(Hint: think of other card games involving the element of chance)
A Bold Quest to Understand Probability…Do you dare?

(Kerry Weinberg & John Williams)

Names_________________________________________________________________

Directions:

•         Get familiar with SET!
1)      Lay out all the cards on the ground and discuss with your group what it
means to find a “set” within the larger group of cards. A set is defined as a group
of 3 cards that are either all the same by color / shape / number / shading, or all
different. Consider the following four possibilities that a set could result from:

Type of Set
4 different / 0 same
3 different / 1 same
2 different / 2 same
1 different / 3 same
Examples are shown below (images from setgame.com): 4
different / 0 same
Type of Set
4 different / 0 same
3 different / 1 same
2 different / 2 same
1 different / 3 same

(The magic rule: If 2 are and 1 is not, then it is not a set!)
Before you indulge yourself in a riveting game of SET, lets take a look at how
many cards make up the deck. It will be important to know this for later on. ;)

There are 4 distinct attributes (color, shape, shading, & number), if 3 attributes
are held constant at a time, there are 3 possible variations that can be created as
a result for each of the 4 attributes. As an example, if we are looking at 2 shaded
ovals, there are 3 different colors that we could see this in. Since a deck of SET
cards has NO repeat cards, the total number of cards in a deck must be:

3x3x3x3 = 81 cards.

Does this make sense to you? How many cards would be in the deck if only 2
shapes, 2 colors, 2 shadings, and 2 numbers were possible for each card?

Now we’re on our way to looking at the number of combinations that are
possible when creating a SET from a full deck of cards (Nice!).

1)       Set the full deck of cards in front of you (face up for now). Draw
your first card, how many cards do you have to choose from?
2)       Now draw your second card. How many cards can you choose on this
turn to be on your way to creating a SET? (hint, it doesn’t matter what this
card is, but now the deck only has 80 cards to choose from)
3)       Now pick a third card, remember, this card must make your 3 cards
a SET. How many cards are left that can make that happen?

Recall the following formula for probability,

, the same can be said about combinations. The total number of combinations possible is
equal to the individual combinations possible at each step multiplied together.

Did you get 6480 possible combinations?! Show how you arrived at this number below:

–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Now hold on, we may have double counted some SETS here. Is a red, purple, green set
really different than a red, green, purple set? Order doesn’t matter here, so there isn’t a
difference between these combinations to us. We need to figure out how many overlaps
there are. Using the 3 color example above, how many ways can those 3 attributes be
ordered? Show them below:

The real number of possible sets, without overlaps, is thus 6480/(# of overlaps possible
when varying one attribute). So how many possible sets are there?

# of possible sets:______________________________________________________

You just found the TOTAL number of possible sets. Now we need to know the probability
of finding each TYPE of set from this total number of possible sets. If you know these
probabilities, you can dominate your friends when you play this game at home!
Knowledge is power!

Lets go through the probability of finding a (4 different, 0 same) SET:

Step 1) Q: How many ways to pick the first card?
A: 81 ways, any card can be chosen.

Step 2) Q: How many ways are there to pick the attributes that are the same on the
second card?
A: Remember the following formula for finding the number of possible
combinations when choosing a sample size from a full set?

(where “n” is the total number of elements being chosen from and ” k” is the number of
elements you are choosing from that set)
In our case, we have 4 attributes and are choosing 0 to be the same, or…

(remember that 0! = 1, it’s an odd rule, but it is true)

Step 3) Q: How many ways to choose the second card?
A: Every attribute must be different, so there are 2 colors, 2 shapes, 2 numbers, and 2
2x2x2x2 = 16 ways Step 4) Q: So how many total possibilities are there
for this type of set?

A: Combinations at each step multiplied together (and divided by overlap #) So # of
possible SETs for this type = (81x1x16)/6 = 216 possibilities. Step 5) Q: What is the
probability of finding this type of set?

A:

Therefore, likelihood of finding this set = 216/1080 = 20%

Now you have seen an example of how to find the probability of finding a specific type of
set. Try to find the probability of finding the other 3 types of sets on your own. Use the
example above as a guide. Show all your work.

• 3 different / 1 same
• 2 different / 2 same

• 1 different / 3 same

Concept Question:

Knowing what you know now, is there a systematic method for searching for sets

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– –––
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– ––––––
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– –––––––––
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
The real game of set involves laying out 12 cards at a time face up and trying to find as
many sets as possible. Any number of people can play together and the person who
finds the most number of sets within the 12 cards wins. Try playing the game with your
new strategy, you should come out on top!
SOLUTIONS

A Bold Quest to Understand Probability…Do you dare?

(Kerry Weinberg & John Williams)

Names_________________________________________________________________

Directions:

•         Get familiar with SET!
1)      Lay out all the cards on the ground and discuss with your group what it
means to find a “set” within the larger group of cards. A set is defined as a group
of 3 cards that are either all the same by color / shape / number / shading, or all
different. Consider the following four possibilities that a set could result from:

Type of Set
4 different / 0 same
3 different / 1 same
2 different / 2 same
1 different / 3 same
Examples are shown below:
Type of Set 4 different / 0 same
4 different / 0 same
3 different / 1 same
2 different / 2 same
1 different / 3 same

(The magic rule: If 2 are and 1 is not, then it is not a set!)
Before you indulge yourself in a riveting game of SET, lets take a look at how
many cards make up the deck. It will be important to know this for later on. ;)

There are 4 distinct attributes (color, shape, shading, & number), if 3 attributes
are held constant at a time, there are 3 possible variations that can be created as
a result for each of the 4 attributes. As an example, if we are looking at 2 shaded
ovals, there are 3 different colors that we could see this in. Since a deck of SET
cards has NO repeat cards, the total number of cards in a deck must be:

3x3x3x3 = 81 cards.

Does this make sense to you? How many cards would be in the deck if only 2
shapes, 2 colors, 2 shadings, and 2 numbers were possible for each card?

Now we’re on our way to looking at the number of combinations that are
possible when creating a SET from a full deck of cards (Nice!).

4)       Set the full deck of cards in front of you (face up for now). Draw
your first card, how many cards do you have to choose from?
5)       Now draw your second card. How many cards can you choose on this
turn to be on your way to creating a SET? (hint, it doesn’t matter what this
card is, but now the deck only has 80 cards to choose from)
6)       Now pick a third card, remember, this card must make your 3 cards
a SET. How many cards are left that can make that happen?

Recall the following formula for probability,

, the same can be said about combinations. The total number of combinations possible is
equal to the individual combinations possible at each step multiplied together.

Did you get 6480 possible combinations?! Show how you arrived at this number below:

81*80*1
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
Now hold on, we may have double counted some SETS here. Is a red, purple, green set
really different than a red, green, purple set? Order doesn’t matter here, so there isn’t a
difference between these combinations to us. We need to figure out how many overlaps
there are. Using the 3 color example above, how many ways can those 3 attributes be
ordered? Show them below:

6 ways to arrange 3 items: 3!=3*2*1

The real number of possible sets, without overlaps, is thus 6480/(# of overlaps possible
when varying one attribute). So how many possible sets are there?

# of possible
sets:____________________81*80*1/6__________________________________

You just found the TOTAL number of possible sets. Now we need to know the probability
of finding each TYPE of set from this total number of possible sets. If you know these
probabilities, you can dominate your friends when you play this game at home!
Knowledge is power!

Lets go through the probability of finding a (4 different, 0 same) SET:

Step 1) Q: How many ways to pick the first card?
A: 81 ways, any card can be chosen.

Step 2) Q: How many ways are there to pick the attributes that are the same on the
second card?
A: Remember the following formula for finding the number of possible
permutations when choosing a sample size from a full set?
(where “n” is the total number of elements being chosen from and ” k” is the number of
elements you are choosing from that set) In our case, we have 4 attributes and are choosing
0 to be the same, or…

(remember that 0! = 1, it’s an odd rule, but it is true)

Step 3) Q: How many ways to choose the second card?
A: Every attribute must be different, so there are 2 colors, 2 shapes, 2 numbers, and 2
2x2x2x2 = 16 ways Step 4) Q: So how many total possibilities are there
for this type of set?

A: Combinations at each step multiplied together (and divided by overlap #) So # of
possible SETs for this type = (81x1x16)/6 = 216 possibilities. Step 5) Q: What is the
probability of finding this type of set?

A:

Therefore, likelihood of finding this set = 216/1080 = 20%

Now you have seen an example of how to find the probability of finding a specific type of
set. Try to find the probability of finding the other 3 types of sets on your own. Use the
example above as a guide. Show all your work.

• 3 different / 1 same

40%
• 2 different / 2 same

30%
•        1

diffe
rent /
3
same

10%

Concept Question:

Knowing what you know now, is there a systematic method for searching for sets

––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– –––
–––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– ––––––
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––– –––––––––
––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
The real game of set involves laying out 12 cards at a time face up and trying to find as
many sets as possible. Any number of people can play together and the person who
finds the most number of sets within the 12 cards wins. Try playing the game with your
new strategy, you should come out on top!

```
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