ATTRIBUTE GAMES AND VENN DIAGRAMS
F. Richard Singer III conceptualstudy.org Edition
date: May 2008
Bridging to Venn Diagrams: Some bridging strategies center on a common principle that has some highly manifest instances and then bridges to a type of instance that is more remote. However a bridging strategy may use any means to relate something that might remote to something manifest. To bridge to the concept of a Venn diagram we start with the basic attribute game and focus on how it can be used with elementary school children. Similar attribute games can also be used at a higher level to bridge to additional concepts. Below are pictures of the tokens that are used for this game. Items vary by the attributes types {size, color, shape}. Sizes are large and small. Colors are blue and red. Shapes are circle and diamond. A larger set of attribute items can also be used, but even college students can benefit by starting with the basic game.
==t t = = t t
Note: There is a template for making attribute tokens at the end of this resource.
This game uses 6 labels for sizes and colors and shapes. It is played on a board with 2 intersecting circles. These circles partition the attribute items into 4 regions. Each will have 2 elements, depending on the values chosen for the labels. As a prelude to the game, place the labels as in the sample diagram to the left. Ask how the tokens for these items should be placed. Expect some doubts, but you are likely to find some who will place in the regions indicated below. Make sure that everyone understands why each item can only be placed in this manner.
Ø
You might choose two
other labels (of different attribute types) and again ask how the tokens should
be placed.
Ø
You might place a
token and ask what the label possibilities are.
SAMPLE DIAGRAM
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Challenge Question: If lbc (large blue circle) goes in the left region of the left circle, which other item can you locate no matter what the labels are? (See appendix 1 for a sample analysis).
Attribute Game Rules: The game involves 2 teams, with 2 or 4 people on each. To start, Team 1 selects labels of different attribute types and places them face down beside the circles. Team 2 chooses any token. Team 1 must put this token in the correct place. Team 2 must place the remaining tokens in the 4 regions. On each trial, Team 1 either verifies the choice or removes the token if it is in the wrong region and gives it back to Team 2. Team 1 collects one point each time they return a token, however if they return a token incorrectly or allow it to be placed incorrectly, they forfeit the game. Team 2 may try a rejected token elsewhere or try a different token. Once Team 2 has placed all the tokens they must either correctly identify the labels or forfeit the game. The goal of the game for Team 2 is to identify the labels giving Team 1 as few points as possible in the process. If they correctly identify the label then Team 1 and Team 2 reverse roles and play is repeated. If there is no forfeit then the team with the most points is the winner. The same rules apply to a games using 3 circles. The game can also be played with more attribute items.
Suggested Use: The purpose of playing and discussing the game is to make the use of the regions extremely manifest. This is also the purpose for either playing or discussing the 3-circle version. For elementary children such activities should probably be spread over more than one session. When using the game in a class, divide the class into groups of six to the extent possible, as teams of size 3 seem to work best. However opponent teams need not be the same size. Have each group play the game and record any observations. Then bring the groups together for a discussion. Then have them play the game once more. Discuss strategies used game playing. For instance, when a token placement is rejected some people try it some other place but others try a different token. Ask questions (such as the sample question) to help reinforce the way items are placed.
If there is an interest then play a more challenging game, using a 3 circles. Otherwise at least have the class chose labels and place pieces. If you only use attributes of different types then each region will have exactly one token. However you can use a larger set of attribute items and more labels. You can also allow using labels of the same type. Introduce the term ‘Venn Diagram’ and discuss the concepts of union and intersection. Illustrate the use of Venn Diagrams with the other situations given later.
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Vegetable Situation: A survey was conducted about the role vegetables in family diets. The most frequently used cooked green vegetables were {Peas, Spinach, Broccoli}. One question asked which of these vegetable the family used on the average at least once a week. The following information was gathered. Broccoli 34, Spinach 43, Peas 56. Of those, 7 used both Broccoli and Spinach, 11 used both Broccoli and Peas, 19 used both Spinach and Peas. Moreover 4 of these used all three vegetables. There were 8 who never used any of these vegetable at least once a week. How many said only Broccoli? How many only Spinach? How many only Peas? How many families were surveyed? How many families used exactly two of theses vegetables at least once a week?
To construct a Venn diagram for the number of families for
this information, begin by placing the 4 in the middle region. Since there are
7 in BS place a 3 in the other part of BS. The 7 and 15 can be placed in a
similar fashion. Since there are 34 in B, there are 20 in broccoli only. There
are 21 in spinach only, 34 in peas only. Adding the numbers in all regions,
gives 112 families in the survey. Taking 3+7+15 gives 25 families who
use exactly two of theses vegetables at least once a week.
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Notation: The notation BS is short for BÇS, i.e. for the intersection of the sets B and S.
Reference: For more examples of this type, along with explanations use any of the following links.
http://www.chaselink.com/tune/
http://regentsprep.org/Regents/math/venn/PracVenn.htm
http://www.purplemath.com/modules/venndiag4.htm
http://www.beva.org/maen503/week2/venn_diagram_examples.htm
http://www.math.tamu.edu/~kahlig/venn/venn.html
Other Bridges: The attribute game can be used as a starting point to bridge to some other topics, including some that can be used in college level courses in Boolean Algebra and Informal Reasoning. Below are some ideas that can be used to develop bridging strategies. Some of these are developed in detail in the paper entitled Bridging from the Attribute Game. That paper is available on the Constructivist Learning Section of conceptualstudy.org.
Set Concepts: The set of items in the outside part of the sample diagram can be named as LB, as it is the intersection of set of large items with the set of blue items. It can also be named in terms of the labels S and R. The set of items that are not small is called the complement of the set of small items. Using the symbol ‘/’ for complements we can use /S/R to name this set. It can also be named in terms of the complement of the union of S with R, i.e as /(SÈR). This gives an example of DeMorgan’s Law that says that /(SÈR) = /S/R. As another example the subsets of R are SR and /SR, i.e. we have R = SRÈ/SR. This can be deduced using the distributive law along with the complement law and the multiplicative identity law.
Strategy Design: Coupled with the well known game Twenty Questions, this game can also be used as a bridge to one type of strategy design concepts. In Twenty Questions (with no reason to use less questions), an ideal question would eliminate half of the possibilities, regardless of whether the answer was yes or no. If there was a million possibilities, using such ideal questions would yield the answer. Recall that in an attribute game only an incorrect placement gives the opposing team a point. So if the placement is incorrect, we want it to eliminate as many alternatives as possible. For the basic game, there is a strategy that will result in never giving the other team more than 3 points, no matter how much luck goes against your team.
To illustrate this strategy, denote the values of the circles as X and Y.
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Suppose srd was initially place in XY. This gives the 6 indicated possibilities for the labels, where SR means X = S and Y = R, etc. Chose a piece that differs in one way from srd, say lrd. Use it until it is placed. It must go in XY or /XY or X/Y. If lrdÏXY then the zeros indicate that 2 possibilities have been eliminated. In the worst case, you may get 2 rejections before placing it X/Y. Now either sbdÎXY or lrcÎXY, so at most there will be 1 more rejection. |
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srdÎXY |
SR |
SD |
RS |
RD |
DS |
DR |
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lrdÏXY |
1 |
1 |
1 |
0 |
1 |
0 |
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lrdÏ/XY |
0 |
0 |
1 |
1 |
1 |
1 |
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lrdÎX/Y |
0 |
0 |
1 |
0 |
1 |
0 |
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sbdÏXY |
1 |
0 |
1 |
1 |
0 |
1 |
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lrcÎXY |
1 |
0 |
1 |
0 |
0 |
0 |
This can also provide a simple bridge to the concept of expected cost. Using this strategy the probability of it giving up 3 points is 1/6, the probability of it giving up 2 points is 1/3, the probability of it giving up 1 point is 1/6,the probability of it giving up 0 points is 1/6. Thus the expected cost can be calculated as 3·1/6+2·1/3+1·1/3+0·1/6 = 1½.
Appendix 1 Analysis of the
Challenge Question
Challenge Question: If lbc goes in the left region of the left circle, which other item can you locate no matter what the labels are?
Jan: The labels might be Large and Red. This would put both lrc and lrd in the intersection. However they might be Blue and Small, putting sbc and sbd in the intersection. So different labels that put lbc in the left part of the left circle put those 4 items in different regions.
Bob: The 6
possibilities for the label pairs with lbc in the left region of the left circle
are indicated in the table below. With each label pair I used an 0 to indicate
that an item could not be in the intersection and a 1 to indicate it must be in
the intersection. For everything but srd, there was both a way it would be and
a way it would not be. However in each case srd was in the right region of the
right circle.
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= |
= |
t |
t |
= |
= |
t |
t |
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Large Red |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
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Large Diamond |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
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Blue Small |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
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Blue Diamond |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
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Circle Small |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
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Circle Red |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
0 |
Kay: We do not need to consider possible label pair or any items except srd. We can deduce that srd belongs in the right part of the right circle as follows. Items inside of a circle must share an attribute. Since lbc is in the left circle, srd cannot be in the left circle. Items outside of a circle must share an attribute. Since lbc is outside of the right circle, srd must be inside of the right circle. Thus srd is in the right region of the right circle.
To make these
attribute item tokens,
print, laminate, cut out.
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Large |
Blue |
Circle |
Green |
Square |
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Small |
Red |
Diamond |
Yellow |
Triangle |