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# Halfplay. Playing, mathematizing and problem-solving with simple rational numbers

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## Abstract

A game like Halfplay is thought to build relations to rational numbers in a human way. It is obvious that natural numbers (N) take an important part in many games. Why rational numbers (Q) should not be integrated too? Skill, co-construction, fortune, frankness, probability, ease, social relations and sometimes conflicts are characteristics of playing. Halfplay generates such experiences. These experiences offer fundaments and materials for effective processes of mathematizing and realistic problem solving with rational numbers. You come to like symbolic, mental and social relations between rules and the meanings of actions.
Halfplay
Playing, mathematisizing and problem-solving with simple rational numbers
Stefan L. Meyer
24.04.09
Meyer, S. (2010). Das Halbespiel -eine ganze Sache. Darstellungen und Operationen mit Bruchzahlen am Zahlenstrahl
spielerisch erfahren (4.-7. Klasse). Praxis der Mathematik in der Schule, (32), 913. Verfügbar unter:
https://www.researchgate.net/publication/355718202_Das_Halbespiel_-
_eine_ganze_Sache_Darstellungen_und_Operationen_mit_Bruchzahlen_am_Zahlenstrahl_spielerisch_erfahren
University of Applied Sciences of Special Needs Education
Schaffhauserstrasse 239
Postbox 5850
CH-8050 Zürich
Switzerland
2
1
Halfplay
Introduction
A game like Halfplay is thought to build relations to rational
numbers in a human way. It is obvious that natural numbers (N)
take an important part in many games. Why rational numbers (Q)
should not be integrated too? Skill, co-construction, fortune,
frankness, probability, ease, social relations and sometimes
conflicts are characteristics of playing. Halfplay generates such
experiences. These experiences offer fundaments and materials
for effective processes of mathematizing and realistic problem
solving with rational numbers. You come to like symbolic, mental
and social relations between rules and the meanings of actions.
Rules of the game
One form of Halfplay is similar to Chutes and Ladders. Another
form is linear with improper or mixed fractions. The players
themselves determine the meaning and forms of the game.You
play at dice and move a figure. Fix the rules before. One point of
the dice is equal to ½. Make short work of this rule.
Shuffle the “cards of fortune” and put them face down on a pile. If
you arrive on a blue (gray) field, take the card of fortune on the
Enlarge the playground-copy on A3-size or others.
Talk with children and be aware if Halfplay is something
meaningful for them. Ask about the interest and the insights of the
game. Try to check the level of comprehension of the rational
numbers when children are playing and talking. Probably children
might be motivated to create a more sophisticated (or easier)
game. Perhaps they want to change the cover and the order of
the fields. Let them do it.
These are important elements of the relations with the content.
There are infinite rational numbers and variations. Experience
and theory prove that “every beginning is difficult”, specially with
rational numbers. Halfplay wants to give to the beginnings and
the exercises a natural educational fundament and a level-
oriented didactical conversation. Halfplay connects early
experiences of dividing into halves with playing with halves of
numbers. Distinguish consciously the playtime from the math-
lessons! Confusions damage the playtime as well as the
seriousness of the mathematical conversations.
Observe the children during the playtime and play with them.
Listen to their questions, conflicts or standpoints. Try to be a
discrete coach, if needed a referee. Note freely. This can serve
data very effectively for future math-lessons.
Select with the children which topics of the game has to be
mathematized in the math-lesson. Children might reinvent new
tasks or develop insight and new arithmetic rules, for example:
one point of the dice is equal to 1 ½ or to 2 ½ . Or they invent
Thirdplay”. Let them solve their real problem-situations with the
methods of mathematizing and problem-solving. They
reconstruct experiences using symbol representations,
reasoning, conjecturing and proving. When invited to think
metacognitively, they discover increasingly abstract
relationships.
Be free to use and change Halfplay as an educational or
scientific object. -Respect the copyright if you have commercial
interests with Halfplay or similar games.
Stefan L. Meyer, HfH, 24.04.2009
Halfplay ½ linear, improper fractions (part 1)
Start 2
2
2
12
32
42
52
62
72
82
9
2
10 2
11 2
12 2
13 2
14 2
15 2
16 2
17 2
18 2
19
2
20 2
21 2
22 2
23 2
24 2
25 2
26 2
27 2
28 2
29
2
30 2
31 2
32 2
33 2
34 2
35 2
36 2
37 2
38 2
39
2
40 2
41 2
42 2
43 2
44 2
45 2
46 2
47 2
48 2
49
2
50 2
51 2
52 2
53 2
54 2
55 2
56 2
57 2
58 2
59
2
60 2
61 2
62 2
63 2
64 2
65 2
66 2
69
2
67 2
68
Halfplay ½ linear, improper fractions (part 2)
END
2
70 2
71 2
72 2
73 2
74 2
75 2
76 2
77 2
78 2
79
2
80 2
81 2
82 2
83 2
84 2
85 2
86 2
87 2
88 2
89
2
90 2
91 2
92 2
93
Halfplay ½ linear (part 1)
30 30 ½ 31 31 ½ 32 32 ½ 33 33 ½ 34 34 ½
25 25 ½ 26 26 ½ 27 27 ½ 28 28 ½ 29 29 ½
20 20 ½ 21 21 ½ 22 22 ½ 23 23 ½ 24 24 ½
15 15 ½ 16 16 ½ 17 17 ½ 18 18 ½ 19 19 ½
10 10 ½ 11 11 ½ 12 12 ½ 13 13 ½ 14 14 ½
55 ½ 66 ½ 77 ½ 88 ½ 99 ½
Start ½ 11 ½ 22 ½ 33 ½ 44 ½
Halfplay ½ linear (part 2)
45 45 ½ 46 46 ½ 47 END
40 40 ½ 41 41 ½ 42 42 ½ 43 43 ½ 44 44 ½
35 35 ½ 36 36 ½ 37 37 ½ 38 38 ½ 39 39 ½
Halfplay ½
END 47 46 ½ 46 45 ½ 45 44 ½ 44 43 ½ 43 42 ½ 42
36 36 ½ 37 37 ½ 38 38 ½ 39 39 ½ 40 40 ½ 41 41 ½
35 ½ 35 34 ½ 34 33 ½ 33 32 ½ 32 31 ½ 31 30 ½ 30
24 24 ½ 25 25 ½ 26 26 ½ 27 27 ½ 28 28 ½ 29 29 ½
23 ½ 23 22 ½ 22 21 ½ 21 20 ½ 20 19 ½ 19 18 ½ 18
12 12 ½ 13 13 ½ 14 14 ½ 15 15 ½ 16 16 ½ 17 17 ½
11 ½ 11 10 ½ 10 9 ½ 98 ½ 87 ½ 76 ½ 6
Start ½ 11 ½ 22 ½ 33 ½ 44 ½ 55 ½
Cards of fortune (enlarge on A3 and cut it out)
One half
ahead One half back Three wholes
back Play at dice
once again Go to 20 ½
Two halves
back Six wholes
back Play at dice
twice again Go to 10 ½
Five halves
back Nine wholes
back Go to 23 ½ Go to 19 ½
Ten halves