A thinking game for children, teaching food pyramid by solving an integerprogramming problem
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Abstract: Teaching mathematic to children by using thinking games is a usual method that is used
by teachers. In this paper a new thinking game is introduced. The goal of the game is to encourage
children to practice the basic arithmetic operations and to learn the concept of food pyramid, by
playing this game they also solve an integer programming problem, that is to solve a given problem
in mathematical model, without knowing the concept of integer programming.
1. Introduction
In this paper a new game is introduced. In the second section the food pyramid is described. The
third section is contained the description of game and its rules. And in the next sections the
relation of this game with integer programming is mentioned and also the goals and benefits of
game is illustrated.
2. Food pyramid
The game is based on food pyramid. A food guide pyramid is a triangular or pyramid-shaped
nutrition guide divided into sections to show the recommended intake for each food group[1].
According to this there are five main groups that are: 1-bread, 2- vegetables, 3- fruit, 4- milk, and
5- meat. In our daily diet we have to eat 15-26 unit of food [1].
In Fig 1 the amount of each group for daily diet is mentioned. For more information about the
importance of food pyramid and healthy diet see [1].
Figure 1. The food pyramid and the amount of each food group [1]
3. The game
In this section we introduce the game
The needed tools for this game are: 1- the board of game, 2- Three dices, and 3- 150 marbles
And we need 4 players.
At the beginning of the game by throwing the dice, the order of players is determined and each
player chooses her pyramid on the board of game. See Figure 2.
Figure 2: The board of game
In the first section, each player fixes the amount of food that wants to eat, that is done by
throwing three dices and then using the four basic operation of arithmetic on the achieved
numbers.
For example if someone has 3, 3, 6, it is possible to achieve 21 by (6 × 3) + 3.
This value fixes the amount of food that the player wants to eat and it should be between 15 and
26 because the amount of daily food according to food pyramid is between 15 to 26 units.
After determining the amount of food, each player should take the marbles equal to specified
value.
In the second section the game begins by the first player. In each round the player achieved 2
values by dices and by using the basic operations achieved a value then she should select a level
of her pyramid and put the marbles on that level.
The number of pieces for each level is written. For example the amount of vegetable level is
between 3 and 5. Each player in her round could put less than 3 and not more than 5 on her
vegetable level.
The game continues in this manner, and in each round player fills her grids according to the
constraints.
The winner of the game is someone who uses all her marbles and fills the levels according to the
constraints.
Table1: amount of each food group for daily diet
Food groups
Daily Amount-unit
Bread
6-11
Vegetables
3-5
Fruit
2-4
Meat
2-3
Milk
2-3
4. Relation between this game and integer programming
In the following a brief introduction to linear programming and then the relation of this game to
Integer programming problem is described.
Linear programming
Linear programming is to maximize or minimize a linear function according to some constraints.
Linear programming was introduced in 1947.
A linear programming problem is defined as follow:
min 𝑧 = 𝑐1 𝑥1 + 𝑐2 𝑥2 + ⋯ + 𝑐𝑛 𝑥𝑛
𝑠. 𝑡. 𝑎11 𝑥1 + ⋯ + 𝑎1𝑛 𝑥𝑛 ≤ 𝑏1
… … ….
𝑎𝑚1 𝑥1 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 ≤ 𝑏𝑚
0 ≤ 𝑥𝑖 𝑖 = 1, … , 𝑛
In this concept 𝑧 = 𝑐1 𝑥1 + 𝑐2 𝑥2 + ⋯ + 𝑐𝑛 𝑥𝑛 is the goal function that should be minimized. 𝑋 =
{𝑥1 , … , 𝑥𝑛 } are the variables that should be chosen such that all of the none equalities be
satisfied. A value for 𝑋 that satisfied all the none equalities is called feasible, the goal is to find a
feasible value that minimizes 𝑧.
There are various methods for solving this problem. If all the variables are restricted to be integer
the problem in known as integer programming. That is known as NP-hard problem. That means
it is very hard to solve this problem [2].
According to the game in the first section each player determines the amount of food that is a
fixed value called 𝐶. The variables are the amount of each food group. The goal of the game is to
use all the marbles that is equal to minimize 𝐶 − (𝑥1 + 𝑥2 + ⋯ + 𝑥5 ), According to table 1 the
constraints of this game are:
6 ≤ 𝑥1 ≤ 11
3 ≤ 𝑥2 ≤ 5
2 ≤ 𝑥3 ≤ 4
2 ≤ 𝑥4 ≤ 3
2 ≤ 𝑥5 ≤ 3
So by playing the second section of the game, players solve their integer-programming problem.
5. The educational goals and benefits of this game
As it is said the main ability for playing this game is to knowing the 4 basic arithmetic
operations. So the first goal is to encourage the player to practice the basic arithmetic operations.
The second goal is to teach them the concept of food pyramid, using healthy diet is essential for
everybody.
And the main goal is to solve the integer-programming problem; players solve this without
knowing the concept of integer programming.
By playing this game player would try to achieve creative methods for winning, as it is said
solving integer programming problem is very hard, this may help to achieve new methods for
solving the integer-programming problem that is very important problem in operation research.
6. Conclusion
In this paper a new thinking game is described for students to help practicing arithmetic
operations and also learning the concept of food pyramid.
By playing this game an integer programming problem is solved by player. In the future work
the way of winning strategies of this game would be analyses, may be mathematicians could use
the heuristic methods of creative students to find new methods for these problems.
References
[1] http://en.wikipedia.org/wiki/Food_guide_pyramid
[2] Alexander Schrijver (1998). Theory of linear and integer programming. John Wiley and Sons