# Download e-book Mathematics in fun and in earnest

Try them with your students throughout the month of March and share the fun with others on social media. Since pi is so closely related to measuring circles, primary students can celebrate Pi Day by exploring measurement. Students can investigate how to measure big things and small things around school, and then compare their measurements to measurements from a mystery school.

If you use this activity to celebrate Pi Day, challenge older students to measure not only length and height but also to measure all the way around such objects as round tables, playground equipment, and other circular items found in real life. Enjoy this delicious problem that comes from a classic Foxtrot comic and looks at pi from all angles.

Studying historical puzzles can give students a perspective on the usefulness of math as a tool and on the creative aspects of problem solving. Di Day offers a wonderful opportunity for such inquiry.

## The Project Gutenberg eBook of Amusements In Mathematics, by Henry Ernest Dudeney.

Terrel Trotter Jr. How students develop a conceptual understanding of the formulas of the circumference and area of a circle by exploring the number pi. Several activities are suggested in this article. Why does the same value of pi appear in both formulas for the circumference and area of a circle? This activity presents interesting ways to approach pi. It illustrates a way in which students can estimate the ratio of the area of the circle to the radius square, and also helps students understand why the same value of pi appears in both formulas for the circumference and the area of a circle.

Why is it not possible to recite pi backwards? This and more fun questions to ask your students on Pi Day! Do your algebra 1 students know that pi, because it is a ratio, is also a slope? Before you have students explore these unwrapped circles, have them predict what the graph of the unwrapped circles will look like.

Will it be linear?

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Or will it be curved like a circle? Either before or after students construct their graph, they can grapple with what slope and rate of change would mean in this context. Does their answer change after this investigation? This pi day, why not have students get in a pi fight? Okta, the octopus, hangs out near different coordinates on a unit circle, and students must choose an angle and distance to fire a pie at Okta.

Many new ideas appearing in one area can be very useful in neighbouring fields.

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Certain desirable conditions should be fulfilled: real reasonable accessibility to ideas without superfluous technicalities, from an expert who truly desires and knows how to communicate his knowledge to professional mathematicians unfamiliar with his field. In my opinion this is one of the most difficult tasks in the popularization of our mathematical knowledge.

This difficulty and how far we are from the ideal are quite obvious when one attends our general International Congresses of Mathematicians and becomes aware of the failures of so many of our most outstanding mathematicians to give a moderately useful glimpse of what they have done to a wide audience of professional mathematicians. It is quite common to follow the easy path to talk to those how many? Teachers and university students. The general trends of contemporary mathematics should be made accessible to them by experts really competent both in the field and in the necessary skills to communicate at this level.

Also the main highlights of the history and evolution of a field can be of great use to illuminate it for those who approach it for the first time. Other professionals inside and outside the academic world. There are many aspects of mathematics they perhaps do not use in their work that could throw some new light on their way of thinking and solving their own problems.

High school students. The most important aspects of the history, evolution and applications of each one of the topics they are exposed to. The lives of the most important men and women of mathematics. The cultural impacts of mathematics on the human history.

General public. Avoiding technicalities one should try to transmit as much as possible of the impact and methods of mathematical thinking about some special subjects. The biographies of dead and living mathematicians of deep interest. Applications, ideas and facts they should know as part of the human culture. Small children.

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With a correct awareness of their possible interest and capacity, by those who know how to communicate with them with great enthusiasm. Through exhibits, competitions, games…. In the paper [3] I have tried to analyze in some detail the relationships between mathematics and games by looking at the intrinsic nature of these activities, at their respective modes of practice and at the ways they have impinged upon each other historically with great impact.

Here I will briefly summarize some of the conclusions reached there in order to continue later with the implications this situation has for the popularization of mathematics. Mathematics is a science, i. Mathematics is an instrument for the exploration of the world, thanks to the systematic examination to which it subjects the different structures it detects in the universe and to its peculiar way of abstracting and expanding them.

Mathematics has been a model of thought, through its qualities of objectivity, consistency and soberness, which give it a preeminent place among the different human ways of acquiring information. But mathematics is also a game.

When one looks at the main features by which sociologists characterize games cf. It performs a certain function in human development, and is certainly not solely and perhaps not mainly for children. A game is not a joke, it has to be taken seriously, the worst game spoiler is the one who does not do this.

The structure of games and of mathematics, the way in which they create a new order, a new life, through the acceptance of certain objects and rules defining them and through the consistent adherence to this set of rules, is strikingly similar.

On the other hand, if one looks at the ways in which one is introduced, becomes acquainted and finally reaches a certain degree of expertise in games and in mathematics, one cannot fail to see a strong similarity, which is certainly not surprising when one considers the common features of games and mathematics in their nature and structure.

Given the strong similarities of mathematics and games, one should a priori expect a deep intertwining between them along history. The impact of games and their ludic spirit on mathematics and the presence of mathematics in games has in fact been very powerful. It has very often been the case in the history of mathematics that an interesting question made in a gamelike manner or an ingenious observation about an apparently innocent situation has given rise to new modes of thinking.

What is the plane figure of minimal area such that a needle of length one can be continuously inverted inside it? Also one can see the strong and important implications of the game known as The Tower of Hanoi in an interesting paper by A. Hinz [5], which also appeared very recently in this journal.

Similar examples can easily be found in ancient as well as in modern mathematics. The list of mathematical objects that have come to existence motivated by the spirit of games would certainly be without end. If games have given rise to interesting, deep, useful and important mathematics, it is no less true that the richness of mathematical themes present in classical and modern games is impressive. The best way to perceive this is to look through the classical works by Lucas or Ball Ball and Coxeter and through the bibliographical compilations made by W. Schaaf on the recent literature on games, published by the NCTM.

Still more interesting is the fact that there is so much deep mathematics with the flavour of games. Among modern examples one can select a few in which this is particularly obvious. After these considerations it becomes quite easy to understand why so many among the greatest mathematicians have been so fond of games and in some cases have become great practitioners of some of them. Among others there is a Jew who plays extraordinarily well. Many names of famous mathematicians could be mentioned here in connection with the ludic spirit of mathematics: Fibonacci, Cardano, Pascal, Daniel Bernoulli, Gauss, Hamilton, Hilbert, Einstein, von Neumann….

After what we have seen about the similarity in their intrinsic nature, structure, spirit and ways of practice of games and mathematics it should be quite clear that:. Games do not need boring systematic introductions before arriving to something interesting, as is often the case with our presentation of mathematics.

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## Undergraduate

Good games and puzzles can avoid the effect of the psychological blocks that straight mathematical presentations tend to cause not only in children, but also in many adults, very often because of previous unpleasant mathematical experiences. Games place everybody initially in a situation of equality, in which not so much depends on previous performance and knowledge as in mathematics. Games are better at fostering ingenuity, imagination, phantasy, experimentation, manipulation,… since they constitute much more clearly a free activity quite open to us.

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All these reasons can explain the great success of the really good writers of mathematical recreations along many centuries, and one can easily agree with Berlekamp, Conway and Guy, when in the dedication of their excellent work on games [1] they write:. There are, of course, good and bad games and puzzles. What are the characteristics of a stimulating game from our point of view? I will point out some of the general traits that in my opinion help a game to be good:.

The great master of popularization of mathematics in the second half of our century, Martin Gardner, has assessed the situation quite rightly, talking about the use of games for mathematical teaching.