Centuries of Numbers Reasoning by ACC Art Books

THE ARCHIPELAGO AND . . .

These are the foundations of the conceptual structure of LUMI. Graphically portrayed by a small archipelago of seven islands, each representing a diﬀerent area of development, the islands are all connected to each other, as growth is a global process involving various cognitive, emotional, and social dominions, all of which are interconnected.

The products in the LUMI range belong to seven distinct—but interconnected—areas to stimulate integrated and global learning. The contents of the activity books and games have been created based on the specific goals indicated and can be summarized in seven large categories which are transversal to the age range and represented graphically by the seven islands in the archipelago:

. . . ITS INHABITANTS

You will also find some friendly companions among the animals living on the islands, who have taken on the role of accompanying our explorers, based on their age group.

Logi Uniq

This hermit crab is a sweet little animal. His strong shell means he can hide away whenever there’s a predator about. Logi is the perfect companion for our youngest adventurers, protecting them as they gradually explore the world around them, before they can eventually shake off their shells and wander off to new horizons, confident in their newfound skills.

Another resident of the LUMI archipelago is the sea turtle. As soon as they can move, these brave creatures head for the sea, never looking back. Uniq is the perfect companion for our young but intrepid explorers as they enthusiastically head off on a voyage of discovery of basic skills, encouraged by their conquests in physical, linguistic, and social skills.

Mega Ido

Our third friendly animal guide is the whale. Since the beginning of time, whales have always fascinated humankind, representing a symbol of humanity itself and of a search for knowledge. Mega’s fellow adventurers are strong and determined, ready to leave dry land behind and sail the seas to the very abysses of knowledge, on a discovery of their own individuality.

And finally, everyone in the LUMI archipelago has an invaluable friend in the seagull. Free, independent, determined, curious, and loyal to the group, Ido the seagull accompanies our more mature adventurers as they leave the safety of the lido to explore the more complex areas of the world of knowledge, transforming themselves into young but skilled leaders.

AN EDUCATIONAL VISION

You’d be forgiven for thinking of mathematics as complicated, nothing to do with everyday life; some of you may even think that they’re no good at it! But this is simply not possible: math is a human activity, it is a part of us, and it changes over time.

It certainly isn’t just about numbers! Even the youngest of children can have fun learning about mathematical thought, engaging in math activities, even in the most unexpected daily situations: during snack time, in the kitchen, chatting or playing with friends. . . . After all:

“[...] math isn’t something you do just by doing . . . math; doing math means taking on a certain attitude while doing other things. A certain way of ‘seeing’ the world, of ‘reading’ reality, of interpreting events. [...] If a child is open to it, spontaneous math is everywhere to be found.”

Angeli, D’Amore, Di Nunzio, Fascinelli, 2011 7 - 9 FOR CHILDREN OF YEARS

The activities in this book are based on the ideas of Bruno D’Amore and the Research and Experimentation in the Didactics and Application of Mathematics group work that he founded in 1984 in the Faculty of Math of the University of Bologna. We hope that this book will inspire curiosity about math in all children (and their parents) who read it.

ADVICE FOR PARENTS

When looking at a child’s math results, in order to encourage them, remember that:

During the learning process it is important to start with and validate any knowledge, including any spontaneous knowledge, that the child may already have on the subject.

Learning sometimes starts with misunderstanding. And this is normal. You cannot learn what you already know: you can only learn what you don’t know yet.

All “mistakes” are useful. Misunderstandings reveal the way someone looks at things: they are useful clues to better understand the “mistaken” person’s thought processes.

If you believe your child has made a mistake, try to understand their reasoning behind it before intervening with your own way of problemsolving.

Try to be welcoming and inclusive: there may be lots of different ways to undertake an activity, and just as many to represent and manipulate mathematical objects. It is important to take them all on board.

Not only does every child have his own way of learning, but learning itself is flexible and evolves over time.

Everyone has the right to the time they require to do, try, experiment, scribble.

If they don’t understand today, it doesn’t mean they’ll never understand.

Sometimes those activities that don’t seem mathematical at all are actually very mathematical indeed: in fact, there are various contexts in which we speak of math, and different ways of doing it.

Egyptians and Fractions

I have always enjoyed wandering among the objects in the BRITISH MUSEUM . But the most exciting time was when I was eight and I saw the MATHEMATICAL PAPYRUS for the first time. This 3550-YEAR-OLD papyrus is 6½ feet long and 12½ inches wide and holds 84 mathematical problems, many of which are on FRACTIONS , just like the ones our teacher gave us yesterday for homework. The papyrus was commissioned by Pharaoh APOPHIS , who asked his scribe AHMES to copy an even older document: full of mathematical challenges for any students learning to become scribes, but to teach them in a fun way.

What surprised me the most was the title: “Inquiry into the knowledge of all things, mysteries . . . all secrets.” That hadn’t been copied from an older document. It was the pharaoh himself who demanded this title, which suggests something that had never occurred to me: that we can use math to reveal secrets.

And also, to become the king of one of the most advanced populations of the world! If you look closely, the construction of the pyramids, the division of the fields, the accounting of trade passing on the ships along the Nile are all activities that have to do with fractions. STRANGE, RIGHT? There was, however, something a bit strange in that papyrus: I couldn’t recognize the numbers! I thought perhaps it was the fact that it was written in HIERATIC , an italic type of hieroglyphs, but I just could not understand which ones were the numbers. How can you do fractions without numbers? Only when I read the captions of the museum did I discover that the ANCIENT EGYPTIANS WROTE NUMBERS DIFFERENTLY . So, if you want to open your mind to the “knowledge of all existing things,” as the pharaoh used to say, then do what I did: PLAY WITH EGYPTIAN NUMBERS

COUNTING EXPLAINING

Egyptian Numbers

One of the ways in which Egyptians used to write numbers was HIEROGLYPHIC WRITING . Find out how it works: 1 10 100 1,000 10,000 100,000

This is the key:

Let’s have some examples:

This number corresponds to: 10+10+1+1, which is 22

This number corresponds to: 1000+100+100+10+10+100 , which is 1320

NOW YOU TRY! Follow the previous key and FIND OUT which numbers are written here below in hieroglyphs.

Now help AHMES the scribe to calm ISHME and ADAD . These two Assyrians are to visit the pharaoh, but are now arguing about whether the numbers they have in front of them are the same or not.

Let’s think this through. Who is right: Adad, or Ishme? WRITE in the space below the number you find using today’s symbols.

“The numbers are the same because the symbols are the same.”

“But if I tell you that the symbols are in different places? Something must be different!”

Who is right, ADAD or ISHME ?

Write your reasoning below, or discuss it with a friend.

NOW LET’S WORK IT BACKWARDS!

The number on the right has the same digits as the number on the left, but they are inverted. Try writing in the space below these two numbers using hieroglyphs.

And what about how we write numbers today? Is the position of the digits important? In what way?

ADAD

ISHME

The Eye of Horus and Fractions

UMAI has just finished plowing his fields, and it is time to sow. To decide how to distribute the seeds, he uses Eye of HORUS fractions:

This is how UMAI decided to cultivate his field. At the moment, he has only planted onions.

UMAI ’s neighbor is also cultivating his field following the fractions hidden in the EYE OF HORUS

INDICATE which fractions correspond to each part of the three fields on the right.

COUNTING EXPLAINING MEASURING

Muhammad’s Scales

BAGHDAD’ s fruit sellers have an infallible method to work out how many of a certain fruit weigh the same as one of another kind. And they learned this the time that MUHAMMAD resolved the melon-and-pomegranate problem with his scales. This was his reasoning:

The scales must be balanced to start. On one side are two melons and a pomegranate; on the other are one melon and three pomegranates.

The pomegranates on the right and left weigh the same, so we can remove one of them from each side.

Just as we did before with the pomegranates, we can now also remove one melon from each side.

Can you finish Muhammad’s work and say how many pomegranates weigh the same as one melon?

ONE MELON WEIGHS HOW MANY POMEGRANATES?

Now you can become a seller too. ANSWER the questions under the scales.

1) How many figs to a pomegranate?

2) How many dates to a pomegranate?

3) How many dates to a fig?

COUNTING

EXPLAINING DESIGNING

Math Windows

MASHRABIYAS are typical architectural elements found in Arab buildings. They allow air into the rooms and are characterized by repetitive geometrical motifs. The artisans have called MUHAMMAD to come and help them decide whether some new ornaments they are designing could be used for the windows. LOOK AT the stars in the three figures.

Now DRAW FIGURE 4 under here.

1) How many red stars will be in FIGURE 4 ?

FIGURE

The artisans show AL-KHWĀRISMĪ a series of triangular tiles.

In FIGURE 1 there are SIX TILES , and each subsequent figure has more tiles than the last. The artisans cannot understand how to continue decorating the palace and ask AL-KHWĀRISMĪ for advice. YOU CAN TRY TO HELP THEM, TOO.

1) How many triangular tiles will there be in FIGURE 8 ?

2) And how many in FIGURE 12 ?

3) The artisans ask: “How did you do that? What is the rule? Come on, tell us, so we can write it down and never forget it.”

WRITE down the rule for the right number of tiles in the artisans’ general ledger:

4) One of the apprentices asks for help: “If I have 72 tiles, how can I know what figures I can make to use them all?” HELP HIM understand by explaining the rule:

These are the first three figures of another sequence designed by the artisans:

5) Draw FIGURE 6 below.

6) How many green squares will FIGURE 7 have? And FIGURE 10 ?

8) How many green squares will FIGURE 32 have?

7) How many green squares will have to be added to each subsequent figure?

FIGURE

FIGURE 1

FIGURE 2

FIGURE 3 FIGURE 4

Hopscotch

Today we can relax a bit. As I stroll leisurely around the city, I pause in Piazza San Giovanni to watch two children playing hopscotch. Their names are ELSA and TOMÀ .

Elsa throws a stone in front of her. The stone falls on SQUARE 3 , so Elsa jumps THREE TIMES and lands on SQUARE 3 in ROW A

Tomà throws his stone, which lands on SQUARE 2 and he jumps forward TWICE along ROW B Look where they have landed.

Without moving, Elsa and Tomà throw their stones again. Elsa lands on SQUARE 5 in ROW A, and Tomà, throwing far away, lands on SQUARE 6 in ROW B . DRAW where the two friends are now.

1) TOMÀ has landed in row at square

2) ELSA has landed in row at square

Elsa now must land in the NEXT SQUARE . She throws her stone and makes it! Tomà, on the other hand, can already make his way back. He throws the stone and JUMPS FORWARD TWO SPACES .

DRAW where the two friends are now.

3) TOMÀ has landed in row at square

4) ELSA has landed in row at square

I like ELSA and TOMÀ a lot and suggest a new game. I draw a grid on the ground and pick up my two dice: a red one with numbers from 1 to 6 and a yellow one with letters from A to F . Every time I throw the dice, Elsa and Tomà have to put a FOOT or a HAND in the right square. YOU TRY , too. WATCH the dice throws and DRAW feet and hands in the correct squares.

E F 1 2 3 4 5 6

The children really enjoyed this new game. I give them my dice as a present, and they leave me, smiling: “Thank you, JOACHIM ! We will go straight to our friends to teach them how to play.” TRY to play the game with your friends, too.

MEASURING DESIGNING

The battle rages between the army and the revolutionaries in the Bastille district. But help is on the way! How can we understand what is going on in the streets and in the squares?

The only things that we know for sure are that: the revolutionaries are hiding in the streets and squares that have the same AREA but a different PERIMETER ; the army is hiding in a place with the same PERIMETER but a different AREA ; and the fighting is going on in the streets and squares that have the same AREA and PERIMETER !

Use the ruler that you cut out from the back of the book to MEASURE the dimensions of the streets and squares:

1) In which streets and squares are the revolutionaries hiding?

2) Where is the army hiding?

3) Where are they fighting?

ACTIVITIES, TEXTS, AND STORYTELLING

Agnese Del Zozzo and Marzia Garzetti both have mathematics degrees; and after years of selfemployment in the world of education, they began researching the teaching and communication of mathematics. They are currently PhD students at the universities of Trento and Bolzano, respectively.

Tecnoscienza is a group of authors and educators who have been involved for more than 15 years in the dissemination of science, technology, mathematics, and the environment for numerous institutions, such as museums and businesses. Their books have been published in more than 20 countries and are designed to stimulate thoughts, actions, and emotions.

ILLUSTRATIONS

Arianna Bellucci, studied Entertainment Design at the Nemo Academy of Digital Arts in Florence, Italy, and has worked as a 2D artist in the videogame industry, mainly with LKA studio on the release of “The Town of Light.” She is a member of Fuffa, an illustration studio based in Tuscany that was created by a team of four friends passionate about illustration and books. Together they make picture, activity, and interactive books, as well as puzzles and toys. Between them, they have years of experience in the editorial field, and they share the same goal: to create something beautiful.