5 Italian fish
Paolo en Lisa walk through the library, giggling and
chuckling.
‘W e’ll see if the cook can solve one of our ancient sums’.
‘I bet he will have quite a bit of trouble with it’.
They find a problem by the Italian mathematician Philipo
Calandri from 1491. It goes something like this:
The head of a fish weighs 1/ 3 of the whole fish,
the tail weighs 1/ 4 and the body weighs 300
grams. How much does the whole fish weigh?
1
How much does the fish weigh? Give a smart
approximation.
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Fish sticks?
2 The cook is quite clever! He makes a drawing first: ‘This
is the fish. The head on the left, the tail on the right.
And now we write in the numbers ...’
3 Find out how much the fish weighs exactly.
draft
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4
The body of a larger specimen weighs 750 gram.
What is the weight of this fish?
Draw a rectangular bar on the right first.
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An eel has a much longer body
proportionally. The head weighs only
1/ part of the whole fish and the
12
tail just 1/8 part.
5
If an eel’s body weighs 1520 grams, what is its total weight?
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6
Here is another eel:
How heavy is its body?
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+
= 720 gram
notes
Appendix
Venice
‘Quite handy, those bars the cook used,’ Paolo says, ’I have to remember that
‘But maybe mister Calandri also used a clever method’ Lisa replies.
She looks at the problem again:
The head of a fish weighs 1/3 of the whole fish,
his tail weighs 1/4 and his body weighs 300 grams.
What does the whole fish weigh?
‘Maybe he solved the problem in a very different way ... but on the other hand
maybe not ... we’ll have to find out!’
One snap of the fingers, a flash, and the children are seated in a 15th century
gondola in Venice.
‘Welcome to Venice!’ the captain says cheerfully.
‘Who ... uhm ... who are you?’ Lisa can only stutter.
‘I am Luca Pacioli, and I teach mathematics at the university’.
‘That must be a clever man’, Paolo thinks to himself.
During the first part of the boat trip he tells the children the first part of
Calandri’s calculations:
Assume the fish weighs 120 grams.
Then the head weighs 40 grams,
the tail weighs 30 grams
and the body 50 grams.
7 How might the rest of the solution method go?
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325
Then mister Pacioli tells them how Calandri proceeds:
The body’s true weight, 300 grams, is 6 times as much as the
answer we found, 50 grams.
So the assumed weight, 120 gram, must to be multiplied by 6.
Answer: the total weight is 720 grams, the head weighs 240 grams,
the body weighs 300 grams, and the tail weighs 180 grams.
‘Wow, that’s quite different from the approach we use, don’t you think?’
Lisa nods, she has to agree with Paolo.
8
‘Why does he start with 120?’ Paolo wonders. What do you think?
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handwriting by Paolo Dagomari (ca. 1281-1370) who solved the same fish problem many years earlier
326
Appendix
9 ‘And then Calandri probably uses the 120 gram to calculate the weight of the
head, the tail and the body.’ Show how he does that:
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10 ‘Finally he multiplies the assumed weight with 6. Of course!’
Why does Paolo say ‘of course’, do you think?
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draft
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A false rule?
‘Calandri’s solution method is known as the ‘rule of false position’. Did you
know that the Egyptions also used this rule 4000 years ago?’
Lisa and Paolo shake their head. They have almost completed the gondola
trip through Venice, so mister Pacioli quickly talks on.
‘The rule works like this: you assume that a certain conveniently chosen
number, the ‘starting number’, is the solution. With this starting number
you calculate the information in the problem. Most likely the starting
number will not give you the right answer, and so you need to adjust your
initial choice. This is done by looking how many times the result of your
calculation fits into the given value. That is the number with which to
multiply the starting number, and then you have the solution.’
Which name would you give to this rule?
327
Back on the quay Paolo asks: ‘Do you think Calandri’s method is handy at
all, Lisa?’
Lisa shrugs: ‘I’ll have to compare it with my own method first’.
11
Try to solve the next problem in both ways: your own method on the left
(bar, shortened notations, ....), and with the rule of false position on the
right.
The head of a fish weighs 1/4 of the whole
fish, the tail weighs 1/5 and the body weighs
440 gram. How much does the fish weigh?
your own method
Calandri’s method
choose a starting number: …………
notes
328
Appendix
12
Here you see problems 5 and 6 once again. Investigate whether Calandri’s
method also works for these problems.
problem 6
problem 5
An eel has a much longer body proportionally. The head weighs only 1/12 part
of the whole fish and the tail just 1/8
part.
Here is another eel:
How heavy is its body?
If an eel’s body weighs 1520 grams, what is its
total weight?
notes
329
T h e rule o f false po sitio n is also us ed w ith tw o
starting num b e rs instead o f o ne . O ften th e
firs t atte m pt w ill result in a value w h ich is too
lo w , and th e se cond attem pt a w h ich is too
h ig h . T h at’s w h y th is m eth od is also calle d th e
‘rule of surplus and d eficie ncy’. I t s tates h ow
th e surplus and th e de ficie ncy can b e us ed to
find th e so lution. B ut w e w o n’t do th at here.
S um m ary
M ath pro b lem s lik e th e C aland ri fis h pro blem can b e
s olved in d iffere nt w ays. O ne stud e nt m ay d ecide to
d r aw a d iagram first, ano th e r stud ent m ig ht pre fer
to try d iffere nt num b e rs, and yet anoth er o ne m ay
ch oo se to so lve th e pro b lem by r easoning .
A lo ng tim e ag o people used the s o-calle d rule o f
false po sitio n: th ink of a po ssib le so lution, calculate
th e pro b le m thro ugh a nd d e pe nding o n h ow m uch
yo ur answ er d iffer s from w h at it o ugh t to b e , th e
s olutio n m ust b e ad jus te d .
13
W h at do you th ink o f th e r ule o f false position
(use fulness , conv enience)?
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330
Appendix
6
Number riddles
Egypt
Lisa and Paolo are back in Itapisuma; their time travel is over.
“We have to go back to school soon !” Lisa can hardly believe it.
“But we looked up a few number riddles for you. Did you know that long ago
people solved mathematical puzzles like Calandri’s fish problem for fun?
Sort of as a hobby. . Even in ancient Egypt!”
Let’s see if you can solve them …
1
“A quantity whose half is added to it becomes 16. What is the quantity?”
Clue 1: try a number
Clue 2: draw a strip first
notes
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“In the past people used a lot of words in mathematics, you have seen that a
few times already. I think that’s pretty inconvenient, çause sometimes I
can hardly see what the task is!
Don’t you think it’s a hassle?”
”That’s because ‘they didnt use symbols for their calculations at that time,
like +, -, x, :, and =”, Paolo explains. “But we do!”
2 Try to rewrite the problem in task 1 in a shorter and clearer way.
331
In the unit Fancy Fair you learned that the unknown can be represented
by a letter. With this letter you can construct an equation to solve the
problem. The Egyptians called the unknown quantity ‘hau’, which meant
’heap’, and they used a hieroglyphic in the shape of a scroll to symbolize
it:
In this passage from Papyrus Rhind you see problem 24:
A quantity whose seventh part is added to it becomes 19.
3 Where do you see the number 19? Put a red frame around it.
(you need to know, l means 1 en Ι means 10).
4 Which equation can you construct for
this problem? Solve the equation too.
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notes
Appendix
Paolo has also selected problem 33 from the Papyrus Rhind:
A quantity whose 2/3 part, 1/2 part , and 1/7 part are
added to it becomes 33. What is this quantity?
5
“Can you construct an equation for this one, too?” Lisa wonders.
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notes
333
Comparison with Diophantus
“Approximately 250 years after Christ there lived a Greek mathemati-cian
named Diophantus of Alexandria. His book Arithmetica consisted of 13
volumes, but only 6 of these have been recovered. What makes
Arithmetica so special is the fact that it contains abbreviations and
symbols for the first time. And look, the numbers 1 through 10 are
written with letters.”
Paolo also discovers that it is not easy to
determine the very beginning of equation
solving.
Especially the development of notations took a long time.
“Thank goodness those people invented symbols, otherwise we might still
have been calculating with words nowadays!” exclaims Lisa.
“Diophantus described different types of riddles. For each type he gave an
example and the standard method for solving it. For instance, a riddle on
difference: “to divide a given number into two numbers with a certain
difference between them”.
6
Which two numbers is Diophantos looking for in the following case?
Divide the number 100 into two numbers, such that the
difference between those numbers is 30.
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kladblaadje
334
Appendix
“According to Diophantus, what should you do next?”
Lisa sits down at the table, and together they read on.
Namethe smallest of the two numbers ς, then the bigger one is ς
+ 30, and the sum 2ς + 30 = 100.”
7 How does Diofantos’ solution continue? You may write the letter s
instead of ς.
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8 “A number riddle on proportions? That’s old stuff for us.”
Lisa is no longer interested.
“No, it looks kind of different”, replies Paolo.
To divide a given number (60) into two numbers that satisfy a
certain proportion (3:1).
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kladblaadje
335
”Here you see how Diofantos does it”.
Name the smaller number ς, then the larger number is 3ς, and
then the sum is 4ς = 60, ς = 15.
“Perhaps you can use this method for the other problems as well?” Lisa
wonders.
“You can’t possibly compare them, they are so different!”
Lisa disagrees.
“Take Calandri’s fish problem, for example, the one you can solve with the
rule of false position. I think it contains an unknown quantity, too.”
9 Choose a task from section 5 and find out whether Lisa’s hunch is right.
Think carefully which quantity you should represent by a letter (because one
quantity is more convenient than the other …).
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notes
336
Appendix
10
Now make up a riddle of your own which requires an equation to solve it.
Calculate it through yourself first, on one of the empty pages in the back.
Write the riddle down below and ask another student to solve it.
Riddle: _____________________________________________________
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notes
Solution: ____________________________________________________
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338