The document describes a math experiment conducted by a group of students at a school in Greece. The students chose to teach a chapter on polynomials in English. They translated the chapter from Greek to English and presented it to their class. They had help from their math and English teachers. They also compared a Greek math textbook to an American one used in US colleges. The goals of the experiment were to encourage collaborative learning, study interdisciplinary topics between math and English, and provide a first experience with research projects.

1. Maths in English
An experiment at school…
ΠΡΟΣΤΠΟ ΠΕΙΡΑΜΑΣΙΚΟ ΛΤΚΕΙΟ ΒΑΡΒΑΚΕΙΟ Τ ΢ΥΟΛΗ΢
΢χ. Έτος 2011-2012

2. ΣΟ ΠΕΙΡΑΜΑ ΜΑ΢
 Μϋςα ςτην τϊξη του Β2 μύα ομϊδα μαθητών/τριών αποφαςύςαμε να
κϊνουμε ϋνα πεύραμα με τύτλο «Μαθηματικϊ ςτα Αγγλικϊ». Επιλϋξαμε
το κεφϊλαιο των Μαθηματικών «Πολυώνυμα», το οπούο διδαςκόμαςτε
εκεύνο το διϊςτημα, το μεταφρϊςαμε ςτα Αγγλικϊ και το παρουςιϊςαμε
ςτην τϊξη ςτα Αγγλικϊ. Βοηθούσ ςτο πεύραμϊ μασ εύχαμε τον καθηγητό
των Μαθηματικών Δρ Γ. Κόςςυβα και την καθηγότρια των Αγγλικών Δρ
Λ. Νιτςοπούλου. Βοηθητικό υλικό εύχαμε ϋνα βιβλύο Μαθηματικών ςτα
Αγγλικϊ, που διδϊςκεται ςε Αμερικανικϊ κολλϋγια μϋςησ εκπαύδευςησ
και ϋνα γλωςςϊρι μαθηματικών όρων που ϋφτιαξε η καθηγότρια κ.
Νιτςοπούλου. Στη ςυνϋχεια, μαζύ με ομϊδα μαθητών/τριών από το Β4
προχωρόςαμε ςε ςύγκριςη των δύο ςχολικών βιβλύων (Ελληνικό και
Αμερικανικό) και παραθϋςαμε τα αποτελϋςματα τησ ςύγκριςησ.

3. ΣΚΟΠΟΙ
 Η ανϊπτυξη και η ενθϊρρυνςη τησ ομαδοςυνεργατικόσ
προςϋγγιςη τησ μαθηςιακόσ διδαςκαλύασ.
 Η μελϋτη τησ διαθεματικότητασ (Μαθηματικϊ- Αγγλικϊ ).
 Η βιωματικό μϊθηςη με την λύςη αςκόςεων ςτα αγγλικϊ.
 Η ενθϊρρυνςη τησ δημιουργικότητασ των μαθητών ςτην
ψηφιακό τεχνολογύα.
 Μια πρώτη προςϋγγιςη τησ ερευνητικόσ εργαςύασ(project)
ςτο πλαύςιο του αναλυτικού προγρϊμματοσ.
Η διερευνητικό προςϋγγιςη τησ μϊθηςησ.
Η διαπολιτιςμικό ςκϋψη και προςϋγγιςη των
μαθητών/τριών με την ςύγκριςη Αμερικανικού-
Ελληνικού ςχολικού βιβλύου Μαθηματικών .

4. ΣΤΟΧΟΙ
 Η ανϊπτυξη τησ διαμεςολαβητικόσ ικανότητασ των
μαθητών /τριών
 Η διαπολιτιςμικό προςϋγγιςη του μαθόματοσ μϋςα από
τη ςύγκριςη Ελληνικού – Αμερικανικού βιβλύου
μαθηματικών.

5. USABILITY OF POLYNOMIALS

6. Polynomials are also used to model
the trajectory of a cannonball

7. TAYLOR SERIES
 In mathematics, a Taylor series is a representation of a function as an infinite
sum of terms that are calculated from the values of the function's derivatives at
a single point.
 The concept of a Taylor series was formally introduced by the English
mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then
that series is also called a Maclaurin series, named after the Scottish
mathematician Colin Maclaurin, who made extensive use of this special case of
Taylor series in the 18th century.
 It is common practice to approximate a function by using a finite number of
terms of its Taylor series. Taylor's theorem gives quantitative estimates on the
error in this approximation. Any finite number of initial terms of the Taylor
series of a function is called a Taylor polynomial. The Taylor series of a function
is the limit of that function's Taylor polynomials, provided that the limit exists.
A function may not be equal to its Taylor series, even if its Taylor series
converges at every point. A function that is equal to its Taylor series in an open
interval (or a disc in the complex plane) is known as an analytic function.

8. Degrees of Polynomials
(x
3
3x
2
1) ( x
3
3x
2
1) x
3
x
3
3x
2
3x
2
1 1 0 (zero Polynomial)
2 2 2 2
(5 x 2x 1) ( 5x 3x 2) 5x 2x 1 5x 3x 2 5x 3
(Polynomial of degree 1)
3 2 3 3 3 2
(2 x x 1) ( 2x 2x 3) 2x x2 1 2x 2x 3 x 2x 2
(Polynomial of degree 2)
3 2 3 2 3 2 3 2
(x 2x 5x 7) (4 x 5x 3) x 2x 5x 7 4x 5x 3
3 2 3 2
(1 4) x (2 5) x 5x (7 3) 5x 3x 5x 10
3 2 3 2 3 2 3 2
(x 2x 5x 7) (4 x 5x 3) x 2x 5x 7 4x 5x 3
3 2 3 2
(1 4) x (2 5) x 5x (7 3) 5x 3x 5x 10
(Polynomial of degree 3)

9. DEFINITIONS
Definition of a monomial
A monomial x is an algebraic expression in the form of ax n , where a
is any real number and n is a non-negative integer
Definition of a polynomial in x (polynomial)
A polynomial in x is an algebraic expression of the form
n n 1 2 1
an x an 1x ... a2 x a1 x a 0 , where a n 0

10. EQUATION BETWEEN 2
POLYNOMIALS Two polynomials
n n 1 2 1
an x an 1x ... a2 x a1 x a0
m m 1 2 1
bm x bm 1 x ... b 2 x b1 x a0
m n
are equal whenever
a0 b 0 , a1 b1 , ..., a n b n και a n 1
an 2
... am 0

11. NUMERICAL VALUE OF A
POLYNOMIAL
In order to The
find the value expression
of the that forms is
polynomial , called
we replace the numerical
X with a value of the
certain real polynomial
number r for X=r

12. ARITHMETICAL PRICE OF THE
POLYNOMIAL 3 2
Let a polynomial P ( 1) ( 1) 2 ( 1) 4 ( 1) 1 0
n n 1
P ( x) an x an Ix ......a 1 x a0
If we put in the place of X a real number ρ , then the real number
n n 1
P( ) n n
1 ... a1 0
this result,is called arithmetic price,or simpler
price of the polynomial for x=ρ
Then the ρ symbol is called root of the polynomial.For example, the price of the polynomial
3 2
P ( x) x 2x 4x 1
For x=1  P (1) 3
1 2 1
2
4 1 1 6 but
For x=-1  P ( 1) ( 1)
3
2 ( 1)
2
4 ( 1) 1 0 which means that
-1 is the root of the polynomial P(x)

13. In order to find X(x) and U(x) we
follow a certain procedure
 1.Make the shape of the division and write the 2
polynomials
 2.Find the first term of the quotient by finding the
first term of the dividend with the first term X of the
division
 3. Multiply with x-3 and subtract the product from
the dividend. So we find the first partial difference
 4.Repeat 2 and 3 with a new dividend we find the
second partial difference -4x-1

14. Adding • Simply combine like
terms
polynomials
Subtracting • Same as adding just
change same signs
polynomials
• Use the distributive
Multiplying property

15. William George Horner
 William George Horner (1786 – 22 September 1837)
was a British mathematician and schoolmaster. The
invention of the zoetrope, in 1834 and under a different
name (Daedaleum), has been attributed to him.

16. Horner’s Work
 Horner published a mode of solving numerical equations of any degree, now
known as Horner's method. According to Augustus De Morgan, he first made it
known in a paper read before the Royal Society, 1st July 1819, by Davies Gilbert,
headed A New Method of Solving Numerical Equations of all Orders by
Continuous Approximation, and published in the Philosophical Transactions
for the same year. But this version of the history is comprehensively denied by
later historians. De Morgan's advocacy of Horner's priority in discovery led to
"Horner's method" being so called in textbooks, but this is a misnomer. Not
only did the 1819 paper not contain that method, but it also appeared in an 1820
paper by Theodore Holdred, being published by Horner only in 1830; and the
method was by no means novel, having appeared in the work of the Chinese
mathematician Zhu Shijie centuries before, and also in the work of Paolo
Ruffini.[4]
 The method was republished by Horner in the Ladies' Diary for 1838, and a
simpler and more extended version appeared in vol. i. of the Mathematician,
1843

17. CONSTRUCTION OF HORNER’S
TABLE
For the construction of the table we follow the next steps:
We fill the first line with the coefficients of the polynomial P x and the first
place of the third line with the first coefficient of P x .
Afterwards the table is completed as following:
Every item on the second line results by multiplying the immediate past item
of the third line with p .
Every other item of the third line results by adding the corresponding items
from the first and second lines.
The last item of the third line is the difference of the division between P x
and x p , namely the price of the polynomial P x for x p . The other items of
the third line are the coefficients of the quotient of the said division.
Let’s work right now on the Horner Configuration to find the quotient and the
5 4
difference of the division of P x 3x 3x 6 x 13 to x 2 .

18. THE HORNER CONFIGURATION
Let’s say we have P x 3x
3
8x
2
7x 2 . Consequently, we define the division
P x : x p .
The Horner Configuration is a different way of performing the operation of dividing
polynomials and can be visually depicted by the following board:
Coefficients of P x
3 -8 7 2 p
3p 3p 8 p 3p 8 p 7 p
3 3p 8 3p 8 p 7 3p 8 p 7 p 2
Coefficients of the quotient Difference

19. Abilities
 The degree of the product of the 2 non-zero polynomials
is equal to the total of the degrees of the Z polynomials.
 For every pair of polynomial D(x) and δ(x) with δ(x) 0
there are 2unique polynomials X(x) and U(x),so as:
 D(x)=δ(x)X(x)+U(x)
 Where U(x) is either the zero polynomial or is of less
degree than the δ(x)

20. Just like the division of natural
numbers
D(x) •Is called dividend
δ(x) •Is called divisor
X(x) •Is called quotient

21. COMPARISON AND CONTRAST
BETWEEN THE AMERICAN AND THE
GREEK MATHS BOOK
Differences
The English book includes It also shows the goals and
more thorough examples It includes more exercises reasons for learning
thus making it much and a wider variety of polynomials and their
easier for the student to them. practical application in
understand . real life.

22. More differences…
It contains useful tips It combines both algebra
for study and and geometry, revealing
technology. all polynomial’s aspects.
It is easier on the eyes It is more interesting
since it contains images making mathematics
that make it more more approachable
attractive. towards the students.

23. Similarities between the American and
the Greek Maths book
 They have the same terminology.
 The difficulty of the exercises is gradually increasing.
 They both provide students with the answers of the
exercises.
 They also contain examples that show how to solve the
exercises step by step.

24. Bibliography :, Άλγεβρα β’ λυκεύου, Larson’s algebra for
college students, www.wikipedia.org
Foundations, Cambridge University Press, 1991
Edit&presentation Supervisor:Κωνςταντύνοσ Ζόκοσ
Project made by:Βαςύλησ Zωγόπουλοσ ,Νεοκλόσ
Καςιμἀτησ , Γεωργύα Θεοδωρακοπούλου ,Γιϊννησ
Έξαρχοσ, του Β2 τμόματοσ του Προτύπου Πειραματικού
ΓΕΛ τησ Βαρβακεύου Σχολόσ
Many thanks to Our teachers Dr Λύλιαν Νιτςοπούλου
και Dr Γεώργιο Κόςςυβα who helped us in this project