unidad didáctica expresiones algebraicas - educastur.princast

UNIDAD DIDÁCTICA
EXPRESIONES ALGEBRAICAS
------------------------------------------------
1

ÍNDICE
1. INTRODUCCIÓN ---------------------------------------------------------------------- 3
2. CONTENIDOS ------------------------------------------------------------------------- 3
3. OBJETIVOS ---------------------------------------------------------------------------- 4
4. METODOLOGÍA ---------------------------------------------------------------------- 4
5. SECUENCIACIÓN DE LAS SESIONES. MATERIALES
CURRICULARES DE CADA SESIÓN --------------------------------------------- 6
6. COMPETENCIAS BÁSICAS QUE SE TRABAJAN
EN LA UNIDAD DIDÁCTICA -------------------------------------------------------- 32
7. MATERIALES Y ESPACIOS NECESARIOS -------------------------------- 33
8. EVALUACIÓN. CRITERIOS DE EVALUACIÓN Y DE
CALIFICACIÓN ------------------------------------------------------------------------- 33
9. EN LA CLASE DE INGLÉS ------------------------------------------------------ 35
2

INTRODUCCION
Esta unidad está dirigida a 2º curso de E.S.O. Los alumnos han trabajado esta
unidad en 1ª, dentro del programa bilingüe y conocen algunos conceptos
básicos del tema, conviene repasarlos para profundizar en el lenguaje
algebraico y la jerarquía de operaciones, que ya han trabajado en el bloque
números. La importancia de la unidad radica en que se ha diseñado para que el
alumno adquiera el dominio del álgebra básica para la posterior solución de
problemas matemáticos que puedan presentarse en la vida diaria.
De acuerdo con el Decreto 74/2007 del 14 de junio, esta unidad está en el
bloque III de contenidos para 2º de ESO titulado “Algebra” .
La unidad didáctica se compone de un total de 14 sesiones lectivas.
Al finalizar esta unidad los alumnos deberían ser capaces de operar
correctamente con expresiones algebraicas. Siendo conscientes de los errores
sistemáticos que suelen cometer al utilizar el álgebra Utilizar las propiedades
de las operaciones, la jerarquía y las reglas de uso de los paréntesis en
expresiones algebraicas. Utilizar el lenguaje algebraico para expresar
situaciones de la vida diaria.
CONTENIDOS
Los contenidos a desarrollar son los siguientes:
.Repaso de situaciones matemáticas en las que se use el lenguaje algebraico.
Monomios. Elementos de un monomio. Valor numérico de un monomio
Operaciones con monomios. Suma , resta, multiplicación y división.
Polinomios. Valor numérico de un polinomio.
Operaciones con polinomios. Suma, resta, multiplicación.
Igualdades notables.
Contenidos específicos relacionados con el idioma inglés:
Los alumnos deberán familiarizarse con todo el vocabulario en inglés propio del
tema. En objetivos se hace una lista con las palabras que deberán conocer.
Aunque este tema consiste básicamente en conocer el lenguaje algebraico y
operar correctamente para luego poder resolver ecuaciones y problemas de la
vida cotidiana, aquí aparecen expresiones como increased by , decreased by,
more than, less than, triple, double, like terms, unlike terms y los ordinales fith,
sixths,
En los ejemplos se usarán mucho expresiones como How much y How many
( contables e incontables), muchos adjetivos y sustantivos, genitivo sajón ya
que en cada expresión algebraica se puede introducir el vocabulario que nos
interesa que aprendan y utilizar todo el vocabulario relacionado con un
determinado campo semántico, por ejemplo las relaciones familiares en las
expresiones algebraicas en las que aparecen edades.
3

OBJETIVOS:
Objetivos específicos de Matemáticas:
Simbolizar en lenguaje algebraico enunciados dados de forma verbal, y
viceversa.
Conocer perfectamente todo el vocabulario que se utiliza en algebra ya que los
contenidos de esta unidad conforman el cimiento para determinar y organizar
parte de los contenidos de la asignatura.
Objetivos lingüísticos y comunicativos:
Al finalizar esta unidad los alumnos deberían:
− Ser capaces de entender instrucciones sencillas en inglés, tanto orales
como escritas, para realizar las actividades propuestas.
− Usar, leer y escribir correctamente las palabras más frecuentes propias
del tema. En su vocabulario incluirán al menos estas palabras.
Variable, unknown, terms, like terms, unlike terms, coefficient, grade,
monomial, polynomial, expand brackets, cancel, simplify, convert, one half,
third, fourth, fifths, right side, left side, sixths, twentieths, double, triple.
− Ampliar campos semánticos (. árboles, frutas, miembros de la familia,)
− Adverbios de tiempo en las expresiones de edades: in five years time, in
a couple of days, two years ago.
− Ser capaces de formular preguntas para obtener información usando
contables e incontables. ( How much, how many)
METODOLOGÍA
Se pretende una metodología activa, intuitiva y motivadora que despierte
interés y fomente el aprendizaje por el descubrimiento de los conceptos a
partir de los conocimientos y experiencias personales.
Basándose en el hecho de que si el alumno descubre los conceptos por sí
mismo, éstos se asientan de manera más duradera en su estructura lógica, se
pretenden diseñar y elaborar actividades para que los estudiantes descubran
los conceptos y no sólo los almacenen.
Se fomentarán clases activas, en las que desarrollen sus habilidades
Las actividades han de estimularles a preguntar, reflexionar y a expresar su
pensamiento verbalmente.
Al trabajar las actividades se pretende que el alumnado relacione, el mayor
número de conceptos posibles.
Las actividades a proponer tienen varios niveles en su desarrollo hacia la
solución, que permitan un ritmo diferente según el alumnado.
4

Se promoverán agrupaciones diversas y se utilizarán distintos recursos.
( bibliográficos, audiovisuales, uso de calculadoras y páginas web )
El álgebra ha de ser usada en diferentes contextos: juegos, situaciones
personales, familiares, ciencia…
Al final de la tarea, la profesora puede intervenir facilitando la síntesis y la
elaboración de conclusiones finales a partir de las que hayan podido obtener
los estudiantes aisladamente.
Reflexionar sobre lo que se va obteniendo durante las clases o sobre las
razones por las que no se logra avanzar.
Utilizar los errores de los alumnos en los aprendizajes de nuevos conceptos,
poniendo en conflicto los erróneos sustentados por los alumnos con las
nociones correctas mediante preguntas o actividades adecuadas.
Las actividades propuestas seguirán la secuencia:
Hacer-discutir-descubrir-exponer, expresándose ,a veces, oralmente (puesta
en común) y ,a veces, por escrito.
Debido a la complejidad del lenguaje algebraico en la mayoría de las
actividades aparece un soporte teórico y bastantes ejercicios cuyo grado de
dificultad es gradual.
Aunque parezca excesiva la cantidad de actividades relacionadas con
operaciones algebraicas, es necesaria para afianzar su seguridad operando y
posteriormente garantizar el correcto planteamiento de los problemas.
5

SECUENCIACION DE LAS SESIONES. MATERIALES CURRICULARES DE
CADA SESION.
La unidad didáctica se compone de un total de 14 sesiones lectivas distribuidas
así:
Session number Session title Activities
Session 1
Brainstorm.Using Activity 1
algebra.
Writing algebraic
expressions
Session 2 Words used in Activities 2 and 3
algebra.. Monomials.
Parts of a monomial..
Session 3 Operations with Activities 4 and 5
monomial.
. Operations.(addition
and subtraction)
Session 4 Operations with Activities 6 and 7
monomial. Operations
(multiplication and
division)
Session 5 Substitution Activity 8 Game 2
Session 6 Substitution Activity 8
Game 3
Session 7
Polynomials.
To read the theory
Activity 9
Identifying polynomials Exercise 1
and its parts.
Session 8
. Identifying polynomials Activity 9
and its parts.
Exercises 2 and 3
Session 9
Classifying polynomial Activity 10
Simplifying algebraic Activity 11
expressions
Session 10 Simplifying algebraic Activity 11
expressions
To read the theory
Expanding brackets Activity 12
Exercises 1 and 2
Session 11
Expanding brackets
Operations
polynomials.
with
Session12 . Operations with
polynomials.
Perfect Squares.
Activity 12
Exercises 3 and 4
Activity 13
Exercises 1 to 4
Activity 13
Exercise 5
Exercise 6
Session 13 . Perfect Squares. Activity 13
Exercise 7 to 10
Session 14 Multiplying polynomials Activity 14
6

ACTIVIDADES
Using algebra
Algebra is an area of mathematics where letters are used to represent numbers.
Activity 1
1.- Brainstorm .-The students will look for some Spanish word
related with algebra and will talk about the importance of
other cultures.
2.- They will think about the different algebraic expressions they already know:
-Generalizing the evolution of a numerical series.
Example: 1,3,5,7… n+2
-Expressing the relation between different magnitudes (formulae)
Example:
9 C
F = + 32
5
Distance=velocity x time
Area= l 2
( they will write different formulas)
-Solving mathematical problems.
Example 1.-(Students will conclude the sentences)
Peter has planted a certain number of sunflowers seeds, but you don´t know
how many. Let x stand for the unknown number. Peter has planted ……….
Example 2.-
Ann spent 4.5 € on a sandwich. Jane spent a on a salad.
How much did they spend in total? They spent 4.5+a in total.
Example 3.-
Mark has x CDs in his collection. Susan has y CDs in her collection.
How many CDs do they have altogether? They have ……. CDs in their
collections..
Example 4.-
Adrian has 14 sweets. He ate some of them. How many sweets did he have
left?
He has left.(They complete the sentences using algebraic expressions
and pronouns)
7

Writing algebraic expressions
An algebraic expression is a mathematical phrase which contains numbers,
operators, (add, subtract, multiply, divide), and at least one variable (like x, y).
Shorthand used in algebra:
a means 1xa or 1a or a 1
-a means -1xa or -1a or -a 1
4a
a
2
ab
abc
3ab
a 2
a 3
means 4xa or (a+a+a+a)
1
means a ÷ 2 or a
2
means axb
means axbxc
means 3xaxb or (ab+ab+ab)
means axa
means axaxa
5a 2 means 5xaxa or (a 2 + a 2 + a 2 + a 2 + a 2 )
(3a) 2
means 3ax3a or 3x3xaxa
4a 2 b 3 3
means 4xaxaxbxb or (a 2 3
b + a 2 3
b + a 2 3
b + a 2 3
b + a 2
b )
Activity 2
1. - Students will try to write the following expressions:
4 less than p is p-4
9 more than x is x+9
A number x increased by 8 is x+8
The sum of a number b and 3 is b+3
A number q decreased by 5 is q-5.
Six times x is 6x
The triple of m 3m is 3x
Twice y is 2y
The double of r is 2r
x
A quarter of x is 4
8

Half m is 2
m
6x
Six sevenths of x is or 7
A fifth of t is 5
t
6 x
7
20 1 x
20 % of x is x or x or ( remember to simplify )
100 5 5
m times m is mxm or m 2
p times q is pxq or pq
five times t cubed is 5xtxtxt or is 5t 3
4
2t 2
v means 2xtxtxvxvxvxv
2.-They will do the previous exercise in reverse.
9

Words used in algebra
Variable: When letters are used in place of different numbers they are called
variables , unknowns or indeterminates..
Constant : Anything that has a fixed value ( can´t be changed ) is called a c
constant. 5, -4, 2
1
are constants, because their values do not change.
Term: A Term is a single unit containing one or more variables, often with
a constant in front or a constant on its own.
2
a
5x, 3ab, 4x y,
− pq,
7and
are examples of terms.
b
Coefficient: The number in front of a term is called the coefficient of the term.
In the term 5x, 5 is the coefficient of x.
In the term -4pq, -4 is the coefficient of pq.
In the term –y, -1 is the coefficient of y ( as –y=-1y).
Expression: A collection of terms separated by plus signs or minus signs is called
an expression.
2
3x −2x + 4 is an expression with three terms.
Like terms: Terms that use the same letter or arrangement of letters, are called “like
terms” . The only difference is the coefficient (number in front) of the term.
3x, 2x and –x are like terms.
5ab,4ab and –ab are like terms
8 x 2
y , 3 x 2 y and -x 2 y are like terms.
(The powers of each letter must be the same)
Unlike terms: Terms that are not the same.
3a and 5b are unlike terms.
4xy and 4x are unlike terms.
x 2 y and xy 2 are unlike terms.
Note : The letters in a term are usually written in alphabetical order.
For example, we would write 4ab rather than 4ba
However. the order of the letters within the term is not important.
10

Identifying monomials and parts of a monomial
Monomial.- An algebraic expression consisting of only one term
The exponent on a term tells you the grade or the"degree" of the term ( never a
negative whole number)
The degree of monomial is the sum of the exponents of the variables in the
monomial.
5 degree zero.
4x degree one.
3x 2 degree two.
4x 5
degree five
xyzp degree four
4xyz degree three
Activity 3
1.-Write several examples of monomials
2.-Fill lin the gaps:
Monomial 6x -2x a b
coefficient
Variable
2 4
2
3 xy
x 8
1
4
ab
Grade
3.- Complete:
The grade of a monomial is….
In the monomial 3x 4 , 3 is …
A negative number never appears in …
11

Operations with monomials. Combining terms
You can simplify expressions by adding o subtracting like terms.(Term with the
same variable)
3b and 5b are like term, so they can be added.
If a variable has no number in front of it, the value of that variable is 1
8xy 10xy have the same variable xy , so we add them.
6t and 4v are unlike terms as they have different variables. So, you cannot
subtract to simplify this expression.
It is often best to group like terms together first, and then simplify:
2x 2 + 3x – 4 – x 2 + x + 9 = (2x 2 – x 2 ) + (3x + x) + (–4 + 9) = x 2 + 4x + 5
Terms can be combined ONLY IF they have the exact same variable part
Monomials that differ only their numerical coefficients are said to be like terms.
12

Activity 4
1.- Fill in the gaps:
4x and 3
4x and 3y
Not like terms
The second term has no
variable.
4x and 3x 2 The second term has the
same variable, but
different degree.
4x and 4y
4x and 3x
2.-Group like terms. Write each expression in the correct box.
2,5 a 3 b
-6ab 3
Now the variables match
and the degrees match
4
ab 2
a
3
b 4
2a
2ab
3
b 4
2
b 2
4
5ab
a 2 ab
6cab 2 a
9ab
4a
2
b 2
3
b 4
15a
3
b 4
13

Addition and subtraction
Activity 5
1-Simplify the following:
1) b + b + b + b 9) b-5b
2) a + 3s 10) 4a-8a-6a-a
3) 9w – 2w 11) 4a-6b+a
4) 4r + 7r 12) 3x-5x-x
5) d + 2d + 5d 13) 5b-4g+4b-6b
6) 4t + 3t – 5t 14) 9g-7g-4g
7) 9a – a – 5a 15) 6m-m+9b-5b-4m
8) 6b-7b-9b 16) 8f+5y-2f-4y+3fy
2.- Complete:
1) 9a - 3a = 10 ) 5x + x - 8x = 19) 9a 2 - 3a 2 =
2) 11x + 3x = 11) -9x - 3x = 20) 4zx + zx - 9zx =
3) 4x + x = 12) x + 3 - 2x +1 = 21) a 4 b - 12a 4 b - 2a 4 b =
4) 4z + 3z -7z = 13) -7x -10x + x = 22) -y 2 - 5y 2 =
5) 3b - 6b - 2b = 14) -7x -2x - 9x =
1
23) 2x+ x =
3
6) 2b - 5b = 15) 3x 2 - 3 - 12x 2 + 2 =
2
24) x- x =
7) ab - 3ab = 16) x 2 + 2x 2 + 10x -19x -8 +3
=
8) 3a - 5b + 3a - 6b = 17) -5x 3 -7 - 3x 3 +3 =
9) 10x + 3x - 6x = 18 ) -2x 3 + 4x 3 - 10x - 11 =
3
2 1
25) x- x + x =
3 5
2 2 2 3 2
26) x − x + x =
3 4
3 3 3 2 3
27) x − x − x =
5 25
14

Multiplication
When multiplying terms, remember the following steps:
1) First multiply the numbers.
2) Then multiply the variables (the powers property)
3) Write the numeral first followed by the variable.
4) Write the variables, or letter, in alphabetical order.
Warning: Don't get careless and confuse multiplication and addition. This may
sound like a silly thing to say, but it is the most commonly-made mistake (after
confusing the order of operations):
(x)(x) = x 2
(multiplication)
x + x = 2x (addition)
" x 2 " DOES NOT EQUAL " 2x "
For example if you have something like x 3 + x 2 , DO NOT try to say that it is
equal to x 5 or 5x. If you have something like 2x + x, DO NOT say that this is
equal to 2x 2 .
When multiplying the variable x by another term, it´s very easy to confuse ×(
multiplication for ) with x (letter). So, leave out the multiplication sign. For
example, write 4xa as 4a.
7ax6b= 42ab
7x6=42. Then multiply the variables axb, so 7ax6b=42ab
8 ab 3 3a 2 b 5 =24a 3 b 8
Activity 6
1.- Complete:
4x 5x = 2e x 3 3a 2 x 4a
x (-3x) = 3d x 4d 2d x 5d
9x (-10x 5 ) = 2w x 5w 4b x 6b
-4x 3 (- 9x) = 5 x 3d 3c x 6c
3ab 3a = g x 7g 3g x 2g 2
(-3x) 6x 2 = 2 x 5d 2t x 3t 4
4 ab(-9)ab 2 = 2q x 7m 3mx4mzp
5z 6z(-4z)= 6 x 4b 2mxmx5p
20a 4 b 2 5 a 5 b= 8q x 2g 4hx8kx2p
-9rst10r 2 s 2 t 2 = 9s x 5 3mx2mxm
2.- Complete ( Remember how to multiply fractions):
2
4 2
18x. x
( )
=
abc − 4 ab =
3
− 2 2
6x x =
3 − 2
ab 25a
b
3 =
3
5
15

Division of monomials
Divide top and bottom by common factors.
The method we use to simplify algebraic fractions is the same that we use to
simplify ordinary fractions
10ab
=5a ( divide the top and bottom by the same monomial, 2b )
2b
4z
1
= (This is an algebraic expression but not a monomial, because the
2
8z
2z
variable is not elevated to a positive number)
Activity 7
1.- Simplify each of the following:
15ab
1)
3b
81x
2) 27
14ab
3)
2ab
18b
4)
9
2.- Complete::
1) 6x 5 : 2x =
5)
6)
6pq
2pq
14xy
7xy
3
x
7)
3
x
4ab
8)
4ab
20x 2 2 2
2 3
y 18a
b ab c
9)
13) 17)
2
2
xy
9b
4ab
3
3
64a
b 4a
b
10) 14) 18)
2
2
8a
8a
11)
12)
12x
2 y
15)
4x
2
6
24ab
a b
16)
8ab
4ab
2ab
2 19)
8ab
2
14xy
z
7xy
2 3
12x
my
9xmy
2
10x
y
20)
2
4xy
2) 4x 4 : (-2x) =
3) -15x 2 : (-5x) =
4) -21x 6 : (- 7x) =
5) 20a 4 b :10 a =
6) (-72x 9 ) : 4x 2 =
7) 15a 5 b 7 :(-3) a 3 b 3 =
8) 125z 8 : 5z 6 =
16

Evaluating expressions (substitution)
In algebra when we replace letters with numbers when evaluation expressions
we call it substitution. When you are substituting numbers in an expression, it is
good idea to put a bracket around the number that replaces the letter.
Activity 8
1.- Substitute the values for each letter and use the rules of operations to help
you work out the answers to the questions below:
a=3 b = 2 c = 1 d = 8 e = 0 f = 9 g = 10
1) cg=
2) bc + 7=
3) 5d +a-g=
4) 2c 2 -e=
5) f ÷ a + g + ce=
6) g 2 - bc=
7) (b + c) 2 +2d=
8)
2(a + b) × (2f – 5) =
17

2.- GAME 3 in a line
Rules:
Take turns throwing the dice. Put a counter on a square which matches your
dice score. The first person to get 3 in a line is the winner.
Substitution
d + 3 = 8 7 – d = 4 10 ÷ d = 5 4 + d = 5
d x 3 = 9 2 x d = 12 d ÷ 4 = 1 d x d = 4
d x 2 = 10 10 – d = 9 d – 3 = 3 d x d = 16
d - 2 = 4 d x d = 25 9 - d = 6 8 ÷ d = 2
4a – 11 = 9 5(4 - t) = 10 2 g = 8 2 m 2 = 18
18

2u – 1 = u d x 2 = 8 6( 9 - w) = 24 6z + 1 = 5z + 4
4y = y + 3 7– a = 0.5
2
8 = 4(3y – 4) 3 (b – 1) = 6
d + 4 = 7 d x d = 1 6 + d = 7 d ÷ 2 = 3
2 d – 1 = d d 2 = 25 (h 2 -4)=9 5 x (d – 2) = 5
d – 1 = 4 5 d = 10 d – 1 = 5 4 d = 24
3 (d – 1) = 9 3 + d = 8 5 – d = 4 10 – d = 6
19

3.- Game
Take turns rolling the dice. Substitute the number on the dice for the letter in the
formula. Complete the operation, get the solution and move forward that
number of squares. The winner is the first person to go twice around the board
and reach the chequered flag.
20

Polynomial
A polynomial is an expression constructed from variables (also known as
indeterminates) and constants, using the operations of addition,
subtraction, multiplication, and constant non-negative whole number
exponents
This year we will work with univariate polynomials
When a term contains both a number and a variable, the number is called the
"coefficient". The coefficient on the leading term is called the "leading"
coefficient.
In the above example, the coefficient of the leading term is 4; the coefficient of
the second term is 3 and -7 is the constant term.
The highest power in a one-variable polynomial is called its order, or degree.
For instance, the leading term in the above polynomial is a "second-degree
term" or "a term of degree two". The second term is a "first degree" term. The
degree of the leading term tells you the degree of the whole polynomial; the
polynomial above is a "second-degree polynomial".
Polynomials are usually written this way, with the terms written in "decreasing"
order; that is, with the largest exponent first, the second highest next, and so
forth, until you get down to the number which is alone (the constant).
Some polynomials don´t have a constant.
Notice the exponents on the terms. The first term has an exponent of 2; the
second term has an "understood" exponent of 1; and the last term doesn't have
any variable at all. Any term that doesn't have a variable in it is called a
"constant" term because, no matter what value you may put in for the variable x,
that constant term will never change. In the picture above, no matter what x
might be, 7 will always be just 7.
21

The first term in the polynomial, when it is written in decreasing order, is also
the term with the biggest exponent, and is called the "leading term".
The exponent on a term tells you the "degree" of the term. Here are a couple
more examples:
The "poly" in "polynomial" means "many”. The term "polynomial" should only
refer to sums of many terms, but the term is used to refer to anything from one
term to the sum of a zillion terms. However, the shorter polynomials do have
their own names: monomial, binomial trinomial.
• a one-term polynomial, such as 2x or 4x 2 , may also be called a
"monomial" ("mono" meaning "one")
• a two-term polynomial, such as 2x + y or x 2 – 4, may also be called a
"binomial" ("bi" meaning "two")
• a three-term polynomial, such as 2x + y + z or x 4 + 4x 2 – 4, may also be
called a "trinomial" ("tri" meaning "three")
3
4
x − + 7x
2
is not a polynomial, because its second term involves division by
x
3 2
the variable x and also because its third term contains an exponent that is not a
whole number.
22

Identifying polynomial and its parts
Activity 9
1.- Complete:
Polynomial
Terms
2x 5 – 5x 3 – 10x
+9
10x 3 – 14x 2 + 3x
22– x 2 – x
Leading term
Leading coefficient
Constant
Degree
2.- Complete:
• 2x 5 – 5x 3 – 10x + 9
This polynomial has ………… ..terms, including a
…………………………………………term, a third-degree term, a
………………………………….term, and a constant term. This is a …….….-
degree polynomial.
• 7x 4 + 6x 2 + x
This polynomial has …………..terms, including a
-degree term, a
-degree term, and a first-degree term. There is no ……………………….
3.- Try to summarize and explain the parts of a polynomial.
23

Classifying polynomial
Activity 10
1.- Classify each expression as monomial, binomial or trinomial. Complete the
sentences:
• . – 5x 3 – 10x + 9 is a ……………… because it has
………………………………
• 10x + 8 is
……………………………………………………………………….
• 3x 2 − 2x
+ 1
is a
…………………………………………………………………….
• 45ª is a
……………………………………………………………………………….
• 34d 6 is a
……………………………………………………………………………….
• 9-x+2+3x is
a…………………………………………………………………………….
• 10x 3 – 14x 2 + 3x is
a……………………………………………………………………
2.- Match each of the algebraic expressions with their names.
2x-14 trinomial
3x 2 − 2x
+ 1
other polynomial
3 + 2
x
2x-5ax+8xy+4
4x-5x+9x
3x- 2 x
monomial
binomial
none of those.
24

Simplifying algebraic expressions06-2008 All Rights Reserved
Activity 11 (Students will see these examples before working on their own)
Example 1: 4x 2 + 5x – 8 – x 2 + x + 9
Remember that there is 1 in front of a variable expression with no written
coefficient, as is shown in red below:
(4x 2 – x 2 )+(5x + x) + (–8 + 9) = (4x 2 – 1x 2 ) + (5x + 1x) + (–8 + 9) = 3x 2 + 6x + 1
Example 2: 10x 3 – 14x 2 + 3x – 4x 3 + 4x – 6
10x 3 – 14x 2 + 3x – 4x 3 + 4x– 6 =(10x 3 –4x 3 )+(–14x 2 )+(3x+4x)–6=6x 3 –14x 2 +7x–6
Warning: When moving the terms around, remember that the terms' signs move
with them. Don't confuse yourself by leaving orphaned "plus" and "minus" signs
behind.
Example 3: 25 – (x + 3 – x 2 )
The first thing you need to do is to take the negative through the parentheses:
25 – (x + 3 – x 2 ) = 25 – x – 3 + x 2 = x 2 – x + 25 – 3 = x 2 – x + 22
Example 4: x + 2(x – [3x – 8] + 3)
This is just an order of operations problem with a variable in it. If you work
carefully from the inside out, paying careful attention to your "minus" signs, then
you should be fine:
x + 2(x – [3x – 8] + 3) = x + 2(x – [3x – 8] + 3 = x + 2(x – 3x + 8 + 3) =
x + 2(–2x + 11) = x – 4x + 22 = –3x + 22
Simplify:
1.- [(5x – 4) – 2x] – [(10x – 7) – (3x – 8)]
2.- –4y – [3x + (3y – 2x + {2y – 7} ) – 4x + 5]
3.-
x x − 2 x
− − 1 +
2 5 10
4.- Students will write different expressions to simplify in their notebooks.
25

Expanding brackets
In algebra, many expressions include brackets. A term ( number or variable)
outside the brackets means that you must multiply each term inside the
brackets by the outside term.
To expand brackets we must multiply each term in one bracket by each term in
the
other and then join the like terms
You can check brackets expansions by comparing them to areas. Calculate the
area of each part of the rectangle and add both areas.
5(x+y) = 5• (x+y)
5(X+Y) = 5X+5Y.
Bracketed expressions
Sometimes you will need to multiply bracketed expressions by each other.
For example: (a+b)(c+d)
This means (a+b) multiplied by (c+d).
Look at the rectangles:
The area of the whole rectangle is (a+b)(c+d)
It is the same as the sum of the four separate areas
So: (a+b)(c+d)=ac+ad+bc+bd
Notice that each term in the first bracket is multiplied by each term in the
second bracket.
You can also think of the area of the rectangle as the sum of two separate
parts:
(a+b)(c+d)= a(c+d)+b(c+d)
Think of multiplying each term in
the first bracket by the whole
of the second bracket.
This is called multiplying out
the brackets.
26

Activity 12
1.-Expand the brackets:
1.).4(x-y)
2) 8c(d + e).
3.) m (4x - 2y - 6z).
4) 2x(x-y)
2.- Expand the brackets and simplify :
1) 4x(2x-3)-4(3x-2)
2) x(3x-1)+2x(x-3)
3) 5c(c-2)-4a(2a-5)
4) x 2 (x-3x 3 2
)-2x(x − 3x
)
5) abc(a 2 -bc)+a 2 (a-b)
6) xyz(z-y)+xz(y-x)
3.-Remove the brackets and simplify:
1) (x + 8) · (x + 2) =
2) (x + 4) · (x - 1) =
3) (x - 6) · (x - 2) =
4) (x - 1) · (x - 3) =
5) (x - 5) · (x - 1) =
=
4.- Draw different coloured rectangles showing what you have done in the
previous exercise.
27

Operations with polynomials
Activity 13
1.- Complete these operation squares:
+ 2c+3d 8c+2d
4c+5d
3c+d
+ -c-3d -3c+d
5c+4d
c-4d
2.- Subtract binomials in the column from the ones in the row.
- c+3d 8c+2d
4c+5d
3c+d
-
a 2 -ab
a 2 -10ab
-4a 2 -ab
3a 2 +ab
3- Fill in the gaps:
+
5a+8b
4a-b
7a+5b
-6a+9b
+
5a 2 +8ab
2a 2 +10ab
6a 2 -ab
3a 2 -ab
4.-Subtract binomials in the column from the ones in the row
-
8t-3u
15t-u
5t-4u
2t+6u
28

5.- Complete:
x 2c+3d 8c+2d
4c+5d
3c+d
x -c-2d 4c-2d
-c+5d
3c-d
Special identities:
Examples:
( 2x + 3 ) 2 = 4x 2 +12x +9
( x + 7) · (x –7 )= x 2 – 49
(x-4) 2 = x 2 –8x+16
Remember:
(a+b) 2 = (a + b)·(a + b) = a 2 + ab + ba + b 2 = a 2 + 2ab + b 2
(a-b) 2 = (a -b )·(a –b ) = a 2 – ab –ba + b 2 = a 2 - 2ab + b 2
(a+b)·(a-b)= a 2 –ab + ba –b 2 = a 2 –b 2
6.- Calculate:
x a+2 a-2
a-2
a+2
7- Calculate:
a) (2x+1) 2 =
f) (2x+5y) 2 =
b) (3a-2b) 2 g) ( 3x
− 2)( 3x
+ 1)=
=
c) (2-3x)(2+3x)= 3
3
h) x − 3
x + 3
=
2
2
d) (3-5x) 2 2
=
4x 3
i) + =
3 4
e) (2a-3b)(2a+3b)=
5
5
j) x − 2y
x + 2y
=
2
2
29

8.-Complete:
2
2
a) x + 2xy
+ y = ( + ) 2 2
c) b − 2b
+ 1 = ( - ) 2
b) 25x 2 + 10x
+ 1 = ( + ) 2 2
d) x − 16 = (x+4) ( - )
9.- Simplify:
2
x − 9
a) =
2
x − 6x
+ 9
x − 2
b) =
2
x − 4x
+ 4
10.- Factorize and simplify:
4x
+ 4y
a) =
2
x + xy
3
m
b) =
2 3
m + m
9a
2 + 6a
+ 1
c) =
3a
+ 1
2x
+ 3
d) =
2
4x
− 9
2
x − xy
c) =
2 2
x − y
4x
d) =
4x
+ 8y
30

Multiplying polynomial
The rules are the same as in the multiplication of binomials.
Multiply each term in the first expression by the whole of the second expression.
It´s sometimes is useful to do it in the same way as arithmetic multiplication
Example:
11.- Multiply:
3
2
x − 5x
+ 1 x + 2
1) ( )( ) =
x
x
x
2
2
2) ( 2 − 5 + 1)( 3 − 2) =
x
x
x
3 2
3
3) ( 4 − 6 − 3)( 4 − 5) =
x
x
x
3
2
4) ( 2 − 5 + 1)( 3 − 4) =
31

COMPETENCIAS BASICAS QUE SE TRABAJAN EN LA UNIDAD
DIDACTICA
En esta unidad se trabajan las siguientes competencias básicas:
Competencia matemática.-
La competencia matemática es evidente. Se trabaja el uso de las operaciones
básicas con cálculos mentales y de operaciones combinadas.
Competencia en comunicación lingüística.
Se trabaja a través de la lectura comprensiva de los problemas, de los que los
alumnos deberán extraer los datos que les conduzcan a su planteamiento y
posterior resolución.
En general, en la realización de todas las actividades ( con lectura e
interpretación de la información por parte de cada alumno de los ejercicios que
le toque hacer) se fomenta el desarrollo de esta competencia.
Competencia en el conocimiento y la interacción con el mundo físico.
En la primera actividad los alumnos buscarán palabras de origen árabe y
recogerán información sobre la gran aportación de los árabes a las
matemáticas.
Tratamiento de la información y competencia digital.
Se les propone algunas páginas web para que realicen ejercicios interactivos
on line.
Competencia social y ciudadana
Esta competencia se potenciara a través de actividades en grupos pequeños y
también de la interpretación y análisis de las expresiones matemáticas tratando
de que desarrollen una actitud crítica.
Competencia artística
Con la elaboración de sus materiales en los juegos, en las balanzas de
ecuaciones, tales como el mural para la clase ( hechos en cartulina a los que
pueden añadir dibujos)
Competencia para aprender a aprender
La lectura de los , el desafío de su solución, el uso métodos intuitivos y
deductivos, el ensayo error, el análisis crítico de los resultados les permite
desarrollar y consolidar hábitos de disciplina, estudio y trabajo individual y en
equipo.
Competencia autonomía e iniciativa personal
Ser conscientes de su mejora en el aprendizaje les lleva a manifestar una
actitud positiva ante la resolución de problemas y mostrar confianza en su
propia capacidad para enfrentarse a ellos con éxito y adquirir un nivel de
autoestima adecuado que le permita disfrutar de los aspectos creativos,
manipulativas, estéticos y utilitarios de las matemáticas.
32

MATERIALES NECESARIOS
Fotocopias de las actividades para trabajar en clase.
Ordenador y cañón para visionar un video.
Útiles de dibujo: cartulinas, tijeras, pinturas, etc. ( lo hacen en la clase de
plástica) para construir sus tableros con expresiones algebraicas.
Calculadoras.
Conexión a Internet para visitar páginas web
EVALUACIÓN. CRITERIOS DE EVALUACIÓN
− Realizar operaciones con monomios y polinomios (suma, resta,
multiplicación,)
− Saber utilizar letras como representación de números;
− Trasladar situaciones reales a expresiones algebraicas.
− Resolver ecuaciones sencillas de primer grado con una incógnita.
− Valorar la precisión del lenguaje algebraico utilizado para expresar todo tipo
de informaciones que contengan cantidades, medidas, relaciones
numéricas.
− Planificar la estrategia de resolución del problema y utilizar tablas, gráficos,
esquemas o representaciones de tipo simbólico cuando se requiera
− Exponer, utilizando un lenguaje matemático preciso en forma oral o escrita,
los razonamientos y estrategias seguidas en la resolución, así como admitir
y valorar las de los demás.
CRITERIOS DE CALIFICACION.
Al final del proceso de evaluación se procede a la calificación de los alumnos,
es necesario tener fijados unos criterios de calificación claros y precisos, con
indicación del peso de los instrumentos de evaluación empleados.
Para la calificación se emplearán los procedimientos e instrumentos de
evaluación indicados en el apartado anterior, asignando, en la calificación, el
siguiente peso a cada uno:
Observación sistemática del 20%
alumno/a
Trabajo del alumno/a (individual o/y 10%
grupo)
Pruebas escritas 70%
En las pruebas escritas figurará la calificación correspondiente a cada uno de
los problemas y/o cuestiones que contenga.
En los problemas se valora:
Planteamiento 50%
25%
Resolución
Discusión 25%
En las cuestiones teóricas se valora:
33

Respuesta correcta 60%
Correcta expresión 10%
Razonamiento 30%
En las cuestiones prácticas se valora:
Desarrollo correcto 80%
Justificación 20%
BIBLIOGRAFIA:
− Libros y material utilizados en la elaboración de esta unidad didáctica.
− Revise GCSE . Mathematics. Letts.
− Bitesize revision. Maths. High Level. BBC.
− Scottish Secondary Mathematics. Heinemann
− GCSE Mathematics. Heinemann.
− Edco maths 2. Junior Certificate.
− 2005RevisionGuide.ks3Maths.Collins.
− Anaya 1º, 2º y 3º ESO.
− www.bbc.co.uk/schools/gcsebitesize/maths
34

EN EL AULA DE INGLÉS
COMPETENCIA EN COMUNICACIÓN LINGÜÍSTICA
• Adquisición de vocabulario referido a números (ordinales, cardinales,
fracciones), comparativos y superlativos de adjetivos, nombres contables
y no contables.
• Lectura comprensiva de problemas matemáticos.
• Expresión oral y escrita sobre problemas y expresiones matemáticas.
• Cuidado en la pronunciación, ritmo y entonación de sus producciones
orales.
ACTIVIDADES
Exercise 1(Vocabulary)
Write these expressions in figures:
• Six sevenths of x
• A fifth of t
• Eight times three is twenty-four.
• The double of h
• Sixteen fours are sixty-four
• A quarter of x
• Ten twos are twenty.
Exercise 2 (Reading)
Do you know how to say these expressions? Read them aloud.
• 4/5 x
• 1/7 x
• 3t
• 7/9 h
35

Exercise 3(Grammar)
Read these problems. Can you correct the mistakes?
• You can do two kilos of cheese with fourty-fourth
litres of milk. Peter have got eighty-sixth litres of milk.
How many cheese can he do?
• Maria drink four pot of teas a day. There is three
quarter of a litre in a teapot. ¿How much litres of tea
do Maria drink in four days?
Exercise 4(Speaking – oral interaction)
Can you work out the answers? Discuss with your classmate and explain how
to solve the problems.
Exercise 5 (Writing)
Invent another problem similar to the ones in exercise 3. Read it aloud.
36

Exercise 6 ( Speaking)
Compare all the problems. Work in pairs and decide which one is the easiest,
the most difficult, the most original… Vote for the best and tell the rest of your
partners why you have chosen that problem.
The most difficult The easiest The most original The best
37

Exercise 7
Write a problem using the words in the box
First marathon the two fifths my best friend and me
the double tired thirsty a pain in my right leg a bicycle
Self-evaluation
• What have you learnt?
• Was this unit interesting?
• Was it difficult?
38