AICOLE
Ana Rodríguez García
1. CONTEXTO
1.1 El centro
1.1.1 Identificación, situación, tipo, características principales, señas de identidad
El IES Cantabria es un centro público situado en Santander y que depende de la Consejería de
Educación de Cantabria. En el centro se pueden cursar los siguientes estudios: ESO, ESO bilingüe,
Bachillerato y varios grados de FP de la familia de sanidad.
No se ha podido acceder al Proyecto Educativo de Centro (PEC) en el que se presenta el ideario
específico del IES así como el tipo de alumnado que acoge (no se encuentra en su página web).
Sin embargo, a partir de la localización se puede esperar que el alumnado de ESO y Bachillerato
provenga de familias de clase media.
1.1.2 Programa de bilingüismo en el centro. Breve explicación
El IES Cantabria comenzó con el programa bilingüe del MEC-British Council en el año 2004 en 1º
de ESO y desde el curso 2007/2008 el programa se extiende a todos los cursos de la ESO. Se
puede escoger cursar ESO fuera del programa bilingüe. La mayor parte de los alumnos que
ingresan en 1º ESO provienen del CEIP Eloy Villanueva (Santander) que cuenta con el programa
bilingüe del MEC-British Council desde el curso 96/97.
Los alumnos que quieran entrar en el programa bilingüe en cualquiera de los cursos han de
demostrar el suficiente nivel de inglés. Como referencia para 1º de ESO debe demostrar un nivel
A2.
Las materias que se imparten en inglés en la ESO son: Ciencias de la naturaleza, Geografía e
Historia, música, Bilogía y Geología, Educación Física y Física y Química. La asignatura de Inglés
(5 horas semanales) sigue un currículo adaptado a su nivel competencial. Al finalizar la ESO, los
alumnos realizan las pruebas del I.G.C.S.E (International General Certificate of Secondary
Education).
El programa bilingüe no continua como tal en bachillerato, aunque en la asignatura de Inglés se
imparte un currículo adaptado al nivel de conocimiento de estos alumnos con 5 horas
semanales.
El IES cuenta con un profesor nativo, una asesora lingüística nativa en el ámbito científico y el
nivel mínimo exigido para impartir asignaturas en inglés en Cantabria es el B2. No se ha
encontrado información sobre la organización del departamento bilingüe.
1.2 Los alumnos
1.2.1 Edad, curso
Alumnos de 2º ESO (13-14 años).
1.2.2 Nivel de competencia L2
Los alumnos que entran en el centro en 1ºESO provienen en un 70% de una educación con un
programa bilingüe (MEC) desde infantil y poseen un nivel A2 alto. Al resto de alumnos que
ingresan en 1ºESO en sección bilingüe se les exige un nivel A2.
El alumnado de 2ºESO posee, tras el paso por 1ºESO, como mínimo un nivel A2+. Elementos del
currículo: objetivo, contenido, estándar
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Las sesiones propuestas pertenecen a la asignatura de Matemáticas ya que es la especialidad
que estoy cursando, aunque en el programa del centro esta materia se imparte en castellano.
El objetivo didáctico es reconocer el significado geométrico y aritmético del Teorema de
Pitágoras a través del inglés.
El contenido específico de las sesiones es el siguiente:
- Teorema de Pitágoras: significado geométrico y aritmético
- Aplicaciones del teorema de Pitágoras:
o Cálculo de un lado de un triángulo rectángulo conociendo los otros dos.
o Identificación de triángulos rectángulos a partir de las medidas de sus lados.
Los estándares de aprendizaje evaluables asociados a este contenido son:
- Dadas las longitudes de los lados de un triángulo, reconoce si es o no rectángulo.
- Calcula el lado desconocido de un triángulo rectángulo conocidos los otros dos.
- Aplica el teorema de Pitágoras en la resolución de problemas geométricos sencillos.
Estas sesiones contribuyen principalmente a la adquisición de la competencia matemática y
competencias básicas en ciencia y tecnología, competencia lingüística, así como a la conciencia
y expresiones culturales gracias a la historia y relevancia del teorema. Además, se escogen
actividades que favorezcan de forma transversal la competencia aprender a aprender, las
competencias sociales y cívicas, y la competencia sentido de iniciativa y espíritu emprendedor.
El contenido, criterios de evaluación y estándares de evaluación asociados a la unidad didáctica
completa en la que se encuentran las sesiones se muestran en la tabla 1.
Contenidos
- Triángulos rectángulos.
- El teorema de Pitágoras. Justificación geométrica y aplicaciones.
Criterios de evaluación
Estándares de aprendizaje evaluables
1. Reconocer el significado aritmético del Teorema de Pitágoras 1.1 Comprende los significados aritmético y geométrico
(cuadrados de números, ternas pitagóricas) y el significado
del Teorema de Pitágoras y los utiliza para la búsqueda
geométrico (áreas de cuadrados construidos sobre los lados) y de ternas pitagóricas o la comprobación del teorema
emplearlo para resolver problemas geométricos.
construyendo otros polígonos sobre los lados del
Se trata de comprobar el empleo del teorema de Pitágoras para triángulo rectángulo.
obtener medidas y comprobar relaciones entre figuras, así como 1.2 Aplica el teorema de Pitágoras para calcular
para resolver triángulos y áreas de polígonos regulares en
longitudes desconocidas en la resolución de triángulos y
diferentes contextos.
áreas de polígonos regulares, en contextos geométricos
- Competencia matemática
o en contextos reales.
- Conciencia y expresiones culturales
Tabla 1: Tabla adaptada del “Decreto 38/2015, de 22 de mayo, que establece el currículo de la Educación Secundaria
Obligatoria y del Bachillerato en la Comunidad Autónoma de Cantabria.”
2. CONTEXTUALIZACIÓN DE LA ACTIVIDAD DENTRO DE LA UNIDAD DIDÁCTICA
Las sesiones presentadas en este trabajo son las primeras de la Unidad Didáctica (UD) “Teorema
de Pitágoras”, que es asimismo la primera unidad del bloque de contenido de Geometría (Bloque
3) del Currículo de Cantabria (Decreto 38/215). Se trata por tanto de la primera toma de
contacto en la asignatura con la geometría desde el curso pasado.
La UD cuenta con un total de 6 sesiones de 55 minutos. Se presentan aquí las primeras 3 sesiones
destinadas a repasar los contenidos básicos necesarios vistos en el curso anterior (triángulos, su
clasificación y triángulos rectángulos) y a presentar el Teorema de Pitágoras con sus aplicaciones
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más inmediatas, estableciendo así los cimientos para poder desarrollar las siguientes sesiones.
También se plantea la tarea final de la UD.
Es la primera vez que se aborda el Teorema de Pitágoras, que será desarrollado y aplicado con
mayor profundidad en 3º y 4 ESO. El desarrollo de las sesiones se ha organizado teniendo en
cuenta la taxonomía de Bloom y las etapas del acto didáctico en matemáticas1i (comprender,
enunciar, memorizar y aplicar).
Así, las primeras actividades están enfocadas en recordar los conocimientos previos de
geometría y comunicación necesarios para adquirir el nuevo contenido (input). A continuación,
se busca provocar una experiencia en la que descubran el Teorema de Pitágoras mediante la
manipulación física del concepto (creación de figuras en 2D) y reflexiones individuales guiadas
con preguntas. Una vez “descubierto” el Teorema y con la comprensión del concepto, se enuncia
el Teorema de Pitágoras de forma rigurosa en lenguaje matemático.
En la segunda sesión, para adquirir e integrar el conocimiento, se proponen ejercicios que
desarrollan LOTS en los que se usa directamente el Teorema para las dos aplicaciones explicadas.
Los ejercicios del final de la sesión van aumentando en complejidad exigiendo mayor reflexión
y profundizando así en el conocimiento (HOTS).
La tercera sesión está dedicada a aplicar el Teorema en problemas contextualizados que exigen
evaluar determinar una estrategia de resolución del problema argumentada (HOTS). Los
alumnos comprueban la aplicabilidad de lo aprendido. Las sesiones 4 y 5 (no desarrolladas en
este trabajo) seguirían el mismo esquema que las sesiones 2 y 3, con otras aplicaciones del
Teorema en polígonos regulares y ejercicios en contextos reales.
La tarea final, en la sesión 6, es la de plantear, crear y resolver un problema propio que cumpla
unas ciertas condiciones.
En esta secuenciación están integrados los conocimientos lingüísticos a través de los elementos
de repaso y apuntes (ver anexo) 6+que recopilan las estructuras y términos claves, así como las
interacciones verbales en grupo y por pares, la lectura de un texto y el visionado de una canción.
1
Fernández Bravo, J.A. (2007). Números en color. Acción y reacción en la enseñanza-aprendizaje de la
matemática. Madrid. Editorial CCS.
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3. LEARNING OUTCOMES
The aim of the sessions is to study the Pythagorean Theorem (or Pythagoras’ Theorem, PT)
through the medium of English.
The teaching objectives are:
- To understand the Pythagorean Theorem (geometrically and arithmetically)
- To use the Pythagorean Theorem for finding missing sides in a right triangle
- To identify a right triangle from the measurements of its sides
The learning outcomes: By the end of these sessions, learners will be able to:
- Identify the sides of a right triangle: hypotenuse and short sides
- Determine if a triangle is a right triangle given the side lengths.
- Find the hypotenuse in a right triangle given the short sides
- Find a short side in a right triangle given the other two sides
- Apply the Pythagorean Theorem to solve simple problems in real contexts
3.1
Content
The content:
- Elements of a right triangle: how to label and names of the sides (hypotenuse and short
sides)
- Pythagorean Theorem
- Identification of right triangles from the measurements of their sides (the converse of
the Pythagorean Theorem)
- Theorem’s application to find the missing side of a right triangle given the other sides
3.2 Cognitive
The structure of the sessions (activities and their sequencing) aims to encourage these thinking
skills:
-
understand the relationship between the areas (and lengths) of the squares on a right
triangle
understand the logic of proving a theorem
problem-solving using already known concepts
check properties and conditions before applying formulas
discuss and express ideas
create a problem of their own
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3.3 Communication
Although every subject has a specialized vocabulary, Mathematics could be considered as a
language on their own. Learners need both old and new specific vocabulary and phrases for the
sessions.
The objective of the two first activities is to refresh key terms and phrases and introduce some
new ones. The rest of the activities and explanations focus also on reinforcing and developing
this knowledge. Discussions between pairs and the whole class are also planned for improving
speaking skills. Besides, there is a reading focused activity and some listening activities. Learners
are also asked to justify their answers with words and not only numbers.
3.3.1 Key terms
Two types of key terms are distinguished: the ones already seen in previous sessions and the
new ones.
- Nouns/adjectives:
New:
 hypotenuse, leg, short side
Already seen:
 triangle, vertex, degree, angle, acute angle, obtuse angle, right angle, right triangle,
isosceles triangle, equilateral triangle, base, height/altitude, square root, square,
perpendicular, length, x squared
- Verbs:
 to check
 to test
 to substitute
 to calculate
 to work out
 to solve for / to find
 to sum
 to measure
 to make up
 to prove/ to demonstrate
- Phrases:
 X is equal to Y / X equals Y
 X is no equal to Y / X doesn’t equal Y
 to square a number
 to find the square root of a number
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3.3.2 Structures
The main structures needed for the lessons are indicated below. Most of then should have been
seen in Maths and English classes before the lesson. Nevertheless, the activities are conceived
to keep working on them and help with their use.
-
Present simple: to talk about the properties, elements, and classification of triangles.
Present perfect: to talk about what they have done/understood/listened to in an activity
Question forms:
1st type conditional to make hypothesis: “If a triangle has a right angle then it is a right
triangle”
Comparatives “this angle measures more than 90⁰”, “b side is bigger/shorter than c side”
Superlatives: “the hypothenuse is the longest side in a right triangle”
Future tenses (debates and problem creating)
3.3.3 Language skills
The basic language skills are contemplated in the lesson’s plan: reading, writing, listening and
speaking. More specifically learners will work on:
-
Expressing their doubts and questions about the content
Making hypothesis
Expressing mathematical answers precisely
Understanding instructions
Understanding the situation of the problems
3.4 Context/ Culture/ Community
The Pythagorean Theorem is one of the most famous theorems in Mathematics and it represents
also an indispensable cultural knowledge in our society. Some of the history of Pythagoras and
the Theorem will be learned as well as the importance of its applications to solve everyday
problems.
4. CLIL ACTIVITY
4.1 Final task in mind. Brief explanation
At the end of the unit (here we only develop the first three sessions) students are asked to create
individually their own problem in which the Pythagorean Theorem is needed. They will have
already seen multiple examples of how the Theorem is used in real life.
The problem must be typed, use appropriate mathematical vocabulary, include units of
measurement and show how the Pythagorean Theorem is used in everyday life. The solution
with all the steps must be provided as well. A model 2D or 3D must be created to illustrate either
the problem formulation or the solution. Finally, students must write a reflection on the task.
4.2 Activities
The table below shows the activities programmed for the three sessions (brief explanation, type
of grouping and language skills involved). The details of each activity are provided in the annexe.
The activities are sequenced to facilitate scaffolding.
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Type of grouping
and language
skills*
L, S
Introduction. Show students photos and/or objects with triangular shapes. Ask students different Whole
questions and have a class discussion about what they remember/know about triangles and class
1st Session: Review, right triangles, and PT's demonstration
Activity 1
polygons. Possible questions may be:
- Which geometric shapes have in common these two images?
- What do you know about triangles? How many sides do they have? And angles? And vertex?
- Are they any differences or similarities between the triangles?
Activity 2 Vocabulary and concept review. Give students the worksheet to review triangles, key vocabulary,
and structures. The worksheet (see annexe) contains shapes labelled, a gap-fill theory exercise and
a review of concepts needed as input for new knowledge (PT).
Activity 3 Discovering the mystery of right triangles. Tell students that right triangles hide a mystery and
that they are going to discover it. Give students the worksheet with the instructions to follow to
construct the elements needed to prove geometrically the PT (without mentioning PT).
Provide also oral instructions if needed and prepare visual material to show if the worksheet
images are not enough.
The skills needed to construct the triangles and squares should have been taught in Design class
(coordination with design teacher needed). They will work in pairs and at the end of the exercise
some questions are asked to help with the reflection: e-g.
- Which is the area of …. and……?
- Based on your previous answers, can you find a relationship between the areas of the a,b,c
squares?
Activity 4 Discussion about the discoveries. Ask students to share their discoveries, doubts and reflections.
The worksheet provides sentences to help them express the mathematical ideaTeacher's PT explanation. Explain the PT on the blackboard and with visual support. Give students the
worksheet with theory.
Activity 5 Who was Pythagoras? Give learners the worksheet with an adapted text about Pythagoras and his
wife Theano. will read a short text about the history of the PT and will answer some questions.
2nd Session: PT's applications
Possible doubts about the last class will be discussed. Ask students if they have any questions or doubts
about the last lesson and correct text questions.
Activity 6 Pythagorean Theorem's song. Tell students about how universally know is PT and show them a
song about PT. It's a song made for young learners, easy to follow and it will help also with the
pronunciation of the new vocabulary. Play it twice and the second time provides them with the
lyrics so they can read (and sing if they want). Ask their opinion about the song.
Teacher's explanation. Explain the applications of PT with visual support if needed. Give the students the
worksheet with the activities 7-10.
Activity 7 Testing if a triangle is right-angled. Simple exercises with an example given.
Activity 8 Find the missing side (hypothenuse). Basic exercises with an example given.
Activity 9 Find the missing side (leg/short side). Basic exercises. In this exercise there is no example,
learners will need to apply the same reasoning to solve the unknown side. If needed, an example
will be explained on the board.
Activity
Mixed exercises. With these exercises, learners will need to apply the content from previous
10
activities, but complex reasoning is needed (not just applying formulas). Some of the exercises
are to be done for homework
3rd Session: Using PT to solve real problems and make up new ones
Possible doubts about the last class will be discussed. Ask students if they have any questions or doubts
about the last lesson and correct activity 10.
Activity Solving real problems. Give students the worksheet with problems in real contexts where the PT
11
is needed with increasing difficulty. They will work in pairs
+ 3 more sessions where the PT is used to solve problems with other regular polygons and more
real-life problems.
Individual R,W
Pairs
L, R,
S,W
Groups
L,S
Whole
class
L
Individual R,W
Whole
class
L, S
Whole
class
L
L
Pairs
Pairs
Pairs
R,W
R,W
R,W
Individual
Whole
class
L, S
Pairs
R,W
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Final task: Making up a problem. Students have to make up their own problem on the condition
that the PT has to be used. The instructions for the project are:
- The problem must be typed, use appropriate mathematical vocabulary, include units of
measurement and show how the Pythagorean Theorem is used in everyday life.
- The solution with all the steps must be provided as well.
- A model 2D or 3D must be created to illustrate either the problem formulation or the solution.
- A final reflection is also compulsory
Students will be given the evaluation grid.
* L = listening, R= reading, W = writing and S = speaking
Individual R,W
4.3 Materials/Aids
The main material needed is:
-
-
Computer + projector + loudspeaker
Blackboard
Colour paper + scissors
Worksheets (see annexe)
Audiovisual material:
o The Pythagorean Theorem’s song (https://www.youtube.com/watch?v=l8bnZh8Zuc)
o Pythagorean Theorem proofs
(https://www.youtube.com/watch?v=CAkMUdeB06o,
https://www.youtube.com/watch?v=uOTs2ck1_jU)
Teaching resources sites on the internet, e.g:
o http://www.shodor.org/interactivate/lessons/
o https://nrich.maths.org/
4.4 Cooperation
4.4.1 English teacher
The required linguistic knowledge is defined in chapter 4.3. Cooperation between Maths teacher
and English teacher is essential and must take place throughout the whole academic year.
Some weeks before starting the unit, the lessons’ plan is shared and discussed with the English
teacher. It will help the Maths teacher to get a feedback, and adapt the content if necessary,
and English teacher to identify if some of the linguistic knowledge (other than the specific
mathematical vocabulary) needs to be reinforced.
For these lessons, in particular, the collaboration of the English teacher will be requested to
prepare vocabulary for the reading of Pythagoras text and the song.
4.4.2 Cross-curricular aspects
Design class: Activity 3 requires drawing right triangles and squares using square, bevel and/or
compass. Students will need to use skills developed in the design class and cooperation with
Design teacher is needed to assure this content is worked before the lesson
History class: Students read about Pythagoras in Ancient Greek. Cooperation with History class
teacher would be positive.
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5. EVALUATION
5.1 Criteria
The assessment criteria are:
-
Identify the sides of a right triangle: hypotenuse and short sides.
Determine if a triangle is a right triangle given the side lengths.
Apply the geometric interpretation of the Pythagorean theorem.
Find the length of the missing side of a right triangle given the other two sides.
Apply the Pythagorean Theorem to solve simple problems in real contexts.
Apply mathematical vocabulary to describe problem resolution processes
5.2 Instruments
This unit’s assessment will be specified through the following instruments:
-
-
Initial diagnostic assessment to identify previous knowledge (geometry concepts and
specific vocabulary) with the introduction conversation (activity 1) and the review of
concepts and vocabulary (activity 2).
Continuous formative assessment through the time dedicated at the beginning of each
session to discuss doubts and ask questions
Continuous formative assessment through the correction of exercises in class.
Continuous formative assessment through the teacher’s journal
Summative formative assessment through the evaluation of the student’s notebooks
Summative formative assessment of the final task
Summative formative final assessment through an exam with exercises similar to those
done in class (LOTS and HOTS).
At the beginning of the unit, students will be informed about the summative assessment tools.
The assessment grids for the evaluation of the notebooks and the final task will be also given.
6. CONCLUSIÓN
Así como el refrán dice “del dicho al hecho hay un buen trecho”, de la teoría a la práctica se
podría decir lo mismo. Este proyecto ha servido para dar un primer paso en ese trecho, si bien
aún queda el camino de la aplicación real en el aula.
La realización de la actividad ha supuesto un desafío al combinar la aplicación de la metodología
Aicole y el hecho de ser la primera vez que me enfrento a planificar clases y actividades. Una de
las mayores dificultades ha sido el idioma, mi inglés se estaba oxidando. Me ha servido de
recordatorio sobre la necesidad de revisar y mejorar el lenguaje cotidiano (BICS), e investigar y
aprender el lenguaje específico académico de las matemáticas (CALP). Sin embargo, el hecho de
planificar con una visión Clil considero que es de especial importancia en matemáticas ya que
las Matemáticas pueden considerarse per se un idioma. Por otro lado, escoger un contenido y
sobre todo definir los límites de lo que quería abarcar me ha tomado más tiempo del que
imaginaba.
Mi autoevaluación es la siguiente:
-
Entre los puntos débiles destacaría no explicar con más detalle las técnicas de andamiaje
en cada ejercicio. Debería revisar con mayor detalle las competencias de un alumno con
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A2 para verificar que las instrucciones y contenidos son lo suficientemente claros y a la
vez promueven aprendizaje. Debería incluir más actividades que permitieran abordar
los conceptos desde otro punto de vista. Sería necesario revisar con alguien, con mayor
nivel de inglés, los textos para corregir los fallos y eliminar expresiones “españolizadas”.
Además, podría sería deseable sintetizar el desarrollo de las ideas
-
En cuanto a los puntos fuertes, se podría señalar la secuenciación de las actividades que
buscar facilitar el aprendizaje siguiendo los pasos del proceso cognitivo. También
considero que tanto la actividad de “descubrimiento” del Teorema como la actividad
final pueden ser especialmente interesantes para el aprendizaje y el alumnado.
7. ANEXO
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Anexo
This document explains briefly the activities planned for the lessons. The text in blue means it is
destined for the students, it’s going to be part of the worksheets provided to the students.
Activity 1: Introduction
These images are shown in the class, students are asked some questions to brainstorm what
they know and remember about triangles, specifically about right triangles (also called rightangled triangles). The teacher will guide the conversation.
Some of the possible questions may be:






What geometric shapes have in common these two images?
Are they any differences or similarities between the triangles?
What do you remember about triangles?
How many sides has a triangle? And angles? And vertex?
How can we classify triangles?
What is a right triangle?
Activity 2: Reviewing vocabulary and concepts
This activity will help students remember the main concepts and vocabulary needed for the
lesson. The key terms of hypotenuse and legs (or short sides) are also introduced.
A triangle is a polygon with three sides, three angles and three vertices (attention! vertices is
the plural of vertex).
In a triangle we use capital letters for the vertices, and we name the opposite side of each vertex
with the same letter in small letters. The triangle below is the ABC triangle. 1
1
Esta parte del ejercicio ha sido adaptada del documento “student” de Ma Josep Sanz Espuny recuperado
de http://www.xtec.cat/sgfp/llicencies/200809/memories/2024/
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Anexo

How do we classify angles? And triangles? Please fill in the blanks with the correct term.
There are more words than gaps so you will not use all the words provided.
If an angle measures more than 90⁰, then it is a(n) ………… angle
If an angle measures less than 90⁰, then it is a(n) ………… angle
If an angle measures more than 90⁰ it is a(n) ………… angle
The three angles of a triangle always add to ……… degrees
………………. has three equal sides
………………. has two equal sides
By ……….
………………. has no equal sides
Classify
triangles
………………. has three angles <90⁰
By ……….
………………. has one angle = 90⁰
………………. has one angle > 90⁰
The segments crossing the sides show equal sides .
The square is an angle means a right angle.
acute triangle
equilateral triangle
obtuse triangle
right triangle
happy triangle
isosceles triangle
wrong angle
180
obtuse angle
acute angle
really cute angle
right angle
side
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angle
90
scalene triangle
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Anexo
 Sometimes a triangle has two names, for example:
CDE triangle has a right angle and two equal sides
 It is a right isosceles triangle
 Perimeter and area
A
The perimeter of a polygon is defined as the
distance around it.
c
b
h
In this example the perimeter of ABC triangle is
B
C
perimeter = a +b +c
a
The area of a triangle is one half of base times height.
=
×
×
in this example
=
×a×h
Remember, every side of the triangle can be a base, and from every vertex you can
draw the line perpendicular to a line containing the base - that's the height of the
triangle. Every triangle has three heights, which are also called altitudes.
 Now we are going to focus on right triangles (or right-angled triangle) and learn some new
vocabulary
- The longest side of a right triangle is called the hypotenuse
and is always the side opposite the right angle.
B
c
a
- The other sides are called legs or short sides.
hypotenuse
leg /
short side
C
b
leg /
short side
A
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Anexo
Activity 3: Discovering the mystery of right triangles
Students will be guided orally and with the support of this worksheet, the aim is to discover the
relationship between the areas of the squares along the sides in a right triangle. They will
follow/read the instructions to create the geometrical elements needed to prove the PT.
Right triangles are not only the triangles which are never wrong (excuse me for the bad joke),
but they have also a mystery hidden in their sides and we are going to discover it. Frequently
mathematical “mysteries” can be solved without numbers and this one is one of those. Work in
pairs and follow the instructions to find the secret of right triangles!!!
1. Draw a right triangle (for example one whose sides measure 3 cm, 4 cm and
5 cm. You can take a look at the steps seen in Design class if you need. Name
and write the sides as in the figure below.
c
a
b
2. Draw squares along the hypotenuse and the two legs. We will name them the a, b and c
squares.
c
=
a
=
b
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Anexo
3. Take the scissors and carefully cut all the figures and make 7 more triangles. By the end, you
will have 8 equal right triangles and 3 squares.
4. Arrange the squares and triangles like in the figure below.
5. Answer the questions
- Which is the area of the squares a, b and c?
- Which is the area of the triangle ABC?
- Which is the area of the big two squares?
- Based on your previous answers, can you find a relationship between the areas of the a,b,c
squares?
Activity 4: Discussion about the discoveries
Let’s bring our ideas together. You can look at the following sentences to express yourself
 The area of the …… square is bigger/smaller than the area of the …….. square
 The area of the ……is equal to the area of the …….
 We have found that……..
 If the areas of……. sum, then……………
After this activity, the Pythagorean Theorem will be explained by the teacher, who will use visual
support. Besides, students will have the following material.
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Anexo
Pythagorean Theorem (also called Pythagoras’ Theorem)
The mystery we have solved is called the
Pythagorean Theorem (also called Pythagoras’
Theorem).
=
=
c
a
b
The Pythagorean Theorem says that in a right
triangle the area of the square on the hypothenuse
is equal to the sum of the areas of the squares on
the two shorter sides (legs).
+
=
In the example:
+
=
=
It is a relationship between areas, but it can also be
a relationship between the lengths of the sides.
Remember this theorem can only be used when we
have a right triangle.
There are several methods to prove the Pythagorean Theorem, more than 300!!! We have just
used one in activity 3. In this video, we can see a very simple one with water!
https://www.youtube.com/watch?v=CAkMUdeB06o
Or this one, with sugars! https://www.youtube.com/watch?v=uOTs2ck1_jU
Activity 5: Who was Pythagoras?2
Read and answer the questions below:
Pythagoras was born on the island of Samos (Greece) in 569
BC. His father, Mnesarchus, was a merchant and his mother
Pythais, was a native of Samos. Young Pythagoras spent most
of his early years in Samos but travelled to many places with
his father. He was intelligent and well-educated. Pythagoras
was also fond of poetry.
Pythagoras also made important discoveries in music,
astronomy and medicine, and created the Pythagorean
Academy, but he is remembered today for his famous
theorem in geometry, the ‘Pythagoras’s Theorem’.
Nevertheless, there is no proof that he discovered it. It is
believed that he learned the theorem during his studies in
Egypt. The Egyptians probably knew of the relationship for a
thousand years before Pythagoras. Before the Egyptians,
there is proof that Babylonians also knew this relationship.
2
Esta actividad ha sido recuperada y adaptada del repositorio “libredisposicion.es”
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Anexo
Pythagoras married Theano.
Theano was the first female mathematician in history. She was
born in Crotona (Greece in c.VI BC). Her father was Midon, a very
rich man who sent Theano to study with Pythagoras.
Theano wrote about Mathematics, Physics and Medicine, but her
most important contribution was the theorem about the Golden
Proportion, that you will study the next course.
When Pythagoras died due to a rebellion that destroyed the
Academy, Theano and her daughters spread the mathematical
knowledge of the Academy throughout Greece and Egypt.
Answer the following questions:
-
Pythagoras died in 495 BC, at what age did he die?
What were the Pythagoras’ interests?
At that time, it was not usual for a woman to study maths, why do you think Theano was
able to get a good education?
Was Pythagoras the first to this property of right triangles?
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Anexo
SECOND SESSION
Doubts
Activity 6. The Pythagorean Theorem’s song
Pythagorean Theorem is one of the most famous and popular theorems. Why? Because it is very
useful to solve a multitude of problems. We will explain its applications later, firstly we are going
to watch and listen to a very curious song about the Pythagorean Theorem.
https://www.youtube.com/watch?v=l8-bnZh8Zuc
For the second listening, students will have provided the lyrics of the song.
Colin Dodds - Pythagorean Theorem (Math Song)
Look at a right triangle
With a 90 degree right angle
Across from the right angle is the hypotenuse
It's no surprise the hypotenuse is the longest side
Now how do you find the hypotenuse's length
If you know the length of the two other sides?
Let's take you back to ol' Ancient Greece
Pythagorus is gonna show you why:
a squared plus b squared equals c squared
Where c is the length of the hypotenuse
a squared plus b squared equals c squared
Where a and b are the length of the other sides
The Pythagorean Theorem
Is a delicate calculation
To find the hypotenuse take the square root
Of the sum of the two other sides' squares...and then compute
How do you find the hypotenuse's length
If you know the length of the two other sides?
Let's take you back to ol' Ancient Greece
Pythagorus is gonna show you why:
a squared plus b squared equals c squared
Where c is the length of the hypotenuse
a squared plus b squared equals c squared
Where a and b are the length of the other sides
----Song and video by Colin Dodds
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Anexo
Teacher’s explanation about the applications of the Pythagorean Theorem:
- Identification of right triangles from the measurements of their sides (the converse of
the Pythagorean Theorem)
- Theorem’s application to find the missing side of a right triangle given the other sides
Activity 7. Testing if a triangle is right-angled 3
Are the following triangles right-angled triangles?
Look at the example and work out if exercise b) and c) triangles are right-angled trianglesa)
If a triangle is a right-angled triangle the Pythagoras’
Theorem must work. We have to check if it does with this
particular triangle.
( ℎ
) =ℎ
1) + ( ℎ
=
+
= 8 = 64
+
= 5 + 6 = 25 + 36 = 61
8 is not equal to 5 + 6
so ABC is not a right-angled triangle.
b)
c)
Activity 8: Finding the length of the hypotenuse4
You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if
you know the length of the two legs. Look at the example and find the length of the hypotenuse
on triangles a) and b):
1st step: Write down the formula
=
+
nd
2 step: Substitute
= 3 +4
rd
3 step: Calculate
= 9 + 16
= 25
Calculate the square root of 25
=5
3
Ejercicio obtenido del documento “student” de Ma Josep Sanz Espuny recuperado de
http://www.xtec.cat/sgfp/llicencies/200809/memories/2024/
4
(Idem)
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Anexo
a)
b)
Activity 9: Finding the length of the legs
In this exercise there is no example, learners will need to apply the same reasoning to solve the
unknown side. If needed, an example will be explained on the board.
Applying the same reasoning of the previous activity find the length of the missing sides. Pay
attention to which side is missing in the equation to solve it for.
Activity 10: Mixed exercises
Solve these exercises. Start drawing the triangle, labelling its sides and angles as shown in
activity 2, and then follow the steps.
1)In a right triangle the length of the two short sides is 6 cm and 9 cm, find the length of the
hypotenuse.
2) In a right triangle the hypotenuse length is 23 cm, find one of the short sides, knowing that
the other one is 15 cm long.
3) How long is the diagonal of a square whose side is 7 cm length?
4) In a triangle, one of the sides measures 4 units and the other 6 units. How long is the third
side?
5) Find the missing side of this isosceles right-angled triangle.
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Anexo
THIRD SESSION
Doubts
Activity 11. Solving real problems applying Pythagora’s Theorem5
1. To get from point A to point B you must avoid walking through a pond. To avoid the pond,
you must walk 34 meters south and 41 meters east. How many meters would be saved if it were
possible to walk across the pond?
2. A suitcase measures 24 cm long and 18 cm high. What is the diagonal length of the suitcase?
3. In a computer catalogue, a computer monitor is listed as being 19 inches. This distance is the
diagonal distance across the screen. If the screen measures 10 inches in height, what is the
actual width of the screen?
4.Oscar's dog house is shaped like a tent. The slanted sides are both 1 meter long and the bottom
of the house is 1.5 meters across. What is the height of his dog’s house at its tallest point?
++++ other problems
Final task. Making up a problem
At the end of the unit (here we only develop the first three sessions) students are asked to create
individually their own problem in which the Pythagorean Theorem is needed. They will have
already seen multiple examples of how the Theorem is used in real life.
The problem must be typed, use appropriate mathematical vocabulary, include units of
measurement and show how the Pythagorean Theorem is used in everyday life. The solution
with all the steps must be provided as well. A model 2D or 3D must be created to illustrate either
the problem formulation or the solution. Finally, students must write a reflection on the task.
5
Ejercicio obtenido del documento “student” de Ma Josep Sanz Espuny recuperado de
http://www.xtec.cat/sgfp/llicencies/200809/memories/2024/
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