Secuencia AICLE Matemáticas

1. From Flatland to Spaceland
MATHEMATICS. 2 ESO
by Rafael Cabezuelo Vivo
nd
May 2011

2. IDENTIFICACIÓN DEL MATERIAL AICLE
TÍTULO
FROM FLATLAND TO SPACELAND
NIVEL LINGÜÍSTICO
SEGÚN MCER
A2.1
IDIOMA
INGLÉS
ÁREA/ MATERIA
MATEMÁTICAS
NÚCLEO TEMÁTICO
GEOMETRÍA
La unidad pretende pasar de las figuras planas a los
GUÍON TEMÁTICO (no cuerpos tridimensionales, haciendo un recorrido por las
principales características de éstos. Se utiliza lo aprendido
más de 100 palabras) para describir objetos de uso cotidiano. Se incluye una
autoevaluación.
FORMATO
PDF
CORRESPONDENCIA
2º ESO
CURRICULAR
AUTORÍA
RAFAEL CABEZUELO VIVO
TEMPORALIZACIÓN
APROXIMADA
6 SESIONES más una tarea final y una autoevaluación de
contenidos y destrezas.
COMPETENCIAS
BÁSICAS
Lingüística: Mediante la lectura comprensiva de textos, la
resolución de problemas y el uso de descripciones.
Matemática: Con la interpretación y descripción de la
realidad a través del pensamiento matemático, aplicando los
conocimientos a objetos y realidades cercanas.
Interacción con el mundo: A través de las estructuras
geométricas, desarrollando la visión espacial y relacionando
las formas estudiadas con objetos cotidianos.
Aprender a aprender: Por medio de la perseverancia, la
sistematización, la autonomía y la habilidad para comunicar
con eficacia los resultados del propio trabajo.
OBSERVACIONES
Los contenidos de las sesiones pueden exceder de una hora
de clase real, especialmente cuando se llevan a cabo algún
‘role play’, trabajo grupal y/o manual. Las dificultades
matemáticas pueden también suponer una necesidad de
adaptación de la duración de cada actividad.
Las actividades de postarea, al final de cada sesión o grupo
de sesiones podían utilizarse todas como actividades
finales, junto a la ficha de autoevaluación. Además, cada
sesión puede utilizarse de forma independiente.
Secuencia AICLE 2o ESO
2
From Flatland to Spaceland

3. TABLA DE PROGRAMACIÓN AICLE
OBJETIVOS DE
ETAPA
CONTENIDOS DE
CURSO
TEMA O
SUBTEMA
MODELOS
DISCURSIVOS
TAREAS
• Identificar las formas y relaciones espaciales que se presentan en la vida
cotidiana, analizar las propiedades y relaciones geométricas implicadas y ser
sensible a la belleza que generan al tiempo que estimulan la creatividad y la
imaginación.
• Integrar los conocimientos matemáticos en el conjunto de saberes que se van
adquiriendo desde las distintas áreas de modo que puedan emplearse de forma
creativa, analítica y crítica.
• Poliedros y cuerpos de revolución. Desarrollos planos y elementos
característicos. Clasificación atendiendo a distintos criterios. Utilización de
propiedades, regularidades y relaciones para resolver problemas del mundo
físico.
• Volúmenes de cuerpos geométricos. Estimación y cálculo de longitudes,
superficies y volúmenes.
• Utilización de procedimientos para analizar los poliedros u obtener otros.
•
•
•
•
Geometría
Figuras planas
Cuerpos geométricos
Poliedros
•
•
•
•
Cuerpos de revolución
Áreas
Volúmenes
Historia de la Matemática
• Comparar objetos reales con las figuras estudiadas
• Analizar y describir objetos de uso cotidiano
• Explicar procesos de cálculo
• Análisis de las figuras planas como tarea previa de comprensión de los cuerpos
sólidos.
• Construcción y análisis de los cuerpos platónicos.
• Cálculo de superficie y volumen de los principales poliedros y cuerpos de
revolución (prismas, pirámides, cilindros y conos).
• Descripción y análisis geométrico de un objeto cotidiano.
FUNCIONES:
Hacer descripciones, explicar cálculos.
CONTENIDOS
LINGÜÍSTICOS
CRITERIOS DE
EVALUACIÓN
Secuencia AICLE 2o ESO
ESTRUCTURAS:
I think that…, The object/building is like a..., From my point of view…
On one hand… on the other hand..., I (don't) agree with you, etc.
a times/plus/minus/divided by b, one third (fractions), is equal to, etc.
LÉXICO:
apex, apothem, cone, cross section, cube, cubic, cylinder, dimension,
dodecahedron, dodecahedron, edge, face, figure, flat, flatland, height, hexagon,
icosahedron, irregular, n-sided, oblique, octahedron, pentagon, perimeter, pi (π),
plane, platonic solids, polygon, polyhedron, prism, pyramid, radius, rectangle,
regular, right, shape, side, slant length, solid, space, square, square root, squared,
straight, surface-area, tetrahedron, triangle, vertex, volume, etc.
•
•
•
•
•
Relacionar a Escher con la geometría.
Conocer y distinguir los cuerpos platónicos.
Conocer y distinguir los principales poliedros y cuerpos de revolución.
Calcular la fórmula de Euler
Calcular el área y el volumen de prismas, pirámides, conos y cilindros.
3
From Flatland to Spaceland

4. Image 1
Image 2
What can you tell me about these images?
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From Flatland to Spaceland

5. Session 1
Pretask
1. Word cloud. Look at the words and the images above. First, listen and repeat the
words. Then, fill in the gaps in the texts below. Gaps with the same letter must be
filled with the same word.
Image 1 shows a a)_____________ that can be drawn on a b)_____________
surface called a c)_____________ (it is like on an endless piece of paper). Our
world has three d)_____________, but there are only two d)_____________ on
a plane that are length and height, or x and y.
A e)_____________ is a 2-dimensional shape made of f)_____________lines.
Image 1 shows a e)_____________ named g)_____________, do you know
more of them? h)_____________, i)_____________ and _____________.
Image 2 shows a j)_____________, which is a three-dimensional
k)_____________. k)_____________ Geometry is the geometry of threedimensional l)_____________, the kind of k)_____________ we live in. It is
called three-dimensional, or 3-D because there are three dimensions: width,
depth and height or x, y and z.
A m)_____________ is a j)_____________that has 12 n)_____________ (from
Greek -dodeca- meaning 12). Each face has 5 o)_____________, and is
actually a pentagon. When we say m)_____________ we often mean regular
m)_____________ (in other words all n)_____________ are the same size and
shape), but it doesn't have to be. If you have more than one m)_____________
they are called p)_____________. It is one of the five platonic
k)_____________.
More
k)_____________
are
q)_____________,
r)_____________, etc.
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From Flatland to Spaceland

6. Task: Living in Flatland
2. Listening.
You are going to listen a text from the book "Flatland: A Romance of Many Dimensions". It
is an 1884 science fiction novel by the English schoolmaster Edwin Abbott Abbott. Listen
carefully and underline the words that you hear:
flatland
space
figures
triangles
heptagons
sheep
surface
readers
squares
paper
sinking
world
rising
countrymen
lower
pentagons
curved lines
edges
my universe
border
3. Now answer these questions about the text and Flatland:
What kind of figures can you find in Flatland?
Name as many of these figures as you can remember. It doesn't matter if they don't
appear in the text:
These are the characteristics of certain plane figures. Which figures are we talking about?
They are 2-dimensional shapes.
They are made of straight lines.
The shape is "closed" (all the lines connect up)
4. Make questions using the information you can find in this website:
http://www.mathsisfun.com/geometry/polygons.html
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From Flatland to Spaceland

7. Statements
Questions
Polygon comes from Greek. PolyWhere does the word polygon come from? or
means "many" and -gon means "angle". What is the origin of the word polygon?
It is an Icosagon.
All sides has the same lenght and all
angles are also equal.
It is a polygon with, at least, one
internal angle greater than 180º.
Another name is Tetragon.
5. From the same book now read this text:
"Place a penny on the middle of one of your tables in
Space; and leaning over it, look down upon it. It will
appear a circle.
But now, drawing back to
the edge of the table,
gradually lower your eye
(thus bringing yourself
more and more into the
condition of the inhabitants
of Flatland), and you will find the
penny becoming more and more
oval to your view ; and at last
when you have placed your eye
exactly on the edge of the table
(so that you are, as it were, actually a Flatland citizen) the
penny will then have ceased to appear oval at all, and will
have become, so far as you can see, a straight line."
What is the writer describing?
The editor of the book tells you to make three drawings as an illustration to make the text
clear. Use the empty boxes provided next to the text.
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From Flatland to Spaceland

8. What I Learned
•
•
•
•
•
6. True or False.
The plural of polyhedron is polyhedrons.
Flatland is a two-dimensioned country.
There are lots of dodecahedra living in Flatland.
Figures in Flatland can rise above or sink below the surface.
A polygon is a closed plane figure made of stright lines.
T/F
T/F
T/F
T/F
T/F
7. Classify these plane figures in the category they straight best (it can be more than
one). Use X to select the category:
Figure
Name
Polygons
Not a
Polygon Regular Irregular Concave Convex
Triangle
8. Colour the regular
polygons
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From Flatland to Spaceland

9. Session 2
Pretask
1. Vocabulary activation. Listen and repeat. Then match pictures and words:
2
1
5
4
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3
6
From Flatland to Spaceland

10. 2. Answer these questions in groups of four:
•
•
•
•
What does the building in the second picture
look like?
How many sides do you think the dice has?
What is the main difference between the third
and the sixth picture?
Do you think that solid geometry is important
in our lifes? Why?
I think that…
The object/building is like a...
From my point of view…
On one hand… on the other hand...
I agree with you / I don’t with you
Because...
Task: Moving to Spaceland
3. Video
You are going to see a video about how solids
can be seen in a place like Flatland. The narrator
is Maurits Cornelis Escher (1898-1972), most
commonly known as M. C. Escher. He was a
fascinating artist whose compositions are
worldwide famous. Escher became fascinated by
the mosaics and symmetries in Alhambra, when
he visited it in 1922.
Watch the video in
this link: http://www.dimensions-math.org/Dim_E.htm
(you may need to download the video before watching)
Now read these questions about the video before watching it
again. It is the time to resolve any doubt you can have, so ask
your teacher anything you don't understand.
•
Complete the following table with the names, number of
faces, vertices and edges for each polyhedron.
Icosahedron
Octahedron
Dodecahedron
Tetrahedron
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Hexahedron
10
Cube
From Flatland to Spaceland

11. Picture
•
Name
Faces
Vertices
Edges
Can you identify the polyhedron by its plane section? Write them here in order of
appearance (from 5'10" to 6'20").
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From Flatland to Spaceland

12. •
In the video there is a second and colorful method to explain polyhedra to our flat
friends, the lizards. Underline the correct answer.
Stereographic projection
Solid inflation
Face colouring
•
Can you identify the polyhedron by it's plane proyection? Write them here in order of
appearance (from 10'20" to 12'20").
•
The Greek philosophers attributed a magical importance to these 5 solids, associating
one of the fundamental elements from which the world is formed to each of them. What
is the name for these figures? Underline the correct answer.
Fantastic Solids
Platonic Solids
Plutonic Solids
4. Do it yourself
Now it's your time. The best way to understand _________ ← Fill in from the last question
solids is to build them. For this you have to follow the
instructions given in this website (use models with tabs):
http://www.mathsisfun.com/geometry/model-constructiontips.html
Some games (role-playing games) use these solids as
dice. To make them you have to write a number on each
face, but you have to follow this simple rule:
Opposite faces must always add up to the same value!!
(use a regular cubic dice to check it)
5. Counting Faces, Vertices and Edges.
If you count the number of faces (the flat surfaces), vertices (corner points), and edges of
a polyhedron, you can discover an interesting thing:
The number of faces plus the number of vertices
minus the number of edges equals 2 . This can be
written neatly as a little equation:
Euler's Formula
Secuencia AICLE 2o ESO
F + V - E = 2
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From Flatland to Spaceland

13. It is known as the "Euler's Formula", and is very useful to make sure you have counted
correctly! Now it is your turn! Check Euler's Formula for the platonic solids, and now you
can use your models to count everything!!
Name
Image
Faces
Vertices
Edges
F + V - E
Dodecahedron
Tetrahedron
Icosahedron
Hexahedron
Octahedron
What I Learned
6. Fill in the blanks. Use the words given to complete this summary.
In this lesson on three-dimensional solids, you've seen a lot of _____________. But there
are five special _____________, known collectively as the _____________, that are
different from all the others.
What makes the _____________ special? Well, two things, actually.
1. They are the only polyhedra whose _____________ are all exactly the same.
Every _____________ is identical to every other _____________. For instance,
a cube is a Platonic solid because all six of its _____________ are congruent
_____________.
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From Flatland to Spaceland

14. 2. The same number of _____________ meet at each _____________. Every
_____________ has the same number of adjacent _____________ as every
other _____________. For example, three equilateral triangles meet at each
_____________ of a _____________.
No other _____________ satisfy both of these conditions. Consider a pentagonal prism. It
satisfies the second condition because three _____________ meet at each
_____________, but it violates the first condition because the _____________ are not
identical; some are _____________and some are _____________.
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From Flatland to Spaceland

15. Session 3
Pretask
In the following sessions (3 to 6) you are going to work with 3D solids.
1. Vocabulary activation. Listen and repeat. Then write the appropiate words under
each picture (one word can be written under more than one picture):
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From Flatland to Spaceland

16. 2. Video: You are going to see a video about some properties of solids.
http://www.brightstorm.com/math/geometry/volume/3-d-solid-properties
Now you have to say if the following statements are true or false. You can review the video
at home if you need it.
•
•
•
•
•
A vertex is a line where three or more faces intersect.
Where two faces intersect you create an edge.
Bases are polyhedra.
The bases are lateral faces.
Right and Oblique prisms have the same volume and surface area.
(With the same bases and height)
T/F
T/F
T/F
T/F
T/F
Task: Square goes upward Flatland
3. Reading: Adapted from Flatland (p73-74.)
In section sixteen a stranger named Sphere tries to reveal to the main character, Square,
the mysteries of Spaceland.
Sphere. Tell me, Mr. Mathematician ; if a Point
moves Northward, and leaves a luminous tail,
what name would you give to the tail?
Square. A straight Line with two extremities.
Sphere. Now the line moves parallel to itself, East
and West, so that every point in it leaves behind it
the tail of a Straight Line. What name will you give
to this Figure?
Square. A Square, with four sides and four
angles.
Sphere. Now open your imagination a little, and
imagine a Square in Flatland, moving parallel to
itself upward.
Square. What? Northward?
Sphere. No, not Northward ; upward ; out of
Flatland altogether. I mean that every Point in you
(because you are a Square), in your inside,
passes upwards through Space. Each Point
describes a Straight Line of its own.
I was now impatient and under a strong temptation
to launch my visitor into Space, or out of Flatland,
anywhere, so that I could get rid of him. Instead I
replied:
Square. And what is the nature of this Figure? I
hope you can describe it in the language of
Flatland.
Sphere. Oh, certainly. It is all plain and simple,
but you must not speak of the result as being a
Figure, but as a Solid. But I will describe it to you.
- We began with a single Point, which of course
being itself a Point has only one terminal Point.
Secuencia AICLE 2o ESO
- One Point produces a Line with two terminal
Points.
- One Line produces a Square with four terminal
Points.
Now you can answer to your own question: I, 2, 4,
are evidently in Geometrical Progression. What is
the next number?
Square. Eight.
Sphere. Exactly. The Square produces something
which we call a Cube with eight terminal Points.
Now are you convinced?
Square. And has this Creature sides, as well as
angles or what you call "terminal Points"?
Sphere. Of course; and we call them faces.
Square. And how many faces or sides will I
generate by the motion of my inside in an
"upward" direction, and whom you call a Cube?
Sphere. How can you ask? And you are a
mathematician! The side of anything is always, if I
may so say, one Dimension behind the thing.
Consequently, as there is no Dimension behind a
Point, a Point has 0 sides ; a Line, has 2 sides (for
the Points of a Line may be called by courtesy, its
sides) ; a Square has 4 sides ; 0, 2, 4 ; what kind
of Progression do you call that? What is the next
number?
Square. Arithmetical. Six.
Sphere. Exactly. Then you see you have
answered your own question. The Cube which
you will generate will be bounded by six sides.
You see it all now, eh?
16
From Flatland to Spaceland

17. 4. Text attack!
Write the following questions in chronological order, then match them with the correct
answers. Find the odd answer:
•
•
•
•
•
•
•
•
•
Where does the square need to move to create a
cube?
Why is Square impatient?
Sphere sum up the whole process again, write it
in the correct order:
How is the square obtained?
How can Square know the number of faces in a
cube?
How can Square know the number of vertices in a
cube?
What does Square understand when Sphere tells
him that a square shoud move upwards to
construct a cube? Why?
What does Square want to do with Sphere?
What would be the name of the cube faces in
Flatland?
•
•
•
•
•
•
•
•
•
•
Secuencia AICLE 2o ESO
Here is how the figures are formed:
Point → Line → Square → Cube.
Square understand northward instead of upward,
beacuse he lives in Flatland and it is difficult for him
to undertstan the three-dimensional space.
The square needs to move upwards to create a
cube.
Becouse he can't understand the three-dimensional
space very well.
He wants to launch Sphere into space or out of
Flatland.
Square launches Sphere into space, out of Flatland.
Because they are in arithmetical progression, each
number is two more than the previous one. So 0 for a
point, 2 for a line, 4 for a square, ... and 6 for a cube.
The name would be sides.
Because they are in geometrical progression, each
number is double of the previous one. So 1 for a
point, 2 for a line, 4 for a square, ... and 8 for a cube.
The line has to move parallel to itself, East and West.
17
From Flatland to Spaceland

18. 1.
How is the square obtained?
The line has to move parallel to itself,
East and West.
2.
3.
4.
5.
6.
7.
8.
9.
This is the odd answer:
Can you write a title for this chapter of the story?
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From Flatland to Spaceland

19. What solid do you obtain if the square moves upward a
length greater than (or less than) the size of the square?
5. Your turn!
Now you have to make your own story. You are going to work in groups of four. The
teacher will dictate the beginning of a text, in which you will have to describe how a
pentagon from Flatland can be transformed into a pyramid. Then, when the teacher claps
one member of the group will continue the text by writing another short paragraph about
the transformation. Each time the teacher claps, you will pass the paper to a new group
member who will write the next section of text. Continue like this until the circle is
complete. The member of the group who wrote the first paragraph will also write the last
one. When you finish, choose a spokesperson to read your text out loud to the rest
of the class.
Teacher’s dictation
Sphere. Imagine, my friend, that you are now a pentagon in Flatland...
Student's 1 text:
Student's 2 text:
Student's 3 text:
Student's 4 text:
Student's 1 text -last paragraph-:
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From Flatland to Spaceland

20. Session 4
Task: Prisms Surface-Area & Volume
1. Reading: Adapted from www.mathsisfun.com
If you make the cross section of a cube you will get a
square, and the cross section of this building is a
triangle ...
de
pt
h
A cross section is the shape you get when cutting
straight across an object.
height
These are the three dimensions that a solid has: width,
depth and height
width
A solid that has the exact same polygon as its cross section all
along its length is called a
Prism
The bases of the prism will be also the same polygon. According
to the cross section or the base of a prism it can be named...
Triangular prism
Square prism
Rectangular prism
Pentagonal prism
and so on...
A square prim that has edges of equal length can be called a cube
(or hexahedron) and each face will be a square. Do you remember
the platonic solids? So a cube is just a special type of square prism,
and a square prism is just a special type of rectangular prism, and
They are all cuboids!
If the cross section of a prism is a regular polygon
(Equilateral Triangle, square, regular pentagon, regular
hexagon, etc), then you have a Regular Prism.
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From Flatland to Spaceland

21. 2. Complete the following table with actual prismatic objects:
Figure
Base Poligon
Cross section
Name
Regular
3. Calculate. Surface-area of a prism.
The surface area of a prism is the sum of the area of all its faces. As the bases are
polygons you will need to remember how to calculate their area. It is measured in squared
units (f.i. m2, ft2)
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From Flatland to Spaceland

22. You will need to calculate the area of one base (Ab).
s = 5 in (side)
a = 3,44 in (apothem)
H = 10 in (height)
p⋅a 5⋅s⋅a
=
=
2
2
p stands for perimeter
Ab =
=
5⋅5⋅3.44 86
= = 43 in²
2
2
Then you will have to calculate the area of one face
(Af), which will allways be a cuadrilateral.
A f =s⋅H =5⋅10= 50 in²
There are two bases and five lateral faces (in this example), so the
total surface area will be:
Atotal =2⋅Ab+5⋅A f =2⋅43+5⋅50=86+250= 336 in²
Now calculate the following prisms' surface areas:
Prism
Base area
Face area
Surface area
Regular prism
Base side: 5 in
Height: 12 in
Base sides:
3 x 10 in
Height:
17 in
4. Calculate. Volume of a prism.
The Volume of a prism is simply the area of one of its bases times the height of the prism.
It doesn't matter if it is a right or an oblique prism. It is measured in cubic units (f.i. cm 3,
in3). For the previous example we can calculate the volume in this way:
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From Flatland to Spaceland

23. We already know the area of the base. Do you remember it? Ab =
p⋅a
=
2
Can it be that easy? I think you can do it on your own: V =Ab⋅H =
⋅
in²
=
in³
Complete the following table:
Prism
Base area
Height
Volume
Regular prism
Base side: 3 in
Apothem: 2 in
Height: 24 in
Regular prism
Base side: 4 in
Apothem: 3,5 in
Height: 11 in
5. Search.
Bullring in Montoro has a prismatic shape. Can you tell
me how many sides the base has? What is the name for
this n-sided polygon?
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From Flatland to Spaceland

24. Session 5
Task: Pyramids Surface-Area & Volume
1. Video: You are going to see a video about regular pyramids surface-area.
http://www.brightstorm.com/math/geometry/area/surface-area-of-pyramids
Now you have to select the correct answer to the following questions. You can review the
video at home if you need it. Then make groups of four to construct the questions and the
answers.
.
h?
e side
t lengt
e slan
th of th triangular
g
is th
the len
f the
What
ngth is the height o
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sla
- The
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eight.
f the h
slant le
o
- The
length
is the
s.
th
face
What is
nt leng
the firs
The sla
t calcu
- T he m
lations
an calc
that th
Wha
- Th e m
ulate th
e man
t sha
an calc
makes
e surfa
pes
- Th e m
?
u
c
- The
do th
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e fac
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gle firs
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the len
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t.
e surfa
f
f the
gt
ce o f t
- The faces o the pyra
pyra
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f the
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face
mid
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h
p
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have
the p yramid h ave a tri
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angu
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na
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.
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2. Calculate: Calculate the surface area of these pyramids.
To calculate the slant length you need to remember the Pythagorean Theorem. If you cut
the pyramid by the apothem of the base and the apex (the top point) you can see a right
triangle:
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25. Imagine a regular square
pyramid. Let's calculate its
surface area:
Base = 10 in
Height = 12 in
Cutting the pyramid we have an
isosceles triangle, that can be
divided into two right triangles.
Get focus on ABC, which is a
right triangle with a 90º angle at
C.
AC = One leg of the right triangle = Heigth of the pyramid = 12 in
CB = Other leg of the right triangle = one half of the base of the pyramid = 5 in
AB = Hypotenuse of the right triangle = Slant length of the pyramid = Unknown
AB 2= AC 2+CB 2 ;
AB=√ AC 2+CB 2=√ 122+5 2=√ 144+25=√ 169= 13 in ← Slant length
Area of the base: As it is a square its surface is:
Area of one face: It is the area of a triangle:
Ab =base⋅base=10⋅10= 100 in
Af =
2
base⋅height 10⋅13
=
= 65 in 2
2
2
There are one base and four lateral faces, so the total surface-area will be:
Atotal =Ab +4⋅A f =100+4⋅65=100+260= 360 in 2
Now calculate the following pyramids' surface areas:
Pyramid
Base area Slant length Face area Surface-Area
Tetrahedron
(regular pyramid)
Base side: 3 in
Regular pyramid
Base side: 3.2 in
Height: 6 in
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From Flatland to Spaceland

26. 3. Calculate: Calculate the voume of these pyramids.
The volume of a pyramid is very easy, you just have to calculate one third of the base area
times the height of the pyramid. Let's make the previous example:
We already know the area of the base. Do you remember it? Ab =base⋅base=
1
1
Can it be that easy? V = ⋅Ab⋅H = ⋅
3
3
⋅
=
in 2
in³
Complete the following table:
Pyramid
Base area
Height
Volume
Regular pyramid
Base side: 4.5 in
Apothem: 3.1 in
Height: 12 in
Regular pyramid
Base side: 10 in
Apothem: 8.66 in
Height: 16 in
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From Flatland to Spaceland

27. Session 6
Task: Smooth down the Prism and the Pyramid
1. Reading:
When the base of a prism changes from a polygon to a circle then you get a __________.
Doing the same with a pyramid what you get is a _____________. The volume can be
calculated in a similar way, but the surface-area is slightly different. Here you can see the
formulas:
Cylinder
Cone
2 Bases: Ab =π⋅r 2
1 Side: A s=2⋅π⋅r⋅h
Surface area
Base: Ab =π⋅r 2
Side: A s=π⋅r⋅s=π⋅r⋅√ h 2+r 2
TOTAL:
Atotal =2⋅Ab+ As=2⋅π⋅r⋅ r+h)
(
Volume
TOTAL:
Atotal =Ab +As =π⋅r⋅(r +s)
1
1
V = ⋅Ab⋅h= ⋅π⋅r 2⋅h
3
3
2
V =Ab⋅h=π⋅r ⋅h
2. It's your turn!
A sheet cylinder:
Look at a sheet of paper, how many cylinders can be
made using its dimensions?
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From Flatland to Spaceland

28. Calculate the surface area and the volume of the obtained cylinders, and then compare
the values.
The volume of a bucket
Can you tink of the shape of a bucket? What is this shape like?
D
You will need the diameter or radius of both bases:
D = 2.30 dm
d
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d = 1.80 dm
height = 4.22 dm
Think the best way to calculate the volume (1 dm 3 = 1 l). You
could need Thales' theorem to find one missing data.
28
From Flatland to Spaceland

29. What I Learned
From session 3 to 6 you have learned a lot of
think about polyhedra and non-polyhedra 3d
solids. Let's see what you can remember.
3. Word cloud
Match the images with the words:
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From Flatland to Spaceland

30. 4. Do it yourself
Most of our buildings have a polyhedra, cylinder or cone shape. Can you analyze one of
them. Look at this church. You will have to calculate its surface-area and its volume. Notice
that is full of Polyhedra. Use convenient units for all dimensions and calculus.
Cut the different parts of the church and paste them on the next table:


30 ft
40 ft
20 ft
17,3 ft


50 ft
120 ft
50 ft
100 ft
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31. Tower:
Tower roof:
Type:
Type:
Base polygon:
Base
polygon:
Side:
Side:
Apothem:
Apothem:
Surface-area:
Height:
Volume:
Height:
Surface-area:
Roof:
Type:
Base polygon:
Side:
Volume:
Height:
Surface-area:
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Volume:
From Flatland to Spaceland

32. Building:
Type:
Base polygon:
Side:
Height:
Surface-area:
Volume:
Now complete the table:
Surface-area
Volume
Building
Roof
Tower
Tower Roof
WHOLE CHURCH
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33. HOMEWORK. FINAL SUMMARY
Final Task: Geometry around
1. Description
Find a cool geometrical object at home. You have to create a presentation with the image
of the object. You will have to explain later a few things about it to the rest of your
classmates.
•
•
•
•
•
•
•
•
What kind of geometrical object does it look like?
Which are its faces, bases, edges, vertices or apex?
Does it check Euler's Formula?
Is it simple (just one shape) or complex (like the church)?
What are its dimensions (you can sketch the object)?
What are its surface-area and volume?
How did you make the calculations?
What is it for?
Use your own pictures and text to make an eye-catching presentation. Remember not to
overload the slides, it is better to use single phrases and good pictures to show what you
want to say. Think about the student at the end of the class, and use large texts and
adecuate colours.
Please, don't forget to bring
your object with you!!
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34. ASSESSMENT WORKSHEET.
NAME:
DATE:
Your task is reflecting on what you have learned. Read the following statements about
skills and knowledge you have learned during the project. Please, circle one of these
options:
YES NO NOT YET.
Self-assessment chart
I CAN
I KNOW
Organize vocabulary into categories
Take notes from a listening or a video
Get valuable information from different sources
Describe images and pictures
Summarize the main ideas from a text
Participate in a role-play
Understand plane figures
Calculate Euler's formula with a polyhedron
Calculate surface-area and volume of a solid
Analyse and describe geometrically an object
YES
YES
YES
YES
YES
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
NO
NO
NO
NO
NO
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
NOT YET
What is a platonic solid and their caractheristics
Escher was very interested in geometry
The concept of volume and surface-area
What are the main polyhedra and how to identify
them
The difference between the main polyhedra and
non-polyhedra
The difference between the main polygons and
non-polygons
YES NO NOT YET
YES NO NOT YET
YES NO NOT YET
YES NO NOT YET
YES NO NOT YET
YES NO NOT YET
Feedback
CONTENTS
DEVELOPED SKILLS
SUGGESTIONS FOR
IMPROVEMENT
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From Flatland to Spaceland