ISOMETRIES IN THE
COORDINATE PLANE
Docente: Cutillo Silvia
Presso ITC Caruso NA
Prof.ssa Castagneto Laura
Liceo Scientifico Labriola Na
Content
Isometries in the coordinate plane
Studentage 15-16 y.o.
Teaching aims:
to enable learners to understand that different types of isometries
to develop their ability to perform and describe transformations of a
2-D shape in all 4 quadrants of a Cartesian plane
to increase their skills to observe regularities in different examples
and to deduce mathematical laws
Learning outcomes
MOST LEARNERS SHOULD
KNOW...
BE ABLE TO …
BE AWARE …
demonstrate an understanding
of transformations (translations,
rotations, and reflections) of 2-D
shapes in all four quadrants of
the Cartesian plane
carry
out
transformations
(translations, reflections and
rotations) on a Cartesian plane.
Model with mathematics.
definitions
of
rotations,
reflections, and translations in
terms
of
angles,
circles,
perpendicular lines, parallel
lines, and line segments.
describe the effect of dilations,
translations,rotations,and
reflections on two-dimensional
figures using coordinates.
Reason
abstractly
quantitatively
transformations as functions
that take points in the plane as
inputs and give other points as
outputs
to extend their findings to
understand that a series of
transformations
can
be
calculated using multiple rules
Look for and express regularity
in repeated reasoning.
Given two 2-D shapes, describe
transformations that could be
used to take one shape to the
other and vice versa.
and
Competence
Description
Communicative
Can describe different types of plane transformations
Linguistic
Can use language to observe and analyse
Artistic
Can recognise symmetries in monuments, paintings
Digital
Can use Geogebra
Mathematical
Can reason and solve problems mathematically
Social
Can cooperate with others
GLOSSARY
Function
Let A and B sets, a function from A to B is a rule in which to
every element x of A assigns an unique element in B
Transformation of a plane
Let A the set of points in the plane. A transformation in the
plane is a one-to-one mapping from A to A
Isometry
An isometry is a transformation of the planethat preserves
the distance between points
Rotation (turn)
Rotation is defined by stating the centre of rotation,
amount of turning in degrees and the direction of rotation
(clockwise or anticlockwise)
Translation (slide)
A translation is defined by specifying the distance and the
direction of a movement
Reflection(flip)
reflection over a line is a transformation in which each
point of the original figure has an image that is the same
distance from the line of reflection as the original point but
is on the opposite side of the line.
Wordlist
Function
Funzione
Collinear point
Puntallineati
Range
Immagine
x coordinate
ascissa
Symmetric
points
with respect to a
point (line)
Punti simmetrici
rispetto ad un punto
(una retta)
y coordinate
ordinata
X -axis
Asse delle ascisse
Midpoint of a line
segment
Punto medio di un
segmento
Invariant point
Punto unito
Y-axis
Asse
ordinate
Quadrant
Quadrante
Rotation
Rotazione
delle
Transformations
in the plane
coordinate
Trasformazioni del
piano
Axis of symmetry
Asse di simmetria
Half line
Semiretta
Reflection across a
line
Simmetria assiale
Point reflection
Simmetria centrale
Centre of
symmetry
Centro di
simmetria
Symmetric figure
with rispect to a
point
Figura simmetrica
rispetto ad un
punto
Transformation of
a plane
Trasformazioni
piane
Symmetric figure
with rispect to a
line
Figura simmetrica
rispetto ad una
retta
Translation
Traslazione
Perpendicular
bisector in a
line segment
Asse
di
segmento
Perpendicular line
through a point to
a given line
Perpendicolare da
un punto ad una
retta data
un
Horizontal
component of a
vector
Componente di un
vettore secondo
l’asse x
Vertical
component of a
vector
Componente di un
vettore secondo
l’asse y
Vector norm
Modulo
vettore
di
un
Composition of
isometries
Composizione
isometrie
di
Ordered pair
Coppia ordinata
Distance from a
point to a line
Distanza di un
puntoda una retta
Congruent figures
Figurecongruenti
Isometry
Isometria
Lunghezza
Lenght
Opposite
oppoto
Curly brackets
Parentesi graffe
Resources
 A little mirror
 Worksheets
 Interactive whiteboard
 PC ( to use Geogebra )
 Colored paper
 Squared math paper
 PPT presentations
Procedure
Groups Activity using worksheet and/or simple objects from everyday life
Whole class
Class discussion on preceeding activity
Summary of the discussion by drawing mind map or other visuals
Glossary writing
Individually
Writing observations and feedback on worksheets
Students are divided in groups with the help of English
teacher in order to balance both language and mathematical skills (i.e. the
best students in mathematics, the best English speaking people should not
belong to all to one or few groups )
Lesson 1
Activating prior knowledge
Lesson 2 -3
Introduction to plane transformation
Reflection
Learning outcomes
to carry out reflections
to write the coordinate of image of a 2D shape after
a reflection of the shape has been completed
Lesson 4-5
Translation and Rotation
Learning outcomes
to carry out translations and rotations
to write the coordinate of image of a 2D shape
after a translation/rotation of the shape has
Lesson 6
Consecutive plane transformations
Learning outcomes
•
been completed
•
•
To perform consecutive transformations on a 2-D
shape using Geogebra and identify coordinates of the
vertices of the image
to describe a sequence of transformations to carry a
certain figure onto another
to use the new vocabulary
Lesson 1
Activating prior knowledge
Learners are encouraged to produce content language at the start of the lesson.
It is expected that students will have prior knowledge/experience related to the
concepts and skills identified below. It may be necessary to pre-assess in order to
determine if time needs to be spent on conceptual activities that help students
develop a deeper understanding of these ideas.
 plotting points on a coordinate plane
 - congruence of geometric figures and the correspondence of their vertices,
sides, and angles
 interpreting and sketching views from different perspectives
 calculate the perimeter and area of fundamental geometric plane figures
LESSON 1
•Activating prior knowledge
•Developing communication skills
Cognitive skills: Remembering,
Defining
Learning skills: cooperating with
others, organizing information
•Building specific vocabulary
Students are asked to complete worksheets and write the a glossary of key words
they can find on the worksheet
Class discussion and preparation of a poster with key words to be hanged in the
classroom
Reflection
Material : 1 plane mirror with a support, 2
pencils, 1 piece of blank paper, 1 centimeter
ruler
1. Hold a pencil in a vertical position about
10 cm in front of a flat mirror, as in the
photograph. Look into the mirror and you
will see an image of the pencil.
2. Moving a second pencil around behind the
mirror, it will be possible to have a complete
view of the object: its lower part as reflected
in the mirror and the top with a direct view,
3. What is the appropriate distance that
makes it possible ?
( Use a centimeter ruler to measure these
distances: (a) From the first pencil to the
mirror surface and (b) From the mirror
surface to the image).
4. Try using other distances from the first
pencil to the mirror surface.
Rotation
Explore Rotational
Symmetry by Making Rose
Windows
A photo of a rose window will be given to
each group. The task is to determine how
many degrees each rose window rotates
before it appears the same.
Translation
A picture similar to the following will be given to
each group.
Students have to answer to the following questions :
How many birds you have to translate to obtain this
image? Describe the translations
http://www.tess-elation.co.uk/birds---an-introduction
LESSON 6
Activity 1
http://www.geogebra.org/en/wiki/index.php/Transform
ations_English