geometrymeasurementrme workshop seameo qitepoctober 2013
Geometry
&
Measurem
ent
SEAMEO QITEP in MATHEMATICS
Yogyakarta, October 5th & 7th 2013
ILHAM RIZKIANTO
Ilham_rizkianto@uny.ac.id
I see what you mean
Kanisza Figure
Central feature of geometry
A central feature of geometry is
learning to ‘see’, that is, to discern,
geometrical objects and relationships,
and to become aware of relationships
as properties that objects may or may
not satisy.
Let’s move the bookcase
Imagine Considering
moving a bookcase
from one room to
another, but not being
sure if it would fit
through the doorway.
If you did not have a
ruler, how might you
find out without
actually moving it?
The essence of
measurement
The key aspect of any measuring is to
make a comparisons. One way to do
this is to decide on a unit and to have
some way of replicating the unit (or
breaking it up into smaller sub units)
and juxtaposing this with the thing to
be measured.
The connection
Mathematics is not only connected to the world of
numbers.
In Geometry, the issue is to understand the space
around us. It is related to the two- & three- dimensional
world and the related shapes & figures.
Measurement is aimed at quantifying our physical
environment. The emphasize in this process makes
measurement the connecting link between arithmetic and
geometry.
arithmetic
measurement
geometry
Activity 1: Live up the paper
Lay the first sheet down on
the table. Lay the second one
partly on the top of the first,
so that the top-left corner of
the upper piece coincides with
the top-right corner of the
lower piece and the bottomleft corner of the upper piece
coincides with the left edge of
the lower piece.
Repeat this with several more
pieces. What do you notice?
Activity 1: Live up the paper
You may discover that some pieces
coincide. You may be surprised at the
shape you see emerging. This shape
will tell you about angles involved.
Now, try to find at least 4
questions that could be used as
follow up questions for this
activity.
?
Geometry tidbits
Every teacher should have his/her own
special collection of geometric tidbits –
short little puzzles, problems, and
curiosities in geometry to warm up the
class, to gain attention, to involve, to
challenge, to maintain interest, or
simply to give a change of pace.
Geometry tidbits 1
Move just three dots to form an arrow
pointing down instead of up.
Geometry tidbits 2
A solid has this for both its top and
front view. Draw its side view.
Geometry tidbits 3
How many rectangles are in this
figure?
Activity 2: Popcorn Holder
Take two pieces of the paper.
Take the first piece and curl it in
potrait orientation so it forms a
tall, thin cylindrical shape. Take
the other one, do the same but in
landscape orientation so it forms
a shorter and fatter cylindrical
shape.
If you were to fill each of these
cylinders with popcorn, which
one do you think would hold
more? Or, do you think they
would hold the same amount?
Activity 2: Popcorn Holder
This activity is intended to explore the
volume of cylinders with the same
lateral areas and to see the connection
between volume and area.
Now, try to find at least 4
questions that could be used as
follow up questions for this
activity.
?
The core teaching principles of RME
Interactivity
Reality
Level
Activity
Guidance
Intertwinment
Activity 3: Tangram
Mark points at the following
coordinates: (0,0), (0,2), (1,1), (1,3),
(2,2), (2,4), (3,1), (3,3), (4,0), (4,4)
Connect the following points with a line
segment to create a tangram set: (0,0)
and (4,4), (0,2) and (2,4), (1,1) and
(1,3), (1,3) and (4,0), (2,4) and (3,3)
Cut out the tangram pieces and create
an animal or object using all seven
pieces
Activity 3: Tangram
There are 13 possible tangrams
altogether that are in the form of
polygon. Of these 13 tangrams, one is
a triangle, six are quadrilaterals,
two are pentagons, and four are
hexagons.
Try to find all those 13 tangrams.
Activity 3: Tangram
BIG IDEA: How a shape can be
composed and decomposed, or its
relationship to other shapes, provides
insights into the properties of the shape
The geometric thinking involved has to
do with the invariance of area under
moving of the pieces, as well as trying to
imagine a silhouette shape as
decomposed into the puzzle pieces.
Activity 4: Make your own system
Construct your own measurement system “from scratch” according to
following outline:
1. Choose some commonly available object to define the basic
measurement unit. Measure your height with this unit.
2. Define related units of linear measure that are more convenient
for dealing with much bigger and much smaller things. Use them to
specify the distance from Yogyakarta to your home town, the length of
football field, the width of A4 paper,and the thickness of the RME book.
3. Define related units to measure area and volume. Use them to
specify the area of A4 paper and one much larger object of your choice, and
the volume of a bottle for dispenser and one much smaller object of your
choice.
4. Make conversion tables that relate your system to the metric
system.
5. Compare your system to the metric system. In what ways is it
ase send
written
activity
better?the
In what
ways result
is it notof
as this
good?
word ducument to
ilham.rizkianto@yahoo.com
REFLECTION
What have
you learned
?
One quote for you
“Everybody is a genius.
But if you judge a fish by
its ability to climb a tree,
it will live its whole life
believing that it is
Terima Kasih
ขอบคคณ
ຂອບໃຈ
cảm ơn-
អរគគ ណ
Salamat po
&
Measurem
ent
SEAMEO QITEP in MATHEMATICS
Yogyakarta, October 5th & 7th 2013
ILHAM RIZKIANTO
Ilham_rizkianto@uny.ac.id
I see what you mean
Kanisza Figure
Central feature of geometry
A central feature of geometry is
learning to ‘see’, that is, to discern,
geometrical objects and relationships,
and to become aware of relationships
as properties that objects may or may
not satisy.
Let’s move the bookcase
Imagine Considering
moving a bookcase
from one room to
another, but not being
sure if it would fit
through the doorway.
If you did not have a
ruler, how might you
find out without
actually moving it?
The essence of
measurement
The key aspect of any measuring is to
make a comparisons. One way to do
this is to decide on a unit and to have
some way of replicating the unit (or
breaking it up into smaller sub units)
and juxtaposing this with the thing to
be measured.
The connection
Mathematics is not only connected to the world of
numbers.
In Geometry, the issue is to understand the space
around us. It is related to the two- & three- dimensional
world and the related shapes & figures.
Measurement is aimed at quantifying our physical
environment. The emphasize in this process makes
measurement the connecting link between arithmetic and
geometry.
arithmetic
measurement
geometry
Activity 1: Live up the paper
Lay the first sheet down on
the table. Lay the second one
partly on the top of the first,
so that the top-left corner of
the upper piece coincides with
the top-right corner of the
lower piece and the bottomleft corner of the upper piece
coincides with the left edge of
the lower piece.
Repeat this with several more
pieces. What do you notice?
Activity 1: Live up the paper
You may discover that some pieces
coincide. You may be surprised at the
shape you see emerging. This shape
will tell you about angles involved.
Now, try to find at least 4
questions that could be used as
follow up questions for this
activity.
?
Geometry tidbits
Every teacher should have his/her own
special collection of geometric tidbits –
short little puzzles, problems, and
curiosities in geometry to warm up the
class, to gain attention, to involve, to
challenge, to maintain interest, or
simply to give a change of pace.
Geometry tidbits 1
Move just three dots to form an arrow
pointing down instead of up.
Geometry tidbits 2
A solid has this for both its top and
front view. Draw its side view.
Geometry tidbits 3
How many rectangles are in this
figure?
Activity 2: Popcorn Holder
Take two pieces of the paper.
Take the first piece and curl it in
potrait orientation so it forms a
tall, thin cylindrical shape. Take
the other one, do the same but in
landscape orientation so it forms
a shorter and fatter cylindrical
shape.
If you were to fill each of these
cylinders with popcorn, which
one do you think would hold
more? Or, do you think they
would hold the same amount?
Activity 2: Popcorn Holder
This activity is intended to explore the
volume of cylinders with the same
lateral areas and to see the connection
between volume and area.
Now, try to find at least 4
questions that could be used as
follow up questions for this
activity.
?
The core teaching principles of RME
Interactivity
Reality
Level
Activity
Guidance
Intertwinment
Activity 3: Tangram
Mark points at the following
coordinates: (0,0), (0,2), (1,1), (1,3),
(2,2), (2,4), (3,1), (3,3), (4,0), (4,4)
Connect the following points with a line
segment to create a tangram set: (0,0)
and (4,4), (0,2) and (2,4), (1,1) and
(1,3), (1,3) and (4,0), (2,4) and (3,3)
Cut out the tangram pieces and create
an animal or object using all seven
pieces
Activity 3: Tangram
There are 13 possible tangrams
altogether that are in the form of
polygon. Of these 13 tangrams, one is
a triangle, six are quadrilaterals,
two are pentagons, and four are
hexagons.
Try to find all those 13 tangrams.
Activity 3: Tangram
BIG IDEA: How a shape can be
composed and decomposed, or its
relationship to other shapes, provides
insights into the properties of the shape
The geometric thinking involved has to
do with the invariance of area under
moving of the pieces, as well as trying to
imagine a silhouette shape as
decomposed into the puzzle pieces.
Activity 4: Make your own system
Construct your own measurement system “from scratch” according to
following outline:
1. Choose some commonly available object to define the basic
measurement unit. Measure your height with this unit.
2. Define related units of linear measure that are more convenient
for dealing with much bigger and much smaller things. Use them to
specify the distance from Yogyakarta to your home town, the length of
football field, the width of A4 paper,and the thickness of the RME book.
3. Define related units to measure area and volume. Use them to
specify the area of A4 paper and one much larger object of your choice, and
the volume of a bottle for dispenser and one much smaller object of your
choice.
4. Make conversion tables that relate your system to the metric
system.
5. Compare your system to the metric system. In what ways is it
ase send
written
activity
better?the
In what
ways result
is it notof
as this
good?
word ducument to
ilham.rizkianto@yahoo.com
REFLECTION
What have
you learned
?
One quote for you
“Everybody is a genius.
But if you judge a fish by
its ability to climb a tree,
it will live its whole life
believing that it is
Terima Kasih
ขอบคคณ
ຂອບໃຈ
cảm ơn-
អរគគ ណ
Salamat po