Geometry & Measurement

SEAMEO QITEP in MATHEMATICSYogyakarta, October 5th & 7th 2013

ILHAM RIZKIANTO

Kanisza Figure

I see what you mean

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.

Central feature of geometry

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?

Let’s move the bookcase

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 essence of measurement

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.

The connection

arithmeticmeasurement

geometry

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 bottom-left 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.

Activity 1: Live up the paper

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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

Move just three dots to form an arrow pointing down instead of up.

Geometry tidbits 1

A solid has this for both its top and front view. Draw its side view.

Geometry tidbits 2

How many rectangles are in this figure?

Geometry tidbits 3

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.

Activity 2: Popcorn Holder

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The core teaching principles of RME

Activity

Reality Level

Intertwinment

Interactivity

Guidance

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

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 shapeThe 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 3: Tangram

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 better? In what ways is it not as good?

Activity 4: Make your own system

Please send the written result of this activity in word ducument to

ilham.rizkianto@yahoo.com

What have you learned ?

REFLECTION

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 stupid.”― Albert Einstein

Terima Kasihขอบคุ�ณຂອບໃຈcảm ơn-

អរគុ�ណSalamat po