Transcript Geometry

EMSE 3123
Math and Science in Education
Geometry
Presented by
Frank H. Osborne, Ph. D.
© 2015
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Geometry
• Geometry is the only domain in the Common
Core Standards that spans all grade levels.
• Lack of understanding or weakness in
knowledge of geometry is a very serious
problem.
• Teachers of elementary children need to pay
attention to teaching geometry or working it
into lessons about other areas of math and
science.
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Geometry
• The main points of content in geometry in
grades K-3 are:
– Identifying and describing shapes
– Analyzing, comparing, creating and composing
shapes
– Reasoning with shapes and their attributes
• In grade 4, children begin learning about lines
and angles
• In grade 5, graphing begins
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Pre-Number Concepts
Earlier we learned that
• Children need practice in working with a
variety of manipulatives together.
• The manipulatives are sorted according
to a particular attribute.
• Attributes can be size, color, shape,
thickness, length, etc.
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Example
• Use colored attribute blocks or shapes.
• Pick out all with a given shape.
• Later activities can concentrate on two
attributes (e.g. red with four corners)
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Example
• Use attribute blocks or shapes to
construct the shape below. Have the
students duplicate the shape.
• Ask them to tell you the names of each of
the parts they used.
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Shapes
• Shapes can be two-dimensional (flat) or
three-dimensional (solid). Flat shapes are
also known as planar because they lie in
one plane.
• Children learn to identify and describe
shapes (squares, circles, triangles,
rectangles, hexagons, cubes, cones,
cylinders, and spheres).
• They also study the effects of rotating,
moving or positioning shapes.
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Two-dimensional Shapes
• Some common planar shapes
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Three-dimensional Shapes
• Some common solid shapes
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Naming Shapes
• Children learn to name shapes correctly
despite their orientation or overall size.
• The rectangle is still the same even
though it has been rotated 90‫ﹾ‬.
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Naming Shapes
• Children learn to name shapes correctly
despite their orientation or overall size.
• All of these are triangles despite size or
shape.
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Naming and Positioning Shapes
• Children describe shapes of objects in the
environment using shape terminology.
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A door is a rectangle.
They use orange cones to mark the street.
A box is a rectangular prism.
Ice cream is put into a cone.
Earth is a sphere. So is a ball.
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Naming and Positioning Shapes
• Children describe positions of objects
using above, below, beside, in front of,
behind and next to.
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Naming and Positioning Shapes
• Children describe positions of objects
using above, below, beside, in front of,
behind and next to.
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Naming and Positioning Shapes
• Children describe positions of objects
using above, below, beside, in front of,
behind and next to.
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Working with Shapes
• Children learn the terminology of shapes.
• Later on, develop their vocabulary by
using the term “vertex” instead of
“corner.”
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How Many?
• How many sides and corners does each have?
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How Many?
• How many sides and corners does each have?
• The circle has a curve, but not sides because
sides are straight lines.
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Working with Shapes
• Solid shapes also have terminology.
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Attributes of Solid Figures
• What attributes does each have?
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Attributes of Solid Figures
• What attributes does each have?
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Creating with Shapes
• Children model shapes in the world by
building models out of components.
• Also try three-dimensional objects like
bocks, balls and sticks, and others.
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Patterns of Shapes
• Given the pattern on the left, complete
the pattern on the right.
• Activities like these are most important
for Kindergarten and First Grade but
can be modified for use in higher grades.
• Children should also be able to sort
things and also place them in sequence.
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Reasoning with Shapes and their Attributes
• First grade
• Distinguish between defining attributes and
non-defining attributes.
– Defining Attributes: A triangle is a closed figure
with three sides an three angles.
– Non-defining Attributes: Color, orientation,
overall size.
• What are defining attributes of the other
shapes?
• Internalize attributes by building and
drawing shapes.
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Reasoning with Shapes and their Attributes
• First grade
• Compose two-dimensional shapes
• Three-dimensional shapes can be treated in a
similar fashion.
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Reasoning with Shapes and their Attributes
• First grade
• Partition a rectangle into equal units that can
be counted.
• Divide a circle into halves and quarters.
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Reasoning with Shapes and their Attributes
• First grade
• Partition circles and rectangles into two and
four equal shares.
– Describe the shares using the words halves,
fourths, and quarters, and use the phrases half of,
fourth of, and quarter of.
– Describe the whole as two of or four of the shares
• Understand for these examples that the
decomposition into more and more equal
shares creates smaller shares.
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Reasoning with Shapes and their Attributes
• Second grade
• Students can recognize and draw shapes
having specified attributes, such as a given
number of angles or a given number of equal
faces.
• Identify triangles, quadrilaterals, pentagons,
hexagons and cubes.
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Reasoning with Shapes and their Attributes
• Properties of quadrilaterals
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Reasoning with Shapes and their Attributes
• Second grade
• Partition a rectangle into rows and columns
of same-size squares and count to find the
total number of them.
• Partition circles and rectangles into two,
three, or four equal shares, describe the
shares using the words halves, thirds, half of,
a third of, etc., and describe the whole as two
halves, three thirds, four fourths.
• Recognize that equal shares of identical
wholes need not have the same shape.
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Reasoning with Shapes and their Attributes
• Second grade
• Recognize that equal shares of identical
wholes need not have the same shape.
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Reasoning with Shapes and their Attributes
• Third grade
• Understand that shapes in different categories
(e.g., rhombuses, rectangles, and others) may
share attributes (e.g., having four sides), and
that the shared attributes can define a larger
category (e.g., quadrilaterals).
• Recognize rhombuses, rectangles, and squares
as examples of quadrilaterals, and draw
examples of quadrilaterals that do not belong to
any of these subcategories.
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Reasoning with Shapes and their Attributes
• Third grade
• Partition shapes into parts with equal areas.
• Express the area of each part as a unit
fraction of the whole. For example, partition
a shape into 4 equal parts with equal area, and
describe the area of each part of 1/4 of the area
of the shape.
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Lines, Angles and Shapes
• In fourth grade
• Draw and identify lines and angles, and
classify shapes by properties of their lines
and angles.
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Lines, Angles and Shapes
• In fourth grade
• Draw and identify lines and angles, and
classify shapes by properties of their lines
and angles.
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Lines, Angles and Shapes
• In fourth grade
• Identify parallel lines in plane figures.
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Lines, Angles and Shapes
• In fourth grade
• Identify types of triangles.
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Lines, Angles and Shapes
• In fourth grade
• Identify symmetry.
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Graphing on the Coordinate Plane
• In fifth grade
• Graphing on the coordinate plane is used to
solve real-world and mathematical problems.
• We develop the coordinate plane starting
with the number line.
• The number line extends from -∞ to +∞ and
is called the x-axis.
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Graphing on the Coordinate Plane
• In fifth grade
• We can also make the
number line vertical.
• We call this the y-axis.
• It is perpendicular to
the x-axis.
• We arrange the two
perpendicular axes in
such a way that they
cross at the point of
zero (0) on each one.
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Graphing on the Coordinate Plane
• In fifth grade
• The x-axis and the y-axis cross each other at
the point where each axis equals zero. This is
called the origin.
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Graphing on the Coordinate Plane
• In fifth grade
• Each location on the coordinate plane has its
own location. These are given as (x, y) pairs.
• To locate the coordinates, you count from the
origin.
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For positive x values count to the right.
For negative x values count to the left.
For positive y values count up.
For negative y values count down.
• All points on the coordinate plane can be
located this way.
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Graphing on the Coordinate Plane
• We can plot some
points on the grid.
• A (2, 1)
– Right 2, Up 1
• B (-2, 1)
– Left 2, Up 1
• C (-2, -1)
– Left 2, Down 1
• D (2, -1)
– Right 2, Down 1
• Result is a little
4x2 rectangle.
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Graphing on the Coordinate Plane
• The axes divide the
grid into four
quadrants as shown.
• Quadrant I: x and y
are both positive
• Quadrant II: x is
negative, y is positive
• Quadrant III: x and y
are both negative
• Quadrant IV: x is
positive, y is negative
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Graphing on the Coordinate Plane
• The Grade 5
standards indicate
that much graphing
and problemsolving work should
be done in the first
quadrant of the
grid.
• At this point, we do
not need the shaded
parts of the grid.
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Graphing on the Coordinate Plane
• So we enlarge the first quadrant and use only
it for our graphing.
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Graphing on the Coordinate Plane
• We can find the area of a rectangle as shown.
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Hierarchy of Plane Figures
• As an example, we study quadrilaterals.
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The End
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