Geometry

Presented byFrank H. Osborne, Ph. D.

© 2015

EMSE 3123Math and Science in Education

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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 90ْ.

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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.– 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.– 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 problem-solving 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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