1. EMSE 3123 Math and Science in Education
Geometry
Presented by Frank H. Osborne, Ph. D. © 2015

2. 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.

3. 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

4. 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.

5. 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)

6. 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.

7. 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.

8. Two-dimensional Shapes
Some common planar shapes

9. Three-dimensional Shapes
Some common solid shapes

10. 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.

11. Naming Shapes
Children learn to name shapes correctly despite their orientation or overall size.
All of these are triangles despite size or shape.

12. 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.

13. Naming and Positioning Shapes
Children describe positions of objects using above, below, beside, in front of, behind and next to.

14. Naming and Positioning Shapes
Children describe positions of objects using above, below, beside, in front of, behind and next to.

15. Naming and Positioning Shapes
Children describe positions of objects using above, below, beside, in front of, behind and next to.

16. Working with Shapes Children learn the terminology of shapes.
Later on, develop their vocabulary by using the term “vertex” instead of “corner.”

17. How Many?
How many sides and corners does each have?

18. How Many? How many sides and corners does each have?
The circle has a curve, but not sides because sides are straight lines.

19. Working with Shapes
Solid shapes also have terminology.

20. Attributes of Solid Figures
What attributes does each have?

21. Attributes of Solid Figures
What attributes does each have?

22. 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.

23. 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.

24. 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.

25. Reasoning with Shapes and their Attributes
First grade
Compose two-dimensional shapes
Three-dimensional shapes can be treated in a similar fashion.

26. 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.

27. 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.

28. 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.

29. Reasoning with Shapes and their Attributes
Properties of quadrilaterals

30. 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.

31. Reasoning with Shapes and their Attributes
Second grade
Recognize that equal shares of identical wholes need not have the same shape.

32. 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.

33. 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.

34. Lines, Angles and Shapes
In fourth grade
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

35. Lines, Angles and Shapes
In fourth grade
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

36. Lines, Angles and Shapes
In fourth grade
Identify parallel lines in plane figures.

37. Lines, Angles and Shapes
In fourth grade
Identify types of triangles.

38. Lines, Angles and Shapes
In fourth grade
Identify symmetry.

39. 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.

40. 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.

41. 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.

42. 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.

43. 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.

44. 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

45. 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.

46. Graphing on the Coordinate Plane
So we enlarge the first quadrant and use only it for our graphing.

47. Graphing on the Coordinate Plane
We can find the area of a rectangle as shown.

48. Hierarchy of Plane Figures
As an example, we study quadrilaterals.

49. The End