1. Shape and Space Grades 4–6

2. Program for the Day Welcome and Introductions Black Box Geometry
The Net Game OR Straw and String Construction
Cheese Cubes
Morning Break
Compass and Straight Edge Constructions
Cut a Shape
Create a Copy
Transform a Shape – Tessellations

3. Program for the Day Lunch Cartoons Where is the Rectangle? Transform L
Fold an Icosahedron
Mathematics Debates
Reflection and Continuing the Journey

4. Answers Yes. No. I don’t understand. Please ask again in another way.
I don’t know. Please tell me how I can find that out.

5. Curriculum Outcomes: Black Box Geometry
4.4 Describe and construct rectangular and triangular prisms. [C, CN, R, V]
5.5 Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are parallel, intersecting, perpendicular, vertical and horizontal [C, CN, R, T, V]
5.6 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their attributes. [C, R, V]
6.5 Describe and compare the sides and angles of regular and irregular polygons. [C, PS, R, V]

6. Some Types of Properties
2-D
Sides
Angles
Diagonals
Symmetry
Concavity
3-D
Edges
Vertices
Faces
Symmetry

7. Translations with Visualization as the Mediator
From To
Object
Representation
Language

8. Poly - gon Many angles (or corners) Poly - hedron Many faces

9. 2-D Poly - gon
3-D Poly - hedron
prism - fixed cross-section, cross-sections parallel to base are all congruent shapes, base and top are parallel.
pyramid – apex, cross-sections parallel to base are all geometrically similar shapes
triangle
quadrilateral
tetra - 4
penta - 5
hexa - 6
hepta - 7
octa - 8
nona - 9
deca - 10
dodeca - 12
icosa - 20

10. Classification (defined by properties)
Hierarchy (shapes can have more than one name)

11. Attributes
e.g., thick, pointy, curved, big, long, yellow.
Properties
e.g., has three sides; has four right angles; has diagonals of equal length; has six faces.

12. Quadrilaterals
Planar (flat), closed (“no gaps, no extras”) shapes with four straight sides.

13. has four straight sides, planar and closed so is a quadrilateral
also has a pair of parallel sides so is at least a trapezium
actually has two pairs of parallel sides so is a parallelogram
has right angles so is a rectangle
has all sides equal in length so is a square

14. has four straight sides, planar and closed so is a quadrilateral
also has two pairs of adjacent sides equal in length so is a kite
actually has all sides equal in length so is a rhombus
has a right angle as well so is a square

15. Van Hiele Levels
Level 1. Visual. Children at this level identify and operate on shapes according to appearance. They use the idea of congruency of visual properties and identification is based on these visual properties such as “it is a cube because it looks like a box” or “it is a rectangle because it looks like a door.” At this stage, little attention is given to properties of the shapes. Research done in the United States found that at least half the Grade 6 children were still operating at Level 1.

16. Van Hiele Levels
Level 2. Descriptive/Analytic. At this level the properties of shapes assume precedence with children characterizing shapes by their properties. The focus is on relationships within classes rather than relationships between. A cube is now a cube because it is three dimensional with all faces the same sized squares. Students should be operating at this level when they enter high school and at the early stages of it as a minimum. Only 44 percent of students in the US were operating consistently at Level 2 at Grade 9. Other studies have shown that 40 percent of students complete high school below Level 2.

17. Van Hiele Levels
Level 3. Abstract/Relational. Students are able to deal with abstract definitions, understanding the idea of necessary and sufficient conditions, recognizing a hierarchy, and reasoning about the properties of classes of figures. Logical argument is part of this level. Internationally, most geometry curriculum strive to attain this level.
Clements, 1994, in Grouws: Handbook of Mathematics Education, NCTM

18. Curriculum Outcomes: The Net Game
4.4 Describe and construct rectangular and triangular prisms. [C, CN, R, V]

19. The Net Game 2 1 3 6 5 4

20. Curriculum Outcomes: Straw and String Construction
4.4 Describe and construct rectangular prisms and triangular prisms. [C, CN, R, V]
5.5 Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are parallel, intersecting, perpendicular, vertical and horizontal. [C, CN, R, T, V]

21. Three Basic Types of 3-D Construction
a solid construction using moulding type material or solid material that can be cut
a net construction where faces are joined
a framework construction focusing on the edges.

22. Pentominoes

23. Pentominoes

24. Curriculum Outcomes: Cheese Cubes
5.5 Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are parallel, intersecting, perpendicular, vertical and horizontal. [C, CN, R, T, V]
5.6 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their attributes. [C, R, V]

25. Curriculum Outcomes: Compass and Straight Edge Construction
6.4 Construct and compare triangles, including scalene, isosceles, equilateral, right, obtuse and acute, in different orientations. [C, PS, R, V]

26. Dynamic Geometry Software
Cabri Geometry
The original – designed in France at Grenoble University
Has been used from preschool children to PhD students in mathematics
Geometer Sketchpad
US designed
Used widely in schools, particularly in North America

29. What are the big ideas of Shape and Space?

30. Curriculum Outcomes: Cut a Shape
4.5 Demonstrate an understanding of symmetry by identifying symmetrical 2-D shapes, creating symmetrical 2-D shapes and drawing one or more lines of symmetry in a 2-D shape.
[C, CN, V]
5.6 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their properties. [C, R, V]

33. Curriculum Outcomes: Create a Copy
4.5 Demonstrate an understanding of symmetry by identifying symmetrical 2-D shapes, creating symmetrical 2-D shapes and drawing one or more lines of symmetry in a 2-D shape. [C, CN, V]
5.7 Perform a single transformation (translation, rotation or reflection) of a 2-D shape (with and without technology) and draw and describe the image. [C, CN, T, V]
6.7 Perform a combination of successive transformations of 2-D shapes to create a design, and identify and describe the transformations. [C, CN, T, V]

34. Curriculum Outcomes: Transform a Shape - Tessellations
5.7 Perform a single transformation (translation, rotation or reflection) of a 2-D shape (with and without technology) and draw and describe the image. [C, CN, T, V]
5.8 Identify a single transformation including a translation, a rotation and a reflection of 2-D shapes. [C, T, V]
6.6 Perform a combination of translation(s), rotation(s) and/or reflections(s) on a single 2-D shape, with and without technology, and draw and describe the image. [C, CN, PS, T, V]
6.7 Perform a combination of successive transformations of 2-D shapes to create a design, and identify and describe the transformations. [C, CN, T, V]

35. Tessellations

39. .

42. Curriculum Outcomes: Cartoons
5.7 Perform a single transformation (translation, rotation or reflection) of a 2-D shape (with and without technology) and draw and describe the image.
[C, CN, T, V]

45. Dilation about a Point

46. Dilation about a Horizontal Line

47. Dilation about a Vertical Line

48. Curriculum Outcomes: Where is the Rectangle?
6.8 Identify and plot points in the first quadrant of a Cartesian plane using whole number ordered pairs. [C, CN, V]

49. Curriculum Outcomes: Transform L
5.7 Perform a single transformation (translation, rotation or reflection) of a 2-D shape (with and without technology) and draw and describe the image. [C, CN, T, V]
6.6 Perform a combination of translation(s), rotation(s) and/or reflections(s) on a single 2-D shape, with and without technology, and draw and describe the image.
[C, CN, PS, T, V]
6.8 Identify and plot points in the first quadrant of a Cartesian plane using whole number ordered pairs. [C, CN, V]
6.9 Perform and describe a single transformation of a 2-D shape in the first quadrant of a Cartesian plane (limited to whole number vertices). [C, CN, PS, T, V]

50. Curriculum Outcomes: Fold an Icosahedron
4.4 Describe and construct rectangular prisms and triangular prisms. [C, CN, R, V]
5.6 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their attributes.
[C, R, V]
6.1 Demonstrate an understanding of angles by classifying angles according to their measurement and estimating the measure of angles using 45 degrees, 90 degrees and 180 degrees as reference angles. [C, CN, ME, V]

51. Curriculum Outcomes: Mathematical Debates
5.5 Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are parallel, intersecting, perpendicular, vertical and horizontal. [C, CN, R, T, V]
5.6 Identify and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms and rhombuses, according to their attributes. [C, R, V]
6.4 Construct and compare triangles, including scalene, isosceles, equilateral, right, obtuse and acute, in different orientations.
[C, PS, R, V]
6.5 Describe and compare the sides and angles of regular and irregular polygons. [C, PS, R, V]
6.6 Perform a combination of translation(s), rotations(s) and/or reflection(s) on a single 2-D shape, with and without technology, and draw and describe the image. [C, CN, PS, T, V]

52. Mathematics Debates For each of the statements, decide whether it is:
always true
sometimes true – and specify conditions when it is true
never true.
You must be able to justify your answer.

53. All triangles tessellate.
All quadrilaterals tessellate.
A 2-D shape whose diagonals bisect each other is a rectangle.
A 2-D shape with diagonals equal in length, intersecting at right angles, is a square.
If I make a single planar cut through a cube, the only shapes I can make are a square, a rectangle, and a triangle.
The diagonals of a cube intersect at right angles.
There are only five solids (Platonic solids) which are regular in every respect, so they have the same regular polygon for all faces and all angles between faces are the same

54. Always true
Fold arms
Sometimes true – and when it is true
Clasp hands
Never true
Hands flat on table
Don’t think?
Hands on head – massage brain.

55. Complete your big picture ideas of Shape and Space