Shape and Space
Grades 4–6
1
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
2
Program for the Day
Lunch
• Cartoons
• Where is the Rectangle?
• Transform L
• Fold an Icosahedron
• Mathematics Debates
Reflection and Continuing the Journey
3
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.
4
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]
5
Some Types of Properties
2-D
3-D
•
•
•
•
•
•
•
•
•
Sides
Angles
Diagonals
Symmetry
Concavity
Edges
Vertices
Faces
Symmetry
6
Translations with Visualization as
the Mediator
From
Object
Representation
Language
To
Object
Representation
Language
7
Poly - gon
Many angles (or corners)
Poly - hedron
Many
faces
8
2-D
3-D
Poly - gon
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 penta hexa hepta octa nona deca dodeca icosa -
4
5
6
7
8
9
10
12
20
9
Classification (defined by properties)
Hierarchy (shapes can have more than one
name)
10
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.
11
Quadrilaterals
Planar (flat), closed (“no gaps, no extras”)
shapes with four straight sides.
12
• 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
13
• 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
14
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.
15
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.
16
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
17
Curriculum Outcomes:
The Net Game
4.4 Describe and construct rectangular and
triangular prisms. [C, CN, R, V]
18
The Net Game
1
2
3
4
5
6
19
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]
20
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.
21
Platonic Solids
Tetrahedron has three edges meeting at each of its four
vertices, four triangular faces, six edges.
Fire
Hexahedron (cube) has three edges meeting at each of its
eight vertices, six square faces, twelve edges.
Earth
Octahedron has four edges meeting at each of its six
vertices, eight triangular faces, twelve edges.
Air
Dodecahedron has three edges meeting at each of its
twenty vertices, twelve pentagonal faces, thirty edges.
Water
Icosahedron has five edges meeting at each of its twelve
vertices, twenty triangular faces, thirty edges.
The whole cosmos
22
23
24
25
26
27
28
Pentominoes
29
Pentominoes
30
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]
31
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]
32
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
33
34
35
Big Picture Ideas of
Shape and Space
36
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]
37
38
39
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]
40
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]
41
Tessellations
42
43
44
45
.
46
47
48
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]
49
50
51
Dilation about a Point
52
Dilation about a Horizontal Line
53
Dilation about a Vertical Line
54
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]
55
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]
56
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]
57
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]
58
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.
59
A.
B.
C.
All triangles tessellate.
All quadrilaterals tessellate.
A 2-D shape whose diagonals bisect each other is a
rectangle.
D. A 2-D shape with diagonals equal in length, intersecting at
right angles, is a square.
E. If I make a single planar cut through a cube, the only
shapes I can make are a square, a rectangle, and a
triangle.
F. The diagonals of a cube intersect at right angles.
G. 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
60
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.
61
Complete your big picture ideas of
Shape and Space
62