The document describes several instructional materials for teaching mathematics concepts: 1. Grid board, modified geoboard, fraction slider, and number slider are used to teach perimeter, area, fractions, and integers. 2. Algebra tiles are used to model linear expressions, solve equations, and simplify polynomials. 3. Fraction pie relates fractions to circle circumference and parallelogram perimeter. 4. A powerpoint on perimeter and area teaches calculating these values for polygons and circles. 5. Models of the platonic solids, sphere, and archimedean solids are used to investigate their properties like surface area and volume.

1. Instructional Materials
in Mathematics
By Mary Caryl Yaun

2. Grid Board

3. Grid Board
Pedagogical Uses
• Perimeter of Plane Figures
• Area of Plane Figures
• Coordinate System
• Graphing Functions

4. Grid Board
Objectives
This instructional material will help the
students:
1.To better understand the concepts of
perimeter and area of plane figures
2.To differentiate the difference between
perimeter and area of plane figures
3.To solve for the perimeter and area of
plane figures.

5. Grid Board
How to Use
PERIMETER of Plane Figures
1.Determine the desired figure to illustrate.
2.Use white board pen to shade the
corresponding square units of the desired
figure.
3.Count the number of sides in the boundary
of the figure. The resulting number will be
the perimeter of the figure.

6. Grid Board
How to Use
AREA of Plane Figures
1.Determine the desired figure to illustrate.
2.Use white board pen to shade the
corresponding square units of the desired
figure.
3.Count the number of square units of the
entire figure. The resulting number will be
the area of the figure.

7. Modified GeoBoard

8. Modified GeoBoard
Pedagogical Uses
• Plane Figures
• Transformations
• Similarity
• Coordination
• Counting
• Right Angles
• Pattern
• Classification
• Scaling
• Position
• Congruence
• Area
• Perimeter

9. Modified GeoBoard
Objective
This instructional material will help the
students to:
1. form different plane figures both regular
and irregular
2. find the perimeter of regular and irregular
polygons
3.find the area of regular and irregular
polygons.

10. Modified GeoBoard
How to Use
Perimeter of Regular and Irregular Polygon
1. Using a white board pen, draw a polygon
on the modified geoboard.
2. Count the number of points around the
polygon. The resulting number is the
perimeter of the polygon.

11. Modified GeoBoard
How to Use
Area of Regular Polygon
1. Using a white board pen, draw a regular
polygon on the modified geoboard.
2. Count the square units inside the
polygon. The resulting number is the area
of the regular polygon.

12. Modified GeoBoard
How to Use
Area of Arbitrary Polygon
1. Using a white board pen, draw an
arbitrary polygon on the modified
geoboard.
2. Count the number of points that touches
the boundary lines.
3.Count the points inside the polygon.

13. Modified GeoBoard
How to Use
Area of Arbitrary Polygon
4. Divide the number of boundary points by
2.
5. Add the quotient in #4 with the number of
points inside the polygon.
6. Subtract 1 from the sum in #5. The
resulting difference will be the area of the
arbitrary polygon.

14. Fraction Slider

15. Fraction Slider
Pedagogical Uses
• Addition of Fractions
• Subtraction of Fractions

16. Fraction Slider
Objectives
This instructional material will help the
students to:
1. understand the concept of adding and
subtracting of fractions
2. perform addition of fractions
3. perform subtraction of fractions.

17. Fraction Slider
How to Use
Addition and Subtraction of Fractions
1. Take the fraction bars that correspond to
the given fraction
2. Slide in the 1st fraction bar (the bigger
number). Align its left side to the origin.

18. Fraction Slider
How to Use
Addition and Subtraction of Fractions
3. From the ending point of the 1st bar, slide
in the 2nd fraction bar in the direction
indicated by the 2nd number or addend
(left if the number is negative; right if the
number is positive).
4. The ending point of the 2nd bar is the
answer.

19. Number Slider

20. Number Slider
Pedagogical Uses
• Addition of Integers
• Subtraction of Integers

21. Number Slider
Objectives
This instructional material will help the
students to:
1. understand the process of adding and
subtracting integers
2. perform addition of integers
3. perform subtraction of integers.

22. Number Slider
How to Use
Addition and Subtraction of Integers
1. Attach the blank number line to the
fraction slider.
2. Write the necessary numbers on the
blank number line as well as on the
number bars.

23. Number Slider
How to Use
Addition and Subtraction of Integers
3. Take the number bars that corresponds to
the given.
4. Slide in the 1st number bar. Align it to the
origin (towards the left if the 1st number is
negative; towards the right if the 1st
number is positive).

24. Number Slider
How to Use
Addition and Subtraction of Integers
5. Slide in the 2nd number bar. Align it to the
ending point of the 1st number bar
(towards the left if the 2nd number is
negative; towards the right if the 2nd
number is positive).

25. Number Slide
How to Use
Addition and Subtraction of Integers
6. The ending point of the 2nd number bar is
the answer.

26. Algebra Tiles

27. Algebra Tiles
Pedagogical Uses
• Addition and Subtraction of Integers
• Modeling Linear Expressions
• Solving Linear Equations
• Simplifyings Polynomials
• Solving Equations for Unknown Variable
• Multiplication and Division of Polynomials
• Completing the Square
• Investigations

28. Algebra Tiles
Objectives
This instructional material will help the
students to:
1. Associate linear expressions with
concrete objects, specifically tiles
2. Solve addition and subtraction of integers
using tangible materials
3. Simplify polynomials using
representations.

29. Algebra Tiles
Each tile represents an area:
Area of large square = x(x) = x2
Area of rectangle = 1(x) = x
Area of small square = 1(1) = 1
x
x
x
1
1
1

30. Algebra Tiles
Positive Algebra Tiles Negative Algebra Tiles
x2
Tiles
x Tiles
Units Tiles

31. Algebra Tiles
How to Use
Additive Inverse
- a combined positive and negative tile of
the same area produces a zero pair.
= 0
= 0

32. Algebra Tiles
Addition of Integers
2 + 1 =
3 + (-1) =
+
+
= 3
= 2

33. Algebra Tiles
Subtraction of Integers
-5 - (-2) =
3 - (-4) = -
+ = 7
-
= - 3
+

34. Algebra Tiles
Modeling Linear Expressions
3x
3x - 6

35. Algebra Tiles
Simplifying Polynomials
Simplify 2x + 4 + x +2
= = 3x + 6
=

36. Algebra Tiles
Finding the Value of x
2x - 4 = 8

37. Algebra Tiles
Finding the Value of x
x = 2

38. Algebra Tiles
Substitution
Evaluate 3 + 2x if x = 4
11

39. Fraction Pie

40. Fraction Pie
Pedagogical Uses
• Identifying Fractions
• Circumference of a Circle
• Area of a Circle
• Perimeter of a Parallelogram

41. Fraction Pie
Objectives
This instructional material will help the
students to:
1. find the circumference of the circle
2. find the perimeter of the parallelogram
using the circumference of a circle
3. explain the relationship between the circle
and parallelogram.

42. Fraction Pie
How to Use
Finding the relationship between a circle
and a parallelogram
The radius of a circle
is the height of the
parallelogram and
the base of a
parallelogram is the
circumference of a
circle

43. Perimeter
and Area

44. Perimeter and Area
Pedagogical Uses
Distance
Polygons
Perimeter of polygons
Area of polygons

45. Perimeter and Area
Objective
This power point presentation will
help the students to:
1. measure the perimeter of polygons
2. measure the area of polygons

46. Perimeter
It is the length of the boundary of a
closed figure.
-Concise Mathematics Dictionary
It is measured by adding all the
sides of the plane figure.

47. Perimeter
Example:
P – perimeter
S – side
5
10
5
10
Let:

48. Area
A measure of the extent of a surface, or
the part of surface enclosed by some
specified boundary.
-Concise Mathematics Dictionary
It is measured by the number of
squares inside the figure.

49. Area
Example:
A – area
S – side
5
10
Let :

50. Triangle
Perimeter Area
h
a
b
c

51. Example
10
5
12
4

52. Example
5
10
12
4

53. Square
Perimeter Area
a

54. Example
3

55. Rectangle
Perimeter Area
a
b

56. Example
3
13

57. Parallelogram
Perimeter Area
a h
b

58. Example
7 5
11

59. Rhombus
Perimeter Area
m
n
h
a

60. Example
8
11
6

61. Trapezoid
Perimeter Area
h
c
a
b
d

62. Example
3
4
5
7
4

63. Example
3
4
5
7
4

64. Circle
Circumference Area
d
r
d = diameter
r = radius
π = 3.14
Let:

65. Example
4

66. Platonic Solids

67. Mensuration of Solid Figures
Platonic Solids Plus Sphere

68. Platonic Solids
 The term platonic solids refers to the
regular polyhedra. In geometry, a
polyhedron (the word is a Greek
neologism meaning many seats) is a
solid bounded by plane surfaces, which
are called the faces; the intersection
of three or more edges is called a
vertex.
 What distinguishes regular polyhedra
from all others is the fact that all of
their faces are congruent with one
another.

69. History
 Pythagoras, in 5th
BCE, knew of the
tetrahedron, hexahedron or cube, and
dodecahedron. He learned these solids from
Egyptian Sacred Geometry.
 Plato in his ‘Theaetetus’ dialogue, a
discussion around the question ”What is
knowledge?” that dates to about 369 BCE, added
the octahedron and the icosahedron.
Collectively, these 5 solids became known as
the Platonic Solids, after this ancient Greek
philosopher.

70. History
 Plato speculated that these 5 solids were the
shapes of the fundamental components of the
physical universe, what he called the “Theory
of Everything”.
 In this theory, the world was composed of
entirely of 4 elements: fire, air, water and
earth and each of the elements were made up
of tiny fundamental particles. This was the
precursor to the atomic theory. The shapes
or particles that he chose for the elements
were the Platonic Solids and the intuitive
justification for these associations were:

71. History
1. The tetrahedron was the shape of fire, because fire
is sharp and stabbing
2. The octahedron was like air, its miniscule
components being so smooth that one could barely
feel them.
3. Water was made up of icosahedra, which are the most
smooth and round of the Platonic Solids that flows
out of one’s hands when picked up as if made of
tiny balls.
4. Earth consisted of hexahedron, which are solid and
sturdy and highly un-spherical.
5. The dodecahedron left unmatched to one of the four
elements, had 12 faces that Plato decided
corresponded with the 12 constellations of the
Zodiac and was the symbol of ether or universe or
spirit, writing “God used this solid for the whole
universe, embroidering figures on it”.

72. Tetrahedron
 It has:
• 4 faces
(triangle)
• 4 vertices
• 6 edges

73. Tetrahedron
Let:
a = Edge
S = Surface Area
V = Volume
a

74. Tetrahedron
5
Example:

75. Hexahedron or Cube
 It has:
• 6 faces (squares)
• 8 vertices
• 12 edges

76. Hexahedron or Cube
Let:
a = Edge
S = Surface Area
V = Volume
D = Diagonal
a

77. Hexahedron or Cube
Example:
5

78. Octahedron
 It has:
• 8 faces (triangles)
• 6 vertices
• 12 edges

79. Octahedron
Let:
a = Edge
S = Surface Area
V = Volume
a

80. Octahedron
Example:
5

81. Dodecahedron
 It has:
• 12 faces (pentagons)
• 20 vertices
• 30 edges

82. Dodecahedron
Let:
a = Edge
S = Surface Area
V = Volume
a

83. Dodecahedron
Example:
5

84. Icosahedron
 It has:
• 20 faces (triangles)
• 12 vertices
• 30 edges

85. Icosahedron
Let:
a = Edge
S = Surface Area
V = Volume
a

86. Icosahedron
Example:
5

87. Sphere
A sphere is a solid described y the
revolution of a semi circle about a
fixed diameter.

88. Sphere
Properties of a sphere:
A sphere has a center.
All the points on the surface of the sphere
are equidistant from the center.
The distance between the center and any
point on the surface of the sphere is the
radius of the sphere.

89. Sphere
Let:
r = Radius
S = Surface Area
V = Volume

90. Sphere
Example:
5

91. Archimedean Solids

92. Archimedean Solids
Pedagogical Uses
• Polygons
• Volume
• Surface Area
• Tessellation
• Mathematical Investigation

93. Archimedean Solids
Objectives
The model will help the students to:
1. identify the relationship between the
platonic and archimedean solids
2. investigate the surface area of the
archimedean solids

94. Archimedean Solids
How to Use
1. Use it as a model to investigate the
relationship between the archimedean and
platonic solids.
2. Allow the students to measure the surface
area of the archimedean solids.

95. Geometry in Metro Cebu

96. Horizons 101

97. Magellan’s Cross

98. Fort San Pedro

99. Fort San Pedro

100. Plaza
Independe
ncia

101. Mariners’ Court

102. Compania
Compania
Maritima
Maritima

103. glesia ni Cristo
Crown Regency

104. Thank you
Photos Credit to:
Fate Jacaban