Instructional Materials in MathematicsBy Mary Caryl Yaun

Grid Board

Grid Board

Pedagogical Uses

• Perimeter of Plane Figures

• Area of Plane Figures

• Coordinate System

• Graphing Functions

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.

Grid Board

How to Use

PERIMETER of Plane Figures1.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.

Grid Board

How to Use

AREA of Plane Figures1.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.

Modified GeoBoard

Modified GeoBoard

Pedagogical Uses

• Plane Figures• Transformations• Similarity• Coordination• Counting• Right Angles• Pattern

• Classification• Scaling• Position• Congruence• Area• Perimeter

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.

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.

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.

Modified GeoBoard

How to Use

Area of Arbitrary Polygon1. 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.

Modified GeoBoard

How to Use

Area of Arbitrary Polygon4. 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.

Fraction Slider

Fraction Slider

Pedagogical Uses

• Addition of Fractions

• Subtraction of Fractions

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.

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.

Fraction Slider

How to Use

Addition and Subtraction of Fractions3. 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.

Number Slider

Number Slider

Pedagogical Uses

• Addition of Integers

• Subtraction of Integers

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.

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.

Number Slider

How to Use

Addition and Subtraction of Integers3. 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).

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).

Number Slide

How to Use

Addition and Subtraction of Integers

6. The ending point of the 2nd number bar is the answer.

Algebra Tiles

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

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.

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

Algebra TilesPositive Algebra Tiles Negative Algebra Tiles

x2 Tiles

x Tiles

Units Tiles

Algebra Tiles

How to Use

Additive Inverse

- a combined positive and negative tile of the same area produces a zero pair.

= 0

= 0

Algebra Tiles

Addition of Integers

2 + 1 =

3 + (-1) =

+

+

= 3

= 2

Algebra Tiles

Subtraction of Integers

-5 - (-2) =

3 - (-4) = -

+ = 7

-= - 3

+

Algebra Tiles

Modeling Linear Expressions

3x

3x - 6

Algebra Tiles

Simplifying Polynomials

Simplify 2x + 4 + x +2

= = 3x + 6

=

Algebra Tiles

Finding the Value of x

2x - 4 = 8

Algebra Tiles

Finding the Value of x

x = 2

Algebra Tiles

Substitution

Evaluate 3 + 2x if x = 4

11

Fraction Pie

Fraction Pie

Pedagogical Uses

• Identifying Fractions

• Circumference of a Circle

• Area of a Circle

• Perimeter of a Parallelogram

Fraction Pie

Objectives

This instructional material will help the students to:

1. find the circumference of the circle2. find the perimeter of the parallelogram

using the circumference of a circle3. explain the relationship between the circle

and parallelogram.

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

Perimeter and Area

Perimeter and AreaPedagogical Uses

DistancePolygonsPerimeter of polygonsArea of polygons

Perimeter and Area

Objective

This power point presentation will help the students to:1. measure the perimeter of polygons2. measure the area of polygons

PerimeterIt 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.

PerimeterExample:

P – perimeterS – side

5

10

5

10

Let:

AreaA 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.

AreaExample:

A – areaS – side

5

10

Let :

Triangle

Perimeter Area

ha

b

c

Example

105

12

4

Example

510

12

4

Square

Perimeter Area

a

Example

3

Rectangle

Perimeter Area

a

b

Example

3

13

Parallelogram

Perimeter Area

a h

b

Example

7 5

11

Rhombus

Perimeter Area

m

n

h

a

Example

8

11

6

Trapezoid

Perimeter Area

hc

a

b

d

Example

34

5

7

4

Example

34

5

7

4

Circle

Circumference Area

d

r

d = diameterr = radiusπ = 3.14

Let:

Example

4

Platonic Solids

Mensuration of Solid Figures

Platonic Solids Plus Sphere

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.

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.

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:

History1. The tetrahedron was the shape of fire, because fire

is sharp and stabbing2. 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”.

Tetrahedron It has:

• 4 faces (triangle)

• 4 vertices• 6 edges

TetrahedronLet: a = Edge

S = Surface Area V = Volume

a

Tetrahedron

5

Example:

Hexahedron or Cube It has:

• 6 faces (squares)• 8 vertices• 12 edges

Hexahedron or CubeLet:

a = Edge S = Surface Area V = Volume D = Diagonal

a

Hexahedron or CubeExample:

5

Octahedron It has:

• 8 faces (triangles)• 6 vertices• 12 edges

OctahedronLet:

a = Edge S = Surface Area V = Volume

a

OctahedronExample:

5

Dodecahedron It has:

• 12 faces (pentagons)• 20 vertices• 30 edges

DodecahedronLet:

a = Edge S = Surface Area V = Volume

a

DodecahedronExample:

5

Icosahedron It has:

• 20 faces (triangles)• 12 vertices• 30 edges

IcosahedronLet:

a = Edge S = Surface Area V = Volume

a

IcosahedronExample:

5

SphereA sphere is a solid described y the revolution of a semi circle about a fixed diameter.

SphereProperties 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.

SphereLet:

r = Radius S = Surface Area V = Volume

SphereExample:

5

Archimedean Solids

Archimedean Solids

Pedagogical Uses

• Polygons

• Volume

• Surface Area

• Tessellation

• Mathematical Investigation

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

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.

Geometry in Metro Cebu

Horizons 101

Magellan’s Cross

Fort San Pedro

Fort San Pedro

Plaza

Independencia

Mariners’ Court

Compania

Compania

Maritima

Maritima

Iglesia ni Cristo

Crown Regency

Thank you

Photos Credit to: Fate Jacaban