Instrumentation in mathematics (Instructional Materials)

INSTRUMENTATION in MATHEMATICS

By: Lara Katrina Kaamino

Manipulatives in Mathematics

manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it

The use of manipulatives provides a way for students to learn concepts in a developmentally appropriate, hands-on and an experiencing way

These are examples of manipulatives in mathematics

Content (Manipulatives)

Fraction Law of signed number Geoboard (with PPT) Platonic Solids (with PPT) Archimedean Solids Pie Chart Algebra Tiles

Fraction

Thinking Blocks Manipulative

Materials Used:

Chipboard Glue Colored paper Transparent tape Ruler Pencil Scissors

Objectives

To help student develop the ability of solving fraction in an easy and interesting way

To motivate students to learn fraction with the use of this manipulative

Pedagogical use: This instructional material is

composed of two manipulatives: (1) board and (2) tiles

Thinking Blocks Fraction Manipulative teaches students how to model and solve word problems involving fractions and whole numbers.

Thinking Blocks Fraction Manipulative is ideal for students who are learning model drawing strategies. Thinking Blocks Fraction manipulative is best suited for students ages 10+.

Procedure:1. Identify the total number of

parts (fraction) then build a model using the same tile color.

Example: The whole is

55

45

35

15

25

55

2. If you are asked to add and then simply put the corresponding number of tile (numerator) on top of the tile you constructed in step 1.

15

25

55

45

35

15

25

15

25

Law of Signed Number

Law of Signed Number Flash Cards

Materials used:

Chipboard Glue Colored paper Transparent tape Marker Ruler Pencil Scissors

Objectives

To help students master the basic operation

To motivate students learn the law of signed number with the use of this manipulative

To develop speed an accuracy in solving equation with the basic operation

Pedagogical use: Law of signed number flashcards can be use in drills and practices. This can help students develop speed and accuracy in performing the basic operation.

Procedure: Just like the typical flashcards,

the teacher flashes the given equation and the students should immediately answer the equation presented.

The signs and the operation can also be change as well as the given numbers

Geoboard

Improvised Geoboard

Materials used:

Chipboard Colored paper Glue Pencil Ruler Scissors Puncher for the dots

Objectives

To help students solve the area and perimeter of a given polygon in a fun and interesting way

To aid the teacher in making the students understand the lesson on area and perimeter of a polygon in an easy manner

Pedagogical use: Improvised geoboard is

a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons

It consist of a physical board with a certain number of dots used in finding the area and perimeter of a certain polygon

Procedure: Just like the typical geoboard, the

teacher will create a polygon out of the dots but the only difference is that instead of using a rubber band the teacher will use white board pen to draw a polygon on the improvised geoboard.

PowerPoint Presentations

Area and Perimeter of

Plane Figures

A square is a figure with 4 equal sides.

Square

d

Area =¿𝑎2

Perimeter = 4(side) =4a

diagonal=

a

A rectangle is any quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (90°).

Rectangle

Area= base x altitude

= ba or ab

Perimeter= 2a+ 2b

=2(a+b)

diagonal=

b

ad

A Rhombus is a shape with 4 equal straight sides. Opposite sides are parallel, and opposite angles are equal. And the two diagonals of a rhombus bisect each other at right angles.

Rhombus

D

A

90 °

𝑑1

𝑑2

B

C

Given Diagonalsand : Area=

Given side a and one angle: Area=

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel

Parallelogram

𝜃

A B

C

𝑑1 𝑑2

D

Given diagonals and and an included angle : Area=

Given two side a and b and one angle A:Area=ab sin A

A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides parallel.

Trapezoid

a

b

Area=

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same.

Triangle

Given base b and height h:Area=

Give three sides: Area = where s= semi perimeter

s=

h

b

ac

𝜃

Given two sides and an included angle: Area=

Given three angles A, B, C, and one side a:

Area =

Circle is a set of all points in a plane that are at a given distance from a given point, the center; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.

Circle

Area = x

r= radius

Circumference= 2

Area and Perimeter of

Irregular Figures

How will you find the Area and Perimeter of this Irregular Figure?

a e

cb

g

d

f

h XYZ

a e

cb

g

d

f

h

Perimeter

P= a+b+c+d+e+f+g+h

1cm 1 cm

4 cm

1cm

6cm

1c

m

4cm4cm

P = 1+1+4+1+1+4+6+4 =22

XYZ

Area1.Find the area

of Plane Y, X and Z

2.The sum of the area of plane Y, X and Z is the total area of the Irregular Figure

a e

cb

g

d

f

h ij

Z

Y

X

𝐴𝑋= h𝑎

𝐴𝑍=𝑒𝑓𝐴𝑌=𝑐𝑖

𝐴𝑋𝑌𝑍=𝐴𝑋+𝐴𝑌+ 𝐴𝑍

1 cm 1 cm

4 cm

1 cm

6 cm

1 c

m

4 cm

4cm 3 cm3 cm

Z

Y

X = 1 = 4

= 1 = 4

= 4 = 12

= 4 + 12 + 4 = 20

Using Geoboard in Finding

the Area and Perimeter of an Irregular Figure

A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons. It consists of a physical board with a certain number of nails half driven in, around which are wrapped rubber bands.

Using Geoboard find the perimeter and area of the irregular figure

In finding the perimeter:Count the number of nails surrounding the shape formed by your rubber band and the total number of the nails corresponds the perimeter of the shape

GEOBOARD

Perimeter= 22

In finding the area:

Every four nails of the shape formed by the rubber band serves as one unit. Count the number of the units inside the shape and the total units corresponds to the area of the shape.

Area =20

Find the area and perimeter and area of the following figures using your Geoboard:

Platonic Solids and Archimedean Solids

Miniature Platonic Solids

Miniature Archimedean Solids

Materials used:

Chipboard Double adhesive tape Colored paper Glue Scissors Poster paint Pattern from

http://www.korthalsaltes.com/

Objectives

To determine the different faces and forms of the platonic and Archimedean solids through this models

To develop students understanding about polyhedra

Pedagogical use:This is use as models in presenting the Platonic and Archimedean solids.

Platonic SolidsSurface Area and Volume

History of the Platonic Solids

Platonic solids were known to humans much earlier than the time of Plato. There are carved stones (dated approximately 2000 BC) that have been discovered in Scotland. Some of them are carved with lines corresponding to the edges of regular polyhedra.

The name “Platonic solids” for regular polyhedra comes from the Greek philosopher Plato (427 - 347 BC) who associated them with the “elements” and the cosmos in his book Timaeus.

“Elements,” in ancient beliefs, were the four objects that constructed the physical world; these elements are fire, air, earth, and water. Plato suggested that the geometric forms of the smallest particles of these elements are regular polyhedra.

There are five regular polyhedra that were discovered by the ancient Greeks

The Pythagoreans knew of the tetrahedron, the cube, and the dodecahedron; the mathematician Theaetetus added the octahedron and the icosahedron

Symbolism from Plato:› Octahedron =

air› Tetrahedron =

fire› Cube = earth› Icosahedron =

water› Dodecahedron

= the universe

Platonic Solids

Tetrahedron

The tetrahedron is bounded by four equilateral triangles. It has the smallest volume for its surface and represents the property of dryness. It corresponds to fire.

Number of faces: 4Number of vertices: 4Number of edges: 6

Tetrahedron: Area and Volume

𝐴=√3×𝑎2

𝑉=√212

𝑎2

Cube

The cube or hexahedron is bounded by six squares. The hexahedron, standing firmly on its base, corresponds to the stable earth


Cube: Area and Volume

𝐴𝐿=4×𝑎2

𝐴𝑇=6×𝑎2

𝑉=𝑎3

Area

Volume

Octahedron

The octahedron is bounded by eight equilateral triangles. It rotates freely when held by two opposite vertices and corresponds to air


Octahedron : Area and Volume

𝐴=2√3×𝑎2

𝑉=√23×𝑎2

Dodecahedron

The dodecahedron is bounded by twelve equilateral pentagons. It corresponds to the universe because the zodiac has twelve signs corresponding to the twelve faces of the dodecahedron.


Dodecahedron: Area and Volume

𝐴=30×𝑎×𝑎𝑝

𝑉=14

(15+7 √5 )𝑎3

Icosahedron

The icosahedron is bounded by twenty equilateral triangles. It has the largest volume for its surface area and represents the property of wetness. The icosahedron corresponds to water.


Icosahedron: Area and Volume

𝐴=5×√3×𝑎2

𝑉=512

(3+√5 )𝑎3

Pie Chart

Pie Chat Manipulative

Materials used:

Foil Chipboard Scissors Glue Double adhesive tape Pencil Compass

Objective

To show the relationship between the perimeter of the circle and a parallelogram

To develop students ability in analyzing a certain concept

Pedagogical use: This is can be use in fraction and finding the relationship between the circle and parallelogram

Algebra Tiles

Algebra Tiles Manipulative

Materials used:

Chipboard Scissors Glue Double adhesive tape Pencil Compass

Objectives

To help students learn how to represent and solve algebra problem with the use of the algebra tiles

To motivate students in solving algebraic equation in a fun and interesting waY

Pedagogical use:Algebra tiles are known as mathematical manipulatives that allow students to better understand ways of algebraic thinking and the concepts of algebra. These tiles have proven to provide concrete models for elementary school, middle school, high school, and college-level introductory algebra students.

Procedure:Use tiles to represent variables and constants, learn how to represent and solve algebra problem. Solve equations, substitute in variable expressions, and expand and factor. Flip tiles, remove zero pairs, copy and arrange, and make your way toward a better understanding of algebra.

EXAMPLE:

From : http://mathbits.com/MathBits/AlgebraTiles/AlgebraTiles/AlgebraTiles.html

Instrumentation in mathematics (Instructional Materials)

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