CHAPTER EIGHT
IMPROVISATION
8.1 Objectives
By the end of this chapter, you should be able to:
i. identify plane and solid figures
ii. name these figures
iii. enumerate the properties of some of these shapes
iv. modeling of these shapes
v. Define improvisation, learning aid, its classification, and its criteria for selecting learning aids.
8.2 Introduction
In this chapter, you will learn the learning aids and its criteria of selecting the aids. Also, discussion is made on various plane and solid shapes and their properties. Materials needed of modeling plane and solid shapes were discussed with practical.
8.3 Meaning of Improvisation
Improvisation is the use of alternate resources to enhance education in the absence or scarcity of a specific first-hand teaching tool. It is also the act of supplying instructional materials from our region when there is a scarcity or absence of conventional ones. It might be defined as the sourcing, selection, and deployment of relevant instructional aspects of teaching and learning in order to realize educational goals and objectives in a meaningful manner.
Instructional materials are the classroom resources utilized by mathematics teachers. These are materials utilized by both instructors and students to facilitate effective teaching and learning. It is utilized to enhance the teaching and learning of mathematics. They aid both the instructor and the student by concretizing what is taught and learned. These materials exist in various forms and may represent actual items. For example, maps, charts, photos, etc. These materials may appeal to the hearing, sight, or both senses.
Local resources that may be utilized to create mathematics teaching/learning aids are those that can be obtained from home or school contexts. These materials are readily available and inexpensive, although the necessary instruments can be rented from other sources. Materials include cardboard string, a ruler, plywood, nail drawing paper, graph paper, a thin iron rod, letter steeds, beads, gum, a blade, a knife, a saw, mathematics set tools, a hammer, etc.
8.4 What is learning aids?
The process of acquiring new understanding, knowledge, behaviors, skills, values, attitudes, and preferences is the definition of learning. It is any change in behavior that is relatively permanent and results from practices or experience. According to some mathematics educators, a learning aid in mathematics education is anything designed to improve mathematics learning and retention by learners. In a simple term it is any form of materials that can aid or speed up the process of learning mathematics with or without any assistance of a second person (mathematics teacher).
8.5 Classification of Learning aids
There are various classifications of learning aids in mathematics as listed below:
i. Traditional Learning Aids: This includes learning through books, periodicals, journals, projects, thesis, handbook, chalkboard, etc.
ii. Visual Learning Aids: VLA includes posters, models, figures, charts, graphs, etc. It also includes graphics, such as diagrams, cut-outs, globes, objects, cartons, bulletin boards, pictures, map, and others.
iii. Mechanical Learning Aids: This includes an audio learning machine, tape recorder, radio, projector, film strips.
iv. Audio-visual Learning Aids: This type of learning aids which includes video, cassettes, film, television, photographic slides
v. Visual Material Learning Aids: This includes charts, organization charts, flow charts etc.
vi. Mathematical Games: games puzzles in mathematics to stimulate interest and encourage self-thinking and learning in the students
vii. Mathematical Laboratory
8.6 Criteria for Selecting Learning Aids
The following criteria are suggested in the selection of learning aids for mathematics teaching-learning:
i. Relevance: Learning aid must be relevant to the subject matter and serve as a means of assisting the student learning on the particular subject matter presented. It should be considered from the point of view of actual helpfulness to the learner.
ii. Learner’s ability: Before choosing a learning aid for a planned subject matter, the intellectual ability of the class is taken into consideration, so that it is not too advance or too simple. Furthermore, variety in types of aid is necessary because of the individual differences among students.
iii. Availability: Materials should be available and the aids should be numerous enough to permit selection by both teacher and learner; their helpfulness or lack of it should be looked into.
iv. Simplicity: Learning aid should not be too numerous. The teacher should not attempt to use many aids so as not to confuse the students by their multiplicity and rapid changes. An adequate and varied an indiscriminate and free use of them.
v. Cost: Learning aid should be economical. They should not be too expensive and too difficult to prepare, so that much time and energy would not be wasted on the part of the teacher.
vi. Accuracy: It should not be mere attraction. They should not attract attention to themselves but should increase interest in any comprehension of the subject matter to be learned.
vii. Usable: Learning aid should be readily usable. They should not take too much time and to keep them ready for use. They should also be able to last for considerable length of time without losing any of effectiveness.
viii. Objectivity: It should be adapted to the subject matter and to the goals to be secured through the mastery of subject matter.
ix. Size: Before choosing a learning aid for a planned lesson, you must considerable the size of the aid and also class size.
x. Teachers ’ability: If the learning aid is relevant to the subject matter but the teacher may not be able to operate or explain its application. In this situation, the student will not acquire the right knowledge from the lesson. Teacher must consider his/her ability before choosing any aid.
xi. Complexity: Some learning aids are very complex to explain even when the teacher known it. This type of aid is not easy to be comprehended by learners, so teacher should endeavor to avoid these complex learning aids.
xii. Durability: Some learning aid can be used in one or more attempts and require replacement due the materials used to make it, and storage facility. Such are not usually expensive and care must be taken to choose a more permanent and durable be stored and
8.7 Needs of Learning Aids
Mathematics teachers use a variety of aids to make the process of teaching and learning simple, interesting, no abstractness and effective which makes it easier to teach and learn even the most difficult concepts in the subject. The following are some of benefits of using learning aids:
i. Learning aid makes sense and saves time as students learn very quickly by watching and doing rather than reading. Therefore, teachers use aids.
ii. There are many students with a tendency to forget easily. Such students can get benefits with learning aids.
iii. Using learning aids will assist in understanding the concept easily and also grasp it completely.
iv. Learning aids enhance the conceptual thinking of students
v. Using learning aids createsan environment for motivation and interest in the learning process and does not learning concepts boring.
vi. With using of learning aids, the learners can learn with accuracy and even faster.
vii. It is proved that learning with visual representation stays in memory for a longer time than with textual representation. It impacts better with direct experience.
8.8 Practical Lesson
Below provide locally made learning aids and their objective in a lesson.
Topic 1: parallelograms on the same base and within the same parallel are equal in area.
Material: cardboard and drawing board procedure:
Draw on a piece of paper two parallelograms on the same base and within the same parallel. Pin the diagram on a drawing board having the same area with the parallelogram ABCD separately
Result: Turn the parallelogram ABCD
Over and place it on the parallelogram ABEF, you will discover that ABCD and ABEF are equal in area.
Topic 2: The Exterior angle of a triangle is equal to the sum of the interior opposite angles.
Material: Card Board and drawing board
Procedure:
Draw a triangle on a piece of paper and produce one of the lengths to obtain an exterior angle. Pin diagram on a drawing board. Mark out the interior opposite angles cut out two angles using a separate paper. Arrange the two angles on the exterior angles and the angles are found to be equal to the exterior angle. See diagram below:
Result: The sum of angles A and B when arrange on angles BCD are found to be equal.
Topic 3: To calculate areas of (i) square (ii) rectangle (iii) trapezium (iv) parallelogram (v) regular polygon
Procedure: Rule the plywood like graph paper with square of unit length. Nail each point where two lines meet. What you obtain is known as a Geo board and can be applied in a number of ways.
A GEO BOARD
To calculate the area of a square of length 4 unit string to carve out the area and count the number of square holes to give the area in square units.
For the area of trapezium, count the number of unit squares. Combine two half square to give the required area. In the trapezium given here the area = 8sq. units
ii. For the of parallelogram, count the number og square and combine two half units and add up. In this parallelogram the area = 6square units. Similar method is used for area of polygon etc.
Topic4: Longitude and Latitude
Objective: To prepare a learning aid for teaching latitude and longitude
Material: Palm frond for weaving basket or iron rod
Procedure: Make two small circle of the same radius from the material by tying. Make a third and bigger circle to represent the equator. Make another six circle of the same radius and bigger
than any of the first three circles Erect a rod longer in one of the six circles, ensuring that the length of the rod is longer than the diameter. Tie the remaining five circles in a vertical form at uniform intervals there represent the longitudes, fix the equator and the other circles as latitudes as shown below and tie the joints with string.
Note: when this material is not available, you can use iron or flex wire bent into this shape.
Topic5: Rectangular base pyramid.
Objective: To prepare a model for rectangle base pyramid
Procedure: Draw the figure on the cardboard as shown in the figure below and fold it up to close, by using gum after cutting through the edges.
Below see, before and after:
a. Before folding.
b. After folding (Rectangular pyramid)
Topic 6: Place value.
Objective: To prepare model of abacus for counting leading to place value
Procedure: Construct a rectangle frame of size 30cm by 20cm from the plank.
The thickness of the wood may be two centimeters. Choose two opposite frame and punch them in three places of equal intervals Arrange the beads on string on the frame as shown below:
Abacus
Topic7: plane shapes (circle, triangle, and square)
Objective: to identify various sorts of shapes I Their qualities and (ii) the representation of various planar shapes.
The material is either cardboard or plywood.
On cardboard, these basic forms are traced or created and then cut out.
Meaning and Scopes of planar forms
Generally speaking, plane forms are flat figures that are also referred to as two-dimensional (2-D) shapes. This category includes materials such as table tops, chalkboards, cardboard, ceiling board, circular objects, etc.
This lesson focuses on the geometric forms circle, polygon, and quadrilateral.
Circle: the location of (or the route traced by) a moving point that keeps the same distance from a fixed point. The measured distance around this route is known as its circumference. Other qualities of a circle are clued.
Radius, diameter, arc, chord, sector, segment
Polygon: A polygon is any plane shape bounded by straight line segment and it must not cross one another so that a polygon has only interior angles. Examples of polygons:
Pentagon Hexagon
Quadrilateral- it is a plane shape enclosed by focus line segments. Generally quadrilateral is a four sided shape.
Examples of quadrilateral are the parallelogram, rhombus, kite, rectangle, square, and trapezium.
Students Activity
1. Define a circle.
2. Outline all the properties of a circle. Student Activity
3. List 10 local materials that can be used to prepare learning Aid
4. Name the material you can use locally to prepare the following learning aids
(i) Ruler
(ii) Elliptical shape
(iii) Parabola frustum of a core
(iv) Cylinder
(v) Triangle prism
(vi) A hexagon
(vii) Sphere
5. Describe fully how you will prepare a cuboid using local material.
Parallelogram: is a quadrilateral (four side figure) made up of both pairs of opposite side that are parallel.
§ Opposite side are equal
§ Opposite side are equal in length and parallel
§ The diagonal bisects each other.
§ Each diagonal bisects and formed into congruent triangles
§ Allied angle are supplementary i.e. <A+<B= 180o
§ <B+<C=180o <C+<D=180o
Trapezium-is a quadrilateral having a pair of equal opposite side parallel. If the knowledge of trapezia is useful in dealing with cross-section of cones, pyramids prisms and other 3-D shapes
Special kinds of parallelogram.
There are special kinds of parallelogram. These are rhombus, rectangle and square.
A square is a parallelogram, which has four equal sides. A square has the following angle:
(i) The angles are all 90o
(ii) All the sides are equal
(iii) Diagonals are equal
(iv) Diagonals bisect each other at night angles
(v) Diagonals bisect the angles.
<A=<B=<C=<D=90o
AD=DC=CB=AB
The rectangle is a parallelogram in which all four angles are right angles (90o). The rectangle has the following properties
(i) The angles are (90o)
(ii) Diagonals are equal in length
(iii) Opposite sides are equal
Rhombus is a parallelogram in which a pair of adjacent sides is equal. A rhombus has the following properties
(i) Sides are all equal
(ii) The diagonals bisect each other at right angles
(iii) The diagonals bisect the angles
Kite-it is quadrilateral with two pairs of equal adjacent sides. It should be mooted that in a1 kite the opposite sides are not equal and it is not a regular quadrilateral because neither the angles nor the sides are equal.
Generally, a kite has the following properties
i. It has 2 pairs of equal adjacent sides AB=AD and CB=CD
ii. The diagonal BD and AC meet each other at right <AOB=90o
iii. The correct diagonal cuts it into two isosceles Ds
Topics 8: SOLID SHAPES
Objectives: kinds and properties of solid shapes
ii. Modeling of solid shapes.
Introduction
In your secondary school days, you might have been learnt different kinds of shapes. Today you will look into solid shapes of 3-dimensional figures. Generally, solid are hard object figures. They have the following properties.
(i) They conform to various shapes or possess number of surfaces
(ii) They are three dimensional
(iii) The cannot be squeezed
(iv) They are not fluid
Examples of solid objects include, stone, cube, cone, cuboids, cylinder, blocks, tine of different shapes etc. Properties of solid shapes
Cuboids- it is a 3-dimenional solid with six rectangular sides. They are shapes like cartons of chalk; wooden boxes, cement blocks, match boxes, textbooks, empty packet of sugar. A cuboid has 6 faces, 8 vertices and 12 edges.
Cube-it resembles cuboid in all respects that cuboid has longer dimensions than cube. They are solid objects having 6 equal faces, 12 edges and 8 vertices. Examples are: cube of sugar, Maggi cube, dice etc.
Cylinder- is round object having round faces (curricular faces) and flat tops. Examples include: tin of milk, a lead pencil, a water pipe, a drum, a cylindrical wood etc.
Spheres- there are shapes that look like the shapes of egg, balloons, tomatoes and water pots.
Cones-it is circular based object having only one vertex, one circular edge and two faces. It can be referred to as a circular pyramid. Examples include the sharpened end of a drawing pencil, a kerosene funnel, and a top of salt measure.
Tetrahedron-it is a solid object having four faces, six edges
ii. Modeling of Solid Shapes
Material: Making these models you need calendar, cellotapes, gum, pair of scissors cardboard, wood glue, nails etc.
Activity: You ask the students to bring along all the requirements for laboratory practical. You now cut out the shapes using scissors or blade and a cardboard. Each time you are cutting a shape, let the student observed the process from beginning to an end. Give every individual student opportunity to do the same thing processes and give necessary assistance and motivation for self-learning. As a teacher, try to demonstrate different shapes on how to form geometrical shapes.
8.9 Summary
In this chapter, you have learned:
§ Improvisation is an act of using local materials to construct instructional aids in teaching mathematics. Local materials are materials obtained at home or school environment, such as empty bottle, mathematical set instruments, cardboard papers, ply wood, nail, ruler, iron rod, knife, saw, etc.
§ Learning aids is any form of materials that can aid or speed up learning of mathematics with or without any assistance of second person (mathematics teacher)
§ Instructional materials are those resource materials used by mathematics teachers in the classroom
§ Classification of learning aids are traditional, visual, mechanical, audio-visual, mathematics games, mathematical laboratory
§ Criteria for selection of learning aid relevance, learner ability, usage, availability, accuracy, durability, class size, teacher ability and complexity
§ Practical lesson was discussed by defining the plane shapes, solid shapes, their properties and modeling.
Student Activity
1. Define a polygon
a. Quadrilateral
b. A parallelogram
c. A trapezium
d. Square
e. A rectangle
f. A kite
g. A rhombus
2. Outline the properties of
a. Rhombus
b. A square
c. A kite
d. Polygon
e. Quadrilateral
3. Draw the following solid shapes.
a. Cuboids
b. Cone
c. Cylinder
d. Tetrahedron
References
National Teachers’ Institute, Kaduna (NCE/DLS) Mathematics 2
National Teachers’ Institute & National Open University of Nigeria: PED 243 Measurements and Shapes.