CHAPTER SIX
MAHEMATICS LABORATORY
6.1 Objectives
By the end of this chapter, you should be able to:
i. define mathematics laboratory
ii. state the functions of mathematics laboratory
iii. enumerate facilities in mathematics laboratory
iv. state some mathematics topics in mathematics laboratory
v. design and general layout of mathematics laboratory
vi. perform activities of mathematics laboratory lesson
6.2 Introduction
The wrong perceptions about mathematics by most secondary school students in particular and the public in general that mathematics is difficult, needs to be thrashed. This is attributed to abstract nature of the subject, among others. Practical tasks in mathematics will be conducted in mathematics laboratory to enable you achieve optimum potential through the use of the laboratory facilities. In this chapter will deals with following sub-units: concept of mathematics laboratory, facilities needed in laboratory, laboratory lessons and its functions in mathematics lessons.
6.3 Meaning of Mathematics Laboratory
The mathematics laboratory is a space where teachers and students may experiment and investigate mathematical patterns and concepts. There is a collection of games, puzzles, and other teaching and learning resources in this area. At other terms, a mathematics laboratory is a space in a school that has the necessary equipment for the practical instruction and study of mathematics. It gives the necessary experiences for the acquisition of mathematical ideas, principles, and generalizations. It is a resource centre designed for the growth of mathematical activities in schools that students and teachers utilize to investigate the world of mathematics.
6.4 Designs and Basic Format
Since mathematics lab activities need students or teachers to explore the world of mathematics in order to learn, discover, and create an interest in the topic, you are persuaded that it is essential in your school. These are recommended laboratory design and overall layout:
6.4.1 Room Accommodation
The recommended design and basic arrangement of the laboratory should provide seating for around 30 students at a time. Ensure the cleanliness of the laboratory and its equipment, and offer at least two escape doors.
6.4.2 Equipment and Materials
The minimal materials required to be retained in the laboratory may comprise all important equipment, raw materials, and other items necessary to properly carry out the tasks. The number of various items may vary from school to school based on the size of the class. Sharp objects should not be carelessly left out in the open, as stock records are crucial to the efficient running of a laboratory. Some of the necessary resources are provided in the subsequent unit.
6.4.3 Human Resources
It is preferred that a person with a Bachelor of Education (Mathematics) or above be placed in charge of the mathematics laboratory. He/she is required to have specialized knowledge and an interest in the subject in order to perform practical work. A laboratory attendant or laboratory assistant with the required qualifications and subject-matter expertise is an asset. The teacher and laboratory assistant must closely monitor the pupils' activities to verify that their behaviour conforms to laboratory norms and regulations.
6.4.4 Allocation of time for activities
15 to 20 percent of the total available time for mathematics should be allocated to activities. In the timetable, slots for laboratory tasks can be allocated appropriately.
6.5 Infrastructure at the Mathematics Laboratory
The purpose of the mathematics laboratory is to make learning mathematics enjoyable and engaging, and to encourage individual involvement in such activities. The sort of experiment required is determined by the materials necessary for the goal. Stones, bottle tops, sticks, beads, empty tins, graph papers, duplicating sheets, cardboard, sheets, rubber beads, instruments in mathematics sets, clocks, iron rod, cello tape, plaster, wire point and paint brush, sand, glue, number charts, balance scale, thermometer, beakers, measuring cylinder, screw driver, knives, models, pliers, nails, razor blades, shears, magnetic compass, geo-board, hammer.
These materials are obtained within the environment of the teacher and student. They are always available and not too expensive, so the materials are group into two. Improvise and purchase. Some of laboratory instruments used in determining some concepts are as follow:
Length – Scale ruler, tape rule
Mass – spring balance
Volume – cylinder, beakers, burette, pipettes, volumetric flasks and conceal
Weight – weighting scale or weighting balance
Temperature – thermometer
Time – clock, stop watch
Mass and weight are frequently used interchangeably. The two names, however, have different connotations. The following table shows the differences between the two values:
S/N | MASS | WEIGHT |
1. | the amount of material in a thing | the gravitational force acting on an object |
2.. | From one location to the next, the object's mass is constant | The area affects the weight of items |
3. | Grams are used to measure mass | Weight is measured in Newton |
Activity 1
1. What is mathematics laboratory
2. Give five activities that you can be used in mathematics laboratory
3. Give 10 materials that can be used in mathematics laboratory
4. Explain the design and general layout in mathematics laboratory
6.6 Functions of Mathematics Laboratory
Some of the ways in which activities in mathematics lab could contribute to mathematics learning of the subject are:
i. It assists in taking care of individual differences, and enrich students with adequate mathematical skills
ii. It is a resource centre because it serves as a research centre which could lead the teacher and student to discover or verify some mathematical laws, formulae and theorems.
iii. Collaborative work in a mathematical laboratory usually encourages sharing of knowledge and cross breeding of ideas
iv. Mathematics laboratory enables the students to know and familiarize themselves with names of the equipment kept there and their functions.
v. Mathematics laboratory helps to eliminate or minimize the abstract nature of mathematics concepts since concrete or visual materials are used to teach/learn such concepts.
vi. It assists in creating motivation and building positive attitude, confidence and reducing anxiety towards mathematics.
vii. It helps the students to relate the mathematical concept learnt to real life situation through various practical activities.
viii. An activity involves both the mind and hands of the student working together which facilitates cognition.
ix. It provides opportunity to students to repeat an activity several times. They can revisit and rethink on a problem and its solution.
6.7 Types of Laboratory lessons
Lessons in mathematics laboratory solely depend on the objective of the individual carrying out the experiment. Below are some of the stated lessons that can take place in mathematical laboratory.
Numbers and Operations
§ Use of charts to teach number pattern such as even, odd, prime number, multiplication factors multiple
§ Use of counters to teach counting and basic operation
§ Use of abacus place values
§ Materials in shopping corner in the laboratory for teaching buying/selling
§ Verify the result of the product of two numbers of the same signs and those of opposite signs
§ Compute Pascal’s triangle for the coefficient of binomial expansion of power 20.
Statistics and Probability
§ Tossing a coin for prediction purpose and finding the probability of an event
§ Collect grades in two subjects and find their coefficient of correlation.
§ Collect original data from a survey to determine what variable are related.
Ratio and Proportion
§ Compare the ratios of a sizes of similar triangles and the ratio of their areas
§ Compare the volumes of a cube to the one whose size is of length five times the first by pouring and deducing the ratio of their volume
Geometry
§ To carry out paper folding activities to find;
- The midpoint of line segment
- The perpendicular bisector of a line segment
- The bisector of an angle
- The perpendicular to a line from a point given outside
- The perpendicular to a line from a point on the line
§ To carry out activities using geo board to;
- Find the area of triangle
- Find area of any polygon by completing the rectangle
- Obtain a square on a given line segment
- Given an area, obtain different polygons of the same area
§ Calculate the angles of elevation of an object on top of a tree using clinometers
§ Calculating the distance between two points on the parallel of attitude
§ Sketching different types of curves
§ Measurement of
- Length
- Volume/capacity
- Weight/mass
- Time
- Temperature
- Angles
§ Using materials to drive the formula for
- Perimeter of polygons (rectangles and triangles)
- Areas of rectangles/square and triangle
- Circumference and area of a circle
- Constant π
Trigonometry
§ Obtain a trigonometrically ratio table of sines, cosines and tangent of angles 30, 60 and 45 from equilateral and isosceles triangles
Deduce by cutting right angled triangles of sides in multiple of 3, 4 and 5 that a triangle of side 3n and 5n is a right-angled triangle where n is a natural number.
6.8 Preparing for a laboratory lesson
Laboratory lesson refers to the laboratory method of instruction. This approach requires a laboratory equipped with mathematics-related equipment and other important teaching tools. For instance, equipment pertaining to geometry, trigonometry, mathematical models, charts, balances, different wooden or hardboard figures and forms, graph paper, etc.
This approach is founded on the adage "learning by doing." It is a strategy that helps pupils to discover mathematical facts via activities. It proceeds from the tangible to the abstract. Laboratory approach is a technique that stimulates students' actions and motivates them to create discoveries.
The following approach must be followed by teachers while organizing mathematics laboratory lessons.
§ Aim of the practical work: the teacher clearly states the aims of the practical work or experiment to be carry out by the students.
§ Provided materials and instruments: the students are provided with the necessary materials and instruments.
§ Provide clear instructions: that is clear procedure of the experiment.
§ Draw the conclusions: The students are required to draw the conclusions as per the aim of the experiment.
Example 1
Derivation of the formula for the volume of a cone
Aims: To derive the formula for the volume for the volume of a cone
Material and Instruments: cone and cylinder of the same diameter and height at least 3 sets of varying dimensions, sawdust, water and sand.
Procedure: students to do the following activity.
i. Consider each pair of cones and cylinders that are the same height and diameter.
ii. Record the height and diameter.
iii. Count how many times the cone is dumped into the cylinder and record the number in a table column.
iv. Sand, water, or sawdust should be poured into the cone until it is completely full.
v. Convert the other two sets of cones and cylinders to use in the experiment, and record the reading as before.
S/N | DIAMETER OF CONE/CYLINDER | HEIGHT IF CONE/ CYLINDER | NO OF MEASURE OF CONE TO FILL THE CYLINDER |
1 | 3cm | 6cm | 3 |
2 | 5cm | 7cm | 3 |
3 | 6cm | 10cm | 3 |
Drawing Conclusions
Each time, irrespective of the variations in diameter and height it takes 3 measures of cone to fill the cylinder.
Volume of cone = 1/3 volume of cylinder
But volume of cylinder = πr2h.
Volume of cone = 1/3 πr2h.
Example 2
Sum of three angles of a triangle is 180 degrees. How do we prove this in the laboratory?
Aims: To prove that sum of the three angles, of a triangle is equal to two right angles or 180 degrees
Materials and instruments: Cardboard sheet, pencil scale, triangle, scissors, razor blade.
Procedure: In the laboratory, students will be given cardboard sheet each and then they are told how to draw triangle and cut out separately with the help of scissors.
Observations: Students will measure the angles of the triangles drawn and write these in a tabular form.
FIGURE NO | MEASURE OF DIFFERENCE ANGLES | ANGLES | ||
1 | A | B | C | A+B+C |
1 | 90 | 60 | 30 | 180 |
2 | 120 | 30 | 30 | 180 |
3 | 60 | 60 | 60 | 180 |
Calculation: after calculating the angles of several triangles represented on cardboard sheets. We compute their total and draw a conclusion. Students will be able to draw the inductive conclusion that the sum of a triangle's three angles is 180 degrees by computing the triangle's three angles in this manner.
Example 3
Topic: Congruent Triangles
Objective: At the end of this lesson, students should be able to:
Identify the properties congruent triangles
Materials: from paper cut triangles of the same size and different sizes, using ruler and protractor
Procedure: The significance of triangular congruency is explained to the pupils by the teacher.
The term "congruent" refers to two or more triangles that are identical in every way. When one is pulled up, it may be put perfectly atop the other since their sides and angles are both equal. additionally have equal areas. Below are the conditions
1. The three side are equal to another triangle, represented by side, side, side (SSS)
2. Two sides are equal to two side of another triangle and the included angle, represented by (SAS)
3. Two angles of are triangle equal to another triangle and any one side of the triangles, represented by (AAS).
Step II: Teacher provides much paper cutting of triangles and call on some students to identify pairs of congruent triangles. The students should do this by actually placing one triangle out on another until they get those that fit into one another without any left over. After a series of such exercise, you should ask them to compare:
i. The length of corresponding sides by actual measuring ruler and
ii. The size of corresponding angles by using a protractor.
Step III: Teacher should demonstrate with the paper cutting and actual measurements to establish the condition for congruent of triangles.
Example 4
Topic: Similar Triangles
Objective: At the conclusion of this lesson, students should be able to recognize triangles that are similar.
Triangles are said to be similar, or to have the same shapes, when their respective angles are equivalent to those of other triangles. This doesn't mean that their sides are congruent or equal; rather, it just means that the ratios of each side are the same. For e.g s ABC and XYZ are similar then A = X, B = Y, C = Z
Also the ratio of the corresponding sides is equal.
i.e AB = AC = BC. Similar s is provided with right angled triangle whose lengths are (3,4,5),
XY XZ XY
(9, 12, 15) and (5, 12, 13), (10, 24, 26) the students should identify them as similar triangles by comparing their angles. Let students calculate their areas. For the (5, 12, 13) then area in ½.5.12 = 30 sq units while that of (10, 24, 26) is ½.10.24 = 120 sq units. The formula 1/2bl is used because the s are right angle s. the ratio of pairs of corresponding sides is ½, the square of ratio of the corresponding sides is ½, the square of this ¼. The ratio of the area is 30/120 = ¼, so ratio of areas = square of ratio of the corresponding sides. Consider the second pair of s use
have the ratio of area = ½.3.4 = 1/9,
the ratio of the corresponding sides 3/9 = 4/12 = 5/15 = 1/3
½.9.12
Square of ratio of corresponding side = 1/9.
By generalization the students will learn that if two triangles are similar, the ratio of their areas is equal to the square of the ratio of the length of corresponding sides. You may end the lab lesson by giving the summary of conditions of similar triangles that students should take note.
i. When two angles from one triangle are equivalent to two angles from another.
ii. If their matching sides' ratios are equal.
iii. If the ratio of the triangles' areas to their squares of matching sides is the same. Equiangular triangles are included in the category of similar triangles.
Example 5
Topic: Perimeter of plane shapes
Objective: The student should be able to obtain of the perimeters of plane shapes.
Material: Cardboard, thread, ruler
Let your students obtain the perimeters of rectangle, square parallelogram, quadrilaterals and triangles and irregular shapes.
The perimeter can be measured by string on irregular shape the surface while we use the string to trace the other edges till we get back to our starting point.
Perimeter of plane shapes, they can use ruler to measure each side directly and sum up the measurement obtained.
Examples: find the perimeters of the following shape whose dimensions are shown.
1. 2. 3.
Solution
1. Perimeter = 7+8+9 = 24 units
2. Per = 5+17+5+17 = 44 units
3. Perimeter = 5+7+3+6 = 21 units
6.9 Mathematics Laboratory Based Approach (MLBA)
This is learner centered activities carried out in the mathematics lab for teaching-learning, which include experimentation, exploration, inquiry, discussion, project work and cooperative learning. Students manipulate the materials in the learning process and the teacher plays the role of a facilitator. The approach has merits and demerits and itemized as follows:
Merits of MLBA
i. The approach is founded on the notion of learning by doing.
ii. It is based on the student's self-pacing
iii. It clarifies key concepts, principles, etc.
iv. It fosters self-confidence and teaches pupils the value of hard work.
v. The students are instructed in the usage of various laboratory equipment.
vi. It helps pupils acquire a habit of scientific inquiry and research
vii. This approach introduces mathematics as an applied topic viii.
viii. It affords pupils opportunities for social contact and teamwork.
ix. It is student-centered and engages pupils since they actively participate in the learning process.
Demerits
i. This method can be used for a small class only
ii. It requires a lot of planning and organization
iii. This method it is not possible to make progress quickly.
iv. This method it is suitable only for certain topics.
v. This method requires laboratory equipped with different apparatus
vi. Not all teachers can use this method effectively.
vii. It is an expensive method because not all schools can adopt the method
viii. This method has very little of theoretical part in it.
Activity 2
1. Enumerate five functions of mathematics laboratory
2. Give five activities that you can be used in mathematics laboratory
3. Give 10 materials that can be used in mathematics laboratory
4. Explain five (5) merits of mathematics laboratory-based approach
5. List instructional materials, used in measuring
a. Length b. Volume/Capacity c. Weight/Mass d. Temperature e. Time
6. Choose any of the topic and prepare a mathematics laboratory (practical lessons) to be carried out in a named class
7. Give two activities of laboratory lesson on statistics and probability.
8. State three functions of a mathematics laboratory.
9. Prepare a mathematical laboratory lesson for finding the curved surface area of a cylinder in terms of a rectangle.
10. Draw and list the properties of the following solid shapes (i) Cuboids (ii) Cone (iii) Cylinder (iv) Tetrahedron
11. Given a cardboard sheet, cello tape and a pair of scissors, explain how you would make (i) a cube (ii) square based pyramid
12. What instructional materials will you see in the teaching of the concept of similarity as applied to triangles in mathematical laboratory lesson?
13. Discuss the use of graph paper in the teaching of similar triangles.
14. Why is it necessary to use instructional materials in teaching congruence of triangles?
15. What is significance of the order of the labeling two congruent
16. List the three test for the congruency of s
6.10 Summary
§ Mathematics laboratory is a room set aside in school build where materials/equipment are kept for teaching and learning mathematics in a practical manner
§ Activities in mathematics laboratory are carryout either individual or group
§ The functions of mathematics laboratory are aimed at making teaching-learning of mathematics interesting, joyful, confidence and permanent in mind of learners and reduce mathematics- anxiety. It encourages students to become autonomous learners and allows individual students to learn at his or her space.
§ Design and general layout provide functional mathematics lab, both materialand human resources
§ Some laboratory lessons that can be presented through MLBA, which include: area of plane shapes using geo-board, volume of figures, tossing of dices, making of models and charts.
References
Niger State College of Education (2010). A Handbook in Mathematics for Tertiary Institution in Nigeria, 1(1) Publisher: NSCOE, Minna
National Teachers’ Institute, Kaduna (NCE/DLS) Mathematics 2
NCCE, (2012) Minimum Standard Mathematics