ON TEACHING ELEMENTARY MATHEMATICS. 479. 



square centimetres in a square inch can be obtained by division. To how 

 many places of decimals may the result be regarded as accurate ? 



Construction of paper models of solids to illustrate the notions of 

 surface iind volume. 



Measurement of volume should be illustrated by cubical bricks. 

 Cubes of 1 inch and 1 centimetre can be obtained cheaply. Volumes of 

 rectangular solids, prisms, cylinders, and cones should be measured where 

 possible, and the results verified by displacement of water if access to a 

 physics laboratory is to be had. Measui'ements of area and volume form 

 a useful introduction to the notion of an algebraic formula. 



As a pupil advances in elementary algebra, geometrical illustrations 

 may be employed with advantage, e.g., the verification with squared paper 

 of the formult^ corresponding to the propositions of Euclid, Book IL, 

 graphs, the solution of quadratic equations with ruler and compasses. 



(Scheme submitted hy Prof. Pebkt; tJiis Scheme is intended to accompany 

 a Course of Arithmetic, Algebra, and E.vperimental Science.) 



Practice in decimals, using scales for measuring such distances as 

 3'22 inches, or 12"5 centimetres. 



Contracted and approximate methods of multiplying and dividing 

 numbers ; using rough checks in arithmetical work ; evaluating formula. 



Mensuration, — Testing experimentally the rule for the length of the 

 circumference of a circle, using strings or a tape measure round cylinders, 

 or by rolling a disc or sphere, or in other ways ; inventing methods of 

 measuring approximately the lengths of curves ; testing the rules for 

 the areas of a triangle, rectangle, parallelogram, circle, ellipse, surface of 

 cylinder, surface of cone, ikc, using scales and squared paper ; proposi- 

 tions in Euclid relating to areas tested by squared paper, also by arith- 

 metical work on actual measurements ; the determination of the areas of 

 an irregular plane figure (1) by using Simpson's or other well-known rules 

 for the case where a number of equidistant ordinates or widths are given ; 

 (2) by the use of squared paper when equidistant ordinates are not given, 

 finding such ordinates ; (3) weighing a piece of cardboard and comparing 

 with the weight of a square piece ; (4) counting squares on squared paper 

 to verify rules. Rules for volumes of prisms, cylinders, cones, spheres, 

 and rings, verified by actual experiment ; for example, by filling vessels 

 with water, or by weighing objects of these shapes made of material of 

 known density, or by allowing such objects to cause water to overflow 

 from a vessel. 



The determination of the volume of an irregular solid by each of the 

 three methods for an irregular area, the process being first to obtain an 

 irregular plane figure in which the varying ordinates or widths represent 

 the varying cross-sections of the solid ; volumes of frustra of pyramids 

 and cones ; computation of weights from volumes when densities are 

 given. 



Stating a mensuration rule as an algebraic formula. In such a formula 

 any one of the quantities may be the unknown one, the others beino- 

 known. Numerical exercises in mensuration. The expei'imental woi'k in 

 this subject ought to be taken up in connection with practice in weighing 

 and measuring generally, finding specific gravities, illustrations of the 

 principle of Archimedes, the displacement of floating bodies, and other ele- 

 mentary scientific work. A good teacher will not overdo this experimental 



