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The geometry work contained in this chapter should be commenced not later than the seventh year of school, and should be continued throughout the remainder of the grammar-school course.

1251. No formal definitions of lines, angles, etc., should be given at the beginning. After drawing angles of various sizes and with lines of different lengths, the pupils will be able to understand that “an angle is the difference in direction of two straight lines that meet in one point, or that would meet if produced."

1255. The semi-circular protractor is better than the common rectangular one for beginners, as they see more clearly by using the former that an angle is measured by the arc of a circle. Two protractors are printed on a fly-leaf in the back of the textbook, for the use of such pupils as cannot procure others. Protractors made of stout manilla paper can be obtained from the Milton Bradley Co., New York, at one cent each in quantities. A large protractor is needed for blackboard use. This can be made of pasteboard; or wooden ones can be bought of the Keuffel & Esser Co., New York.

Many scholars that are able to measure an angle one of whose sides is horizontal, Fig. 1, find it difficult at first to ascertain the number of degrees in an angle formed by two oblique lines, Figs. 2 and 3. They should be permitted to discover the method

for themselves. All that is necessary, is to place the center (A) of the base of the protractor on the vertex of the angle, Figs. 1-3, and the edge of the protractor on one of the sides, the other side cutting the circumference. In Figs. 2 and 3, the number of degrees in the angle XAZ is determined by the number of degrees

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in the arc BM, and the upper row of figures is used, having the zero mark at B. In Fig. 1, the number of degrees in the angle is measured by the arc CM, which requires the use of the lower row of figures.

1256. The average class will find the 100 exercises to Art. 1269, inclusive, sufficient for the first year's work. This will give three per week, and leave some time for review. Pupils should work the exercises at home without any preliminary discussion in class. After the exercises are brought in, they should be done on the blackboard, at which time the mistakes made can be pointed out. While first-class drawing cannot be expected from the instruments used by school-children, the teacher should exact the best work possible under the circumstances. A hard pencil, kept sharp, is necessary to secure the requisite fineness of line.

1. In drawing an angle, commence at the vertex.

This exercise is given to remove the impression sometimes formed, that the size of an angle depends upon the length of the lines, instead of their greater or less difference in direction.

In this and all other exercises, the pupils should be encouraged

to commence occasionally with an oblique line. No two results should be exactly alike. If two pupils compare notes, it should be for the purpose of producing a different drawing. One pupil's angle may have its vertex at the right, another at the left; one vertex may be above, another below ; etc. The better the teaching, the greater will be the variety of results in exercises that permit of variety.

3. It is expected that the pupils will see for themselves that each arc will contain 1 of 360°.

4. Using the ruler, draw the first line of any convenient length and in any direction. Placing A of the protractor at either end, mark off 45°, being careful to use the proper row of figures. Remove the protractor; place the ruler so that its edge just touches the end of the line and the 45° point, and draw the second line. This latter should not be of the same length as the first, unless for some good reason; so that pupils will not consider that the lines forming an angle should be equally long.

Write the number of degrees in each angle.

5. The teacher should not inform the pupils in advance how many degrees they will find in the second angle. They should measure it for themselves, using the protraetor.

In drawing these angles, the figure in the book should not be followed. The second line should be drawn to the left in some cases; the lower angle may be made 60°; etc.

When two lines meet to form two angles, it is not at all necessary that the point of meeting should be at the center of one line.

1257. Pupils should be taught that horizontal lines are lines parallel to the surface of still water. Floating straws are horizontal, and may point in any direction. A spirit level is used by the carpenter to determine whether or not a beam, for instance, is horizontal. A vertical line is one that has the direction of a plumb line, which is used by a mason to ascertain if a wall is perpendicular.

In drawings, however, lines that will be horizontal when the paper is placed upon the wall, are called horizontal lines; and lines that will be vertical when the paper is placed upon the wall, are called vertical lines.

6. The perpendicular need not be drawn to the center of either of the others, nor need it always be drawn above. The teacher should encourage variety.

8. The pupils should draw these lines, and mark in each angle the number of degrees it contains. Encourage the greatest possible variety in the size of the angles and the direction of the lines.

9. While pupils may be able by this time to give the result without drawing the angles and measuring the second one, the teacher should not fail to give them the necessary practice in constructing angles of a given number of degrees, and in measuring the contents of others.

Many scholars make as ridiculous mistakes in the measurement of angles as they do in their work in numbers, frequently reading the wrong figures, and figures from the wrong row marking an angle of 45°, for instance, 135°; etc. They should learn to “ approximate” the size of an angle, as well as to “estimate " the probable answer to an arithmetical problem. An acute angle should not be marked as containing over 90°; etc.

10. Having learned by observation that the sum of two adjacent angles is 180°, the pupils should now discover that the sum of any number of angles formed on one side of a straight line is 180°. When they have learned this from drawing the first exercise, they may be permitted to calculate the result in the other two, especially as the protractors are not marked for fractions of a degree.

The first exercise should show the same variety in the work of the different pupils as has been recommended for previous work.

12. In constructing a square, the protractor is used to erect a perpendicular at each corner. These perpendiculars are made

equal to each other and to the original line. A fourth line is drawn. The accuracy of the work may be tested by measuring, with the protractor, the two upper angles.

The base lines used by different pupils should be of different lengths. The pupils should be permitted, also, to construct the square in their own way. Some may erect a perpendicular at one end of the given line, and at the extremity of the second line erect another perpendicular. Some pupils may not measure the first line, drawing the second and third lines lightly of indefinite length, and using compasses to make them equal to the first. In this case, the light lines should not be erased; but the square should be marked off by heavier lines.

It is a good practice to have the pupils give a written description of their method of working one of these exercises, which should be accepted as a regular composition. The language should be correct; the proper technical terms should be employed; and there should be sufficient detail to enable any one not familiar with the work to understand just how it was done.

13. Pupils should be permitted to learn for themselves from this exercise and from 14, that vertical, or opposite, angles are equal.

17. After drawing the required lines, the scholar should mark in each angle its contents in degrees.

19. This exercise should enable the pupil to see that the sum of all of the angles formed about a point will be 360°.

20. The teacher should not give unnecessary assistance. If the scholars have a few days in which to work out an exercise, they should find no difficulty in managing this.

The word “adjacent" in geometry is applied to each of the two angles formed by one straight line meeting another. In 19, the two lower angles are adjacent; but none of the upper three angles is adjacent to any other, because three straight lines are used to construct two angles in each case. No two angles in 20 are adjacent, and no two are vertical.

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