how many sides? 13. A triangle has how many sides? 14. The four sides of a square are of the same length. 15. Two sides of the oblong are longer than the other two sides. 16. A triangle has how many corners? 17. The square and the oblong have each how many corners? 18. Cut a square out of paper. 19. Fold it to make an oblong. 20. Fold it to make a triangle.

LESSON X-FRACTIONS

One Half. 1. To cut an apple in half, we cut it "through the middle." 2. There are two halves in one apple. 3. The halves are of the same size. 4. If I eat one half of an apple, how much of the apple will be left? 5. If I cut two apples in half, how many halves shall I have? 6. Draw an apple on the board. Using a crayon for a knife, cut it into halves. 7. Draw a pie, a loaf, an orange, etc., on the board and cut them in the same manner. 8. How many halves are there in an orange? A loaf? A pie? 9. How many halves are there in anything? Using a crayon, cut it

10. Draw a line on the board. into halves. 11. Which half is the longer? 12. Show where you would cut the ruler in order to cut it into halves. 13. Stand halfway from the desk to the door. 14. I have two apples. How can I give one half of them to Alice? 15. Draw two apples on the board and divide them thus : 010. 16. Show in the same way one half of a group of four apples, six apples, eight apples, ten apples, and twelve apples, using the arrangement of groups given on the flash cards. 17. What is one half of six apples? 18. Place eight boys in a line in front of the class. 19. Send one

half of the boys to their seats. 20. Four boys are one half of how many boys? 21. Send one half of the remaining

boys to their seats.

many boys? One Fourth.

22. Two boys are one half of how

1. Show one half of a circle. 2. Show one half of one of these halves. 3. Show one half of the other half of the circle. 4. The circle is now cut into four equal parts. 5. Each part is one fourth of a circle. 6. One fourth of a circle is the same as a quarter of a circle. 7. There are four quarters or fourths in a circle. 8. There are two fourths in one half of a circle. 9. In the same way show the fourths of a square; of an oblong; of a line. 10. If I cut a pie into fourths, and eat one of the fourths, how many fourths will be left? 11. Take twelve blocks and divide them into four equal groups. 12. How many blocks are one fourth of twelve blocks? One Third. Treat in a similar manner the fraction one third. Use the foot rule and the yard stick in illustrating one third. Show one third of lines, circles, squares, and oblongs. By means of diagrams lead the pupils to see that one third of a circle is less than one half of a circle. Using six objects, show what is meant by one half of six; and with six similar objects show what is meant by one third of six. Show also what is meant by two thirds of six.

Draw figures to illustrate one fourth, three fourths, one third, two thirds, etc., using squares, circles, lines, and oblongs, and have the pupils recognize the fraction by the shaded areas. With objects have pupils find one half, one fourth, and one third of numbers through twelve.

Using the groups on the flash cards, have the pupils show one half of ten oranges; one half of four apples, thus:

Require exact statements with reference to these fractional parts.

At the completion of this work the pupils should have a very clear conception of what is meant by one half, one third, and one fourth. They should be able to show this by means of simple drawings. They should also be able to show what is meant by one half of six apples, etc. They should have memorized some of the simple facts, such as two apples are one half of four apples; three boys are one half of six boys; one half of eight blocks is four blocks; etc.

LESSON XI-DIFFERENCE

The purpose of this lesson is to lead the pupils to measure the excess of one quantity over another quantity, and to acquaint them with the language forms used to express this difference. A new operation is involved in this, namely, subtraction. ILLUSTRATION. Problems: 1. Harry, Walter, and Ethel may stand in line in front of the class. 2. How many children are standing in line, Edna? 3. How many girls are standing in line, Fred? 4. How many boys are standing in line, Mary? 5. Of which are there more, girls or boys? 6. How can I make the number of boys and girls the same? 7. Who can tell me another way of making the number of girls and boys the same? (The teacher should lead the pupils to see that the number of boys and girls may be made the same, (a) by sending one of the boys to his seat, or (b) by having another girl stand in line.) 8. There are now two boys and two girls standing in line. If I place another girl in the line, how many girls will there be then? 9. Rose may stand in line with the others. There are now two boys and

three girls standing in line. 10. How many more girls are there than boys? Answer: There is one more girl than boys. 11. Show this by taking away one girl. 12. Return the girl to her place. Show the same by adding one boy. Measure differences between the number of blocks in two groups.

Draw on the board two circles to represent two plates. One of them is the teacher's, and the other the pupil's. Place three oranges on one plate, and one on the other plate. Lead the pupils to see how many more oranges there are on one plate than on the other. The number that must be added to the smaller quantity or taken from the larger quantity, in order to make both quantities the same, is the measure of difference. The pupils should be able to get this difference by both methods. Make no attempt to have the pupils memorize these results.

LESSON XII-NUMBER STORIES

ILLUSTRATION.

Problems: 1. Harry, take five blocks

2. Mary, tell blocks out of 3. Grace, you

out of the box and put them on the table. what Harry did. Story: Harry took five the box and placed them on the table. may take away two of the blocks that are on the table. 4. Fred may tell us a story about what he saw. Story: There were five blocks on the table. Grace took two of the blocks away. There are three blocks left on the table. 5. Walter may take two of the blocks away. 6. Martha, tell a story about what you saw. Story: There were

three blocks on the table. Walter took two of the blocks away. There is one block left on the table. 7. George may take one block away. 8. Lottie, tell a story about what you saw. Story: There was one block on the table.

George took it away. There are no blocks left on the table. 9. Alice may place three blocks on the table. 10. Fred may tell what he saw. Story: There were no blocks on the table. Alice placed three blocks on the table. There are three blocks on the table. 11. Jane may place two more blocks on the table. 12. Richard may tell a story about what he saw. Story: There were three blocks on the table. Jane placed two more blocks on the table. There are now five blocks on the table. 13. Ruth may place one more block on the table. 14. Jessie may tell a story about what she saw. Story: There were five blocks on the table. Ruth placed one more block on the table. There are now six blocks on the table. 15. We shall call the blocks apples, and the table a tree. Mary may tell us how many apples are on the tree. 16. Willie may pick two of the apples. 17. Frank, tell a story about what Willie did. Story: There were six apples on a tree. Willie picked two of them. There are four apples left on the tree. 18. Who wants to pick two more apples off the tree? Vera may do so. 19. Edna may tell a story about what was done. 20. James may pick one half of the apples that are left. 21. Mary may tell what she saw. 22. Fred may pick the rest of the apples.

We shall pretend that these two blocks are birds, and the table a fence. 1. Willie, tell a story about what you see. Story: I see two birds sitting on a fence. 2. Frank, show two more birds coming to the fence. 3. Grace, tell the story about these birds. Story: There were two birds sitting on a fence. Two more birds came to the fence. There are four birds on the fence. 4. Lucy may make two of the birds fly away. 5. David may tell a story about the birds. Story: There were four birds on a fence. Two