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1. Any side of any one of the above figures is parallel to the opposite side of the same figure. Hence the figures are called parallelograms.

2. Each of the above figures has four sides. Hence the figures are called quadrilaterals.

3. If all the sides of a figure are equal, the figure is said to be equilateral.

4. If all the angles of a parallelogram are right angles (angles of 90°) the figure is said to be rectangular.

5. Which of the above figures are equilateral ? 6. Which of the above figures are rectangular? 7. Which of the above figures are not equilateral ? 8. Which of the above figures are not rectangular? 9. Which of the above figures are parallelograms? 10. Which of the above figures are quadrilaterals ? 11. Can you draw a quadrilateral that is not a parallelo

gram?

12. Is any one of the above figures an equilateral rectangular parallelogram?

13. Observe that every rectangular parallelogram has four right angles.

14. In a rhomboid or a rhombus two of the angles are less than right angles and two of them are greater than right angles. Convince yourself by cutting a rhomboid from paper and comparing it with rectangular figures that two of the angles of a rhomboid are as much less than two right angles as the other two are greater than two right angles.

123. MISCELLANEOUS REVIEWS. 1. If one of the angles of a rhombus is an angle of 80 degrees, what is the number of degrees in each of the other angles ?

2. Draw a rhomboid one of whose angles is an angle of 70; give the number of degrees in each of the other angles.

3. An oblong has four right angles. The angles of a rhomboid are together equal to how many right angles ?

4. If an oblong is a feet long and b feet wide, the number of square feet in the area is ab.*

If the side of a square is a, the number of square feet in its area is

5. If a rectangular solid is a feet long, 6 feet wide, and c feet thick, the number of cubic feet in its solid contents is abc. If the side of a cube is a feet, the number of cubic feet in its solid contents is

6. If a man earns b dollars each week and spends c dollars, in one week he will save dollars; in 7 weeks he will save

dollars. 7. A framed picture, on the inside of the frame, is 18 in. by 22 in.; the frame is 4 inches wide. How many inches in the outside perimeter of the frame?

8. Think of two fields : one is 9 rd. by 16 rd. ; the other is 12 rd. by 12 rd. How do the square rods of the two fields compare? How much more fence would be required to inclose one field than the other ?

* This means the product of a and b. Observe that it is the number a (not a feet) that we multiply by the number b (not b feet). While it is probably true (see foot-note, p. 41) that the multiplicand always expresses measured quantity, it is also true that we often find the product of two factors mechanically. Indeed this is what we usually do in all multiplication of abstract numbers. In this case we find the product of a and b and know from former observations that this numher equals the number of square feet in the oblong.

DIVISION.

124. Division is (1) the process of finding how many times one number is contained in another number; or (2), it is finding one of the equal parts of a number.

Note 1. — The word number as used above, stands for measured magnitude.

125. The dividend is the number (of things) to be divided.

NOTE. — Since in multiplication the multiplicand and product must always be considered concrete (see foot-note, p. 41), then in division, the dividend, and either the divisor or the quotient, must be so regarded.

126. The divisor is the number by which we divide.

NOTE. — The word number as used in Art. 126 may stand for measured magnitude or for pure number, according to the aspect of the division problem. In the problem 324 = 6, if we sire to find how many times 6 is contained in 324, the 6 stands for measured magnitude - a number of things. But if we desire to find one sixth of 324, then the 6 is pure number, and is the ratio of the dividend to the required quotient.

127. The quotient is the number obtained by dividing.

NOTE. — If the divisor is pure number the quotient represents measured magnitude. If the divisor represents measured magnitude the quotient is pure number.

128. The sign, +, which is read divided by, indicates that the number before the sign is a dividend and the number following the sign, a divisor.

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No. 3.

No. 4. 2 bush.) 246 bush.

2)246 bush. 123

123 bush.
No. 5.

No. 6.
2a)6ab + 8ac - 12 a 2)6 ab + 8ac – 12a
3b + 40 6

3ab + fac 6 a 1. In Example No. 1, we are required to find in 1565 dollars. *

2. In Example No. 2, we are required to find - of 1565 dollars. +

3. In Example No. 3, we are required 4. In Example No. 4, we are required 5. In Example No. 5, we are required 6. In Example No. 6, we are required

NOTE. — Let it be observed that all the examples given on this page, indeed all division problems, may be regarded as requirements to find how many times one number of things is contained in another number of like things. Referring to Example No. 2 given above: If one were required to find one fifth of 1565 silver dollars, he might first take 5 dollars from the 1565 dollars, and put one of the dollars taken in each of five places. He might then take another five dollars from the number of dollars to be divided, and put one dollar with each of the dollars first taken. In this manner he would continue to distribute fives of dollars until all the dollars had been placed in the five piles. He would then count the dollars in each pile. Observe, then, that one fifth of 1565 dollars is as many dollars as $5 is contained times in $ 1565. It is contained 313 times; hence one fifth of 1565 dollars is 313 dollars.

It is not deemed advisable to attempt such an explanation as the foregoing with young pupils; but the more mature and thoughtful pupils may now learn that it is possible to solve all division problems by one thought process — finding how many times one number of things is contained in another number of like things. But if this method is adopted great care must be taken both in understanding the conditions of the problems and in the interpretation of the results obtained.

* Fill the blank with the words, how many times five dollars are contained. + Fill the blank with the words, one fifth.

Division — Simple Numbers. 130. Find the quotient of 576 divided by 4. “ Short Division."

Explanation No. 1. 4)576 One fourth of 5 hundred is 1 hundred with a remain144 der of 1 hundred; 1 hundred equals 10 tens; 10 tens

plus 7 tens are 17 tens. One fourth of 17 tens is 4 tens with a remainder of 1 ten; 1 ten equals 10 units; 10 units plus 6 units are 16 units. One fourth of 16 units is 4 units. Hence one fourth of 576 is 144.

Explanation No. 2. Four is contained in 5 hundred, 1 hundred times, with a remainder of 1 hundred; 1 hundred equals 10 tens; 10 tens and 7 tens are 17 tens. Four is contained in 17 tens, 4 tens (40) times with a remainder of i ten; 1 ten equals 10 units; 10 units and 6 units are 16 units. Four is contained in 16 units 4 times.

Hence 4 is contained in 576, 144 times.

131. Find the quotient of 8675 divided by 25. “ Long Division.”

Explanation No. 1. 25)8675(347 One twenty-fifth of 86 hundred is 3 hundred, 75

with a remainder of 11 hundred ; 11 hundred 117

equal 110 tens. 110 tens plus 7 tens equal 117

tens. One twenty-fifth of 117 tens is 4 tens, 100

with a remainder of 17 tens; 17 tens equal 170 175

units; 170 units plus 5 units equal 175 units. 175 One twenty-fifth of 175 units is 7 units.

Hence one twenty-fifth of 8675 is 347. TO THE PUPIL. Make another explanation of this process similar to Explanation No. 2, under Art. 130.

132. PROBLEMS. 1. 93492_49

6. 5904: 328 2. 92169: 77

7. 7693: 157 3. 72855:45

8. 8190: 546 4. 34694_38

9. 12960: 864 5. 54875+25

10. 10950-438 (a) Find the sum of the ten quotients.

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