122. Rectangle and Circle Graphs.-Rectangles of the same width but of different lengths are often used for comparisons, as in the graph above. Such graphs are also called bar graphs. Graphs are made by dividing a rectangle into different-size parts for comparisons. Above is a bar graph showing the number of bushels of wheat raised in eight states during the year 1916. On the opposite page is found an illustration of a circle graph.

EXERCISES

1. Explain the graph above. About how many millions and fraction of a million bushels of wheat were raised in each of the states mentioned? State each as a one-place decimal. Why can you not tell exactly? Can you tell exactly enough to give a useful comparison?

2. The number of bushels of wheat raised in Pennsylvania in 1916 is what per cent of that raised in Kansas? in Minnesota? in Illinois? in Iowa?

3. Make a bar graph for the following: the average wheat yield per acre for 1898-1907 was 32.6 bu. in Great Britain; 28.4 bu. in Germany; 20.8 bu. in France; 13.9 bu. in Austria-Hungary; 9.3 bu. in Russia. Change each number to a whole number. What happens to fractions less than .5? to those .5 or more? If uncertain, see Art. 24. In the graph letin. represent 1 bu.

4. The circular graph to the right gives the use of income of the American Telegraph and Telephone Company for 1911. What is the sum of the angles at the centre of a circle? What should be the number of degrees in the

angle of each sector in the graph? Measure each to see

if the graph is correct.

19%

INTEREST &
DIVIDENDS

20% MATERIAL &

50% SALARIES &

WAGES

RENT

6% SURPLUS

5% TAXES

5. Make a bar or a circular graph for the following: The number of cotton bales produced in 1915 were

Is it better to represent 100,000 bales by 1 in., or by a smaller length? What length will you use?

6. Bring to school papers or magazines containing graphs of some of the forms studied. Explain what each graph shows. Does the graph show this more clearly than a column of figures would?

Hold a number contest, using operations with whole. numbers.

123. Continuous Line Graph.-Line graphs, such as the one above, are the most common as well as the most useful graphs. The two heavy lines, at the bottom and at the left, are called the co-ordinate axes, or merely the axes of the graph. Paper ruled as that in the picture is called co-ordinate paper.

EXERCISES

1. State how much is represented in the above graph by one of the squares taken the horizontal way; taken the vertical way.

State the population of the United States for the

year 1820; 1845; 1885; 1897.

3. State the year in which the population of the United States was 10,000,000; 35,000,000; 25,000,000.

4. Note an irregularity in the curve. What historical reason is there for this?

5. The accompanying readings were taken one winter's day on a Fahrenheit thermometer. To graph the data,

8 A.M. + 24° -9 A.M. + 28°

10 A.M. + 29° 11 A.M. + 32° 12 noon +35° 1 P.M. + 36° 2 P.M. +35° 3 P.M. + 33° 4 P.M. + 30° 5 P.M.

as to leave 20

Next draw the

bottom of the

first draw the vertical axis to the left on the co-ordinate paper so squares to the right of it. horizontal axis near the paper so as to leave at least 40 squares above this line. Let each square along the vertical axis represent 1°. Write along the vertical axis to the left at the proper places 0°, 10°, etc. Let 2 squares along the horizontal axis represent 1 hr. of time. Begin at the intersection of the axes and write below the horizontal axis 8, 9, etc., through 5. Place a mark on the 8-hour line opposite 24°; one on the 9-hour line opposite 28°; and so on. Be very careful to locate each point accurately. Use a sharp pencil to join all the points with a smooth curve, as on the opposite page.

27°

6. Suppose the temperature changes were continuous. Read from your graph the temperatures at 8.30; at 11.15; at 3.30; at 2.45.

7. About when was the temperature 25°? 24.5° ? freezing? highest?

8. Take readings on your thermometer at home or at the school, at the same time daily for a week or two weeks. Graph your data as you have done the above.

VII

GRAPHS OF EQUATIONS

124. Constants and Variables. In finding the circumference of a circle, the diameter is multiplied by 22. For each circle the same number, 22, is used. Such numbers are called constants. The diameters and circumferences vary for different circles and are called variables.

125. Equations with Two Variables.-Equations containing two variables arise often. Such are

the circumference of a circle in terms of its diameter;

the velocity in feet a falling body gains during t seconds;

S = 16t2,

the space in feet a body falls during t seconds.

(3)

126. Independent and Dependent Variables.-As different values are given to t in (2), different values will be obtained for v. The value of v, then, depends upon the value of t. For this reason the literal numbers to which numerical values are assigned, as t, are called independent variables. A literal number, as v, whose value depends upon the value of the other variable is called the dependent variable.