3. Ten boys in a class were racing with each other to see which one could make the best record in arithmetic. Their records were as follows:

(1) Make a line graph of each boy's record.

(2) Make a bar graph of each boy's record.

(3) Find each boy's average for the week and make a line graph of these averages.

D. RAPID CALCULATION IN ADDITION

I. Addition of Integers

A good exercise to increase speed in addition is to count by 2's, by 3's, and by all the other numbers up to 9, beginning with different numbers.

1. (a) Begin at 0, 1, and 2 and count by 3's to the first number greater than 100.

(b) Time yourself on each counting and see how much you can reduce the number of minutes after counting five times.

2. (a) Begin at 0, 1, 2, and 3 and count by 4's to the first number greater than 100.

(b) Time yourself as before.

3. Spend ten minutes every day for at least two weeks counting by 5's, 6's, 7's, 8's, and 9's, beginning first with zero and then with other numbers.

4. Suggestions for rapid and accurate addition:

(a) Write small neat figures.

(b) Space the figures well, neither too close together nor too far apart.

(c) Keep the columns straight.

(d) In adding think the results only. Do not repeat the separate numbers to yourself.

(e) Look for combinations of numbers whose sum is 10. (f) Always check by adding the columns in the opposite direction.

5. Drill on the following until you can give all the results correctly in 45 seconds or less:

17. Some accountants use another method of checking called "casting out nines."

The following problem shows how it is used:

as you do it, thus: 7 +4

In each number, add the digits, dropping out the 9's as you do it, and write the remainder opposite each number.

In the first number, drop the 5 and 4 and write the remainder 3. In the fourth number, think 7 + 8 15; drop 9, leaving 6; 6+7 13; drop 9, leaving 4.

=

=

=

=

6, the

Add these remainders, dropping out 9's 11; drop 9 leaving 2. 2+1+3 final remainder. From the sum drop the 4 and 5, which leaves 3+3 = 6 as the remainder. The remainder from the sum must equal the final remainder from the numbers added.

Casting out nines is not an absolute proof that the addition is correct, but it is a fairly good proof.

18. (a) For practice in addition, pupils may write columns of three- and four-place figures and add.

(b) A better drill may be had by using the Courtis Standard Tests. These are sets of cards with well-graded exercises in addition, subtraction, multiplication, and division, with standards of time set for each. With these tests are cards on which pupils keep graphs of their records in both accuracy and speed.

19. A time or speed graph may be made similar to the ones which measured the pupils' accuracy.

The three graphs show the time spent by a pupil in preparing his arithmetic lesson for three weeks.

(a) State the time for each day of each week.

(b) On which day of each week was he the slowest? (c) On which day was he the quickest?