be able to answer readily, regarding each combination, such questions as the following for the combination 5 and 4: How many are 5 and 4? How many are 35 and 4? 65 and 4? 95 and 4? If a 4 is added to a number ending in 5, the answer will end in what number? Before taking up the addition columns, it is expected that the teacher will give oral drills of this kind upon all of the combinations of a lesson after they have been studied through Step B. Step C. As the combinations are arranged in the several lessons, the sum of the first combination is made the lower number in the second combination, the sum of the second combination is made the lower number in the third combination, etc. This arrangement is for the purpose of constructing addition columns involving the combinations of the lesson. With the combinations on the board for reference, arranged as in Lesson A, unite them to form short columns, thus: In the combination 2 and 3 in the lesson, let the pointer rest on 3. Pass it to the 3 in the first of these columns. The pupil will recognize the Move the pointer the pointer to the When the sum has 432I 6 2 3 2 3 2 3 3 4 5 2 5 combination and will give the sum, 5. to the 4. If the pupil hesitates, pass combination 5 and 4 in the lesson. been given, return to the column and add. The pupil will soon see that the 2 and 3 in the column stand in the same relation to 4 as the 5 does in the combination in the lesson. If in the addition of a column the pupil fails to add correctly any given combination, say 17 and 6, do not ask him what 7 and 6 make, but drop the column and take up the study of the combination 7 and 6 through Steps A and B. Return to the column, and if the drill on the single combination was thorough, the pupil will be able to add the column. Step C consists in adding the columns as indicated in Step C, pp. 48, 49. The pupil should begin the addition of a column by naming the sum of the combination at the foot of the column. The columns should be added with a regular cadence. Perfect knowledge of the combinations, together with right habits of work, will keep the pupil from resorting to serial counting as a means of finding the sums. For the fixing of these habits, much of the work on addition for the first few lessons should be oral. SUBTRACTION Each lesson in addition is followed by a corresponding lesson in subtraction, and the same combinations are involved in both. Subtraction is the operation of finding the difference between two quantities. This difference may be found by taking from the greater of the two quantities an amount equal to the lesser quantity; or it may be found by adding to the lesser quantity such an amount as will make it equal to the greater. The amount remaining in the one case is the amount added in the other case. Applied to the solution of problems in subtraction, these methods are as follows: Problem 1: Fred has 5 marbles. Walter has 3 marbles. How many more marbles has Fred than Walter? Problem 2: Fred had 5 marbles. He lost 3 of them. How many marbles has he left? By the first method, a quantity equal in amount to the number of marbles Walter has is taken from the number of marbles Fred has, and the remaining number is the difference, or answer. This is expressed in the language of subtraction: Three marbles from 5 marbles leave 2 marbles; or 5 marbles less 3 marbles are 2 marbles. By the second method, such an amount is added to the smaller quantity as will make it equal to the larger quantity. This is expressed in some such language as: "3 marbles and how many marbles are 5 marbles?" Or, "3 marbles and what are 5 marbles?" Or, "3 marbles and 2 marbles are 5 marbles," — a form embodying the answer. The answer, 2 marbles, is part of the number fact involved in the addition form previously learned, when 3 marbles and 2 marbles were added to make 5 marbles. In the second problem, in which the difference between a given quantity and a part of itself is required, there is no essential difference in the way of finding the part remaining. The method of subtraction recommended for use in connection with the exercises of this text is the second of the above methods, which is the so-called "Austrian method." It is also known as the "additive method," and is sometimes spoken of as the "computors' method" or the "method of making change." Its advantage lies in the fact that the number facts of addition are used to find the differences in subtraction. After the addition combinations have been memorized, no new number facts are necessary in order to perform the operation of subtraction. The pupil has only to learn how to apply to a new mode of expression the knowledge that he has already acquired. This he learns to do without much difficulty. 3 The addition combination 2 is read, 2 and 3 are 5. The corresponding subtraction 5 2 is read, 2 and what are 5, 3 or 2 and how many are 5, or simply 2 and 3 are 5. Subtract thus: 5 and 4 are 9; 3 and 2 are 5. Write the 4 under the 5 and the 2 under the 3. That is, supply the figure required in each column in order to obtain the sum at the top. 59 - 35 24 If the figure in the subtrahend represents an amount larger than the corresponding figure in the minuend, sub52 tract thus: 29, 9 and 3 are 12; carry 1 to 2 as in addi 23 tion, making it 3; 3 and 2 are 5. The answer is 23. This method of subtraction may also be explained as follows: 5 15 25 Compare -3, -13, -23. If both the minuend and the subtrahend are increased by the same amount, the difference is not changed. The steps necessary to find the difference may be explained thus: It is evident that there is no number (excepting a negative quantity) which added to 9 makes 2; so 10 is added to the 2, changing that number to 12. Nine and 3 make 12. Ten must, therefore, be added to the subtrahend to equalize this change. This is done by increasing the next lower number by 1. This changes the 2 to 3. Three and 2 make 5. The answer is 23. This may be given: 9 from 12 leaves 3. 3 from 5 leaves 2, if the teacher prefers to use this language form to express the operation. The two language forms should not be confused. The 2 in the minuend is increased by the addition of 10, and the lower number in the next combination is increased by 1 to equalize this change, as above. Again, the minuend is the sum of two numbers. The subtrahend is one of the numbers. The other number is a number which added to the subtrahend will make the minuend. () difference, or other addend. 27 subtrahend, or given addend. 72 minuend, or sum of addends. Explanation: 7 and 5 make 12. 72 minuend. - 27 subtrahend. 45 difference. Write 5 as the units' figure in the missing addend, and add the 1 ten to the 2 tens of the other addend. 3 and 4 make 7. 45 is the missing addend. The method of subtraction involving the "borrowing" of 1 ten from the next place in the minuend, etc., should not be used in the exercises of this text, as the combinations would thereby be changed. Do not permit pupils to acquire the habit of making these changes in the written work. MULTIPLICATION Facility in multiplication and division, as in addition and subtraction, is acquired only by much practice. Each lesson in multiplication consists of a few number facts which are to be memorized perfectly, and then used until accuracy and facility in handling these have been acquired. The number facts of each lesson should become perfect reflexes before the new facts of the succeeding lesson are introduced. Much drill work is provided for in the text. This should be supplemented by similar material in case the pupils have not acquired the desired skill in handling the facts of each lesson. No provision has been made for the mastery of the "tables" as such. Each fact must finally be known without reference to the table to which it belongs. A table of products and quotients is given near the close of the chapter on multiplication and division. It is for reference merely. Such multipliers as 20, 30, etc., are introduced before multipliers like 23, 34, etc., since the |