PROBLEM: If 40 men can build a fence in 30 days, how many days will it take 100 men to build the same fence ? It will not take 100 men as long as 40 men. It will take only as long. of 30 days = 12 days. Hence, it will take 100 men 12 days to build the same fence. PROBLEM: Mr. Brown was mixing 4 grades of coffee while I was in his store this morning. He had some worth 31 cents per pound, some 19 cents, some 40 cents, some 42 cents. He used equal quantities of each kind. He wants to make a profit of 6 cents on each pound of the mixture. How will he sell it per pound? SOLUTION: A 4-lb. mixture is worth 31 +19 +40 +42, or 132 cents. One pound of the mixture is worth of 132 cents, or 33 cents. To make a profit of 6 cents on a pound, he must sell it for 33+6, or 39 cents. SHORT REVIEW: 1. What is the difference between addition and multiplication? 2. What is an addend? a sum ? 3. Give two ways of proving an addition. 4. Name the difficulties you have found in long division. 5. The remainder always has what name? 6. Define what you mean by divisor, dividend, quotient, remainder, short division, long division. 7. How can you tell when a quotient figure is too large too small? right? A. H.-9 PART THREE. FACTORING-LEAST COMMON CANCELLATION. "What is really wanted is to light up the spirit that is within a boy. In some sense and in some effectual degree, there is in every boy the material of good work in the world; in every boy, not only in those who are brilliant, not only in those who are quick, but in those who are solid, and even in those who are dull." - Gladstone. DEFINITIONS. T to HE pupil should be made to feel that he has accepted nothing without a clear and demonstrative reason, the entire exclusion of authority, tradition, prejudice or selfinterest. BAIN. IN education the process of self-development should be encouraged to the uttermost. Children should be led to make their own investigations, and to draw their own inferences. They should be told as little as possible, and induced to discover as much as possible. Any piece of knowledge which the pupil has himself acquired, any problem which he has himself solved, becomes by virtue of the conquest much more thoroughly his than it could else -SPENCER, "Education." be. FACTORING. TALK: Pupils should learn to factor numbers readily if they are to become expert in the operations of arithmetic. This knowledge is very essential in finding the Greatest Common Divisor and Least Common Multiple; also in fractions, percentage and its applications, and in the work in ratio and proportion. Give the children a thorough knowledge of factoring and they will perform many of the problems they are given without the use of pencil. Have them memorize the factors of the numbers to one hundred that they meet most frequently. It should be part of their stock in trade. The factors of a number are the numbers which, when multiplied together, equal that number. The factors of 9 are 3 and 3, because 3 × 3 = 9. A factor of a number, then, is a divisor of the number. 2 is a factor of 10, since it is a divisor of 10. Because we can divide any number by itself and 1 or unity, we seldom mention them as factors. Factoring a number is separating it into its factors. Some numbers, like 3, 5, 11, etc., can only be divided by themselves and unity. As they cannot otherwise be factored, we call them prime numbers. Those numbers that can be factored, like 6, 10, 18, 22, etc., are called composite numbers. They are called so from the fact that they are thought of as composed of other numbers multiplied together. An exact divisor of a number is one that is contained in it an exact number of times, with no remainder. 3 is an exact divisor of 18, since 3 is contained in 18 exactly six times. An even number is one that is exactly divisible by 2. 4, 6, 8, 10, 12, etc., are even numbers. All numbers that end in 2, 4, 6, 8, or 0 are even. An odd number is one that is not exactly divisible by 2, as 1, 3, 5, 7, 9, etc. All numbers that end in 1, 3, 5, 7, or 9 are odd, and that means that they cannot be divided by 2 or any multiple of 2. Teach the child to always recognize, at a glance, an even or odd number. Ask him how he knows. Any number that is a prime number is a prime factor. 5 and 2 are factors of 10 and because they are prime numbers they are also prime factors. Likewise, factors that are composite numbers are composite factors. 6 and 4 are composite factors of 24, and they are composite numbers. The prime factors of 24 are three twos and one three; of 36, two twos and two threes; and of 28, two twos and one seven. |