FRACTIONS 99. To separate a number of objects into groups, each having the same number. The number of ones in each line is divided into two equal parts. Each of the two equal parts of any number is one half (also written) the number. Two halves of any number together equal the whole of it. TO THE TEACHER. Have pupils separate different even numbers of objects into two equal parts, and drill them in telling the whole number, the number in each half, the number in two halves. Have pupils put even numbers of dashes, dots, circles, squares, etc., on paper or blackboard, separating each number into two equal groups. 100. INTEGRAL AND FRACTIONAL UNITS Study Recitation We have found that if a number of objects is separated into two equal parts, each group is of the whole. If each object is divided into 2 equal parts, each part is of the object. A -B If the line is divided into two equal parts, each of the parts, ac and BC, is of the whole line. If the square is divided into two rectangles by the dotted line, each rectangle is what part of the square? If the circle is divided into two equal parts by the dotted line, each part is what part of the circle? One half of the unit 1 is 1. There are 5 lines in this row; each line is what part of the whole number of lines, or of the 5 lines combined into 1 line ? A unit is a single thing, as: 1 line; 1 dot; 1 dollar; 1. 1 of the 5 lines is a unit. If the 5 lines are combined, end to end, so as to make 1 line, as then this line becomes a unit because it is now 1 line instead of 5 separate lines. If we now separate the line into 5 equal parts, as then each part becomes a unit again, but is a smaller unit than the whole line from which it was taken. Each of these equal units is a fractional part of the large unit, the whole line, and is of it. It is called a fractional unit. line is what part of the 4 lines? Why is each line a unit? If we combine the 4 lines into 1 line , that line is a unit. Why? If now we divide it into 4 equal parts, of the whole line each part becomes what fractional unit. ? 3 lines, each of the same length as 1 of the fractional units, are of the whole line, or 3 times the fractional unit of the line. 4 lines, each of the same length as 1 of the fractional units, are of the whole line, or 4 times the fractional unit of the line. If the number 1 is divided into 2 equal parts, each part is a fractional unit, . 3 halves means 3 of such fractional units. If the number 1 is divided into 4 equal parts, each of these parts is the fractional unit, . 2 of these parts are 2 of the fractional units, or 2. 3 of these parts are 3 of the fractional units, or . such parts are 7 of these fractional units, or 1. 3 2. 7 If the number 1 is divided into 5 equal parts, each part is the fractional unit. 4 of these parts are 4 of the fractional units, or . When we speak of 7 horses, what is the unit? When we speak of, what is the unit? What kind of a unit? Answer: A fractional unit. When we speak of 9 dollars, what is the unit? What kind of a unit? How many of these units are there in ? How many of these units are there in ? When we speak of 2 bushels and 3 bushels, what is the unit in each case? What is the sum of these units? What is their difference? When we speak of and, what is the unit in each case? What is the sum of the units? What is the difference of the units? If we say 3 desks, what is the unit? 2 times 3 desks are how many desks? How many units are there in this number? What is 1 of the units? If we say 2 times 3, what is the product? How many units are there in the product? What is each unit? If we say 2 times, how many 8ths are there? What is the unit? When we speak of a number of things, no matter what kind, 1 thing of that kind is the unit. Show what is meant by this statement. it If we speak of a number, as: 4, 5, 7, without giving any name, what is the unit ? 7 When we write a fraction, as: 2, 3, 2, 1 of the kind named is the unit; as,,, . 12. TO THE TEACHER. Have pupils draw lines 12 inches long on the board and then divide the lines into equal parts and tell what fraction of the whole line each of the equal parts is; what fraction of the whole line two or more of the parts are; and what fraction one or more of the equal parts is of two or more parts. Question thoroughly to see that pupils understand the nature of the fractional unit and of the fraction. REVIEW 101. A fraction is one or more fractional units: 5 7 3' 6' 6' 6° A fractional unit is made by dividing a unit into equal parts; it is one of the equal parts of a unit: 1, 4, 1, 1. A fraction is written with two numbers, one above and the other below a line: The denominator of a fraction is the number below the line, and shows into how many parts the unit is divided. It tells the unit. If we write the denominator of any fraction with 1 for its numerator, we have the fractional unit of the fraction. ILLUSTRATION: . Writing the denominator 6 with 1 for its numerator thus, , we have the fractional unit. There are 5 of these fractional units in g. The numerator is the number above the line, and shows how many fractional units there are in the fraction. ILLUSTRATION: 3. The numerator shows that three fractional units are taken. The denominator shows that the fractional unit is, or that the unit has been divided into 8 equal parts. |