1. Common Fractions © Math As A Second Language All Rights Reserved next #6 Taking the Fear out of Math 1÷3 1 3

2. next © Math As A Second Language All Rights Reserved Inventing Common Fractions next You have probably noticed how much more convenient it is to write “5 inches” than to write “5 of what it takes 12 of to equal 1 foot”. So our first abbreviation is to replace “5 of what it takes 12 of to equal 1 foot” by “5 twelfths of a foot”.

3. © Math As A Second Language All Rights Reserved A “twelfth” means “1 of what it takes 12 of to equal the given unit”. So for example… next --- 5 twelfths of a foot means 5 of what it takes 12 of to equal 1 foot. Definition --- 5 twelfths of a dozen means 5 of what it takes 12 of to equal 1 dozen.

4. next © Math As A Second Language All Rights Reserved Since we use numbers as adjectives when we do arithmetic, we want to invent a mathematical symbol to represent a twelfth. next The symbol we invent to denote one twelfth is called a common fraction, and it is written as 1 / 12. The top number (1) is called the numerator and the bottom number (12) is called the denominator.

5. © Math As A Second Language All Rights Reserved Notice that the word “numerator” suggests the word “enumerate” which means “to count”. “To count” suggests “how many” and “how many” suggests an adjective. Hence, in terms of our adjective/noun theme, the numerator is the adjective. 1 next Notes note 1 There is a tendency for students to define the numerator as being the top. This masks the true meaning of the numerator. In fact if “numerator” simply meant “top”, most likely we would have not replaced the simpler word “top” by the more cumbersome word “numerator”.

6. next © Math As A Second Language All Rights Reserved Notice that the word “denominator” suggests the word “denomination” which suggests the size of the quantity, and the size is a noun. Hence, in terms of our adjective/noun theme, the denominator is the noun. next Notes 5 12 5 is the adjective. 12 is the noun. 5 is the numerator. 12 is denominator.

7. next © Math As A Second Language All Rights Reserved The important point is that just as the word “inch” is a noun, so is the word “twelfths”. next In a similar way, we can define the following common fractions… which we read as “a (or, one) half” and which means 1 of what it takes 2 of to equal the whole. 1 2

8. © Math As A Second Language All Rights Reserved next which we read as “a (or, one) third” and it means 1 of what it takes 3 of to equal the whole. 1 3 which we read as “a (or, one) fourth” and it means 1 of what it takes 4 of to equal the whole. 1 4

9. © Math As A Second Language All Rights Reserved next which we read as “a (or, one) fifth” and it means 1 of what it takes 5 of to equal the whole. 1 5 which we read as “a (or, one) sixth” and it means 1 of what it takes 6 of to equal the whole. 1 6

10. © Math As A Second Language All Rights Reserved The above common fractions are called unit fractions because they behave the same way as other units. For example, when we count “1, 2, 3,...” the numbers are assumed to be modifying a particular unit. Therefore, 1, 2, 3 can refer to “1 half, 2 halves, 3 halves,...” or “1 third, 2 thirds, 3 thirds,...” or “1 fourth, 2 fourths, 3 fourths,... ” next

11. © Math As A Second Language All Rights Reserved An Important Connection between Division and Common Fractions next When we say to take 1 of what it takes 5 of to equal a given unit, it means the same thing as dividing the given unit by 5. In other words, taking a fifth of a number means the same thing as dividing the number by 5. For example, 1 / 5 of 30 means the same thing as 30 ÷ 5.

12. © Math As A Second Language All Rights Reserved More Notation next In the same way that we may think of 3 apples as 3 × 1 apple, we may think of 3 fifths as 3 × 1 fifth, and we write it as 3 / 5. In this context, 3 / 5 of 30 means 3 × 1 / 5 of 30 or 3 × 6 or 18. To take a fractional part of a number we divide the number by the denominator (to find the size of each part) and then multiply by the numerator (the number of parts we are taking).

13. © Math As A Second Language All Rights Reserved next For example, to take 4 / 7 of 56 we would first divide 56 by 7 to obtain 8, and we would then multiply 8 by 4 to obtain 32. In terms of a picture, we may think of 56 as being represented by a rectangle (which we personify by referring to it as a “corn bread”) corn bread

14. next © Math As A Second Language All Rights Reserved next We then divide the corn bread into 7 equally-sized pieces to obtain… Thus, 56 is represented by our corn bread. 56 88888888 8888 And finally, we take 4 of the pieces. next

15. © Math As A Second Language All Rights Reserved There is a tendency for some teachers (and some textbooks as well) to define 4 / 7 by saying it means to divide the given unit into 7 parts of equal size and then to take 4 of these equal parts. next Important Note for the Teacher Roughly speaking, they say it means to take “4 out of 7”.

16. next © Math As A Second Language All Rights Reserved This is not a problem as long as the numerator is not greater than the denominator. However, it can raise sort of a mystical question when the numerator is greater than the denominator. next For example, if we define 8 / 7 as meaning to divide the given unit into 7 parts of equal size and then take 8 of these parts, it raises the serious question as to whether we can take 8 parts from a group that has only 7 parts.

17. © Math As A Second Language All Rights Reserved However this problem is avoided if we use the adjective/noun way of defining a common fraction. Namely, we define 4 / 7 by saying that we are taking 4 of what it takes 7 of to equal the given unit. next In a similar way, 8 / 7 means that we take 8 of what it takes 7 of to equal the given unit. In that way, we see that it is equal to the entire given unit (that is, 7 sevenths) plus 1 more part.

18. © Math As A Second Language All Rights Reserved While most of us might not have thought about it in that way, the ordinary ruler is a very nice example of the “marriage” between arithmetic and geometry. next The ruler is basically a straight line (geometry) with equally spaced points (again, geometry) marked on it. The points are then given names such as 1, 2, 3, etc (arithmetic). An Application of Geometry to Arithmetic

19. next © Math As A Second Language All Rights Reserved In essence, the ruler is a model for the number line where geometric points are given arithmetical names. 2 next note 2 Notice that name “number line” itself indicates a combination of arithmetic (number) and geometry (line). One of the constructions that’s described in Euclid’s elements is how to divide a line segment of any length into any number of equally sized pieces. next

20. © Math As A Second Language All Rights Reserved Most of us are aware of the simple case of dividing a piece of string or a sheet of paper into two pieces of equal size. next Namely, we essentially fold it in half by placing the ends together.

21. © Math As A Second Language All Rights Reserved Let’s look at Euclid’s way to divide a piece of string (of any given length) into 5 equally sized pieces. 3 next Suppose you want to divide the line segment AB into 5 pieces of equal size. AB note 3 We choose 5 simply for illustrative purposes. The same concept would work for obtaining any number of equally-sized pieces. next

22. © Math As A Second Language All Rights Reserved Step 1: Through the point A draw a line of any length of your of your choosing. next AB

23. © Math As A Second Language All Rights Reserved Step 2: Pick any size length and on the line you chose, mark that length off 5 consecutive times. next Label the points you obtain in this way C, D, E, F, and G. AB C D E F G

24. © Math As A Second Language All Rights Reserved next note 4 There is a subtle but important difference between writing AC and AC. Namely when we write AC we are referring to the set of points that constitute the line segment AC. However when we write AC we are referring to the length of the line segment AC. Thus, to be precise, we do not write AC = CD because these two line segments do not consist of the same points. However what is true is that the length of these two segments are the same; and to indicate this we write AC = CD. By construction the line segment AG is divided into 5 pieces of equal length. That is, AC = CD = DE = EF = FG. 4

25. next © Math As A Second Language All Rights Reserved Step 3: Draw the line segment GB. next AB C D E F G

26. © Math As A Second Language All Rights Reserved Step 4: Through each of the points C, D, E, and F draw lines that are parallel to GB, and label the points at which these lines intersect AB by H, I, J, and K. next KJ I H AB C D E F G

27. © Math As A Second Language All Rights Reserved next And the fact that the points on the line segment AG are equally spaced means that the line segment AB has also been divided into 5 pieces of equal length. That is, AH = HI = IJ = JK = KB In most text books, the “whole” is usually a circle (either a pie or a pizza). However, it is much easier to divide a line segment into pieces of equal length (5 pieces) than it is to divide a circle into 5 pieces of equal size. Students might find it to be an enjoyable activity to practice the above construction.

28. next © Math As A Second Language All Rights Reserved Moreover, our corn bread is a “thick” number line, and the same Euclidian construction easily divides the corn bread into any number of equal parts. next KJ I H AB C D E F G Students seem to visualize a two dimensional cornbread more easily than the one dimensional number line.

29. © Math As A Second Language All Rights Reserved next By now you should be getting the idea that when treated in terms of the adjective/noun theme, the arithmetic of fractions is a special application of the arithmetic of whole numbers. Final Note All we have done is defined units that are a fractional part of other units and expressed these new units as common fractions.

30. next © Math As A Second Language All Rights Reserved next This is why it so important for students to internalize the arithmetic of whole numbers. Final Note If the students’ knowledge of arithmetic consists of rote learning, it is very likely that serious problems will arise when these students encounter the arithmetic of fractions.

31. next We will get a clearer insight to the arithmetic of fractions in our next presentations. © Math As A Second Language All Rights Reserved Common Fractions 1515