1. Comparing Fractions © Math As A Second Language All Rights Reserved next #6T Taking the Fear out of Math 7 15 2 3

2. next © Math As A Second Language All Rights Reserved Preface next Once we have internalized the meaning of a common fraction, there are times when we can tell, simply by looking, which of two fractions is greater. For example, students who have not internalized the definition often believe that 7 / 15 is greater than 2 / 3 because 7 is greater than 2, and 15 is greater than 3.

3. © Math As A Second Language All Rights Reserved next However, if we have internalized what a common fraction means we see at once that 7 / 15 is less than half while 2 / 3 is more than half. In terms of manipulatives that are easy for even the youngest students to understand, imagine that there are 15 pencils on the table and they take 7 of them (that is, they have taken 7 pencils). Classroom Note

4. next © Math As A Second Language All Rights Reserved next They will be able to see that there are more pencils left (8) than the number of pieces they have taken (7). So they have taken less than half of the pencils 7 pencils 8 pencils

5. © Math As A Second Language All Rights Reserved next And if there are 3 pencils on the table and they take 2 of them (that is, they have taken 2 / 3 of the pencils), they have taken more than they have left on the table and hence they have taken more than half of the pencils. 2 pencils1 pencil

6. © Math As A Second Language All Rights Reserved next However, there are times when it is not as easy to determine by sight which of the two fractions names the greater amount. For example, while it is relatively easy to see that both 2 / 5 and 3 / 8 are less than 1 / 2, it is not as easy to see which of the two fractions is the greater one. So we will soon develop a technique that works all the time.

7. © Math As A Second Language All Rights Reserved Rational Numbers next The study of common fractions begins with what happens when we divide two whole numbers. In such a situation, the quotient might be a whole number, but it doesn’t have to be. So to extend the whole number system, we agree to define the quotient of two whole numbers (provided the divisor is not 0) to be a rational number.

8. © Math As A Second Language All Rights Reserved Rational Numbers next Thus, every whole number is a rational number (for example, 6 is also 6 ÷ 1) but not every rational number is a whole number. Although 5 ÷ 3 is not a whole number, it is a rational number which we may denote by the common fraction 5 / 3.

9. © Math As A Second Language All Rights Reserved next In the previous lesson, we introduced the concept of unit fractions. From a non-mathematical point of view, notice that the two statements “John is taller than Bill” and “Bill is shorter than John” do not sound the same. Yet, both say the same thing from a different point of emphasis.

10. © Math As A Second Language All Rights Reserved next The same relationship exists between non zero whole numbers and a special set of common fractions called unit fractions. We “complement” the set of numbers 1, 2, 3, 4, 5, 6 etc., with a new set of numbers, 1 / 1, 1 / 2, 1 / 3, 1 / 4, 1 / 5, 1 / 6 etc. To see the connection between these two sets of numbers visually, consider the “corn bread” (i.e., rectangle) that appears below.

11. © Math As A Second Language All Rights Reserved next Looking at the rectangle, it is impossible to tell whether we started with the smaller rectangle and marked it off 3 times to obtain the larger rectangle, or whether we started with the larger rectangle and divided it into 3 smaller pieces of equal size.

12. © Math As A Second Language All Rights Reserved next In terms of our corn bread model, if we put in the units, we do not confuse 1 piece with 1 corn bread. However, once the units are omitted it is impossible to know without being told what the 1 is modifying. 1 piece 1 Corn Bread 1 1

13. next © Math As A Second Language All Rights Reserved next For example, if the adjective 1 is modifying “piece”, it means that the adjective 3 is modifying the (whole) “corn bread” Corn Bread 1 piece 3 pieces

14. next © Math As A Second Language All Rights Reserved next However, if we started with the corn bread as our unit and used the adjective 1 to modify it, then each piece of the corn bread is modified by the adjective 1 / 3 because each piece is 1 of what it takes 3 of to equal the whole corn bread. Corn Bread 1/31/3 1/31/3 1/31/3

15. next © Math As A Second Language All Rights Reserved What we’re calling our “corn breads” can represent any amount. next Notes Thus, if our “corn bread” represented $24, 1 / 3 of the “corn bread” would represent $8 because $24 ÷ 3 = $8. The fact that 3 × 8 = 24 allows us to say that 24 is 3 times 8 and equivalently that 8 is 1 / 3 of 24. next

16. © Math As A Second Language All Rights Reserved Thus, in much the same way that “John is taller than Bill”, and “Bill is shorter than John” are two different ways of transmitting the same information, “28 ÷ 4 = 7” and “ 1 / 4 of 28 = 7 ” are also two different ways of transmitting the same information. 1 next Notes note 1We should be cautious here. 4 pens at $7 each is not the same “event” as 7 pens at $4 each even though the cost is the same in both cases. In this context, 1 / 4 of 28 = 7 is an answer to 28 ÷ 4 = ___. And 1/7 of 28 = 4 is an answer to 28 ÷ 7 = ___. Hence, 1 / 4 of 28 = 7 and 1 / 7 of 28 = 4 are both correct ways of saying that the product of 4 and 7 is 28.

17. next © Math As A Second Language All Rights Reserved An ordinary ruler, marked off in inches, uses other unit fractions as well. next For example, if the ruler is marked off in eighths of an inch. We can use 1 / 8 as a unit fraction, meaning that any measurement on the ruler can be given in terms of the number of inches and also in terms of the number of 1 / 8 of an inch. 1 inch 1/81/8 2/82/8 3/83/8 4/84/8 5/85/8 6/86/8 7/87/8 8/88/8 8 / 8 = 1

18. next © Math As A Second Language All Rights Reserved That is, just as we can count 1, 2, 3, 4, 5 etc., we can also count, 1 eighth, 2 eighths, 3 eighths, 4 eighths, 5 eights, etc., or in more mathematical terms, 1 × 1 / 8, 2 × 1 / 8, 3 × 1 / 8, 4 × 1 / 8, 5 × 1 / 8, etc. next In this context, as shown in the figure below, the shaded region which we call 5 / 8 is the same size as 5 × 1 / 8. In this form, it is easy to see why 5 is the called the numerator (it counts the number of pieces) and why the denominator is referred to as “eighths”. 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8 1/81/8

19. © Math As A Second Language All Rights Reserved This leads to a conceptual problem when we define 9 / 8. Namely, if there are only 8 pieces, then you can’t take 9 of them. However, you can take 9 of what it would take 8 of to make the whole. next There is a tendency to define 5 / 8 by saying we divide the unit into 8 pieces and then take 5 of the pieces. Notes 1 inch 9/89/8 10 / 8 1/81/8 2/82/8 3/83/8 4/84/8 5/85/8 6/86/8 7/87/8 8/88/8

20. next © Math As A Second Language All Rights Reserved The ancient Greek mathematicians knew how to divide a line segment into any number of pieces of equal length 2. next Viewing the original length as representing 1 unit, and assuming that the segment was divided into n pieces of equal length, each of the smaller pieces was named by the unit fraction 1 / n. note 2 The concept of dividing a line segment into any number of pieces of equal length gave rise to the concept of the number line. In this context, we may think of the corn breads we’ve drawn as being “thick” number lines. Thus when we talk about dividing the corn bread into 5 pieces of equal size, it corresponds to the ancient Greeks dividing a line segment into 5 pieces of equal length. next Historical Geometrical Note

21. next © Math As A Second Language All Rights Reserved The ancient Egyptians were enamored with unit fractions, and they were particularly interested in expressing all common fractions as sums of unit fractions. next For example, they were intrigued by such facts as 5 / 6 could be written as the sum of the two unit fractions, 1 / 2 and 1 / 3 ( 1 / 2 + 1 / 3 = 5 / 6 ) and in that context they viewed unit fractions as the “building blocks” for obtaining all common fractions. Historical Geometrical Note

22. next © Math As A Second Language All Rights Reserved This is an interesting way of illustrating that 1 + 1 can equal 5 if the numbers are not modifying the same noun. next More specifically… 1 half + 1 third = 5 sixths Anecdotal Aside

23. next In our next presentation, we will discuss equivalent fractions. © Math As A Second Language All Rights Reserved Comparing Fractions 1515 1818