1. 6 Chapter Rational Numbers and Proportional Reasoning
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 6-1 The Set of Rational Numbers
Students will be able to understand and explain
• Different representations for rational numbers.
• Equal fractions, equivalent fractions, and the simplest form of fractions.
• Ordering of rational numbers.
• Denseness property of rational numbers.

3. Definition
Numerator, Denominator
numerator
denominator

4. Uses of Rational Numbers

5. Rational Number Models

6. Meaning of a Fraction To understand the meaning of any fraction, using
the parts-to-whole model, we must consider each of the following:
The whole being considered.
The number b of equal-size parts into which the whole is divided.
The number a of parts of the whole that are selected.

7. Definition Proper fraction A fraction where Examples are
Improper fraction
A fraction such that
Examples are

8. Number Line Model
What numbers are represented on the number line?

9. Equivalent or Equal Fractions
Equivalent fractions are numbers that represent the same point on a number line.

10. Fraction Strips

11. Fundamental Law of Fractions
Let be any fraction and n a nonzero whole
number, then

12. Example
Find a value for x such that

13. Simplifying Fractions
A rational number is in simplest form if b > 0 and GCD(a,b) = 1; that is, if a and b have no common factor greater than 1.

14. Example
Write each of the following fractions in simplest form if they are not already so: a. b. c. d.
cannot be simplified

15. Example (continued) e. f. g.
cannot be simplified

16. Equality of Fractions Show that
Method 1: Simplify both fractions to the same simplest form.

17. Equality of Fractions Show that
Method 2: Rewrite both fractions with the same least common denominator.
LCM(42, 35) = 210, so

18. Equality of Fractions
Method 3: Rewrite both fractions with a common denominator.
A common multiple of 42 and 35 is 42 · 35 = 1470.

19. Equality of Fractions Two fractions and , d  0 are equal if and
only if ad = bc.

20. Ordering Rational Numbers
If a, b, and c are integers and b > 0, then if and only if a > c.
If a, b, c, and d are integers
and b > 0, d > 0, then
if and only if ad > bc.

21. Denseness of Rational Numbers
Given rational numbers there is another rational number between these two numbers.

22. Example Find two fractions between Because is between
Now find two fractions equal to respectively, but with greater denominators.

23. Example (continued)
are all between
so they are between

24. Example Show that the sequence …, is increasing.
Because the nth term of the sequence is the next term is
We need to show that for all positive integers n,

25. Example (continued) The inequality is true, if, and only if,
which is true for all n.
Thus, the sequence is increasing.

26. Denseness of Rational Numbers
Let be any rational numbers with positive denominators, where Then