1. Unit 1B: Rational Numbers Mathematics 9 Miss Wilkinson October 2, 2013

2. Introduction Rational Numbers are numbers that can be expressed in the form where a & b are integers and b ≠ 0. Rational numbers are parts of a whole, expressed as a fraction (1/4), decimal (0.25), or percent (25%). Examples:

3. Fun Fact: Who Was Correct? The ancient greek mathematician Pythagoras believed that all numbers were rational (could be written as a fraction). One of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational. The discovery of irrational numbers is said to have been shocking to the Pythagoreans, and Hippasus is supposed to have drowned at sea, apparently as a punishment from the gods.

4. Part 1: Comparing and Ordering Rational Numbers OBJECTIVES In part 1, we will learn to reduce, compare and order rational numbers, Be able to express rational numbers in fractional form with a common denominator or decimal form. Learn to compare rational numbers using a number line and identify rational numbers between two given rational numbers.

5. Reducing Fractions We reduce a fraction to lowest terms by finding an equivalent fraction in which the numerator and denominator are as small as possible. To reduce fractions, find the greatest common factor for the numerator and denominator. Examples: 1. 2. 3.

6. Compare & Order Fractions We can compare rational numbers by expressing them all as fractions with a common denominator or by expressing them as decimals. A) To compare fractions, express each pair of fractions with the common denominator. To find a common denominator, determine the lowest common multiple (LCM) of the given denominators.

7. Ex: Which is greater → or ? Step 1: The LCM of 6 and 9 is 18. Step 2: We know that 6 divides into 18 three times so we will multiply the numerator of the first term by three, and 9 divides into 18 two times, so we will multiply the numerator of the second term by two. Step 2: Re-write and as: and Step 3: Compare the numerators. therefore, < <

8. Practice Compare the fractions by circling the correct symbol. Remember: The fractions must have the same denominator to be able to compare! 1. 2. 3.

9. B) We can also compare fractions by converting them to decimals. To convert fractions to decimals, divide the denominator by the numerator. Note: A fraction is essentially a division operation. Example: means 3 ÷ 4 = 0.75 1. =2. =

10. C) We can compare decimals by converting decimals to fractions. To do this, write the number as you would read it. Example: 0.05 is read as 5 hundredths. Therefore, put 5 over 100 and reduce to lowest term.

11. Practice Convert the following decimals into fractions. Remember to reduce to lowest terms! 1. 0.3 = 2. - 0.48=3. 0.024 =

12. Rational Numbers on a Number Line OBJECTIVE FOCUS: learn to compare rational numbers using a number line and identify rational numbers between two given rational numbers.

13. Example 1

14. Example 2

15. Checkpoint 1