2. Lesson Menu Main Idea and New Vocabulary NGSSS Key Concept:Irrational Numbers Example 1:Classify Numbers Example 2:Classify Numbers Example 3:Classify Numbers Example 4:Compare Real Numbers Example 5:Compare Real Numbers Example 6:Compare Real Numbers Example 7:Order Real Numbers Example 8:Real-World Example Five-Minute Check

3. Main Idea/Vocabulary Compare mathematical expressions involving real numbers. irrational number real number

4. NGSSS MA.8.A.6.2 Make reasonable approximations of square roots and mathematical expressions that include square roots, and use them to estimate solutions to problems and to compare mathematical expressions involving real numbers and radical expressions.

5. Key Concept

6. Example 1 Classify Numbers Name all sets of numbers to which the real number 0.090909… belongs. Answer: The decimal ends in a repeating pattern. It is a rational number because it is equivalent to.

7. Example 1 CYP A.whole B.rational C.whole, integer, rational D.irrational Name all sets of numbers to which the real number 0.454545… belongs.

8. Example 2 Classify Numbers Name all sets of numbers to which the real number belongs. Answer: Since = 5, it is a whole number, an integer, and a rational number.

9. Example 2 CYP A.whole B.whole, integer C.whole, integer, rational D.irrational Name all sets of numbers to which the real number belongs.

10. Example 3 Classify Numbers Name all sets of numbers to which the real number belongs. Answer: –3.464101615… Since the decimal does not terminate or repeat, it is an irrational number.

11. A.whole B.integer, rational C.whole, integer, rational D.irrational Example 3 CYP Name all sets of numbers to which the real number belongs.

12. Example 4 Compare Real Numbers Write each number as a decimal. Replace with, or = to make a true statement.

13. Example 4 Compare Real Numbers Since 3.872983346… is less than 3.875,. Answer:

14. Example 4 CYP A.< B.> C.= Replace with, or = to make 4 a true statement.

15. Example 5 Compare Real Numbers Write as a decimal. ≈ 3.224903099… = 3.222222222… Replace with, or = to make a true statement.

16. Example 5 Compare Real Numbers Since 3.222222222… is less than 3.224903099…, Answer:

17. Example 5 CYP A.< B.> C.= Replace with, or = to make a true statement.

18. Example 6 Compare Real Numbers Write each number as a decimal. 12.3% = 0.123 = 0.1 Replace with, or = to make 12.3% ● a true statement.

19. Example 6 Compare Real Numbers Answer: 12.3% > Since 0.123 is greater than 0.1, 12.3% >.

20. Example 6 CYP A.< B.> C.= Replace with, or = to make 90% a true statement.

21. Example 7 Write each number as a decimal. Then order the decimals. 3 = 3.00 Order Real Numbers Order from least to greatest. Verify your answer by graphing on a number line.

22. Example 7 Order Real Numbers Answer: From least to greatest, the order is 3,,, and 4.

23. Example 7 CYP Order from least to greatest. A. B. C. D.

24. Example 8 BEACH VOLLEYBALL The time in seconds that it takes an object to fall d feet is About how many seconds would it take for a volleyball thrown 32 feet up in the air to fall from its highest point to the sand?

25. Example 8 Use a calculator to approximate the time it would take for a volleyball to fall to the sand. Answer: It would take a volleyball thrown 32 feet up in the air about 1.4 seconds to fall to the sand. Replace d with 32. ≈ 1.4 s Use a calculator.

26. Example 8 CYP A.about 70 s B.about 9.7 s C.about 2.5 s D.about 1.1 s PHYSICAL SCIENCE The time in seconds that it takes an object to fall d feet is About how many seconds longer would it take for a ball dropped from 100 feet to fall to the ground than a ball dropped from 30 feet?

27. A.whole B.integer C.whole, integer D.whole, integer, rational Name all sets of numbers to which the real number 286 belongs. Five Minute Check 1

28. A.integer B. integer, rational C.whole, integer, rational D.irrational Five Minute Check 2 Name all sets of numbers to which the real number belongs.

29. A.< B.> C.= Five Minute Check 3 Replace with, or = to make a true statement.

30. A.< B.> C.= Five Minute Check 4 Replace with, or = to make a true statement.

31. A.sometimes B.always C.never Are irrational numbers sometimes, always, or never rational numbers? Five Minute Check 5

32. A.real B.fractions C.rational D.negative integers Five Minute Check 6 To which set does NOT belong?