Warm Up

1. Warm Up Simplify each expression. 1. 16 19 2. 25 4. 3. 20 Tell which two whole numbers each square root falls between. 5 and 6 6. 2 and 3 5. 3 and 4 7. 8. 9 and 10

2. Real Numbers Vocabulary, Rational, Irrational, Classifying

3. 3 36 + 7 = 3(6) + 7 3 36 + 7 Evaluating Expressions Involving Square Roots Evaluate the expression. Evaluate the square root. = 18 + 7 Multiply. = 25 Add.

4. 2 25 + 4 2 25 + 4 = 2(5) + 4 Try This! Evaluate the expression. Evaluate the square root. = 10 + 4 Multiply. = 14 Add.

5. Vocabulary Natural numbersare the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are whole numbers and their opposites: –3, –2, –1, 0, 1, 2, 3, … Rational numberscan be expressed in the form , where a and b are both integers and b ≠ 0: , , .

6. Vocabulary (continued) Repeating decimalsare rational numbers in decimal form that have a block of one or more digits that repeat continuously: 1.3, 0.6, 2.14 Irrational numberscannot be expressed in the form . They include square roots of whole numbers that are not perfect squares and nonterminating decimals that do not repeat: , ,  Terminating decimalsare rational numbers in decimal form that have a finite number of digits: 1.5, 2.75, 4.0

7. Real Numbers Rational Numbers (Q) Irrational Numbers Integers (Z) Whole Numbers (W) Natural Numbers (N)

8. Estimating Real Numbers The square roots of many numbers like , are not whole numbers. A calculator can approximate the value of as 3.872983346... Without a calculator, you can use square roots of perfect squares to help estimate the square roots of other numbers.

9. 55 is between 7 and 8 because 55 is between 49 and 64. Estimating Square Roots of Numbers Each square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 55? 55 49 < 55 72 = 49 64 > 55 82 = 64

10. – is between –9 and –10 because 90 is between 81 and 100. 90 Estimating Square Roots of Numbers Each square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 90? – 90 81 < 90 –92 = 81 100 > 90 –102 = 100

11. 80 is between 8 and 9 because 80 is between 64 and 81. TRY THIS! Each square root is between two integers. Name the integers. Explain your answer. Think: What are perfect squares close to 80? 80 64 < 80 82 = 64 81 > 80 92 = 81

12. – is between –6 and –7 because 45 is between 36 and 49. 45 Try This! Each square root is between two integers. Name the integers. Think: What are perfect squares close to 45? – 45 36 < 45 –62 = 36 49 > 45 –72 = 49

13. 1 Understand the problem Problem-Solving Application As part of her art project, Shonda will need to make a square covered in glitter. Her tube of glitter covers 13 square inches. What is the greatest side length Shonda’s square can have? The answer will be the side length of the square. List the important information: • The tube of glitter can cover an area of 13 square inches.

14. Make a Plan The side length of the square is because = 13.  Because 13 is not a perfect square, is not a whole number. Estimate to the nearest tenth. Find the two whole numbers that is between. Because 13 is between the perfect squares 9 and 16. is between and , or between 3 and 4. 2 Continued

15. Because 13 is closer to 16 than to 9, is closer to 4 than to 3. 3 4 You can use a guess-and-check method to estimate . . Estimating Real Numbers

16. 3 Solve is greater than 3.6. is less than 3.7. Because 13 is closer to 12.96 than to 13.69, is closer to 3.6 than to 3.7.  3.6 Estimating Real Numbers Guess 3.6: 3.62 = 12.96 too low Guess 3.7: 3.72 = 13.69 too high 3.6 3.7 4 3

17. 4 Look Back Continued A square with a side length of 3.6 inches would have an area of 12.96 square inches. Because 12.96 is close to 13, 3.6 inches is a reasonable estimate.

18. Use a guess and check method to estimate . is greater than 6.1. is less than 6.2. Try This! What if…? Nancy decides to buy more wildflower seeds and now has enough to cover 38 ft2. What is the side length of a square garden with an area of 38 ft2 to the nearest tenth? Guess 6.1 6.12 = 37.21 too low Guess 6.2 6.22 = 38.44 too high A square garden with a side length of 6.2 ft would have an area of 38.44 ft2. 38.44 ft is close to 38, so 6.2 is a reasonable answer.

19. Real numbers All numbers that can be represented on a number line are called real numbers and can be classified according to their characteristics.

20. Classifying Real Numbers Write all classifications that apply to each Real number. A. –32 32 can be written as a fraction and a decimal. –32 = – = –32.0 rational number, integer, terminating decimal B. 5 5 can be written as a fraction and a decimal. 5 = = 5.0 rational number, integer, whole number, natural number, terminating decimal C. The digits continue with no pattern. irrational number

21. 9 = 3 81 3 9 3 = = 3 Classifying Real Numbers Write all classifications that apply to each Real number. 9 A. rational number, integer, whole number, natural number, terminating decimal –35.9 –35.9 is a terminating decimal. B. rational number, terminating decimal C. rational number, integer, whole number, natural number, terminating decimal

22. 7 can be written as a repeating decimal. 4 9 3a. 7 67  9 = 7.444… = 7.4 49 3c. = 3.16227766… Try This! Write all classifications that apply to each real number. rational number, repeating decimal 3b. –12 -12 can be written as a fraction and a decimal. –12 = – = –12.0 rational number, terminating decimal, integer The digits continue with no pattern. irrational number

23. 7. –3.89 Lesson Quiz Find each square root. 1 2 3 7 -8 3. 4. 1. 12 2. – 5. The area of a square piece of cloth is 68 in2. How long is each side of the piece of cloth? Round your answer to the nearest tenth of an inch. 8.2 in. Write all classifications that apply to each real number. rational, integer, whole number, natural number, terminating decimal 6. 1 rational, repeating decimal 8. irrational