Estimating Square Roots and Solving Problems

1. Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

2. 3. 8+ 144 4. 7 289 Warm Up Find the two square roots of each number. Evaluate each expression. 1. 144 2. 256 12 16 20 119

3. Problem of the Day A pyramid of blocks is built in layers. The bottom layer has 62, or 36, blocks. The next layer has 52 blocks, and so on until the top layer has 1 block. How many blocks are there in all? 91 blocks

4. 3-6 Learn to estimate square roots and solve problems using square roots.

5. 55 is between 7 and 8 because 55 is between 49 and 64. Additional Example 1A: Estimating Square Roots of Numbers The 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

6. – is between –9 and –10 because 90 is between 81 and 100. 90 Additional Example 1B: Estimating Square Roots of Numbers Continued The 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

7. 80 is between 8 and 9 because 80 is between 64 and 81. Check It Out: Example 1A The 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

8. – is between –6 and –7 because 45 is between 36 and 49. 45 Check It Out: Example 1B The square root is between two integers. Name the integers. Think: What are perfect squares close to 45? – 45 36 < 45 (–6)2 = 36 49 > 45 (–7)2 = 49

9. The length of each side of the square is √500 . Additional Example 2: Application You want to sew a fringe on a square tablecloth with an area of 500 square inches. Calculate the length of each side of the tablecloth and the length of fringe you will need to the nearest tenth of an inch. 484 < 500 < 529 List the perfect squares nearest 500. Find the square roots of the perfect squares. The number will be between 22 and 23. The length of each side of the table is about 22.4 in., and you will need about 89.6 in. of fringe.

10. The length of each side of the square is √168 . < √144 < √168 √169 12 < < 13 √168 √168  13 Check It Out: Example 2 A tent was advertised in the newspaper as having an enclosed square area of 168 ft2. What is the approximate length of the sides of the square area? Round your answer to the nearest foot. 144 < 168 < 169 List the perfect squares nearest 168. Find the square roots of 144 and 169. 168 is closer to 169 than to 144. Each side of the tent is about 13 feet long.

11. Additional Example 3: Approximating Square Roots to the Nearest Hundredth Approximate to the nearest hundredth. Step 1: Find the value of the whole number. Find the perfect squares nearest 141. 121 < 141 < 144 Find the square roots of the perfect squares. The number will be between 11 and 12. The whole number part of the answer is 11.

12. Additional Example 3 Continued Approximate to the nearest hundredth. Step 2: Find the value of the decimal. Find the difference between the given number, 141, and the lower perfect square. 141 – 121 = 20 Find the difference between the greater perfect square and the lower perfect square. 144 – 121 = 23 20 23 Write the difference as a ratio. Divide to find the approximate decimal value. 20  3 ≈ 0.869

13. The approximate value of to the nearest hundredth is 11.87. Additional Example 3 Continued Approximate to the nearest hundredth. Step 3: Find the approximate value. Combine the whole number and decimal. 11 + 0.869 = 11.869 Round to the nearest hundredth. 11.869 ≈ 11.87

14. Check It Out: Example 3 Approximate to the nearest hundredth. Step 1: Find the value of the whole number. Find the perfect squares nearest 240. 225 < 240 < 256 Find the square roots of the perfect squares. The number will be between 15 and 16. The whole number part of the answer is 15.

15. Check It Out: Example 3 Continued Approximate to the nearest hundredth. Step 2: Find the value of the decimal. Find the difference between the given number, 240, and the lower perfect square. 240 – 225 = 15 Find the difference between the greater perfect square and the lower perfect square. 256 – 225 = 31 15 31 Write the difference as a ratio. Divide to find the approximate decimal value. 15  31 ≈ 0.484

16. The approximate value of to the nearest hundredth is 15.48. Check It Out: Example 3 Continued Approximate to the nearest hundredth. Step 3: Find the approximate value. Combine the whole number and decimal. 15 + 0.484 = 15.484 Round to the nearest hundredth. 15.484 ≈ 15.48

17. Additional Example 4: Using a Calculator to Estimate the Value of a Square Root Use a calculator to find 600. Round to the nearest tenth. Using a calculator, 600 ≈ 24.49489742…. Rounded, 600 is 24.5.

18. Using a calculator, 800 ≈ 28.2842712…. Check It Out: Example 4 Use a calculator to find 800. Round to the nearest tenth. Rounded, 800 is 28.3.

19. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

20. Lesson Quiz Each square root is between two integers. Name the two integers. 1. 27 2. – 456 5 and 6 –22 and –21 Use a calculator to find each value. Round to the nearest tenth. 9.4 35.0 3. 89 4. 1223 5. A square field has an area of 2000 square feet. To the nearest foot, how much fencing would be needed to enclose the field? 179 ft

21. Lesson Quiz for Student Response Systems 1. Name the two integers the square root is between. A. 5 and 6 B. 6 and 7 C.8 and 9 D.38 and 40

22. Lesson Quiz for Student Response Systems 2. Use a calculator to round to the nearest tenth. A. 2.3 B. 4.5 C.6.7 D.9.2

23. Lesson Quiz for Student Response Systems 3. Name the two integers the square root is between. A. –2 and –3 B. –10 and –11 C. –13 and –14 D. –15 and –16