Estimating Square Roots Lesson 2.2 Core Focus on Geometry Estimating Square Roots

Warm-Up Find each value. 1. 2. 3. 6 11 9

Estimating Square Roots Lesson 2.2 Estimating Square Roots Estimate the values of square roots.

Vocabulary Rational Number A number that can be written as a fraction with integers in the numerator and denominator. Examples: Irrational Numbers Numbers that cannot be expressed as a fraction with integers in the numerator and denominator. Good to Know! Irrational numbers can also be recognized in decimal form when the decimal does not repeat or terminate.

Example 1 Determine the two positive integers that each root lies between. a. b. 35 is between the perfect 83 is between the perfect squares 25 and 36. squares 81 and 100.

Good to Know! When estimating square roots, you can provide an even more accurate estimate than the closest two integers by determining which integer the perfect square is closer to. Estimate . The number 12 is approximately halfway between the perfect squares 9 and 16. This means will be about halfway between 3 and 4. A good estimate would be .

Estimating Square Roots Determine which two perfect squares the number under the radical sign falls between. Determine the integers that represent the square root of those perfect squares. Write a statement using integer values: “ falls between ___ and ___”. Determine if the value of the square root is close to the first or second integer. Estimate its value. Good to Know! A calculator is a helpful tool when working with square roots of non-perfect squares. Estimating gives you a good idea of the approximate value, but a calculator will give you a better approximation.

Explore! Calculate It! Step 1 Locate the button on a calculator. Use your calculator to find the value of 29² using the button. x² x² Step 2 What is another way to find 29² using a calculator? Check your answer to Step 1 using this method. Step 3 Locate the button on a calculator. Describe where it is located in relation to the button on the calculator. x² Step 4 Find the value of . Some calculators require you to press the button before entering the number. Other calculators require you to enter the number first and then press the button. Experiment with a calculator and find which key strokes are necessary. Step 5 How does your answer for compare to your work done in Steps 1 and 2? Is 841 a perfect square? Step 6 Use a calculator to find the value of . Write the entire number that shows on the calculator screen.

Explore! Calculate It! Step 7 is an irrational number. Irrational numbers are decimals that do not repeat or terminate. Why does the number on the calculator screen end? Step 8 Instructions will often state, “Round to the nearest _________.” The solution you get after rounding is an approximation. Round to the nearest tenth. Step 9 For each of the following, estimate the value of the square root to the nearest tenth. Then use a calculator to find the value of each square root to the nearest tenth. a. b. c.

Example 2 Use a number line to approximate the value of to the nearest tenth. Use a calculator to find its value to the nearest tenth. 24 falls between perfect squares 16 and 25. is between 4 and 5. is very close to 5, so a good estimation would be  4.9. Calculator method.  4.898979486 Round to the nearest tenth.  4.898979486  4.9

Communication Prompt Why is it important to be able to estimate the value of a square root?

Exit Problems Determine the two positive integers that each square root falls between. 1. 2. 3. Solve for x. Round to the nearest tenth. x2 – 18 = 24 3 and 4 5 and 6 x = ±6.5