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TargetFundamentals™ Essential Math Skills Pre-Algebra Skill 3 The student will simplify computations using the inverse relationships of squaring and finding square roots to solve problems. Skill: INSTRUCTIONAL PREPARATION Duplicate the following (one per student unless otherwise indicated): • To Square or Not to Square reference sheet • Where’s the Root? worksheet Prepare an overhead transparency of the following: • To Square or Not to Square reference sheet • Where’s the Root? worksheet RECALL Before beginning the Review component, facilitate a discussion based on the following questions: ¾How is an equation solved? (The variable is isolated using inverse operations.) ¾What is the inverse operation of addition? (Subtraction) ¾What is the inverse operation of multiplication? (Division) REVIEW 1. To begin, write the fraction 2 on the transparency and ask the following questions: 4 ¾What does it mean to simplify a fraction? (Remove common factors larger than 1 from the numerator and denominator.) ¾Can this fraction be simplified? If so, to what? (Yes, by dividing 2 out of the numerator and denominator, leaving 1 ) 2 ¾Why do we bother to simplify fractions? (To make it easier to solve problems containing fractions.) TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-1 Last printed 8/19/04 Explain that expressions containing square roots can often be simplified, for the same reason—to make it easier to solve problems. Today they will be simplifying expressions containing either the square root or the square of a number, and they will be using the same concept of inverse operations with which they are already familiar. 2. Distribute the To Square or Not to Square reference sheet and display the transparency. Have the students read through the “Words to Know” section located at the bottom of the reference sheet. Point out the difference between the terms, “radical” and “radicand,” emphasizing that the radical is the square root symbol, and the radicand is the number inside the radical. Answer any questions about these terms. Direct attention to the “Inverse Operations” box and explain how squaring and taking the square root are inverse operations, so long as the numbers are nonnegative. Squaring the square root will return the original number, as will taking the square root of the square of the original number. Work through the first example in the section, showing how both cases return the original number (3), because it is positive. Emphasize that care must be taken when there is a negative number in the radicand. Ask the following questions: ¾If you multiply a positive number by itself, what kind of number results? (A positive number) ¾What do you get when you multiply a negative number by itself? (A positive number) ¾Will you always get a positive result when squaring a number? (Yes, unless the number is 0, which squared is still 0) Explain to the students that there are numbers that will give negative numbers when squared, and they may learn about them in later math courses. For now, whenever the radicand is negative, they can conclude that the answer will not be a real number. Work through the second example, asking these questions: ¾According to the order of operations, what do we do first? (Square the -4) ¾After we do that, what is the result? ( 16 ) ¾What’s the answer? (4) (Point out the list of perfect squares on the right of the reference sheet, and tell the students that they can refer to it when working today’s problems.) ¾Why isn’t the answer the same as the original number? (Because there are two numbers whose square is 16, 4 and -4.) Remind them that a radical by itself means to take the positive of the two numbers whose square is the radicand, +4 in this case. To get -4, the expression would be - 16 . Have the students work in pairs to evaluate the four expressions in the next box. When they have finished, have volunteers write their answers on the transparency. The answers are: 7, 5, 1, and “not a real number.” Answer any questions. TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-2 Last printed 8/19/04 3. Direct attention to the “Simplifying Square Roots” section. Explain that the key to this is factoring, like with simplifying fractions. The difference is that here, they’re looking for factors that are perfect squares. If the radicand has a perfect square factor, then they can break up the root into the product of two roots, one with the perfect square radicand. The radicand of the other root should have no perfect square factors larger than 1. Then, the perfect square root can be simplified. Work through the two examples, showing how the original radicand was factored, and how the perfect square root was simplified. Have student pairs work the two sample problems. When they have completed, have volunteers write their answers on the transparency. The answers are: 7 2 and 3 11 . If you feel that the students need more practice, then dictate additional problems for them to solve on the reverse side of their reference sheets. 4. Direct attention to the “Estimating Square Roots” section. Explain that it’s important to be able to estimate an answer to a problem, to verify that you’re on the right track and to see if the final answer makes sense. One way of estimating square roots is to find the two consecutive perfect squares that are on either side of the radicand. The value of the square root is then between the square roots of the two perfect squares. Work through the example and then have student pairs solve the two sample problems as before. The answers are: 41 is between 6 and 7; 83 is between 9 and 10. Answer any questions about the lesson. 5. Distribute the Where’s the Root? worksheet, to be completed independently. Allow adequate time for task completion. Ask volunteers to display their solutions on the transparency, and verify their results using the Teacher’s Answer Key. WRAP-UP • To conclude this lesson, have the students write a response to the following prompts, using complete sentences, in their math journals or on a sheet of notebook paper. Allow adequate time for task completion and then ask volunteers to share their responses with the class. ¾When simplifying a radical, what are you looking for? (Two factors one of which is a perfect square) ¾What is one way to estimate a square root? (Find the two consecutive perfect squares between which it lies; the square root lies between the roots of the two perfect squares.) TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-3 Last printed 8/19/04 To Square or Not to Square Inverse Operations • If x ≥ 0, then • If x ≥ 0, then • If x < 0, then d xi 2 Perfect Squares 12 = 1, 1 = 1 62 = 36, 36 = 6 x2 = x 2 2 = 4, 4 = 2 7 2 = 49, 49 = 7 x2 = -x 32 = 9, 9 = 3 82 = 64, 64 = 8 =3 4 2 = 16, 16 = 4 9 2 = 81, 81 = 9 b g 52 = 25, 25 = 5 102 = 100, 100 = 10 32 = 9 = 3, b-4g 2 =x d 3i 2 = 16 = 4 = - -4 Use the inverse properties to evaluate these expressions: 49 52 Simplifying Square Roots • • • Find a perfect square factor in the radicand Write the root as a product of two roots Simplify the root of the perfect square 50 = 25 2 = 5 2 24 = 4 6 = 2 6 Estimating Square Roots • • If the radicand is not a perfect square, then find the two perfect squares surrounding the radicand. The square root is between the roots of the perfect squares 53 is between 49 and 64 53 is between 7 ( 49 ) and 8 ( 64 ) Simplify: 98 -1 (-1) 2 Estimate: 99 41 83 WORDS TO KNOW Perfect square – The product of an integer and itself 1, 4, 9, 16, 25, 36, … Radical sign ( ) – A symbol that indicates the square root Radicand – The number under the radical sign Square root of a number – The number when multiplied by itself will equal the number in the radical 36 = 6, 64 = 8 TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-4 Last printed 8/19/04 Name ________________________________________________________________________ Where’s the Root? Directions: For Problems 1-8, simplify each expression. If the answer is not an integer, then express your answers in simplified radical form and give the two integers between which the answer lies. 2. 112 4. - 72 28 6. d2 31i -4 8. (-3) 2 1. 81 3. d 7i 5. 7. 2 2 Directions: For Problems 9 and 10, read each problem carefully and solve. 9. What is the area of a square with side lengths measuring 3 units long? 10. Substitute the values a = -3 and b = 4 into this equation and solve for c: c2 = a2 + b2. TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-5 Last printed 8/19/04 TEACHER’S ANSWER KEY Where’s the Root? 1. 2. 3. 4. 9 11 7 -6 2 , between -9 and -8 5. 2 7, between 5 and 6 6. 4 • 31 = 124 7. Not a real number 8. 3 9. 3 square units 10. c2 = (-3)2 + 42 = 9 + 16 = 25 c2 = 25 c = ±5 TargetFundamentals™ 2004 Evans Newton Incorporated MP.3-6 Last printed 8/19/04