Cambium Learning Voyager

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1 Product Sample For questions or more information, contact: Cambium Learning Voyager Dallas Parkway, Ste. 400 Dallas, TX

2 9CHAPTER Objective 5 Solve quadratic equations by completing the square. Objective 5 Pretest Students complete the Objective 5 Pretest on the same day as the Objective 4 Posttest. Using the Results Score the pretest and update the class record card. If the majority of students do not demonstrate mastery of the concepts, use the 3-Day Instructional Plan A for Objective 5. If the majority of students demonstrate mastery of the concepts, use the 3-Day Instructional Plan B for Objective 5. Chapter 9 Objective 5 Pretest Name Date Solve each quadratic equation using square roots. 1. x = 64. (x 3) = 16 x = ± 64 x 3 = 4 or x 3 = 4 x = ±8 x = 7 or x = 1 Solve each quadratic equation by completing the square. 3. x + 6x + 3 = 0 4. x 8x + 8 = 0 (x + 6x + 9) 6 = 0 (x 8x + 16) 8 = 0 (x + 6x + 9) = 6 (x 8x + 16) = 8 (x + 3) = 6 (x 4) = 8 x + 3 = ± 6 x 4 = ± 8 x = 3 ± 6 x = 4 ± 8 x = 4 ± 13 Chapter 9 Objective 5 Inside Algebra Copyright 011 Cambium Learning Sopris West. All rights reserved. 844 Chapter 9 Objective 5

3 Objective 5 Goals and Activities Objective 5 Goals The following activities, when used with the instructional plans on pages 846 and 847, enable students to: Solve the equation (x 7) = 81 to get x = or 16 Solve the equation (x + 1) 5 = 0 to get x = 1 ± 5 Objective 5 Activities Concept Development Activity CD 1 Using Algebra Tiles to Complete a Square, page 848 Practice Activity PA 1 Solving Equations by Groups, page 850 Progress-Monitoring Activities PM 1 Apply Skills 1, page 851 PM Apply Skills, page 85 Problem-Solving Activity PS 1 Completing the Square With Negative Middle Terms, page 853 Ongoing Assessment Posttest Objective 5, page 855 Review Chapter 9 Review, page 856 CD = Concept Development PM = Progress Monitoring PS = Problem Solving PA = Practice Activity = Includes Problem Solving Chapter 9 Objective 5 845

9 Instructional CHAPTER Objective 5 Plans 3-Day Instructional Plan A Use the 3-Day Instructional Plan A when pretest results indicate that students would benefit from a slower pace.

ACCELERATE DIFFERENTIATE Day 1 CD 1 Using Algebra Tiles to Complete a Square PA 1 Solving Equations by Groups CD 1 Using Algebra Tiles to Complete a Square PM 1 Apply Skills 1 Day PS 1 Completing the

4 9 Instructional CHAPTER Objective 5 Plans 3-Day Instructional Plan A Use the 3-Day Instructional Plan A when pretest results indicate that students would benefit from a slower pace. This plan is used when the majority of students need more time or did not demonstrate mastery on the pretest. Students who do not demonstrate mastery should begin on the differentiated path. ACCELERATE DIFFERENTIATE Day 1 CD 1 Using Algebra Tiles to Complete a Square PA 1 Solving Equations by Groups CD 1 Using Algebra Tiles to Complete a Square PM 1 Apply Skills 1 Day PS 1 Completing the Square With Negative Middle Terms PA 1 Solving Equations by Groups PM Apply Skills Day 3 Posttest Objective 5 Review Chapter 9 Review End of Chapter Chapter Test Differentiation Retest CD = Concept Development PM = Progress Monitoring PS = Problem Solving PA = Practice Activity = Includes Problem Solving 846 Chapter 9 Objective 5

3-Day Instructional Plan B Use the 3-Day Instructional Plan B when pretest results indicate that students can move through the activities at a faster pace.

CD 1 Using Algebra Tiles to Complete a Square Day 1 PA 1 Solving Equations by Groups PM 1 Apply Skills 1 Day PS 1 Completing the Square With Negative Middle Terms PM

5 3-Day Instructional Plan B Use the 3-Day Instructional Plan B when pretest results indicate that students can move through the activities at a faster pace. This plan is ideal when the majority of students demonstrate mastery on the pretest. CD 1 Using Algebra Tiles to Complete a Square Day 1 PA 1 Solving Equations by Groups PM 1 Apply Skills 1 Day PS 1 Completing the Square With Negative Middle Terms PM Apply Skills Day 3 Posttest Objective 5 Review Chapter 9 Review End of Chapter Chapter Test Differentiation Retest CD = Concept Development PM = Progress Monitoring PS = Problem Solving PA = Practice Activity = Includes Problem Solving Chapter 9 Objective 5 847

Objective 5 Concept Development Activity CD 1 Using Algebra Tiles to Complete a Square Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B.

6 Objective 5 Concept Development Activity CD 1 Using Algebra Tiles to Complete a Square Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. In this activity, students factor quadratic trinomials by using algebra tiles to complete the square. MATERIALS Algebra tiles, one set per pair of students Overhead algebra tiles DIRECTIONS 1. Review the following terms with students: difference of squares A binomial of the form a b = (a + b)(a b) factor A monomial that evenly divides a value perfect square The product of a monomial with itself, for example, x, 16a, or 49 quadratic formula x = b ± b 4ac where a ax + bx + c = 0 quadratic trinomial A polynomial of the form ax + bx + c. Write x + 4x + 1 = 0 on the board. 3. Group students in pairs, and tell them to attempt to solve this quadratic equation using any method they have learned, for example, area rugs, factoring, perfect squares, or difference of squares. 4. When most pairs of students realize they are not having any success, give a set of algebra tiles to each pair of students. 5. Have students attempt to build a rectangle that represents x + 4x + 1 = 0 and is as close to a square as possible. Tell them there may appear to be a piece missing from a perfect square, but that is okay. 6. After a reasonable amount of time, if there are any students who are confident in their answer, allow them to present it to the class using a set of overhead algebra tiles. 7. Discuss the following term with students: completing the square Adding to or subtracting from a quadratic equation to make it into a perfect square trinomial; a method used to find the solutions of a quadratic equation 8. Tell students you will present a solution to the class and ask for their input. Show students the tiles below, and explain that they will represent the units in the problem. 1 x x x x x 1 unit 1 x unit 4 x units 9. Show that x + 4x + 1 = 0 can be represented by building the diagram shown below. Ask students to identify what is missing from the diagram that would complete the square. x x 1 x x x Draw the answer by showing that with three more single unit squares, the diagram would be a complete square. 10. Have students discuss what the dimensions of the new complete square would be. If necessary, point out that the dimensions would be (x + ) by (x + ). 11. Write (x + )(x + ) on the board. Ask students if they can think of another way to write this expression in a more concise manner. (x + ) 1. Remind students that the original diagram was not a complete square so they must subtract the three missing pieces from the new expression. Thus, the expression changes to (x + ) 3. = Includes Problem Solving 848 Chapter 9 Objective 5

7 13. Have the pairs of students attempt to solve the equation (x + ) 3. You may need to remind them that to remove a square they can use a square root. If students are having difficulty, demonstrate an example on the board. (x + ) 3 = 0 (x + ) = 3 (x + ) = ± 3 (x + ) = ± 3 x = ± Point out that this expression represents two answers. Review how to simplify an expression containing ±. Discuss how the answers are irrational and would have to be rounded to be written without the radical sign. 15. Write more problems on the board, and have the pairs of students attempt to solve them using algebra tiles and the concept of completing the square. Sample problems: x + 6x + 3 = 0 x = 3 ± 6 x + 10x + 5 = 0 x = 5 ± 0 or x = 5 ± 5 x + 8x + = 0 x = 4 ± 14 NEXT STEPS Differentiate 3-Day Instructional Plan A: PA 1, page 850 Students who are on the accelerated path, for additional practice PM 1, page 851 Students who are on the differentiated path, to assess progress 3-Day Instructional Plan B: PA 1, page 850 All students, for additional practice Chapter 9 Objective 5 849

8 Objective 5 Practice Activity PA 1 Solving Equations by Groups Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. In this activity, students solve equations by completing the square. materials Blackline Master 105 Name Date SOLVING EQUATIONS BY GROUPS a. (x + 5) 9 = 0 x =, 8 b. (x 7) 4 = 0 x = 5, 9 c. x + 6x + = 0 x = 3 ± 7 d. x + 14x 5 = 0 x = 7 ± 54 or 7 ± 3 6 e. x + 16x + 13 = 0 x = 8 ± 51 f. x + 4x 9 = 0 x = ± 13 g. x + 1x + 7 = 0 x = 6 ± Directions 1. Review the following term with students: completing the square Adding to or subtracting from a quadratic equation to make it into a perfect square trinomial; a method used to find the solutions of a quadratic equation. Divide the class into groups of four. 3. Make an overhead transparency of Blackline Master 105, Solving Equations by Groups. Alternatively, write the equations from the Blackline Master on the board or overhead. 4. Assign one equation to each group. Have each group solve the equation by completing the square. 5. When all groups are finished, have them trade equations with another group and verify the work is correct. 6. Ask for a volunteer from each group to put the group s equation and the appropriate work on the board. Have the class discuss each equation and solution. h. x + 8x + 11 = 0 x = 4 ± 5 i. x + 0x + 31 = 0 x = 10 ± 69 Inside Algebra Blackline Master 105 NEXT STEPS Differentiate 3-Day Instructional Plan A: PM 1, page 851 Students who are on the accelerated path, to assess progress PM, page 85 Students who are on the differentiated path, to assess progress 3-Day Instructional Plan B: PM 1, page 851 All students, to assess progress 850 Chapter 9 Objective 5

Objective 5 Progress-Monitoring Activities PM 1 Apply Skills 1 Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. MATERIALS Interactive Text, page 360 DIRECTIONS 1.

9 Objective 5 Progress-Monitoring Activities PM 1 Apply Skills 1 Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. MATERIALS Interactive Text, page 360 DIRECTIONS 1. Have students turn to Interactive Text, page 360, Apply Skills 1.. Remind students of the key term: completing the square. 3. Monitor student work, and provide feedback as necessary. Watch for: Do students understand that not all expressions can be factored? progress monitoring Name Date A pply S kills 1 Solve the equations using the method of completing the square. Work in pairs or groups. 1. x + 1x + 5 = 0. a + 6a + 1 = 0 x + 1x = 0 a + 6a = 0 (x + 6) 31 = 0 (a + 3) 8 = 0 (x + 6) = 31 (a + 3) = 8 x + 6 = ± 31 a + 3 = ± 4 x = 6 ± 31 a + 3 = ± a = 3 ± 3. x + 8x + 3 = 0 4. x + 14x + 5 = 0 x + 8x = 0 x + 14x = 0 (x + 4) 13 = 0 (x + 7) 44 = 0 (x + 4) = 13 (x + 7) = 44 x + 4 = ± 13 x + 7 = ± 4 11 x = 4 ± 13 x + 7 = ± 11 x = 7 ± x 16x + 8 = 0 6. x + x 4 = 0 x 16x = 0 x + x = 0 (x 8) 56 = 0 (x + 1) 5 = 0 (x 8) = 56 (x + 1) = 5 x 8 = ± 4 14 x + 1 = ± 5 x 8 = ± 14 x = 1 ± 5 x = 8 ± Chapter 9 Objective 5 PM 1 Inside Algebra Copyright 011 Cambium Learning Sopris West. All rights reserved. Do students remember to halve the middle number to find the square root? NEXT STEPS Differentiate 3-Day Instructional Plan A: PS 1, page 853 Students who are on the accelerated path, to develop problem-solving skills PA 1, page 850 Students who are on the differentiated path, for additional practice 3-Day Instructional Plan B: PS 1, page 853 All students, to develop problemsolving skills Chapter 9 Objective 5 851

10 Objective 5 Progress-Monitoring Activities PM Apply Skills Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. MATERIALS Interactive Text, page 361 DIRECTIONS 1. Have students turn to Interactive Text, page 361, Apply Skills.. Remind students of the key term: completing the square. 3. Monitor student work, and provide feedback as necessary. Watch for: Do students correctly handle negative numbers? Do students check their answer by substituting into the original equation? Copyright 011 Cambium Learning Sopris West. All rights reserved. Name Date A pply S kills Solve each of the equations by using the method of completing the square. Work in pairs or groups. 1. c + 4c + = 0. x 1x + 5 = 0 c + 4c + 4 = 0 x 1x = 0 (c + ) = 0 (x 6) 31 = 0 (c + ) = (x 6) = 31 c + = ± x 6 = ± 31 c = ± x = 6 ± x + 10x + 1 = 0 4. b b 9 = 0 x + 10x = 0 b b = 0 (x + 5) 13 = 0 (b 1) 10 = 0 (x + 5) = 13 (b 1) = 10 x + 5 = ± 13 b 1 = ± 10 x = 5 ± 13 b = 1 ± x 6x 1 = 0 6. x + 5x + = 0 x 6x = 0 x + 5 x + 5 x = 0 (x 3) 1 = 0 ½x + 5 (x 3) = 1 ¼ 17 4 = 0 x 3 = ± 1 ½ x + 5 ¼ 17 = 4 x = 3 ± 1 x + 5 = ± 17 4 x + 5 = ± 17 5 ± 17 x = Inside Algebra Chapter 9 Objective 5 PM 361 progress monitoring NEXT STEPS Differentiate 3-Day Instructional Plans A and B: Objective 5 Posttest, page 855 All students 85 Chapter 9 Objective 5

Objective 5 Problem-Solving Activity PS 1 Completing the Square With Negative Middle Terms Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B.

Materials Blackline Master 107 Name Date C O M P L E T I N G T H E S Q U A R E W I T H NEGATIVE MIDDLE TERMS a. x 6x + 4 = 0 x = 3 ± 5 b. x 14x 1 = 0 x = 7 ± 50 or 7 ± 5 c. x 16x + 3 = 0 x = 8 ± 61 d.

Review the following term with students: completing the square Adding to or subtracting from a quadratic equation to make it into a perfect square trinomial; a method used to find the solutions of a

11 Objective 5 Problem-Solving Activity PS 1 Completing the Square With Negative Middle Terms Use with 3-Day Instructional Plan A or 3-Day Instructional Plan B. In this activity, students solve equations with negative middle terms using the method of completing the square. Materials Blackline Master 107 Name Date C O M P L E T I N G T H E S Q U A R E W I T H NEGATIVE MIDDLE TERMS a. x 6x + 4 = 0 x = 3 ± 5 b. x 14x 1 = 0 x = 7 ± 50 or 7 ± 5 c. x 16x + 3 = 0 x = 8 ± 61 d. x 4x 7 = 0 x = ± 11 e. x 1x + 11 = 0 x = 1, 11 f. x 8x + 5 = 0 x = 4 ± 11 g. x 0x + 1 = 0 x = 10 ± DIRECTIONS 1. Review the following term with students: completing the square Adding to or subtracting from a quadratic equation to make it into a perfect square trinomial; a method used to find the solutions of a quadratic equation. Divide the class into groups of four. 3. Make an overhead transparency of Blackline Master 107, Completing the Square With Negative Middle Terms. Alternatively, write the equations from the Blackline Master on the board or overhead. 4. Assign one equation to each group. Have each group solve the equation by completing the square. 5. When all groups are finished, have them trade equations with another group and verify the work is correct. 6. Ask for a volunteer from each group to put the group s equation and the appropriate work on the board. Have the class discuss each equation and solution. Inside Algebra Blackline Master 107 NEXT STEPS Differentiate 3-Day Instructional Plans A and B: PM, page 85 All students, to assess progress = Includes Problem Solving Chapter 9 Objective 5 853

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9CHAPTER Objective 5 Ongoing Assessment Objective 5 Posttest Discuss with students the key concepts in Objective 5. Following the discussion, administer the Objective 5 Posttest to all students.

13 9CHAPTER Objective 5 Ongoing Assessment Objective 5 Posttest Discuss with students the key concepts in Objective 5. Following the discussion, administer the Objective 5 Posttest to all students. Using the Results Score the posttest and update the class record card. Provide reinforcement for students who do not demonstrate mastery of the concepts through individual or small-group reteaching of key concepts. Name Date Solve each quadratic equation using square roots. 1. x = 36. (x + 4) = 11 x = ± 36 x + 4 = 11 or x + 4 = 11 x = ±6 x = 7 or x = 15 Solve each quadratic equation by completing the square. 3. x 6x + = 0 4. x + 10x + 1 = 0 (x 6x + 9) 7 = 0 (x + 10x + 5) 13 = 0 (x 6x + 9) = 7 (x + 10x + 5) = 13 (x 3) = 7 (x + 5) = 13 x 3 = ± 7 x + 5 = ± 13 x = 3 ± 7 x = 5 ± 13 Chapter 9 Objective 5 Posttest Copyright 011 Cambium Learning Sopris West. All rights reserved. Inside Algebra Chapter 9 Objective Chapter 9 Objective 5 855

TEACHER GUIDE VOLUME 1. Larry Bradsby. Chapter 3, Objective 2

TEACHER GUIDE VOLUME 1. Larry Bradsby. Chapter 3, Objective 2 TEACHER GUIDE VOLUME Larry Bradsby Chapter 3, Objective 2 Solving Linear Equations In this chapter, students begin to solve basic linear equations using addition, subtraction, multiplication, division,