42 Supporting Mathematical Discourse

Supporting Mathematical Discourse with Pearson High School Mathematics Common CorePearson High School Mathematics Common Core was designed to create learning environments where students engage in meaningful mathematical discourse. Through this discourse, students become mathematical thinkers and develop proficiency with the Standards for Mathematical Practice. In this walk-through, you will find suggestions for supporting and enhancing the mathematical discourse in your classroom.

QR codes link to professional development videos for the program. Each video opens with an explanation of the pedagogy of one of the five steps of the lesson by lead author Dr. Randall Charles, followed by a classroom look at implementing Pearson High School Mathematics Common Core. To view the videos, scan each QR code using a QR code reader app on a smartphone or tablet.

1 InteractIve LearnIngThe purpose of the solve It! is engage students in a discussion around the problem presented. Students are encouraged to work in pairs or small groups to not just solve the problem presented, but defend their solution.

Use the probing questions in the Teacher’s Editions as needed to help move students along with their solutions.

During the sharing phase of the lesson, try some of these techniques to enhance the quality of discourse:

• Revoice students’ answers. By restating a student’s explanations, you can clarify the explanation for other students in the class, and, even more important, can help the student confirm and solidify his or her thinking.

• Prompt for participation. Ask other students what they think about a response or an explanation to draw more students into the discussion.

546 Chapter 9 Quadratic Functions and Equations

As a cat walks along the railing of a balcony, it knocks a flowerpot off the railing. The function h ( t ) 5 216t 2 1 c gives the height h of the flowerpot after t seconds when it falls from a height of c feet. How long will it take the flowerpot to reach the ground? Explain your reasoning.

64 ft

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Supporting Mathematical Discourse 43

2 Guided instructionDuring the Guided Instruction phase of the lesson, the classroom discourse may shift to more of a sense-making focus. The guiding questions in the Teacher’s Edition can provide scaffolding to support student understanding of new concepts being presented in the problems.

The visually rich offer opportunities to confirm or clarify students’ understandings of the problem.

Spend some time having students describe the visual representation. Ask questions, such as “Will this happen in all cases?” “Can you think of an example where it wouldn’t be the case?” or “Can you predict what will happen if...?” that can lead students to see the structure of mathematics.

The Got It? also offers opportunities for sense-making discourse. In addition to the questions in the Teacher’s Edition, ask questions that can help students determine the mathematical correctness of solutions. Ask, for example, “Is it true for all cases?” or “How can you prove that?”

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Problem 2

Got It?

584 Chapter 9 Quadratic Functions and Equations

When the radicand in the quadratic formula is not a perfect square, you can use a calculator to approximate the solutions of an equation.

Finding Approximate Solutions

Sports In the shot put, an athlete throws a heavy metal ball through the air. The arc of the ball can be modeled by the equation y 5 20.04x2 1 0.84x 1 2, where x is the horizontal distance, in meters, from the athlete and y is the height, in meters, of the ball. How far from the athlete will the ball land?

0 5 20.04x 2 1 0.84x 1 2 Substitute 0 for y in the given equation.

x 5 2b "b 2 2 4ac2a Use the quadratic formula.

x 520.84 "0.842 2 4(20.04)(2)

2(20.04) Substitute 20.04 for a, 0.84 for b, and 2 for c.

x 5 20.84 "1.025620.08 Simplify.

x 5 20.84 1 "1.025620.08 or x 5 20.84 2 "1.0256

20.08 Write as two equations.

x < 22.16 or x < 23.16 Simplify.

Only the positive answer makes sense in this situation. The ball will land about 23.16 m from the athlete.

2. A batter strikes a baseball. The equation y 5 20.005x 2 1 0.7x 1 3.5 models its path, where x is the horizontal distance, in feet, the ball travels and y is the height, in feet, of the ball. How far from the batter will the ball land? Round to the nearest tenth of a foot.

There are many methods for solving a quadratic equation.

Method When to UseGraphing Use if you have a graphing calculator handy.Square roots Use if the equation has no x-term.Factoring Use if you can factor the equation easily.Completing the square Use if the coefficient of x 2 is 1, but you cannot easily factor the

equation.Quadratic formula Use if the equation cannot be factored easily or at all.

Why do you substitute 0 for y?When the ball hits the ground, its height will be 0.

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44 Supporting Mathematical Discourse

3 Lesson CheCkWith the Lesson Check, the discourse is primarily written, rather than oral. Students are asked to reason about concepts, or justify a solution.

Lesson Check

Problem 4

Got It?

586 Chapter 9 Quadratic Functions and Equations

Using the Discriminant

How many real-number solutions does 2x 2 2 3x 5 25 have?

2x 2 2 3x 1 5 5 0

b 2 2 4ac 5 (23)2 2 4(2)(5) 5 231

Because the discriminant is negative, the equation has no real-number solutions.

Draw a conclusion.

Evaluate the discriminant by substituting 2 for a, 23 for b, and 5 for c.

Write the equation in standard form.

4. a. How many real-number solutions does 6x 2 2 5x 5 7 have? b. Reasoning If a is positive and c is negative, how many real-number

solutions will the equation ax 2 1 bx 1 c 5 0 have? Explain.

Can you solve this problem another way?Yes. You could actually solve the equation to find any solutions. However, you only need to know the number of solutions, so use the discriminant.

Do you know HOW?Use the quadratic formula to solve each equation. If necessary, round answers to the nearest hundredth.

1. 23x 2 2 11x 1 4 5 0

2. 7x 2 2 2x 5 8

3. How many real-number solutions does the equation 22x 2 1 8x 2 5 5 0 have?

Do you UNDERSTAND? 4. Vocabulary Explain how the discriminant of the

equation ax 2 1 bx 1 c 5 0 is related to the number of x-intercepts of the graph of y 5 ax 2 1 bx 1 c.

5. Reasoning What method would you use to solve the equation x 2 1 9x 1 c 5 0 if c 5 14? If c 5 7? Explain.

6. Writing Explain how completing the square is used to derive the quadratic formula.

Practice and Problem-Solving Exercises

Use the quadratic formula to solve each equation.

7. 2x 2 1 5x 1 3 5 0 8. 5x 2 1 16x 2 84 5 0 9. 4x 2 1 7x 2 15 5 0

10. 3x 2 2 41x 5 2110 11. 18x 2 2 45x 2 50 5 0 12. 3x 2 1 44x 5 296

13. 3x 2 1 19x 5 154 14. 2x 2 2 x 2 120 5 0 15. 5x 2 2 47x 5 156

PracticeA See Problem 1.

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

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After students have completed the Lesson Check, think about having students share their answers with classmates.

As students are sharing, ask questions to help students critique the reasoning of others. Ask, for example, “Do you agree with that solution?” or “What do you think about that solution?”

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Supporting Mathematical Discourse 45

4 PracticeThe Practice phase of the lesson is another opportunity for students to build their discourse in writing. Many of the exercises require students to explain their thinking, reason about solutions, or find an error in a solution.

5 assess and remediateAs you assign Differentiated Activities to your students based on their Lesson Quiz scores, think about having students complete activities that maximize interaction with mathematical discourse.

580 Chapter 9 Quadratic Functions and Equations

32. Think About a Plan A park is installing a rectangular reflecting pool surrounded by a concrete walkway of uniform width. The reflecting pool will measure 42 ft by 26 ft. There is enough concrete to cover 460 ft2 for the walkway. What is the maximum width x of the walkway?

• How can drawing a diagram help you solve this problem? • How can you write an expression in terms of x for the area of the walkway?

33. Landscaping A school is fencing in a rectangular area for a playground. It plans to enclose the playground using fencing on three sides, as shown at the right. The school has budgeted enough money for 75 ft of fencing material and would like to make a playground with an area of 600 ft2.

a. Let w represent the width of the playground. Write an expression in terms of w for the length of the playground.

b. Write and solve an equation to find the width w. Round to the nearest tenth of a foot.

c. What should the length of the playground be?

Solve each equation. If necessary, round to the nearest hundredth. If there is no real-number solution, write no solution.

34. q 2 1 3q 1 1 5 0 35. s 2 1 5s 5 211 36. w 2 1 7w 2 40 5 0

37. z 2 2 8z 5 213 38. 4p 2 2 40p 1 56 5 0 39. m 2 1 4m 1 13 5 28

40. 2p 2 2 15p 1 8 5 43 41. 3r 2 2 27r 5 3 42. s 2 1 9s 1 20 5 0

43. Error Analysis A classmate was completing the square to solve 4x 2 1 10x 5 8. For her first step she wrote 4x 2 1 10x 1 25 5 8 1 25. What was her error?

44. Reasoning Explain why completing the square is a better strategy for solving x2 2 7x 2 9 5 0 than graphing or factoring.

45. Open-Ended Write a quadratic equation and solve it by completing the square. Show your work.

Use each graph to estimate the values of x for which f (x) 5 5. Then write and solve an equation to find the values of x such that f (x) 5 5. Round to the nearest hundredth.

46. f (x) 5 x 2 2 2x 2 1 47. f (x) 5 212 x 2 1 2x 1 6

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Points Differentiated Remediation

InterventionOn-levelExtension

5 Assess & RemediateAssign the Lesson Quiz. Appropriate intervention, practice, or enrichment is automatically generated based on student performance.

581A Lesson Resources

Intervention•Reteaching(2pages)Providesreteachingandpracticeexercisesforthekeylessonconcepts.Usewithstrugglingstudentsorabsentstudents.

• English Language Learner SupportHelpsstudentsdevelopandreinforcemathematicalvocabularyandkeyconcepts.

Lesson Resources Differentiated Remediation

All-in-One Resources/OnlineEnglish Language Learner Support

All-in-One Resources/OnlineReteaching

0–23–4

5

9-5

Additional Instructional SupportAlgebra 1 CompanionStudentscanusetheAlgebra 1 Companionworktext(4pages)asyouteachthelesson.UsetheCompaniontosupport

• NewVocabulary

• KeyConcepts

• GotItforeachProblem

• LessonCheck

ELL SupportUse Manipulatives Writex2 1 4x 5 0ontheboard.Useanoverheadprojectandoverheadalgebratilestomodelcompletingthesquare.Thinkaloudasyoumanipulatethetiles.Writethesolution,x2 1 4x 1 4 5 (x 1 2)2ontheboard.Pointtothetilesandbeexplicitabouthowyouarrivedatthesolution.Thendoanotherexamplewithstudentsfollowingalongattheirseatswiththeirowntiles.Finally,writex2 1 2x 5 0ontheboard.Monitorstudentsastheyusealgebratilestocompletethesquaretosolvetheequation.

5 Assess & RemediateLesson Quiz 1. Whatisthevalueofcsuchthat

x2 1 10x 1 cisaperfectsquaretrinomial?

2. Whatarethesolutionsoftheequationx2 2 6x 5 27?

3. Findthevertexofy 5 x2 1 14x 2 40bycompletingthesquare.

4. Thecombinedareaofthe3windowpanesandframeshownbelowis924in.2.Theframeisofuniformwidth.Whatisthesidelength,x,ofeachsquarewindowframe?

5. Do you UnDERStAnD?Whatwillhappenifyoutrytousecompletingthesquaretosolveanequationoftheformx 2 1 bx 1 c 5 0inwhichc . Qb2R

2?

Answers to lesson quiz

1. 25 2. 23,9 3. (27,289) 4. 10in. 5. Therewillbenorealsolutions;after

youhavecompletedthesquare,therightsideoftheequationwillbenegative.

PrescriPtion for remediAtionUsethestudentworkontheLessonQuiztoprescribeadifferentiatedreviewassignment.

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Name Class Date

Prentice Hall Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

49

You have learned to square binomials. Notice how the coeffi cient of the a term is related to the constant value in every perfect-square trinomial.

(a 1 1)2 5 (a 1 1)(a 1 1) 5 a2 1 2a 1 1 S Q22R

25 1

(a 2 1)2 5 (a 2 1)(a 2 1) 5 a2 2 2a 1 1 S Q222 R

25 1

(a 2 2)2 5 (a 2 2)(a 2 2) 5 a2 2 4a 1 4 S Q242 R

25 4

(a 1 3)2 5 (a 1 3)(a 1 3) 5 a2 1 6a 1 9 S Q62R

25 9

In each case, half the coeffi cient of the a term squared equals the constant term. You can use this pattern to fi nd the value that makes a trinomial a perfect square.

Problem

What is the value of c such that x2 2 14x 1 c is a perfect-square trinomial?

Th e coeffi cient of the x term is 214. Using the pattern, c 5 Q2142 R

2 or 49.

So, x2 2 14x 1 49 is a perfect-square trinomial.

Exercises

Find the value of c such that each expression is a perfect-square trinomial.

1. a2 1 8a 1 c 2. x2 2 16x 1 c 3. m2 1 20m 1 c

4. p2 2 14p 1 c 5. y2 2 10y 1 c 6. b2 1 18b 1 c

7. d2 1 12d 1 c 8. n2 2 n 1 c 9. w2 1 3w 1 c

9-5 ReteachingCompleting the Square

16 64 100

49 25 81

36 14

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Prentice Hall Algebra 1 • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

41

Name Class Date

9-5 ELL SupportCompleting the Square

Th ere are two sets of note cards below that show Kris how to fi nd the solutions of the equation g2 2 4g 5 45. Th e set on the left explains the thinking. Th e set on the right shows the steps. Write the thinking and the steps in the correct order.

Think Cards Write Cards

Think Write

Add 2 to each side. (g 2 2)2 5 49

g 5 9 or g 5 25

g 2 2 5 7 or g 2 2 5 27

g2 2 4g 1 4 5 45 1 4

g 2 2 5 4!49

(g 2 2)2 5 45 1 4

Write as two equations.

Simplify the right side.

Add Qb2R25 4 to each side.

Find square roots of each side.

Write g2 2 4g 1 4 as a square.

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6 g 5 9 or g 5 25

g 2 2 5 7 or g 2 2 5 27

g 2 2 5 w!49

(g 2 2)2 5 49

(g 2 2)2 5 45 1 4

g2 2 4g 1 4 5 45 1 4First, add Qb2R25 4 to each side.

Second, write g2 2 4g 1 4 as a square.

Third, simplify the right side.

Next, fi nd square roots of each side.

Then, write as two equations.

Finally, add 2 to each side.

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Chapter 9 270

9-5 Completing the Square

Underline the correct word to complete each sentence about square roots.

1. Every positive number has one / two real square roots.

2. The square root symbol is also known as a radical / radicand .

3. The square roots of 121 are and .

Vocabulary Builder

perfect square (noun) PUR fi kt skwehr

Definition: A perfect square is a number that can be written as the square of an integer.

Example: 16 is a perfect square because 16 5 42. The expression x2 1 10x 1 25 is

a perfect square trinomial because x2 1 10x 1 25 5 (x 1 5)2.

Nonexample: 27 is not a perfect square because there is no integer that you can square to get 27.

Use Your Vocabulary

Write each number as the square of another number. Then determine whether the original number is a perfect square.

Number x 2 Perfect Square?

4. 144 2 Yes / No

5. 49 2 Yes / No

6. 8 aÄ b

2 Yes / No

7. 6 aÄ b

2 Yes / No

8. 4 2

Yes / No

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When selecting exercises to go over in class, focus on those that can promote classroom discourse. As students share their solutions, ask students questions, such as, “How did you reach that solution?” Does that always work?” or “Can you think of a case when it doesn’t work?”

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