Solving Absolute Value Equations Graphically Class

Home , Absolute value (algebra)

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” LESSON esn 2 .2 Lesson 77 preview LessonPerformance the Task. V-shaped path and an absolute value equation. Then and why situation this represented can be by a View Engage the online. photo section the Discuss asan alsoknown disjunction, “or” statement. expression, thenwrite two related equations witha Possible answer: Isolate theabsolute value Essential or nosolution. equations make senseandcontain more thanonesolution, why solutionstoExplain toofabsolute avariety apartner value Objective Language Practices Mathematical problems. Also A-REI.B.3, A-REI.D.11 Create equations andinequalitiesinonevariable andusethemto solve The student isexpectedto: Common Core Math Standards Equations Solving Absolute Value an absolute value equation? PERFORMANCE TASK PREVIEW: LESSON ENGAGE COMMON COMMON CORE CORE MP.6 Precision A-CED.A.1 2 Q .2 uestion:

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2. 1.

B A “or.” To why see there two can solutions, be you can with adisjunction, amathematical statement created by aconnecting two other statements with word the Absolute value equations differfrom linear equations inthat may they have two solutions. This is indicated Essential Question: 2.2 Name Module 2 Module Reflect

If so, what would each like look graphically? Is it possible for an absolute value equation to have no solutions? one solution? Why not three or four? Why might most you absolute expect value equations to have two solutions?

with thehorizontal linepassingthrough thevertex willhave 1solution. exactly will nothave points andtheequation ofintersection willhave nosolutions. Agraph oraboveabsolute value function, adownward- Yes; yes; Agraph withthehorizontal lineentirely below anupward- horizontal lineat mosttwo times. solutions. at Looking thisgraphically, anabsolute value graph a intersect can absolute value beeitherpositive can ornegative. So, theabsoluteIf value expression isnotequalto zero, theexpression insidean Explore Solve the equation the Solve Write solution the to equation this as adisjunction: Plot ƒ(x) function the x and mark points the where graphs the g(x) function the The points are (2, = or x or 2 Equations Absolute Value Solving How canyou solve anabsolute value equation? Solving AbsoluteSolving Value Equations Graphically = = 2as ahorizontal line on same the grid, 2

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78 = a, ⎟ x ⎜ If two values If values two on a = a on ⎟ x ⎜ 1 Plot each side as if it were a were each side as if it Plot So you can remove the absolute the absolute can remove So you Sample answer: The solutions are solutions are The Sample answer: How do you solve an absolute value equation equation value absolute an solve do you How graphically? Why is it important to isolate the absolute the absolute isolate to important it is Why absolute an when solving expression value How do you interpret the solutions to an an to the solutions interpret do you How like equation value absolute Why do you write the solutions to the absolute the absolute to the solutions write do you Why a disjunction? as equation value Equations Value Solving Absolute EXPLORE EXPLORE EXPLAIN EXPLAIN INTEGRATE TECHNOLOGY INTEGRATE AVOID COMMON ERRORS COMMON AVOID QUESTIONING STRATEGIES QUESTIONING QUESTIONING STRATEGIES QUESTIONING of the variable both satisfy an equation, then one or then an equation, both satisfy of the variable the other can be correct. equation? value Solving Absolute Value Equations Equations Value Solving Absolute Graphically the graphing completing of the option have Students the in online. book either activity or Solving Absolute Value Equations Equations Value Solving Absolute Algebraically value the absolute isolate not may students Some step a first as the equation side of one on expression of the importance Stress the equation. when solving the in form is the equation so this that step line? number separate function of x, and find the x-coordinates separate of the intersection points. value bars and rewrite the expression as a the expression bars and rewrite value disjunction. the same distance from 0 on either side of the from the same distance number line. which has the solution x = a or x = –a. the solution which has DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 14/05/14 2:46 PM

2 3 0 0 1 2 0 _ 1 -2 -7 2 3 - - 2 - 8 = = = - - 3 3 x x 2 - - − 5 4 16 x - - −6 −2 is positive or negative, so you must work work must so you negative, or positive is

–a. 5 2 = = - = x 6 78 24 4 6 - or 4 4 or or - - =

1 ⎟ a OR x 7 7 3 12 21 3x =

⎜ = 2 or x 3 or 6 ======

0 -2 x ⎟ ⎟ x - 5 - 5 - 5

-20 4 3 _ 4x 4x - 6 = 1 ⎜ ⎜ = - -22 3x = + 2 = x x 2 or 1 - Solving Absolute Value Equations Equations Value Solving Absolute Algebraically or x

= 8 3 _ -4 5 = 2 = =

3 + 10 ⎟ ⎟

is often used to express the solutions to absolute value equations, and and equations, value absolute to the solutions used express to often is - 2 = 19 - ⎟

= 6 ⎟

- 6 = 2 Solve each absolute value equation algebraically. Graph the Graph algebraically. value equation Solve each absolute line. a number on solutions = 20 = 18 or ⎟

- 6 - 6 ⎟ − 2

- 5 Integrate Mathematical Practices Mathematical Integrate + 2 = 8

3x 3x + 3x ⎟ + 2 = 20 or ⎜

⎜ ⎜ is a positive number), then x number), a positive a is a (where + 2 4x x x ⎜ x ⎜ 3x

x ⎜ = 1 2 x ⎜

3x -2

−2 _ ⎟ 3

⎜

Example 1 PROFESSIONAL DEVELOPMENT PROFESSIONAL , Practice MP.6 Mathematical address to opportunity an provides lesson This precisely. communicate and precision” to “attend to students which calls for them, graphing both by equations value absolute to find the solutions Students a learn that Students algebra. through and technology, without and with disjunction they use the properties of algebra to accurately and efficiently find the solutions to to efficiently find the solutions and accurately to algebra they of use the properties equations. value types absolute of various ]]

. You cannot know from theoriginal from know cannot x. You of values the in rule for here a disjunction the use of Notice bars value the absolute inside whether the expression equation x. finish isolating to both possibilities through Solvex. for both sides. 2 from Subtract equations. two Rewrite as 3x 4. both sides. 2 to Add 3 Turn Your Module Module 2 3. Solve each absolute value equation algebraically. Graph the solutions on a on the solutions Graph algebraically. value equation Solve each absolute line. number Solve for x. equations. two Rewrite as 4x sides. all 5 to four Add 4x 3. by both sides Divide   To solve absolute value equations algebraically, first isolate the absolute value expression value the absolute isolate first algebraically, equations value absolute solve To 1 Explain equation the same way you would isolate a variable. Then use the Then rule: variable. a isolate would you way the same equation If

A2_MNLESE385894_U1M02L2 78

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” negative. 0is x=–2,because neither positive nor solution, and 0is0,you candividebothsidesby dto get affect the solution? the daffect does esn 2 .2 Lesson 79 onebe solution. For example, absolutethe value expression equalszero, there will must no solution. be Explain to students that when equation not does have two solutions, there then Some students may that think ifan absolute value ⎜ than TwoSolutions Absolute Value Equations withFewer d The firststep isto addcto bothsidesto get negative number value expression isequalto zero orequalto a points. andfunction count then number the of intersection equation. Graph eachsideof equation the as a AVOID COMMON ERRORS INTEGRATE TECHNOLOGY QUESTIONING STRATEGIES EXPLAIN ax +b ⎜ ax +b the numberthe of solutions to an absolute value A graphing calculator to check used can be d In absolute the value expression fewer than two solutions? When an does absolute value equation have ⎟ ⎜ =0. ⎟ ax +b 0. Because the product ofanumber theproduct =0.Because ⎟ –c=for nonzero variables, how 2 It does not affect it. doesnotaffect It ⎜ 3x +6 when theabsolute ⎟ =0has one

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© Houghton Mifflin Harcourt Publishing Company of solutions. solutions. of switch roles and repeat exercise the using an equation that has adifferent number explanation about conjecture the whether is correct or incorrect. Have students and give an example. Then have other the student solve example the and write an to write aconjecture about what of type absolute value equation has no solutions, have two solutions, one solution, or no solution. For example, one instruct student Have students work inpairs to brainstorm of types absolute value equations that Peer-to-Peer Activity COLLABORATIVE LEARNING oue 2 Module zero is its own opposite. When absolute the value expression is to equal apositive number. When absolute the value isto equal zero, there is asingle solution because You have that seen absolute value equations have two solutions isolated the when absolute value expression Explain2 5. If solved. equation so, be solution. finishthe can If write not, “no solution.” Isolate absolute the expression value Divide sides both by 2. Add 4to sides. both as Rewrite one equation. Multiply sides both by   Your Turn Subtract 2from sides. both d obt ie. Add 3to sides. both bouevle r ee eaie No Solution Absolute values are never negative. sides by Divide both −5. solution absolute because value is never negative. Example 2

No solution

3 3 _ −5 5 3 ⎜ ⎜ ⎜ __ _

__ 2 1 2 1 2 1 ⎜ 2x

x x

⎜ x x + 5 + 5 + 5

− 4 + 1 Isolate absolute the expression value “no solution.” If solved. equation so,if the be solution. finishthe can If write not, ⎟ ⎟ ⎟ ⎟ ⎟

= =

+ 7=5 − 3= 12 + 2= - -2 _ 3 2

Absolute Value Equations withFewer than TwoSolutions −3 _ 3 5 .

_ 3 5

⎜ ⎜ in each equationin each to determine ifthe 2x 2x 2x − 4 − 4 − 5 − 2x 4 x ⎟ ⎟ ⎜ ⎜

x x = = = = = + 1 + 1 6.

⎟ ⎟ in each equationin each to determine 2 4 0 0 0 79

= = 10 x 9

9 is −2

⎜ ⎜ ⎜

_ _ _ 3 4 3 4 4 3 equal toequal anegative number, there is no

x x

x - 2 - 2 − 2 ⎟ ⎟ ⎟

+ 7= = 0 = = 0 _ 3 2

esn 2 Lesson 16/05/14 4:31AM 80

= c. ⎟ ax + b ⎜ Both processes are similar are Both processes value the absolute Isolate How do you solve a linear absolute value value absolute a linear solve do you How equation? How is the process of solving a linear absolute absolute a linear solving of the process is How a solving of the process like equation value Discuss with students how to solve an an solve to how Discuss students with Equations Value Solving Absolute ELABORATE ELABORATE PEER TO PEER ACTIVITY PEER TO PEER THE SUMMARIZE LESSON QUESTIONING STRATEGIES QUESTIONING INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES expression; write resulting equation as the equation resulting write expression; disjunction and solve linear equations; of two each equation. equation? linear regular Have students work in pairs. Have one student write write student one Have in pairs. work students Have the partner solve have and equation value absolute an the solution(s) why partner The then explains it. the repeat and roles switch sense. Students makes use the to phrase students Encourage process. “This the statement and zero” from “distance true.” the equation makes integer negative/positive MP.8 Focus on Patterns Focus absolute value equation of the form the form of equation value absolute initially, except that you isolate the absolute value value the absolute isolate you that except initially, in the case of variable the but isolate in one case, is the the process there, From the linear equation. each partsame for of the disjunction of the two equation. value the absolute for linear equations Students should routinely rewrite the next step as a as step the next rewrite routinely should Students the form of equation a compound or disjunction, each part thenax + b = c or solve ax + b = –c and the equation. of DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 14/05/14 2:46 PM

80 and solve and solve bars value the absolute , then remove Describe, in your own words, the basic steps to solving absolute absolute solving to the steps basic words, own Describe, in your if the function opens upward and y ≥ k if the function opens upward

Discuss how the range of the absolute value function differs from the range of of the range from function differs value the absolute of the range Discuss how

ax + b = c the form of equations two as = c. (3) Rewrite the equation ⎟ follow this solving plan: (1) Write the original equation; then (2) isolate then (2) isolate the original equation; (1) Write plan: this solving follow Elaborate The range of a non-constant linear function is all real numbers. The range of an range The linear function of a non-constant numbers. range real is all The function value absolute is y The solution to a mathematical equation is not simply any value of the variable of the variable value is not simply any equation a mathematical to solution The in the equation works that one value Supplying only true. the equation makes that which is incorrect. works, that the only value it is implies that function opens downward. Because the graph of a linear function a of a linear Because the graph is a line, function downward. opens value of an absolute Because the graph intersect line will it only once. horizontal all. at or not twice, line can intersect it once, a horizontal V, function is a Isolate the absolute value expression. If the absolute value expression is equal to is equal to expression If value the absolute expression. value the absolute Isolate If the absolute case. negative and both the positive for solve number, a positive zero is equal to expression value the equation. There is one solution. If the absolute value expression is equal to a is equal to expression is one solution. If value absolute the There the equation. is no solution. then there number, negative Essential QuestionEssential Check-In

Discussion Why is important to solve both equations in the disjunction arising from an absolute value value absolute an from arising in the disjunction both equations solve to important is Why will work the variable for the solution knowing it, solve and one pick just not Why equation? ? the equation into backed when plugged a linear function. Graphically, how does this explain why a linear equation always has exactly one exactly one has always equation a linear why does this explain how function. Graphically, a linear one, have can equation value while absolute an solution value equations and how many solutions to expect. to solutions many how and equations value ax + b

DIFFERENTIATE INSTRUCTION DIFFERENTIATE ⎜ Some students may need help in deciding whether absolute value equations have have equations value whether in deciding absolute need help may students Some they that suggest to want may You solutions. two or solution, one solutions, no always the form will have the equal sign. It side of one on expression value the absolute Critical Thinking Critical solution(s) in the original problem. solution(s) . There may be 0, 1, or 2 be 0, 1, or may x. There for each equation and (4) solve = –c; and ax + b “ using the answer write solutions, two are there (5) If solutions.

Module 2 9. 8. 7.

A2_MNLESE385894_U1M02L2 80

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” to equation. original the that x-value the of an intersection point is asolution When solving equations graphically, remind students thatverifying solutions both make equation the true. substituting values the into equation original the and Focus onReasoning esn 2 .2 Lesson 81 MP.2 ASSIGNMENT GUIDE PRACTICES INTEGRATE MATHEMATICAL EVALUATE Fewer than Two Solutions Absolute Value Equations with Example 2 Algebraically AbsoluteSolving Value Equations Example 1 Graphically AbsoluteSolving Value Equations Explore Concepts andSkills Remind students to check solutions their by Practice Exercises 9–16 Exercises 5–8 Exercise 1–4 CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U1M02L2 81

Exercise © Houghton Mifflin Harcourt Publishing Company 23–25 18, 21 5–16 1–4 22 20 19 17 oue 2 Module 1. following absolute the Solve equations value by graphing. 5. number line. absolute each Solve equation value algebraically. Graph solutions the on a 3. x 2x ⎜ ⎜ −2 x x Evaluate: Homework andPractice - - x 2x = 0 or = 0 3 8 − 3 ⎜ ⎟ - = 3 r 2x or = 3 x

−4 = 3 8 3 3 2 3 3 3 2 2 Depth ofKnowledge (D.O.K.) + 5 _ - 3 2 ⎟ Strategic Thinking Strategic Thinking Strategic Thinking Skills/Concepts Skills/Concepts

r or

+ 2=5 4 - Strategic Thinking Strategic Thinking Strategic Thinking Skills/Concepts - ⎟ - - 2

or x 4 + 4=2 8 4 4 8 0 - - x 8 4 4 8 0 y = 6 - = y 1 4 −6 4 x = = 8 0 x x − −3 _ 3 2

1 2 81 3 MP.6 MP.3 MP.6 MP.6 MP.4 MP.4 MP.6 MP.5 COMMON 4. 2. 6. CORE Precision Logic Precision Precision Modeling Modeling Precision UsingTools -

x (

2 ⎜ ⎜

x No solution - _ - 3 1 _ _ ⎜ 3 1 2 3 24 = x

8 ) x ( 8 x Mathematical Practices

+ 1 ( x + −3 orx _ − 2 3 1 - + - -

) 4 ⎟ 4 4

20 ⎟

x 4 ) + 5=9

- - - ⎟ = 3 12 = = =

+ 3=2 4 4 8 8 4 4 8 0 0 −3 or −1 or 3 - y y 16 = 4 4 1 or - 12 8 8 ( x _ x 3 1

)

8 x ( • Extra Practice Extra • Help and Hints • Homework Online • _ + 3 1

) 4 x x - = = = 4 -3 −7 − esn 2 Lesson 21 0 14/05/14 3:10PM 82

the solution points when they graph when they graph points the solution give examples of absolute absolute of examples to give students Ask Equations Value Solving Absolute AVOID COMMON ERRORS COMMON AVOID MATHEMATICAL INTEGRATE PRACTICES Students may erroneously include points on the on points include erroneously may Students between graph process the solution that students Remind solutions. value absolute an to 2 solutions 0, 1, or gives solutions. many infinitely not equation, MP.3 Focus on Critical Thinking on Critical Focus value equations that have no solutions. Suggest that that Suggest solutions. no have that equations value no with equation an of a graph how they think about any show not should will graph look. This solution graph. empty an is so it points, DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-B;CA-B 14/05/14 2:46 PM

0 6 2 9 _ 4 5 -1 −5 −1 Lesson 2 = = 1 2 = - = -9 2 4 3

⎟ x

- x

- 2 7

2 7 _

1 2 3 _ 2 − 6 = + + 4 x x 4 2

- - 2 1 _ − 1 2 _ 1 ⎜ 0 6 6 0 or or or or ⇢ - 1 - -

- 2 2 5 _ -3 −8 −7 8 8 2 - 2 = 5 -1 - 1 or - - = = 1 = 3 -2

2 + 10 = ⎟ ⎟ 3 -

x = 3 = = 7 = = - = -5 -

⎟

= = 0 = 0 ⎟ + 3 = 6 ⎟ ⎟

- ⎟

x x

- 3 = 2 7 4

− 6 = 1 1 2 ⎟ ⎟

⎟ _ x 2 7 4 _

10 ⎟ 2 1 10 _ - − 6 − 6 - - + − 3 − 3 - + 4 = 0 + 5 −x − 3 + 3 + 4 + 4 5 −x− 6 x x x + 4

x ⎜ - −x

−x ⎜ ⎜ x x

2x ⎜ ⎜ 2 1 ⎜ _ 1 2 2x 2x 6 1 2 _ 6 _ 2x 12 12 ⎜ ⎜ ⎜ ⎜ ⎜ - −3 -

No solution

5 - 5 7 −3 −8 7 −8

12. 14. 10. 8.

6 82 5 6 6 4 5 4 5 4 4 3

3 3 3

2 3 -1 3 4 3 8 _

2 2 1 = = =

x 1 1 2x

- 3

0 0 2 x 1 2 - 1 1

- - 2 -4 or or or 2 2 - - - = 0 = 0 =

= 6

3 3 ⎟ ⎟ −2 1

)

- 3 - 1 3 - 10 3 3 5 1

_ _ _

3 + 2 = = 4

+ 7 = 5 + 6 = 6

= -1 4 4 + 2 = 1

+ 3 ⎟ + 4 + 4

⎟ ⎟ - ⎟ ⎟ = = = ⎟ ⎟ - -

x x x x 5 3 ⎜ ⎜ 5 5 − 3 2 2 + 2 + 2 - - 4 + 4 - - 4 - - 3 = - x + 4

x x 6 6 x 6 2x ⎜ 0 ⎜

⎜ x

⎜ ( ⎜ 3x 3x - - - 1 4 2 ⎜ No solution 2 ⎜ _ 3 2 4 1 2

No solution

11. Module 2 13. Solve the absolute value equations. Solve the absolute 9. in the following equations to determine if they determine to in the equations following value expressions the absolute Isolate solution”. “no not, write If can find the solution(s). graph be and so, solved. If 7.

A2_MNLESE385894_U1M02L2 82

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” solution nonnegative before proceed with the they thatthem absolute the value expression should be the original real-world original the problem. 2 solutions to associated the equation, depending on remember that of may type function this have 0,1,or with aV-shaped help diagram. This will them absolute value equation applies, have students start Focus onModeling esn 2 .2 Lesson 83 rewriting form inthe them for students donot who solve equations these by first When solving absolute equations algebraically, watch MP.4 AVOID COMMON ERRORS PRACTICES INTEGRATE MATHEMATICAL When modeling aproblem an inwhich steps. ⎜ ax +b ⎟ =c.Remind CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U1M02L2 83

2x 7.5 seconds 12.5seconds. andagainafter The isat leadgoose v after shooting begins isd begins shooting after The distance dinfeet that fliesintseconds theflock

then distance oftheleadbird from thephotographer is distance equals75feet. ⎜ The buoys shore from theleft-hand placed shouldbe and330ft at 150ft

⎜ the buoys?the riverthe bottom is 30feet below surface. the How far from left-hand the shore should you place down to river the bottom. Suppose you are harbormaster the and you want to place buoys where at ahorizontal distance of 75feet from photographer. the after times the photographing the is that begins goose lead the a steady second. 30feet per Write and solve an equation to find photographstaking until it at is flying is 100feet past. The flock is 300feet horizontally goose lead the from her, and continues formation. The photographer photographs taking starts when of is approaching geese A flock aphotographer, in flying buoys?the riverthe bottom is 30feet below surface. the How far from left-hand the shore should you place down to river the bottom. Suppose you are harbormaster the and you want to place buoys where A ship aground risks running bottom ifthe of its keel (its lowest point under water) the reaches distance to left-hand the shore (infeet). d(h) The bottom of ariver makes aV-shape that with absolute the modeled can be value function, ⎜

- _ 1 5 30t 2 2x 30t ( 6

⎜ x h + 7 + 5 = - 300 + 7= 4 or2x - 240 d - 300=75 - ( _ - 5 1 x t 40 ⎟

⎜ ) ) 5 30t

h = 4 = = -3or 2x - 3

0 = − 240 ⎟ t ⎟ - d

⎜ = 12.5 = 375 = 75 ⎟ -48 30t d(h) - _

2 3 + 40 4 or

h ⎟ 2 =6

- 300 − 48,where dis depth the of river the bottom (infeet) and his horizontal the = 330 or = = - 80 -30 -30 3 a horizontal distance of75feet from thephotographer after or or or ⎟ .Find thetimeswhenthis 120 + 7= ( - t x 2 )

= -11 = 30t . The horizontal = 30t.The 160 - -4 - - 300= h _ 11 1 2 200 = 150

30t

t = 7.5 = 225 0 -75 83 280 16. 320

-5 -5 - 2 360 ⎜ ⎜ ⎜ -3x -3x -3x -3x 400 + 2 + 2 + 2 -3x - + 2= 1 ⎟ 440

x - ⎟ ⎟

= = -2 = 0 = 0 2 = _ 2 3 0 480

-2 0 . h 1 esn 2 Lesson 2 6/8/15 12:14PM 84

gives numerical numerical ” gives solving the equation solving 0, students should should c > 0, students = c, where ⎟ ax + b equations of the of equations value absolute solving When ⎜ Equations Value Solving Absolute AVOID COMMON ERRORS COMMON AVOID SMALL GROUP ACTIVITY GROUP SMALL MATHEMATICAL INTEGRATE PRACTICES Watch for students who are confused by nested nested confused by who are students for Watch to students Remind equations. value absolute the each part of for disjunctions write carefully solution the same using appropriate, as solution, equation. value absolute a single they use for process Have students work in small groups to make a poster a poster make to in small groups work students Have an solving for the steps apply to how showing a different group each Give equation. value absolute its present each group have Then solve. to equation a volunteer for ask and the class, of the rest to poster each step. explain to the group from MP.2 Focus on Reasoning Focus form – recognize that the equation states that a negative a negative that states theequation that recognize is This number. a positive equalto is value absolute absolute of the definition because of possible not so whilevalue, “ the to solutions not are these answers answers, verify to students want may You original equation. will points. be There intersection no graphing. this by DO NOT EDIT--Changes must be made through “File info” “File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/8/15 12:14 PM

1 3 __

3, - 3 = = 1 = 3x 3x 1. x = - 4 =

⎟ 2 or or or + 24,

− 4 ⎟ 3 5 _

-1 + x

⎜ ⎟ = 2 = 5 = = 5.

− 3 ⎟ ⎟ x 16 with two equations using the positive using the positive equations with two = 3x

3x - 3 = 2 ⎟ − ⎜

- 3 - 3 1 2 _ x 30 feet. = 30 feet. ⎜

3x − 3 2 _ - 3 3x 3x ⎜

⎜

x − 1 2 _ ⎜ 84 =

- The student tried to replace the absolute value value the absolute replace tried to student The equation the number of on the other values and negative this number was side of the equal sign. However, a positive like and cannot be treated negative expression value absolute isolated The number. this number and therefore a negative is equal to has no solution. equation

-2.

1 5 3 3 _ _ =

-1 3 19 7 3

_ = - = - = -3 = = = x x x x

x 4

= 8 or x F. B. D. - H. 3 are given by by given = 3 are x satisfy the equation that x that are a distance of 5 from of 5 from a distance are that

or or 30

= -4 = gives 5 gives 7 3 3 7

_ _ While attempting to solve the equation −3 the equation solve to attempting While

⎟

5 3 7 -

_ - 24 3 =

16 ⎟ = = = =

= + ⎟ ⎟

x 4 - 4 ⎟ 4 4 Find the points where a circle centered at at centered a circle where the points Find - 3 x

- ⎜ - - -

16 x

x ⎜

⎟ the Error 3 1 x x _

5 3

_ 3 1 4 ⎜ ⎜ - is the distance along the width of the house. Where should should Where the house. of the width along the distance x is = = = = x - -3 ⎜

x x x x Focus on Higher Order Thinking on Higher Order Focus x

axis. Use an absolute absolute an Use the x-axis. 5 crosses of a radius 0) with 3 2 ⎜ _

- Use the model function to solve for x when h(x) for the model functionUse solve to No Solution. Terry does not have a spot on his roof that is 30 feet is 30 feet that a spot on his roof does not have Terry Solution.No high.

The points are (-2, 0) and (8, 0). are points The The points on the x-axis points The the center of the circle at x at of the circle the center Solving Terry is trying to place a satellite dish on the roof of his house at the at house his of the roof on dish trying is a satellite place to Terry the height and wide, 32 feet is house His 30 feet. of height recommended be can described the function the h(x) roof of by A. Explain where where Terry place the dish? place Terry up with the following results. Explain the error and find solution: the correct and the error Explain results. the following with up Geometry Select the value or values of x of Select values or the value value equation and the fact that all points on a circle are the are a circle on all the fact points that and equation value the center. from (the radius) distance same (3, E. G. C. -3

H.O.T. H.O.T.

21.

22. Module 2 Module 19.

20.

A2_MNLESE385894_U1M02L2 84

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or fewer places. V-shaped andthisgraph alineintwo canintersect no intersection points,no intersection andthat thegraph off esn 2 .2 Lesson 85 graph of looks. function the possible number of intersection points and how the related functions. Conjectures should include the of related the functions ƒ of f on this, conjectures shouldincludethat thegraphs pointsintersection oftherelated Based functions. equations. havethey for learned solving absolute value Have students compare and contrast methods the ⎜ to solution Ask students what to with apartner discuss the JOURNAL PEER TO PEER DISCUSSION ax +b ( x ) = ⎟ =care thex-coordinates ofthe ⎜ ax +b ⎜ ⎜ ax +b ax +b ⎟ andg ⎟ ⎟ =cmeans interms of graph the =cand graphs the of their ( ( x x ) ) =ccanhave two, one, or = The solutionsto ⎜ ax +b ⎟ and g and ( ( x x ) =c. ) is CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough “File info” A2_MNLESE385894_U1M02L2.indd 85

Answers willvary. answer: Sample time, distance, height, length,speed There are two only onepath possible solutionsbecause produced solutions. Follow eachpossiblesolutionpathifneeded. andusemore disjunctions

⎜ Justify YourReasoning Communicate Mathematical Ideas Check for Reasonableness negative answer for an absolute value equation not make sense? 3 explain what algebraic properties make it possible to do so. the numberthe of possible solutions. Use your knowledge of solving absolute value equations to solve equation. this Justify ⎜ ⎜ ⎜ 2x x x - 2

⎜ - + 5 ⎜

(

⎜ 2x 3 2x 2 ⎟ ⎟ ⎟ - + = 5

- + -3 ⎜ ⎜ ⎜ 5 2x 2x 2x 5 2x ⎜ 5 5 ) ⎟ ⎟ x

⎜ ⎜ ⎜ ⎟

= + + + - x x x

- 2 - + x - - - 10 2x - 5 5 5 3 3 5 x ⎟ ⎟ ⎟ ⎟ ⎟ 2 2 2

-7 2 ======x ⎟ ⎟ ⎟

= = = -7 Distributive Property = = -7 4 10 13 13 13 10 8 13

_ _ _

7 7 2 2 11 This absolute value equation has nested absolute values. 2

For what of type real-world quantities would the or x or or or or or or or

Solve absolute this value equation and

x x No solution 2x 2x ⎜ ⎜ 2x 2x - = -9 = -18 + + + 2 5 = - = - 85 5 5 = -13 ⎟ ⎟

= -7 - _ _ 7 2 2 3 3

= -10 Addition ofEquality Property Subtraction Property ofEquality Property Subtraction Division Property of Equality Property Division Definition ofabsolute value Lesson 2 Lesson 19/03/14 12:13PM 86 5 feet 5 feet in their equation in their equation

, and the snowball , and

. Thus, the ratio the ratio . Thus, 2 5 _ 5 24 _

. Then discuss how solving solving discuss . Then how

5 24 _ . Explain that the snowball rises that . Explain

5 2 _ Discuss with students how solving for the for solving how Discuss students with Equations Value Solving Absolute .

5 2 _ AVOID COMMON ERRORS COMMON AVOID MATHEMATICAL INTEGRATE PRACTICES Some students may use the ratio use the ratio may students Some on Communication Focus MP.3 north for every 2 feet west it runs every it north for west 2 feet is Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: does not answer the problem. The difference difference The the problem. for answer e does not the the distance equal is to between distances the two apart. are two instead of of instead on the left, е - 6 = - on eastward position solves for the distance of the of the distance for solves position eastward е - 6 = the right, snowball on 19/03/14 12:13 PM © Houghton Mifflin Harcourt Publishing Company to ) Lesson Lesson 2 6, 12 ( feet apart. feet

3 5 __ = 9

1 5 __ – 1

4 5 __

) 86 − 6 e (

2 5 _

feet apart.

3 5 _ − 5 24 _ 9

= 5 6 _

= ⎟

n(e) e 5 -axis to the snowball on the the snowball to the y-axis on from ft. distance The

48 _ ⎜ 4 5 _ − 6 = - e = 12 ⎟

6 5

_ = or

) ⎟ ⎟ − So the fragments are 10 ft. So are the fragments

1 5 _ − 6 5 − 6 − 6 54 _

e e e ⎜ or or ( ⎜ ⎜

5 5 5

=1 24 24 54 _ _ _

5 5 2 2 2 5

_ _ _ gives the distance from from the distance x gives of the value 6 from . Subtracting

) 4 5 _ = = = = =

⎟ e

, 12

− 6 = . That distance is 4 is distance . That 5 4 − 6 _ e ) e ⎜

10 ( west wall. One fragment flies off to the northeast, moving 2 feet east for every for east 2 feet flies moving the northeast, to off fragment wall. One -west x, 12

EXTENSION ACTIVITY EXTENSION ( be Have students try to find an alternate solution method using the formula for the for the formula method using solution try alternate find students to an Have to the right the snowball on for find the coordinates should Student a line. of slope left– 4 6 is

The fragments are are fragments The

n(e)

Module Module 2 The fragments can be described as two lines originating at the child’s the child’s at lines originating can be described as two fragments The function. value a single absolute by and then be replaced coordinates, n(e) 5 feet north of travel, and the other moves 2 feet west for every 5 feet north of travel. every travel. for north of 5 feet west 2 feet moves the other and travel, north of 5 feet both fragments n(e), of position, describes the function that northward value absolute an Write are apart far How they the school door are. of east far how a function of as when theythe strike fragments the wall? A snowball comes apart as a child throws it north, resulting in two halves traveling away away traveling halves in two resulting north, it throws a child as apart A snowball comes an along thedoor, school of east 6 feet and south 12 feet standing is child The the child. from east Lesson Lesson Task Performance To find where the fragments strike the school wall, solve for the eastward the eastward for solve the school wall, strike the fragments find where To north 12 feet are of the child. position when the fragments n(e)

A2_MNLESE385894_U1M02L2.indd 86

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT