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Ramp Up to Algebra Curriculum Updated 08/09 Katie Livermore Christine Buckman Ramp Up to Algebra The objective of the Ramp Up to Algebra course is to prepare students for success in Integrated Algebra and on the Integrated Algebra Regents Examination. It is expected that students will take this course in the 9th grade. This curriculum is designed for students that are not performing at grade level in the area of mathematics. This is a one year course, two periods per day. Completion will provide students the prerequisite skills to ensure success with the further study of mathematics. Successful completion will allow them to progress to the New York State level of mathematics; Integrated Algebra. Throughout this curriculum the text referred to is Ramp Up to Algebra, provided by America's Choice. In order for many of the ~"" cent standards to be addressed, additional lessons and materials were added. While the writers realize that the order in which the topics are taught is open to individual interpretation, it is recommended that teachers follow the sequence as laid out in this document. This allows students to learn and review the basic skills required for concepts later in the course. Students should be awarded one math credit and one elective credit upon: • Completion of the course as outlined in this curriculum • A cumulative final examination • An average grade of at least 65%** • Adherence to the Poughkeepsie High School attendance policy **Students final grade for the course will be an average of the four quarter grades and their grade on the final examination. Ramp Up to Algebra Sequencing and Pacing Guide Unit Number Number of Title Days ,,, ,, 1 The Number System (RU Unit #2) 2 Negatives (RU Unit 115) 17 3 Factors and Fractions (RU Unit #4) 22 17 .. 4 Exponents & Polynomials (not in RU) 14 5 Using Equations to Solve Problems (RU Unit #7 & 8) 17 6 Ratio and Proportionality (RU Unit 116) 21 7 GeomelIy and Measure (RU Unit #3) 15 8 Graphing (RU Unit #7) 14 9 Foundations of Algebm (RU Unit #1) 20 Total Days 157 ,, In the unit outlines below, (*J indicates a lesson Ihal is being added 10 the Ramp Up 10 Algebra curriculum, in this 08/09 updale. Ramp Up 10 Algebra - Unit 1 - The Number System (RU Unit #2) D ay Goal ! To understan d decimal Ramp Up Lesson Intro !o Algebra Lesson (Plus extra worksheets) #! i Exponents Review Place Value : Worksheet; numbers usmg place values 2 3 Place Values and Rounding Worksheet; To use #2 number Number lines to Lines represent" order, and estimate whole numbers, fractions, and decimals To #3 identify Classifying and Numbers represent Real Numbers Worksheet numbers, whole numbers. integers, and rational numbers usmg number 4 ,, 114 Rational Numbers NYSAlgebra Standard Digits) Place A.N.! Value, Power : Identify and .pplytbe • ofTen . (Multiples of . properties of real numbers (closure, Ten), Expanded commutative, Fonn, associative. : distributive, : Decimals, identity, inverse) Tenths, Hundredths, etc. Fractions Interval, A.N.! Inequality (same as above) «, », Rounding Objective 7.0 Worksheet natural lines and diagrams To convert fractions Word Wall Suggestions Fractions aod Decimals Skills Review Worksheet Naturnl A.N.1 (same as above) Numbers, Wbole Numbers, Integers, Rational Numbers, Numerator, Denominator, Classifying (describing), Venn Diagrams, Between) Irrational Equivalent, Fraction Bar I (means to A.N.l (same as abeve) to Always, Sometimes, Never Activity on Number Systems and Operations tenninaHn g or repeating deCimals, to convert decimals to fractions. and to identify rational numbers 5 To write H, and Fractions represent and equivalent Decimals fractions Comparing Decimals Worksheet; divide numerator by denominator), Mixed Number, Improper Fraction, Terminating Decimal, Repeating Decimal, Irrational Numbers, Unit Fraction A.N,1 Simplest Form, (same as above) Compare Comparing on : Fractions numbers • Worksheet', lines and to order fractions and decimals 6 7 To use rational numbers to measure length and weight To add and subtract whole numbers and , decimals #6 Measuring Precise, A.N.5 Approximatio Solve algebraic ns, Density problems arising with Rational Numbers from situations that involve #7 Riddle Worksheets Adding and Subtracting On the Number Line (2); Objective 4,0 Worksheet; Adding, Operation, Subtraction, Commutative Property, Associative SimnlitviM • Property, fractions, decimals, percents (decreaselincreas e and discount), and proportionality!di reet variatlon A.RP, I Recognize that mathematical ideas can be supported by a variety of strateRies using number lines; 8 9 To use mental strategies and stand methods to add and subtract To multiply and divide whole numbers and decimals using number lines and area models; #8 Adding and Subtracting Objective 20.0 Worksheet multidigit numbers II To use the "guess and check" Identity Property, Inverse Operations, Sum, Difference Mental Strategies, Intro. Unit 1, Assignment 9 and Dividing Evaluating Expressions #9 Multiplying #10 Multiplying Multidigit Numbers Intro. Unit 1, Assignment lOA & lOB - Simplifying Expressions MUltiple Representation Activity on Division for Problem Solving To use mental strategies and the standard method to multiolv 10 To divide Numerical Expressions Worksheet; Regrouping, Carrying Repeated Addition, Repeated Subtraction, Dividend. Divisor, Quotient A.CN.2 Understand the corresponding ,.. .." edures for ,. similar problems or mathematical concepts Count on, Doubling, Halving, Count Back, Estimating, Area (Rectangle & Parallelogram IA=bh), Patterns, Conjectures #11 Dividing Multidigit Numbers #12 Estimating Square Roots Objective 6.0 Worksheet Multiple Representation Activity - Laws of Exponents for Multiplication and Division Base Ten Blocks info. Intro. Unit 10, Lesson #5 Square Roots Short Division, Long Division, Lining Up Places A.CN.2 Square, Square Root, Guess and Check, A.N.I (same as above) (same as above) • strategy to estimate Accuracy, Area \Square; A=s) square roots 12 To represent percents on. number line from Ito #13 Percents and Number Lines Fractions, Convert, Decimals, and Peroents Worksheet; Percent, % Intro. Unit 4, Lesson #1 100%~ Percents to with decimals and fractions Decimals; Decimals to Percents from Oto Multiple Representation Activity on Percents --.Deeimals-->Fract ions "of'means Finding a Percent "multiplIed ofaNumber Worksheets (2) bt' I 13 To calculate and use AN.5 (same as above) #14 Percents and Decimals A.N5 (same as above) decimal equivalent sof percents I l~ To solve 15 problems 16 that , involve Intro. Unit 4, Lesson 112 Percent ofa Number #15 Applying Percent Inlro. Unit 4, Lesson #3 Mixed Review • Intro. Unit 4, Lesson #7 Percentages - Sales • Tax percent Intro. Unit 4, Lesson #9 Percentages - Sales Tax Intro. Unit 4 • Sales Tax, Interest. Percent-off, : Discount, A.N.5 (same as above) Lesson #8 Percentages ~ Discounts Intra. Unit 4, Lesson #10 Percentages - . Simple Interest Intra. Unit 4, Lesson #11 Percentages , Commissions • • • · , Percentagesi Mixed Problems • • ·· • • • · 17 To review the number system concepts #16 The Unit in Review Objective 17.0 Worksheet Intro. Unit 4, Lesscn#13 Percentages ~ Review · ,. Intro. Unit 4, • Lesson #12 . ··• ,~-- All ofthe above slllnilards ··· I • • ·• Ramp Up to Algebra-Unit 2-Negatives (RU Unit #5) Day Goal 1 To recognize the need for negative numbers. and to place Ramp Up Lesson 6: Extending the Number Line negative Intro to Algebra Lesson (plus extra worksheets) Worksheet: Developing skills in Algebra Book A Graphing integers sheet (page 21) numbers on the number line 2 3 4-S ! To compare 7: Putting positive and Numbers in Order negative numbers and zero, using is less than.(<) is less than or equalto,(:s) is greater than,(» and is greater than or equal to(?) To multiply 13: Multiplying and divide and Dividing positive and negative numbers To use the 8: Adding with number line to Negative add positive Numbers and negative numbers Worksheet: Developing skills in Algebra Book A Directed Distance sheet (page 19) Intro Unit 1 Integers packet Multiplicationassignment #6 Dividing integers~ Assignment # 11 Intro Unit 1 Integers packet Addition Assignment #5 Worksheet: Developing skills in Algebra Book A Addition of Integers Word Wall Suggestions • Negative numbers • Positive nwnbers • origin NYS Algebra Standard A.N.l Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, invers~) A.CM.12 • positive direction Understand and • negative use appropriate direction language, • ascending representations, • descending and tenninology • (the inequality when describing symbols) objects, relationships, mathematical solutions, and rationale A.PS.3 Observe and explain patterns to formulate generalizations and coni ectures A.CM.S • Addend Communicate • sum logical arguments clearly, showing why a result makes sense and why the reasoning is valid 6 7 To use the : 9: Subtracting number line to willi Negative subtract Numbers positive and negative numbers To understand and use equivalence relationships between adding and subtracting I Pages 27, 29, 31, and 33 Riddle sheet #6 • difl"erence "Wh.t do you call a cow that won't give milk.?'" ACM.8 Reflect on strntegies of others in relation to one~s own strategy Worksheet: Developing skills in Algebra Book A Subtracting Integers Page 35 10: Adding and Subtracting Riddle sheet #5 ACN,1 Understand and make connections among multiple representations of the same mathematical idea "Where do chickens go to work?" Worksheet Developing skills in Algebra ,, ;--------- ,, ,, 8 ,, , ,, To model addition and subtraction of positive and negative numbers Book Adding and Subtracting Integers Page 37 A,R.l Use physical Objects, diagrams. charts, tables, graphs, symbols, equations. or object created using technology as representations of mathematical 11: Balloon Model , concepts 9 12: Reviewing To review what you have Addition and learned about Subtraction Intro Unit I Integers packet Working with I A,CNA Understand how : concepts, ,, ,, 1 posjti~e and ,, mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics A.N.! numbers ,, , , , , , 10 14: Mixed operations To calculate expressions , involving operstion. using L'le ,, distributive property 11 , To apply the : number properties to positive and negative : numbers ,, ,, ,, 15: Number Properties 13 To revIew working with 16: Progress Check ,, ,, , do" Intra Unit 1 Integers Packet Quiz #2 positive and negative , (See above) Intro Unit 5 Distributive Property packet Lesson #11 Simplifying expressIons Joke Ul1 'Whata ' , Matador tries to ,, , ,, , 12, , Intra Unit I Integers packet Worlcing with Integers- Assignment #8 : mOre than one l procedures. and Integers- Assignment #7 : negative I , A.CN.2 Understand the corresponding procedures for similar problems or mathematical concepts , ,, , ,, , A.PS.! , Use a variety of , problem solving strategies to understand new numbers : mathematical 14 17: Learning from the Progress To review and learn from the Progress Check Check Intra Unit 5 Distributive Property packet Quiz 2 Unit 5 The Distributive Property , ~"""~ ......~ .... ,, , content A.PS,g Determine ,, information ,, ,, required to solve a problem, choose methods for obtaining the information~ and define parameters for I acceptable , , 15 To Graph with 18: It's Cold Up positive and There negative numbers : solutions A.CN.5 Understand how quantitative models connect to various 16 To solve word 19: Word problems Problems involving positive and physical models and .!epresentations ACNA (See above) negative numbers *17 To review the mathematical concepts of the writ All standards from previous lessons in the unit Ramp Up to Algebra-Unit 3-Factors and Fractions (RU Unit #4) Day I Goal To define factor and multiple, and In understand how factors and multiples appear in the multiplication tabl~ a..•.l ... n : the number line Ramp Up Lesson I: Multiples and Factors Intro to Algebra Lesson (plus extra worksheets) • • • • Word Wall Suggestions NYS Algebra Standard multiple factor factorization product A.CM.2 Use m.athematical representations to communicate, with appropriate accuracy. including numerical tables, formulas, functions, equations, charts, graphs, Venn diagramsl and other 2 To identify prime numbers and composite numbers 2: Prime and Composite Numbers • pnme • composite • distinct diagrams. ACM.3 Present organized mathematical ideas with the use of appropriate standard notations. 3 To writc any 3: Prime natural Factorization number greater than I as a product of including the use ofsymbols and other representations when sharing an idea in verbal : and written : form ACM .pnme factorization Communicate logical • Fundamental arguments Theorem of clearly, Arithmetic its prime factors • Factor tree 4 To define and understand common multiples 4: Common Multiples 5 To understand and use the relationships between the greatest common factor (GCF) of two numbers and their least common multiple (LCM) To review factors, multiples, pnme numbers, pnme factorization, greatest common factor, and least common multiple To solve realworld problems involving factors and multiples To mUltiply a fraction by a whole number, and to write 5: GCF and LCM i 6 7 8 • Common multiple • Least common multiple (LCM) • Greatest common factor (GCF) 6: Reviewing Multiples and Factors 7: Making Sizes that Fit 10: Multiplying bya Whole Number • Inverse showing why a result makes sense and why the reasoning is valid. ACM.5 (see above) AN.I Identify and apply properties of real numbers ACN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics A.CN.6 Recognize and apply mathematics to situations in the outside world AN.I (see above) equivaJent fractions using division 9 To multiply II: Multiplying Fractions • horizontal • vertical AN.I (.eeabove) To add and subtract 8: Adding and Subtracting AN.! ($eeabove) :fractions with Fractions • Equivalent fractions • fraction • numel1ltor • denominator • mixed nwnber • improper fraction • simplest form fractions, and to represent products as areas of rectan~les 10 common denominators 11 12, 13 To add lhletions that have different denominators To review the concepts of 9: Adding with Different Denominators 12: Progress Check AN. I (see above) ACM.2 (seeahove) fractions and adding, subtracting, 14 and multiplying fractions To subtract 13: Finding fractions with Differences different denominators, to see subtraction as the inverse of . • ascending order • descending order • Inverse equations A.N.I (see above) addition, and to use subtraction to order fractions 15 To review 14: Addition and addition and subtraction of fractions and the inverse Subtraction as Inverses relationshin AN.! (see above) 16 17 . 18 19 20 21 22 between these operations To divide with fractions To calculate with fraction in order to solve problems involving measurement and calculations of length and distance To apply the number properties to operations with fractions TO,review work based on the skills and concepts discussed in the unit To reinforce understanding of the topics covered in the previous lesson To calculate with fractions in problems involving unit fractions and enlargements To review the work of the unit and prepare for the final assessment 16: Dividing Fractions 15: Shortest Distance • reciprocals A.N.l (see above) A.R4 Select appropriate representations to solve problem situations 17: Mixed Operations A.N.I (see above) 18: Progress Check A.N.I (see above) 19: Leanning From the Progress Check All standards from previous lessons in the unit 20: Shortest Time A.RA (see above) 21: The Unit in Review All standards from previous lessons in the unit Ramp Up to Algebra-Unit 4---Exponents & Polynomials (not in RU) Day Goal Ramp Up Lesson *1 To apply properties Not in RU of exponents involving products Intro to Algebra Lesson (plus extra worksheets) Arlington ProjectExponents and Their Properties Find a Match Word Wan Suggestions Power, exponent, base, Product of Powers Property, NYS Algebra Standard A.A.12. Multiply and divide monomial expressIons Powerofa with a common Power base, using the Property, Powerofa exponent properties of Product Property *2 *3 To apply properties of exponents involving quotients To use zero and Not in RU Not in RU negative exponents Arlington ProjectExponents Quotient of a Powers A.A.12. Same as above Property, and Their Powerofa Properties Quotient Property Can you Build This? Arlington reciprocal Project-Zero A.A.12. Same as above and Negative Exponents How did the Light Dress up for the Costume Party? Why did the Farmer Open a Bakery? *4 To convert numbers Not in RU How did Slugger McFist Get a Black Eye? Intro. Lesson Standard A.NA. • from standard form #8 - Scientific Fonn, Scientific Notation; Notation to scientific notation Scientific Notation Worksheet . Multiple Understand and use scientific notation to compute products and quotients of numbers Representatio n Activity on Scientific Notation *5 To convert numbers from scientific notation to standard form . A.N.4. Understand and Not in RU use scientific notation to compute products and quotients of numbers *6 To introduce and add polynomials NotinRU Arlington Monomial, Project- binomial~ Combining Like Tenns trinomial. polynomial, degree, *7 To subtract polynomials Not in RU WestSe. Addition of Monomials and Polynomials • Arlington ProjectCombining Uk. Terms A.A.!3. Add, subtract, and multiply monomials and polynomials leading coefficient AA13. Add, subtract, and multiply monomials and , polynomials West SeaSubtraction of Monomials and Polynomials --.g • To multiply monomials by monomials Not in RU Arlington ProjectMUltiplying a Polynomial AA.I3. Add, subtract, and multiply monomials and bya Monomial polynomials West SeaMultipHcation of Monomials *9 To multiply a monomial by a polynomial Not in RU Arlington ProjectMUltiplying a Polynomial bya A.A. \3. Add, subtract, and multiply monomials and polynomials Monomial West SeaMultiplication of Polynomials bya Monomial *\0 *11 *12 To mUltiply a binomial by a binomial To multiply a binomial by a polynomial To divide a monomial by a monomial NotinRU Arlington ProjectMultiplying Polynomials Not in RU West SeaMultiplication of Binomials Arlington A.A.13. Add, subtract, and multiply monomials and polynomials Project- A.A.!3. Add, subtract, Multiplying and mUltiply Polynomials monomials and polynomials A.A.14. Divide a polynomial by a monomial or Not in RU binomiaL where *13 To divide a polynomial by a monomial Not in RU Division of Monomials and Polynomials the quolienl has no remainder A.A. 14. Divide a polynomial by a monomial or binomial. where the quoLient has no remainder Not in RU To review concepts ofexponents and polynomials ,,• *14 West SeaAddition and Subtraction of Polynomials ,, ,, West Sea~ Properties of , ,, Exponents ,, ,, M ___ West Sea ' Multiplication : of Polynomials ,, All of the standards above ,, , , , , , I Name _____________________________ Date _____________________ A. Write as factors. Then, rewrite with exponents. Factors _____ .~ _________ Exponents _____________ Factors ________________ Exponent. ____________ Factors ________________ Exponents ____________ B. Tell whether each equation is true or false. the right side to nl.ake a true equation. ]f false, correct 1. r'. r' = ,9 ___________________________________ 2. x 5 • 3xy = X 2 3 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ___ y 3. 4n'p' np2 = 4n'pZ ____________________________ C. Multiply or divide. d' 2. 1. ~ );,0. a2 5. nz.. n- 1 3 7. x XS 8. a- 2 b3 a- 3 b D. Find each number named in scientific notation. 1. 1.44 56 x 10" 2. 6.23 Cha:pter 10· Exponents and Functions x 10-' 3. 2.06 X 10- 2 Name _ _ _ _ _ _ _ _ _ _ _ _ __ Date _ _ _ _ _ _ _ _ __ ~':Pivi:SIP"'ia;id'R\.Ile'<()f'EXpQnfJnts;>" ',' , ',··.:Eiif!r6is~;!)7' LeSSOf/S 10,3 to to,5 A. Multiply or divide. 1. x S • x7 2. 0 6 Ii' 3. b4 cZ • bS 4 alOif a5 b& 5, 3x 3y2 • 2xy2 4 6 6. 20n4 c 4n c· B. Find the missing term. = 1. e' • • 2a 3 3. 5. a 2 b 3 • _.... ,16 =: = IOa7 a 4b 4 *x2.=xs 2. ~ 4. 6y' l8y' • n 2x S = n 3x 5 6. C. Find the missing term. a' 1. . . = ~ ~ ~ • ~ ~ ~ .~ 0 ~ ~ 0 E ~ •2 c » Q "1> Y 3 . ~=4"s 2n' .. 4. 24x 10 • = 4x9 6. ~=x3l y ~ , " ~ 0 ~ D 0 @ 1" 2. ~ = y6 ~ 1 1• (13 ~ 8 I " 1 Chapter 10· Exponents and Funclions 57 Narn~ ~"'~''---," .. _'''':-''c:,-=::"-;c~c:.'-,,.'~C"'"-'.'~'"._ _--'----' , ".- .' ." ": ''', Date _ _ _ _ _ _ _ _ _ _ _ __ .. - , ' . '111' •.. 8x+ 5y+ "17x= -9x+ 5y 1. 9x+ 4x= 14. 3.5y- 7.2y= 2. 15. 17x+ X= A.7y- 2.3y= 3. m+ (4m) = 16. 30 + 5c - 90 = 4. 7x- 8x= 17. 2x- 9x+ 7 = 5. 18. 140-190= 7x-8-11x= 6. -0+ 90= 19. 3x- 3y- 9x+ 7y= 7. 6xy+5xy= 20. 17x+ 4 - 3x= 8. "9m- m = 21. 3x-7y- 12y= 9. 150+ (110) = 22. 110-130+ 150= 10. '14x+ 13x= 23. 17x + 50 - 3x - 40 = 11. 5x'y+ 13x'y= 24. 6x+ 9y+ 2x- 8y+ 5 ~ 12. 21xy+C9xy)= 25. 3xy + 4xy + 5x' Y + 6xy' = 13. 17x+ 1 ~ 26. "25y - 17y + 6xy - 3xy ~ Published by lnst[lJCIIonoI Fair, Copyright protecled. I Page 12 Q-7<12<!-1788-J Mgetxo J.. - Name ________________________ Date ___~ ______________ I .p , ,:Jj • What's Not to Like? I Simplify ea~h expre~ion by combining like terrrls. Circle the expression in each ; problem that does not belong. Place the letter above the problem number below. ~ ~ ~~ ~ ~~ ~~ ~~ 1. A. Sf + 3r + 91 - lOr E. r+t-8r+ 13t I. -r+4t+ 10f+8r 2, D. 12x- 3y+ x+ 2V E. 3 (4x-3y) + x+ 3y F. 4 (4x- 20 - 3x+ 7V 3. E, 4 (y - 7x) - V O. 7 (4x + y) + 7y I, -30x - (2x) W. 4x+ y- 7x- lOy 4, U.6(x- v)-3(3x+)I) V. 3 (3x- y) - 6y 5. Q. 3 (r-1) -4r+ 5 X. 2 (3 - 2r) - 4 (2 - " Z. -r + 7 + 3r - 9 - 2r M. 10 (x + )I) + x + y N, "(x + y) - 2 (x + 0 B, 3 (0 + 2b) - (b + 2(1) 7, A. 3 (2b - a) - (20 - b) 8, I. '0 (a- b) -2 (a-b) + 8 (a-b) U. 6(a- b)- 4 (0- b) + (0- b) O. (0- b) -(0- b) + (0- b) 9. R. 3 (x - y) - 2 (y - x) 10. L ~4 C. 2 (a + 2b) - (0 - b) S.2(x-y)-3(y-x) (X+ 2 (5xy- x) T. 3 (y - x) - 2 N. -2 (3x + 3 (lOxy+ 3:<) M. -4 (x+ 5 (3xy+ x» Two expressions in each problem are 2 5 8 1 4 7 Page 13 10 ex - y) 3 6 9 r Dote _ _ _ _ _ _ _ _ _ _ __ Nome _ _ _ _ __ Exponents 1. Write in exponential form. 4· x· x· y. Y· y=4x2y3 The cube of c - 4 6. 2. =(C_4)3 The quotient of 3 and 1I1e cube of y+2 mn*mn-mn·mn 8, (-x) (-x) (-x) 4. 5(0+1)(0+1)(0+1) 5. (a + b) squared 9. 3' ab' ab' ab' ab 10. The square of = = ='3 II. Evaluate each expression if x '1 , Y 2, Z 5X 2 j2 = 5 • x· x· z· z = 5 • ~1 • ~1 • -3 • "3 = 45 6. lOzS 3. 4y"z 5, -(xyz) 10. llx2 Page 26 x2 y - 3 .ame ________________________ Date _________________ Adding and Subtracting Polynomials (x" +2x'- 8x) - (2x 2 + 7x- 5) = x3 + 2x" -8x+2x"-7x+ 5 '" x" + 4x" -15x+ 5 1. (4x + 2) + (x - I) = 2. (5a-2b+4)+(2a+b+2)= 3. (30 + 2b) - (a - b) = 4. (x" + y2 - 5. (40 2 - ab) - (x" _ V' + ab) = 5ab - 6ti) + (lOab - 6a 2 - 8b') = 6. (4x2 -2x-3)-(5x-4)= 7, (4a 2 - 4ab - ti) + (0 2 - ti) + (2ab + a 2 + b 2 ) = 8, (4x' -6x2 + 3x- 1) - (8.0 + 4x2 - 2x+ 3) = Q, (a + 2b) + (3b- 4e) + (50- 7e) + 3b= 10. (x 2 _ 2XV + y2) _ (x2 _ 2XV + V') = 11. ex+ 3)1) + (3x- y) - (x- V) = 12. (2x' + Sy' - 22) - ex' - 0 - 22) + (4x 2 - Sy') = 13. (2x+3)+(2x"+x-5)= 14. (2y+ 3x-4) + (9- 8V-5x) + (3x+ 4y- 2) = 15. c20 + 8) - (3y2- 4y- 6) = 16. (7y+ 4x+ 9) - (6x- 8y+ 11) = Find the perimeter. 2x-6 17, 18. , x+3 x+3 2x- 6 x-2 3x-7 Page 27 Name _________________________ 0016 ________________________ Monomial Quiz Polly N. Omlal did not understand the rules of exponents when she completed the Monomial Quiz. Find and correct the 10 errors Polly made. Monomlol Quiz 1. (4C)2 i: i Name: =8e' Polly 1l. (_xy2)3 (2x' yi =-4i'y" 2. 4(e)' =4c2 12. x 2 (2xt) (4z5) = -3'x3Z6 3. (2cPb) (40ti) =B0 3b3 13. (3pq2r) (g32 r) _- pq4r' 4. 14. (-x) (2):0 (3xyz) = -6x3 y2 z (4pq) (_p2 q 3) = -4p 3 q ' 5. 2x (_xy) (_y') = 2x2 y3 15. 2x2 (xy2)' (XZ2)2 = 2X4 y' Z4 6. (<looi = 16be' 16. 7. -abc' (edl = -0005d 3 17. (-x) (_x 5 ) = 8. -2 (3x)' (xy) (2x) 2 = -12x3 Y 9. 0(20 2)3 =60 6 10. 35 (2s1)2 = -12s2f lB. (2u)' (u'v)' (>0 = 2u'v"w :I' (5x2) (7xy3) = '35x 3V3 19. (-x 3 y) (6xy)2 = -6x' V3 20. (<Iff') (2rf) (t2) = sr< f' Explain to Polly how to calculate the power of a power, for example, (5x' l Page 28 Name _ _ _ _ _ _ _ _ _ _ _ _ __ Date _ _ _ _ _ _ _ _ _ _ _ ___ Multiplying a Polynomial by a Monomial -20 2 (9 - 0 - 402 ) = -2a' • 9 - (20'· a) (20' • 4a') =-lSa' + 2a 3 + 80' (x + 2) (2X2) = 2x' • x + 2X2 • 2 = 2i' + 4x' l. 2 (x" - xy+ Sy') = ';, = 2_ -2n (4 + 5n3 ) 3. c'd (c 2 cfl + 2cd' + a) = 2xy' (2 - x- x2 y) = 4. 5_ (a' - 3ab - 2b 2 ) (2ab) 6_ an (Sn 2 - 7_ ew' z - 2wz + z) (-z') = 2n) = i: = ,, ., B_ -3ati' (a 3 b' - 2a'b) = 9. 4x'y(9x' - 6xy2 - 7) = 10_ -6k'rrf (2k - 3m + 4km - k'm') = 11. -n' (n + 4n') = 12_ (4x' - 7x) (-x) = 2x'(i'-2x'+Bx-S) = 13. 14. (6x 3 ) (3x' - 1) = 15_ (6x- sx" + 8) (3x) = 16_ -Sx' (2x 3 + 3x' - 7x + 9) = Find the area. 17_ 18. x+4 2xL----1 A triangle has a base length (b) of 2x + 4 and a height (h) of Sy. 1 (Area = - bh) _I 2 Page 29 , .• '(,s' - s+ 3) - 2 (8 2 - 8 + 3) i" S' 52 S· 5 + 5' 3 - 2 • 5· - 2 (-5) - 2 • 3 ~ 53 52 + 35 - 25 2 + 25 - 6 = 5' -35 2 + 55- 6 · 1. (z - 3)(z+ 3) = 'I! " " i: 2, (3t-2)(t-3)= 11. (2x- 1) (x' + x+ 3) = 3. (0+5) (0+ 5) = 4. (0+ b)(2x+ y) = 6, (4x - 5)(4x + 5) = 7, (1.6n - 9)(O.2n - 5) = 8. (2e + d) (c' + 2c + 2d) = 17, (x 2 - 3) (2x' + 3x+ 5) = Page 30 F Name ______________________ · " Date _____________________ Multiplying Binomials Using FOIL 1 I. 4. II I 1. 2. 3. 4. first Quter Inner last (x+ 5)(x-3) =x' x+ x ('3) + 5' x+ 5 ('3) = X' - 3x+ 5x- 15 = X' + 2x- 15 I " I W I il .,, 2. 1. (X+ 2)(X+ 3) = , ' 10. (2x+ 5)(4x- 3) = , 'I ·• . .' 2. (y+ 7)(y+ 4) = 11. (n - 7) (3n - 2) = 3. (x- 8)(x+4) = 12. (5x+ 2) (3x-7) = 4. (x- 8)(x-4) = 13. (4x+5)(2x-3)= 5. (y- 4)(y+ 5) = 14. (-x - 4) (4 + 3x) = (x- 9)(x- 2) = 15. ex + 2y)(2x+ 3y) = 7. (2x+4)(x+3)= 16. (6x - y) (3x - 8, (3x+ 2)(2x+ 5) = 17. (4x + y)(3x - 4y) = 6. 9, (4x- 9) (3x + 1) = 20 = 18. (50 + 3b)(40 - b) = Page 31 Dole _ _ _ _ _ _ _ _ _ _ __ (x+2) (x~4) = x2-4x+ 2x- 8 = x'- 2x- 8 F ColumnA I I l Column B 1. (x-4)(x+ 3) I. 25x 2 + 70x + 49 2. (x+ 2)(x- 6) T. 6x 2 -7x- 3 3. (3x + 1)(2x - 3) O. x 2 -x-12 4. (x+ 3) (6x-1) A. 24x2 + x-lO 5. (5x+ 7) (5x+ 7) l. O.6x' + 15.2x + 5 6. (4x-l)(4x+ 1) R. x2-4x-12 7. (O.2x - 3) (O,3x - 2) M. 6x 2 + 17x- 3 I !,, a 8. (3x + 1)(O.2x+ 5) S. 24x2 + 58x+ 35 9. (4x + 5)(6x + 7) I. O.06X2 - 1.3x + 6 10. (8x - 5) (3x + 2) N, 16x' - 1 All but one of the answers Is 32561471089 What is special about the factored form of answer N - 16x' - 1? _ __ Page 32 Name _________________________ Dare ________________________ Special Products (2x+ 5) (2x- 5)" 4x' - 10x+ lOx- 25 =4x' (0 + b)(o - b)" 0' - b' , II 1. (x+ 3)(x- 3) = 10, (2X+ 9) (2x- 9) = 2, (y - lOHy + 10) = 11. (7x - 5)(7x+ 5) = 3, (0+4) (0 -4) = 12, (x+ y) (x- y) = 13, (5x- y) (5x+ y) = 4, (x+ 7) (x- 7) = 5, (2X+ 1)(2x-l) " 6, (5y - 6)(5y + 6)" 7, (4x+3)(4x-3)= 8, (3n+ 7)(3n-7)= 9, (3e + 4)(3e - 4) = 14, (2x+ lly)(2x-l1y)" 15, (3x - 7y) (3x+ 7y) " Thinking About It 19, The product of the sum and difference of two terms is equal to the _____ of the squares of the terms, 20. Why is the product of a sum ond difference of two terms a binomial and not a trinomial? ________'____________________ Page 33 "] Name _________________________ Date _ _ _ _ _ _ _ _ _ _ _ _ __ Squaring Binomials 1, (x-sf= s, (x-si= I, 4, (Sn + 1i = 11, (4x- vi= 12, (6x- 5vi = , !i , i: 5. (y- lOi = 13. (3V-5z)2= 14, (70 + 2bi = " 8. (20+ 5)2 = 16. (50 + 3bi = Page 34 Name: _ _ _ _ _ _ _ _ _ _ _ __ Dare; ---- Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1 Wben we want to express a product of the same number like3· 3·3·3 we can usc a shortcut notation 4 3 ,-~ ..... The exponent teUs how many times the base is used as a faetur in the producl Exercise #1: Write each of the following in the form ofan expanded product. (al xl = (bl 4' = (c) (2x)' = (d) {x+5}' = Exud:se #2; Express each of the following with an equivalent expression involving exponents. (a) z·z-z·z·z= (b) 6-6-6 = (d) x'x' y. y. y= (e) (x+ y}(x+ y)(x+ y)= Exercise #3: Consider the product shown below: (al Write both parts oftbi, product as extended products. (b) Write the product x4 involving an exponent x 3 as an expanded product and in tenns of an equivalent expression Exercise #4: Express each of the following product<; as a single variable raised to a power. (a) X' -,,' (b) x··x' (d) I-l-y" Al'8e\># I,UIlitN6,,·~Algc:b&l-Ll Tru:::Adiagtoo A.lg1:bto\ Projeo:t. LaGrangC\-ith;. NY 12540 \ <i ! We also must be able to divide monomial ~pressions that have the same base. understanding how thi. process wow i.the following: The key to Any quantity divided by itself, except for zero, is equal to I. , Exercise #5: Consider the quotient x 3 • X (a) Fill in the following: x' = x' . _ __ (c) Rewrite the quotient as the product ofwo fiaCtiODS. one of them being equal to I. (b) Rewrite the nwnerator of the quotient using (a). (d) Simplify the quotient using the Mulliplicotive Identity Property of Real Numbers. Exut:.ise #6: Fill in the blanks in the box below: ExPoNENT PROPERTIES For any real numbers a and b~ Exerch'e #7: Write each product or quotient in its simplest form. (a). 0 4 ·a4 ;;:: (b) i'·x'= (e) 34 _3 2 ;;:: Cd) y2'i= (h) (i) 2'·2·2'= (1) YS''''''y= (g) (x')' = (k) A!gebTa l.lfni!~-~ Ala<bla-L! The Arl~oll IJgd;ni P'rojr.::t, l..aOrangeville, NY t254D 6y' = z, = " 0) Ce) .. lOy' (1) b' = b' :;?y5 xSy2 = (i'l' = .' Name: ,_ _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ _ __ Exponents and Their Properties - Multiplying and Dividing Monomials Algebra 1 Homework Skin 17. Express the product with exponents. I. a·a·,a·b·b= 2.(2.)(2.)(2.) = 3. (2x){2.)y.y= 18. 19. Express the product in simplest form. 4. h' ·b= Reasonlng SimplifY. 5. l·y') ;;; 6. 2 l X 'X 7. 1'/,4 8. y.y= 4 x' 20. ",= 21. z(2z)'(2z)= c>d ·x4 = 'n= 2 9. 0 10. ";.•'= 'a = 23. 7 , x-·x -x• 24. II. Z4 'Z4;;;; Express the quotient in simplest fonn. 13. x' = 25. x' Determine True or False for each. State the reason for your answer. 15. 27. 16. 28. Name: ____________ --------------------Exponenlll alld Their Properties - Multiplying alld Dividing Monomials D~ Additional Practice Problems Reasoning Skill Find the product. I. x'·x' ; 2. y'Z'y= 3. y'.y'.y'; 4. a·a·a·a= af, _0 4 5. 6. 7. o 4' 20. -; 2J. -; 22. -= 4 x' x " X' Simplify. - ; 33. --; r 24. -= 4x 13 _,2 , 34. a.b3 -= 35. (ab)' ---= ab' y' 23. 4x' 32. b' =: 8. X 4 .x,5= 9. l-y'::::: 10. / . / =; 11. w'~'W'W2= 12. p~.p3.p~ B. X4·~·X= h'·ll= 15. X4 25. 26. yz.y::. 17. x-x-x"'" 18. mlJ _m17 :::;:; ;c4y2 36. x'y' 37. 4x4l 2xy' = ; 38. xiJ 'x!;:::: I' 39. --= z' -= z, z' -= 5' 28. -= 29. p' --= p' 30. 3l. b" ·b' = 40. 5 = x' x' " _Xl = 16. t' " 27. 14. 19. . = ,'J .f7 ::::::; rS ,,.3 Find each quotient X'ly7 z= xlyzJ x4 / , x4y2 41. ---; -= 42. ~--= x" 43. 6' 6' x lO -= xy' x'-y' x', 1,' = Alg.;bm l, Uoit.tl<i-Quadntle Algcbm -Lt TheJufu>gtoo Algl:bra Proj~ ~11e, NY 1t540 \ 'I 'I Name; _ _ _ _ _ _ _ _ _ _ _ __ Date; ---- Zero and Negative Exponents Algebra 1 In our last lesson we leamed how to simplify products and quotients of monomials using Jaws of exponents with positive integers. But, zero and negative exponents are also possible, ExercIse #1: Recall that ~ = x..... x (a) Usit\g this exponent law, simplifY each ofthe following. (b) What must eaeb of these quantities equal, assuming none of the variables equals zero? Exercise #2: Simplify each of the following: (al 1250 = (b) (c) 5xo = (2y)" = We can investigate negative exponents in a very similar fashion to the zero exponent The key is to define a negative exponent in such a way that our fundamental rules for exponents don't need to change. 2 Exercise #3: Consider the quotient -~y x (a) Write this quotient using the exponent Jaw from Exercise # I. < (b) Write this quotient in its simplest form without a negative exponent Ex£rcise #4: Rewrite each ex.pression in simplest terms without the use ofnegative exp'onents. (b) x-' = (c) 2-3 = (d) y-10 = \~5 Exercise #5: Rewrite each ofthe foUowing monomials without the use of negative exponents. (d) y_,-, ~ x NEGATIVE ANI> ZERO ExpONEN'IS [r a is any integer- and (l) ,r ¢ 0 then c~1.. (3) x' - ~ ,·=1 5 Exercise #6: Which of the following is equivalent to x_s Y-l ? x y l (I) -= l. (3) , x'y' I (4) ...,-;; xy (2)L i' Exercise #1: Rewrite the following expressions without negative or zero exponents. (1)) 4' = (e) (g) 3x· ~ (i) ~, = x -r' = (I) (-1)· = , -2 (I) r t = Exercise #8: Evaluate each of the foUowing expressions using the -values a = -1 , b::::: 2 and c:z.: 3. Use the Sl'ORE feature on your calculator to aid you. (b) (aber' = Name: -------------------- Date: _ _ _ _ _ __ Zero and Negative Exponents Algebra 1 Homework SldIIs For problems 1 tbrough 36, rewrite without zero or 17. -3' = 33. x' 2y-' = negative exponents. 34. -3; y'" = 19. (-3f' = xOy-3 35, ~= 36, 2x-'y-4 = -I1'-' = 21. ( 2, 6, 2-4 = 22. /1)-' = l"3 1 7. Z-, = 1 23. 1-6 = Use the STORE feature on your cakulator to belp evaluate the following. 37, y-' for y=2 24. (-5" = 8. • = 4 9. (-3r'= 26. -2-' = II. :5x-4= x' 12. y-J = 27. (-2f' = , 40, (x+3r' forx=-4 28. (-2r' = 29, (-T'r' = . 30. 15. T' = 41. forx=-1,y=2 4 2 42. ( x, y ')' for x=-,y=-~ 3 43, 16. (16x'y-'j' = x~y 7 _ 2 4 x""x:Y fo(x=-,y=- S 3 A1getna 1, U!litfl6~~ AlgdJI'lI-ll Thc~AJgdlta l'tvjed..1..eGmn~r~ NY 11l4n 1'\ 1 Reasoning Fill ill III. missing 0 for oach of Ihe foRowing. 55. Evaluate each of the following products: (a) 2' ·2-' = 2 (b) 5 .,'= 63. (-4)" =0 (0) 10'" ·10' = 46. _I ,,[J-> 25 56. Which ofthe following is correct? Find the value of x that makes each statement true. 48. 6-'=~ 66.2'·2'=2" 49. uP 1 10,000 Explain why the other choice is incorrect. True or False Write the answer to each of the following as a single number. 57. 52. [-1+(5+2)"]'= 58. GJ' =2 (~r =-~ 59. (-Zr' =.!. 4 A1gebn t, Unit 116 -~ A,lgebr./. - Ll '1'b<! A~ll Algebrn ~ l.!!Gr.mgC'olUe, NY 11540 Name' _ _ _ _ _ _ _ _ _ _ _ __ Date; Combining Like Terms Algebra 1 We have already seen the process of combining like terms when solving linear equations. In this -lesson we will broaden our understanding of what constitutes like terms and how to combine them. F"ust. we review the reasoning process behind combining like linear terms. Exercise #1: Fill in tho blanks for each with the real number property that justifies the particular step. (I) (I) _ _ _ _ _ _ _ _ _ __ 6x+2y+3x+4y~6x+3x+2y+4y (2) ~(6+3)x+(2+4)y (2) =9x+6y Exercise #2: Combine each of the following like tenns using the Distributive Property, (a) 2x+7x= (b) -5x'y+2?y= Clearly like terms are those monomials in an expression that have the same variables raised to the sarne power. We should be able to combine them mentally by fIrst identifying like teons and then summing aU coefficients of those tenus. Erercise #3: Combine an like terms in the following expressions. (a) 8x+4-6x-7 (b) 4y'-20+3y'+S (d) h'+3x-7+5x'-8x+3 (el -3? +9x-6+4x' -2x-8 Exercise #4: Which of the following expressions cannot be simplified? (1) 3x+6x (3) 3x+6y (2) 6y-3y (4) 2x' + 7x' Algebrn 1, u.ut #6 - Qu,adt.w.e AlgcbA - LJ TM Arlington Algch!a: Projcd, La~1k, NY l2540 (e) 3w-2(3-5w) We will oftentimes be asked to combine terms either in sums or differences. Differences can be particularly tricky because subtntction is not commutative, mea.ning the order in which you do the subtraction will change the result. Eurcise #5: Which ofthe foUowingrepresents the sum of (3x1 -3x+8) and (_Sx 2 +4x+2) 1 (I) -8x' -x+1O (3). 2x' -x+l0 (2) -2x' + x+ 10 (4) 8x'-7x+6 (I) 3x'+3x+3 (3) Hx' + 13x+ 3 (2) -3x' +3x+9 (4) 3x' +13x-9 "ExercJse#7: When (4~-8x-3) is subtrncted from (il-2x+l) the result is (1) -3x' +6.>:+4 (3) 5x'+6x+4 (2) 3x' +6x-2 (4) -3x' -6x-2 - - - - - - - - - - Additional Classroom Practice - - - - - - - - - Simplify by combining. l. 5x+2y-I!x+7y= 2. (2x-4)+(5x+9)= ). (2x-Y)-(5x+y) = 4. 3x2 +4x+2+2x2-5x-1 = 5. 3x2-5x+1-4~ +2x+3= 6. 5x z +8x~(3x2- - 2x) = 7. (x'-2x)-(3x'-7x) = 8. From 4x 2 + 2x- 3 subtract 9. Add to. Subtract 4a 2 + 2ab+ 3b7 _a 2 _3ab+b 2 3/ +4y-5 5y'-Zy+! Xl - AJgcl!!a I, Unit fJ<i -QwdnI.I.kAJgwra-Ll The Ailingtoo Algebra ho.l«t, UlGtangevilk!, NY IZS41J 3. 12. How much less than 5x 2 - 3):+ 2 is Y+5? Name:~~ _ _ _ _ _ _ _ _ __ nate: _~~_ _ __ Combining Like Terms Algebra 1 Homework t5.combill~ 2x+4.-3;c+2.5+~ SkiD Combine as indicated. 16. I. 9%+7+2.1'-4= 2. 4x-3-(3x-2)~ 3. (x' -4x)+(3x-5) ~ 4. 4t-3t' +5+{-2t' +3t-5); 5. (12t-9) -(51-3)= 6. eombiIic: a 2 -Sa+25;-a2 +6a+9 19. Subtract 5x+3y from 4x+ y (0-60)-(40+80)= 20. From 5.1'+3y subtract 4x+y 7. (3XY' +3xy-4x'y)-2x'Y+ xy' = Applications 8. (3x-4x')+(-4x-.') = 21. Represent the perimeter of a sqWIJ:e whose side length is given by the binomial 4x+6. 9. 1+3y'-5y'+6y'= 10. Sx 2 _4xy+6y 2 _xi + 3xy+ 2y2 +xy;:;; II. a1. +3'a+ 5+ 2a'2 -4a-l-·5;: +20= 12. _r2 s2 +2i2:VZ ~6r2$Z _5t'2v2 :::: 13. a+8c+5b-c-Sa-5b :::: 22. Represent the perimeter of a rectangle whose width is y and whose length is the binomial 2y-7. - 23. Write !he length of the arc ABC as a binomial involvirig x, y, and z. A 2xy-z B 14. [-4y+(1O-5y))+2- Y= lxy+2z Alzd,Im I, Unit /J6 - ~ Algd:IR - 1.) Tb¢.Arllngtac Algdlra rroja:t, ~e«iU<:. NY 12540 c 24, Express the perimeter orthe triangle os. 29, binomial, HoW much greater than x 2 - xy is Sx"+IO>:v? .<-4y Sx+8y 30. What expression must be added to 3x2 ~ 5x + 4 to give the result 7xl -5x-<i'J 2:ct-3y 25. The perimeter of a triangle is given by the expression 12x" -4.<+ 15, Find the third side of the triangle if the other two sides 31. From the swn of 6x-5a.nd 2x+4subttact 3x-9 measure 4);? +3 and 5x-4. 32, Subtract the sum of 2):."1' 3x+4 and x 2 +2x-1 from 6x2 -2x+l Reasolling 26. RecaU that two expressions are additive inverses if their sum is equal to zero. Find the additive inverse for each of the following: (a) 7x-4 (b) c' -4c+5 27, What is -3a+5b decreased by 9a+ 2b 28. Byhowmucbdoes 4x-3exceed 7x+51 AJgclmll, UIDt&'6-~Al.6ebra-U 'l'hc Arlu,gtooAlgdlmProJeet. u~ NY 12S40 Name: Date: _ _ _ _ __ ---------------- Multiplying a Polynomial by a Monomial Algebra 1 In the previous lessons. you've worked with monomials and their exponent properties. In thls. lesson we will begin to work with polynomials, or expressions that contain more than one monomial. The most common polynomillls are binomials (those with two monomial lenos) and ttinomials (those with three monomial terms). First, we review the important real number properties associated with multiplying monomials. Exerdselll: Fill in the blanks for each of!he fullowing wi!h !he real number property that justifies !he particular step. (1),_ _ _ _ _ _ _ __ (2)_ _ _ _ _ _ _ __ (2) (3) Exponent Property of Multiplication (3) Exucise #2: Simplify each of the following products using reat number pcoperties like in Exercise #1, (b) (-4lz)(zyz') = ClearlYt we would like to be able to do this mUltiplication without going through each of these steps. It should be clear from the last exercise tbat you can simply multiply the coefficients together and then add powers on like bases, Exercise #3: Find the following products. (a) (4x'l)(zxy')= (b) (5r 2s)(2r$')= (e) (-3pt')(-6 p2t') = We now need to be able to multiply polynomials by monomials, You have actually done this before. as the foUowing exercise ~.U illustrate. Exel'cise#4: Rewrite the following without parentheses by applying the Distributive Property, (b) -3(2x-7)= Algebra t, UM Mi - Q.\llIdtsIi.: Algebra - f.A The Adinglon AIgdna rwj<:a, ~grnlle. NY 12540 (e) -(6-3x) = Multiplying monomials with variables over polynomials uses the Distributive Property in the same way. Exercise. #5: Rewrite the following products without parentheses by app~ying the Distributive Property. (a) 2:r(3x+4) Additional Classroom Exercises Filld the product: I. (2x'y)(-x'Yl= 12. 2:r(5x+7)= 2. (3yl){2y'l) = 13. 5ab( 4a'b+ 2<1b-4,,) = 3. (5ab'c)( 4a'b'c) = 14. 4xl (5x-4+2x 2 ):::: 4. (-2x'Y')( -.g2l = 15. 2xy' (3x' +4.g _ y2) = 5. (7 p'r't)(3pr4t') = Distribute and Combine Like Terms: 6. (-4r'x')(3r4x) = 1. 3(h-5)= 8. -6(x+4) = 9. 3x(2x+9) = 10. 5x(x-3)= II. -(x'-4x+7)= Algebrn 1, Um! C6 _ Quadrati" Algclm!. - III The: Arllngl.Ol:l Algebrn I"roj«t. L:iGr:mpille, NY 12$40 16. 5x(3x-2)-x(I-3x) = 11. 3x(2x-1)+2(2x-1) = 18. x(x+5}-2{x+5)= 19. x(x-3)-3(x-3)= 20. x(x+y)-y(x+y) = Name' _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ __ Multiplying a Polynomial by a Monomial Algebra 1 Homework Skill Find the -product: I. (6:xy)(-Zz)= 2. (5a')(-5 a)= 3. (-72)(+ 4. (2r 2-,') ( -4r'.) = 5. Hd )(Z4ad) = 6. (-2x)' = 7. (3a1b)' = 8. 5x(2.1.'-8) = 9. 6x(3x-tJ= 13. Distribute and Combine Like Terms: 17. 4-3(2x+5)= 18. (x'-3x+7)-(x'-6x-2)= 19. (3x'+8x-5)-(-2.'+4x-lO) = 20. x(x+ 7)+4(x+ 7)= 21. 2x(3x-4)+5(3x-4) = 22. a(a-b)+b(a-b) = 23. 2x'(x+4)-3x(5x+l)= 24. 4x(4%+3)+3(4x+3)= 2 2(.'-2X+4)= 15. 3x(2x-9) = 16. 5xj2x'-3x+7)= , . AJgebra I, Unit lUi - QIOOMie Algdml. .. U . l'b¢Arlinj,'lOIi AIgclmi I'n'lj~ LaOral!g6ilk, NY 12540 Applications 25. If the length ofa square can be represented by the monomial 3x then: <a) Express the perimeter of the square as. monomial in terms of x. (b) Express the area of the square as a monomial in tenus of x. 26. The width ofa rectangle is represented by w. The length is three more than twiee the width. (a) Express the perimeter cflbe rectangle as a binomial in terms ofw. (b) Express the area of the rectangle as a binomial in terms ofw. Reasoning 27. Simplify each of the following ifpoosible. ifnot possible, explain why, (e) x' -i' = (d) x' i' ~ 28. Determine each of the following products by writing them out in an expanded product form, The first is done as an illustration fur you. (xl)' = (b) =;(2+2+2 29. Fill in the blank for the foUowing Exponent Property: Algcbnt,Otmlf6-~~-LA The Arlingtoa Algebra. ~"" ~~IIe, NY 12m (d) (x')' ~ Nwne' ____________________________ Dale' _ _ _ _ _ __ . Multiplying Polynomials Algebra 1 In the last lesson we worked extensively with multiplying polynomials by tnODOmialS. In this lesson we will generalize this process so that we may multiply pelyoomials by pelyoomials. The first exercise wiU illustrate the real number properties associated with this process. Exuclse #1: Fill in the blanks below with the real number property that justifies each step. (I) (x+2)(x+4)=x(x+4)+2(x+4) (2) (3) =: x'x+4·x+2'x+2·4 (4) Exttrdse #2: Using real number properties~ find the products given below. (a) (2x+4)(3x-I)= (b) (x+7)(x-5)= (0) (2y-3)(4y-6) Multiplying two linear binomials is such an important skin that a mnemonic has been developed to help remember it: FOIL - Multiply the First. Outer, Inner, and Last terms of the two binomials together and then combine the like terms. Exercise #3: Multiply the following binomials together either using a method as in Execcise #2 or by «FOILing" the two binomials. <ol (x+4)(x+l)= (b) (y+3){y-5)= (0) (2x-7)(3x+2) = Exercise #4: Which of the foHowing is equivalent to (X_4)2? (l)x'+16 (2) x'-16 Alllcbtll: 1, l1nk#6 -~ A1gdmt- LS Tb<;o Artmgto,l(l AIgdI~ ProjOd. I..4GtMgeviiff., NY 12540 (d) (x-S)(x+S) = . . We can also multiply polynomials together that have more than just two tenns. Each term in the first polynomial must multiply each term in the second polynomial for the distribudoo property to occur. Exuds. #5: Find tha following produol by distributing the binomial over the trinootiaL Since multiplication of these higher powered polynomials can become confusing, it is he1pful to use a multiplication table to carry out the product. Exercise #6: Use the following table to help evaluate the following product -4x (X-2)(3x 2 -4x+7)= x 7 I. • -2 .• Additional Classroom Exercises L (x+5)(x-2)= 7. (3x+l)2 = 2. (2X+S){X+3) = 8- (x+6)(x-6)= 3. (x-2)(x+3)= 9. (2x+I)(h-I)= 4. (lx-S)(2x+4) = 10. (4x+5)(4x-5) = 5. (x-5)' = I!. (3x-Zl(2x2 +5x-l) 6. (2x-3)2 = Algebra I. Unitll4-QlW;l.ra.ticAlgw.a-LS TIm ArlingtOD /Jgdml ~ r..~Jl~lle. NY 115((1 Name: _ _ _ _ _ _ _ _ _ _ __ . Date: ---- Multiplying Polynomials Algebra 1 Homework Skill Fmd each or the following products in simplest form. . 1. (x-2)(x-3)= 17. (x+4}(x-4)= 2. (y+6)(y-l) = 18. (x-1)(x+7)= 3. (0+5)(0+3)= 19. (y+2)(y-2)= 4. (r+4)(r+5)= 20. (4x+3)(4%-3)= 5. (2):- 3)(x+ 5) = 21. (5+ %)(5-.) = 6. (2.-9)(.1:+ 3)= 22. (3- y)(3+ y)= 7. (y-7)(y-2)= 23. (4.1:+1)(4%-1)= 8. (20+5)(30+1)= 24. (x+2)' = 9. (%-3)(%+8)= 25. (x-6)' = 10. (x+5}(%-9)= 26. (4):+1)' = 11. (3x+W)(x-5)= 12. (5+x)(x+7)= 27. (3x-2)' = 2 28. (6x+1) = 13. (6-x)(4+x)= 14. (3-2x)(4+3x)= 15. (9+x)(8+x)= .. 16. (2-x)(3-5x)= Applications 29. If the side length of a square is given by the binomial (4x - 3) then wbich of the following gives the square's area? (1) 8x+6 (2) 16x'-9 Algcblll. I, Utti:t#6- QWtdm1ie Algwm-LS AJ~ ~ LaGWlgt::Vi1k, NY 12540 The ArliogtOn (3) 16x' + 24.+9 Reasoning 30, Find the products of the followingpolynornials: (bJ (x-3)(x' +2>:+9) = <OJ (3x+5)(2.'-4x+3) = 3 L Consider the following expression: (x+ 2)' (a) Rewrite the expression as a produ¢t of three binomials. (b) Evaluate this product by multiplying the last two binomials in part (a) to fonn a trinomial and then multiply this trinomial by the fU'St binomial. 32. Rewrite the following expression withQut the use of parentheses. Keep in mind that you must multiply the binomials together flISt and then ~rfonn the subtraction. (x-3)(x-5)-(x+I)(x-4) = Algebra t, unlilio - Qwldmie Afg" - L5 Thl!: AtliJlitoo A1~r.t PrtIjtd:,LAGrangeviIJe, NY lZ54() ;! I 0 , FIND A MATCH Solve any equation in the top block and find the solution in the botlom block, Transfer Ihe word from the top box to the corresponding bottom box, Keep working and you will get another joke, , ® A , HER (2 U2)2 (-k 2)2 @ ® BROTHER , (-3u3)iT AND I@ (9 u 2 k 2)2 THAT (-k 2)3 (-4U 2)3 (7 k)2 I~@ , 3 {4u k)2 i TOO ® , ® HER @ C2 u 2 k 4 )3 REASON @ @ A HE (2u 4 I I® THE (-2k)5 / IF C6k4)2 @ CANNOT - @ (5u k 8)3 US k 3 )2 (-8 @ BE FALLS (u 2 k 3 )s @ TRIPS ~, (-3 u k)5 ! HER " i@ (-u 10 k)21 CANNOri@ ~ ", HELP @ i (-u k)9 (-2uk)7 BROTHERi@ LADY ~ ASSIST --+------11 -----+- - - - - 1 - - - - - - 1 :J) iT (10u 2 k)3 r2uk4)4 36k e -8u6k'2 U'5 -64u 6 -27u 9 -k 6 k4 1000u S k 3 -32k S 16u4 k 1S 81 u 4 k4 4u 4 u 20k 2 '64u 10k 6 49k 2 -u 9 k 9 -243u 5 k 5 16u 6 k 2 I, m 6'~ Q", ~ :£ ;f:::; ~:c "m g ,;:j C , 0" ~> ON 'N <'~ ..... '" W Rl DIRECTIONS: Figure oullhe value of each expression below and find it next to a dot. Connect the dots in the same order as the exercises are numbered. Be caratullo . lift your pencil and begin again each time you see the Inslruction "LIFT PENCIL." @"O n:t> iil::n Can You Build This? ~" ~6 "m eO: -32. ~5? g:iE 0-81 Sf~ ::t "0 -1 1 -0 0 9 0 ~ I',j -27 -0 0 4 1 - 0 216 125 0 0.0081 00.01 .- • ®Gf Lift Pencil CD (-2}3 ® (0.5)2 8. -1 16 • 0 0 .16 -64 125 256 64 9 49 0 .25 .CD 72 ® 24 ® (-6)2 036 100 2387' • -64 343 -4 • • 272 125 0 2197' 16 -, 625 10,000' .- 0.0001 • -27 -0 -1 32 ®GY @ (-10)4 0 -8 YOU WILL CREATE AN INTERESTING STRUCTURE. CAN YOU BUILD IT? 343 0 64 ® (O,1f Lltt Pencil CD (-2)6 ® (-8)' ® (~y ® (0.3)4 Litt Pencil , '_J CD CfsY ® (iY ® @ 43 (0.01)2 ® (-100)2 l.ift Pencil CD (-1)7 ® (~y ® (-O.1t @16 1 LIft Pencil CD 13 3 ® (-0.09)2 ® (_1)'9 ® (-2)6 ® (ly !top '8 , .,:' ,i ,: .~. ".,.. --...... \, '. ~~, ; Write a fraction (or 1) lor each power. For each set of exeR:ises. there is one extra answer. Write the lel1er 01 this answer in the corresponding box at the right 1 .. Answers 7-2 2-3 ®~ @ 4~ 3-2 ©~ ® • · 2 ), . . 4-3 ®~ g-2 '®2~ 2-5 3-4 i • · 12-2 5 • • , • I i, 9-3 460 ®2doo 1oo-'l 1 81 ® 10.~0 ® 11 Q) 1 32 © 5;2 _1 10 ® @ 1,000,000 1 5-4 (f) scio © 15° @6~5 @ 1~8 1 . 8-2 10-5 4-4 CD 1 256 @ .L @ 1 100,000 ® 196 64 1 9 r3 ®t25 ®t E 96 ® 11. ® 8\ 15-2 ® 3!3 © 1 ® 3cio 2-5 ®6~ . 16-1 ®i2 @ 1~9 ® t~ 11° CD2~ • , 10 13-2 6° 1000- .f, @ 5-3 • 1 Answers 10-1 20-2 @ 8 a-3 ®4~ 4 6 2 7 2-7 15-1 · 181315110 \411 1 \9\61 1~ 10'-4 3 7 t CD lObo ®6~3 CD 1 CD 7~9 PRE·Al.GEBRA WITH PIZZAZZ' @Creative Publicaliolls 63 Why Did The Farmer Open A Bakery? TO ANSWER THIS QUESTION: Express each product below as a single power 01 10 or 8. Draw a straight line connecting each exercise with its answer. Each line will cross a number and a letter. The number tells you where to put the letter in the row 01 bo)(es at the bottom 01 the page. 104 '103 10-4 • 10- 2 8- 3 • , 102 .10. 10-2 '103 • , ) "" ,10- 5 • 105 • 8' 8- 2 • 8- 7 • 8- 5 • 8- 6 • 84 .8-8 .8- 12 ® (j) @ • 1 CD 6 • 10 .8- 1 .8- 7 @ .1O~7 .10- 6 .8- 2 CD .10 • .8 2 ® @ ® • .107 @® 8·8 • 84 • 83 • 1 2 3 4 5 6 .104 @ 103 .103 • 10- 8 • 10 • 104 • 10'-9 • 8- 6 '8- 1 • 8 CD • 106 '10- 2 • 4 • 87 • ".'/0< 88- 1 .8- 2 • 8- 5 • 3 • .87 .8- 3 .10- 5 .103 7 B 9 10 11 12 13 M 15 16 17 • PRE·ALGEBRA WITH PIZZAZZ! © Crcah'iD Pu!)liClll1()flS 65 8i @-o . ,,:Jl it. rn ' ~f: >Q "m eCll How Did Slugger McFist Get A BLACK EYE? TO ANSWER THIS QUESTION: Express any quotient belpw as a decimal numeral and find this numeral in the code key, Notice the letter next to it. Print this letter in the box at the bottom of the page that contains the exercise number, Keep working and you will discover the answer to the title question. iVl\ 10-8 CD 10 5 -i'- 10 2 = .'" §~ ® 10 2 10 5 = ~:J:l !;i -I -!- I " ~ I:j @10- 6 .;.10 2 = @) 10- 1 .;. 10-3 = @10 2 + 10- 7 = ® 10 6 .;. 10 = CD 10-3 + 10-3 = III ~ 10-9 = @ 104 ® @) 10 10 _ 10 20 @.2Q.. @ 0.0011 E @104_ 106 0.01 @J.!L= 2 10lH 10:- @ 3 10 103 @10 15 100lA = 1000lS .;. @1O- 5 .;. 10-2 _ ® \,,~. 0.11 T 110 10-1 10 0.00011 G 10-1 10-5 _ 10 5 0.00001 @ 10-5 _ 4 10-7 .;. 10 14 = 10 = 10-3 = @Ho.;. 106 = ~/ I 0.0000011 C _ @ 10-2 = ® 10 .:!.Q. 0.0000000001 U 0.00000001 10-3 103 - 100,000 I L 10,000,000 IB 1,000,000,000 IV ,:,.) ~ Chapter 2 ALGEBRAIC VARlABLES &. EXPRESSIONS 1. Simplify each ""Pression: c4 ,c:"CS a d Sp" q'. p , 3q ,J.' b 6x' 2y' '3x' e 10' . l(r' , 10' c x' ;.(1. X4 f 2 c (a + hJ' (a+ ·l· 2"3 2. Simplify each expression: i, • • 4x'y' b 3, d 2,,/ The expression br' ., x ----or x (iY is equivelent to: (1) x" (3) x' ,to (4) x' (2) each a (2x)' b (3y')' 33 ( (4x'y')' Cbaptu2 ALGEBRAIC VARIABLES & EXPRESSIONS Properties of Exponents (continued) 5. Simplify.rM.. c'd" using only poSitive exponen~. (1) c"d" (3) c'd' (2)+ c (4)+ c " . 6. Simplify the fullowing expression using only positive e::<ponents: 7. When simplified, the e.xpreSSion( 223)-2 becomes: (1) ~. 3. -'" (3) 9{ ~' (2) (-~)' (4) ~7 • is equal to xa-'l. or,(J when x tF 0, What conclusion docs a zerO exponent suggest to you? 8. Zr x . 9. Simplify: (' 'y . x 'X (x')' (1) 1 10. - Simplify: (3) 0 (2) ~ (4) x' (1) ~ (3d (2) t (4) 2x 2x'"" 3x' 3 ... 34 . Chapter 2 ALGEBRAIC VARIAJ3LES &.EXPRESSIONS 1. Fmd the ,um of 6,c, 9i' and 2. Add: 3, 4. 7. What is th.sum offlx + 7y, -12>: + 3z, and -4y + 621 8. Add: Combine: (4. - 5b) + (2a - 8b) + (. + b) 9. Express the sum of24c2 + 23c'+ 8 and 31c 14c - 11 3S :a trinomial. Find the sum of -Bab, -Sab and -Sab. 10. Add: (1) 26.b (2) ab 5. 17.' - 19. - 3 16.' + 23a - 11 -19R' + 7R - 6 -13R' - 8R + 6 -3x + Sy, -4x - 6y, and '.Ix , 4y. (3) -26ab (4) Wab Find the total of 6x' 2x' + 51. 6. -4,c. 31. 8x' + 71', and 11. Combine: (91 + 7) + (3)" - 5) FInd the sum; + (-21 + 1) 12. Fmd the sum: (-2a - 3b + 5) + (6. + 8b - 6) + (. - b - 3) , : (3x - 5y + 4) + (-5x - y -I) + (-6<+ 3y + 1) 35 Chapter 2 ALGEBRAIC VARIABLES & EXPRESSIONS (-3x - 2y) (1. + 3y) (62 + 3x 2) - (3:c' - 4x + 5) 2. From (4a' + a-I) subtract (a' - 3. + 4). 7. Subtract 3. Combine: 8. Express as a trinomial: (2a + b) c (a + b) lSb' + 6b-7 -3b' + 5b + S (7<'- 6c-5) - (2<' + 5c + 1) 4. Subtract: -3:c' + 5")' + 6y' 9. Subtract (8)<' + 6. + 3) from (-2><' + 5. - 9) 8><' + 9")' - 8y' 5. From (-13><'1) suhtract (-6x'y'). 10. Combine: (7x + y) (6x + 2y) (x - 4y) 36 Chapter 2 ALGEBRAIC VARIABLES & EXPRESSIONS Addition and Subtraction of Polynomials 1. 2. 3. , The expression (3x + 2y) - (2x - 3y) - (x - y) when simplified is: (1) 6x + 6y (3) -2y (2) x + 6y (4) 6y The sum of (-7x' + 3x + 2) and (-2x' -3x -2) is: If 3al - (1) -5x'+ 6x + 4 (3) -5x' (2) -9x' (4) 7a + 6 is subtracted from 4a2 - -9x' + 4 3a + 4, the result is: (1) 7.' 10. + 10 (3) .'+4.-2 (2) .' - 10. - 2 (4) -.' 4. + 2 4. From the sum of (2x - y) and (3x + z) subtract (x - y - z). 5. Fmd the perimeter of the following in terms of x. (4x + 1) (3x - 1) (7x + 1) 37 Chapter 2 ALGEBRAIC VARlABLES & EXPRESSIONS 1. Find the product of8x' and 6x'. -3ab by -2a'b 2. Multiply: 3. What does (1) -21y' 71 times -3/ equal? (2) -21y" (4) 21/ 4. 5. 6. Multiply: (1) 4x'y' (2) 5x'y' 7. What does -6r times 3x" equal? (1) -18x' (3) 18x6 (2) -18x' (4) 18xB 8. Find the product of lOx and lOx. 9. Multiply: 10. What is the product of~x4y and 11. The expression -4x (-4JC) is equal to: 12x'y by 3xY' (3) 211' xyby4xy lxl? (3) 4xy (4) 5xy Find the product of _9a 4 and _3a 3. What is the product of~xy and (1) -16x' (3) 8x' (2) 16x' (4) 16x' 12. The product (-5x') (4x') is equal to: (1) -2Ox' (2) 2Ox' 38 (3) -20x' (4) 20x' Chaprer2 ALGEBRAIC VARIABLES & EXPRESSIONS 1. 2. Multiply. Find the product: 5x(7x - 3) . What does 10(3x' + 2x 1) equal? 7. 7.(3. + b) 8. (1) sOx' + 20x - 1 (3) 30x' + lOx + 10 (2) 30x' + 2x - 1 (4) 30x' + 20x - 10 Multiply: 12a{6. - 3) ..;,>-< 2x(4x' - 7x + 2) 3. Multiply: 4. The product of8y(4y - 2) is: (1) 12/ 16y (3) 32y' - 16y (2) 32y - 16 5. Find the product 10. Multiply: 9(40 + 3) 4y(-4 + 4y) (4) 16y' 11. The product of 20(2 + 3x') is: 6x(5 - 2x) Multiply: (1) 30x - 12x' (3) 30x - 8x' (2) 30x' - 12x 9. (4) 30x' - 8x (1) 40x + 60x' (3) 40 + 60x' (2) 40 + 60x (4) 40x' + 60x -_. 6. Multiply: 12. Multiply: !x<4x + 6) 39 (lOx' + 25x + 5) Chapter 2 ALGEBRAIC VARIABLES &- EXPRESSIONS 1. Express the product (X+ 3)(x - 2) as a trinomial. 7. Simplitjr. 2. Multiply 3. - 5 by 7. ·3. 8. Hnd the product of 8x - 7y and 8x + 7y. 3. Fmd the product of 4. + 5 and 4. + 5. 9. (2x +3)' Express the product (lOx - 3)(3x - 4) as a trinomial 4. Express (6x - 2)(3x - 1) as. trinomial. 10. Multiply 5. - 7 times 2. + 4. 5. "What is the product of 2a + band 2a - h? 11. Express as a trinomial: (7. + 2b){3. + 2b) 6. 12. If (7% + 6)(11" - 8) is written in the form, ~ + bx + (;, what is the value ofc? If(4" - 5){2x + 3) is written in the form, ar + bx + c, what is the value b? 40 Cbapter2 ALGEBRAIC VARIABLES & EXPRESSIONS 1. (00)(700) Multiply: (1) 7a'l:l (2) 2. Sa'b' If (3x - 2)(4x + 2) is written in (1) 0 (2) -2 3. 4. 5. (3) 700 (4) 8ab a:< + bx + c form, what is the value of c;-? (3) 12 (4) -4 If (x - 3)(x + 3) is written in a:2 + bx ... c form, what is the value ofb? When (2x - SimpllfY: (1) 1 (3) -9 (2) 0 (4) 9 3Y IS expanded, the result is (1) 4x' -9 (3) 4x' - 12x + 9 (2)4x+9 (4) 4x' + 12x - 9 ~ (15:<' + 5x - 10) (1) 3x' + 5:<' - 2x (3) 3x' + x' - 2x (2) 3x' +x' - 2 (4) 3x' +Ix - 2 41 Chapter 2 ALGEBRAIC VARIABLES &. EXPRESSIONS 1. Fmd the quotient of -iSx'" and 2>:'. 7. Divide: 2. Divide: 8. Divide: 3. Divide: (24x' - 18x' + 12x) + 6x 9. Fmd the quotient of:: 4. Divide: (30x' - 15"') + 5x' 10. Divide: 5. Find the quotient: 72x' ,. 24x' 11. Divide: (14x'+22x'-1Sx')bJI.(2x') (-24x'Y) by (3x') + 2x - 8 x-2 (lOO.'b") by 12x 6. 12. Divide: 42 Divide: - 18x' + 9x) by (9x) Ramp Up to Algebra Unil 5 Using Equations to Solve Problems (RU Unit #8) Day I Goal To translate verbal phrases into mathematical RampUp Lessions Not in RU Intro to Algebra Lesson (plus extra worksheets) Intro Unit 3-Algebra Algebra #6 Translating word expressions- words to symbols expressIOns (multiple worksheets provided) Word Wall Suggestions NYS Algebra Standard A.A. I Translate a quantitative verbal phrase Expression Also, categorize words representing into an operations algebraic (i.e. the expression Addition category would contain words such as total, altogether, more than, etc) 2 To translate algebraic language into verbal expressions Not in RU Translating Algebraic Language into Verbal Expressions (worksheet) A.A.2 Write a verbal expression that matches a given mathematical expression 4 To solve equations Not in RU usmg addition and subtraction Intro Unit 3-Algebra # 1 Solving Equations (addition/subtraction) Joke # 13 "How did the Vikings send secret messages?" opposite A.A.22 Solve all types of linear equations in one variable Developing Skills in 5 To solve equations Not in RU usmg multiplication and division 6 To solve twa step Not in RU Algebra, Book A "",59 Intro Unit 3-Algebra #2 Solving Equations (multiplication/division) Joke # 14 "What do you get if you cross a chicken with a cement mixer?" Developing Skills in Algebra, book A pp.63 Intro Unit 3-Algebra #3 Mixed problems A.A.22 Solve all types of linear equations in one variable A.A.22 Solve all lake # 16 "What do you call a crate of Mallard Ducks?'" equations 7,8,9 To solve equations requiring more than one step Not in RU Intro Unit J-Algebra #4 and #S Solving Equations-more than I, step- loke ftl9 "Who wrote the book 'I Didn't Do It'?", Joke#20 UWho wrote the book typcs of linear equations jn one variable A.A.22 Solve all types of linear equations in one variable 'Terrible Wealher~t.. , 10ke #21 "Who wrote , the book 'Grocery Packing at the oj() Not in To graph linear RU inequalities in one variable $uoermarkel'1" Intra Unit 10 Lesson 1- Graphing Inequalities Graph of an A.A.24, inequality Solve linear inequalites in one variable Arlington Project- Intra to Inequalities *11 *12 To solve and graph linear inequalities (no division by negatives) Not in RU To solve and Not in graph linear RU inequalities (with division by negatives) Why Did the Kangaroo See a Psychiatrist? Intra Unit 10 Lesson 2- Solving Inequalities A.A.24, Solve linear inequalites in one variable Arlington ProjectSolving Linear Inequalities Arlington ProjectGraphing the Solution to a Linear Inequality Intro Unit 10 Lesson 2- Solving Inequal1ties Intro Unit 10 Lesson 3- Inequalities Arlington ProjectSolving Linear AA,24, Solve linear inequaUtes in one variable Arlington ProjectGraphing the Solution to a Linear Inequality , Why was the Photographer Arrested? Why did They Try to Build a House on Orgo's Head? Get the Message *13 To graph optional compound inequalities in one variable *14 To solve anf optional graph compound inequalities in one variable *15 To translate Not in verbal RU sentences into mathematical equations and inequalities *16 *17 To write and solve equations and inequalities Review Not in RU West Sea- Solving Linear Inequalities West Sea- One Variable Inequality Intro Unit 3-Algebra #7--Translating work expressions into equations-number problems intro Unit 3-Algebra #8--Writing and solving equations Not in Review Exercises Unit RU #3, Joke #22 "Who Compound inequality A.A.24. Solve linear inequalites in one variable A.A.24. Solve linear inequalites in one variable A.A.4 Translate verbal sentences into mathematical equations or inequalities A.A.6 Analyze and solve verbal problems whose solutions requires solving a linear equation in one variable All previous standards wrote the book 'Ihe French Che.f'!". Joke #23 "What do you call it when you cut up your credit cards']I', Chapter #3 Algebraic Equations and Inequalities 1005457 Name ______________________ Da~ ______________ Inequalities (All$wcr 10# 08Q07S) Solve each inequality. , I. 2 + 2, ,,; 5(, + 8) 2. 2n + 15 ,,; 3 4. 5v-52:-48 5. 7. -21 ,,; -6(0 + 12) 8. -5p + 10 > III 9. -25 ,. 4 10. 4(d + 20) :> -22 II. -113 ::. 9j - 15 12. m -+5;;'5 -2 13. 14. -98 < JO + -120 15. -8(h - 10) ;;, 16 17. 18. 3u + 2 ,. 12u - 30 I u -18 ,,; - - 19 9 ". 4q+2()";6q+l1 6. 109 - 21 '" 6g - 26 + 12y -+4;;'-7 8 16. 2k + 23 :S 3k + 8 r - + 7 < -16 -3 :19. 11 :S 4(d - 4) 20. 7y + 6 :S 81 21. 10v + 11 ,. 83 23. -8e - 9 > 124 24. -23 - j ,. -9(j + 7) 26. 31+22';4t-5 27. -23 < 13 " 22. -78 2: -12p - 12 25. -7(m + 9) 2: 37 + 2m http://www.edhelper.com/mathiinequahties5.htm + -2n . MathAI Quiz#9 1. Graph the inequalities. Draw the graphs to the right of each inequality. A)X>9 B)X<8 C)y >S D)-5>X E)2<X 2. Solve and Graph the following inequalities. F) 3x-2>-1O H)3x-I<14 NAME: Date: I) 6x-2> IO J) 4x+l<-23 Name_~_. _ _ _ _ _ _ _ _ _ __ Date _ _ _ _ _ _ _ __ Lesson 7.1 A. What i. wrong with the graph of each inequality? Write your answer on the lines. 1. x> 6 IllIII$)II. 1 234 567 8 2. x s; -3 1"1111)11 -8 -7 -6 -5 -4 -3 -2 -1 B. Write the inequality described by the graph on Ihe number line. 1. • 1 1 1 1 o ~ 1 1 • 234567 2."11$11111' o 1 2 3 4 5 6 7 IIII~.ll" 3• • -7 -6 -5 ,; 0 < • ~ • i ,; • i 4. c ~ < ~ c 0 2 ~ • ~ ! Il i;' "l" ~ ~ 8 I I $ I -3 -2 -} I -5 -4 -3 -2 ··1 1 I 0 I 0 1 2 • ~m ·c "J! .. ~4 5. .. I I • -3 -2-1 1 I 0 2 3 4 I I • ~ j ~ " "ij, ~ II Chapter 7 • Inequalities 37 ~HeiDer ...... com· Name Oat. ----- Inequalities {AlI$wer ID P 0&\4J15} Solve each inequality. l. 2 + 2, s 5(, + 8) 2. 2n + 15 S 3 3. 4q+20S6q+1l 4. 5v - 5 <0 -48 5. 6. 109 - 21 < 6g - 26 u -18 S -9 - I 19 7. ·21 S -6(e + 12) 8. -5p + 10 ". III 10. 4(d + 20) ". -22 II. -113 <0 9j - 15 9. ·25 ". 4 + 12y I 12. m -+5<05 ,i -2 13. I , 14. ·98 < 10 + ·12e 15. ·8(h - 10) <0 16 17. 18. 3" + 2 > 12u - 30 -+4<0-7 8 16. 2k + 23 S 3k + 8 r - + 7 < ·16 -3 19. 11 S 4(d - 4) 20. 7y + 6 22. ·78 2: -12p - 12 23. -8e - 9 ,. 124 24. ·23 • j ". 25. -7(m + 9) <0 37 + 2m 26. 31 + 22 27. -23 < 13 + ·20 http://www.•dh.!peLcomlmatblineq".litiesS.htm S; S; 81 4t - 5 21. 10v+·l1>83 -90 + 7) · MathAI Quiz#9 Date: 1. Graph the inequalities. Draw the graphs to the right of each inequality. A)X>9 B)X<8 C)y NAME: ~5 D)-5>X E)2<X 2, Solve and Graph the following inequalities. F) 3x-2>-IO G)3x>-12 H) 3x-1 <14 Date _ _ _ _ _ _ Name _ _ _ _ __ ~ __ A. What Is wrong with the graph of each tnequality? Write your answer on the lines. 1. x> 6 1111111$11" 1 2 3 4 5 678 -_._ 2. x '" -3 tlllll$II" -8-7 -6 -5 -4 -3 -2-1 B. Write the inequality described by the graph on the number line. 1. • I o I I I $I 1 ! .. 2 3 4 5 6 7 2.4lli$lllll. o 1 2 3 456 7 3• • 1 1 1 1 1 . 1 1 " ~ ~ 1• ~ .l!' ,& ~ " • ~ c •• ~ ~ a ~ Q t8 4. • .. -7-6-5-4-3-2-1 0 1 1 ~ I 1 1 -5 -4 -3 -2 -1 0 2 I • • ci s ~ •!• 5. .. 1 1 I -3 -2-1 I 1 1 1 I 0 1 2 3 4 • ------_._.._- • 0 11 0 "§• 0 Chapter 7 • Inequalllies 37 I) 6x-2> 10 J) 4x+I<-23 Nrune _____________________________ Date ______________ Lessons 7.210 7.4 A. Name a number in the solution by looking at the graph. Then check the number. 1. a + 1> -2 II I I $ I I I I I. -0 -4 -3 -2 -1 0 1 2 Z.2y-4<& " I I I I , I I .. 1 234 5 6 7 8 Number in solution _________ NumbeT in solution ______ Check: Check: B. Solve each inequality. Then, graph the solution. Check with a pOint from the graph of the solution. 1. -Sm > 10 • I I I I I I I I I I I I • Check: 38 Chapter 7 ¥ Inequalities 2. 3w - 3 oS 6 • I I I I I I I I I I I I .. Check: / INEQUALITIES RULES :5 means ______ <means > means > means ___ ~ _ _ Solving Inequalities When dealing with an inequality, your fITst step is to _ _" Once you do that, you need to solve for it, just like you would an equation. That means _ _~~_ _ like terms, undoing _ _~~_ _ by subtracting and __ _ ____ by adding. Finally, you will _ both sides to get the variable by itself, unless it is in a fraction, in which case you will_. Once you have an answer, it is time to put the inequality sign back into the problem. There is one VERY important rule about the signs: if you multiply or divide by a ~__ the variable alone, you have to switch the . ____ ~ number to get of the Nams ______________________________________ J4otoring Through J4ath Which state builds more cars and trucks than any other state in the United States? (Hint: This state is the only state touched by four of the five Great Lakes and the only state divided into two parts.) To find out, solve Ihe compound inequalities for I below. Match each compound inequality in Column A 10 ils graph in Column B. Read down the column of written letters 10 discover Ihe answer. Column B ColumnA 1. <3;',-4>4 H. G. 4.6-2x>200r8-xSO 5. -12,,2x-6<4 I. 6. 2> ax - 14 > 26 A. C. Ii. ' ; "It B. R. • I ! t I I I ., : "6 I I. "II I I I ., • ; Ii. , I n 'i Compound inequalities I I I I I I e I I I I I I •• I I I I I I I I , , • I I !II I : I e I , I I I • • I I I : ,i • I • ; t"'x $ I I • X I •• I I ,)IoX I I e I l)ox I • I I ,""X • I • hi X i -10-9..t!-7-6·!5-4·3-2-10: 123456'1 & 910 ~Ei I ' • I I I I I • I I I I • I I IIV.. ,X I -10-9-3-7-6-S-4-3-2A 012: 3 4 5 6 7 8: 910 , , I I I I I I I I • , • I • I I I »ox I I • I: J I I I I J , I " I I )oX ·10.g~~~~444-10t234567a9tO 0(, «, I I I I I • I G j • , e I I , I 't'+x 404~~~~444-1012345G78910 .... ; ; : I ; , I I • ; •• : j I •• -10·9"a·1-&·S-4..3-'l·10 1 2 3 4 5 6 7 8 910 Answer: _ _ _ _ _ _ _ _ _ _ _ _ _ ___ III I 1 2' 3456'18910 -10.g..a-7~ 0$-4 -34·10 1 2: 3 4 S 6 7 S 9: 10 ... O• III ~10.g-8-7.(j_5_4-3-2·1 x I FSl22010 Algebra Made Simple. «) Frank Schaffer Publications, IflG. Name' _____________________________________ Hole-in-One ,, , What do you get when you cross a card with a game of golf? , , To find out, solve each equation for x. To spell out the answer. write the letter of the corresponding problem above the given answer. \ T. x + 4" 14 E. 26 -x> 34 A. x-122:7 H. 65 -x.,;63 \ \ N. x-12>35 I. -x-18<4 \ , O. B -X" 10 ,, ,•, , "'• \ H. -6x" 18 '. "\ I , I' I' E. &-3;'15 A. -5+x;'S , j" ,, C, -4x - B" 16 E. -x - 13 > 1 .. J,_._!' ,, I .... I -Q,. N. 15 + 17."; -19 L. x-12:S; 13 '" ",N x ;;, 11 x> 47 x < 10 .;;,2 x:Z: 19 x~6 x> -6 x <:-8 x> -3 x:> -:2 x> -22 x .,; 25 x <-14 ,.. , xs-2 • FS1220l!) Algebra Made Simple . .., Frank:"""'""::,~PublJ::"':ti:.n:',,~1nC:-.---------D~~~~~~~~~~Di Nrune _______________________ Inequalities (Answer ID /I 0473182) Solve each inequality. I. 234 < 26y 2. -21t :5 -540 3. 5d 2: 7 4. 5. 437 < -2c 6. 8 :5 J - 9 7 7. q -14 2: p - < 15.4 --20 8. -30! < 317 9. b -17.6 :5 - 13 10. 18z 2: 20 11. 289 :5 -23, 12. 12h < 84 13. 14. 15. -24n < -29 e 18 :5 -4 16. -14m :5 -14 u --12 5 :5 17. k -10 < - 16 19. b - 18. g - :5 3 -2 20. 181 :5 8z 21. 6! 2: 170 , 24. -29 > -3w ;, 6.3 19 22. 355 < 17x 23. - -11 2: 5 http://www.edhelper.com!math/inequalities4.htm 211512005 .HeW~· Nrune _______________________ Dale _ _ __ Inequalities (Answer ID H0887017) Solve each inequality. I. -3.4 5 14.2 + f 2. h - 2.2 " 11.4 3. -7 > w + 15 4. 12 - m > 23 5. -25.9 - k > 6.1 6. b + 16.3 " -12.3 7. 2 > g + (-9) 8. 10. -17 "-11 +e 11. 27 - v 5 14 12. x - (-1.7) < 4.3 13. z + 19 > 8 14. 4+n<13 15. 3.2 5 Y - 30.7 16. -20.9 5 1 + 5.9 17. b + (-18) " -8 18. 22 5 -24 - n 19. a + 10.2 5 15.8 20. 15 " 9 - r 21. y + 18 > 5 22. 16 - 1 " 29 23. 12 5 6 + w 24. 19 < c + (-9) 25. -20 > 13 + z 26. j + 14.5 " 3.2 27. -4.7 < m + 2.9 28. -10-,57 29. x - 21.7 5 13.6 30. 18.9 5 e - 5.4 31. 19 - d < 20 32. 8 5 26 - u 33. 11 .::;; v + 7 34. P + (-17) 5 -5 35. -17.1 > 16.2 + f 36. g - 30.4 > 3.5 37. -11 5 3 + k 38. h + 15 " 7 39. 16"-26-q 40. n - (-5) < 13 41. 13.3 5 x + (-19.1) 42. 29.8 5 -14.1 - z 17 ~ -28 - u http://W.NW.edhelper.com/mathiinequalities2.htm 9. -6 5 d + 10 2/15/2005 Graph the following inequalities. 8) -I:ox>4 .' " 9) 2>X<6 . 10) . . ,. '. -, ',,' -4>X<5 .. 11) 3<X<8 12) . -10>X<-2 13) -4>X<-1 14) 3<X>7 NAME: DATE: Math Al HWOlll/!' Inequalities Solve and graph each equation. 1) -2x-l> 5 2) -3y-6 > 12 3)5y+3>3 4)-10 < 4y 5)15 < -5y 6) 2y-Sy > 9 7) -lOy < 20 Dole _ _ _ _ _ _ _ _ _ _ __ Nome _ _ _ _ _ _ _ _ _ _ __ Compound Inequalities: Solve and Graph 2x '6 3-4as5 3 - 3 - 4as5 - 3 2"'2 x",3 -4 -3 -2 _1 :2 0 3 <1 01 "'!-l 'Isy and -5 -4 -3 -2 -, -i i 0 i 2 .4 :$ 5 y<3 .11111111.111(11101111)10 -5 -4 -3 -2 -1 0 1 2 4 ,3 5 7. '6';;-2zs4 t,;; 'lor t s '3 ~IIIIIIIIIIIIIIIIIIIII" ~IIIIIIIIIIIIIIIIIIIII. -11)""1...!1-1·~-5"'·l-2-1 Il I , l " 5" 7 & 9 10 ~IIIIIIIIIIIIIIIIIIIII> 3. 0< -1 y+3-3<6-3 Y'" 'I .1lI-~"-7·64-e:·3-2-1 5 .1 I I I I I 1 10 i I I I I I I I I t I I .. 2sy+3andy+3<6-3 2-3sy+3-3 5 5 2sy+3<6 J. 50 + 1 - 1 < '4 1 50 < '5 '40 <1 '4 - '4 x>'3Iorlx='3 -5 50+ 1 <'4 or x + 1 S '3 or x + l! 3 ," 3" $ ~ 7 111111 ~IIIIIIIIIIIIIIIIIIIII> 9. 3<2x+ 1 <7 ~IIIIIIIIIIIIIIIIIIIII _Hlw,·fI_1_6_S_4_J_2_1 I) 1 :< 3 , :; .. '1 fI 911) .. ~IIIIIIIIIIIIIIIIIIIII _10_9_11-_1_6.5-4.3_2_10:123 <f S. ~ '1 .. 9 19 .. 10. '8<2x+4,;;-2 4. -2<31-2< JO .. 111111111111111111111> .IO·'·B-1-a4-4.3~·10 I) ~IIIIIIIIIIIIIIIIIIIII> -10-9-8-1-6-5-,(-J-2-)!) , '1 J ' )23A5'IS')O 5. -(x- 2) l! 3 ~ .. 7 a 910 11. -6'; 3 - 2 (X+ 4) ,; 3 ~IIIIIIIIIIIIIIIIIIIII. ~IIIIIIIIIIIIIIIIIIIII" -IO·9~-7~·I-~-l4-101234567e'JO 12. 4-3x"';'8or3x- J s8 6. 3x-7< 11 or9x-4>x+4 ~IIIIIIIIIIIIIIIIIIIII .. .. 111111111111111111111 .. Page 25 Nama ________________________ Data ________________________ Basic Inequalities: Solve and Graph 6 < 3 (1 - 8) 6<3-35 6-3<3-3-35 3 "3 '3s <"'3 'l~s '1 $< "111111111011111111111 • -10 -'I -Il -7 -6 -0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 15x- 2 < 3x- 11 8, 1. x+4>12 • I I I I I I I I 1,1 I I I I I I I I I I I .... I I I I I I I I I I I 1 I I III I 1 I I .. 9, 2 (t + 3) <3 (t + 2) 2, 32 > "4 (4y) .111111111111111111111 .... 111111111111111111111 .. 3, 3y+ 1 < 13 10, .. I I I I 1 I 1 I I I I I I I I I I 1 I 1 I. ~G~~-1~~~-14.10123. 4. 1 5(11)910 • I I I 1I I I 1I 1I I 111I 1I 1I I • wro~~~~~~4~.IOI 1 _lU_,!wJ)_1-4_5_<l.-3_1_'1'1 1:1 J 01 S (, 1 II" 10 2 - ~ <'1 .. ~3'S'la9W x- 1.5 < 0,5 (x + 4) 11. 10-<2z+ 182 2 .111111111111111111111' 5. 15x-2(x-4»3 IIIIIIIIIIIIIIII~ _10_9_6w1_4~""'_l_1_1 I) I 2:>" S ~ fill. 1",910 12. "3 (2m - 8) < 2 (m + 14) .111111111111111111111 . . . 111111111111111111111 .. -HI.9.6_;.~_S_4..J_2_1 0 11 3 4 ~ 6 1 I 910 6, "2x - 5> 6 -1O*9.e_1"~_~.4-J_2_1 13, 0 l ' 1 4 5 6., II" 10 2x + 3 < 6x - 1 .111111111111111111111 • • 11111111111,111111111. -IO-9-1I.1-4·~-4-J·'2-1 11 1 1 3" 5 4 1 " 9 HI -11l.·),.-r-6-$4-3-2.1 \I 1 2 J .. 11.1 e " 10 14. 3x-2;e7x-l0 7, "3m+6(m-2»9 .111111111111111111111 • • 111111111111111111111. -\(1 -9...g ·1 -6 -5 -4 -3 -2 -I II I 11: 3 , S ~ 7 a 9 I'D -10 -9 -e -1 Paga 24 -6 -5 ..... -3 ·2 _I 0 1 2 3 4 S (I 1 8 9 10 Name' _ _ _ _ _ _ _ _ _ _ _ __ Introduction to Inequalities Algebra I For any two real numbers a and h, one of three possibilities exists: either a is less than b (a<b), a is equal to b (a=b). or a is greater than h{a> b), Wbencomparing the values of two numbers on a real number line~ the larger one is to the right of the smaller one, If the two numbers have the same value, then their graphs are at the same location on the real number tine. Exercise 111: Slate whether the given inequality represents a true (1) statement or a mise (F) statement (a) 7>5 3 12 (e) ->- (i) fi.ij" JIT (b) -7>-2 (f) -2.6> -4.1 (j) JO >-./4 (e) 0<-4 (g) ,,>Ji (k) -,JI,,-I ·1 I (d) ->_. {Ill --J35 > -6 (I) 1-21<1-~ 2 5 20 4 Recall: a ~ b means that "a is less than or equal to b." and a ~ b equal to b." m~s that "0 is greater than or Exercise #2: State whether the given inequality represents a true (T) statement or a false (F) statement 12 (aJ -22:-6 (b) 3.,;£ 5 30 (e) -75,-5 _ (d) -4<:-3_ Exercise #3: Circle each replacement for x from the accompanying list that makes the inequality true. (a) x<:-5 -14,-1l,-8,-5,-2,-1,0,3,7,10 -14,-11,-8,-5,-2,-1,0,3,7,10 (e) IxI,,3 -8, -7, -6, - 5, -4, - 3, -2, -1.5, -I, -0.6, 0, I, 1.25,2, 3, JO, 4, 5, 5.8, 6 (d) Ixl <: 2 -8, -7, -6, - 5, -4, - 3, - 2, -1.5, -1, -0.6, 0, 1, 1.25, 2, 3, $, 4, 5, 5.8, 6 (e) x<5 m, .fij, Ji4, .fi5, .fi6, m, .J28 5 (f) x> 3 0')'5' 2'6'9' 12' ,3,4 7 3 10 15 20 2 AJgebra 1, UnitH! -Alge.bm'l'- F~- (,9 Arlington High School. UGraIlg¢viUe, NY 17540 When we graph solutions to an inequality on a real number line. we darken aU numbers on the number line that make the inequality true. We call the set ofall solutions the solution set. Exercise #4: Match each inequality with the grapb ofits solution set a._IIIIIII+I!I!!II· -5 -4 -3 -2 -1 0 2. x>-3 I 4! I I I I !' I ill I ! I I I I I • h. -5 -4 -3 -2 -I 0 1 2 3 4 5 6 7 8 9 , I I <\l I I I I I I I I I I I I'" -5 -4 -3 -2 -I 0 1 2 3 4 5 6 7 8 9 3. :£>2 c. 4. d. • x~2 I I I I I I I -5 -4 -3 -2 -1 0 5. x<2 6, ,,;;2 e. • ·5I -4I ·3+-2I , I H f 8. 1-<1>2_ g. 1-<1<2_ h. , I I I •I I I I I I I • 1 2 3 4 5 6 7 8 9 I I I I I I I I I I I • I I I I $ I I I I I I I • I I I $ I I I I I I I • -I 0 -5 -4 -3 -2 -I 0 7. 2 3 4 5 6 7 8 9 " ·5 -4 -3 -2 -I 0 4 f I I <jl j I 2 3 4 5 6 1 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 I I <jl I I I I ·5 -4 ·3 ·2 ·1 0 I I 1 2 3 4 5 6 7 8 9 • Exercise #5: Gcaph the solution set for each inequality 00 the accompanying number line. (a) ,';;-1 111111111111111111111 • -10 (b) x>4 ,«6 5 10 111111111111111111111 • -10 (e) o ·5 o -5 5 10 - 111111111111111111111 ' -10 -5 o 5 111111111111111111111 • -10 (el Ixl>1 -5 o 10 111111111111111111111 • -10 -5 o 5 10 AJgrol'\ll.l1Jtitffl-~F~-L9 "!be A~Q AJgcbta Ptojo>;t, LaGnngtville, NY 12540 3'1 Dote' _ _ _ _ __ Name: _ _ _ _ _ _ _ _ _ _ __ Introduction to Inequalities Algebra 1 Homework Skills 1. 'State whether the given inequality represents a. true (T) statement or a false (F) statement 5 -J9 S;-3 (a) 8<6 (e) 1.:?5S 4 (i) (b) -11>-12 (I) -3;;'-5 (j) 1-2j>j-q (e) -3<0 (g) 1<$4 (1<) 2 3 Cd) ->- 3 4 (h) -Ji'i <-3 1-31<111 (1) .JO:>'O _ 2. Graph the solution set for each inequality on the accompanying number line. I I I I 1I I I I I 1I I I I I 1I I I I • Ca) ,$-3 -10 o -5 5 10 I I I I I I I I I I I I I I 1I I I I I I • (b) x:>.7 -10 o -5 5 10 1 I I I I I I I I 1 I 1 1 1 1- I I I I I I • (e) x>O -10 o -5 5 10 I I I I I I I I I I II I I II I I I II • . 1 (e) x>52 o -s -10 5 10 I I I I I I Ii I I II I I II I I I II • -10 o -5 5 10 I I I I I I I I I I II I I II I I I I I • (I) x S; -1.5 -10 o -5 5 lO Applications 3. The price Charged by the U.S. Postal Service to mail an envelope first class exceeds $0.41 if the weight of the envelope is greater than one Qunc;c. Graph all numbers of ounces for which the cost to mail an envelope first class exceeds $0.41 using the number line below. I I I I " ~54~l...z-101234S AlgdmII, Unil: fl-AJgdIraii:: ~- 13 ArlingtotI High SdJ.oot. Lu.GtmgevllJe, NY lasro Reasoning 4. Graph the sollltion set for each inequality on the accompanying number line. (a) Ixj,,4 I I I I I ·10 (b) 1*2.5 1 1 1 I 1 -10 (0) Ixj>L5 -5 I I II I I I I -5 -5 Ixj<5 I I I I -10 1 1 1 11 1 5 0 1 III 1 1 1 5 0 1 1 11 1 • 10 5 I I I I I I IIII 0 -5 I 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 -10 (d) 1 • 10 I 1 I 1 • 10 I I I I I I I I I • 5 10 5. Classify ""chof the following as either true (T) or false (F). (a) For any ceal numbers a. h. and c, if a <: b and b < c, then a <: c. (b) For any real numbers (1. h. and c, if a <:h. then a+c <: b+c. (0) For any real numhcIS a aod b, ifa d, then 101 <101. 6. The follOWing exercise investigates a property of inequalities that will be needed later in the course. (aJ Complete the last column in the fullowing cbart by placing «<" or ">U in the given circle in order to make a true statement. original true operation on both sides final true statement ofthe inequality ""'tement multiply by -2 2<3 4>2 -3<-1 (b) Based upon your work in the table in part f--..__.... (a), complete the fullowing property of 4> --(j inequalities by selecting the correct phrase to fill in the bhmk 9<12 multiply by -I -40-6 -40 2 mUltiply by --4 I2U4 divide by -2 -2()3 divide by-3 -3U-4 MULTIPLICATION PROPERTY OF INEQUALITIES When multiplying or dividing both side<) oran inequality by any negative number. the inequality sign must remain the same in order to keep a true statement. be changed to = be reversed Algebra I, Ulli! #1- Algtbniic FouDdaIioas - 1.9 The Artiogtoa Algebra Project, ~gcvilJe, NY 12540 ., Name: _ _ _ _ _ _ _ _ _ _ __ Dale: _ _ _ _ _ __ Solving Linear Inequalities Algebra 1 Solving inequalities. like solving equations. consists of finding aU values ofthe variable that make the inequality Ime. Burchelll: Consider the linear inequality 2.<+ 3 > 11. (a) Circle each of the following values of x that lie in the solution set of this inequality. x;=-2 x=4 x=1T (b) Solve the linear equation 2x+ 3= I r. x=4.1 (c) What is the solution set of the linear inequality 2. + 3 > II? . Exercise #2: Which ofme following represen~ the solution set of the inequality 11 ~4x + 31 (I) xs2 (3)x22 (2) x5-2 (4) .<2-2 Linear inequalities may be solved in .an almost identical fashion to linear equalities, The key difference comes when mUltiplying or dividing both sides of the inequality by a negative number. Exercise #3: Place an inequality symbol, > or <, between each of lhe foHowing sets of numbers. ~rdse #4: Solve each of the following inequalities. Remember to reverse the inequality if you divide or multiply by a negative number. (aJ 2.>:-5>13 A,lgWra I, Unit 113 - r..iImr Algo:bm - (,9 I (b) --x+5$·-7 2 The ArlitlgtOlt A4dm Project. Lt:Gtang1MU~. NY 1254{l (c) -7x+5533 Exercise #5: Find the solution set of each of the following linear inequalities. (0) 3(2r-5)-4(Zx+l)S;-26 (aJ 3(2x-4»Z(x+4) Exercise #0: Represent each sentence as an algebraic inequality; then, solve the inequality. (a) twice n is grea.terthan the sum of4 and 6. (OJ The sum ofg lUld Z is at least I!. (e) The productof4 andy is at most 21 more thany. (d) Twice !he sum ofx and I is less !han 18. Exercise fl7: Considerthe linear inequality lr+ I> 3x-2. (a) Enter the following two functions into your calculator (b) Graph each of these linear equations Oil the to compare the two sides of the inequality for the grid below. given ofvalues ofi. Then fill in the fable below. y 1'; =3x+l and Yz =3x-2 r; =3x+l 1; =3x-2 3x+l>3x 2 I, (TIF) , , I x ,, ·5 "I, ·2 , 0 , , , i 3 6 , "-1 (cj Considering your answers to part <aj and (0), what is the solution set of the inequauty 3x + 1> 3x - 2 '! AI~ I, Uo.itl13 -Li-.r~(1l-L<J 'i'bcArliDglOO Algebra ~ Le~tIMUe, NY 12549 x (d) JustifY your answer algebraicaUy. Namc: _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ _ __ Solving Linear Inequa6ties Algebra 1 Homework Skills 1. Which of the following represents the solution set of the inequality 4x +6" 267 (1)"$ 5 (3),,;'5 (2),,$8 , (4) ..,;'8 2. Which ofthe following represents til. solution set ofthe inequality -3x+6 <: 27? (I) ;c<-7 (3),,>7 (2) ,,5,-7 (4) <>-7 3. Which of the following is the solution set oCthe inequality 1O:S 4x-14? (1) ,,5,6 (3)";'-1 (2) ,,;,6 (4) .«-1 4. Find the solution set for each ofthe foIiowing inequalities: (a) 12>3x-18 (b) O.75y+6<:9 (e) IOm-20>3m+1 (d) 3(x-4)+2$x+12 (e) 4.... 4;'2(2<+2) (I) 9z-(6-z);'12z-10 Algcbm 1, Unit #J - Linear Mgdm - Ll} TheArl~Algdn:a Projf:cl, ~Ik, NY t1S4O U3 5. Represent each sentence as an algebraic inequality; th~ solve the inequality. (a) The product ofnine and x is greater than six more than the product of three and x. (b) Four times the difference of x and :; is less than or equal to the sum of x and 15. Reasoning 6. Consider the linear inequality .!. x - 2;>.!..r+ 3. 2 2 (a) Enter the following two functions into your calculator to compare the two sides of the inequality for the given of values of x. Fill in the table below. (b) Graph each of these linear equations on the grid below. y 1 y;. =-x+3 2 x 1 y., =-x-2 2 1 1 -x-2>-x+3 2 2 1 Yz =-x+3 . 2 (TIF) x ,• -4 · -2 I . · 0 · 4 6 (c) Considering your answers to part (a) and (b), what is (d) Justify your answer algebraically. the solution set of the inequality'! x - 2;"! x + 3 '1 2 2 7, Which ofthe following inequalities has aU real nwnbers as its solution set? (1) 3>+2>21<+7 (3) 4x+8>:4x+16 (2) 7x+3>7x+IO (4) 5,-4>21<+3 8. Which ofthe following inequalities has no values in its solution set (it's empty)'? (1) 6<+2>6x-1 (3) 8x+7<3x+9 (2) 4.:<+8;'4.<+8 (4) 2x+6<2<-10 Algebra I, Untllll-Linmr A1gebfl1~ [3 The Arliug\Oll Algebn ~ ~evil.h; NY 12540 Name: D.Ie: ---- Graphing tbe Solution of a Linear Inequality Algebra 1 In this lesson we will rearn to communicate our solution sets both algebraically and graphically_ First we will review trow to graph inequalities. Exncis< #1: Graph each inequality below on the number line provided. (.j .;'3 II1111I11111III111111 • ·10 (b) .«-2 ·5 0 5 10 111111111111111111111 • o ·5 ·10 10 5 111111111111111111111 • (oj -4<x';S o ·5 ·10 10 5 Exercise #2: Write the inequality represented by each graph below. 00 00 1111111$11111111111.1-s 0 5 H) ~IO ~ 111111.11111111111.111. -10 -s 0 5 10 11111.1111111111111<111. -10 -5 () 5 Exercise #3: A car tire company recommends the pressure. p, for a particular tire should be at least 30 pounds per square inch and less than 36 pounds per square inch. (a) Represent the recommended tire pressures as an algebraic inequality. (b) Graph the inequality that represents the recommended tire pressures, • 1 1+-+-11,-+-+-+-11-+--1--+1-1--+-+-+. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Algcbnt I. t1citMl-Lincar A!gdml- t.lO TbeAl'llApnAfgebra Pl'ojM, ~ NY 12540 to Exercise #4: SQlve each inequality and graph its solution set. If no solution exists, then so state. (a) 12x-14<9x+13 (b) 2(x+9»3x+1O (e) 4(x+3)<4x+9 ( (d) 3x+13';5(x+2) ) (e) 5(2x+ll):?3(x+2) " (I) O.IOx+40>O.12x+25 ) ) Exercise #5: Whlch of the followuig represents the solution set of 16 >-3x+ 10? (I) j I I I I I I I I I I 1$1 II I I I I I • -10 ·5 0 :) 10 (2) 1111111101111111111 Ii' ·HI .5 0 5 (3) ""11111111011111111 -10 (4) Hi .$ 0 5 HI 11111111$111111111111' ·HI.) 0510 Eurclse #6: Translate each verbal sentence into an algebraic inequality. Then solve and graph the solution set (a) Eight is less !han four times a number added to ten. AI&dm I, Unit 10-U_ Algcbru. -U{) TheArlingWa AIgdI:R~ LeGmngCY\1k., NY 12540 (b) Five less than twice a number is at most ~3. N~ ________________________ Date: ___________ Graphing tbe Solution of a Linear Inequality Algebra 1 Homework Skills l. Which of the following inequalities is graphed below? (3)x<-1 (I) >:<2 (2) x';-1 (4) x>-1 111111111$111111 -10 -50S 1I I • .0 2" Which .fthe following inequalities is grapbed below? (1) -3<x,;7 (3) -3';x;;7 1 1 1 1 1 1 1 ill I ·10 (2) -3';x<7 ..s 1 1 I 1 1 1 I. I I I • 0 S lil (4) -3<>:<7 3. Which of the following values ofx 1S not in the solution set graphed below? (1)-5 (3)0 1111.111111111<1>11111' -s 0 5 10 -(0 (4)4 (2)-6 4. Solve each inequality and graph its solution set If no solution exists, then so state. (a) 1-5>:>36 (b) S;;Zx+8 (e) 3(2x+7)<2(3x+l) < (<I) 6x+2-3(x-3}> x-I <: < (I) 0.15x+50 > 0.20>+ 35 (e) 2(4x+3»IO(x-2) ) <: " AlgdImI.UnJt#3-~A1gebl'll-Lt{) Th!: Arlington Mgdn PmJo;t. I..AOningcvilk:, NY 12S«J ) ) < 5. Translate each verbal sentence into an algebraic inequality. Theil solve and graph the solution set. (a) Four times. number decreased by IS is less than six more than the same number. (b) Six times the sum ofa number and 2 is at least 30. (c)Three times Illesum ora number and 7 is at most nine more than five times the number. (d) Five-fourths ofa number increased by six. is greater than one-halfof the same number. Reasoning o. Felicia's Ice Cream Shoppe offers a maximum ofeigbt scoops of icc cream on a single conc. They serve only whole scoops (a customer may not order 2 and a half scoops for example). If a customer just ordered an ice cream cone, which of the foUowing solutions represents the number of scoops the customer may have ordered? Expfain, (1) •I 0 (2) •I 0 . I I I I I I 1 2 3 4 5 6 7 ~ 2 3 4 5 6 7 8 ExPLANATION: Algdmt I. urut ir.J -Lillear~-UO 1be~A1gd)nl.Ptojcet, ~geYilte, I• (3) 8 9 10 • • • ••• • • I I NY 12540 • I <fI I I I 0 I (4) 1 9 I• to • I <fI <fI $ $ 0 I I 2 3 4 5 2 3 4 I I '" I I• 6 7 8 9 10 ••• <II 1 I ' 5 6 7 8 9 10 ~ Why Did The Kangaroo See A Psychiatrist? @"" Find the graph of ths solution set of any Inequality below in the corresponding colurl]n of graphs, Notice the lettsr ne'xl to it. Write this letter in each box that contains the n'umber of that exercise, Keep working and you will discover the answer to the title question. oJ) ~r;; =-r ." ~Ol ~m c CD ~Jl tf »0 G)x < 1 gS "-; ® q $ I lip -3 :r :!I @x,,"1 ~ ®x > 1 @x",1 2 3 @IIIJ -3 ® @s '" x @4 ~I! I I I III -3-2-10123 CD © t 1 @ ®x <-2 ,'f® x :;;, -2 0 @-1 <x @ ®x e$;-2 -, ® ·1 I I 1'1 II -3 -2 -, 0 , 2 3 ®x'£ 1 <.V x> -2 -2 @)x <-1 I I I I I I -:l ""2 I -2'I -3 I' -3 -2 -1 -1 -1 I I 0 1 I I 2 ,3 ~ @x < 3 I $, 2I 'I3 ~ 0 I a , •( jI -3 I -2 i -1 I m q; 0 1 ° I I ~ @O,,"x 2 3 ® •I $ I I I I II -3 -2 -1 0 1 2 3 ® • I I I I $ I I ~ -3 -2 -, 0 1 2 3 ® @x rf I 2 I, j' , 3 ® o 1 t"~i -3 -2 1 1I -3 -2, 1 I -' I I II J$'23 1 0 I I. J1 ~ I, 2 3 ®IIJJ,! -3 I I. 123 1 2 ®IIJ_1 ®.IJJ,! <1])0> x Q) l-i1 ® '• -3I I II 112 3 ~ I I. 11 , , 11 2 I I 0 1 2 f~ I -2 ;;p -1 I I 123 ' 2 -32 @O<x ~ 123 -32 ' -3 @O"'X -2 ~ I $ 23 , I ' 3 6 1615131j1 ~ 7 161156 4 6 1~~55 ~~21616 82113 91318101714 4( ,, ..... -'.,-,,' ' . ~" Why Was The Phofogra.pher Arrested? Solve any inequality below. CIRCLE the letter next 10 the correct answer. Write this leiter in the box at the bottom of the page that conlains the number of that exercise. Keep working and you will discover the answer to the title question, .~1~4~4~.~4~4~4~'~'~l~'~~~l~l~'~l~'~'~'~'~' x+5>9 (E) x> 4 (T)x> 4 4 (D)x<-2 (E}x<4 (S) X "" 5 -6x<-12 (L)x> 2 3x ~ < 15 X ~ 5: 5m n" ~~ ~:; "J: c 1: ~ -N 2'" a~ ::: ..... ~c tv tv (..> _ X < 2 (A) u "" -3 + 4>-3 (E) z > -7 (Ll u (A) z <-7 u-1",,-4 :3;1 cr:; @-5x > -20 (E)x > -2 (R)x "»m" -6x< 12 ~-3 < 12 (V) X> 2 6x x<2 6x <-12 (S) x>-2 ®x-10<-1 (F)x> 9 (P) X ............. < 9 Z m -12 ~ 2 (S) m "" 14 0) m ~ 14 -5x> 20 (D)x>-4 (E) n "" 30 n ~ 30 U"I1" n "" -15 (M) n ;;. -30 n ~-30 n",,15 (R) n (0) n @ - ~ n ;;. -15 (I) n ;;. 30 . n ~ 30 X + 15 < 4 - . @-·h>8 (D)x> -12 (W) X < -12 @-2x ~ -42 (E)x>-11 (P)x;;. 21 v "" -" ii' "'f "' 21 ~t + 13;;. 30 (X) X 8x> -56 (P) x>-7 (N)x <-7 -7.s;-16 (T) d;;. -9 d.s; -9 x>8 "" -30 (S)x> 12 ~-30 (T)x<12 ~ ...-:... (T) t;;. 17 (P)t~ 17 (34a .s; -20 (F) a ;;. -5 (8)a.s; -5 l @ -ik >-1 (P) k > 8 Why Did They Try To Build A House On .Orgo's Head? ~;mm;m;m;~~~w;w~~~· ".i SolVE!" any inequality below and draw a straight line connecting it to the inequality that describes the solution set. The line will cross a number and a lelter. The number leUs you where to put the leiter in the boxes at the bottom ot'the page. Keep working and you will discover the answerto the. title question. ~~~m-m:~m-~~~-m!;l Q) 3x + a > 2 • • x ~ -21 @7x-1<20. @ ® - x>5 @ a - 4x > - 1 2 . ® ® - x>-2 . @ -Sx - 9 ~ -4 • ® - x > -4 ® 63 + 12K < 15 • ® - x"", 7 f.iA\ @ ® -ax + 25 "'" -31. ~ ® CD - x<3 (f) -10 + 2x ~ -52. @ ® ® © - x "",-1 @15>6x-9 Q) @ @ • x<14 ® 48 < 20 - 14x • @) - x~7 @) -so ~ 9x + 3 @ ® @ • x "'" -7 @ 18 - lOX < -22. @ _ x>-9 @7<3x-8 @ . x<5 tf:r -12K - 8 "'" 64 • @ @ @ ® '-!3' @ @ CD • x < 4 @ -17 > -7x - 45 ® ®. x > 4 @ 3x - 42 < 0 • (f) CD • x ~ -11 @ 44 ~ -ax - 4 4 . ® ® . x -6 @ 4x + 12 > -24 • • x < -4 @ -17 "'" -6x + 25 • 11 12 13 14 15 16 17 18" 224 Pf1E·ALGEBnA WITH PIZZAZZ! © Creative PI.lOi!callof'S . . ) ~ . 'Did ~ou Jae.1' .lIout ... ~_A B C D E ~F G H I J - L M N ~- ~ : : K ~. . 0 DIRECTIONS: Solve any inequality below_ In the answer column, find the inequality that describes the solution set and notice the word next to it. Write this word in the box that has the same letter as that exercise. . ' KEEP WORKING AND YOU WILL HEAR ABOUT A COLLEGE EYE DEAL ~~~~~~~~~~~~~ ® 2(3x - 5l > 2x + 6 ® 8(2 + xl ,.;: 3x - 9 .. ) ~ 13x - 7(-2 ® 5(-3x - 1) + x) ;;. 4x - 10 + 7 ,.;: -x + 30 ® 12 + 5x > 2(8x @9x - 6) - 7x 2x ~ 14 - gex - 4) ® -4(3 - 5x) - l1x < 3x IX Ix <4-HAVE: ;;. 22-STUDENTS • X ,.;: -S-CROSS i I X "" -12-COLLEGE . x,,; 2-EYES I x> 6-CONTROL I Ix >4-THE X < i-KNOW ! Ix <3-TO X ,.;: 22-HIS Ix,,",. 2-PROFESSOR I IX +6 Q) 10(x + 2) > -2(6 - 9x) Q) 7(2 + 2x) ~ 4p( - 10) ® 11 + 3(-8 + 5x) < 16x-5 <D -6(7x - 1) < -ax + 9(-3x - 4) ® -9x + 2(4x + 12) ,.;: 4(1 - 3x) - 13 ® 7Cx + 4) + 16 ~ 5x - (lOx - 6) - 6 @ 12(2x + 3) - 3(8 + 7x) > 0 iC - I x<6-WHO x ,.;: -3-0VER '@-3(4x-6)<7-x @ ?• IX ""' ,,; 25-SEEMED 3-ABSOLUTELY IX ""' -25-SUBJECT Ix> 8-NO x> 1-EYED x < -8-HELP x> -4-PUPILS I X < -4-TEACH PRE ·ALGEBRA WI fH P!ZZAZZl © Crealllf(!: Publlcallons 225 · ". let The Message ~Yes DIRECTIONS: For each exercise. determine whether or not the number in braces is a solution of the given open-sentence. Circle the letter in the appropriate column next to each exercise. When you finish, print the circled leiters in the row of boxes at the bottom of the page. FIRST print those from the column marked "Yes," THEN print those from the column marked "No." A ME;SSAGE WILL APPEAR. <D2x+7"'17 I @ 9 + 6s '" 57 @8m-3=19 No {5}W F {8} H A {3}E R ! r.®s4~3~5__n~8______~{HS}~A~W~·. ® 9u + 3 < 24 {2} IE oo14>20-3y {4}8 I <V 17 + ax ;;;. 75 {7}P A. Q!)S5 = 4w + 29 @50-3x=16 @l63;s;;3+Sn. 1@63<3+Sn @5p 15;;;. 60 {9} T {12}L {10} L {10} I NI {lS} E i 1@8d +1= 5d + 10 8' H F I {3} REI {8} 8 A @4y 7 = Y + 17 C!J5"9h = 20 + Sh {7} FR· Q.w79 8m = 34 + m {5} M I @2k + 34>~-70--7-k--f7-4+'}f-E-I-A---" @hSe-7=6+2e {l}O N @3x + 39 ;;;.Sx - 1 {20} T U @3v + 39 > 5v - 1 {20} H M @ 57 - 3g = 93 - 7g {9} H I @8+6x;s;;9x-30 {12} I E 1(@)8x+24=15x-46 {10}E S @6m<7,;;-1------'..,,-{1.f!}f-L-+-O-l . •·~IIIIIIIIIIIIIIIIIIIIIII~ PRt>ALGE8RA WII H PIZZAlL! @C'q":l'tf-; Pu!)hcalloflS 197 Name: ~__~~~______ Date: _--,--- UnillO - Potpourri Introduction to- A1gebra LESSON 1 - GRAPHING INEQUALITIES An illequality is a math sentence with one of the following symbols: EXAMPLE 1: < "less than" < "less than or equal to" > "greater than" > "greater than or equal to" The statement x < 3 means, "some number less than 3." This statement can be put on a number line. .... !& I -~ -\ " I I First, circle the number 3. Second, shade the portion of the number line that is less 111(111 3. 1)0 \ '2." 4 S 10 NOTE; The symbol " < .. and the arrow on the nwnber line point in the same direction. EXAMPLE :Z: Solution: -<I (\) i ( -3 -2 -, 0 EXAMPLE 3: Solution: i. Graph x > -2 on a number line, ( i I 2 " 4 5 Graph x ~ 1 When there is a line under the inequality it means that the point is included in the graph. To show this, darken in the circle. ".II -z -\ •• ltltt)o 0 I '2. 3 4 5 I> 1 eirde is darkened p- \ INTRODUC£ION TO ALGEllM EXAMPLE 4: UNIT 10, P. 2 Which inequality is represented by the graph below? cl ,• •I \ I". 2 3 45 b I \II -2 -I 0 ~l < x < 3 -1 < x :':': 3 a) b) c) -1.:Sx<3 d) -1 ~x.$.3 Since the circle at -1 is open and the circle at 3 is closed (darkened) the correct answer is (b). Solution: . (t. 7< - If. ----'--- ~::I Itt answer: b ..... -3 -2 -I 0 I 2. 3 4 EXERCISES: I) Graph x > 3 on the number line below: """I I , , ,, 4 2 " "'3 -2 .. , 2) I , I I -} -2 -I 0 I > 5 , ,, I \ \~ 23 4 5 Graph X?: O. .,., -3 4) I Graph x:s 1. <t 3) I 0 1 -z , I I • -1 0 1• :1. • , a4 l> 5 Which inequality is represented by the graph below? t • \ • e ~ \ 1 -2 -\ 0 I 2 • 4 :; b <\ j...,. a) b) c) d) 0<x<3 0<"S3 OSx<3 OSxS3 INTRODUcrroN TO Al.GEBRA UNIT 10. p. 3 EXERCISES (Continued) 5) Which is the graph of x < I? , , 61, , .• I I ... -;'-2-10 2.?45 a) •• b) I 0 \ ~ d) 10 -a \ -~ '. .t 'i -I 0 I -2 -I 0 •, 2t I , , •-,.• -,-I •I 0 , 2. i I l> 34 5 I 3 , ... 45 l \ t'" \ Z 3 4 5 , 2. " 4 5 ,,:> 2 a) 0) Graph -2 ~ x <~ \ \ \ \ ~ 4 \ \ \ l b Which inequality is represents x -::: -2 a) b) «2 d) b) 9) i 0 Which inequality represents the graph below? .~ 8) , ...-,-;'-1 ' i Graph x;:::. -I. ... t 7) • 5 , 2. <14- •aw(tI \ I -:!.-~-I 6) c) I 0( \ <:\$""" ...... 11 . . . - 0 -1. -I 0 I 2..:1 4 5 ~\\4Lll'lit _., 1 -I 0 I 2.::1 4 ' .. c) d) ""ll\itl-+ _0 -1-' 0 , 2.:3 4- 5 ~ , l i t ' l·,·--p,. _~-1.-lo 2.34-5 Name: Dare: Introduction to Algebra Unit 10 - Potpourri LESSON 2 - SOLViNG lNEQUALITlES EXAMPLE 1: Solve for x, then graph your solution on the number line. 2,+4<8 Solution: 2x ,,~ 2 Answer: 2 x<2 1-+ ... I -:3'2 -I 0 EXAMPLE 2: Solve for y then graph the solution on the number line. -6 Solution: :s 4 + 5x -6 :s -4 4 + 5x -4 -10 :s 5x 5 5 -2 FLU' TO GRAPH: Z. 3 ~ x x2:..-2 I I -3 -2 -I 0 I • I , , 1 ~ :3 ~ INTRODUCTION TO ALGEBRA *EXAMPLE3: UNIT 10, p. 2 When you multiply or divide by a negative, flip the sign of the inequality. Solve for x, 3 - x > 6. Solution: 3 - x> 6 -3 -J. >-~ X r flip <-3 -s EXAMPLE 4: a) b) EXERCISES: 6< I .. 41 -3 -2 8-2Xl 9+2X<S) Solve for x and graph solution on a number line. Solve for the variable and graph solution on a number line (show all work on a separate sheet of paper. I) -6+5x<4 2) 523x-IO 3) I - Y> 8 4) -6z - 15 < 15 p.5 Name: Introduction to Algebra Date: Unit 10 - Potpourri LESSON 3 - INEQUALITIES EXERCISES: Solve for the indicated variable and graph the solution on a nwnber line. 1) 3x + 4 < x + 6 2) 3(x-l»7 3) 4 - 2x 2: 2x 4) 5 - z < 3 + 3z 5) -2 ( x - 3) > 4 (2< - 6 ) 6) 5 ( 2 - 3x) > -2 (5x) 7) 5:03-2< UNll' IV GJ1APHING L Graph X .. S on Ibe number line. 2. Grnph X > I 8J OD lb. number line. UNIT IV GRAPHING One Variable Inequalily (Continued) 3. Which open sentence is represented by the 6. Whiclt grapb is represented by the open sent"""" X grapb below? :. 31 : : 34 5 6 , : I ) (2) .10234 ,~ (3) (I) (2) X S; 4 X<4 (3)X:'4 (4) X> 4 Which open sentence is represented by the graph below? 7. • (2) (3) 5. X<5 (3)")(:'5 (4) X> 5 Which open sentence is represented by the graph below? I (4) 8. I I I / I I C' . • , • -+ C 34 5 6 I 5 6 3 " , I t/ I I 0 1 2 ~ 3 4 I I I I : -2 .1 0 1 2 o I 1 I I .2 ·1 0 1 '0 j ! 11 •: C 51 I) I :i 3 ? ! I I .1 0 1 2 " 3 " Which graph is represented by the open seotence 2 (1) (2) . <; . I I I ! II) 3 4 I ! 5 6 J Which graph is represented by the open < 31 01 2 3 4 5 6 X S; 5 I ..,;t;sentence X -2 (I) (2) I 1 2 (I)~ 0 I I 2 (4) 4. H cI ·7·6.5'" -3·2 ·1 0 (3) <J '5: X :S.; 6? , I 3I. 4i 5I 6 7: ·1 9 i i ?• 1 2 3 4 5 6 7 ¢I i i i ? 1 ~ 5 6 7 i 1 I ! 3 " (I) (2) XS-6 X<-6 (4) (3) X "' -6 (4) X > -6 82 d ------! 1 2 I I 5' 34 6 I 7 ALGEBRAIC EQUATIONS &INEQUALI- Solving Linear Inequalities (continued) 1. If x is an integer, which is the solution set of -3 .:s x < O? (1) (-3, -2, -I, 0) (3) [-3, -2, -I} (2) (-2, -1) (4) (-2, -I, O} Which inequality is equivalent to 2x 4.:s 14? (1) x. -5 (2) x < +5 (3)x.-9 (4) x < +9 6. 2. Which inequality is equivalent to 2x-3<8? (1) x < ~ (2) x < Ii (3) x' Ii When solved. for y, the solution to 3-4y<y+18is: (1) y.-3 (3) y.7 (2) y<-3 (4) Y < -5 -2(x-3) > 4 (x-9) 3. Find the soluMon set: 7(4-x) <- 14 7. Find the solutuion set: 4. Find the solutuion set: 4(2-x) ::; - 2x 8. Find the solutuion set: -4(x+3) 61 ~ 2 (5-x) Cbapter3 ALGEBRAIC EQPATIONS &INEQPALlTIES Determine if a Given Value is a Solution to a Linear Lq,••tion Is n =7.6 the solution of 3n • 0.2 c 231 6. Is L = g the solution of 2L + 0.7SL + 0.7SL = 28 ? 7. Is x = 100 the solution of l.4x + 4.7x + 5.02. = 111.21 8. 4. Is x = 3000 the solution of .216x = 64S? 9. 5. Is n = -4 the solution of -20 + 6 = 14? 10. Check if the solution set of -20 + 6 ~ 14 is 11 :s -4. 1. 2. 3. Check if the solution set of -3n + 0.2 <: 23 is n -7.6. )0 62 Check if the solution set of 2L + O.75L • 0.75L > 28 contains L Check if the solution set of lAx + 4.7:x + 5.02x :S 11L2 is x Chock if the solution set of .216x > 648 is x > 3000. ~ = 100. 8. Ramp-Up to Algebra- Unit 6 - Ratio and Proportionality (RU Unit #6) Day I Ramp Up Goal Lesson ,i I ,To solv. prohlems using ratios Word Wall Intro To Suggestions Algebra Lesson (plus extra ! worksheets) #2 lotto. Unit 1 Lesson#l Ratio Representing Ratios Part~Part ratio. Part-Whole ratio NYS Algebra Standard A.R.5 Investigate relationships between different representations and to represent l, ratios and !heir I impact on a '-::-~c-7;--=_+;;;;-_ _ _ _-r'_ _ _ _ _--'="'-;~C-_-1l-'g~h·v'o:.",nDroblern : 2 To identifY #3 : Unit Ratio : A.CN.6 unit ratios and equal ratios Unit Ratios and Equal Ratios Recognize and apply mathematics : to situations in Ibe oUillide world 3 To solve ratio problems using ratio tables #4 Ratio Tables Ratio Table A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas~ functions l equations l charts, graphs, Venn , diagrams, and : other diagrams : 4 To use ratio #5 • Regents Exam Proportional A.PS.3 tables to solve--,-,S",o:.:.lv"i",ng>--_mm...i Questi"nsr.:"or"'-~LRel~tionshjp_m -,-,O"b",se",n"'",e.;:a"nd,,--,, problems of proportion 5 Proportional Problems with Ratio Tables 117 Introducing Rares To identifY and represent rates, and to use them to solve I, Lesson#5 ,, 1explain ,, , patterns to formulate generalizations Intro Unit 7 Lesson #5Using Ratios to • Express Rates Quantity, Rate, and coniectures A.M.l Per, Calculate rates Unit Ratio usmg : appropriate units Calculating Rates Worksheet problems 6 : To review : using ratio tables, bar diagrams, and equations to #8 Reviewing Ratio and Arithmetic . Regents Exam Questions for Lesson #8 Whole Number Ratios, Equal Ratios A.CM.12 Use mathematical representations to communicate with appropriate solve ratio and rate problems accuracy> including numerical tables, formulas. functions, equations. charts, graphs, Venn : diagrams, and ; other diagrams 7 To investigate and generalize Scale Factor #9 Enlarging and Reducing enlargement appropriate and reduction representations ofobjects to solve ,, problem ,, 8 A.R.4 Select To apply. scale factor to a quantity situations #10 Scale Factor and Ratio Intro Unit 7 Lesson 6- Similar A.PS.6 Scale Drawings Use a variety of strategies to extend solution Worksheet methods to Similarity; Similarity Ratio To identify the #11 similarity ratio Similarity Ratio for similar polygons and use it to solve prob1ems 10 other problems A.PS.8 Determine information required to solve a problem, choose methods for obtaining the infonnation~ and define ,, ,, ,, , I ,, , parameters for acceptable I 11 solutions ! To identify ! similar triangles and Corresponding #12 Similar Triangles to use their properties to solve formulate generalizations and coni ectuces problems 12 To relate unit price to A.PS.3 Observe and explain patterns to #16 Unit Price proportions Intro Unit 7 Lesson 7 - Unit Buying; Intro Wmksheet - Unit Pricing Exercise Unit Price, Constant of Proportionality A.CNA Understand how concepts, procedures; and : mathematical : results in one area of mathematics can be used to solve problems in other areas of 13 To use ratios #17 Changing Units to convert Unit Conversion ofMeasurernent units of measure, and to learn that the conversion factor is a constant of i prooortionalilY I Worksheet mathematics Conversion Rate \ A.CN.7 , Recognize and ' apply mathematical Regents Exam Questions for Lesson #17 ideas to problem situations that develop outside of 14 To apply ratio and proportionality to solve a problem #22 Distance, Ratio and Proportionality mathematics A.CN.5 Understand how quantitative : models j . 15 I,, ,, To use ratio #23 and proportion Geometry, to solve Ratio, and geometric Proportionality problems connect to various physical models and representations A.PS.4 Use multiple representations to represent and explain problem situations , Intra Unit 10- The Pythagorean A.A.45. Pythagorean Lesson 4- Theorem Theorem to Square Roots Detennine the third side ofa right triangle using the Pythagorean Theorem. given the length ofany two sides *16 ' To use the NotinRU find the , missing side Intra Unit lO- Lesson 6- Pythagorean Theorem ora right triangle Arlington Project- The Pythagorean Theorem What do Two Bullets have , When They Get . Marned? Greek Decoder *17 To use the Pythagorean Theorem to NotinRU What did Lancclot Say to the Beautiful Ellen? Arlington Project- The Converse of the A.A.45. Detemline the third side ofa Pythagorean prove that a triangle is a right triangle right triangle using the Pythagorean Theorem Theorem, given the length ofany two sides "18 ,i To find the i sine~ cosine) and tangent ratios of right triangles "19 • To use Arlington NotinRU i Project- Intra to Trigonometry Trigonometry, Sine Ratio~ Cosine Ratio, Tangent Ratio i A,A.42. Find the sine, cosine, and tangent ratios ,, , ofan angle of ,,, , a right Why Did the Saltine Lock , Itself in the BankV.ult? triangle, given the lengths of the sides A.A.42. Find the sine. cosine, and tangent ratios ofan angle of a right triangle, given the lengths of the sides Arlington Project- Using Trigonometry to Not in RU trigonometric ratios to find the missing side of • right triangle Solve for Missing Sides ! Arlington •Project- Applied , Trig Problems #1 , *20 To use Not in RU trigonometric ratios to find the missing angleofa right triangle , Did you Hear • About ... Arlington Project- Solve for Missing Angles , Arlington Project- Applied Trig Problems #2 , A.A.42. Find the sine, cosine, and tangent ratios of an angle of a right triangle. given • the lengths of the sides Books Never "21 --To review the concepts studied in the unit #24 The Unit in Review Written Solving Algebraic Problems Involving Proportions ACN.2 Understand the corresponding i procedures for Worksheet What Did the Leopard Say After Lunch? West Sea- Right Triangle Trigonometry WestS••Pythagorean Theorem and its AODticalions similar problems or , mathematical , concepts Name: _ _ _ _ _ _ _ _ _ _ __ Dat.: _ _ _ __ The Pythagorean Theorem Algebra 1 From middle school, you should be familiar with the Pythagorean Theorem. It is of great importance in mathematics because of its usefulness in solving fur missing sides of right triangles. THE PYrHAGOREAN TImoREM In. right triangle, the sum of the square. of the lengths of the legs is equal to the square ofthe hypotenuse. In standard fannula form: Exercise #1: [f c represents the length of the hypotenuse of a right triangle and a and b represent the lengths ofthe legs, solve for the length of the missing side in each case. (aJ a=6andb=8 (bJ a=5andc=13 (eJ b=4and c=5 Exercise #2: If c represents the length of the hypotenuse of a right triangle and a and b represent the lengths of the 'egs, solve for the length of the missing side in each case. Express your answers in simplest radical form and then as a decimal to the nearest hundredth. (oj 0=6 andb=4 (b) 0=9 andc=18 Exercise #3: In the diagrams below, find the value ofx. (0) (b) x 5 12 x 10 8 Pythagorean Triples - Any set of t:hree whole numbers, {a, h, c}, that satisfies a1 + b1 = ct is called a Pythagorean Triple, These triples can help us quickly fill in missing sides of right triangles if we can recognize them. three of the most common are shown below: {3,4,5} {5,12,13} {8,IS,l?} Exercise #4: Determine if the following are Pythagorean Triples. Justify your response. (1:» (12,35, 37} (.) {7, 8, 9} The Pythagorean Theorem can be used in many problems because right angles appear in many applied settings. Exen:ise if5: A baseball diamond has the shape or a square measuring 90 feet on each side. Approximate, to the nearest tenth of a foot, the diStance from borne plate to second base. Exercise #6: The length ofa side of an equilateral triangle is 10. Which of the fullowing gives the height Qfthe equilateral triangle (also known as its altitude)? (1) 5 (3) 3.[5 (2) 7.,[2 (4) s,fi Exercise #7: Carlos walks to school from his house by going directly east for a total of 5 miles, then directly north for a total of 3 miles, and then east again for a total of3 miles. Find the shortest distance between Carlos's bouse and his school to the nearest tenth ofa mile. Algtbrn I, unit »8 - Jtigtu Ttiwlglt Trigonomctly - LI The ArlingtOn Algtbm P~ LeGraogcVilk. NY 12540 Name: _ _ _ _ _ _ _ _ _ _ _ __ Date: Pythagorean Theorem Algebra 1 Homework Skills 1. Find the value ofx in the following diagrams. Round to the nearest lenin if necessary. (e) (b) <a) x 8 52 x 15 20 29 48 2. Using the Pythagorean Theorem. fiU in the table. [Assume a and b are the lengths of the legs and c js the length of the hypotenuse.] a 25 I 7 I , , , 60 25 30 9 c b 34 40 3. Which of the follov.-ing represents a Pythagorean Triple'? (1) {5,7,!Ol (3) {21, 72, 75) (2) (IO, 20, 30) (4) {II, 45, 60} Algebl'll I, Unit 88 - 1U&h1 Tri:w&le T~ - t.I IheArlioglon Mgtbnt Proj~ LaOnmgeviU~ NY 12$40 Applications 4. Sameer visited his cousin and had to drive 135 miles north then 90 miles west. At the end of his trip, be pulled out a map to detennine how far from home he was. What is the shortest distance from Sameer·s house to his cousin;s bouse? Round your answer to the nearest mile. 5. Which of the fonowing represents the height of an equilateral triangle if a side of the triangle has a length of8? (1) 4!3 (3) (2) 3.[5 6.fi (4)4 6. In the foHowing isosceles trapezoid, find the value of x. 8 ,, 25 x ', 22 25 • . 7. Consider a square: with the foUowing characteristics and solve: Cal If the perimeter of the square is 20, find the (b) If the length ora diagonal ofllie square is length of a diagonal. Express your answer in 20, ftnd the length of a side to the nearest simplest radical fonn. tenth. ~imll. UrutllR - Rjgh1 Trl4ngkTrig\'u:u:m:.etl)'-Li Tht Adlngton Alg'*'" frojllCf., UGmngIMUe" NY 12540 Name: Date: _ _ _ _ _ __ ----------------The Converse of the Pythagorean Theorem Algebra I In the: Jast lesson, we reviewed how the Pythagorean Theorem can be used to solve for missing side lengIbs of right triangles. The Pythagorean Theorem can b. reversed, called the ""uver,_, to determine if a triangle contal:ns a right angle, Le. is a right t:.riaJigle. More formally. if the side lengths ofa triangle satisfy a 2 +bt =2-. then the triangle must be a right triangle. ExiJrc.ise #1: Determine if the triangles below represent right triangles. Diagrams are not drawn to scale, so no judgment can be made based upon appearance. (a) (h) S 6 Note that the side length substituted for the hypotenuse must be the largest number. Exercise #2: Determine whether each given set of numbers could represent the side lengths of a right triangle. (Hint - You may find the STORE option on your cal.culator helpful for this problem.) (a) {9,12,15} (h) {4, 2Fs, 6} (e) {I, Fl, 4} Exercise #3: Which of the following sets of numbers represents the lengths of the sides of a rigbt triangle? (I) {5,9,lJ} (3) {6,S,IS} (2) {I 5, 36, 39} (4) {H,IS,17} AJgd>ra 1, umt U8-rug.ht Tl'llmgle Tri~-I2 The Arlington AIgeb... ~t, I...aC>Wlpille. NY 12540 Exercise#4: Derennine for what value(s) of x the triangle below would be a right triangle. The figure is not drawn to scale. x+8 x+7 x Exercise #4: The Aloha Fan Company claims to make funs that rotate at least 90'. To test this claim, Jim and Carl position themselves at either end of the rotation cycle as pictured below. According to ': their measurements, is the Aloha Fan Company making a valid claim? 10.1 ft Carl Jim 9.3 ft 5.2 ft Algebra I, UnitQ- RJght TrimlglcTngooomdry'_ U The Arlingtnn Alpal'rojlX1, ~1Ic,. NY 12540 NMOO: _______________________ Date: _ _ _ _ _ __ Tbe Converse of the Pythagorean Theorem Algebra 1 Homework SkiDs For problems I through 6 t detemrine if each given set could represent the lengths of the sides of a right triangle. Justify your answers, Hint - You might find the STORE function useful for these problems. 1. {S, 12, 13} 2. {6, 7.2, 9) 3. {.!.2'4'4 l 5} 4. {Fl,./5,.JlO} s. {,,13, 3, 2.J3} 6 {l.!..5.} . S'2'S ? Which of the following sets ofnumbers could represents the lengths of rhe sides of a right triangle? (1){8, 10, 12l (3) {16, 30, 34) (2) {25,31,40} (4) {19, 20, 22) 8. A right triangle would be formed by using which of the following sets of numberS as the lengths of its sides? (I){9, 40, 41} (3) {IO, 15,20) (2){ I ~ 23, 26) (4) {IS, 2~30) A1gdmll, Unilll3 - fljgbtTriangIe Tri&1lf\Olll\!tlY~U The ArtWgtOft AJg«Ira Proja:\., laGntn~IIe, NY 12540 ApplicatioDS 9. Jacob is building a table and lost his carpenter's square, which is used to fonn right angles. For the lable to be strue_Uy sound, the legs and the table top must fonu right angles, He measures the table top. legs, and the diagonal distance from the bottom of the legs to the opposite comer. From the diagram below. does Jacob's table appear to be structurally sound? Support your answer with mathematical evidence. 8ft 4ft 4ft 9ft Reasoning lO. Determine for what value(s) of x the triangle below would be a right triangle. The figure is not drawn to scale. x+4 x x+2 Algebra !, Unit 113 - Rigln Triangle Tria~ - U AI~ ProJeet.~1I11, NY 12$40 TheAriingtOtt <. i I Name: _ _ _ _ _ _ _ _ _ _ _ _ __ Date: - - - - Similar Right Triangles - Introduction to Trigonometry Algebra 1 Trigonometry is an ancient mathematica1 tool with many applications. even in our modem world. Ancient civilizations used right triangle trigonometry for the purpose of measuring angles and distances in surveying and astronomy, among other fields. When trigonometry was first developed,. it was based on similar right triangles:. We will explore this topic first in the following two exercises. Exercise #1: For each triangIeJ measure the length of each side to the nearest tenth of a centimeter, and then fill out the table below, Round each ratio to the nearest hundredth. When detennining opposite and adjacent sides, refer to the 20' angle. To fill in the small box on the right, use your calculator, in DEGREE MODE. and express the values to the neatest hundredth. ,, Opposite Opposite Adjarent Hypotenuse , Adjacent tan 20' = Hypotenuse Triangle #[ sin20' = Triangle #2 =20' = Exercise #2: Repeat Exercise #1 for the triangles show below that each have l'lIl acute angle of 50·. 50' 50' , Opposite Opposite Adjacent Adjacent Hwotenuse Hypotcnw;e Triangle #1 , Triangle #2 ~bra I. Uni. #8 -1tisht Triangle TrigollOm.ctry - L3 TbeArlU!gtoo~Projeel., ~t,.NY 11S4l} , tan50' = " , i cos50~ , = The Right Triangle Trigonometric Ratios Although we won't prove this fact until a future geometry COUl"Set aU right triangles that have a common aeute angle are similar. Thus, the ratios of their corresponding sides are equal. A very long time ago, these ratios were given names. These trigonometric ratios (trig ratios) will be introduced through the following exereises, each of which refer to the diagram below. c 3 B u----:4:-----'~A In a right triangle: "'no t ~en 0 f 1 leg opposite of the 3llgle anang e "'" sine ofan angle """ leg adjacent to the angle leg opposite of the angle hypotenuse cosine- of an angle "'" "le",gc:ad",J::'';:;<:e!l='::,o:..;tb::e=3llg=le hypotenuse Exercise #3: tanA= tanC~ Exercise #4: sin A "'" sinG= Exercise #5: cos A = cose::::; A Helpful Mnemonic For Remembering the Ratios: SOH-CAH-TOA Sine- is Opposite over Hypotenuse - Cosine is Adjacent over Hypotenuse- Tangent is Opposite over Adjacent Exercise #3: Find each of the following ratios for the right triangle shown below. (a) sinA ~ (b) tanB= B 13 (0) rosA= (0) (d) lanA= cosB~ (I) Ala:d\fll, U!Ul.1i$. -Right Trill.l\glQ Tri~-U "1"heArlillgtOu Algdlra Pttljwt, u~lk, NY t2540 sinB~ 5 C,LL--:-=2--";::" A 1 l'/ame' _ _ _ _ _ _ _ _ _ _ _ __ Da"', _ _ _ __ Similar Right Triangles - Introduction to Trigonometry Algebra 1 Homework Skills For problems 1 - 6, use the triangle to the right to find the given trigonometric ratios. L cosN~ N 15 2. sinN 9 3. tanN- M?---7.::----O>. p 12 4. sinP= 5. cosP= 6. tanP~ 7. Given the right triangle shown, which of the following represents the value of tan A ? (I) 25 A 24 (2) 24 7 (4) 24 25 7 B 8. In the right triangle below, CQ.Q =? (I) 12 5 (2) ~ 12 25 24 c 12 Q S (3).!3. 17 (4).!3. J3 R For problems 9 - 14; use the figure at the right reduce your trig ratios to their simplest form. 9. to determine each trigonometric ratio. Make sure to sinC= 10. cos C = I!. IlmC= 4 12. sinA= 13. cosA= A B 2 14. tllnA= Reasoning Although we will not prove it here, two triangles will always be similar if they have three pairs of congruent angles. This is not true for quadrilaterals or any other polygons. 15, Consider the two right triangles shown below: 35' 35' Based on the information above, are these two right triangles similar? Explain. 16. Why can we say that two right triangles that share an acute angle are similar? Explain. Algd»a I, Unil #8 _ Right Trimgle Trigoll\'!l.'IIdly - LJ The Arlitlgt«u Algebra Projw, Lo.GranglWi1le, NY 12540 31S" Name: ----------------Trigonometry and the Calculator Da!e: _ _ _ _ _ __ Algebra 1 In the previous lesson, we introduced the trigonometric ratios. We also discussed a mnemonic that is helpful to remember the trig ratios: SOH..cAH-TOA. The first exercise reviews how to' write these ratios given the lengths ofthe sides ofa right triangle. Exercise #1: Using the diagram below~ state the value fur each of the following trigonometric ratios. 13 A 12 c (a) sinA= ,5 (c) cosC ~ (d) cosA~ (e) lanA ~ (f) sinC= B Today we will discuss how the graphing calculator is used with trigonometry. However, before we start, it is important that our calculator is the right MODK To change your calculator into DEGREE MODE: . 1. Press the MODE button on your calculator. 2. Use the arrow keys to highlight DEGREE and press ENTER. 3. Exit the menu (this or any other) by hitting QUIT. Now that we are in DEGREE MODE, we can start evaluating some trig ratios without referring to any right triangles whatsoever. The SlN. COS, and TAN buttons are located in the center of the key pad. Exereislt #2: Evaluate SID 30', cos30· and thousandth. Af¢lra I, Uftjtlffl - RlghtTriwg1e Trigoonolctry"- U 'The Arlta&ton AJattra ProjGCt. LaGnugevilk, NY 11541} tan 30', Round any fiQn-exact answer to the nearest Exercise #3: EvaIuate each ofthe following. Round your answers to the nearest Ihousandfh. (a) tan{40') (1)) eo,(20') (e) 'in(63') It is important to remember that each of lh~ answers from EJrercise #3. represent the ratio of two sides of a rigbt triangle. In each case the ratio has already been divided and the calculator is giving the decimal form ofilie ratio. Exercise #4: Could the right triangle below exist with the given measurements? Explain your answer. C ,i 3 40' B 4 A The Inverse Trig Functions - Thus fae. we have been evaluating the sine, cosine. and tangent of angles. By doing so, we have been. finding the ratio of two sides itt a right triangle given an angle. Using the calculator, we can reverse this process and find the angle when given a ratio ofsides. Exercise #5: Consider the following: (a) Evaluate Siu-{!) using your calculator. The screen shot is shown at the below. ,2 30 (0) How .do you interpret tbls: answer? Exercise #9; Find each angle that has the trigonometric ratio given below. Round aU answers to the nearest tenth of a degree. if they are not whole numbers. I (b) oosB=2 (e) sinE=~ 3 Algtbra I, UPlr If3-RighI TriMgle. TriggllOllldty- LA The A!ibgtQl) Algcl!m f'lviw. yGmngeviUC; NY 12;$40 3i7 Name: _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ _ __ Trigonometry aDd the Calculator Algebra 1 Homework Skills For problems t through 6, evaluate each trigonometric function, Round your answers to the nearest thousandth. 1. 'in(SS) 2. oos(4S') 3. tan(20') 4. sm(85') 5. tan(60') 6. sin( 23') For problems 7 through J5. find the angle that has the given trigonometric ratio. Round all non-exact answers to the nearest tenth ofa degree. 4 . A =-I 7• Sill 3 8. cosG=9 8 8 9. tanK=1 10. cosB=11 11. tanR=5 12. oinT =2:. 13. t>nA=3.127 14. ,inB=O.724 15. <05L=0.876 16. If sin If = ~ then mLA is closest to whicb of the following? (1) 32' (3) 24' (2) 76' (4) 56' 17. If tan A =35 then mLA, to the nearest degree, is (I) 74' (3) 24' (2) 18' (4) 55' Algdlra 1, U!lit 1tS-!Ugh! Tr\qle Trigottom<:try - LA 1beMingfOl:I Algtbra li'roj«t. LaGtangeri!le. NY 12S4{l 2 Reasoning 18, Recall from homework problell1ll 7 through 15, that an inverse trigonometric function gives the angle measurement that corresponds to the ratio oftwo sides ofa right triangle. (a) The Screen shots below show what happens when you try to evaluate sin-t (~). VerifY why this happens when you evaluate sin -I [~). I (b) Considering the ract that sfi(Angle) = opposite side, why hypotenuse 'DOMAIN I:Go"t't 0 does it make sense that sin. -1 (~I gives you an ERROR? 2/ 19, Are the triangles below labeled with correct measurements? For each right triangle below, determine if the ratio of the sides accurately corresponds to the angle. (b) 16 30' 4 8 c 20. Consider the following right triangle. (a) Express sin A as a ralio. (b) Find the mLA to the nearest degree. 7 A 3 B Nwne: ____________________________ Date: ______________ Using Trigonometry to Solve for Missing Sides Algebra 1 Right triangle trigonometry was developed in order to find missing sides of right triangles. similar to the Pythagorean Theorem. The key difference, though, is that with trigonometry as long as you have one side ofa right triangle and one of the acute angles. you can then find the other two missing sides, TRIGONOMETRIC RATIOS Recall that in a right triangle with acute angle A. the following ratios are defined: . A sm := opposite hypotenuse cos A =:ad=~",'aee!l=':... hypotenuse tan A == ~o"ppo"",s::i"'O adjacent Exercise #1: For the right triangle below it is known that sin A =.!. Find the value of x. 4 20 A Bxercise #.2: In the right triangle below, find the length of AB to the nearest tenth. C 26 38' B LL_ _ _""---''''"A The key in all of these problems is to properly identify the correct trigonometric ratio to uSe. Exercise#3: For the right triangle below, fmd the length of BC to the nearest tenth. B 65' A AIgebOl I, Unit iI! - Righi Triru:lgle Trigotl~-LS 1k ArliQgtOO Alpbm ~ LaGrangeville, NY 12$40 12 c ExercJse 114: In each triangle below, use the appropriate trig function to soive for the value of;r, Round to the nearest tenth. Triangles are not drawn to scate. (b) (a) 50' 15 42" x 8 Exercise #5: Which of the following would give the length of Be shown below? (I) ISsm(34') (3) C 18 c05(34') 18 18 (2) sin(34'j (4) IStan(34') 34" A B Exercise #6: A laddcr.leans against a building as shown in the picture below. The ladder makes an acute angle with the ground of 72°, If the ladder is 14 feet long" how high, h, does the ladder reacb up the wall? Round your answer to the nearest tenth ofa foot. h Algcm t, Unillf!-ltigb! TriangleTrigomlll'lt1ty-15 The ~ AJgebm f'g;j¢Ct, LilOnngcullc. NY l25® Name: _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ __ Using Trigonometry to Solve for Missing Sides Algebra 1 Homework Skill In problems 1 through 3. determine the trigonometric ratiO' needed to solve for the missing side and then use this ratio to find the missing side. I. In right triangle ABC, mLA = 58' and AB = 8. Find the length of each of the following, Roand your answers to the nearest tenth. C (a) BC (b)AC A 2. In right triangle ABC, mLB=44' and AD= 15. Find tit. length of each your answers to the nearest ten/h. 58' 8 B of the fullowing. Round B WAC 15 (b)BC c A 3. In right triangle ABC, mLC=32' and AD = 24. Find the length of each of the following. Round your answers to the nearest tenth. B (a)AC ~24 :g (b)BC C A 4. Which ofthe following would give the length of hypotenuse PR in the diagram below? (1) 8co,(24') (3)8tan(24') R 24' Q p 8 Applications 5. An isosceles triangle has legs oflength 16 and base angles that measure 4&", Find the beight of the isosceles triangle tu the nearest tenth. Hint - Create Ii right triangle by drawing the height. 16 16 6. Carlos walked 10 miles at an angle of 20· north of due east. To the nearest tenth of a mile, how far east, x, is Carlos from his starting point? N ~ --. .. .. ",,,,"'20' iO miles ' .. ' ' .... ...... y ,-o'==~~====-,;-,-->E x 7, Students are trying to determine the height of the flagpole at Arlington Hlgh. They have measured out a horizontal distance of 40' feet from the flagpole and site the top of it at an angle of eievation of 52'. What is the height, h) of the flagpole? Round your answer to the nearest tenth ora foot. , ,, , ,, ,, " 52· Algebra I, Unit,\lB - Right TrillilgJeTrigon\lrl'letl'y_1..5 'I'M ArlillglDll Algebra Proj<!d., L&Grangevilk, NY 12540 ,, , ,, , ,, , ,, h 1 Name: _ _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ __ Solving For Missing Angles Algebra 1 Today we willleam how to use right triangle trigonometry to find missing angles of a right triangle. In the first exercise, though. we will review bow to solve for a missing side using trigonometry. Burcise #1: Find the length of AS to the nearest tenlh, C 125 32" B -'-----''''-...:>..A 591mg for a Missing Angk; - The process for rmding a missing angle in a right triangle is very similar to that of finding a missing sIde, The key is to identify a trigonometric ratio that can be set up and then use the inverse trigonometric functions to solve fuc that angle. Exercise #2: Solve for mLB to the nearest degree. A 5 3 B C Exercise #3: Find the value of x, in the diagrams below, to the nearest degree. (a) (b) 125 x 40 A!&d!m I, Urulll8 - Right Triqk Tri&')llllmetry - 1..6 lbe ArtiqttJl\ J\Jgd)1II Project. ~lle, NY 125<IU 94 Exercise #4: Find the value of x in the diagrams below. Round your answers to the nearest degree. (a) (0) 10 15 x Exercise #5: A flagpole that is 45~feet high casts a shadow along the ground that is 52~feet long. What is the angle of elevation, A.. of the sun? Round your answer to the nearest degree. ~ •• • ///{If ,• 45 reet 1 I. 52 feet --..j Exercise I#.i: A hot air balloon hovers 75 feet above the groWld. The baUoon is tethered to the ground with a rope that is 125 feet long. At what angle.of elevation, E, is the rope attached to the ground? Round your answer to the nearest degree. 125 feet i 15feet E • Alpm 1, Uoit #&- Rigbt Triwlgle Trigonometry-U'i TheArlingtoo AIgetmt Proja;{, t.aGt:a.nge'\oille, 'NY 11S40 Name: _ _ _ _ _ _ _ _ _ _ _ __ Date: Solving For Missing Angles Algebra 1 Homework Skills 1. For the following right triangles, find the measure of each angle~ x. to the nearest degree: (b) (a) 19 39 11 27 x x (d) (oj 51 21 x 29 x 36 2. Given the following right triangle~ which of the [cHowing is closest to mLA '? A (I) 28' (3) 62' 28 (2) 25' (4) 65' C 3. In the diagram shown) mLN is dosest to (1) 51' (3) IT (2) 54' (4) 39' ub'---1-3- --"" B M 17 p AJ.,g.ebra 1, umtfU!- R.i&bt"lfims:le Ttigofl(lmdfy_ UI The Arlillglt>ll Algdml Pmjed, ~~II¢. NY l2.W'l 21 N Applications 4. An isosceles triangle bas legs measuring 9 feet and a base of 12 feel Find the measure of the base angle, x, to the nearest degree. (Remember~ Right triangle trigonometry can ontl' he used in right triangles.) 9 9 12 5. A skier is going down a slope that measures 7,500 feet long. By the end of the slope. the skier has dropped 2,20(} vertical feet. To the nearest degree, what is the angle, A. of the slope? 2,200 feet A Reasoning 6. Could the fonowing triangle exist with the given measurements? Justify your answer, 24 70 11 • Ngcbm 1, UnitliS -RlgbtTrimlglcc Trigontmlcay-L6 'fbe Arling!nn.Algel::ra Proje.;:t. LaGra:lKevillc, NY 12S4C Date: _ _ _ _ _ __ Name: Applied Trigonometry Problems Algebra 1 Over the last few lessons, we have discussed how to use trigonometIy to solve for the missing sides or angles of a right triangle. Today. we will continue solving such problems in the context of"real life" scenarios. , ' TIlE TIUGONOMETRIC RAnos SiILA= oPP. byp' ad' byp cosA=J; tan A = opp adj Ex<rcis. #1: An ai<Jllane takes off at an angle of 5' from the ground, If the aiIplllne lravelad 100 miles. how far above the ground Is it? Round to the nearest foot. Note that there are 5280 feet in. one mile, [The 5" angle in this problem is called an angle of elevation.] 100 miles 5" Ex£rcise #2~ While walking his dog, Pierre sees the EitTel Tower and notices that the angle of elevation is 24~ to its top. [fPierre is 2215 feet from the middle of the base of the arch how tall is the Eiffel Tower? Round to the nearest foot j I &Jgebr.ll, t/Qi! II 8·RJgb:TriaDgk: T~-L7 The Arlington ~ ~La~lkI, NY 125«1 Exercise #3: Harold is hang gliding off a cliff thot is 120 feet high, He needs 10 IIavel 350 feet horizontally to reacb his destination. To the nearest degree, wbat is his angle of descent, A1 [Note: This angle you are finding is called an angle of depression or an angle of dedination.] --___ u____ u___u_u.... ~ r u_~~::':'1 .'.' ,, 120 It •••/ / / / / . . . . .... 350 ft • 1 Exercise #4: Francisco is trying to reach a window with a ladder that is 15 feet long. Find the angle that the ladder must fonn with the ground in order to reacb a window that is II .feet high, Sketcb a diagram below mat represents this scenario. Round to the nearest degree. Exercise #5: A ladder that is 12 feet long bas its base 5 feet from the edge ofa building against which it js leaning. In order to be stable, the angle the ladder makes with the ground must be less than 60 degrees, Is dUs ladder stable? JusttiY, Exercise #6: A tree casts a shadow that is 32 feet long. Find the height of the tree if the angle of elevation ofilia sun 35.7~. Round your answer to the nearest foot. Nwue: _________________________ Date: _ _ _ _ _ __ Applied Trigonometry Problems Algebra 1 Homework Applications 1. Sitting at the top of a 57 ft. cliff; a lioness sees an elepbant. The angle of depression from the lioness to the elephant is 22·. What is the shortest distance from the lioness to the elephant? Round to the nearest tenth ofa foot . ··························-c~~ 22" 2. An airplane is trying to take off; however. there is an obstruction in the runway. The obstruction is 20 feet high, and the plane is 80 feet from the base of the Object. At what angle must the plane take off to avoid hitting the obstruction? Round to the nearest degree. 3. A 14 foot ladder is leaning against a house. The angle formed by the ladder and the ground is 72' . (a) Determine the distance, d. from the base of the ladder to the house. Round to the nearest foot. (b) Determine the height, h, the ladder reaches up the side of the house. h Round to the nearest foot. AI¢l[ll I, U!Ul:t: 8· Rigbt1'riallglt. Trisnnomcuy -L1 The Arlitlgt(lll Al,&t:M Projrot. LaGnwgmlle. NY IlS<lO 330 4. A 625 foot tong wire is attached to the top of a tower. If the wire makes an angle of 65" with the ground, how taU is the to'wer? Round your answer to the neares1 tenth ofa foot. 625 feet 5. Luke casts a shadow that is 6.3 feet long. Find Luke's height if the angle of elevation to' the sun is 40·. Round to the nearest tenth of. foot 6. A blimp hoven; above the ground at an altitude of 560 feet. Two: points. A and B. located on the ground are shown below with angles of elevation to the blimp of 36" and 52" respectively_ Detennine the distance between the two points, A and B, to the nearest foot. (Note - you will have to use two trigonometric ratios to solve this problem.) s~o feet :s 31 Name: D.le: _ _ _ _ _ __ ----------------- Applied Trigonometry Problems Day 2 Algebra 1 Yesterday we saw "reallifeu examples of when we CQuld use trigonometry to fmd missing distances or angles. Today we wiU revisit this with more "real life"" exrunples. Exercise #1: In many trignnoIllelIy problems, the telmS Angle of Elevallon and Angle or Dep.....lon are used, (a) On the diagram below use x to label the angle of elevation from point A to point B and use y to labeJ the angle ofdepression from point B to point A. B (b) What is true about these two angles? A ,* Exercise #2: A flagpole 30 feet tall casts a shadow 52 feet long. What is the angle of elevation of the sun measured to the nearest degree? ,, """, , ,, , , '~ 30 feet 1 j..--S2feel-+l Eurcise #3: Maria is flying a kite on the bea;;:h. She holds the end of the string 4 feet above ground level and determines the angle ofelevation of the kite to be 54". If the string is 70 feet iong, how high is the kite above the ground to the nearestJoof? Exercise #4: From the top of an 86 foot lighthouse, the angle of depression to a ship in the ocean is 23", How:tar is the ship from the base of the ligbthouse? Round your answer to the nearest foot 1 .. 86 feet 1 Exercise #S: A tower is located 275 feet from a building in the figure shown below. A person from the second story measures an angle of elevatioll to the top of the tower as 42~ and an angle of depression to the bottom of the tower as 26'. Find the height of the tower to the nearest tenth of a foot o o o • 275ft .._ -•• Exercise #6: A helicopter is flying at an elevation of 350 feet, directly above a roadway. Two motorists are driving cars Gil the highway. The angle of depression to one car is 37' and the angle of depression to the other car is 54". How far apart are the cars to the nearest/oot! 54' Name: _ _ _ _ _ _ _ _ _ _ _ _ __ nate: _ _ _ _ _ __ Applied Trigonometry Problems Day 2 Algebra 1 Homework Applications 1. A person standing 60 inches tall casts a shadow 87 inches long. What is the angle of elevation of the sun to the nearest degree? 2. A ship is headed toward a lighthouse which we know is 65 feet high. 1f an observer on the boat measures the angle of elevation"to the top of the tight house to be IS". how far is the boat away from the base oEthe lighthouse) to the nearest/oot! 3, A giant redwood tree casts a shadow that is 532 feet long. Find the height of the tree if the angle of eievation of the sun is32~. Round your answer to the nearest foot 4. What is the measure of the base ang1~ x, of the isosceles triangle shown below? Round your answer to the nearest tenth of a degree, 20 20 36 -::! 'JiJ 5. A person flying a kite originally lets out 232 feet of line. At this point the person observes an angle ofelevation to the kite of48~. The person then lets .out additional line for a total of 312 feet. At this point the person observes an angle of elevation of 53", (a) Find the initial height of the kite, AB, to the nearest/ool. 312 ft 232 (b) Find the final height of the kite,AC, to the nearest/oot. (c) Using your answers from (a) and (b), fmd the amount the kite rose after the line was let out to the nearestjoot. 6. From a point 435 feet from the base of a building it is observed that the angle ofelevation to the top of the building is 24° and the angle ofelevation to the top of a flagpole atop the building is 2"', (a) Find the height ofthe building, h, to the nearest/eel. 00 (b) Find the height to the top of the flagpole, k, to the nearest foal. 00 • 435 • (el Using your answers from (al and (b), find the length of the flagpole to the nearest/wi, u Alswm I, Ornl g. Righi. TriangIt Tri&<!ltCim¢try _ 1.8 The: AIl.!! AI!\.cln'a Proj«t" L\G<Nl:1l.Mlte, f..'Y 11540 h k What Do Two Bullets Have When They Get Married? Work each problem and find your answers at the bottom of the page. Shade out the letter above each correct answer. When you finish. the answer to the title question will remain. G) Find the length of the hypotenuse of each right triangle: A. B. .... ~~ "3 11 em gcm ® A rectangle is 3 meters wide and 10 (j) Each side of a checkerboard measures meters long. How long is the diagonal of the rectangle? 40 em. What is the length of its diagonal? ® A18rectangle is 13 centimeters wide and centimeters long. How long is its ® An inclined ramp rises 4 diagonal? @ 4cm C. meters over a horizon~ distance of 9 meters. ~ 4 m long is the ramp? . A guy wire is attached to an upright pole 6 meters above the ground. If the wire IS anchored to the ground 4 meters from the base of the pole, how long is the wire? ® ... 4m .... 9m A box is 120 cm long and 25 em wide. What is the length of the longest ski pole that could be packed to lie flat in the box? i : , @ The window 01 a burning ® wide A television screen measures 30 em and 22 cm high. What is the building is 24 meters above the ground. The base of a ladder is placed 10 meters from the building. How long must the ladder be to reach the window? diagonal measure of the screen? @'AShiP leaves port and sails 12 kilometers west and then 19 kilometers north. How lar is the ship Irom port? --t 10 m T EE .~~ I~~ ·11 164 H E F B E d3 EOE ",,,, ",,,, ~E ",,, ~~ ~~ <oro "' ·11 "",,11 ~;1; II E~ .... ·11 EE A S B EE E D B EE ~~ ~5 ~~ ~'" ~~ "'''' ~~ ~ID ~~ "'''' <:?~ ~~ PRE··AlGEBRA WITH PIZZAZZI © CreatNe Puhlicalions H I 41 EE Ou EE E<.O v co'" ~'" ",01 II 'II N~ E 0>0> . ~'" ·11 ~- ·11 0'" <oN ·11 00 o · <om ",<.0 (")u'> ·11 11 U G ;~ S « ~ EE ~~ ~) ~~ -EE U U "'N ~~ ·11 0 0 ~~ ,II '" r 11 -<:? '11 Greeli Deader TO DECODE THE MESSAGE AT THE BOTTOM OF THE PAGE: Figure out the length of the missing side of any right triangle below. Find your answer in the answer column and notice the GREEK LETTER next to it. Each time this GREEK LETTER appears in the code, write the letter of that exercise above it. ~ a = 7, b = _ _ , C = 12 () .p @ a = 5, b = , , C = 14 @a=8,b= ,c=V164 Q) a = 4, b = 11, C = _ _ 7l'. u ® a = 12, b = 5, C = ~.._ © a= , b = 7, C = 10 ® a= , b = V4B, C = 13 ® a = _._, b = 12, C = 15 @ a = 10, b II '> 7) € It = __._, C = 16 7 \W)a=l,b= ,c=2 ;',) '.@ a = l,b= l,c= _ _ ~ (!) a = 0.8, b = 0.6, C = _ _ ®a= , b = 1.5, C = 2.5 ' a = _ _ , b = 24, a = a = = 25 ,C = 15 C v156 ~ 12.5 v'3 :.. 1.73 V144 = 12 V137' 11.7 \1'1= 1 v'1ff . 13.1 V49=7 v'51 ' 7.14 V16=4 v'95 . 9.74 v'168:.. 13.0 v'81 = 9 V169 = 13 IJ. {3 \1150:" 12.2 V100 = 10 V121=11 v'2:.1.41 p V4= 2 I< ®a=~_,b=11,c=17 ® ® ® Il a KEEP WORK/NG AND YOU W/LL DECODE THE SECRET MESSAGE. ® ANSWERS v7s, b = v87, b = Vs7, C =__ @ a = __ ~, b = 3, C = 5 SECRET MESSAGE , a K 7T Ie p 7 a U II 1J <I> K 7 t e <; K.p up <; II 771 PRE,ALGEBRA WITH PIZZAZZ' © Creative Publications r + 165 What Did Lancelot Say To The Beautiful Ellen? TO ANSWER THIS QUESTION: Cross out the box containing the answer to each problem. When you finish, write the letters from the boxes that are not crossed out in the boxes at the bottom of the page. <D For each right triangle. fJnd the length of the side that is not given: A 13m 8 E D C ___- - - - -7 E '"3 <0.,," 10 m ® An 18-foot ladder is leaned against ® The bases on a baseball diamond are 90 feet apart. How far is it from home plate to second base? a wall. If Ihe base of the ladder is 6 feeUrom the wall. how high up on the wall will the ladder reach? ® Orgo has let out 50 meters of kite string when he_ observes that his 'kite is directly above Zorna. If ® A quarterback at '\ "1 point A throws the football to 8 / a receiver who catches it at I point 8. How long was the pass? Orgo is 35 met~rs from Zoma, how high is the kite? 25yd Af4:d r o ;S (f) From Canoeville it @) A What is the height of this parallelogram? B What is the area of the pa(allelogram? ~fj7 ~9m-""8m"" is 2.4 kilometers to White Beach and 3.0 kilometers ,-'" LODGE ..&:--"==-TIr w ~ to the lodge. How far is It from White Beach across the lake to the Lodge? CANOEVllLE Ie KW au IT UR AT GR "\1821 yd· v"'fi6km v'i44m '0/136 m "\13.24 kin V842 yd '0/16200' V9sm " 28.7 yd ~ 1.78 km "" 12 m .,;, 11.7m = 1.8 km ,;, 29.0 yd " 127" EA TC UT EA LN GT V105m "\1260' "\1275· "\11275 m \11325 m v'4O "'" 10.2 m = 16.1' " 16.6' 35.7 m "36.4 m "6.32 m ST AS 234 m2 166 PRE·ALGEBRA WITH PIZZAZZ' © Creative'! Publioi!loos ER 204 m~ c, ~ 9.75 m ... · .. ~,-. '--, '- Why Did the Saltine Lock Itself in the Bank Vault? TO ANSWER THIS QUESTION; FOLLOW THESE INSTRUCTIONS: For each exercise, select the correct ratio from the four choices given, Write the letter of the correct choice in the box that contains the number of the exercise. , , , CD sin A ® 1£ @ ..£. B I @sin A @..i ® 1 13 13 ~'" 5 ~I "i4' 3 5 A ® COS A I'i\ 13 ® 5 A i I.!.':!I COS @ 5 @ 4 0::,; - E 12 c! U B ®tanA 5 '~@tanA 3 """'",q ,...._"'..."'...."....\,.. . . . . .""...._''''''.. ."....'''''''~. .\\... 12 ~" .."'"'-"..." ..."'.. " ...".''''....~..., ..... ,..'"''....." ..." ...''!...,......., ....''....' '....''''..."..."'",....."""",...." ..." ,"'"",." . ".... @) sin S @ ~ @ ..£. 5 ®cosB ® tan S 13 ©1£@1£ 13 B 5 n n c!I 12 1f0.1 @ _~ \Cl 3 v2 (j) S'ln A @ 8j ii'~ ~,'!l ~ :r 2'Jl !l:J:j hi ~~ ~ ,~ ® cos A ® ® tan A ,~ ~ ® v'2 1 A 3 ~~...,~...., '...'''..,,... , .....,,..''~" ..." " "..." l<,,,... ......' ''''''\,\\...,,...'''.... ....,,...,,....' '... ~ -s--1 l A~51 . "...." ...." ..." ..."""".,... , .... " ..........." . . ',..." ...,>0.. " ... " ..." ... " .. " ........" "... " . . ., ..., ''''., .. " .. " . ." ... ,~... ''''~,...' ' ' ' '...., ' ' ' ' "~ (; B B @sin'S @ 1 cosS C! ""~..." ...""~" ..." ' ' '....'''...' '...' '....' '....' '...''"""".." ... " ...." ..." ......" ..." .." ... 15 C 'A ...'\,..."..."....'''....''''\,~''...,~...,., V5§ 7 Br-----..VS3" v53 2 C ©L®L 2~A ® ~ 15 ® lL 8 '20' ~ cos A ® ~ @ J§.. @tan A '" ,u @)..L @ .£. @tan S l@sinA j l' V2 5 ,'lOIo,..." 'II,\,,....~...','''''I¢.'\,~,"*-""""" ~~....""..,w••' ...."_''''''...........~''...''..._'~....' .......\ \...''....'~....'_'''''''"'',~'''''''''''''''''..." ..." I '" 20 I 17 17 7 ..._~,...." ....""""'""'..." ...." ..,....." ..".".. C [[ 16 1 B 30 :7 A /"'~ c.. . . . ' . . .".. .... " ...' ...."""""'''.....- ' ' '..' '...- ' '....' ' .... ' .....'''...'''''''''''''......- ' '... ~-'' '~ '~ @>·s(f)v2®_16!@sinA®...L@...L sin 1 v'2 I v2 v'2 ~ ® ~ ~ t1V3\ '" "'" /' 11 cos B ® CD v'2 1 i @tcosA I'i\ v'2 ® E ...L I A ! 0::,; F 1 A/ "6 ® tan B 3 3 v'2 c I ® tan A 1 V2 j.~-.~~T7~~~~~~~~-r~~ t 10 1211615tt8t1611TI3J~t9tt41111122115 [4[211 23 t a1 ~ What Did The Leopard Say After Lunch? TO ANSWER THIS QUESTION: Use the table of trigonometric ratios to do each exercise. Find each answer at the bottom of the page and write the corresponding letter above il. Find the following: © sin 35° ® cos 70" (f) tan 10° ® cos 65 Angle Sin Cos Tan ()" 0.000 1.000 0.000 50 0.087 0.996 0.087 0 1()" 0.174 0.985 0.176 9 15' 0.259 0.966 20" 0.342 0.940 25' 0.423 0.906 3~' 0.500 0.866 35' 0.574 0.819 0.577 0.700 40' 0.643 0.766 0.839 45· 0.707 0.707 1.000 50" 0.766 0.643 1.192 55' 0.819 0.574 1.428 60" 0866 0.500 1.732 650 0.906 0.423 2.145 70' 0.940 0.342 2.747 0, 75' 0.966 0.259 3.732 ~~ ~ 80' 0.985 0.174 5.671 85' 0.996 0.087 9()" 1.000 0.000 B Use the figure at the right for the remaining problems. ,,• a • • AL----c""--DC "@ If m LA = 40 , then ca = ® If ~ = 0.966, then m LA 0 = <D If m LA . 55 then ~ = ® If ~ = 0.707, then m LA = <D If m LA. = 80 ' then cb = ® If ~. = 0.500, then m LA = , 0 , ~., 0 @ If m LA ( " (, I~, ,•• 25 then ~ = 0 = ( , <D If Ii = 1.428, then m LA = ® If m L B = 30 , then ca = CO If ca = 0.996, then m L B = . ® If m LB = 75°, then ~ = ® If : = 0.839, then m L B = ® If m LB = 15° , then c~ = ® If ~ = 0.906, then m LB = 0 '" '"'" "'" ...""" r- •oo '" d d '" " 0> M ~ ~ ~ 0 N <') (") <t <0 d . 168 PAE·ALGEBRA WITH PIZZAZZI © Crea.tive PI.lOlicahcns tb ~, L -~~ .".. 00 d .\ ' ~~ 0 (f) tan 50" Q) sin 85° . "" •0 00 '" 0> 0 d d 0> '" '" 0> 0> ~ to tb OJ") ~ ~ 0.364 ~ 0.268 0.466 '= , ~ ~ ~ ~ ~ ~D~ 0 0 ~ 1~_~~~J ~~ • oo '" '"... '" .... ci"'"'" "" '" ,U., D 'd" ~ 0 0 , ." Did You Hear About. .. . . .... .. .... ..... .......... ... . ............ D E F C B ..: :A " &.IO"IL&.A..1. , , G ~ ... : ~ H J I +.. ''Y'''''' · ... "TTY? 'Y K M L ,,,,,,,,,, ... ,.,,.,.,.,,, ... ,,., 'TTY""Y ~ • ~ :. ... ?• TTYTT",. • 'T'TTTTT+ DIRECTIONS: In any triangle hefow. find the length x, Round it to the nearest 0.1 meter. Find YOlJr answer in the answer column and notice the word ne!<t to it. Write this word in the box that has the same letter as that triangle. KEEP WORKING AND YOU WILL HEAR ABOUT A NOVEL NAME! LJx ® X 30 m X @ '. ® ~X ) ® 8m X @ X 25m 48' X CD ® 16.8 m-ROBINS 13.7 m-BECAUSE 16..7 m-ROOSTER • 123.8m-SO I 46.2 m-NAME , 22.7 m-SUN , I 44,9 m-BEST ! 15.6 m-THE i 87.3 m-WAS: I 3.2 m-BANKS I 110.6 m-GAVE • 98.5 m-WRECKED-r 5.8 m-ROBINSON • 9.9 m-FARMER ' 115.5 m-CREW i I 75m 117.6 m-HE , !12,7m-WHO 15.4 m-PET 95.1 m-THE Q) X ~ X CD A submarine dives at an angle of 13'. How far is it beneath the surface at a pOint 500 meters along the surface from where it submerged? X 11.8 m-HIS ® AI a pOint 20 meters from a flagpole, the angle of elevation of the top of the flagpole is 50". How tall is the Ilagpole? X SOOm 20m PRE· ALGEBRA WITH PIZZAZZ' © Creative Publications 169 Books Never Written , .--- , :." My Ute in the Jungle by 56 Over the Cliff by 60° 3r Q 23° 72· 36° 45° 6° 66° 15° 37· 21° j 77° 29° 29° 34° 62° 53° Catching Butterflies by 6° 45° 45° 29" 53° 53° 29" 6° 45° 55° 6" 56° 6" 34· ABOVE ARE THE TITLES OF THREE "BOOKS NEVER WRITTEN," TO DECODE THE NAMES OF THEIR AUTHORS, FOLLOW THESE DIRECTIONS: In any triangle below, find the measure of the lettered angle to the nearest degree, Each time this measure appears in the code, write the letter above it, Keep working and you will decode the mimes of all three authors. 3 U 7 6 L-J'--_W E O~ DL.1--LJ 4 40 S 14 T ,-r--r-7 G 7 12 L-.L- -W 17 30 ~ 25 20 170 PRE·ALGEBRA WITH PIZZI\ZZl © Crealwe Publicohons ~ j j 20 P 8 10~ H R F z A driveway is built on an incline so that it rises 2 meters ==::a M 23 5 over a distance of 20 meters. What is the degree measure of the slope of the driveway? (See figure below,) 11 15 ~4 J 23 12 2 75 10 NL-l---~ ) Name: Date: Introduc-t-::i.-n-:-t.-A7)~g-Ceb:-r-a--- Unit 10'--""P:-ol"-p-.-u-rri LESSON 4 - SQUARE ROOTS The symbol r means "SQUARE ROOT" EXAMPLE: 14 ~ Solution: "4 = what number squared (times itself) equals 41 , v4= 2 EXAMPLE: 19 = ? Solution: EXERCISES: L J25 = 2. 3. '/-1- = 4, IT ~ What number squared (times itself) equals 9? Find each of the following square roots. 7. 149 = 8, ro = 9. Vl2l = 10, "144 = 5. Ii 64 = 11, 10 = 6. Ji1iii = 12. m 13. ';49 + 2 14. 3 ---------~- +;130 ro ~ = Name: Introduction to Algebra Date: Unit 10 - Potpourri LESSON 5 - SQUARE ROOTS Below is a list of numbers and their squares: Squares Numbers 1 I' = 2 2' = 4 3 3' = 9 4 4' = 16 5 5' = 25 6 6' = 36 7 7' = 49 8 8' = 64 9 9' = 81 10 10' = 100 II 11' = 121 12 12' = EXAMPLE: .144 r53 is closest to what whole number? Solution: 53 is closest to which number under the list of squares? _ __ 53 is between 49 and 64. 53 49 64 '--..J '-..../ 4 C subtract 53 from 49 and 64 II,.J Since 53 is 4 away from 49 and 53 is i 1 away from 64. 53 is dosest to 49. Then, vB is closest to "49 = 7. Answer: 7 INTRODUctION TO ALGEBRA UNIT 10, p. 2 EXAMPLE: Which of the numbers best approximates 0) Solution: b) 4 5 .f.i'7'I 0) 6 d) Remember, approximates means closest to which nU!llber. 27 is betwl"'n lb. ~quares 25 and 36 25~27~36 Since 27 is 2 away from 25 it is closest to 2S so ... >/29 is closest to ,,"'15 = 5 Answer is b EXERCISES: I) "Sis closest to which whole number? 2) To which whale number is 195 closest? 3) Approximate vOf (Pick one below) 4) 6) 6 c) 8 b) 7 d) 10 Which of the fallowing whale numbers best approximates a) 5) a) 4 b) 8 0) 9 ro 7 d) 6 d) 8 Which whole number best approximates .;TI '189 is closest to which of the following whole number? .) 2 b) 9 0) 4 7 UNIT 10'1 p.3 INTRODUCTION TO ALG£BRA EXERCISES (Continued) 7) The ,Illf is closest to which whole number? 8) Which whole number best approximates ,rTI4 ? 9) Which whole number best approximates "59 ? aJ 10 b) 8 c) 1 d) 4 p.IO Name: Introduction to Algebra Date: Unit 10 - Potpourri LESSON 6 - PYTHAGOREAN THEOREM The Pythagorean Tbeorem is used for right tri{mgles oniy. These are triangles that have a 90' angle. A triangle can be labeled as follows: aCross from the 90° angle. It is called the hypo tenus •. <:: = the side b The formula is used when trying to find the third side of a right triangle when the other t\vo sides are given. Formula: EXAMPLE: Given the triangle below, find side c. c 3 4 wherea~3 Solution: and b=4 plug numbers into the formula 9 + 16 = 25 c' =e (what number times itself is 251) Answer: 5 = c p.l\ l~'TRODUCTlON EXAMPLE: TO ALGEBRA UNIT 10, lESSON <i.P 2 Given the triangle below, find side a: b 0:8. ~u.re c=tO { Sol.e t:.. • 0:: "1 do,"~ Q2t ("4" -(,,4 [ +he. opl"'s,k - \00 =-.. + - O'l. ~ at;, EXERCISES: Find tbe missing side of each of tbe triangles below: 3) 'I 15 4 5) c \e. 8 INTRODUCflON TO ALGEBRA EXERCISES - Sct 2: Unit 10, LessDn 6, p. 3 For each right triangle, find the missing length. c a b 17•• =6, b=6 20•• =2,b=6 23.• =7, b=l 26.b=4,c=12 18. 21. 24, 27. a = 12,.b = 16 b = 4, C = 8 a = 6, b = 12 a=8,b=8 19.• = 9, c = 15 22. b=1,o=3 25•• = 9, C = 18 28•• =9, b = 4<l Unit 10. Lesson 6, p. 4 INTRODUCTION TO ALGEBRA NOTE: If a triangle is a right triangle, then the lengths of its sides satisfy tlte Pythagorean Theorem~ EXAMPLE If the lengths of the sides of a triangle are 7, 24, and 25, is the triangle a right trial1gle'? = See If 7' + 24' - 25', _ _ _ _ _ _4 7' = 49 24' 576 25' = 625 49 Thus, the triangle is a right triangle. EXERCISES - So13: + 576 = 625 Tell whether each triangle described is a below is a right triangle. The lengths of the three sides are given, Reeallilia! the hypotenuse is the longest side of a right triangle. The hypotenuse is located across from tne right angle. 1.4.5,6 5. 12, 14. 16 9.11,60,61 2. 6. 8,,10 6. 9,40.41 10. 6,6,9 3, 3, 5. 7 7. ,10,24,25 11. 30,40,60 4. 12. 16. 20 8. 10.20.30 12. 14.48,60 If a:! + b 2 = 0 2 , then the triangle is a right triangle with c the hypotenuse, b p.14 Unic 10, Lesson 6, p. 5 INTRODUCTION TO ALGEBRA Applications of the Pythagorean Theorem EXAMPLE A rectangular field is 50 feet wide by 100 feet long. How long ,s a diagonal path' connecting two opposite corners? Give the answer to the nearest tenth of a foot. a2+D~=c2 50 2 + 1002 = <;2 2,500 + 10,000 = c' 12,500 = c' \112,500 = c 111.800 = c Round to the nearest tenth. _ _ _ _ _ Thus, the diagonal path is approx. 111..8 feet long. EXERCISES· SET 4 Give answers to the nearest tenth. 32. Paul walked 8 miles north and 3 mUes west. How far was he' from his starting point? 34. AT V. screen is 15 inches by 12 inches. What i. lIS diag6nal length? 33. A 12·fool ramp covers 10 feet of ground. How high does it rise? ,35. A 15-foolladder is 5 feel from the base of a building. At whal height does it touch the building? p,15 Chapter 6 TRIGONOMETRIC FUNCTIONS 1. In the accompanying diagram of right triangle CDE, CE" 144, mLD = 90', m LC = 30' and leg ED = 72 em, 'Which equation can no! be used to find leg CD? 144 72 (1) tan 30' = 72 CD (2) cos 30'" CD 144 (3) tan 60' = CD 72 (4) tan 30' = CD 72 135 C~3""O_ _ Chapter 6 TRIGONOMETRIC FUNCTIONS pro ems eow. Use the triangle ABC shown at the right. 1. B 12 6. Write the trig ratio of cos A Write the ratio of sin A , C I ~. ,, , I, , i 7. Is cos A "" sin B? 2. Write the trig ratio .cos B ,i , ,, 3. 8. Write the ratio for tan A . 4. Write the trig ratio tm B ,, 9. What is the wIne of tan A • tan B? I i, 5. Write the ratio for sin B 10. What is the largest positive value of sine of any angle? 11. From the diagtam given below, find the mLF to the nearest degree. , 14. In right triangle ABC,LC is a right angle. If AB = 11 and AC = 7, find mLA , D~: , ,1i sin A =cos B, what is the mLA + mLB? to the nearest degree. F 12. The sine rntio of any acute angle is equal to the cosine ratio of its c?mplement. , , ,, , 15. The value of the sine ratio of an angle can I, neve! be equal to the value of the cosine (True or False) ratio of the same angle. (True or False) =BC = 9 and side AC = 7. Fmd the m LA to the 13. In right triangle ABC, side nearest degree. ,, 16. In reetangle ABCD, diagonal AC is drawn. If CD = 8 and AC =13, find m L ACD to the nearest degree. ,i '--_. 136 , Chapter 6 TRIGONOMETRIC FUNCTIONS Right Triangle Trigonometry, Finding a Side 1. Triangle ABC has a right L. at C. If 6. ma =32 and AB = 8, find the length of In triangle GHI, L. G is • right angle. If rod. 47 and HI • 6. lind. to the nearest integer, the length of GJ. AC to the nearest tenth. .. ·• . , I • 2. hypotenuse DF of rightLl.DEF if 17. 7. If roLF =77, find the length of DE to the nearest integer. ·• In rectangle ABCD, diagonil A<:: forms a 27 degree angle with side AD. If side CD • 4, lind the length ofAC to the nearest integer. ·, ,JI<c 3. From the di~m below, nnd RS to the nearest tenth. x~ • 8. Isosceles A DEF has base DF = 22. Altitude EO is drawn. If mLD = 32, find the length of DE to the nearest integer. i · sh ·· i T • - 4. In right triangle ABC, hypotenuse AB = 12. If mt:.A =70, find BC to the nearest tenth. 9. Fllld the value of hypotenuse HF in RT 5. Given the diagram below, find the length of 10. Side BC in right Ll. ABC =5 and mLC = 90. If mLE = 50, find the length of AC to the nearest inregcr. DE to the nearest tenth. ~~lO ~F Ll. FGH if roLF =16 and HG =14. (Answer to the nearest integer) , I) I 11. In right triangle ABC, mLC = 90", mLA • 70" and AC "78. Find the length of AB nearest integer. Show how you arrived at your answer. 137 to the , I . · ·· Chapter 6 TRIGONOMETRIC FUNCTIONS 1. A building stmd, on level gound. The measure of the angle of elevation of the top of the building, takx:n at a point 400 feet /rom the foot of the building, is 31". Find, ro the nearest fuo~ the height of the building. 4. 2. 5. A 20 foot ladder leans against a house. If the foot of the ladder is 8 feet from the house, find, to the nearest degree, the From the top ofa 100 fuot high pole, an observer measures the angle of depression of a car on the road as 28 degrees. Find, to the nearest fQot, the distance from the cat to the base of the pole. From a point on level ground, the angle of elevation of the top of an 85' pole is 62°, Find, to the neatest integer, the distance from that point to the foot of the pole. measure of the angle that the foot of the ladder makes with the gound. 3. From the top ofa 120 foot lighthouse, the 6. angle of depression of a boat out at sea is 26"'. Find, to the nearest foot, the distance from the boat to the foot of the lighthouse. 138 A 40 foot long wire stretches from the top of a vertical pole to a stake in the ground 18 feet from the foot of the pole. Find, to the nearest degree} the measure of the aOlte angle that the wire makes with the gound. Chapt<r6 TRIGONOMETRIC FUNCTIONS 1. A wire reaches from the top of a 26 meter telephone pole to a point on the ground S 6. The dimensions of • rectnngIe ace 14 coruimetm by 4S centime1=. Fmd, in oent:imet=, the length ofthe diagonal ofthe rectangle. 7. A ABC is a right triangle with the right angle at C. IfAB = 13 and Be = 12, findAC. 8. The hypotenuse ofa right triangle is 25. If meters from the base of the pole. What is the length of the wire to the nearest tenth of a meter? 2. The lengths of the legs of. right triangle are 3 and 6, Fmd, in radical form} the length of the hypotenuse of the right triangle. 3. In the accompanying diagram ofa rectangle ABeD, AB • 6 and Be • 8. What is the ofAC? (1) Express in radical form, the length of one leg of a right triangle if the hypotenuse is 9 and the other leg is 5. 5. Find the length, in radical form, of the hypotenuse ofan isosceles right triangle whose leg equals 3. ..f5 (2) ..J 1025 9. B 4. one leg is 20, the other leg 1s: (3) 15 (4) 45 The length of the hypotenuse ofa right triangle is 7 and the length of one leg is 4. What is the length of the other leg? (1) 11 (3) 3 (2) ..J65 (4) ..J33 10. Which of the following could be the lengths of the sides of a right triangle? (1) 3,5,8 (3) 2,4,6 (4) 5,5,5 (2) 5, 12, 13 139 Cbapter6 TRIGONOMETRIC FUNCTIONS 1. 2. Fllld cos A, if A is a positive acute angle andtanA= 6. If x is a positive acute angle and 7. y. cos x "" 3. 4. 17 ' Find the value of COS x, if tan x :::: j and If cos x =: ~ where x is a positiv-e aCute angle. lind the value of sin x. If A is a positive acute angle, find the value of tan A, if cos A. 8. Find sin x if cos x :::: i and x is a positive acute angle, 9. ]f0' "As 90· and tan A= ~. find the value of cos A. x is a positive acute angle. 5. i. find the value of cos A if A is a positive acute angle. find the value of sin x. If sin x : : : and x is a positive acute angle, find tan x, ]f sin A = 10. Which trig ratio can have a value greater than I? (1) sin x (3) tan x (2) cos x (4) none of these 140 Cbaprer6 TRIGONOMETRIC FUNCTIONS L In triangle ABC, m LC • 90, AC· x, BC = (x + 2) and AB • (x + 3), a Write an equation in tenns of x which can be used to find AC. b Express the equation in part a in standard quadratic fonn 2, In right triangle ABC, AC = x, Be • x + 1 and hypotenuse AB = 2x 1. F'md the length of AC, [Only an algebraic solution will be accepted.) 3. The length of the hypotenuse of a right triangle is 10. The length of the longer leg exceeds the length of the shorter leg by 2, Find the length of the shorterleg, [Only an algebraic solution will be accepted] 4. The hypotenuse of a right triangle is represented by 3x + 4. One leg is represented by x and the other leg is 24. a Find x. b Find the hypotenuse. 141 Cbapter6 TRIGONOMETRIC FUNCTIONS Pythagorean Theorem Application, (continued) The length of the hypotenuse of a right triangle is 13. The length of the shorter leg is seven less than the length of the longer leg. Fmd the length of the longer leg. [Only an algebraic solution will be accepted.] 5. ,. The length of the hypotenuse of a right triangle is 15. If the longer leg is three more than the shorter leg. Find the length of the shorter leg. [Only an algehraic solution will be accepted.] 6. ~ The hypotenuse of a right triangle is 5 and legs are represents by" and " • 1. 7. a F'mdx b F'md the perimeter of the triangle e Find the are. of the triangle , 8. In rect2l1g1e ABCD, two adjacent sides are represented by x and x + 5. If B A 25 x D ~. h .. 5 diagonal AC = 25, lind a the value ofx b the area of rectangle ABCD C . 142 Integrated Algebra I - Right Triangle Trigonometry Uuit Day One To learn the trigonometric ratios and how to use them to solve right triangle Goal, problems Standards: A.A.42 A.A.43 A.A.44 Opening: Draw a right triangle. Deline opposite, adjacent, hypotenuse, making sure thai students understand that «opposite" and "adjacent" are relative to the angle you are referring to. sin A B !!Ill! ~ hyp cos A = adj byp e a tan A ..,; !.I!Jl adj c b A Introduce: SOH CAB TOA Explain: sin ::= sine cos = cosine tau = tangent Draw the diagram below and have students fmd the trig ratios for angles A and B. B sin A = 5 cos A = 13 , C h tan A = 12 A sin B == cos B = _ __ tan B = -- Ask studcDts what they notice (Do they see. a pattern - compare and contrast, Why?) ,,,,ires are meant to train students to draw a right triangle based upon the information These e.. provided. For III, provide a "blank" triangle, wbichstudents will then label with the infurmetion. They will notice that they are not provided with the length of the hypotenuse. Vou may want to "hint" that they might want to use thel'ythagorean Theorem. When they do 118 during work:time, they need to use the pythagorean Theorem again. 1) In right triangle ABC, mLC = 90", BC = 3, and AC = 4. Wh.tissinB? . For #2, students must dntw their own right triangle.) 2) In right triangle ABC, ifmLC = 90', AB = 5, Be =3, and AC= 4, what is the eos A ? Work I'.riod: Students should complete the Arlington Algebra Project, Unit #8 Lesson L3 Homework #1 - 14 (attaclled) NOTE: Exercise #8 in Unit #8, Lesson L3 will require the students to use the Pythagorean Theorem (0 fmd the hypotenuse in order to find the cos Q. Work Period N~: __________________________ Date: ---- Similar Right Triangles - Introduction to Trigonometry Algebra 1 Skills For problems 1 - 6, use the triangle to the right to find the given trigonometric ratios. 1. cosN"'" N 15 2. sinN"" 9 3. tanN= M p 12 4. sinP= 5. cosp= 6. tanP= 1. Given the right triangle shown, wlJich of tile following represents the value of tan A 1 (I) 25 24 (3) ~ A (2) 24 (4) 24 25 7 7 24 B 8. In the right triangle below. cos Q =::? (I) Il: (3) 12 17 (2) 2. (4) 12 13 5 12 Tti. ~ebra t, tlait#l!: -Ri1lht Trigotwmeuy - tJ Th: Arlingtoo I\Jgclna Pl'ojeel, lAGr.mgmlk, NY 11S4O 2S 24 C 12 Q S R For problems 9 - 14, use !he figure at the right to determine each trigonometric mUo. Make..,. In reduce your trig ..tios to !heir simplest fann. 9. sine= 10. COSC~ 11. tanC~ 12. sin A= 13. cosA= 14. !anA= 4 A 2 B Closing: (This ex"",ise from from the Arlington Algebra Project, Unit 11&, Lesson L3 Homework. It is #15.) Consider the two right triangles shown below: 8 10 4 6 Why;' tbesin of35'tbe same in both cases? (Emphasize that the triangles are similar because the sides are in proportion. Therefore, the sine ratios would be equivalent ratios.) Homework: Attached worksheet (Note: Again, the students will need to use the Pythagorean Theorem to solve #5 & 6.) Integrated Algebra I - Trigonometry Unit Introduction to tbe Tril!;onometric Ratios Homework Nwne._________________________ Forquestions#I-4,find •• sinA Date,___________________ b. cos A c. tanA e. cosB = f. tanB ~ d. sinB = h. cos A == e. cosB = c. tan A = f.tanB~ a. sin A d. sin B c. tan A 3. 8 5~ C 12 3) A A 29 20 90" 21 B C 4) f. tanB = b. cos A 2) e. cosB d. sinB a. sinA 1) d. sinB = ~ ~ sin A = ~ b. cos A = c. cos B c. tan A r. 3. ~ sin A = tan B = ~ ~ d. sin B = c A ~ P b. cos A = c. cosS = t~A f. tanB ~ B c. = 5) In AABC, mLC = 90" , AC = 4, and 6) In 8RST, mLS = 90· , RS = 4, and BC = 3. Find sin A. ST = 3. Find sin A. Integrated Algebra I - Rigbt Triangle Trigonometry Unit Day Two Go.I: to learn to use the graphing calculator 1l> determine the sine, cosine and tangent ofa given aeute angle in a right triangle as well as to tind an angle measurement given any two sides of a right triangle. Standards: A.A.42 A.A.43 A.A.44 Opening: Use Exercise #1 in the AIIinglon Algebra Project, Unit #8, Lesson L4 Classwork as • "Do Now" and to launch the opening fur this lesson. (Note: TIrey have labeled the right angle with "B", not "C'') Proceed with Exercises 112 & 3 (in Lesson L4 Classwork) explaining that these answers are the ratios (in decimal form) of me two sides ofme triangle. Exercise #4: Have students look at the diagram and write the equation: ? tan40 ~ d. 4 Empbasize that ifthis equation i. true, then both sides must bo equal. Students should enter tan 40 into their calculators to see that tan 40 ~ .839099631 willen does not equal ¥.. Thus the right triangle does not exist with the given measurement Now introduce the Inverse Trig. Functions following the remainder of I.esson 1.4 Classwork, Work Period: Arlington Algebra Project, Unit 118, Lesson 1.4 Homework Exercises #1-17 (I" page), Closing: Arlington Algehra Project, Unit 118, Lesson !.4 Homework '. Exercise # J8 for discussion. Solicit from students that by definition ofsine or cosine, you cannot have sin-I ~ I, since the hypotenuse is the longest side of a right triangle. Homewf)rk: Attached Worksheet Opening Nwme: ____________________________ Date: ______________ Trigonometry and the Calculator Algebra 1 rn the previous lesson, we introduced the trigonometric ratios. We also discussed a IllnCtnonic that is helpful to remember the trig rauos: SOH'(;AH-TOA. The first exereise review, how to wrhe these ",uos given Ibe lengths of lb. sides of. right triangle. , Exercise, #1: Using the diagram below. stale the value for each of the following trigonometric ratios. 13 A 12 c (a) sinA: (h) IanG= 5 (el (d) rosA = B (e) lanA = cosG= (f) 'inC = Today we wiU discuss how the graphing calculator is used with trigonomctry_ However, before we s~ it is important that our calculator is the right MODE. To change your calculator into DEGREE MODE: I. p{ess the MODE button on your calculator. 2, Use the arrow keys to highlight DEGREE and press ENnl:R 3, Exit the menu (this or any other) by hitting QUIT, Now that we are in DEGREE MODE, we can start evaluating some trig ratios without referring to any right triangles whatsoever. The SIN, COS~ and TAN buttons are located i.n the center of tbe key pad. Exercise #2: Evaluate sin30', cos30~ and tan30·. Round any non-exact answer to the nearest thousandth. sin(30) (;,05(30) .8660254038 .5773502692 tan(30) • Algt'bm I, Uml tUl-Rl.s!lt Triangk- TrigoDOttlCltY-LA The Arliogloo Algebr.t ProjCCl., 1..I~&mIk, MY 12540 .5 Burdse #3: Evaluate each of the fonowing. Round your answers to the nearest thousandth. (a) tan( 40") (1)) oos(20") (0) sin(63") It is important to remember that: each of the answers from Exercise #3, represent the ratio of two sides of a rigltt lrillllgie. In each case the ratio bas already been divided and the oatoatator is giving the d""imalronn of the ratio. Exercise #4: Could the right triangle below exist with the given measurements? Explain your answer, C 3 40' B 4 A The Inverse Trig Functions - Thus far, we have been evaluating the sine, cosine, and tangent of angles, By doing so, we have been finding the ratio of two sides in a right triangle given an angle. Using the calculator. we can reverse this process and find the angle when given a ratio ofsides, Exercise lIS: Consider the following: (a) Evaluate sin-I (~) using your calculator, The screen shot is shown at the below. sin-1(1/2) I (b) How do you interpret this answer? E.Xercise~: 'Find each angle that has the trigonometric ratio given below, Round all answers to lhe nearest tenth of a degree, if they are not whole numbers, 5 2 (.) tanA=- I (b) oosB=2 (e) sinE =3. 3 · Work Period Name: Da!e: _ _ _ _ __ --------------------~ Trigonometry and the Calculator Algebra 1 Skills For problems I through 6, evaluate eaeh trigononretric function. Round your answers to the nearest thousamith. I. sin (55") 2. oos(45") 3. t3ll(2o-) 5. t3ll( 60') 6.•m(23') For problems 7 through 15, find the angle that has me given trigonometric ratio. RQund aU non-exact answers to the nearest tenth ofa degree. 4 8. ensO=9 8 5 13. IllnA=3.127 11. wnR=- 12. sinT=l 14. sinB=O.724 15. oosL=0.876 16. If sin A l::: ~ then mLA is closest to which of the foltowing? 5 (I) 32' (3) 24' (2) 76' (4) 56' 17, If tan A =3.5 then m/-A. (0 the nearest degree, is (I) 74' (3) 24' (2) 18' (4) 55' 9. wnK=1 2 Closing Reasoning 18. Recall trom homework problems 7 through 15, th.t an inverse trigonomelric functio. gives the angle measurement that corresponds to the ratio aftwo sides ofa right triangle. (a) The screen shots below show what bappens when you try to evaluate sin-I (~J. Verify why this ,2 bappens when you evaluate sin ~l (~ J. I . . the !,act ~ tho t sm . ( Angl\; '_) ;; opposite side ) wuy • (b) CODSldenng ERR:DOMAIN hypotenuse IIIQuit NGato does it make sense that sin-I (~) gives you an ERROR? 19. Are the triangles below labeled with correct measurements? For each right triangle below. determine if the ratio of the sides accurately corresponds to the angie. (a) (b) 60' 16 D.... 30· 4 8 c 20. Consider the following right tnangle, (a) Express sin A as a ratio. (b) Find the mLA to the nearest degree" AIgdmtl. Unl! ItS PJO:!t TriMgk Trigoo.oowlry- fA The ArlingtDn Algclma fulja:1. ~vilh; NY Il)~!) 7 A 3 8 ,LI.UUI",nUI n. Integrated Algebra I - Trigonometry Unit Using the Calculator & Trigonometric Ratios Name~ ________________________ Dale__________________ In #1 - 8, use a calculator to find each of the following to the nearest ten-thousandth: 1) tan 5) 10° 2) 6) tan 55° 3) 7) sin 89° 4) tan 36° 8) In each of the following, use a calculator to fmd the measure of LA to the nearest degree. 9) tan A ~ 0.0875 10) sinA~0.I908 II) cos A ~ 0.9397 12) tan A ~ 0.3640 13) sin A ~ 0.8910 14) cos A ~ 0.8545 Integrated Algebra I - Right Triangle Trignnometry Unit Day Three Review homework from prior day. Goal: To apply trigonometric ratio. to find a) the length ofa side ofa right triangle given the measure ofa side aed the measure ofan acute angle b) the measure (to the nearest degree) ofan angle given two sides of • right tri",!gle Standards: A..A.42 A..A.43 A.A.44 Opening: Draw the foUowing problems on the board and show the students how to Solve for ~'x" using trigonometry. 7 x 12 20 x 8 tan20= 5 sin 32 x cosx=-L --2L 7 12 Note: Instruct students to find the decimal equivalent for the trig value firs~ and then "roaed" at the end of the problem. Work Period: Attached (Arlington Algebra Projec~ Unit #8, Lesson L5 Classwork #2, #3 Unit #8, Lesson L6 elas,work #2, #3, #4) Closing: How do we determine which trigonometric ratio to use when solving a right Triangle problem? What is the difference between sin x and sin~! (x) ? Consider the triangle below: l~ f2 ~ sin 45 = Why are these trig values the same? 00.45 = Is this the only case where they are the same? 1 Home-work! See attached handout. (AMSCO Math A Textbook, Page 813 #1 -7 Page 822 #1 - 12) Integrated Algebra I - Right Triangle Trigonometry Unit Day Three - Work Period Nmne'_______________________ ___________________ D.~, TRIGoNOMETlUC RATIOS Recall that in a right triangle with acute angle A, the following ratios are defined: . A =-~_op=pos=ite:.... SUl hypotenuse ~A= adjacent opposite adjacent 1M A.; _... - - - hypotenuse Exercise #2: In the right triangle below. find the length of AB to the nearest tenth. C 26 3g- B LL---""----'o>"A The key in all ofthese problems is to properly identify the coJ"reCt trigonometric ratio to use. h'xel'cise #3: For the right triangle below, find the length of BC to the nearest tetUh, B 65· A 12 c :SOlVing tOr a Missing Aogl~ The process for finding a missing angle in a right triangle is very similar to that of finding a missing side. The key is to identify a trigonometric fi!!tio that can be set up and then . use the inverse trigonometric functions to solve for that angle. . . Exercise U2, Solve foc mLB to the nearest degree. A 5 3 C LL--_-""B Exercise #3: Find the value of x, in the diagrams below. to the nearest degree. (a) (b) 125 2S x 94 40 Exercise #4: Find the value of x in the diagrams below. Round your answers to the nearest degree. (b) (a) 15 x s. 6. 50· s· 7. s. Integrated Algebra I - Right Triangle Trigonometry Unit Day Three Homework - Day Three AMSCO Math A Textbook - Page 813 - 814 Nrune'___________________________ O&e________________________ In 1-9, in each given triangle, find the length of the side marked x to the nearest/oot or the number of degrees contained in the angle marked x 10 the nearest degree. 2. 1. ~x 25· 4. 3. 18· ,<l 10· 55' Integrated Algebra I - Right Triangle Trigonometry Unit Day Three Homework - Day Three AMSCO Math A Textbook - Page 822 Nwue_________________________ Date_ _ _ _ _ _ _ _ _ _ __ In 1-8: In each given triangle, find to the neareSI centimeter tbe lenglh of the side marked x. 2. 1. , " 3. , ~~s0 ~ x 5. 7. 8. x/1 ~ . lOem x In 9-l2: In each given triangle, find to the nearest degree the measure of (he angle marked 9. ",,10' ~ S' 11. IS' ~ 10. n 4' x 12' . 12.~ 12' x Ig' ,I., Integrated Algebra 1- Right Triangle Trigonometry Unit Day Four Review Homework and Class work from prior day. Goal: To review how trigonometry is used to fmd the length of a side or measure ofan angle in a right triangle Standards: A.A.41 A.A.43 A.A.44 Work Period: Students should complete the double-sided worksheet attached. Quiz: Students should complete the 4 trigonometry problems (taken from the Integrated Algebra I Regents) on the attached quiz. Closing: When do we use right triangle trigonometry and when do we use the Pythagorean Theorem to solve for a length of. side of a triangle? • T "': . . . . I&. ............. Finding the length of a side or the measure of an acute angle in a right triangle using trigonometry N.me~ ___________________________ Date~ ____________ Find the measure of the angle indicated or the length of the missing side marked 'x'. Show all work. 2) . I) x 39° 32° x 3) 4) 5 13cm 3 7cri1 gcm 5) ~4 6) x 7) ~x ~ 8 70 em ~x 10) 9 20° 30 m 12) . 11) 13 15 13) 45 25 ~" 50 QUIZ Integrated Algebra I Regents Questions Nmne~ 1) __________________________ Date (Janumy,2009-2 points) The diagram below.how" tight triangle upe. u 8 c 17 15 p Which ratio represent.. the sine of LU? 15 8 (1) 8 (3) 1:5 (2) 2) i~ 8 (4) 17 (January, 2009-2 points) In the right triangle shown iu the diagram below, "..·hat IS the \<illuc or x to the rU!areslwitoie llurni'JCT? x 30' 24 (l) 12 (3) 21 (2) l4 (4) 28 3) (August, 2008 - 2 points) Which equation could be used to lind tlte measure ofone .cute angie in tlte rig"t triangle shown below? A 5 c LL_ _: -_ _.Oo. B 4 (1) s;n A = t (3) cos B = 4 (2) tao A 45 (4) tan B tt 5 ="54 4) (August, 2008 - 2 points) In the diagram of MBC shown below, Be := 10 and AB "" 16, B To the nearest tenth of n degree, wbat ac,ute angle in the triangle? ($ (I) 32.0 (3) 51.3 (2) 38.7 (4) 90.0 the measure of the largest Integrated Algebra I - Right Triangle Trigonometry Unit Day Five Goal, To apply trigonometric mtios to solve verbal problems Standards: A.A.42 A.A.43 A.A.44 Opening: Recall ratios ~~ theGC Remembet • Side opposite L..A with "50H-cAH·TOA", , SIn = Efi'. Hyp Ad] Cos=Hyp b C Side adjatent ra. =Efi'. Ad] !oLA Then show students how to solve the following three problems (on the next page) that involve ladders leaning against buildings. Work Period: Attached handout with verbal problems Closing: Lorraine said to Rosalie: fir can't decide if! am supposed to use the Sine ratio or the cosine ratio to solve this problem.H How should Rosalie answer Lorraine? Homework: Attached worksheeet Int. A1g. I - Trig. Unit - Day 5 - Opening - Solving Verbal P ....bl..... Nwne 1) ______________________ Da~. A 14 foot ladder is leaning against a house. The angle formed by the ladder and the ground is 7Z' • (a) Detennine the distance, d, from the bose or the ladder to the house. Roand to the nearestfoot. r h (b) 2) Detenume, the heigh~ h, the ladder reaches up the side of the house. Round to the nearestfoot. If a 20-foot ladder reaches 18 reet up a wall, what angle does the ladder make with the grollnd~ to the nearest degree? 18 ft 3) A ladder 25 reet long leans against a building. The hottom of the ladder is 9 reet from the base of the building. What angle does the ladder made with the ground, to tne neLlrest degree? Integrated Algebra 1- Trigonometry Unit Day Five - Work Period Name I) Oate._ _ _ _ _ _ _ _ __ A 625foot long wire is ._bed to the top ofa tower. If the wire makes an angle of6S< willi the ground, how tall is the tower? Round your answer to the nearest tenth Qfafoot. 625 foct 2) A surveyor is standiog 118 feet from the base oflli. Washington Monument. The surveyor measures the angle between the ground and the top of the Monument to be 78". Find the height, h, of the Washington Monument to the nearest/oat. 3) While flying a Idle, Betty lets out 300 feet ofstring. which makes an angle of 38' with the ground. Assuming thai the string is stmight, how high ahove the ground is the Idle? (Round your answer to the neAreSt 1\llltlL) 4) A guy wire reaches trom the top ofa pole ro a sta\re in the ground. The slake is 3.5 meters trom the foot of the pole. The wire makes an angle of65' with the . ground. Find to the nearest meter the length of the wire. 5) In a park, a slide is 9 meters long and is built over a horizontal distance of 6 meters along the ground. Find to the nearest degree the measure of the angie that the slide makes with the horizontal. Integrated Algebra I - Trigonometry Unit Day Five - Solving Verbal Problems - Homework _____________________ N~; Dare,___________________ I) Solve for the angle (x) to the nearest degree x 2) 4 • Solve for the length of the side (xl to the nearest tenth 2.5 inches 3) A ladder is leaning against a waiL The foot ofthe ladder is 65 feet from the wall. The ladder makes an angle of74" with the level ground. How high on the wall does the Jadder reach? Round the answer /0 the nearest tenth ofafoot. 4) A 20·foot pole that is leaning against a walI.reaches a point that is 18 feet above the ground. Find to the nearest degree the number ofdegroes contsined in the angle that the pole makes with the ground. 5) While flying aldte, Doris let out 400 feet of string, Assuming that the string is s!retched taut and it makes an angle of48" with the ground. find to the nearesi/ool how high the kite is. 6) In rectangle ABCD, diagonal AC is drawn. If mLBAC ~ 62 and BC ~ 20 find to the nearest integer a) the length of AB b) the length of AC Integrated Algebra I - Right Triangle Trigonometry Unit DaySb: Goal: To solve trigonometric !<Ilia problems involving the angle ofelevation and The angle ofdepression Standards: A.A.44 A.R.6 Opening: . Use the following diagram to introduce the angle ofelevation and The angle ofdepression. A Angle of elevation c R=ill: ParnIlellines cut by a tnmsversal. Use the example below from the Arlington Algebra Projeet, Unit 8, Lesson g Classwork, to demonstnlle the angle ofelevation. Exercise #2: A flagpole 30 feet tall casts a shadow 52 feet long. What is the angle of elevation of the sun measured to the nearest degree? • • Review alternate interior angles Introduce the angle ofdepression "Ill' Use the example below from the Arlington Algebra Project, Unit 8, Lesson 7 Classworl<, to demonstrate the angle ofdepression. Erocise ~3: Harold is bang gliding off a cliff that is 120 feet high. He needs to tmvel 350 reet borlzontolly to reacb his destination. To the nearest degree, what is his angle of deseeot A? [Nole: This angle you are finding is called an .ngle of depr..sion or an angle of declin.ti....) ._- ... --.._.--... --- --... -.~ 120ft • 350 ft , 1 Work Period: See _ched handout Closing: Explain the relationship between the angle of elevation and the aogle of depression. (alternate interior angles) Homework: See attached handout (From WestSea Integrated Algebra Review. Page 138) Integrated Algebra - Trigonometry Unit Day Six - Work Period - Angles of Elevation and Depression Nrune 1) Daoo,__________________ At a point on the ground 40 meters from the foot ofa tree, the angle ofelevation of the top of the tree oonlairu; 42", Find the height of the tree to the trearesl meier. T i .B 2) From the top of a light house 165 reel above sea level. the angle of depression of a boat at sea conlairu; 35", Find to the neareslfOOl the distance from the boat to the foot of the light house. It A 3) From an airplane that is flying at an altitude of3,OOO feet, the angle ofdepression of an airport grotuld signal measures 27':.1, Find to the nearest hundredfoet the distance between the airplane and the aitpnrt signal. 4) A tree casls a sbadow that is 32 feet long, Find the height ofthe tree if the angle of elevation of the SWl is 35.,/", Found your aoswer to the near..t foot. 5) Find to the nearest degree the measure of the angle of elevation of the sun when a woman 150 centimeters tall casts a sbadow 40 centirnerers long:, 6) From the top of an 86 foot lighthouse, find the angle of depression to the nearest degree when the ship' i. 203 feet from the light hoa.e, ' I 86 feet 1 Integrated Algebra I - Right Triangle Trigonometry Unit Angle ofElevatioD & Angle of Depression Homework Nmne~ t) _____________________ A building stands on level ground. The measure of the angle of elevalion of the top of the bullding, taken at a point 400 feet from the foot of the building, is 31°. Find, to the nearest /00(7 the height of the building. 2) A 20 foot ladder leans against a house. Ifth. foot of the ladder is 8 feet from the house, find, /0 the nearest degree. the measure of the angle that the foot of the ladder makes with the ground. . 3) ' From the top ofa 120 foot lighthouse, the angle of depression of a boat out at sea is 260.. Find, to the nearest fOal. the distance from the boat to the foot of the lighthouse. 4) From the top of a 100 foot high pole, an observer measures the angle of depression of a car on the road as 28 degrees. Find, to the nearest [001, the distance from the car to the base of the pole. 5) From a point on level ground, the angle of elevation of the top ofan 85' pole is 62°. Find, to the nearest integer, the distance from that point to the foot of the pole. 6) A 40 foot long wire stretches from the top of a vertical pote to a stake in the ground 18 feet from the foot of the pole. Find. to the nearest degree) the measure of the acute angle that the wire makes with the ground. Integrated Algebra I - Right Triangle Trigonometry Unit Day Seven - Review Goal: To review trigonometric ratios and how to use them to solve problems Standards: A.A.41 A.A.43 A.A.44 Work Period: See attached handout. Integrated Algebra I - Trigonometry Unit Day Seven - Review Name Date In #1 - 6 refer to tlRST and express the value ofeach ratio as a fiaction. 1) sinR= 2) cosR= 3) tanT= _ _ 4) cosT= 5) sin T= 6) t.anR= 17 R s 15 In 117 - 12: In each given triangle, find to the nearest centimeter the length of the side markedx. 7) 8) 40cm x x 42" 18cm 35" 9) 10) x 50cm 41 em 12) 11) x 180m 20cm 12cm~ 24cm 13) Find f<) the nearest meter the nearest meter the height ofa building if its shadow is 42 meters long when the angle of elevation of the sun measures 4~. 14) A 5-foot wire attached to the top of a tent pole reaches a stake in the grollild 3 feet from the foot ofthe pole. Find to the nearest degree the measure of the angle made by the wire with the ground. 15) A ship is headed toward a lighthouse which we know is 65 feet high. If an observer on the boat measures the angle ofelevation to the top of the lighthouse to he IS', how far is the boat away ftom the base of the lighthouse, to the ""west/oot! 16) A giant redwood tree casts a shadow thet is 532 feet long. Find the height of the tree if the angle ofelevation ofthe sun is 32°. Round your answer to the nearest/oot. 17) From a point 435 feet fi:om the bose ofa building it is observed that the angle of elevation to the top of the building is 24· and the angle of elevation to the rop of a flagpole aUlp the building is 21". (Be careful when looking allhe diagram to realize lhallhe flagpole starts at the top 0/the building) a) Find the heigbt of the building, It, ro the nearest/ool. Ii; 'IJ IJ +.--435 • b) Find the height to the top of the flagpole, k, to the nearest foot. oj Using your answers from aJ and b), find the length of the tlagpole to the nearest foot. \ TEST Integrated Algebra - Trigonometry Nrune~ ________________________ Dare,___________________ Complete all problems. Show your work. B Answer questions # 1 - 6 using the diagram on the left 3 1) sin A = 4) sin B = 2) cosA = 5) cos B == 3) tanA = 6) tan B 5 I, , 4 C A For questions #7 - 12, find the missing side (solve for x) 7) ~ 42 .m 8) x 120 65 = 9) 10~ For questions #10 -12. find the missing angle (solve for x) 9 lQ) ~-.-x----n 6 III 12 x 15 12) x 1 12 Solve the following application problems. Drawing a diagram, if not already provided, may be useful. 13) Sitting at lb. top of a 57 ft. cliff, a lioness . - an elephant The angle of depression from Ibe lioness to the elephant is 22". What is the shortest distance from the lioness to the elephant? ·Round to Ibe nearest tenlb of a foot· . 22' 57 ft 14) .' Find to the ne(l.rest meter the height of a ch.urch spire that casts a shadow of 53.0 meters when (he angle of elevation of the sun measures 68.iJ'>. 15) Find. to the nearest Il:nth ofa 1Oo~ the height ofthe 1ree represented in the accompai!ying diagram. (not dralMllo seale) 16) As seen in the accompanying diagram, • person can IravellromNew York City 10 Bullillo by going north 110 miles to Albany and then west 280 miles to Bullillo, Buffalo 280 miles Albany 170 miles x New York City (aJ Ifan engineer wants to design a highway to connect New York City directly to Bullillo, at what angie,) would she need to build the highway? {Find Ihe angle fO the nearesf degree.] (bJ To the nearest mile, how many miles would be saved by traveling d~ectly Irom New York COy 10 Buffi rather than by traveling first to Albany and then to 8ullillo? Ramp Up to Algebra - Unit 7 - Geometry and Measure (RU Unit #3) Day Goal Ramp Up Lesson 1 To learn about one- and two dimensional measw-es,. and to calculate #3 One-and Two Dimensional 2 3 Measures the perimeters of polygons and the areas •of rectangles, triangles, and (Do this lesson last) composite shapes. 4 5 To calcul.te the radius~ #5 Circles diameter, circumference~ • ·· ·· l~. NYS Algebra Standard A.G.I Find the area and/or perimeter of figures composed of polygons and circles or sec'.ors of a circle. .. Radius~ Diameter~ AGJ , (same as Circumference, : above) It Formulas for To find the i #4 area and : Areaofa Polygon perimeter of composite shapes 7 : To apply geometty to real-life problems Multiple Representation Activity on Geometty Multiple Handouts from "lntro" on . circles . andareaofa circle 6 Intro to Word Wall Suggestions Algebra Lesson Length, Width, Multiple Handouts from Base, Height, "Intro" Diagonal, {labeled Ramp Perimeter~ Perpendicular Up) on area and perimeter of triangles, Fonnulas for area and rectangles, perimeter squares, parallelogoams, rhombuses, and trapezoids area, circumference ! Composite Multiple Handouts from ' area "Intro" on composite shapes #6 Putting Geometry to Work ··· Grain silo, · Braid A.G.l (same as above) A.PS.8 Determine information required to solve a problem, choose methods for obtaining !be ! information. , and define I, .. parameters for acceptable solutions. 8 To calculate the volume and surface areaofa rectangular prism #7 Rectangular pnsms To calculate the volwnes and surface areas of cylinders *10 To represent error III measurement asa compound inequality #8 Parallel Solids (Cylinders only) To review the perimeter and area of twodimensional figures and volume and surface area of threedimensional figures 12 To consolidate understanding of the concepts in this unit 13 To apply the concepts of geometry and measure #11 Progress Check 9 II RUGeometry Review Parts 2 &3 Worksheets Arlington Project- Error In Measurement West SeaError in Measurement #12 Learning from the Progress Check #13 Putting Geometry to Work Prism, Cubic Units. Volwne. Surface Area. Faces A.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders. Cylinder AG.2 (same as above) Error in measurement. relative error AM.3. Find the relative error in measuring square and cubic units when error occurs in linear measure All of the above standards All of the above standards APS.8 (same as above) 14 To review the #14 perimeter and area of two dimensionat fignres and Geometry volume and surllice area of threedimensional Applications , A.PS.2 Recognize and understand equivalent representations ofaproblem situation r a mathematical concept figures '15 To review concepts 0 f I geometry All previous standards Name: _ _ _ _ _ _ _ _ _ _ __ Date: _ _ _ _ __ Error in Measurement Algebra 1 Whenever we measure a quantity, length., area, volume, weight. etcetera, our measurements are never exact. In fact, we always must make a choice about rounding our measurements to a certain degree of accuracy. In this rounding, we introduce error. £xercise#l: Manny measures his heigh~ h, to he h ;73 inches. (a) Which of the following eould no/ be Manny's height? (I) 72.8 (3) 73.6 (2)73.2 (4) 72.5 (b) Give an interval that represents the range ofpossible heights for Manny. State your answer in inequality form. Whenever we round, there is a window of'values that our 'actual measurement eQuId be. This is the range of our measurements.. The range wiU Range Table for Rounding always be half of the unit that you are rounding to. The following table summarizes this range. Exercise #2: Professor Wilder lives 3.4 miles north of Red Hook. where the distance is roWlded to the nearest tenth of a miie, Which of the following gives the range of the distance that Professor Wilder lives north cfRed Hook? (I) ll<d <3.6 (3) 33S"d<3.4S (2) 2.% d < 3.9 (4) 3.J5<d';3.45 Propagation of Errors - When rounding QCC\US repeatedly in a problem, it can build (In itself to produce larger errors. Exercise #3: Jimmy wishes to calculate the perimeter of his triangular flower patch. His measurements, rounded to the nearest inch. are shown in the diagram,. What is the range of his actual perimeter? Write your answer as an inequality. 42 inches 50 inches A.lgdmi I.Ullilfl9-M~!-Ul ~.Artinpll AJgd!ra Ptujeet, ~gcvi.lle. NY 12$46 65 inches - Propagation of errors can be even more pronounced when area and volume calculations are performed, Exercise #4: The length and width of a walk-in closet were measured and rounded to the nearest foot The diagram of the room is sbown below, (a} What is the area of the closet calculated using the dimensions given? 14 reet (b) Write an interval (inequality) that expresses the possible areas of this closet. In¢lude units, Heel James is trying to detennine the number of cubic feet of water needed to fill his swimming pool. The pool is in the shape of a rectangular box as shown below, All of these lengths have been rounded to the nearest foot. Exercise #5: (3) What is the volume of the swimming POOL using the dimensions below? lnclude units in your answer. 20 Il 81l (b) Write an interval that expresses the runge of possible volumes, Include units. Al~l.UD.i(N9-~-LII Tb¢ArtlnglOO Alp"" i'roj«t. ~CYiUe, NY {2$40 377 Date; _ _ _ _ _ __ Name: Error in Measurement Algebra 1 Homework Skills 1, Each of the following variable values has been rounded to the nearest iJlIeger. Write an interval that expresses aU possible values of the variable. (.) x =7 (b) y=12 (c) x=-'15 (d) y=-25 2, Each of the following variable values has been rounded to the nearest tenth. Write an interval that expresses all possible values of the variable. (a) x=2.8 (b) x=4.7 (c) y=-3.6 (d) y=-8.4 3. Eacb of the following variable values bas been rounded to the nearest hundredth. Write an interval that expresses all possible values of the variable. (a) x=4.58 (b) y =0.97 (0) X= -3.68 (d) y=--9.32 Applications 4. Jean Ann measures the length of one side of a square and rounds it to the nearest integer as 8 inches. Which of the following gives the minimum possible area of the square? (I) 64 in' (3) 68 in' (2) 72.25 in' (4) 56.25 in' 5. Maria is trying to determine the perimeter of her rectangular garden. She measures the length and width and rounds them to the neatest whole number as shown below. Write an interval that expresses the range of its possible perimeter values> P. 16 1\ AIg¢bm 1, OoilN9-M_t-LtI TbeAriiQgton. ~ra ProJo<;t. LQG!3ngMll,.. NY ~254:0 6. From the previous problem. Maria would like to cover her garden with mulch that costs $1.25 per square foot. (a) Write an interval that ex.presses the range in the possible values fur the gamen?s ~ A,. (b) Write an interval for the possi.ble amount of money, M, that Maria will have to spend to covet her garden with mulch. Round aU values to the nearest penny. 7. Wolfga.n's Pretzel Cmnpany is designing a new container for their prettels that is cylindricaL The company would like the dimensions, rounded to the nearest whole number, to be those shown below. Determine the range of possible volumes for this container. Round the values of the range to the nearest hundredth. RecaU that the volume of a cylinder is V:= g r7. h . 3in ~ 6 in Reasoning 8. Oftentimes students look at the nwnbers 1 and LO as being the same. Assuming that both of these numbers have been rounded (the firSt to the nearest integer, the second to the nearest tenth), which is more accurate? In other words which is closer to the actual value from which: it was rounded? Explain your answer. Algebml, Uoifll!l-M~!-LII 'the ArlingtOn Algebra. Pmj«;t. ~lle, NY 12540 CInlpter8 MEASUREMENT Error in Measurement (continued) 1. If a-calculation is to be made and the answer is to be rounded to the nearest tenth, all prior calculations should be rounded to the near~st (1) ten (3) hundredth (2) tenth (4) hundred • .. 2. A Ndius of a circle is given as 12.34 em. Explain why it would be incorrect to try and find the diameter of the same circle to the neatest thousandth of a em. .];- 3. In a certain problem, all values given are in the hundredths. The answer is to be rounded off to the nearest tenth. Bob rounded all prior calculations to the nearest hundredth. He then rOWlded his final answer to the nearest tenth. Jim rounded all prior calculations to the nearest tenth and then rounded his final answer to the n~st tenth. Did Bob or Jim use the better method of rounding? Explain. ··· I, 4. A fanner stated that the weight of a turkey was 21 pounds. to the nearest pound. VVhich cannot be :m actual weight of the turkey? (1) 20.6 pounds (3) 21.4 pound, (2) 21.0 pounds (4) 21.6 pounds 5. A computer monitor screen is in the shape of a rectangle. To the nearest inch! the length of the computer monitor screen is 13 inches and its vvidth is 10 incl1es. a What is the least possible value of the area of the computer monitor screen to the nearest ten? b 'What is the greatest possible value of the area of the computer rp.onitor screen to the nearest ten? Justify your answers. , 159 ChapterS MEASUREMENT Error in Measurement (continued) - -... ... 6. A square has an area of30 square inches when the area is rounded to the nearest square inch. Which could be the greatest possible value for the side of the sqwue in inches? (3) 5.522 (1) 5.432 (4) 5.523 (2) 5.477 " 7. A room is 15 feet long, 12 feet wide and 8 feet high when the dimensions are to the nearest foot. Assuming the measurements are off by 1%, find to the neares! cubic foot, the • IargMt possible volume of the room. b smallest possible volume of the room. 2" 8. The radius of it circle is measured as 1S em. to the nearest centimeter, The actual radius is 17.6 em. Fmd to the nearest percent? the percent of error in the: a measurement of the radius. b calculation of the area of the circle. 9. The mounting strap below has a number of holes for bolts to pass through, Ryan and G.uy ,have added the measurement specifications to their drawings. Who is using the proper method of dimensioning in their drawing. Explain. Both drawings specifY that all dimensions arc to.OS inches. Ryan's D~ ~ ~-::- 1~ Gary's Drawing~(}.2$ Dla (typical) 02SDia('>Pi GOl) :-:: l ~ 8(98 0<Z0 1001 1.001 f..<-s""--> • 9.00 ~ lU~ • ... 310' 160 4.00 ,, I I 4.00 1.tOO • MEASUREMENT Error in Measurement (continued) 10. Arect:mgular solid has. measure of 15.25 em in length, 1050cm in width ad 6.75 cmin height. a Using the fOrmula V • LWH, find the \roiume ".ing the given values.oove. Round the final answer to the nearest hundredth em'. h Using the fonnula A = LWand the given values .bove, find the area and round off to the nearest hundredth em'. Using the formula V • AH, find the volume using the value A to the nearest hundredth em' and the value ofH given above. Round the final answer for V to the nearest hundredth em). ( Is the answer to (l or b the more accurate. Explain. * 11. A bolt must fit through the hole ofa flat washer as shown in the illustration. The bolt has • diameter of inch! 0,015 inch. The diameter of che hole in che fl.t rcoder washer is 0.297 inch! 0.031 inch. Explain why this is or is not a good design. .lK)LT DIAMETER WASHER DIAMETER nomial 0.250 minimum _ _ __ nomial maximum _ _ __ maximum _ _ __ 0.297 minimum _ _ __ 12. A cube is supposed to have a side of 12.0 em. If the dimension of the side is !: 5%~ complete the table below. theoretical SIDE AREA 12.0 et1"\ 144.0 em' PERCENT ERROR EN VOLUME 1728.000 em' 0% VOLUME +5% em em' em' % -5% em em' em' % 161 Ramp Up to Algebra - Unit 8 - Graphing (RU Unit #7) Day(s) Goal Ramp Up Lesson Intro to Algebra Lesson (plus Word Wall NYS Algebra Sugestions Standard extra worksheets) 1 To define the coordinate plane, x- and y-axes, .. #1 Building the Coordinate Plane Lesson #1 Coordinate plane, origin, Graphing x-aXIS, y-axiS, Points ordered pairs, x coordinate, y coordinate, quadrants Intro. Unit 9 ongm, ordered pairs, quadrants . . A.R.2 Recognize, compare, and use an array of representational forms A.A.36 Write the equation of a line parallel to the x- or v-axis 2 To graph points on the coordinate axis and #2 Constant Ratios and Graphing Intro Unit 9, Lesson #2 Graphing recognIze Intro. Unit 9, relationships that exist Lesson #3 Graphing Set of points, intersection between x A.PS.I Use a variety of problem solving strategies to understand new mathematical content andy A.A.35 Write the equation of a line, given the coordinates of two points on the line 3&4 To graph lines and compare #3 To graph lines and steepness, Intra. Unit 9 coefficient Lesson #4 Reinforcement A.A.39 Determine whether a given point is on a line, given the equation of the line A.G.5 Investigate and I generalize how steepness and to determine if points are on a given line compare the steepness # II Learning changing the coefficients of a function affects its graph Intro. Unit 9 Lesson #5 Graphing Lines from the Progress Check (Worktime #5,6,9,10) A.A.32 Intra. Unit 9 Lesson #6 Graphing Lines Explain slope as a rate of change between dependent and independent variables Intro. Unit 9 Lesson #7 Graphing Lines A.A.4 : Translate : verbal : sentences into ~ mathematical , equations 5&6 To . underStand • the concept ofslopc [ntro. Unit 9 To introduce . Lesson #& the concept • Slope ofa ofslope as Line (given a the steepness line, find the ofa line slope) #4 #5 Graphing Negative Values 7&8 To understand #12 Linear the significance Graphs of the slope and y- #13 intercept Focus on Slope Intro, Unit 9 Lesson #9 Slope ofa Line(given two points, tind the slo Intro. Unit 9 Lesson #10 Slope and Y intercept rise over run, slope, increasing) decreasing, A.A.32 y-intercept, linear A.A.34 Write the j Explain slope as a rate of change between dependent and independent variables equation of. line, given its slope and the coordinates ofa ' point on the line A.A.37 Determine the slope of a line, iven its 9 10 equation in any fonn parallel, A.A.38 To identify #14 perpendicular, Determine if and graph Parallel and negative Perpendicular two lines are parallel and parallel, given perpendicular Lines reciprocal their equations lines in any form Intro. Unit 9, A.GA y mx+b To be able to graph a line Lesson #11 Identify and graph linear, using the Graphing a SiopeLine Using the quadratic Siope(parabolic), Intercept Intercept absolute value, Method Method and exponential functions A.CN.2 II *12,13 To graph and solve systems of linear equations To graph linear inequalities in two variables *14 Review Intro. Unit 9, Lesson #12 Solving a System of Equations Graphically West SeaGraph Linear Inequalities How do Fish Go Into Business? Intra. Unit 9, Lesson #13 Review Understand the corresponding procedures for similar problems or mathematical concepts A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables A.G.6. Graph linear inequalities All previous standards ~ ~'.l!ll Nams ____________.----________ Dots _________________ Graphing linear Equalities 1 y<:--x+ 1 2 3x-4Y$12 1 1.) Graph V ~ l.} Graph V=-- x+ 1 asa 2 dofled lins. 2.) Choose a point in one half-plane and substitute. Try (0. 3): 3 3.} 1 ·0 ... 1 = 3 2 <: - <: ~ 3 y>-x-3 - 4 ~ x - 3 as a solid 4 lins. 2.) Choose a point in one half-plane and substitute. Try (0, 0): O~! '-3~ O~-3""True 1 = False Shade half-plane that does not contain (0, 3). 3.) Shade the half-plane that contains (0, 0). ;i I, , , I I L y> x ... 1 2, 3x- y$6 3. y+5$O 4, Y" 2x- 3 5, x+y<3 6. 2x+ y>-8 II , '. I Page 87 'I , Wi L: lellin' lingle -----.~-.-.--- -----... tn March 1985, "We Are the World;' sung by USA for Africa, was the fastest seiling single ever made in the world. It sold over 800,OOltcoples in a mailer of days. How many days was this? To find out, graph each inequality on a separate sheet of paper. Count the number of graphs you shaded aboYl> each line (ralher than below each line). Your total. will equal the unbelievable number of days, 1.6x-6y;o,12 2. y';-Sx-l 3.2x-y5:4 5. 14x - 7y 5: '28 10. Y ,,-2x +4 Answer: _ _ ~---FS122010 Algebra Made Simple. C Frank SchaIfer Publt<:ations, Inc. --I Qnequality g~aph£:ltl ,, r+J- (YI oj/; GWOl17 Pelood 3) ~rP/t/f- ~ z. ();rn d :3'r2(;f;, 1J!()5f.- uS( a Jefe-raie. Shed rJ(fa~-, 3) /-7hlsh j77e rt',y-/ f;, !/orY.e4lm/( (p -3::)+1 "7 Lj ~ 3lj- '1,/ :?- & 'J). - /0:::J > - 10 'g) -/&~ /' /(P CJ -,1 C1 +J.) > /0 CJ -3[-fj - I} .? I J ([j) - 8. L~ +~) (9 - r7{.-1-rc2) C9 L-IC~ flj 8 t::- L '7 /1 g How Do Fish Go Into Business? Graph any inequality below. Then read the three statements that appear under the coor<linate grid for that exercise. Circle the leller of the statement tnat correctly describes the location of the graph. Write this letter in each box at the bottom of the second page that contains the number of that exercise. @2x-y;;.4 <Dx+y>2 ®X+y",,2 K F Quadrants 1.11. IV; B All a. Quadrants I. II. IV; P Quadrants I. II. III; excludes boundary line. H Quadrants I. III. IV; U All E All four quadrants; R Quadrants I. IIi. IV; Quadrants I. III. IV; includes boundary line. excludes 'boundary line. four quadrants; excludes boundary line. includes boundary line. includes boundary line. four quadrants; includes boundary line. includes boundary line. excludes boundary line. " J, @)-2x + Y < 4 ® X + y;;;o-3 M L All four quadrants; All four quadrants; excludes boundary line. V Quadrants 11.111. IV; excludes boundary line. G Quadrants I. II, III; excludes boundary line. exel udes boundary line. @3x - 2y "" 6 C All four quadrants; includes boundary line. S Quadrants II. III. IV; J Quadrants H. III. IV; T All lour quadrants; D Quadrants I, III, IV; includes boundary line. includes boundary line. includes boundary line. includes boundary line. i .' 242 PRE ALGEBRA WITH PIZZAZZ! @Creafive Publicalions , ~~~~~~~~~~~_PA~e2 ®2x - y < -3' / <V3x+2y>6 ~--. R All four quadrants; o Quadrants I, II, IV; Z All four quadrants; Y Quadrants I, III, IV; L Quadrants I, II, N Quadrants I, II, IV; I All four quadrants; F Quadrants I, II, III: includes boundary line. III; excludes boundary line. excludes boundary line. excludes boundary line. excludes boundary line. excludes boundary line. ,) .. @>x + 2y,;:;; includes boundary line. includes boundary line. All four quadrants; 'excludes boundary line. @x - @-3X-4y> 12 5 G y;;" 0 , R All four quadrants; U Quadrants II, III, IV; S Ouadrants I, II, IV; I All four quadrants; F Ouadrants II, III, IV; W Quadrants I, II, IV; includes boundary line. A Ouadrants II, Ill, IV; includes boundary line. K Quadrants I, II, III; includes boundary line. excludes boundary line. 5 3 2 a 12 11 12 4 11 9 1- includes boundary line. excludes boundary line. includes boundary line. I, S Quadrants I, III, IV; includes boundary line. includes boundary line, , 5 11 10 5 1 7 9 12 6 11 9 2 PRE·ALGEBRA WITH PIZZAZZ' © Crealive Publicatiorts , , , 243 ChapterS COORDINATE GEOMETRY Graph the following inequalities: 1. 2y>3x-6 2. "-21<4 3. 5y+2x~O U2 ChapterS COORDINATE GEOMETRY t Whlch diagram below represents the graph ofgraph of the statement x < 3? (1) (2) 2. 4. -tJ Whlch ordered pair is not in the solution set of the system of inequalities shown in the accompanying graph? (1) (-2,0) (3) (2,0) .9) (4) (3, 4) (0, -2) The graph of which inequality is shown in the accompanying diagram? , 5. !x (2) Y>!x (1) y > 3. +1 (3) y s +1 (4) y < 1x !x set of the system of inequalities shown in the accompanying graph? (1) (5, 2) (3) (1, -5) (2) (2, 0) (4) (-5, 2) +1 +1 -- Whlch diagtam below represents the graph of the statement x > 3? (1) (2) (4)""' l Jllf .J.~ Which ordered pair is in the solution 1§i 121 Ramp Up to Algebra-Unit 9-Foundations of Algebra (RU Unit #1) Day Goal Ramp Up Lesson Intra to Algebra Word Wall NYS Algebra Lesson (Plus extra Suggestions Standard worksheets) 1 To reason 1: Reasoning with • Expression with diagrams Diagrams • Equation • Diagram • Squared number A.N.l IdentifY and apply the properties of real numbers (closure, commutative, associative, distributive, identity, invers~) 2 To leamhow to justifY mathematical statements about odd and even numbers 2: Reasoning with Numbers • justification (list tools) • • • • as always true, sometimes true, or never even odd remainder properties • counter A.RP.l Recognize that mathematical ideas can be supported by a variety of strategies example • divisible fnle 3 To justifY statements about 4 mathematical expresslons In which letters are used to represent numbers To learn the conventions for using numbers and letters in mathematical expreSSIOns that represent numbers 3: Reasoning with Letters • product A.RP.l • expressIOns (see above) • whole number 4: Conventions for Using Numbers and Letters Intro to Algebra Integers packet Unit #1 VariablesAssignment #3 • conventions • substitution A.CM.l2 Understand and use appropriate language, representations, and tenninology when describing objects, relationships, 5 To use parentheses to clarifY expressions 5: Conventions for using Parentheses Integers Unit 1 Order of OperationsAssignment # 1 (Do not include exponents-these will be covered in Unit 2) 6 To learn the number properties 6: The Number Properties Intro Unit 10 Potpourri Fundamental Laws (properties)- Lesson #9 Exercises set 2 Worksheets developing skills in algebra book A p.51 and 53 7 To summarize the number properties and conventions for using numbers and letters To practice using letters in the formulas for area and perimeter of a rectangle 7: Conventions and the Number Properties 8: Using Letters in Formulas Intro Unit 2 Geometry #1 Area and Perimeter of Rectangles and Squares (pp 1-3) Worksheets developing skills in algebra book A p.7 (skip triangle) • • • • • • • To understand and use the distributive property 9: The Distributive Property Intra Unit 5 Lesson #J-Using the Distributive Property of Multiplication over • distributive property of multiplication over addition 8 9 • quantity • identity • commutative • associative (list and give examples of these for addition and multiplication) mathematical solutions, and rationale A.CN.I Understand and make connections among mUltiple representations of the same mathematical idea A.N.I (see above) A.N.I (see above) formula base height length width area perimeter A.CN.4 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics A.N.I (see above) 10 To use the distributive property to solve problems 10: Applying the Distributive Property Division (note: Skip questions that involve negatives or solving algebraically for the variable) Developing skills in algebra Book A, pp.S3 Intro Unit 5 Lesson #11: Simplifying Expressions (note: Skip questions that involve negatives or solving algebraically for the variable) • expanded form A.N.I (see above) • inverse A.N.! (see above) "flash card" game (see enclosed) To learn the Inverse operations and mverse I properties 12 To review definitions, conventions, the number properties, and how they help with mathematical reasoning 13, To self-assess 14 the errors made on the progress check and continue working on similar I problems II 11: The Inverses of Addition and Multiplication 12: Progress Check All standards from previous lessons 13: Learning from the Progress Check All standards from previous lessons 15 To use letters, tables and formulas to 14: Relationships • quantity • variable quantity Between Quantities represent • vary quantities that vary in A.CM.2 Use mathematical representations to communicate with appropriate accuracy, including relation to each other numerical tables, formulas, functions, 16 To represent the relationship 15: Using Graphs to Represent Relationships between two Intro Unit 9 Part 1 Lesson #1 Graphing Points (pp. 1-4) quantities as a • • • • vertical axis horizontal axis interval ordered pair equations, charts, graphs, Venn diagrams, and other diagrams A.CM.2 (see above) I graph 17 To understand the situation described in a word problem 16: Understanding the Problem Situation Intro Unit 3 Lesson #8: Writing and Solving Equations ClassworklExercise (pp 1-4) (NOTE: do not have the students solve them) IS 19 To use letters, diagrams, tables, fonnulas, and graphs to represent problem situations To use letters, diagrams, and 17: Representing Problem Situations IS: Writing Fonnulas to • unit A.PS.S Determine information required to solve a problem, choose methods for obtaining the infonnation, and define parameters for acceptable solutions A.CM.2 (see above) A.CM.2 I (see above) 20 tables to write formulas and answer questions about problem situations To review the mathematical concepts of the unit Answer Questions 19: The Unit in All standards Review from previous lessons in the unit TENTH GRADE, PQST-RAMP-UP, INTEGRATED ALGEBRA CURRICULUM The objective of this Integrated Algebra course is to prepare students for success on the New York State Integrated Algebra Regents Examination, Thls curriculum has been aligned with the New York State Standards, and it fonnally addresses the Content Strands; the Process Strands are not formally addressed. It is recommended that teachers integrate activities~ such as word problems and problem~solvin& throughout the course to address the Process Strands rather than attempting to formally "teach" these skills in isolation. Comprehensive information regarding the standsrds for the Integrated Algebra course can be found at the Web site htm:/Iwww.emsc.nysed.gov/ciailmstlmathstandardslalgebra.htm This course is designed for those tenth graders who successfully completed the 9th grade Ramp Up to Algebra course, as well as for those tenth graders who did not successfully complete the Integrated Algebra class or Integrated Algebra lA course in the 9th grade. The goal of this course is for students to further develop their mathernatical skins in the areas ofalgebra and geometry. This course is a one-year course, though it has been written with the student population it serves in mind. Thus, the teachers delivering this course are aware that the students in this course are typicaUy students who did not perfonn on grade level in mathematics the previous year. Completion will provide students the prerequisite skills to ensure success with further study of mathematics and allow them to progress to Geometry, the next level in New York State. The curriculum utilizes McDougall LitteU's Algebra 1 and the Arlington Algebra Project in order to fully address aU the Content Strands required by New York Slate. VlhlIe the writers recognize the order the topics are taught is at the discretion of the course teacher. it is recommended that teachers follow the sequence as laid out in this document This allows students to Jearn the basic sidHs required for concepts later in the course. In addition 1 if all teachers adhere to the order presented here. transitions will be easier for students who are required to transfer between different classes. For each unit, a pacing guide is given. It is important to note that each row in the table represents one day ofstudy. Additional time has been allotted in each unit to incorporate workshop model activities, review. and assessment. Approximately 3 weeks of unstructured time provides teachers with the flexibility to have cumulative review at the end of the course to better prepare their students for success on the Integrated Algebra Regents Examination. Students should be awarded one math course credit upon the following: • Compietion of the course outlined in this curriculum • An average grade of65% • Adherence to the Poughkeepsie High School attendance policy Final grades are determined by averaging the four quarter grades. Regents Examination grades of 65% or higher will earn students one Regents credit toward their diploma. However, grades earned on Regents Examinations have no bearing on final course averages. Integrated Algebra Curriculum Map Approx, Time Arlington Frame Section No. • Topic! Strand Standard No, ,, , Unit 1: Algebraic Foundations UI Ll UlL2 UI L3 8 days of lessons A,N,2 2,7,1l.l UI L4 Square Root A,N.3 11.2 UI L5 Division Combining Square A,N.3 11.3 AN.6 1.1, 1.2 A,N,6 2,1 A,A,26 A.G.6 , 2. 1 Roots with Add, & SubL assessment Total time allotted: 13 days A,N,I A.N,I Irrational Numbers 5 days for review, activities, and : Real number system Real Number Properties Square Roots and McDougal-Littell • textbook correlation (chapter,section) 2,2,2.3,2,4 2,2,2.4,2.5 U1L6 Evaluating Algebraic Expressions UIL7 More Evaluating Expressions UI L8 UI L9 Absolute Value (solving) Intra to Inequalities , 1 , I Integrated Algebra Curriculum Map Approx. Time , Arlington Section No. Frame Standard No. ' T opicl Strand McDougal-littell textbook correlation I (chaoter.section1 Unit 2: Linear Functions Coordinate Plane &. Equations Slope and Parallel . U2L1 , U2L2 A.G.4 A.A.21 A.A.33 II days of U2 L2,j Writing eqns. of parallel and I oemendicular lines 5 days for review, activities. and assessment Total time allotted: 16 days U2L3 U2L4 U2L5 ! U2L6 U2L1 . , U2L8 Slope as Rate of Change A.A.34 A.A.35 A.A.38 A,A,32 ~ Writing Equations of AA.34 , lines A,A,35 A,A.36 ' Graphing Lines Horizontal &. Vertical Lines ' AA.6 Modeling with i Linear functions Solving Linear A.G.7 i 4.4 U2L9 U2L10 . Lmes ofBest Fit Correlation Coefficient ,,5.5 , · i 4.4 ·· 5.1,5.2,5.3,4.4,4.5 , , , I, ,i i 4,7 ' 7,1 , Systetlls Graphicallv i 4.1,4.2 Lines lessons I ,,, · i . A.s. ",17_-r'-",5~.6 --,----1 A,G.5 A.8.8 5.6 extension i A,GA , 6.5 extension , AG.5"-----"_ _ _ _ __ Integrated Algebra Curriculum Map : Approx. Time Frame Arlington Section No, Topic! Strand • Standard •No, McDougal-Uttell textbook correlation I (chapter,section, • Unit 3: U3L! Linear Functions U3L2 9 day, of lessons 5 days for U3L3 review, activities, and U3L4 assessment U3 L5 Total time • allolled: 14 • days U3 L6 U3 L7 U3LS U3 L9 U3 LIO U3Ll! Solve Simple Linear AA,3 A,A,4 equations Combine like linear AA.22 term, AA,22 Solve linear equations wI variables on both sides A,N,( Solve with distributive property A,A,4 More practice AA22 solving linear equations A.A,6 Linear Word Problems secutive Integer AA,6 Problems Literal Equations AA23 Solving Linear AA24 "ties Graphing Solutions ,iA.G.6 to Linear Inequality AND compound ineaualities A,A,6 Inequality word I problems 3,(,3.2,3,3 3.4 3.3,3,4 3.1-3.4 nfa 3,8 6,1,6.2,6.3 6.4 6,1-6,4 ,, ,, , Integrated Algebra Curriculum Map Approx. Time Frame Arlington Section No. Topicl Strand Standard No. McDougal-Littell textbook correlation I (chaoter.section) Unit 4: Linear Systems U4L1 U4L2 U4L3 7 days of A.PS.5 A.A.10 SRH 7.2 A.A. to 7.3,7.4 00. 936-937 , Elimination lessons U4L4 5 days for reVIew, activitiest and U4L5 assessment U4L6 Total time allotted: 12 days Guess & Check Solve using Substitution Solve using U4L7 More solving linear systems .Iaebraicallv Word problems and linear systems More word problems and linear SYstems Solve Systems of Linear Inequalities A.A. to 7.2,7.3,7.4 A.A.7 7.1-7.4 . A.A.7 7.1-7.4 A.A.7 7.6 Integrated Algebra Curriculum Map Approx, Time Frame Arlington Section No. T opicl Strand Standard No, McDougal·Littell textbook correlation I (chapter,section) Unit 5: Quadratic , U5L1 Functions U5L2 7 days of lessons S days for : U5L3 , U5IA review~ activities, and assessment , U5L5 , ,, Total time allotted: 12 days ,, ,, USL6 U5L7 Properties of Quadratic graph Graphing Quadratic Functions with a graphing calculator Graphing w/o Calculator AG,7 10.1 A,GA AG,1O 10,3 activity A,G,4 10,4 problem solving workshop A.G. \0 Solving Quadratic A,A,27 IOJ Equations _____,..'A~,G""",8_-+=-;;-_ _ _ _-4 Solving Linear A,G,9 10.3 Quadratic systems graphically A,G.8 10.1· 10.3 , Apps. ofQuad, Functions Dav I AG.IO 10,1- 10.3 Apps, of Quad, AG.S AG,IO Functions Day 2 Integrated Algebra Curriculum Map Approx. Time I Arlington Frame : Section No. Unit 6: Quadratic Algebra 15 days of lessons I : U6L1 ' U6L2 I U6L3 · I U6L4 5 days for review, IU6L5 activities, and · U6L6 assessment Topic! Strand ,.::-:: McDougal-Littell textbook correlation (chapter.section) 8.1, 8.2, 8.3 I Standard : No. · I : Exponent Properties , AN.6 . Zero & Negative I Exponents Combining Like A.A.13 Terms : Mult. Polynomial by AAI3 ' Monomial Multiply i A.A13 PolYnomials · AAI4 Greatest Common 9.1 ·· : 9.2 I, 9.2,9.3 · ! · 9.4 Factor Total time allotted: 20 days ·· i U6L1 I U6L8 : U6L9 : ***[end of2"d i Ouarter 1 : Zero Product Law ,U6LIO More with the Zero I U6L11 9.7 ,, · ,9.5 i , . 9.7,9.8 9.4 ' 9.4 d , U6L13 U6L14 , · U6L15 · I ·, U6Ll6 A.G.B I Using Quadratic : Functions to Factor · Solving Incomplete AA.27 I Ouadratic Eauations A.N.2 Solving Quadratic A.A.8 Word problems 'A.A.S I Solving Quadratic , Word problems II Solving linear/quad, A.A.!! Systems U6Ll7 U6 Sl (fllgebraic.llv)I : Solving linear/quad. I Svstems 1I More Complete Factoring Practice 9.4,9.5 --~. I , ,, , 9.1-9.8 (omit 9.6) · · ·, 9.1-9.8 (omit 9.6) , , ni. , I, , · I A.A.19 A.A.20 -.... , I, ·· ! 9.4,9.5 , A.A. 11 · · I, ·, , A.A.27 A.A.27 Product Law I U6LI2 · : Factor Difference of ,AA.19 , two nerfect SQuares Factor Trinomials AA.20 Factor Completely AA20 ni. .i 9,'.9.8 ·,, ----' Integrated Algebra Curriculum Map Approx. Time , Arlington Frame ! Section No. , Topic! Strand , Standard No. • McDougal-Littell I i textbook correlation : (~haQter.section} , Unit 7: I U181 Rational Algebra , U7LI i 11 days of : U7L2 lessons , 5 days for Writing Equivalent i U7L3 review and , , SRH pp. 912-915 I, I nla rational..~xl2ressions U7IA Total time : U7L5 allotted: 16 days , : U7L6 Mult. & Div. , • A.A.l8 rational expressions ,• i, nla Solve Rational U7L8 Solve Rational , i : A.A.!7 i 12.6 ,, A.A.26 Direct Variation Similar Polygons , I, : 3.6 , A.A.26 12.7 A.N.S A.A.26 4.6 12.1 3.6·extension E,uations II , U7L9 ,U7 LIO ,i, 12.6 Eouations I : , , , Rational expressions U7L7 • 12.3. 12.4. 12.5 i extension : 12.5 Add & Subt. A.A.l7 Rational expressions I Add & Subt. i, i Simplifying complex ,i A.A'!6 II ,i AXI I A.A.17 • AAI5 • A.A'!6 AAI8 RatIonal Expressions assessment ,, Operations with numerical fractions Evaluating rational expressions I . , i Integrated Algebra Curriculum Map "Approx. Time Frame U.it8: Right Arlington Section No. Topicl Strand Standard No. U8LI U8 L2 Pythagorean Theorem A.A.45 Triangle Trigonometry 4 days of Converse of Pythagorean Theorem U8L5 Solve: missing sides U8L6 Solve: missing angle A.A.43 U8L7 U8L8 Applied Trig. I Applied Trig. II A.A.44 1essons 2 days for ~ review and assessment Total time allotted: 6 days I • McDougal-littell textbook , ,, , correlation ,, i (chapter.section) ,, I lA-activity A.AA3 A.A.44 Additional lessons E&Fpp. AIO Al3 Additional lessons E&Fpp. AIO Al3 Additional lessons E&Fpp. AIO- AU Integrated Algebra Curriculum Map Approx. Time Frame Arlington Section No. No. textbook correlation Unit 9: U9Ll Measurement U9L2 Intra to Percents More on Percents Percent 9 days of lessons 5 days for reVIew, activitie~ A.GA 8.5,8.6 and assessment pp. Total time allotted: 14 days A.GJ n1. A.M.3 n1. Integrated Algebra Curriculum Map , Approx. Time Frame I Unit 10: I, Statistics , .8: lessonsof days 5 days for review and ,,i Topic! Strand UlOLI Measures of Central Tendency More work wi Mean 5 Number Summary A.S.7 Percentiles Frequency Histogram. Cum. Freq. Hisls. A.S.6 A.8.5 : UlOL2 ,• UlOL3 , UlOL4 UIOLS assessment UlOL6 Total time allotted: 13 I Standard Arlington Section No. UlOL7 UIOL8 ..... _--- : No. Bivariate Data Analysis Statistics on Graphing Calculator A.S.4 A.S.6 A.S.5 A.S.9 A.S.? A.8.S ' MCDougal-Littell textbook correlation I (chapter.section) 13.6 : nla 13.8 and 13.8 activity 13.8 , ,, 13.7 nla Additional lesson M. p. A26-A27 13.?-activity 13.8-activity ,, Integrated Algebra Curriculum Map • Approx, Time :, Frame ,, , , , f-=-c'" Vnit 11: Sets and Counting Theory Arlington Section No. Ull L1 Intro. to Sets AA29 VII L2 Interval Notation & Infinite Sets Subse~ Empty, Complement Union & Intersection Venn Di'l1;I'ams Fundamental Countin2 Principle Permutations & Cnuntill)'( Permutations & Repetition AA.29 review~ Total time i allotted: 13 ,, days Ull L4 UII LS UIIL6 Ull L7 j L McDougal-littell textbook (chapter.sectian) {Extension after 2.11 5 days for activities, and : assessment Standard No. correlation Ull L3 8 days of lessons Topic/ Strand Ull L8 AA.30 A.A.3l A.A.31 AN.? {Extension after 2.1\ {Extension after 2.1\ {Extension after 2.11 SRH n. 930 ' 13.1,SRHp.931 A.N.S 13.2 AN.& 13.2 ,, Integrated Algebra Curriculum Map Approx. Time Frame Arlington Section No., Topic! Strand Standard No. McDougal-littell textbook correlation (chapter.section) Unit 12: Probability UI2 LI Basic Probability Concepts More Complex ProbabilitY Problems Independent Events Dependent events Mutually Exclusive Events Non-mutually exclusive events A.S.18 13.1,13.4 A.S.22 A's.23 A.S.23 A.S.23 A.S.23 13.4 13.4 13.4 13.4 A.S.23 13.4 U12L2 6 days of lessons 4 days for review and Ul2L3 U12L4 UJ2L5 assessment UJ2L6 Total time allotted: 10 days