georgia middle school math series sample content from texts georgia middle school math series sample content from texts about carnegie learning, inc. carnegie learning is focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standard • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com georgia middle school math series sample content from texts carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-022-3 georgia middle school math level 1 student edition georgia middle school math level 1 student edition volume 1 georgia middle school math level 1 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-025-4 georgia middle school math level 2 student edition georgia middle school math level 2 student edition volume 1 georgia middle school math level 2 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-028-5 georgia middle school math level 3 student edition georgia middle school math level 3 student edition volume 1 georgia middle school math level 3 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software availablegeorgia middle school math series sample content from texts 2010 carnegie learning437 grant st., suite 2000 pittsburgh, pa 15219 phone 412.690.2442 customer service phone 877.401.2527 fax 412.690.2444 www.carnegielearning.com copyright 2010 by carnegie learning, inc. all rights reserved. carnegie learning, cognitive tutor, schoolcare, and learning by doing are all registered marks of carnegie learning, inc. all other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. this product or portions thereof is manufactured under license from carnegie mellon university. permission is granted for photocopying rights within licensed sites only. any other usage or reproduction in any form is prohibited without the express consent of the publisher. isbn 978-1-60972-074-2 2010 middle school sampler printed in the united states of america 1-10/2010 hps 2010 carnegie learning2010 carnegie learning this sampler contains an overview of the carnegie learning georgia middle school math series. the foreword describes the layout of the instructional materials and the pedagogical approach. sample lessons from each grade level are included. course table of contents level 1 - grade 6 table of contents ............................................................................................. page 5 level 2 - grade 7 table of contents ............................................................................................. page 12 level 3 - grade 8 table of contents ............................................................................................. page 19 foreword student textbook set .................................................................................................................... . page 27 instructional design ....................................................................................................................... . page 28 process icons ................................................................................................................................ . page 29 lesson opener .............................................................................................................................. . page 30 problem types ............................................................................................................................... . page 31 chapter summary .......................................................................................................................... . page 42 the student assignment and skills practice book ....................................................................... . page 43 teacher’s implementation guide ................................................................................................... . page 43 teacher resources and assessments book ................................................................................. . page 43 level 1 (grade 6) student lesson 7.1: introduction to percents............................................................................... page 47 student lesson 7.2: estimating percents ..................................................................... page 59 student lesson 8.3: unit rates .................................................................................................... page 73 student lesson 8.4: using proportions to solve problems ......................................................... page 81 sampler contents middle school sampler contents l 3 2010 georgia middle school sampler 32010 carnegie learning level 2 (grade 7) student lesson 4.3: discrete versus continuous ………………………………………...………....… page 95 student lesson 6.2: solving equations ……………………….......……………………………....…… page 117 student assignment 6.2 ……………………………............................…………………………....…… page 127 student skills practice 6.2 ……………………………………...............................……………....…… page 131 teacher implementation guide for lesson 6.2 …………..…....…………………………………....… page 141 student lesson 6.3: multiple representations of problem situations ……………………………... page 155 student assignment 6.3 ……………......................................………………………………………... page 161 student skills practice 6.3 …………………………………...........................……………………....… page 165 chapter 6 summary …………...........................................……………………………………....…… page 175 chapter 6 assessments ………………………….....................................…………………....……… page 189 level 3 (grade 8) student lesson 2.4: investigate magnitude through theoretical probability and experimental probability…………………………………………………………………………………. page 221 student lesson 4.3: operations with square roots ….................................................................. page 231 student lesson 4.4: the pythagorean theorem …........................................................................ page 241 student lesson 9.5: writing explicit formulas for arithmetic sequences………........................... page 255 glossary ..............................................................…………………………………………...………… page 268 l sampler contents middle school 4 2010 georgia middle school sampler 4carnegie learning georgia middle school level 1—grade 6 table of contents ga unit ch. key mathematical objective grade 6 standards unit 01: gathering data (5 weeks) 1 collecting, displaying, and analyzing 1.1 data organization and analysis m6d1 1.2 pictographs and bar graphs m6d1 1.3 line plots and stem-and-leaf plots m6d1 1.4 histograms m6d1 1.5 line graphs and double line graphs m6d1 1.6 summary of graphs m6d1 2 designing and implementing an experiment 2.1 design questions for a survey or experiment m6d1 2.2 design questions that are answered by experiment m6d1 2.3 conducting an experiment m6d1 2.4 analyze results of an experiment and draw conclusions m6d1 unit 02: fun and games: extending and applying (3 weeks) 3 factors, multiples, primes, and composites 3.1 factors and multiples m6n1 3.2 physical models of factors and multiples m6n1 3.3 investigating prime and composite numbers m6n1 3.4 investigating divisibility rules m6n1 4 prime factorization and fundamental theorem of arithmetic 4.1 prime factorization and factor trees m6n1 4.2 investigating multiples and least common multiples m6n1 4.3 investigating factors and greatest common factors m6n1 4.4 using gcf and lcm to solve problems m6n1 2010 carnegie learning middle school carnegie learning georgia middle school level 1—grade 6 l 5 2010 georgia middle school sampler 52010 georgia middle school sampler 6 2010 carnegie learning 6 ga unit ch. key mathematical objective grade 6 standards unit 03: fractions, decimals, ratios, and percents (6 weeks) 5 fractions 5.1 modeling parts of a whole m6n1 5.2 fractional representations m6n1 5.3 dividing a whole into fractional parts m6n1 5.4 benchmark fractions m6n1 5.5 equivalent fractions m6n1 5.6 adding and subtracting fractions with like and unlike denominators m6n1 5.7 improper fractions and mixed numbers m6n1 5.8 parts of parts m6n1 5.9 part in a part m6n1 6 decimals 6.1 introduction to decimals m6n1 6.2 compare, order, estimate, and round decimals m6n1 6.3 fraction-decimal equivalents m6n1 6.4 adding and subtracting decimals m6n1 6.5 multiplying decimals m6n1 6.6 dividing decimals m6n1 l carnegie learning georgia middle school level 1—grade 6 middle school2010 georgia middle school sampler 7 2010 carnegie learning 7 middle school carnegie learning georgia middle school level 1—grade 6 l ga unit ch. key mathematical objective grade 6 standards unit 03: fractions, decimals, ratios, and percents (6 weeks) 7 percents 7.1 introduction to percentages m6n1 7.2 estimating percents m6n1 7.3 determine the percent of a number m6n1 7.4 determine the part, whole, or percent of percent problems m6n1 7.5 solving problems using fractions, decimals, and percents: part ii m6n1 8 ratios 8.1 introduction to ratios m6n1; m6a1; m6a2 8.2 using ratios: measurement conversations m6n1; m6m2; m6a1; m6a2 8.3 unit rates m6n1 8.4 using proportions to solve problems m6n1 unit 04: one-step equations (2 weeks) 9 solving equations 9.1 solving problems using proportions and percent equations m6a2; m6a3 9.2 writing expressions for real-world situations m6a2; m6a3 9.3 writing and evaluating algebraic expressions m6a2; m6a3 9.4 solving one-step equations m6a2; m6a3 9.5 evaluating expressions m6a2; m6a32010 georgia middle school sampler 8 2010 carnegie learning 8 l carnegie learning georgia middle school level 1—grade 6 middle school ga unit ch. key mathematical objective grade 6 standards unit 05: symmetry ( 2 weeks) 10 line, reflectional, and rotational symmetry 10.1 what is symmetry? m6g1 10.2 identifying line and reflectional symmetry m6g1 10.3 rotational symmetry m6g1 10.4 degrees of rotation m6g1 10.5 symmetry in art and architecture m6g1 unit 06: scale factor (3 weeks) 11 units of measure 11.1 history of the english system m6m1; m6m2 11.2 metric system m6m1; m6m2 11.3 between system conversions m6m1; m6m2 11.4 using appropriate measures/tools and units m6m1; m6m2 12 similar figures, scale factors, and drawings 12.1 defining similar figures m6m1; m6m2; m6g1; m6a1 12.2 corresponding parts of similar figures m6m1; m6m2; m6g1; m6a1 12.3 calculating parts of similar figures m6m1; m6m2; m6g1; m6a1 12.4 scale drawings and scale factor m6m1; m6m2; m6g1; m6a1 12.5 calculating scale factor m6m1; m6m2; m6g1; m6a1 12.6 creating scale drawings m6m1; m6m2; m6g1; m6a12010 georgia middle school sampler 9 2010 carnegie learning 9 middle school carnegie learning georgia middle school level 1—grade 6 l ga unit ch. key mathematical objective grade 6 standards unit 07: direct proportion (3 weeks) 13 proportionality 13.1 what is a proportion? m6a2 13.2 modeling proportions: part i m6a2a; m6a2b m6a2c 13.3 modeling proportions: part ii m6a2a; m6a2b m6a2c 13.4 solving without cross multiplication m6a2a; m6a2b m6a2c; m6a2d 13.5 define proportions m6a2a; m6a2b m6a2c; m6a2d 14 direct proportion and proportional reasoning 14.1 defining direct proportionality m6a2a; m6a2b m6a2c; m6a2d 14.2 using equations, tables, graphs, and verbal descriptions (multiple representations) to define proportional relationships m6a2a; m6a2b; m6a2c; m6a2d; m6a2e; m6a2f 14.3 graphing and interpreting graphs of direct proportion m6a2a; m6a2b; m6a2c; m6a2d; m6a2e; m6a2f 14.4 using proportions to solve problems part i: ratios m6a2a; m6a2b; m6a2c; m6a2d; m6a2e; m6a2f; m6a2g 14.5 using proportions to solve problems part ii: algebraically m6a2a; m6a2b; m6a2c; m6a2d; m6a2e; m6a2f; m6a2g2010 georgia middle school sampler 10 2010 carnegie learning 10 l carnegie learning georgia middle school level 1—grade 6 middle school ga unit ch. key mathematical objective grade 6 standards unit 08: solids (4 weeks) 15 geometric solids 15.1 nets m6g2 15.2 cube and rectangluar solids m6m3; m6m4; m6g2 15.3 prisms m6m3; m6m4; m6g2 15.4 pyramids m6m3; m6g2 15.5 cylinders m6m3; m6m4; m6g2 15.6 cones m6m3; m6g2 15.7 putting it all together m6m3; m6m4; m6g2 unit 09: games of chance (3 weeks) 16 introduction to probability 16.1 what is probability? m6d2 16.2 representing probabilities as ratios from 0 to 1 m6d2 16.3 calculating the probability of an event m6d2 16.4 calculating theoretical probability m6d2 17 experimental versus theoretical probability 17.1 conducting an experiment part i: predicting the outcome m6d2 17.2 conducting an experiment part ii: using the outcome to make predictions m6d2 17.3 experimental probability m6d2 17.4 designing and implementing a probability experiment m6d2 17.5 theoretical versus experimental probability m6d22010 georgia middle school sampler 11 2010 carnegie learning 11 middle school carnegie learning georgia middle school level 1—grade 6 l ga unit ch. key mathematical objective grade 6 standards unit 10: show what we know (3 weeks) 18 show what you know 18.1 using samples, ratios, and proportions to make predications 18.2 ratios, rates, and mixture problems 18.3 ratios and part-to-whole relationships 18.4 ratios, part-to-part relationships, and direct variation 18.5 using percents 18.6 percents and taxes 19 show what you know 19.1 measures of central tendency 19.2 collecting and analyzing data 19.3 quartiles and box-and-whisker plots2010 georgia middle school sampler 12 2010 carnegie learning 12 l carnegie learning georgia middle school level 2—grade 7 middle school carnegie learning georgia middle school level 2—grade 7 table of contents ga unit ch. key mathematical objective grade 7 standards unit 01: dealing with data (5 weeks) 1 data collection 1.1 formulate questions and collect data m7d1a 1.2 collect data through random sampling m7d1a; m7d1b 1.3 random sampling m7d1a; m7d1b 1.4 sample size m7d1a; m7d1b 1.5 interpret results m7d1a; m7d1b 2 data displays and analysis 2.1 analyze data using measures of center m7d1c; m7d1d; m7d1f 2.2 display and analyze data using the five-number summary m7d1c; m7d1d; m7d1f 2.3 graph and interpret box-and-whisker plots m7d1c; m7d1d; m7d1f 2.4 display and analyze relationships between two variables using scatter plots m7d1c; m7d1d; m7d1f 2.5 collect, display, and analyze data m7d1c; m7d1d; m7d1f 3 comparing data sets 3.1 design a survey or an experiment m7d1e; m7d1g 3.2 conduct a survey or experiment m7d1e; m7d1g2010 georgia middle school sampler 13 2010 carnegie learning 13 middle school carnegie learning georgia middle school level 2—grade 7 l ga unit ch. key mathematical objective grade 7 standards unit 02: patterns and relationships (4 weeks) 4 relationships between two quantities 4.1 the coordinate plane m7a3a; m7a3b; m7d1 4.2 using tables and graphs m7a3b; m7d1 4.3 discrete versus continuous m7a3; m7d1 4.4 interpolation and extrapolation m7a3; m7d1 5 algebraic expressions 5.1 relationships between quantities m7a1a 5.2 simplifying algebraic expressions m7a1b 5.3 using the distributive property to simplify algebraic expressions m7a1b; m7a1c 5.4 multiple representations of equal expressions m7a1 6 linear equations 6.1 visualizing algebra through picture models m7a2a 6.2 solving equations m7a2 6.3 multiple representations of problem situations m7a2b 6.4 multiple representations of problem situations m7a2 6.5 using two-step equations m7a22010 georgia middle school sampler 14 2010 carnegie learning 14 l carnegie learning georgia middle school level 2—grade 7 middle school ga unit ch. key mathematical objective grade 7 standards unit 03: rational reasoning (5 weeks) 7 positive and negative numbers 7.1 introduction to negative numbers m7n1 7.2 models of signed numbers m7n1 7.3 adding and subtracting integers, part 1 m7n1 7.4 adding and subtracting integers, part 2 m7n1 7.5 subtracting integers m7n1 7.6 multiplying and dividing integers m7n1 8 evaluating and simplifying expressions 8.1 number systems and properties m7n2 8.2 identities and inverses m7n2 8.3 order of oerations m7n2 9 linear equations and the coordinate plane 9.1 graphing signed numbers on the coordinate plane m7a3; m7a4 9.2 making sense of negative solutions m7a3; m7a4 9.3 evaluating and solving algebraic expressions m7a3; m7a4 9.4 rate of change m7a3; m7a4 9.5 using multiple representation to solve/analyze problems situations m7a3; m7a42010 georgia middle school sampler 15 2010 carnegie learning 15 middle school carnegie learning georgia middle school level 2—grade 7 l ga unit ch. key mathematical objective grade 7 standards unit 04: flip, slide, and turn (4 weeks) 10 constructing geometric figures 10.1 what is a geometric construction? m7g1 10.2 constructing angles m7g1 10.3 congruent figures and constructing congruent triangles m7g1 10.4 complements, supplements, midpoints, perpendiculars, and perpendicular bisectors m7g1 10.5 angles formed by transversals of parallel and non-parallel lines m7g1 10.6 constructing parallel lines m7g1 11 reflections, translations, and rotations 11.1 translations on the coordinate plane m7g2 11.2 translations by construction m7g2 11.3 reflections on the coordinate plane m7g2 11.4 reflections by construction m7g2 11.5 rotations on the coordinate plane m7g2 11.6 rotations by construction m7g22010 georgia middle school sampler 16 2010 carnegie learning 16 l carnegie learning georgia middle school level 2—grade 7 middle school ga unit ch. key mathematical objective grade 7 standards unit 05: staying in shape (6 weeks) 12 dilations and similar figures 12.1 comparing figures m7g2 12.2 dilating figures m7g2 12.3 dilations, scale drawings, scale models, and scale factors m7g3a 12.4 similarity m7g3a 13 working with similar figures 13.1 perimeters of similar figures m7g3a; m7g3b 13.2 missing parts of similar figures m7g3a; m7g3b 13.3 indirect measurement m7g3a; m7g3b 13.4 indirect measurement, part 2 m7g3a; m7g3b 14 dilations, scale factors, and similar figures 14.1 scale factor on the coordinate plane m7g2 14.2 scale models on the coordinate plane m7g2 14.3 congruent figures m7g2; m7g3 14.4 calculating unknown parts of congruent figures m7g32010 georgia middle school sampler 17 2010 carnegie learning 17 middle school carnegie learning georgia middle school level 2—grade 7 l ga unit ch. key mathematical objective grade 7 standards unit 06: : values that vary (3 weeks) 15 modeling proportional relationships 15.1 proportional relationships m7a3; m7a5 15.2 direct variation m7a3; m7a6 15.3 equation of direct variation m7a3; m7a7 15.4 inverse variation m7a3; m7a8 15.5 equation for inverse variation m7a3; m7a9 16 using proportions to solve problems 16.1 proportions and direct variation m7a3b; m7a3c; m7a3d 16.2 constant of proportionality m7a3 16.3 using proportional relationships to solve problems, part 1 m7a3; m7a5 16.4 using proportional relationships to solve problems, part 2 m7a3; m7a5 unit 07: slices and shadows (3 weeks) 17 building three-dimensional figures 17.1 translating two-dimensional figures through space m7g4 17.2 rotating two-dimensional figures through space m7g4 17.3 stacking congruent or similar two-dimensional figures m7g4 17.4 recognizing solids resulting from translations, rotations, and stacking m7g4 18 slicing three-dimensional figures 18.1 cross sections of three-dimensional solids m7g4 18.2 cross sections of a cube m7g4 18.3 cross sections of a cone, sphere, and cylinder m7g4 18.4 cross sections of a pyramid and a prism m7g4 18.5 associating solids and cross sections m7g42010 georgia middle school sampler 18 2010 carnegie learning 18 l carnegie learning georgia middle school level 2—grade 7 middle school ga unit ch. key mathematical objective grade 7 standards unit 08: show what we know (3 weeks) 19 show what you know 19.1 patterns and sequences 19.2 determining the 10th term of a sequence 19.3 dtermining the nth term of a sequence 19.4 using a sequence to represent a problem situation 19.5 using tables, graphs, and equations, part 1 19.6 using tables, graphs, and equations, part 2 19.7 comparing problem situations algebraically and graphically 20 show what you know 20.1 introduction to probability 20.2 theoretical and experimental probability 20.3 using probability to make predications 20.4 graphing frequencies of outcomes2010 georgia middle school sampler 19 2010 carnegie learning 19 middle school carnegie learning georgia middle school level 3—grade 8 l carnegie learning georgia middle school level 3—grade 8 table of contents ga unit ch. key mathematical objective grade 8 standards unit 01: outcomes and likelihoods (5 weeks) 1 outcomes 1.1 tree diagrams and sample space m8d3a 1.2 independent and dependent events m8d3 1.3 tree diagrams and the counting principle m8d4 1.4 the counting principle m8d5 1.5 using frequency tables to organize sample spaces m8d2b; m8d2a 2 likelihoods 2.1 determine probabilities of simple events m8d3a 2.2 explore “or” statements as compound problems m8d3a 2.3 explore “and” statements as compound problems m8d3a; m8d2b 2.4 investigate magnitude through experimental probability and theoretical probability m8d3a; m8d2b 2.5 determine probability using all the rules m8d3a; m8d2b 2.6 use tables and diagrams to solve everyday probability problems m8d3a; m8d2b 2.7 use an applet to make sense of probability m8d3a; m8d2b 2.8 determine whether a game is fair m8d22010 georgia middle school sampler 20 2010 carnegie learning 20 l carnegie learning georgia middle school level 3—grade 8 middle school ga unit ch. key mathematical objective grade 8 standards unit 02: the powers that be (4 weeks) 3 properties of exponents 3.1 powers and exponents m8n1i 3.2 multiplying and dividing powers m8n1i 3.3 zero and negative exponents m8n1i 3.4 scientific notation m8n1j 3.5 operations with scientific notation m8n1 3.6 identifying the properties of powers m8n1i 4 squares and the pythagorean theorem 4.1 squares and square roots m8n1a; m8n1d; m8n1f; m8n1c 4.2 distinguish between rational and irrational numbers m8n1c; m8n1h 4.3 operations with square roots m8n1g; m8n1e 4.4 the pythagorean theorem m8g2b 4.5 the converse pythagorean theorem m8g2a2010 georgia middle school sampler 21 2010 carnegie learning 21 middle school carnegie learning georgia middle school level 3—grade 8 l ga unit ch. key mathematical objective grade 8 standards unit 03: equal or not (4 weeks) 5 real numbers and their properties 5.1 evaluating algebraic expressions m8a1b 5.2 simplifying algebraic expressions m8a1c 5.3 writing algebraic expressions m8a1a; m8a1b 5.4 evaluating expressions m8a1b 6 solving equations 6.1 visualizing algebra m8a1 6.2 solving one-step equations m8a1c 6.3 solving two-step equations m8a1a; m8a1c; m8a1e; m8a3i 6.4 using two-step equations m8a1a.; m8a1c; m8a1e; m8a3i 6.5 solving more complicated equations m8a1c 7 solving equations, absolute value equations, and inequalities 7.1 solving and graphing inequalities in one variable m8a2a; m8a2c 7.2 solving and graphing absolute value equations m8a1c 7.3 solving absolute value inequalities m8a1c 7.4 solving literal equations m8a1d2010 georgia middle school sampler 22 2010 carnegie learning 22 l carnegie learning georgia middle school level 3—grade 8 middle school ga unit ch. key mathematical objective grade 8 standards unit 04: functional relationships (5 weeks) 8 venn diagrams and set theory 8.1 using logic to solve problems m8a3a 8.2 using logic to solve problems m8a3a 8.3 introduction to sets m8d1a 8.4 introduction to venn diagrams m8d1b; m8d1c 8.5 three-set venn diagrams m8d1 8.6 solving problems using venn diagrams m8d1 9 arithmetic sequences, relations, and functions 9.1 developing sequences of numbers from diagrams and contexts m8a3 9.2 arithmetic sequences m8a3 9.3 writing an arithmetic sequence using recursive notation m8a3 9.4 representing a pattern of growth by a constant value on a graph m8a3 9.5 writing explicit formulas for arithmetic sequences m8a3 10 relations and functions 10.1 describing characteristics of graphs m8a3a; m8a3b 10.2 defining and recognizing functions m8a3a; m8a3b; m8a3c; m8a3.h 10.3 using function notation m8a3a; m8a3b; m8a3c; m8a3h 10.4 stating the domain and range from various representations m8a3b; m8a3d2010 georgia middle school sampler 23 2010 carnegie learning 23 middle school carnegie learning georgia middle school level 3—grade 8 l ga unit ch. key mathematical objective grade 8 standards unit 05: slippery slope (5 weeks) 11 analyzing linear equations 11.1 determing rate of change from a graph m8a4a; m8a3i 11.2 determing rate of change from a table m8a4b 11.3 determing rate of change from a context m8a4c 11.4 determing rate of change from an equation m8a4g; m8a3i 11.5 determing the y-intercept from various representations m8a4b 11.6 writing an equation of a line m8g4 12 writing and graphing linear equations 12.1 writing equations of lines m8a4f 12.2 linear and piecewise functions m8a4f; m8a4g 12.3 writing and graphing an inequality in two variables m8a5c 12.4 other function representations—putting it all together m8a4 13 lines of best fit 13.1 drawing the line of best fit m8d4b 13.2 using lines of best fit m8d4b 13.3 performing an experiment m8d4a 13.4 correlation m8d4 13.5 using technology to determine a linear regression, part 1 m8d4b 13.6 using technology to determine a linear regression, part 2 m8d4b 13.7 scatter plots and non-linear data m8d42010 georgia middle school sampler 24 2010 carnegie learning 24 l carnegie learning georgia middle school level 3—grade 8 middle school ga unit ch. key mathematical objective grade 8 standards unit 06: traversing congruency ( 4 weeks) 14 line and angle relationships 14.1 relationships between two lines m8g1 14.2 angle relationships formed by two intersecting lines m8g1 14.3 angle relationships formed by two lines intersected by a transversal m8g1 14.4 slopes of parallel and perpendicular lines m8g1 15 parallel line relationships 15.1 angle relationships formed by two parallel lines intersected by a transversal m8g1 15.2 segment relationships formed by parallel lines intersected by a transversals m8g1 15.3 uniqueness of a line parallel to a given line through a point m8g1 15.4 constructing parallel lines using the properties of parallel lines cut by a transversal m8g12010 georgia middle school sampler 25 2010 carnegie learning 25 middle school carnegie learning georgia middle school level 3—grade 8 l ga unit ch. key mathematical objective grade 8 standards unit 07: systems (4 weeks) 16 systems of equations 16.1 using a graph to solve a linear system m8a5a; m8a5b; m8a5c 16.2 graphs and solutions of linear systems m8a5 16.3 using substitution to solve a linear system 1 m8a5b 16.4 using substitution to solve a linear system 2 m8a5b 16.5 using linear combinations to solve a linear system 1 m8a5a; m8a5b 16.6 using linear combinations to solve a linear system 2 m8a5a; m8a5b 17 more with systems of equations and inequalities 17.1 using the best method to solve a linear system, part 1 m8a5b; m8a5d 17.2 using the best method to solve a linear system, part 2 m8a5b; m8a5d 17.3 solving linear systems m8a5d 17.4 solving systems of linear inequalities 1 m8a5c 17.5 solving systems of linear inequalities 2 m8a5c2010 georgia middle school sampler 26 2010 carnegie learning 26 l carnegie learning georgia middle school level 3—grade 8 middle school ga unit ch. key mathematical objective grade 8 standards unit 08: show what we know 18 show what you know 18.1 introduction to quadratic functions 18.2 parabolas 18.3 comparing linear and quadratic functions 18.4 using the quadratic formula to solve quadratic equations 18.5 using a vertical motion model 19 show what you know 19.1 polynomials and polynomial functions 19.2 adding and subtracting polynomials 19.3 multiplying and dividing polynomials 19.4 introduction to exponential functions 19.5 exponential growth and decay 19.6 special topic: logic2010 carnegie learning 27 2010 georgia middle school sampler 27 middle school foreword l a revolutionary math program the carnegie learning georgia middle school math series: levels 1–3 provide research-based and engaging instruction to help all middle school students master math concepts and skills. these instructional materials are aligned to the georgia performance standards for mathematics grades 6–8 and were purposefully designed to address the overview, enduring understanding, essential questions, and evidence of learning components outlined in the various unit organizers. students who complete the series will have a solid foundation to be successful in high school mathematics. the carnegie learning georgia middle school math instructional materials are being developed in partnership with the richmond county design team. the richmond county school district is currently field-testing the materials and will provide feedback for enhancements and revisions to be incorporated in the publication of the series for the 2011–2012 school year. the goal of the georgia middle school math texts is to help students make connections between math concepts and to help them understand mathematical relationships. students will build on their prior knowledge and obtain new knowledge by solving problems set in a real-world context related to their interests. students will construct and interpret mathematical models and be asked to explain their reasoning. this learning by doing approach to mathematics engages students as active participants to learn and develop a deep understanding of mathematics. by completing activities in these instructional materials, students will build both conceptual and procedural knowledge. student textbook set the student textbook set contains two volumes: the student textbook and the student assignments and skills practice book. these volumes are consumable and allow students to write their answers within the book. as students complete lessons, they will have record of their understanding of important rules, properties, and key mathematical concepts. forewordinstructional design within each student lesson, questions and instruction are interleaved to engage students as they develop their own understanding. the lessons are structured to provide students with various opportunities to reason, to model, and to expand on explanations about mathematical ideas. the overarching questioning strategy throughout the text is to promote analysis and higher order thinking skills beyond simple “yes” or “no” responses. by explaining problem-solving steps or rationale for a solution, students will internalize the processes and reasoning behind the mathematics. to achieve the learning goals of each lesson, students will respond to questions that ask them to: • look for patterns • estimate • describe • determine • compare and contrast • calculate • write a rule • generalize • explain your reasoning lessons include a variety of problem types for students. these instructional features include lesson openers, step-by-step demonstrations, pre-written student methods, who’s correct, sorting activities, and more. we invite you to explore how our instructional materials thoughtfully lead and support students. it is our intent to instill the idea that math is relevant not because it comes with a rule book that must be followed in a rote manner, but because it provides a common and useful language and means for discussing and solving complex problems in everyday life. l foreword middle school 28 2010 carnegie learning 2010 georgia middle school sampler 28process icons icons are placed throughout each lesson to promote a think, pair, share implementation. middle school foreword l 29 2010 carnegie learning 2010 georgia middle school sampler 292010 georgia middle school sampler 30 2010 carnegie learning 30 l foreword middle school lesson opener each lesson opener is a short paragraph designed to motivate students to think about and to discuss the usefulness and connections of mathematics in a variety of real-world contexts. these two- to three-minute discussions prepare students to start the lesson. sample: grade 6 lesson 8.4 looks can be deceiving2010 georgia middle school sampler 31 2010 carnegie learning 31 middle school foreword l problem type: real-world connections students engage in solving real-world problem scenarios throughout. sample: grade 8 lesson 4.4 the pythagorean theorem2010 georgia middle school sampler 32 2010 carnegie learning 32 l foreword middle school problem type: worked examples worked examples provide students with step-by-step demonstrations of mathematical ideas. sample: grade 7 lesson 7.6 equal groups2010 georgia middle school sampler 33 2010 carnegie learning 33 middle school foreword l problem type: analyzing student methods pre-written student methods allow students the opportunity to analyze viable methods and problem-solving strategies. questions are presented along with the student methods to help students think deeper about the various strategies. it is our intent to foster flexibility and a student’s internal dialog about the mathematics and strategies used to solve problems. sample: grade 6 lesson 5.8 pizzas by the slice—or the rectangle!2010 georgia middle school sampler 34 2010 carnegie learning 34 l foreword middle school problem type: analyzing correct and incorrect reponses these problems shift the focus from simply right or wrong answers to a focus on determining where the error is, why it is an error, and how to correct it. these types of problems will help students analyze their own work for errors and correctness. sample: grade 8 lesson 2.2 you be the teacher!2010 georgia middle school sampler 35 2010 carnegie learning 35 middle school foreword l problem type: who’s correct? who’s correct? problems present statements and situations that may or may not be correct. students determine what is correct and what is incorrect, and then explain their reasoning. these types of problems will help students analyze their own work for errors and correctness. sample: grade 8 lesson 3.6 watch your step! sample: grade 7 lesson 6.3 lost in translation? that depends on the audience2010 georgia middle school sampler 36 2010 carnegie learning 36 l foreword middle school problem type: using manipulatives manipulatives are used throughout to foster a conceptual understanding of mathematical concepts. these activities provide students with opportunities to develop strategies and reasoning that will serve as the foundation for learning more abstract mathematics. sample: grade 7 lesson 6.2 maintaining a balance2010 georgia middle school sampler 37 2010 carnegie learning 37 middle school foreword l problem type: matching, sorting, and exploring students experience various hands-on activities where they match or sort verbal descriptions, tables, and graphs. sample: grade 7 lesson 4.2 traveling the frequent-flyer friendly skies2010 georgia middle school sampler 38 2010 carnegie learning 38 l foreword middle school problem type: using technology step-by-step instructions provide students with opportunities to understand how to use graphing calculators. sample: grade 7 lesson 2.2 box it up!2010 georgia middle school sampler 39 2010 carnegie learning 39 middle school foreword l problem type: graphic organizers students create their own references of key mathematical concepts. sample: grade 8 lesson 3.6 watch your step!2010 georgia middle school sampler 40 2010 carnegie learning 40 l foreword middle school problem type: summarizing two types of lesson summaries are featured. a summary presents an authored review of the major mathematical concept(s) or rule(s) from the lesson so students have a concise, accurate reference for review. a problem summary uses open-ended questions to guide students to generalize mathematical concepts. sample: grade 7 lesson 6.2 maintaining a balance2010 georgia middle school sampler 41 2010 carnegie learning 41 middle school foreword l sample: grade 7 lesson 6.3 lost in translation? that depends on the audience2010 georgia middle school sampler 42 2010 carnegie learning 42 l foreword middle school chapter summary chapter summaries feature key terms and lesson-by-lesson highlights of the key mathematical concepts.2010 georgia middle school sampler 43 2010 carnegie learning 43 middle school foreword l the student assignment and skills practice book the student assignment and skills practice book provides substantial opportunities for students to practice and apply their understanding of mathematical objectives explored in class. there is one assignment and skills practice activity per lesson. the skills practice worksheets provide the teacher with additional problems sets to differentiate practice for students depending on the skills they still need to work on. the first question in each problem set is an example to guide students on how to complete the exercise; it displays the correct answer. teacher’s implementation guide the teacher’s implementation guide is a resource for planning, guiding, and facilitating student learning. additional questions are provided for the teacher to ask during the student work phase and share phase of each lesson. a lesson map provides a warm-up, the essential ideas of the lesson, the standards, the key terms, the learning goals, and any materials needed to facilitate the activities. throughout the lesson, you will find relevant teacher notes. here are some examples: • problem notes provide a description of the mathematical concepts embedded in the task. • key questions are highlighted to spark student thinking and reasoning. • grouping suggestions provide recommendations on which exercise might be best utilized for whole-group discussion, working in pairs, working individually, and so on. • common misconceptions provide insights for anticipating how to help students who may have mislearned or misinterpreted a math concept. also included in the teacher notes are check for understanding questions provided at the end of each lesson. these questions can help you quickly ascertain which students get it and which may need more time to master the mathematical concept or skill. teacher resources and assessments book the second offering within the teacher textbook set is the teacher resources and assessments book. it contains the answer sets for the skills practice worksheets and the assignments. the curriculum offers multiple opportunities to assess student knowledge. each chapter contains a pre-test, a post-test, a mid-chapter test, an end-of-chapter test, and a standardized test.2010 georgia middle school sampler 44 2010 carnegie learning 44 l foreword middle schoolgeorgia middle school math series level 1 2010 carnegie learning 2010 georgia middle school sampler 452010 carnegie learning 2010 georgia middle school sampler 463 2010 carnegie learning 7.1 percents can make or break you! introduction to percents learning goals in this lesson, you will: write fractions, decimals, and percents. model percents on a hundredths grid. explain the similarities and differences of percents, fractions, and decimals. key terms poll percent what are the latest numbers in the polls? the word poll actually has several meanings. one meaning is that it is another word for survey, but polls usually refer to the speciﬁc survey used to gain the opinions of voters during an election process. in almost any election process, candidates and candidate advisors constantly monitor polls to see what the voters’ opinions are toward that candidate. a slip of a candidate’s approval can be costly when voters head to the polls to cast their vote. what other types of polls have you seen? where have you seen polls displayed? discuss your ideas with your group. problem 1 they’re all part of the same family what do these statements mean to you? there is an 80 percent chance of rain tomorrow. you earn 90 percent on a science test. big sale! 20 percent discount on all regular-priced items. your bill at the eat and talk restaurant is 40. below the total, the restaurant adds a 20 percent tip. the star of the high school basketball team makes 80 percent of his free throws. sales tax is 7 percent in richmond county. yuma, arizona, has 90 percent sunny days. 3 7 7.1 introduction to percents 2010 georgia middle school sampler 47chapter 7 percents 4 2010 carnegie learning 7 2010 georgia middle school sampler 48 the term percent is a fraction in which the denominator is 100. percent can also be another name for hundredths. the percent symbol “%” means “out of 100.” therefore: 35 percent means 35 out of 100. 35 percent as a fraction would be 35 ____ 100 . 35 percent as a decimal would be 0.35. you can shade 35 of the 100 squares on the hundredths grid to represent 35 percent. percents, fractions, and decimals can be used interchangeably. 1. write each shaded part of the hundredths grid as a fraction, decimal, and percent. a. b.7.1 introduction to percents 5 2010 carnegie learning 7 2010 georgia middle school sampler 49 c. d. 2. shade the hundredths grids to represent each percent shown. then, write the equivalent fraction and decimal for each percent. a. 44% b. 16% c. 97%chapter 7 percents 6 2010 carnegie learning 7 2010 georgia middle school sampler 50 d. 117% 3. write each decimal as a percent. a. 0.4 b. 0.37 c. 0.7381 d. 0.52 when the denominator is a factor of 100, it is easy to write a fraction as a percent. when the denominator is not a factor of 100, you will need to divide the numerator by the denominator to write that fraction as a percent. 4. write each fraction as a percent. round your answer to the nearest tenth. a. 4 __ 5 b. 3 ___ 10 c. 3 __ 8 d. 1 __ 3 take note remember a percent means how many hundredths.7.1 introduction to percents 7 7 2010 georgia middle school sampler 51 5. how are percents similar to decimals? how are percents and decimals different? 6. how are percents similar to fractions? how are percents and fractions different? 7. label each indicated mark on the number line as a fraction, decimal, and percent. make sure your fractions are in simplest form. a. fraction 1 1.0 0 1 0 0.0 0% 100% decimal percent 1–3 0.66 b. fraction 1–2 7–8 0.125 0.625 0.75 1.0 1 1 0 0 0.0 0% 25% 37.5% 100% decimal percent c. fraction 2–5 0.2 0.5 0.9 1.0 1 1 0 0 0.0 0% 30% 60% 80% 100% decimal percent 1—10 7—10 2010 carnegie learningchapter 7 percents 8 2010 georgia middle school sampler 52 7 problem 2 survey says… 1. one hundred sixth-grade students take a survey that asks about their preferences for a class trip. a. complete the table by representing each of the survey results as a fraction, a decimal, and a percent. make sure your fractions are in simplest form. fraction decimal percent how many days should we plan for the trip? stay overnight two nights 60 out of 100 students stay overnight one night 25 out of 100 students no overnight stay 15 out of 100 students where should we go? philadelphia 35 out of 100 students washington, d.c. 22 out of 100 students new york city 30 out of 100 students atlanta 13 out of 100 students how should we get here? bus 25 out of 100 students airplane 75 out of 100 students are you planning on going on the trip? yes 100 out of 100 students 2010 carnegie learning7.1 introduction to percents 9 7 2010 georgia middle school sampler 53 b. write a summary of the results of the survey using percents. problem 3 do you think you know your percents? it’s time to play the percentage match game. in this game, you will use your knowledge of percents, fractions, and decimals, and your memory. rules of the game: cut out the cards shown. deal all the cards face down in an array. the ﬁrst player chooses any card in the array. that player then tries to ﬁnd the ﬁrst card’s equivalent match in the array. if the player makes a match, he or she receives one point. the two matched cards are then put into the player’s pile. the ﬁrst player then picks again and repeats the process until a match is not found. if the ﬁrst player cannot ﬁnd an equivalent match, it is the second player’s turn. the same process for picking and matching cards described is now followed by the second player. the game continues until all the cards have been paired with an equivalent match. the player with the most points wins! 2010 carnegie learningchapter 7 percents 10 7 2010 georgia middle school sampler 54 2010 carnegie learning7.1 introduction to percents 11 7 2010 georgia middle school sampler 55 3 __ 5 6 ____ 10 0.6 60% 1 __ 8 12.5% 1 ____ 10 1% 0.1 10% 1 __ 5 2 ____ 10 1 __ 4 2 __ 8 3 ____ 10 30% 1 __ 3 33% 2 __ 6 0. ___ 3 1 __ 2 50% 2 __ 3 66.6 __ 6% 3 __ 4 6 __ 8 0.75 75% 2010 carnegie learningchapter 7 percents 12 7 2010 georgia middle school sampler 56 2010 carnegie learning7.1 introduction to percents 13 7 2010 georgia middle school sampler 57 summary percents, fractions, and decimals can be used interchangeably. the chart summarizes equivalent fractions, decimals, and percents. common equivalent fractions, decimals, and percents fraction 1 __ 5 1 __ 4 1 __ 3 2 __ 5 1 __ 2 3 __ 5 2 __ 3 3 __ 4 4 __ 5 decimal 0.2 0.25 0.33 __ 3 0.4 0.5 0.6 0.66 __ 6 0.75 0.8 percent 20% 25% 33 1 __ 3 % 40% 50% 60% 66 2 __ 3 % 75% 80% be prepared to share your methods and solutions. 2010 carnegie learning7 2010 georgia middle school sampler 58 chapter 7 percents 14 2010 carnegie learning15 7.2 estimating percents 2010 carnegie learning 5.1 7.2 what do you think the meaning of the following statement is? “tomorrow, there will be a 30 percent chance of rain.” although this statement seems simple enough, a study showed that its meaning can vary dramatically from person to person. there appears to be three different interpretations of the statement: (1) it will rain 30 percent of the time during the day; (2) only 30 percent of the forecasted area will have rain while the remaining areas will be dry; (3) there is a 30 in 100, or 3 in 10, chance that it will actually rain. what is common with all of these interpretations is that they are all estimates, but that is where the similarities stop. so what do you think “30 percent chance of rain” means? share your ideas with your partner. wacky weather! estimating percents learning goals in this lesson, you will: estimate percents as fractions and decimals. write fractions as percents. identify equivalent forms of fractions, decimals, and percents. order fractions, decimals, and percents. key term benchmark percents 7 2010 georgia middle school sampler 59chapter 7 percents 16 2010 carnegie learning 7 problem 1 estimating with percents 1. what does the saying, “i gave it 100 percent!” mean? when you estimate with percents, it is easier to work with those you are familiar with. you know that 100 percent means one, or the whole, and 50 percent means half. a laptop computer uses an icon of a battery on the toolbar to show how much power the battery has left. when you glance at the icon, you can get a good estimate of how much battery life is left before you need to recharge the battery. 2. estimate how much battery power remains by writing the percent with each battery icon. a. b. c. d. 2010 georgia middle school sampler 607.2 estimating percents 17 2010 carnegie learning 7 e. f. 3. estimate the shaded part of each circle graph shown, and write it as a percent. a. b. c. d. e. f. 2010 georgia middle school sampler 61chapter 7 percents 18 2010 carnegie learning 7 4. estimate the shaded part of each circle graph, and write it as a fraction, a decimal, and a percent. make sure your fraction is in simplest form. a. b. c. d. e. f. 2010 georgia middle school sampler 627.2 estimating percents 19 2010 carnegie learning 7 problem 2 benchmark percents 1. use your calculator to investigate the following percents. a. 1% of 28 b. 10% of 28 c. 1% of 234 d. 10% of 234 e. 1% of 0.85 f. 10% of 0.85 g. 1% of 5.86 h. 10% of 5.86 i. 1% of 98.72 j. 10% of 98.72 k. 1% of 1085.2 l. 10% of 1085.2 m. what patterns do you notice? 2. write a rule to calculate 1% of any number. 3. write a rule to calculate 10% of any number. 2010 georgia middle school sampler 63chapter 7 percents 20 2010 carnegie learning 7 a benchmark percent is a percent that is commonly used, such as 1 percent, 5 percent, 10 percent, 25 percent, 50 percent, and 100 percent. you can use benchmark percents to calculate the percent of any number. 4. state each relationship. a. how is 50 percent related to 100 percent? b. how is 25 percent related to 100 percent? how is 25 percent related to 50 percent? c. how is 10 percent related to 100 percent? how is 10 percent related to 50 percent? d. how is 5 percent related to 10 percent? e. how is 1 percent related to 10 percent? how is 1 percent related to 5 percent? 5. try these percents mentally. calculate the value of each using your knowledge of benchmark decimals. a. 100% of 300 b. 50% of 300 c. 25% of 300 d. 10% of 300 2010 georgia middle school sampler 647.2 estimating percents 21 2010 carnegie learning 7 e. 5% of 300 f. 1% of 300 akuro eats at the eat and talk restaurant and decides to leave a 15 percent tip. akuro says, “i can easily calculate 10 percent of any number, and then calculate half of that, which is equal to ﬁve percent. i can then add those two percent values together to get a sum of 15 percent.” 6. do you think akuro’s method is reasonable? how much should she leave for a tip of 15% on 16.00? 7. what is 15 percent of these restaurant check totals? explain how to calculate your answer. round to the nearest hundredth if necessary. a. 24.00 b. 32.56 c. 47.00 2010 georgia middle school sampler 65chapter 7 percents 22 2010 carnegie learning 7 you can determine any percent of a number in your head by using 10 percent, 5 percent, and 1 percent. 8. how can you use 10 percent, 5 percent, and/or 1 percent to determine each percent given? explain your reasoning. a. 18% b. how can you calculate 25 percent of a number? c. how can you calculate 37 percent of a number? 9. estimate each using 1 percent, 5 percent, and 10 percent. a. 27% of 84 b. 43% of 116 c. 98% of 389 2010 georgia middle school sampler 667.2 estimating percents 23 2010 carnegie learning 7 d. 77% of 1400 e. 12% of 1248 10. about 12 percent of the united states population is left-handed. estimate how many left-handed students are in a class of: a. 100 students. b. 200 students c. 150 students 2010 georgia middle school sampler 67chapter 7 percents 24 2010 carnegie learning 7 2010 georgia middle school sampler 687.2 estimating percents 25 2010 carnegie learning 7 problem 3 ordering fractions, decimals, and percents 1. order these numbers from greatest to least using what you have learned about fractions, decimals, and percents. cut out the cards to help you order the numbers. 33 1 __ 3 % 1 __ 4 13 ___ 50 78% 0.0666… 0.1% 3 __ 4 50 ___ 75 0.098 0.51 3 __ 5 80% 0.98 1.0 27% 198 _____ 200 2010 georgia middle school sampler 69chapter 7 percents 26 2010 carnegie learning 7 26 2010 georgia middle school sampler 717.2 estimating percents 27 2010 carnegie learning 7 summary ways to calculate common equivalent fractions, decimals, and percents fraction decimal percent percent write the percent as a fraction with a denominator of 100. simplify. 28% 28 ____ 100 7 ___ 25 write the percent as a fraction with a denominator of 100. write the fraction as a decimal. 42% 42 ____ 100 0.42 fraction write the fraction as an equivalent fraction with a denominator of 10, 100, 1000… then, write it as a decimal. 7 ___ 20 35 ____ 100 0.35 or divide the numerator by the denominator. 2 __ 9 2 9 0.22 __ 2… write an equivalent fraction with a denominator of 100. write the fraction as a percent. 3 __ 5 60 ____ 100 60% or use division to write the fraction as a decimal, and then a percent. 5 __ 8 5 8 0.625 62.5% decimal write the decimal as a fraction with a denominator of 10, 100, 1000… simplify. 0.28 28 ____ 100 7 ___ 25 write the decimal as a fraction. then, write the fraction as a percent. 0.08 8 ____ 100 8% or write the decimal as a fraction. then, write the equivalent fraction with a denominator of 100. then, write the fraction as a percent. 0.4 4 ___ 10 40 ____ 100 40% or move the decimal point two places to the right and add the % sign. 0.08 8% be prepared to share your solutions and methods. 2010 georgia middle school sampler 71chapter 7 percents 28 2010 carnegie learning 7 2010 middle school sampler 72 2010 georgia middle school sampler 7281 2010 carnegie learning 88 have you ever traveled to another country? if so, you probably had to take your passport with you, and exchange american dollars into foreign currency. exchange rates are generally between the lowest currency denominations of two countries. so when exchanging american dollars, the exchange rate will reﬂect how much the u.s. dollar is worth compared to the foreign currency. some examples of foreign currencies are the euro, the british pound, and the argentine peso. can you name other foreign currencies? do any currencies share a similar name to the u.s. dollar? discuss your answers with your group. 8.3 what is the better buy? unit rates learning goals in this lesson, you will: use unit rates to solve problems. use unit rates to calculate the best buy. calculate unit rates. key term unit rate problem 1 unit rate as you learned previously, a rate is a ratio in which the two quantities being compared are measured in different units. a unit rate is a comparison of two measurements in which the denominator has a value of one unit. the most common place you may have encountered unit rates is at the supermarket. unit rates can help you determine which of two or more items is the best buy. car manufacturers also use unit rates when they advertise how many miles per gallon their car gets in the city or on the highway. 8.3 unit rates 8 2010 georgia middle school sampler 73chapter 8 ratios 82 2010 carnegie learning 8 1. compare the prices for various sizes of popcorn sold at the local movie theater. mega bag 10.24 for 32 oz giant bag 6.00 for 24 oz medium bag 4.48 for 16 oz kid’s bag 2.40 for 8 oz a. what is the unit rate price per ounce for each of the sizes of popcorn? b. what size popcorn is the best buy? explain your reasoning. 2. the local paper published these rates on gas mileage for a few new cars. “avalar can travel 480 miles on 10 gallons of gas.” “sentar can travel 400 miles on 8 gallons of gas.” “comstar can travel 360 miles on 9 gallons of gas.” change each to unit rates so that it reports miles per one gallon of gas. a. avalar b. sentar c. comstar 2010 georgia middle school sampler 748.3 unit rates 83 2010 carnegie learning 8 3. how can unit rates help you make decisions about comparisons? unit rates are also useful when calculating multiple numbers of an item. 4. complete each table. a. pine wood stakes are sold by the yard. pine wood stakes cost 3.49/yd. 1 yd 2 yds 3 yds 5 yds 10 yds 20 yds 25 yds 3.49 b. carpet is sold by the square yard. classroom carpet sells for 10.50/yd. 1 yd 2 40 yd 2 50 yd 2 100 yd 2 10.50 c. pink lady apples are sold by the pound. one pound of pink lady apples costs 2.99. 1 lb 2 lbs 5 lbs 10 lbs 20 lbs 2.99 d. purchases in your county also have a 7 percent sales tax added for every dollar of the purchase price. 1 5 10 20 50 100 0.07 2010 georgia middle school sampler 75chapter 8 ratios 84 2010 carnegie learning 8 5. tony needs a rate table for his tutoring jobs so that he can look up the charge quickly. a. complete the rate table. hours 0.5 1 1.5 2 2.5 3 3.5 4 charge 2.50 b. how can tony use this table to determine what to charge for: i. 6 hours? ii. 7 hours? iii. 7.5 hours? c. tony made 21.25 last weekend. how many hours did he tutor? explain your reasoning. d. if tony made 125 for one week of tutoring over the summer vacation, how many hours did he tutor? 2010 georgia middle school sampler 768.3 unit rates 85 2010 carnegie learning 8 problem 2 using unit rates the yearbook advisors at stewart middle school organize the prep rally and run each year to raise money for the prep yearbook. the prep rally and run takes place over the weekend so students can accumulate miles for two days. each student must ﬁnd sponsors who will pledge a dollar amount for each mile the student runs. 1. paul asks for 1 pledges for every mile he runs. he was able to ﬁnd 35 people to pledge 1 per mile he runs. casey asks for 2 pledges for every mile she runs. she was able to ﬁnd 15 people to pledge 2 per mile she runs. a. complete the rate table for paul to track his pledges. number of miles run 1 2 3 4 5 6 7 8 9 10 money pledged (in dollars) b. complete the rate table for casey to track her pledges. number of miles run 1 2 3 4 5 6 7 8 9 10 money pledged (in dollars) c. if paul raises 525 for the prep rally and race, how many miles did he run? explain your reasoning. d. if casey raises 420 for the prep rally and race, how many miles did she run? explain your reasoning. 2010 georgia middle school sampler 77chapter 8 ratios 86 2010 carnegie learning 8 8 2. in the spring, the gym teachers at stewart middle school sponsor a bike-a-thon to raise money for new sporting equipment. students seek sponsors to pledge a dollar amount for each mile they ride. a. paul can ride 5.5 miles per hour. at this rate, how far will he ride in 5 hours? b. casey can ride 6.25 miles per hour. at this rate, how far will she ride in 5 hours? c. if leticia rides 36.25 miles in 5 hours, what is her rate? d. guadalupe rode 38 miles at 8 miles per hour. how long did she ride? e. emil got a cramp in his leg after riding 19 miles in 2 hours. what was his rate up until he got a leg cramp? f. ichiro was pedaling at 15 miles per hour and could only last for 19.5 miles. how long did he ride? g. what formula can you use to relate the distance, the rate, and the time? 2010 georgia middle school sampler 788.3 unit rates 87 2010 carnegie learning 8 3. guests at a dinner play are seated at three tables. each table is served large, round loaves of bread instead of individual rolls. each person at the table shares the loaves equally. table 1 has six guests and is served two loaves of bread. table 2 has eight guests and is served three loaves of bread. table 3 has 10 guests and is served four loaves of bread. a. at which table does a guest get the most bread? b. how much bread does each guest at each table get? 4. the school cafeteria plans their lunch menus using proportions. the cafeteria manager knows that four pizzas will serve 10 students. a. how many pizzas should they make for 400 students? write a proportion that represents this situation. b. how many pizzas should they make for 240 students? write a proportion that represents this situation. c. how many students will 40 pizzas serve? write a proportion that represents this situation. be prepared to share your solutions and methods. 2010 georgia middle school sampler 79chapter 8 ratios 88 2010 carnegie learning 8 2010 georgia middle school sampler 8089 2010 carnegie learning 88 have you ever seen a shark up close? perhaps you have seen sharks at an aquarium or on the internet. would you say that sharks generally look scary? well, looks can be deceiving. if you encountered a basking shark, you might be startled, but there is nothing to fear. these mighty beasts actually swim around with their mouths wide open looking quite intimidating, but actually, they are just feeding on plankton. unfortunately, these sharks are on the “endangered” list in the north atlantic ocean. have you ever wondered how scientists keep track of endangered species populations? how would you track endangered species? discuss your ideas with your partner. 8.4 looks can be deceiving! using proportions to solve problems learning goals in this lesson, you will: solve proportions using the scaling method. solve proportions using the unit rate method. solve proportions using the means and extremes method. key terms variable solve a proportion scaling method isolate a variable unit rate method inverse operations means and extremes method problem 1 does that shark have its tag? because it is impossible to count each individual animal, marine biologists use a method called the capture-recapture method to estimate the population of certain sea creatures. biologists are interested in effectively managing populations to ensure the long-term survival of endangered species. in certain areas of the world, biologists randomly catch and tag a given number of sharks. after a period of time, such as a month, they recapture a second sample of sharks and count the total number of sharks as well as the number of recaptured tagged sharks. then, the biologists use proportions to estimate the population of sharks living in a certain area. 8 8.4 using proportions to solve problems 2010 georgia middle school sampler 81chapter 8 ratios 90 2010 carnegie learning 8 biologists can set up a proportion to estimate the total number of sharks in an area. original number of tagged sharks _______________________________ total number of sharks in an area number of recaptured tagged sharks number of sharks caught in the second sample although capturing the sharks once is necessary for tagging, it is not necessary to recapture the sharks each time. at times, the tags can be observed through binoculars from a boat or at shore. biologists originally caught and tagged 24 sharks off the coast of cape cod, massachusetts, and then released them back into the bay. the next month, they caught 80 sharks with 8 of the sharks already tagged. to estimate the shark population off the cape cod coast, biologists set up the following proportion: 24 tagged sharks ________________ p total sharks 8 recaptured tagged sharks _________________________ 80 total sharks notice the p in the proportion. the p is a variable, and it is used to represent the unknown total quantity of sharks off the coast of cape cod. a variable is a letter or symbol used to represent a number. let p represent the total shark population off the coast of cape cod. this proportion could have been written several ways. think about equivalent fractions. you can rearrange the numbers in equivalent fraction statements to make more equivalent fraction statements. example 1 example 2 2 __ 3 4 __ 6 5 __ 7 15 ___ 21 6 __ 3 4 __ 2 21 ___ 7 15 ___ 5 2 __ 4 3 __ 6 5 ___ 15 7 ___ 21 2010 georgia middle school sampler 828.4 using proportions to solve problems 91 2010 carnegie learning 8 1. write three more proportions you could use to determine the total shark population off the coast of cape cod. 2. estimate the total shark population using any of the proportions. 3. did any of the proportions seem more efﬁcient than the other proportions? 4. wildlife biologists tag deer in wildlife refuges. they originally tagged 240 deer and released them back into the refuge. the next month, they observed 180 deer, of which 30 deer were tagged. approximately how many deer are in the refuge? show your work. a proportion of the form a __ b c __ d can be written in many different ways. another example is d __ b c __ a or c __ a d __ b . 5. write all the different ways you can rewrite the proportion a __ b c __ d and maintain equality. 2010 georgia middle school sampler 83chapter 8 ratios 92 2010 carnegie learning 8 problem 2 quality control companies use proportions to predict the failure rate of their products, and to try to correct defects for overall improvements of their product. the ready steady battery company tests batteries as they come through the assembly line and then predicts how many of its total production might be defective. on friday, the quality controller tested every tenth battery and found that of the 320 batteries tested, 8 were defective. if the company shipped a total of 3200 batteries how many might be defective? discuss the three methods. method 1 cassie wrote the following proportion: 8 defective batteries __________________ 320 batteries d defective batteries _________________ 3200 batteries she explained her reasoning by saying, “i can scale up by 10 and determine how many batteries might be defective.” 10 8 ____ 320 d _____ 3200 10 d 80 “so, 80 batteries might be defective.” t c m o a i m d take note a quality controller is a person who randomly tests products to ensure an acceptable amount of product can be sold to the public. any product manufacturer will generally have a team of quality controllers. 2010 georgia middle school sampler 848.4 using proportions to solve problems 93 2010 carnegie learning 8 method 2 donald shared his method. he said, “i can take the ratio 8 ____ 320 and change it to a unit rate. i can then scale up that rate.” 8 defective batteries _____________________ 320 total batteries 1 defective battery ____________________ 40 total batteries “one out of every 40 batteries is defective. so, out of 3200 batteries, 80 batteries could be defective because 3200 40 80.” method 3 natalie said, “i have noticed something about proportions. when i write the proportions with a colon like this: (80)320 25,600 8:320 80:3200 (8)3200 25,600 …the two middle numbers have the same product as the two outside numbers. so, i can solve any proportion by setting these two products equal to each other.” 1. which method do you prefer? why? explain your reasoning. 2. try the various proportion-solving methods on these proportions and determine the unknown value. a. 3 granola bars _____________ 420 calories g granola bars _____________ 140 calories b. 8 correct: 15 questions 24 correct: q questions c. d dollars ________ 5 miles 9 _________ 7.5 miles 2010 georgia middle school sampler 85chapter 8 ratios 94 2010 carnegie learning 8 each of these methods have a speciﬁc name. method 1 is called the scaling method. you can either scale up by multiplying, or you can scale down by dividing. method 2 is called the unit rate method. you calculate the unit rate, and then scale up to the rate you need. method 3 is called the means and extremes method. in a proportion, the product of the means equals the product of the extremes. to solve a proportion using this method, ﬁrst, identify the means and extremes. then, set the product of the means equal to the product of the extremes and solve for the missing quantity. to solve a proportion means to determine all the values of the variables that make the proportion true. in general, a proportion can be written two ways: using colons or setting two ratios equal to each other. for any numbers a, b, c, and d where b and d are not zero, extremes a:b c:d means bc ad or a __ b c __ d means extremes bc ad for example, 7 books ________ 14 days 3 books ________ 6 days the means are 3 and 14. the extremes are 7 and 6. (3)(14) (7)(6) 42 42 you can write four different equations using means and extremes. analyze each equation. 3 (7)(6) _____ 14 14 (7)(6) _____ 3 (3)(14) ______ 7 6 (3)(14) ______ 6 7 2010 georgia middle school sampler 868.4 using proportions to solve problems 95 2010 carnegie learning 8 3. why are these equations all true? explain your reasoning. 4. compare these equations to the equation showing the product of the means equal to the product of the extremes. how was the balance of the equation maintained in each? 5. why is it important to maintain balance in equations? in the proportion a __ b c __ d , you can multiply both sides by b to isolate the variable a. when you isolate the variable in an equation, you perform an operation, or operations, to get the variable by itself on one side of the equals sign. multiplication and division are inverse operations. inverse operations are operations that “undo” each other. method 1: b a __ b c __ d b a cb ___ d method 2: another strategy to isolate the variable a is to multiply the means and extremes, and then isolate the variable by performing inverse operations. step 1: bc ad step 2: bc ___ d ad ___ d step 3: bc ___ d a step 4: a bc ___ d 6. describe each step in method 2. 2010 georgia middle school sampler 87chapter 8 ratios 96 2010 carnegie learning 8 7. rewrite the proportion a __ b c __ d to isolate each of the other variables: b, c, and d. explain the strategies you used to isolate each variable. problem 3 using proportions 1. the school store sells computer games for practicing mathematic skills. the table shows how many of each game were sold last year. game fast facts fraction fun percent sense measurement mania number of games sold 120 80 50 150 a. how many total games were sold last year? b. the store would like to order a total of 1000 games this year. about how many of each game should the store order? c. if the store would like to order a total of 240 games this year, about how many of each game should the store order? 2010 georgia middle school sampler 888.4 using proportions to solve problems 97 2010 carnegie learning 8 2. you are making lemonade to sell at the track meet. according to the recipe, you need 12 ounces of lemon juice for every 240 ounces of sugar water. you have 16 ounces of lemon juice. a. how many ounces of sugar water do you need? show your work. b. how many ounces of lemonade can you make? 3. a maintenance company charges a mall owner 45,000 to clean his 180,000 square foot shopping mall. a. how much should a store of 4800 square feet pay? show your work. b. how much should a store of 9200 square feet pay? 4. the national park service has to keep a certain level of bass stocked in a lake. they tagged 60 bass and released them into the lake. two days later, they caught 128 ﬁsh and found that 32 of them were tagged. what is a good estimate of how many bass are in the lake? show your work. 2010 georgia middle school sampler 89chapter 8 ratios 98 2010 carnegie learning 8 5. an astronaut who weighs 85 kilograms on earth weighs 14.2 kilograms on the moon. how much would a person weigh on the moon if they weigh 95 kilograms on earth? round your answer to the nearest tenth. 6. water goes over niagara falls at a rate of 180 million cubic feet every 30 minutes. how much water goes over the falls in 1 minute? 7. malea’s dvd collection consists of the types of dvds shown in the table. a. determine the percent each type of dvd represents by completing the table. category number of movies percent of the collection comedy 15 musicals 3 dramas 10 documentaries 2 animation 10 b. write each ratio. i. comedy to drama ii. documentaries to the whole collection iii. animation to drama 2010 georgia middle school sampler 908.4 using proportions to solve problems 99 2010 carnegie learning 8 8. the value of the u.s. dollar in comparison to the value of foreign currency changes daily. complete the table shown. euro u.s. dollar 1 1.44 1.00 6.00 6 10 9. bottles of water are sold at various prices and in various sizes. write each as a ratio, and then as a unit rate. which bottle is the best buy? explain how you know. bottle 1 bottle 2 bottle 3 bottle 4 0.39 per 12 oz 0.57 per 24.3 oz 1.39 per 128 oz 0.70 per 33.8 oz 2010 georgia middle school sampler 91chapter 8 ratios 100 2010 carnegie learning 8 10. to make 4 cups of fruity granola, the recipe calls for 1.5 cups of raisins, 1 cup of granola, and 2 cups of blueberries. if you want to make 18 cups of fruity granola, how much of each of the ingredients do you need? be prepared to share your solutions and methods. 2010 georgia middle school sampler 92georgia middle school math series level 2 2010 carnegie learning 2010 georgia middle school sampler 932010 carnegie learning 2010 georgia middle school sampler 944.3 discrete versus continuous 175 2010 carnegie learning 4 2010 georgia middle school sampler 95 4.3 to connect or not connect? that is the question… discrete versus continuous learning goals in this lesson, you will: l identify data sets as discrete or continuous. l sketch graphs, tables, and expressions from various scenarios. key terms l continuous data l discrete data the national association of rocketry annually sponsors an event called team america rocketry challenge (tarp). this event challenges middle and high school students to develop and build a rocket to perform certain tasks. this year’s challenge is for students to build a rocket that can carry an egg exactly 825 feet in the air, and then to carry that egg back to the ground without cracking it. do you think that using a coordinate plane might be a way for students to design a rocket that can perform the challenge? discuss your ideas with your group.chapter 4 relationships between two quantities 176 2010 carnegie learning 4 2010 georgia middle school sampler 176 problem 1 match and sort in this activity, you will match a speciﬁc graph to a scenario, following the given steps. 1. cut out each graph. 2. tape each graph in the box with the appropriate scenario. 3. label the axes with the appropriate quantities. 4. cut out the scenarios, and sort them into similar groups. 2010 georgia middle school sampler 964.3 discrete versus continuous 177 2010 carnegie learning 4 2010 georgia middle school sampler 97 a. y x b. y x c. y x d. y xchapter 4 relationships between two quantities 178 2010 carnegie learning 4 2010 georgia middle school sampler 984.3 discrete versus continuous 179 2010 carnegie learning 4 2010 georgia middle school sampler 99 e. y x f. y x g. y x h. y xchapter 4 relationships between two quantities 180 2010 carnegie learning 4 2010 georgia middle school sampler 1004.3 discrete versus continuous 181 2010 carnegie learning 4 2010 georgia middle school sampler 101 i. y x j. y xchapter 4 relationships between two quantities 182 2010 carnegie learning 4 2010 georgia middle school sampler 1024.3 discrete versus continuous 183 2010 carnegie learning 4 2010 georgia middle school sampler 103 1. you buy t-shirts to sell for your school. there is a 25 design charge. what is the total cost for different numbers of t-shirts? 2. a bus leaves school at the end of the day and stops to drops off its ﬁrst passenger.chapter 4 relationships between two quantities 184 2010 carnegie learning 4 2010 georgia middle school sampler 1044.3 discrete versus continuous 185 2010 carnegie learning 4 2010 georgia middle school sampler 105 3. you have fig newtons for your class party. how many fig newtons will each classmate receive (you don’t know how many classmates will show up)? 4. you are drinking your milk through a straw, and then the carton spills over.chapter 4 relationships between two quantities 186 2010 carnegie learning 4 2010 georgia middle school sampler 1064.3 discrete versus continuous 187 2010 carnegie learning 4 2010 georgia middle school sampler 107 5. your telephone calling card charges 0.40 for the ﬁrst minute of calls and 0.40 for each additional minute of calls. 6. the video stores charges 3.00 for dvd rentals. how many dvds can you rent for different amounts of money?chapter 4 relationships between two quantities 188 2010 carnegie learning 4 2010 georgia middle school sampler 1084.3 discrete versus continuous 189 2010 carnegie learning 4 2010 georgia middle school sampler 109 7. you record the temperature for each hour on february 2, 2010. 8. there is a record of your growth chart since you were born.chapter 4 relationships between two quantities 190 2010 carnegie learning 4 2010 georgia middle school sampler 1104.3 discrete versus continuous 191 2010 carnegie learning 4 2010 georgia middle school sampler 111 9. on monday, the rain fell at a steady rate. then, it let up for a few hours before a sudden downpour. finally it let up. 10. you toss a basketball in the air.chapter 4 relationships between two quantities 192 2010 carnegie learning 4 2010 georgia middle school sampler 1124.3 discrete versus continuous 193 2010 carnegie learning 4 2010 georgia middle school sampler 113 4 5a. how did you sort your graphs? b. did your partner sort his or hers the same way? problem 2 continuous and discrete data data in a graph can be described as continuous data or discrete data. continuous data have an inﬁnite number of values. often, these types of data are associated with types of physical measurement. one general way to tell if data are continuous is to determine if it is possible for the data values to be fractions or decimals. if it is possible, then the data are usually continuous. the points on the graph are connected. in other words, for every point on the x-axis, there will a data point on the y-axis. discrete data only occur if a ﬁnite number of values are possible. discrete data usually occurs in cases where there are only a certain number of values, or when you are counting something using whole numbers. the points are not connected with a line in discrete data graphs. 1. use the scenarios and graphs from problem 1 to answer each. a. which situations are continuous? b. which situations are discrete?chapter 4 relationships between two quantities 194 2010 carnegie learning 4 4 2. read each scenario shown and sketch a graph. name the x- and y-axes. you do not need to give values to axes. then, decide whether or not you will connect the points you plotted based on the scenario. finally, determine if the data are discrete or continuous. a. lunches sold for the week of february 8, 2010. y x b. the level of medicine active in the bloodstream over time. y x 2010 georgia middle school sampler 1144.3 discrete versus continuous 195 2010 carnegie learning 4 2010 georgia middle school sampler 115 c. the amount of money collected on the video game comet avoidance for the last seven days. y x 3. compare your graphs to your partner’s. do they look the same? why or why not?chapter 4 relationships between two quantities 196 2010 carnegie learning 4 2010 georgia middle school sampler 116 4. water freezes at 0 celsius or 32 fahrenheit. water boils at 100 celsius or 212 fahrenheit. use the coordinate plane shown to plot the freezing and boiling points of water for both celsius and fahrenheit measurements. don’t forget to name the coordinate plane and label the x- and y-axes. finally, determine if the data set is discrete or continuous. x 20 10 25 50 75 100 125 150 175 200 30 40 50 60 70 80 90 y 225 be prepared to share your solutions and methods.6.2 solving equations 285 2010 carnegie learning 2010 georgia middle school sampler 117 6 6.2 maintaining a balance solving equations learning goals in this lesson, you will: l develop an understanding of equality. l use properties of equality to solve equations represented with algebra tiles. l solve one-step equations. key terms l solve l solution l inverse operation l properties of equality you have heard of keeping your balance, but did you know there is also a machine called a balance? these devices are also called scales and they have a long and storied past. the balance is an instrument used to measure the weight and mass of an object. no one is quite sure who invented the balance, but there have been models found in mesopotamia and egypt that suggest the machine has been around since 5000 b.c.e. the balance consists of a lever and two pans. the way the balance works is that a weight is placed on one side, while the object being weighed is placed in the pan on the other side of the lever. when the weight and the object being weighed are the same, the lever remains “balanced” in a horizontal position. what other scales or balances have you seen? have you seen scales or balances in the nurse’s ofﬁce or supermarket? what do those scales or balances look like? discuss your answers with your group. 6chapter 6 linear equations 286 2010 carnegie learning 6 2010 georgia middle school sampler 118 problem 1 equal or not 1. each representation shows a balance. what will balance one rectangle in each problem? adjustments can be made in each pan as long as the balance is maintained. describe your strategies. a. strategies: what will balance one rectangle? b. strategies: what will balance one rectangle?6.2 solving equations 287 2010 carnegie learning 6 2010 georgia middle school sampler 119 c. strategies: what will balance one rectangle? d. strategies: what will balance one rectangle?chapter 6 linear equations 288 2010 carnegie learning 6 2010 georgia middle school sampler 120 e. strategies: what will balance one rectangle? 2. generalize the strategies for maintaining balance. complete each sentence. a. to maintain balance when you subtract a quantity from one side, you must b. to maintain balance when you add a quantity to one side, you must c. to maintain balance when you multiply a quantity by one side, you must d. to maintain balance when you divide a quantity by one side, you must6.2 solving equations 289 2010 carnegie learning 6 2010 georgia middle school sampler 121 3. look at the pan balance shown. how many will balance ? strategies: problem 2 one step at a time let’s revisit the pan balance representations from problem 1, question 1, parts (a) and (b) and replace and , with variables and numbers. 1. rewrite the representations from problem, 1 question 1, parts (a) and (b) using symbols. let x and 1 unit. then, describe how the strategies you used to determine what balanced 1 rectangle can apply to the equation. determine what balances x. a. b. 2. does the value of x maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work.chapter 6 linear equations 290 2010 carnegie learning 6 2010 georgia middle school sampler 122 3. how is the strategy you used to determine what balances x related to the operation of the original equation? you just solved one-step equations. when you set expressions equivalent to each other and identify the value that replaces the variable to make the equation true, you solve the equation. to solve an equation, you must get the variable by itself on one side of the equation by performing inverse operations. inverse operations are pairs of operations that undo each other. addition and subtraction are inverse operations. for example, if 3 5 8, then 8 5 3. multiplication and division are inverse operations. for example, if 6 2 12, then 12 6 2. a solution to an equation is any variable value that makes that equation true. to determine if your solution is correct, substitute the value of the variable back into the original equation. if the equation remains balanced, then you have calculated the solution of the equation. 4. consider the four equations shown. first, state the inverse operation needed to isolate the variable. then, solve the equation. show your work. finally, check to see if the value of your solution maintains balance in the original equation. a. m 9 12 b. 21 x 86.2 solving equations 291 2010 carnegie learning 6 2010 georgia middle school sampler 123 c. n __ 5 12 d. 4y 24 problem 3 two steps back 1. rewrite the representations from problem 1, question 1, part (c) using symbols. let x and 1 unit. a. describe how the strategies you used to determine what balanced 1 rectangle can apply to the equation. determine what balances x? b. does the value of x maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work.chapter 6 linear equations 292 2010 carnegie learning 6 2010 georgia middle school sampler 124 you just solved a two-step equation. c. describe the order of operations you used in the original equation. d. compare the way you solved the equation to the order of the operations in the original equation. when you isolate a variable or “undo” operations, you must undo them in the reverse of the order of operations. in other words, the operation that comes last in the original equation should be undone ﬁrst. 2. write a sentence to describe how to apply inverse operations to solve each equation. then, solve each equation and verify your solution. a. 4x 15 61 b. 2 7x 166.2 solving equations 293 2010 carnegie learning 6 2010 georgia middle school sampler 125 c. 5 x __ 2 16 d. 17 2x 8 3. solve the riddle. show your work. a. what is a number that when you multiply it by 3 and subtract 5 from the product, you get 28? let x represent the number you are trying to determine. b. what is a number that when you multiply it by 4 and add 15 to the product, you get 79?chapter 6 linear equations 294 2010 carnegie learning 6 2010 georgia middle school sampler 126 c. make a number riddle for your partner to solve. problem 4 summary the properties of equality allow you to balance and solve equations involving any number. properties of equality for all numbers a, b, and c,… addition property of equality if a b, then a c b c. subtraction property of equality if a b, then a c b c. multiplication property of equality if a b, then ac bc. division property of equality if a b and c 0, then a __ c b __ c . be prepared to share your solutions and methods.6 chapter 6 assignments 89 2010 carnegie learning 2010 georgia middle school sampler 127 assignment assignment for lesson 6.2 name _________________________________________ date ___________________ maintaining a balance solving equations 1. madison middle school has a math and science club that holds meetings after school. the club has decided to enter a two-day competition that involves different math and science challenges. the first day of competition involves solving multi-step math problems. teams will receive one point for every problem they get correct. halfway through the day, the madison middle school team has 4 points. after a dinner break, the team does more problems and is able to finish the day with 11 points. a. the representation shows a balance for this situation. the left side of the balance represents the 4 points the team had at midday plus the additional points they got after dinner. the right side of the balance represents the total points they had at the end of the day. what will balance 1 rectangle in this representation? describe your strategy. strategy: i subtracted 4 squares from each side. what will balance 1 rectangle? seven squares will balance 1 rectangle. how many problems did they get correct after dinner? the team got 7 problems correct.6 90 chapter 6 assignments 2010 carnegie learning 2010 georgia middle school sampler 128 b. rewrite the representation from part (a) using symbols. let x and 1 point. then describe how the strategies you used to determine what balanced 1 rectangle can apply to the equation. x 4 11 i can subtract 4 from each side. x 7 c. does the value of x maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work. x 4 11 7 4 11 11 11 yes, x 7 maintains balance in the original equation. d. what does the value x 7 mean in the original situation? the value x 7 is the number of additional problems the team got correct after dinner on the first day of competition.6 2010 carnegie learning 2010 georgia middle school sampler 129 chapter 6 assignments 91 name_____________________________________________ date ____________________ 2. the second day of competition involves all hands-on science problems. because of the complexity of the problems, each team will get 3 points for every science problem that they get correct. recall that the madison middle school team is starting the day with 11 points. by the end of the day the team had 23 points. a. the representation shows a balance for this situation. the left side of the balance represents the 11 points the team had at the start of the day plus the additional points they got from the science competition. the right side of the balance represents the total points they had at the end of the second day. what will balance 1 rectangle in this representation? describe your strategy. strategy: i subtracted 11 squares from each side which left 3 rectangles on the left side and 12 squares on the right side. then, i divided the 12 squares into 3 equal groups of 4. what will balance 1 rectangle? four squares will balance 1 rectangle. how many science problems did they get correct? they got 4 science problems correct. b. rewrite the representation from part (a) using symbols. let x and 1 point. 3x 11 236 92 chapter 6 assignments 2010 carnegie learning 2010 georgia middle school sampler 130 c. solve the equation from part (b). write a sentence to describe how to apply inverse operations to solve the equation. then, solve the equation. i will subtract 11 from each side. then, i will divide each side by 3. 3x 11 23 3x 11 11 23 11 3x 12 x 4 d. does the value of x that you got as your solution maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work. 3x 11 23 3(4) 11 23 12 11 23 23 23 yes. the value of 4 maintains balance in the equation. therefore, the team answered 4 science problems correctly.6 chapter 6 skill practice 165 2010 carnegie learning 2010 georgia middle school sampler 131 6 skills practice skills practice for lesson 6.2 name _________________________________________ date ___________________ maintaining a balance solving equations vocabulary match the term with its definition. 1. solve c 2. inverse operations f 3. solution b 4. addition property of equality a 5. subtraction property of equality g 6. multiplication property of equality d 7. division property of equality e a. if a b, then a c b c b. any value for a variable that makes the equation true c. when you set expressions equivalent to each other and identify the value that replaces the variable to make the equation true d. if a b, then ac bc e. if a b, and c 0, then a __ c b __ c f. pairs of operations that undo each other g. if a b, then a c b c6 166 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 132 problem set determine what will balance one rectangle. explain your solution. then rewrite the representation as an equation (except question 8). 1. i subtracted 5 squares from each side which left 1 rectangle on one side and 4 rectangles on the other side. 1 rectangle 4 squares. x 5 9 2. i divided each side by 3 which left 1 rectangle on one side and 6 squares on the other side. 1 rectangle 6 squares. 3x 18 3. i divided each side by 5 which left 1 rectangle on one side and 2 squares on the other side. 1 rectangle 2 squares. 5x 106 chapter 6 skill practice 167 2010 carnegie learning 2010 georgia middle school sampler 133 6 name_____________________________________________ date ____________________ 4. i subtracted 8 squares from each side which left 4 rectangles on one side and 12 squares on the other side. then, i divided each side by 4 which left 1 rectangle on one side and 3 squares on the other side. 1 rectangle 3 squares. 4x 8 20 5. i subtracted 1 square and 2 rectangles from each side which left 3 squares on one side and 1 rectangle on the other side. 1 rectangle 3 squares. 4 2x 3x 1 6. i divided each side by 4 which left 1 rectangle on one side and 0.5 square on the other side. 1 rectangle 0.5 square. 4x 26 168 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 134 7. i subtracted 1 square and 3 rectangles from each side which left 2 rectangles on one side and 4 squares on the other side. then, i divided each side by 2 which left 1 rectangle on one side and 2 squares on the other side. 1 rectangle 2 squares. 5x 1 5 3x 8. in the first balance, i know that 2 rectangles balance 1 square. so, in the second balance i can replace the 4 squares each with 2 rectangles. i now have 11 rectangles balancing 22 circles. i can divide each side by 11. so, 1 rectangle 2 circles.6 chapter 6 skill practice 169 2010 carnegie learning 2010 georgia middle school sampler 135 name_____________________________________________ date _________________ solve each one-step equation. state the inverse operation you used to isolate the variable and check your solution. 9. n 35 60 subtract 35 from each side. check: n 35 60 25 35 60 n 35 35 60 35 60 60 n 25 10. b 51 19 add 51 to each side. check: b 51 19 70 51 19 b 51 51 19 51 19 19 b 70 11. 42 3x divide each side by 3. check: 42 3x 42 3(14) 42 ___ 3 3x ___ 3 42 42 14 x 12. s __ 9 11 multiply each side by 9. check: s __ 9 11 99 ___ 9 11 9 ( s __ 9 ) 9(11) 11 11 s 99 13. 12 m 29 add 29 to each side. check: 12 m 29 12 41 29 12 29 m 29 29 12 12 41 m6 170 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 136 14. 16x 80 divide each side by 16. check: 16x 80 16(5) 80 16x ____ 16 80 ___ 16 80 80 x 5 15. 67 y 58 subtract 58 from each side. check: 67 y 58 67 9 58 67 58 y 58 58 67 67 9 y 16. 24 d __ 7 multiply each side by 7. check: 24 d __ 7 24 168 ____ 7 7(24) 7 ( d __ 7 ) 24 24 168 d6 chapter 6 skill practice 171 2010 carnegie learning 2010 georgia middle school sampler 137 name_____________________________________________ date ____________________ solve each two-step equation. state the inverse operations, in the correct order of operations, you used to isolate the variable. check your solution. 17. 3w 17 4 add first, and then divide. check: 3w 17 4 3(7) 17 4 3w 17 17 4 17 21 17 4 3w 21 4 4 3w ___ 3 21 ___ 3 w 7 18. 25 8x 65 subtract first, and then divide. check: 25 8x 65 25 8(5) 65 25 25 8x 65 25 25 40 65 8x 40 65 65 8x ___ 8 40 ___ 8 x 5 19. 94 y __ 4 18 add first, and then multiply. check: 94 y __ 4 18 94 448 ____ 4 18 94 18 y __ 4 18 18 94 112 18 112 y __ 4 94 94 4(112) 4 ( y __ 4 ) 448 y6 172 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 138 20. g __ 9 47 64 subtract first, and then multiply. check: g __ 9 47 64 153 ____ 9 47 64 g __ 9 47 47 64 47 17 47 64 g __ 9 17 64 64 9 ( g __ 9 ) 9 (17) g 153 21. 72 5h 22 subtract first, and then divide. check: 72 5h 22 72 5(10) 22 72 22 5h 22 22 72 50 22 50 5h 72 72 50 ___ 5 5h ___ 5 10 h 22. c ___ 20 41 59 add first, and then multiply. check: c ___ 20 41 59 2000 _____ 20 41 59 c ___ 20 41 41 59 41 100 41 59 c ___ 20 100 59 59 20 ( c ___ 20 ) 20(100) c 2000 23. 138 7h 5 subtract first, and then divide. check: 138 7h 5 138 7(19) 5 138 5 7h 5 5 138 133 5 133 7h 138 138 133 ____ 7 7h ___ 7 19 h6 chapter 6 skill practice 173 2010 carnegie learning 2010 georgia middle school sampler 139 6 name_____________________________________________ date ____________________ 24. 16 8 p __ 2 subtract first, and then multiply. check: 16 8 p __ 2 16 8 16 ___ 2 16 8 8 8 p __ 2 16 8 8 8 p __ 2 16 16 2(8) 2 ( p __ 2 ) 16 p solve each riddle using equations. 25. what is the number that when you multiply it by 3 and subtract 12 from the product, you get 12? 3x 12 12 3x 24 x 8 the number is 8. 26. what is the number that when you multiply it by 7 and add 25 to the product, you get 46? 7x 25 46 7x 21 x 3 the number is 3. 27. what is the number that when you multiply it by 4 and add 74 to the product, you get 90? 4x 74 90 4x 16 x 4 the number is 4.6 174 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 140 28. what is the number that when you multiply it by 11 and subtract 30 from the product, you get 91? 11x 30 91 11x 121 x 11 the number is 11. 29. what is the number that when you multiply it by 5 and add 19 to the product, you get 54? 5x 19 54 5x 35 x 7 the number is 7. 30. what is the number that when you multiply it by 9 and subtract 63 from the product, you get 45? 9x 63 45 9x 108 x 12 the number is 12.6.2 solving equations • 2010 carnegie learning 2010 georgia middle school sampler 141 285a 6.2 solving equations learning goals in this lesson, you will: develop an understanding of equality. use properties of equality to solve equations represented with algebra tiles. solve one-step equations. key terms solve solution inverse operation properties of equality mathematics performance standards m7a1 students will represent and evaluate quantities using algebraic expressions. m7a2 students will understand and apply linear equations in one variable. m7a3 students will understand relationships between two variables. essential ideas inverse operations are pairs of operations that undo each other such as addition and subtraction, or multiplication and division. a solution to an equation is any value that makes that equation true. to determine if a solution to an equation is correct, substitute the value of the variable back into the original equation and if the equation remains balanced, the solution is correct. to isolate a variable in an equation, reverse the order of the operations in the original equation. the operation that comes last in the original equation should be undone first. the addition property of equality states if a b, then a c b c. the subtraction property of equality states if a b, then a c b c. the multiplication property of equality states if a b, then ac bc. the division property of equality states if a b and c 0, then a __ c b __ c . overview balances and balance pans are used to illustrate equations. rectangles, squares and circles are placed on two sides of a balance and students manipulate the shapes to determine what balances one rectangle. students will conclude to maintain a balance the same quantity must be subtracted from both sides, the same quantity must be added to both sides, the same quantity must be multiplied to both sides, or the same quantity must be divided into both sides. students use these balance strategies and apply them to one-step and two-step numeric equations containing a single variable. maintaining a balance 6chapter 6 linear equations 6 • 2010 carnegie learning 2010 georgia middle school sampler 142 285b 1. if , what can you conclude? i can conclude 2. if , what can you conclude? i can conclude 3. if , what can you conclude? i can conclude 4. if , what can you conclude? i can conclude 5. how did you determine the answer to question 1? if two smileys equal two stars, the one star must equal one smiley. i subtracted one object from both sides. 6. how did you determine the answer to question 2? if two stars equal four smileys, then one star must equal two smileys. i cut each side in half. 7. how did you determine the answer to question 3? i subtracted one star from each side. 8. how did you determine the answer to question 4? i subtracted one smiley from each side. show the way warm up6.2 solving equations 285 6 • 2010 carnegie learning 2010 georgia middle school sampler 143 opener read and discuss the lesson opener as a class. the concept of how things balance in the real world is introduced as a motivator to the lesson on solving equations. 6.2 solving equations l 285 6 6.2 maintaining a balance solving equations learning goals in this lesson, you will: l develop an understanding of equality. l use properties of equality to solve equations represented with algebra tiles. l solve one-step equations. key terms l solve l solution l inverse operation l properties of equality you have heard of keeping your balance, but did you know there is also a machine called a balance? these devices are also called scales and they have a long and storied past. the balance is an instrument used to measure the weight and mass of an object. no one is quite sure who invented the balance, but there have been models found in mesopotamia and egypt that suggest the machine has been around since 5000 b.c.e. the balance consists of a lever and two pans. the way the balance works is that a weight is placed on one side, while the object being weighed is placed in the pan on the other side of the lever. when the weight and the object being weighed are the same, the lever remains “balanced” in a horizontal position. what other scales or balances have you seen? have you seen scales or balances in the nurse’s ofﬁce or supermarket? what do those scales or balances look like? discuss your answers with your group.chapter 6 linear equations 286 6 • 2010 carnegie learning 2010 georgia middle school sampler 144 l chapter 6 linear equations 286 6 problem 1 equal or not 1. each representation shows a balance. what will balance one rectangle in each problem? adjustments can be made in each pan as long as the balance is maintained. describe your strategies. a. strategies: i subtracted two squares from both sides. what will balance one rectangle? five squares will balance one rectangle. b. strategies: i divided the six squares into two groups of 3. what will balance one rectangle? three squares will balance one rectangle. problem 1 balance pans containing a variety of shapes are used to illustrate equality. students will determine what balances one rectangle. to isolate one rectangle students use basic operations such as addition, subtraction, multiplication, or division on both balance pans and explain their strategies. they will conclude that to maintain a balance, what is done to side of the balance must be done to the other side of the balance. grouping have students complete questions 1 through 3 with a partner. then, have students share their responses as a class.6.2 solving equations 287 6 • 2010 carnegie learning 2010 georgia middle school sampler 145 guiding questions for share phase, question 1 a. how did you know to subtract two squares from each side first? b. how did you know to divide the six squares into two groups of three? why didn’t you divide the six squares into three groups of two? c. how did you know to subtract six squares from both sides first? how did you know to divide the nine squares into three groups of three? d. how did you know to subtract three squares and two rectangles from each side first? how did you know to divide the twelve squares into three groups of four? why didn’t you divide the twelve squares into four groups of three? e. how did you know to subtract two squares and one rectangle from each side first? 6.2 solving equations l 287 6 c. strategies: i subtracted six squares from both sides, which left three rectangles on one side, and nine squares on the other side. then, i divided the nine squares into three equal groups of 3. what will balance one rectangle? three squares will balance one rectangle. d. strategies: i subtracted three squares and two rectangles from both sides, which left three rectangles on one side, and 12 squares on the other side. then, i divided the 12 squares into three equal groups of 4. what will balance one rectangle? four squares will balance one rectangle.chapter 6 linear equations 288 6 • 2010 carnegie learning 2010 georgia middle school sampler 146 l chapter 6 linear equations 288 6 e. strategies: i subtracted two squares and one rectangle from each side, which left one rectangle on one side and three squares on the other side. what will balance one rectangle? three squares will balance one rectangle. 2. generalize the strategies for maintaining balance. complete each sentence. a. to maintain balance when you subtract a quantity from one side, you must subtract the same quantity from the other side. b. to maintain balance when you add a quantity to one side, you must add the same quantity to the other side. c. to maintain balance when you multiply a quantity by one side, you must multiply the other side by the same quantity. d. to maintain balance when you divide a quantity by one side, you must divide the other side by the same quantity. guiding question for share phase, question 2 if you do different things to each side of the balance, what will happen?6.2 solving equations 289 6 • 2010 carnegie learning 2010 georgia middle school sampler 147 guiding questions for share phase, question 3 how do you know two rectangles balance one square? why are you able to replace two rectangles with one square? how did you know to divide twelve circles into three groups of four? why didn’t you divide twelve circles into four groups of three? problem 2 students will revisit the pan balance representations in problem 1, parts (a) and (b) and replace the rectangles and squares with variables and numbers. each rectangle is equivalent to x, and each square is equivalent to one unit. using the same strategies, students will solve a one-step equation. the definition of inverse operations and solution is provided, which empowers students to use these terms when describing the strategies they employ to solve equations. students check their solutions by substituting the value of the variable back into the original equation and verifying it satisfies the equation. students will solve several one-step equations and check their solutions. grouping have students complete questions 1 through 3 with a partner. then, have students share their responses as a class. ask a student to read the information and definitions following question 3. have students complete question 4 with a partner. then, have students share their responses as a class. 6.2 solving equations l 289 6 3. look at the pan balance shown. how many will balance ? strategies: in the ﬁrst balance, i know that one square balances two rectangles. so in the second balance i can replace one square with two rectangles. so, on the left hand side of the second pan balance, i can replace the two squares with four rectangles. then, i can divide the twelve circles by six rectangles. so, one rectangle balances two circles. therefore, two rectangles balance four circles. problem 2 one step at a time let’s revisit the pan balance representations from problem 1, question 1, parts (a) and (b) and replace and , with variables and numbers. 1. rewrite the representations from problem, 1 question 1, parts (a) and (b) using symbols. let x and 1 unit. then, describe how the strategies you used to determine what balanced 1 rectangle can apply to the equation. determine what balances x. a. x 2 7 i can subtract 2 from both sides. x 5 b. 2x 6 i can divide both sides by 2. x 3 2. does the value of x maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work. a. 5 2 7 b. 2 3 6 7 7 6 6 yes. x 5 maintains balance in the original equation in part (a). yes. x 3 maintains balance in the original equation in part (b).chapter 6 linear equations 290 6 • 2010 carnegie learning 2010 georgia middle school sampler 148 l chapter 6 linear equations 290 6 3. how is the strategy you used to determine what balances x related to the operation of the original equation? the strategy i used to determine what balances x is the opposite of the operation performed in the original equation. you just solved one-step equations. when you set expressions equivalent to each other and identify the value that replaces the variable to make the equation true, you solve the equation. to solve an equation, you must get the variable by itself on one side of the equation by performing inverse operations. inverse operations are pairs of operations that undo each other. addition and subtraction are inverse operations. for example, if 3 5 8, then 8 5 3. multiplication and division are inverse operations. for example, if 6 2 12, then 12 6 2. a solution to an equation is any variable value that makes that equation true. to determine if your solution is correct, substitute the value of the variable back into the original equation. if the equation remains balanced, then you have calculated the solution of the equation. 4. consider the four equations shown. first, state the inverse operation needed to isolate the variable. then, solve the equation. show your work. finally, check to see if the value of your solution maintains balance in the original equation. a. m 9 12 subtract 9 from both sides. m 9 12 check: m 9 9 12 9 3 9 12 m 3 12 12 b. 21 x 8 add 8 to both sides. 21 x 8 check: 21 8 x 8 8 21 29 8 29 x 12 12 guiding questions for discuss phase, questions 1 through 3 what did you do to both sides of the balance in problem 1, question 1, part (a)? if you did the same thing to both sides of the equation in this question, what would you do? what value does x equal? what is the variable value? what is the difference between the variable value and the solution? what is the opposite operation? what is the difference between the inverse operation and the opposite operation? guiding questions for share phase, question 4 how do you isolate the variable? what do you need to do to both sides? how do you know if you need to add or subtract? how do you know if you need to multiply or divide? how do you know if the solution is correct? how many steps does it take to solve this problem?6.2 solving equations 291 6 • 2010 carnegie learning 2010 georgia middle school sampler 149 problem 3 students will revisit the pan balance representations in problem 1, part (c) and replace the rectangles and squares with variables and numbers. each rectangle is equivalent to x, and each square is equivalent to one unit. using the same strategies, students solve a two-step equation. students will write and solve several two-step equations in the form of number riddles (word equations) and check their solutions. grouping ask a student to read question 1, and complete all parts of the question as a class. have students complete questions 2 and 3 with a partner. then, have students share their responses as a class. 6.2 solving equations l 291 6 c. n __ 5 12 multiply both sides by 5. n __ 5 12 check: 5 n __ 5 5 12 60 ___ 5 12 n 60 12 12 d. 4y 24 divide both sides by 4. 4y 24 check: 4y ___ 4 24 ___ 4 4 6 24 y 6 24 24 problem 3 two steps back 1. rewrite the representations from problem 1, question 1, part (c) using symbols. let x and 1 unit. a. describe how the strategies you used to determine what balanced 1 rectangle can apply to the equation. determine what balances x? 3x 6 15 i can subtract 6 from both sides. then, i can divide both sides by 3. x 3 b. does the value of x maintain balance in the original equation? substitute the value of x back into the original equation to check. show your work. 3(3) 6 15 9 6 15 15 15 yes. the value of 3 maintains a balance in the original equation.chapter 6 linear equations 292 6 • 2010 carnegie learning 2010 georgia middle school sampler 150 l chapter 6 linear equations 292 6 you just solved a two-step equation. c. describe the order of operations you used in the original equation. the order of operations i used in the original equation is multiplication, and then addition. d. compare the way you solved the equation to the order of the operations in the original equation. i performed the inverse operations, in the reverse order of operations, of those in the original equation. when you isolate a variable or “undo” operations, you must undo them in the reverse of the order of operations. in other words, the operation that comes last in the original equation should be undone ﬁrst. 2. write a sentence to describe how to apply inverse operations to solve each equation. then, solve each equation and verify your solution. a. 4x 15 61 i will add ﬁrst, and then divide. 4x 15 61 check: 4x 15 15 61 15 4(19) 15 61 4x 76 76 15 61 x 19 61 61 b. 2 7x 16 i will subtract ﬁrst, and then divide. 2 7x 16 check: 2 2 7x 16 2 2 7(2) 16 7x 14 2 14 16 7x ___ 7 14 ___ 7 16 16 x 2 guiding questions for share phase, questions 2 and 3 how did you decide what to do to each side first? what operation is done last? how did you know whether to add or subtract? how did you know whether to multiply or divide? how many steps does it take to solve this problem? is there more than one solution? how do you know if the solution is correct?6.2 solving equations 293 6 • 2010 carnegie learning 2010 georgia middle school sampler 151 6.2 solving equations l 293 6 c. 5 x __ 2 16 i will subtract ﬁrst, and then multiply by 2. 5 5 x __ 2 16 5 check: x __ 2 11 5 22 ___ 2 16 2 x __ 2 2.11 5 11 16 x 22 16 16 d. 17 2x 8 i will subtract ﬁrst, and then divide. 17 8 2x 8 8 check: 9 2x 17 2 ( 9 __ 2 ) 8 9 __ 2 2x ___ 2 17 9 8 9 __ 2 x 17 17 3. solve the riddle. show your work. a. what is a number that when you multiply it by 3 and subtract 5 from the product, you get 28? let x represent the number you are trying to determine. 3x 5 28 3x 33 x 11 the number is 11. b. what is a number that when you multiply it by 4 and add 15 to the product, you get 79? 4x 15 79 4x 64 x 16 the number is 16.chapter 6 linear equations 294 6 • 2010 carnegie learning 2010 georgia middle school sampler 152 l chapter 6 linear equations 294 6 c. make a number riddle for your partner to solve. answers will vary. what is a number that when you multiply it by 5 and add 6 to the product, you get 31? 5x 6 31 5x 25 x 5 problem 4 summary the properties of equality allow you to balance and solve equations involving any number. properties of equality for all numbers a, b, and c,… addition property of equality if a b, then a c b c. subtraction property of equality if a b, then a c b c. multiplication property of equality if a b, then ac bc. division property of equality if a b and c 0, then a __ c b __ c . be prepared to share your solutions and methods. problem 4 the properties of equality are given in a table. these properties allow students to balance and solve equations. grouping as a class, discuss the properties and ask students how these properties were used to solve one-step and two-step equations in this lesson.6.2 solving equations 6 • 2010 carnegie learning 2010 georgia middle school sampler 153 294a check students’ understanding solve each equation. 1. 56 10 2x 56 10 2x 56 10 10 10 2x 46 2x 23 x 2. 13 x __ 3 35 13 x __ 3 35 13 13 x __ 3 35 13 x __ 3 22 3 x __ 3 3 22 x 66 3. 6x 25 79 6x 25 79 6x 25 25 79 25 6x 54 6x ___ 6 54 ___ 6 x 9 4. 38 4x 14 38 4x 14 38 14 4x 14 14 24 4x 24 ___ 4 4x ___ 4 6 x follow upchapter 6 linear equations 6 • 2010 carnegie learning 2010 georgia middle school sampler 154 294b notes6.3 multiple representations of problem situations 295 2010 carnegie learning 2010 georgia middle school sampler 155 6 6.3 lost in translation? that depends on the audience multiple representations of problem situations learning goals in this lesson, you will: l use different methods to represent a problem situation. l identify advantages and disadvantages of using a particular representation. l solve equations. key terms l upper bound l lower bound how many languages are used in your school’s bulletins? in some schools, there may be more than ﬁve languages in school bulletins sent home to parents. translation is becoming a popular occupation, with many people earning college degrees and learning special software to translate one language to another. many times, translators need to take into account the location of the audience for whom they are translating. in fact, translators who translate textbooks into spanish often ask publishers where their target audience is located. in other words, the spanish spoken in panama is not quite the same spanish that is spoken in spain. can you think of other languages that are slightly different depending on the location of the audience? here’s a hint: think about the language you speak! name some businesses or locations where translators are needed regularly. discuss your ideas with your partner. 6chapter 6 linear equations 296 6 2010 carnegie learning 2010 georgia middle school sampler 156 problem 1 translation fees ms. jackson works as a consultant, translating technical documents into spanish and french. she charges 95 an hour. 1. name the quantities that are changing in this problem situation. 2. name the quantity that is constant. 3. which quantity depends on the other? 4. complete the table that represents the various consultant projects ms. jackson has translated recently. label the units of measure. time earnings 0 1 10.5 33 3657.50 4750 71.25 92.56.3 multiple representations of problem situations 297 6 2010 carnegie learning 2010 georgia middle school sampler 157 before you create a graph, ﬁrst consider the upper bound and lower bound values for the x-axis and y-axis. the upper bound and the lower bound of the x- and y-axes are the greatest values and least values of the coordinate plane. 5. determine the upper bound, the lower bound, and the intervals using the table in question 4. a. what is the least number of hours that ms. jackson can work? what is the greatest number of hours that ms. jackson has worked? b. what are the least and greatest amounts of money that ms. jackson has earned? c. consider the range of values you wrote in parts (a) and (b), and choose an upper bound (that should be a whole number) that is slightly greater than your greatest value. then, add the lower and upper bound values to the table shown for each quantity. variable quantity lower bound upper bound interval time worked (in hours) earnings (in dollars) d. calculate the difference between the upper and lower bounds for each quantity. then, choose an interval that divides evenly into this number. doing so will ensure even spacing between the grid lines on your graph. add these intervals to the table.chapter 6 linear equations 298 6 2010 carnegie learning 2010 georgia middle school sampler 158 6. label the graph using the bounds and intervals. then, create a graph of the data from the table in question 4 on the coordinate plane. x y 7. should the points be connected by a line? are the data continuous or discrete? explain your reasoning. 8. describe the relationship between the two quantities represented in the graph. 9. use the graph to answer each question. a. approximately how much money would ms. jackson earn if she worked on a project for 57 hours? b. approximately how many hours would it take ms. jackson to earn 750. 10. write an algebraic equation to represent this situation. deﬁne your variables.6.3 multiple representations of problem situations 299 6 2010 carnegie learning 2010 georgia middle school sampler 159 problem 2 who’s correct? 1. ms. jackson worked 15.75 hours translating a technical manual for smith brothers, inc. she received a check for 1496.25. was her check correct? if not, state the correct amount she should have received. explain your reasoning in terms of the equation and the graph you created. 2. ms. jackson worked 52.5 hours translating a year-end report for anthony’s law ofﬁce. she received a check for 4702.50. was her check correct? if not, state the correct amount she should have received. explain your reasoning in terms of the equation and the graph you created. 3. ms. jackson worked for 35 hours translating a product speciﬁcation document for walker industries. she received a check for 2325. was her check correct? if not, state the correct amount she should have received. explain your reasoning in terms of the equation and the graph you created.chapter 6 linear equations 300 6 2010 carnegie learning 2010 georgia middle school sampler 160 problem 3 problem summary 1. in this lesson, the situation was represented in four different ways: as a sentence, as a table, as a graph, and as an equation. explain the advantages and disadvantages of each representation. use complete sentences. 2. how do you know if a value is a solution to an equation? 3. how do you know if a value is a solution by analyzing a graph? be prepared to share your solutions and methods.6 chapter 6 assignments 93 2010 carnegie learning 2010 georgia middle school sampler 161 6 assignment assignment for lesson 6.3 name _________________________________________ date ___________________ lost in translation? multiple representations of problem situations 1. the grove pharmaceutical company is considering buying a new vial filling machine. the new machine can fill 15 vials per minute. a. name the quantities that are changing in this problem situation. the number of minutes the machine is filling and the total number of vials filled. b. name the quantity that is constant. the machine’s speed of 15 vials per minute is constant. c. which quantity depends on the other? the number of vials filled depends on the number of minutes. d. the company will not run the machine all day because of the cost. to decide how long they will run the machine each day, they need to determine how many capsules the machine will fill. complete the table that represents different amounts of time that the machine will run in a typical day. label the units of measure. time amount minutes number of vials 0 0 40 600 60 900 120 1800 150 2250 175 2625 190 2850 250 37506 94 chapter 6 assignments 2010 carnegie learning 2010 georgia middle school sampler 162 e. determine the upper bound, lower bound, and the intervals. i. what is the least number of minutes the machine can run for? what is the maximum number of minutes the company is considering running the machine for? 0 minutes and 250 minutes ii. what are the least and greatest numbers of vials that can be filled? 0 vials and 3750 vials iii. consider the range of values you wrote in parts (i) and (ii) and choose an upper bound that is slightly greater than your greatest value. then, add the lower and upper bound values to the table shown in part (iv) for each quantity. upper bound for minutes: 500 upper bound for filled vials: 5000 iv. calculate the difference between the upper and lower bounds for each quantity. then choose an interval that divides evenly into this number. add these intervals to the table shown. variable quantity lower bound upper bound interval number of minutes 0 500 25 number of filled vials 0 5000 2506 chapter 6 assignments 95 2010 carnegie learning 2010 georgia middle school sampler 163 name_____________________________________________ date ____________________ f. label the graph using the bounds and intervals. then create a graph of the data from the table in part (d). 50 0 100 150 200 250 300 350 400 450 time (in minutes) 1000 500 1500 2000 2500 3000 3500 4000 4500 number of vials y x g. should the points be connected by a line? are the data continuous or discrete? explain your reasoning. yes. the points should be connected because the data are continuous. the machine can run for any part of a minute. h. describe the relationship between the two quantities represented in the graph. for every minute the machine is on, the number of vials filled increases by 15. i. use the graph to answer each question. i. after about how many minutes will 1100 vials be filled? after about 75 minutes 1100 vials will be filled. ii. about how many vials will be filled in 130 minutes? about 2000 vials will be filled in 130 minutes.6 96 chapter 6 assignments 2010 carnegie learning 2010 georgia middle school sampler 164 j. write an algebraic equation to represent the situation. define your variables. the equation is v 15m, where v represents the number of vials filled, and m represents the number of minutes the machine runs for. k. the supervisor of the company decides that he wants the machine to run for 185 minutes. the production manager uses the graph to determine that 2750 vials will be filled. the quality control manager uses the formula and determines that 2775 vials will be filled. which manager is correct? explain your reasoning in terms of the equation and the graph. why do you think there was a discrepancy? 15(185) 2775 the quality control manager is correct. the ordered pair (185, 2750) is not a point on the line on the graph. the production manager used the graph to find the number of vials, but the value was estimated and is difficult to find exactly.6 chapter 6 skill practice 175 2010 carnegie learning 2010 georgia middle school sampler 165 skills practice skills practice for lesson 6.3 name _________________________________________ date ___________________ lost in translation multiple representations of problem situations vocabulary 1. define upper bound and lower bound and explain their importance in creating a graph. the upper bound and lower bound are the greatest and least values of the x- and y-axes of the coordinate plane. finding the upper and lower bounds can help you create a graph with appropriate intervals to best represent the data. problem set write the missing values to complete each table. 1. a 12-cm pencil is shortened by 1 millimeter for each complete 360 turn in a sharpener. number of 360 turns length of pencil (mm) 0 120 6 114 10 110 23 97 30 90 45 75 50 70 60 60 ? that depends on the audience6 176 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 166 2. john rides a bike at 15 miles per hour. time (hours) distance (miles) 0 0 0.5 7.5 0.75 11.25 1 15 1.5 22.5 2 30 2.25 33.75 2.75 41.25 3. a pool heater heats the water at a rate of 1f per hour. time (hours) temperature (f) 0 66 1 67 2 68 5 71 7 73 10 76 12 78 15 81 4. a gardener can plant 35 flowers every hour. time (hours) number of flowers planted 0 0 2 70 3 105 4.2 147 5 175 7 245 8.6 301 10 3506 chapter 6 skill practice 177 2010 carnegie learning 2010 georgia middle school sampler 167 name_____________________________________________ date ____________________ 5. candice gets 34 miles per gallon of gas on the highway. distance (miles) gas used (gallons) 0 0 17 0.5 68 2 204 6 289 8.5 408 12 527 15.5 612 18 6. a pizza costs 8.50 plus 1.00 for each topping. cost of pizza () number of toppings 8.50 0 9.50 1 11.50 3 12.50 4 14.50 6 15.50 7 16.50 8 17.50 96 178 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 168 write the upper and lower bounds for each situation. then, choose an interval that would ensure even spacing on a graph. 7. a 12-cm pencil is shortened by 1 millimeter for each complete 360 turn in a sharpener. variable quantity lower bound upper bound interval number of turns 0 60 10 length of pencil 60 120 10 8. john rides a bike at 15 miles per hour. variable quantity lower bound upper bound interval time 0 3 0.25 distance 0 50 5 9. a pool heater heats the water at a rate of 1f per hour. variable quantity lower bound upper bound interval time 0 15 1 temperature 65 85 1 10. a gardener can plant 35 flowers every hour. variable quantity lower bound upper bound interval time 0 10 1 number of plants 0 350 25 11. candice gets 34 miles per gallon of gas on the highway. variable quantity lower bound upper bound interval distance 0 650 50 gas used 0 20 2 12. a pizza costs 8.50 plus 1.00 for each topping. variable quantity lower bound upper bound interval cost of pizza 8 18 0.50 number of toppings 0 10 16 chapter 6 skill practice 179 2010 carnegie learning 2010 georgia middle school sampler 169 name_____________________________________________ date ____________________ write an algebraic equation to represent each situation. define your variables. 13. a 12-cm pencil is shortened by 1 millimeter for each complete 360 turn in a sharpener. l 120 t let l represent the length of the pencil and n represent the number of turns. 14. john rides a bike at 15 miles per hour. d 15h let d represent the distance traveled and h represent the time in hours. 15. a pool heater heats the water at a rate of 1f per hour. t 66 h let t represent the temperature and h represent the time in hours. 16. a gardener can plant 35 flowers every hour. f 35h let f represent the number of flowers planted and h represent the time in hours. 17. candice gets 34 miles per gallon of gas on the highway. d 34g let g represent the gallons of gas used and d represent the distance traveled in miles. 18. a pizza costs 8.50 plus 1.00 for each topping. c 8.50 n let c represent the cost of the pizza and n represent the number of toppings.6 180 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 170 create a graph for each situation. 19. a 12-cm pencil is shortened by 1 millimeter for each complete 360 turn in a sharpener. 10 60 0 20 30 40 50 60 70 80 90 number of turns length of pencil (mm) 80 70 90 100 110 120 130 140 150 y x 20. john rides a bike at 15 miles per hour. 0.25 0 0.50 0.75 1.0 1.25 1.50 1.75 2.0 2.50 time (hours) distance (miles) 10 5 15 20 25 30 35 40 45 y x6 chapter 6 skill practice 181 2010 carnegie learning 2010 georgia middle school sampler 171 6 name_____________________________________________ date ____________________ 21. a pool heater heats the water at a rate of 1f per hour. 2 0 4 6 8 10 12 14 16 18 time (hours) temperature (degrees f) 70 68 66 72 74 76 78 80 82 84 y x 22. a gardener can plant 35 flowers every hour. 2 1 0 3 4 5 6 7 8 9 10 time (hours) number of flowers 100 50 150 200 250 300 350 400 4506 182 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 172 23. candice gets 34 miles per gallon of gas on the highway. 150 75 0 225 300 375 450 525 600 675 750 distance (miles) gas used (gallons) 4 2 6 8 10 12 14 16 18 24. a pizza costs 8.50 plus 1.00 for each topping. 1 0 2 3 4 number of toppings total cost of pizza () 5 6 7 8 9 10.5 9.5 8.5 11.5 12.5 13.5 14.5 15.5 17.5 y x 16.56 chapter 6 skill practice 183 2010 carnegie learning 2010 georgia middle school sampler 173 name_____________________________________________ date ____________________ answer each question using a table or graph. 25. a 12-cm pencil is shortened by 1 millimeter for each complete 360 turn in a sharpener. susan predicts that after 25 turns of the pencil sharpener, her 12-centimeter pencil will be 10 centimeters long. is she correct? if not, state the correct length that the pencil will be. explain your reasoning in terms of the equation and the graph. susan’s prediction is incorrect. the length will be 9.5 centimeters. 120 25 95 the ordered pair (25, 100) is not a point on the line in the graph. 26. john rides a bike at 15 miles per hour. john is riding his bike to his friend raymond’s house. he told raymond that he would be at his house in an hour and 15 minutes. if raymond lives 18 3 __ 4 miles from john’s house, is john correct? if not, state the correct time. explain your reasoning in terms of the equation and the graph. john is correct. 15(1.25) 18.75 the ordered pair (1.25, 18.75) is a point on the line in the graph. 27. a pool heater heats the water at a rate of 1f per hour. mr. spencer wants to heat his pool to 77f for a pool party at 4:00 pm. the pool is now 66f and he figures he needs to start heating the pool at 7:00 am will the pool be heated in time for the party? if not, state the correct start time for heating the pool. explain your reasoning in terms of the equation and the graph. the pool will not be heated in time. mr. spencer should start heating the pool at 5:00 am. 66 h 77 h 11; 4:00 pm 11 hours 5:00 am. the ordered pair (9, 77) is not a point on the line in the graph.6 184 chapter 6 skill practice 2010 carnegie learning 2010 georgia middle school sampler 174 6 28. candice gets 34 miles per gallon of gas on the highway. candice is driving to a national park that is 306 miles away. her car has a 16-gallon tank and is half full. she thinks she can make it to her destination without stopping for gas. is she correct? if not, state the correct amount of gas she will need. explain your reasoning in terms of the equation and graph. no, she is not correct. she can drive for 272 miles on 8 gallons of gas. 306 ____ 34 candice needs 9 gallons of gas. the ordered pair (306, 8) is not a point on the line in the graph. 29. a gardener can plant 35 flowers every hour. the gardener needs to provide an estimate for how many flowers he can plant during an 8-hour shift. his estimate states that he can plant 280 flowers in 8 hours. is his estimate a good one? if not, state the number of flowers he would be able to plant. explain your reasoning in terms of the equation and the graph. his estimate is correct. 35(8) 280 the ordered pair (8, 280) is a point along an imaginary line in the graph. 30. a pizza costs 8.50 plus 1.00 for each topping. kiki wants to order a pizza with pepperoni and mushrooms. she has 10. assuming no extra charges, does she have enough money? if not, state the amount that she will need to buy this pizza. explain your reasoning in terms of the equation and graph. kiki does not have enough money for a pizza with two toppings. she would need 10.50. 8.50 2 10.50 the ordered pair (2, 10) is not a point along an imaginary line in the graph.chapter 6 summary 317 2010 carnegie learning 2010 georgia middle school sampler 175 6 chapter 6 summary key terms l equation (6.1) l solved (6.2) l inverse operation (6.2) l solution (6.2) l upper bound (6.3) l lower bound (6.3) use picture algebra to represent and solve problems draw a box to represent an unknown amount. underneath that box, draw a combination of additional boxes and amounts to represent the other unknown amount(s) in terms of the first. write the total the boxes should equal when combined. then, subtract any known amounts from the total, and divide by the number of remaining boxes to solve for the unknown amount. example brianna has three plus two times more rubber bracelets than hannah. together, the girls have 33 bracelets. hannah: brianna: 33 3 33 3 30; 30 3 10 hannah has 10 bracelets, and brianna has 3 2(10), or 23 bracelets. 6.1 properties l properties of equality (6.2)chapter 6 linear equations 318 6 2010 carnegie learning 2010 georgia middle school sampler 176 use an equation to illustrate a relationship between two quantities you can represent an algebra picture as a mathematical sentence using operations and an equals sign (). an equation is a mathematical sentence you create by placing an equals sign between two expressions. one way to write an equation is to think about writing the equation with words. then, write an expression for each part of the word equation, assigning a variable to represent an unknown amount. finally, place the equals sign between the two expressions to write the equation. example sam and grant share a paper route. they deliver a total of 500 papers each morning. sam delivers 124 more papers than grant does. the papers sam delivers the papers grant delivers 500. let g represent the number of papers grant delivers. sam’s papers g 124; grant’s papers g. the equation is g g 124 500. 6.1chapter 6 summary 319 6 2010 carnegie learning 2010 georgia middle school sampler 177 6 develop an understanding of equality and use properties of equality using a balance model a representation is shown on a balance. to maintain balance when you subtract a quantity from one side, you must subtract the same quantity from the other side. to maintain balance when you add a quantity to one side, you must add the same quantity to the other side. to maintain balance when you multiply by a quantity on one side, you must multiply by the same quantity on the other side. to maintain balance when you divide by a quantity on one side, you must divide by the same quantity on the other side. example subtract two squares from each side, which leaves three rectangles on one side and 12 squares on the other side. then divide each side by 3, which leaves one rectangle on one side and four squares on the other side: 1 rectangle 4 squares. 3x 2 14 x 4 6.2chapter 6 linear equations 320 6 2010 carnegie learning 2010 georgia middle school sampler 178 6 solve one-step equations when you set expressions equivalent to each other and identify the value that replaces the variable to make the equation true, you solve the equation. to solve a one-step equation, you must get the variable by itself on one side of the equation by performing an inverse operation. inverse operations are pairs of operations that undo each other. a solution to an equation is any value for a variable that makes the equation true. to determine if your solution is correct, substitute the value of the variable back into the original equation. if the equation remains balanced, then you have calculated the solution of the equation. example solve the equation n 28 11 for n. add 28 to each side. check the solution. n 28 11 39 28 11 n 28 28 11 28 11 11 n 39 solve two-step equations to solve a two-step equation, you must get the variable by itself by performing two inverse operations. perform the inverse operations in the reverse order of operations found in the original equation. to determine if your solution is correct, substitute the value of the variable back into the original equation. if the equation remains balanced, then you have calculated the solution of the equation. example the solution for the equation b __ 5 19 25 for b is shown. subtract first, and then multiply. check the solution. b __ 5 19 25 b __ 5 19 19 25 19 30 ___ 5 19 25 b __ 5 6 6 19 25 5 ( b __ 5 ) 5(6) 25 25 b 30 6.2 6.2chapter 6 summary 321 6 2010 carnegie learning 2010 georgia middle school sampler 179 6 use a table to represent a one-step problem situation to create a table to represent a problem situation, first decide which quantities change, which remain constant, and which quantity depends on the other. label the independent quantity in the left column (or top row) and the dependent quantity in the right column (or bottom row). label the units of measure for each quantity. then, choose several values for the independent quantity, and calculate the corresponding dependent quantity. example jeannie can jump 75 times per minute with her jump rope. the table shown lists the various number of times jeannie can jump in the numbers of minutes shown. time (minutes) 0 3 5 6 7 10 number of jumps 0 225 375 450 525 750 6.3chapter 6 linear equations 322 6 2010 carnegie learning 2010 georgia middle school sampler 180 6 use a graph to represent a one-step problem situation to create a graph to represent a problem situation, first decide which quantities change, which remain constant, and which quantity depends on the other. the dependent quantity is written along the y-axis and the independent quantity is written along the x-axis. then, determine the upper and lower bounds, or greatest and least values, of the x- and y-axes. calculate the difference between the upper and lower bounds for each quantity, and choose an interval that divides evenly into this number to ensure even spacing between the grid lines on your graph. plot input/output points on the graph, and connect them in a line if they represent continuous data. example the number of times jeannie can jump in a certain amount of time is shown in the table and graphed on the coordinate plane. variable quantity lower bound upper bound interval time (in minutes) 0 10 1 number of jumps 0 750 50 x 2 1 100 200 300 400 500 600 700 3 4 5 6 7 8 9 y 800 900 time (in minutes) number of jumps 6.3chapter 6 summary 323 6 2010 carnegie learning 2010 georgia middle school sampler 181 6 use a one-step equation to represent a problem situation write a problem situation that is written in sentence form and requires only one operation to solve as a one-step equation. define unknown amounts in the situation as variables. example let m represent the number of minutes jeannie jumps rope. let j represent the total number of jumps she makes. the equation is j 75m. use the equation and graph to determine if a solution is correct substitute a value for the independent variable into the equation to see if the solution given is correct. you can also look at the graph to see if the solution given is represented as a point along the line in the graph of the equation. example jeannie wants to join a rope jumping contest. she must be able to jump 300 times in four minutes to qualify. she thinks she can make the requirement. by substituting the number of times jeannie can jump in 4 minutes, she determined she can qualify for the rope jumping contest. 75(4) 300. the ordered pair (4, 300) is a point along the line of the graph, so jeannie qualifies for the rope jumping contest. 6.3 6.3chapter 6 linear equations 324 6 2010 carnegie learning 2010 georgia middle school sampler 182 6 use a table to represent a two-step problem situation to create a table to represent a problem situation, first decide which quantities change, which remain constant, and which quantity depends on the other. label the independent quantity in the left column (or top row), the constant quality in the middle column (or row), and the dependent quantity in the right column (or bottom row). label the units of measure for each quantity. then, choose several values for the independent quantity, and determine the corresponding dependent quantity. example a floral shop charges 2 per stem of flowers in an arrangement. it also charges a 4.50 delivery fee. number of flowers (stem) delivery fee (in dollars) total cost of arrangement (in dollars) 10 4.50 24.50 12 4.50 28.50 15 4.50 34.50 17 4.50 38.50 20 4.50 44.50 6.4chapter 6 summary 325 6 2010 carnegie learning 2010 georgia middle school sampler 183 6 use a graph to represent a two-step problem situation to create a graph to represent a problem situation, first decide which quantities change, which remain constant, and which quantity depends on the other. the dependent quantity is written along the y-axis and the independent quantity is written along the x-axis. then, determine the upper and lower bounds, or greatest and least values, of the x- and y-axes. calculate the difference between the upper and lower bounds for each quantity, and choose an interval that divides evenly into this number to ensure even spacing between the grid lines on your graph. plot ordered pairs on the graph, and connect them in a line if they represent continuous data. example variable quantity lower bound upper bound interval number of flowers 10 20 1 total cost of arrangement 5 50 5 x 11 10 5 10 15 20 25 30 35 12 13 14 15 16 17 18 19 y 40 45 number of flowers total cost of arrangement () 6.4chapter 6 linear equations 326 6 2010 carnegie learning 2010 georgia middle school sampler 184 6 use a two-step equation to represent a problem situation write a problem situation that is written in sentence form and requires two operations to solve as a two-step equation. define unknown amounts in the situation as variables. example let c represent the total cost of an arrangement, and f represent the number of flowers in the arrangement. the equation is c 2f 4.50. use the equation and graph to determine if a solution is correct substitute a value for the independent variable into the equation to see if the solution given is correct. you can also look at the graph to see if the solution given is represented as a point along the line in the graph of the equation. example victor takes an order for two dozen roses to be delivered tomorrow. he tells the customer that the arrangement will cost 54.50. victor's calculation is incorrect. the customer will owe 52.50. you can verify that victor's calculation is incorrect by writing an equation. 2(24) 4.50 52.50 the ordered pair (24, 54.50) is not a point along the line in the graph. 6.4 6.4chapter 6 summary 327 6 2010 carnegie learning 2010 georgia middle school sampler 185 6 solve two-step equations to solve a two-step equation, you must get the variable by itself by performing two inverse operations. perform the inverse operations in the reverse order of operations found in the original equation. to determine if your solution is correct, substitute the value of the variable back into the original equation. if the equation remains balanced, then you have calculated the solution of the equation. example 9x 14 94 check the solution. 9x 14 14 94 14 9(12) 14 94 9x 108 108 14 94 9x ___ 9 108 ____ 9 94 94 x 12 6.4chapter 6 linear equations 328 6 2010 carnegie learning 2010 georgia middle school sampler 186 6 write and use two-step equations look at a table of values for a situation. decide which set of values is the independent quantity, and which is the dependent quantity. calculate the difference between consecutive independent quantities in the dependent quantity. multiply this amount by the independent quantity in the equation. define the variables, and write the equation. solve to answer questions about the situation. example the table shows the cost to rent a car. number of days total cost u rent it 3 240 4 269 5 298 6 327 7 356 write an equation that represents the total cost, t, of renting a car from u rent it in terms of the number of days rented, d. 269 240 29 298 269 29 327 298 29 356 327 29 t 240 29(d 3) 6.5chapter 6 summary 329 6 2010 carnegie learning 2010 georgia middle school sampler 187 6 6.5 compare two problem situations substitute the same value into two similar equations to compare their outcomes. example ronald needs to hire a caterer for the school picnic. he is comparing catering companies to find the most affordable option. the total cost, t, for a picnic lunch from callie’s catering in terms of the number of students, s, is represented by the equation t 120 5(s 20). the total cost, t, for a picnic lunch from paco’s picnics in terms of the number of students, s, is represented by the equation t 100 7.50(s 15). calculate the total cost for each company if ronald estimates 200 students. ronald can determine which company is more affordable for providing catering for 200 students. total cost from callie’s catering t 120 5(s 20) 120 5(200 20) 120 900 1020 total cost from paco’s picnics t 100 7.50(s 15) 100 7.50(200 15) 100 1387.50 1487.50 for 200 students, callie’s catering is more affordable.chapter 6 linear equations 330 6 2010 carnegie learning 2010 georgia middle school sampler 1886 chapter 6 assessments 133 2010 carnegie learning 2010 georgia middle school sampler 189 6 pre-test name _____________________________________________ date ____________________ 1. together, sarah and tulsi raised 260 for the bike-a-thon fundraiser. tulsi raised three times as much as sarah. let represent the amount sarah raised. a. complete the picture to represent the situation. 260 sarah: tulsi: b. determine the amount raised by each girl using the picture. explain your reasoning. i divided 260 by 4 to get 65. each box represents 65. sarah raised 65 and tulsi raised 195. c. write an equation to represent the situation. let b represent the amount sarah raised. b 3b 260 2. carolee makes quilts to sell at an arts and crafts fair. she charges 75 per quilt. a. name the quantity that is constant. the amount carolee charges per quilt, 75, is constant. b. which quantity depends on the other? the amount carolee earns depends on the number of quilts she sells. c. write an algebraic equation to represent this situation. define your variables. use your equation to determine how much carolee will earn if she sells 15 quilts. e 75q, where e represents total earnings, in dollars, and q represents the number of quilts made e 75(15); e 1125; carolee will earn 1125 for 15 quilts.6 134 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 190 pre-test page 2 3. dan runs a landscaping business. he charges 30 per hour plus a 25 fee for the initial visit to the site. a. write an algebraic equation to represent this situation. define your variables. t 30h 25, where t represents total earnings, in dollars, and h represents the number of hours worked b. dan worked 9 hours on a landscaping job. he received a check for 255. is this amount correct? if not, state the correct amount he should have received. 30(9) 25 ? 255 295 255 dan should have received a check for 295. 4. use the pan balance to complete parts (a) and (b). a. what will balance 1 rectangle? explain your strategy. four squares will balance 1 rectangle. i subtracted 5 squares from each side, which left 2 rectangles on one side and 8 squares on the other side. then, i divided 8 squares into 2 equal groups of 4. b. use symbols to rewrite the representation shown on the balance. let x and 1 unit. then determine what balances x. tell your strategies for solving. 2x 5 13 2x 8 subtract 5 from each side. x 4 divide each side by 2.6 chapter 6 assessments 135 2010 carnegie learning 2010 georgia middle school sampler 191 name _____________________________________________ date ____________________ pre-test page 3 5. meredith is having a deck built onto her house. the table shows the possible dimensions of her deck. width (in feet) 10 11 12 13 14 15 16 length (in feet) 16.5 18.5 20.5 22.5 24.5 26.5 28.5 a. let l represent the length of the deck. write an equation that represents the length of the deck in terms of the width, w. l 2w 3.5 b. calculate the length of the deck if the width is 20 feet. l 2(20) 3.5 36.5 a deck that is 20 feet wide will be 36.5 feet long. c. if the length of a deck is 25 feet long, what is the width? 25 2x 3.5, so x 14.25 a deck that is 25 feet long will be 14.25 feet wide. d. calculate the area of a deck that measures 15.5 feet wide. l 2(15.5) 3.5 27.5 a 27.5 15.5 426.25 the area of a deck that is 15.5 feet wide will be 426.25 square feet.6 136 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 1926 chapter 6 assessments 137 2010 carnegie learning 2010 georgia middle school sampler 193 6 post-test name _____________________________________________ date ____________________ 1. together, padmi and tori collected 149 cans at the river clean up. tori collected 23 fewer cans than padmi. let represent the number of cans padmi collected. a. complete the picture to represent the situation. 23 padmi: tori: 149 b. determine the number collected by each girl using the picture. explain your reasoning. i added 23 to 149 to get 172. then i divided 172 by 2 to get 86. padmi collected 86 cans. tori collected 86 23 63 cans. c. write an equation to represent the situation. let c represent the amount padmi collected. c c 23 149 2. to earn some extra money, ben mows lawns for people in his neighborhood. he earns 18 per lawn. a. name the quantity that is constant. the amount ben earns per lawn, 18, is constant. b. which quantity depends on the other? the amount ben earns depends on the number of lawns mowed. c. write an algebraic equation to represent this situation. define your variables. use your equation to determine how much ben will earn for 21 lawns. t 18s, where t represents total earnings, in dollars, and s represents the number of lawns mowed t 18(21); t 378; ben will earn 378 for mowing 21 lawns.6 138 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 194 post-test page 2 3. xenia runs a computer installation and repair business. she charges 80 per hour plus an initial fee of 50. a. write an algebraic equation to represent this situation. define your variables. m 80h 50, where m represents total earnings, in dollars, and h represents the number of hours worked b. xenia worked 3 hours on a computer repair job. she received a check for 290. is this amount correct? if not, state the correct amount she should have received. 80(3) 50 ? 290 290 290 xenia should have been paid 290, so the check amount is correct. 4. use the pan balance to complete parts (a) and (b). a. what will balance 1 rectangle? explain your strategy. 3 squares will balance 1 rectangle. i subtracted 2 squares from each side, which left 3 rectangles on one side and 9 squares on the other side. then, i divided 9 squares into 3 equal groups of 3. b. use symbols to rewrite the representation shown on the balance. let x and 1 unit. then determine what balances x. tell your strategies for solving. 3x 2 11 3x 9 subtract 2 from each side. x 3 divide each side by 3.6 chapter 6 assessments 139 2010 carnegie learning 2010 georgia middle school sampler 195 name _____________________________________________ date ____________________ post-test page 3 5. juan is designing a garden for his backyard. the table shows the possible dimensions of his garden. width (in feet) 4 5 6 7 8 9 10 length (in feet) 7 8 9 10 11 12 13 a. let l represent the length of the garden. write an equation that represents the length of the deck in terms of the width, w. l w 3 b. calculate the length of the garden if the width is 15 meters. l 15 3 18 a deck that is 15 meters wide will be 18 meters long. c. if the length of a garden is 20 meters long, what is the width? 20 w 3; w 17 a deck that is 20 meters long will be 17 meters wide. d. calculate the area of a garden that measures 7.5 meters wide. l 7.5 3 10.5 a 10.5 7.5 78.75 the area of a garden that is 7.5 meters wide will be 78.75 square meters.6 140 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 1966 chapter 6 assessments 141 2010 carnegie learning 2010 georgia middle school sampler 197 mid-chapter test name _____________________________________________ date ____________________ 1. kara and trey rode their bicycles 51 miles altogether. kara rode her bike 13 more miles than trey. let represent the number of miles trey rode his bicycle. a. complete the picture to represent the situation. kara: trey: 51 13 b. use the picture to determine the number of miles each person rode their bicycle. explain your reasoning. i subtracted 13 from 51 to get 38, and then divided 38 by 2 to get 19. trey rode his bicycle 19 miles and kara rode her bicycle 32 miles. c. write an equation to represent the situation. let d represent the miles trey rode his bicycle. d d 13 51 2. ian and max have 528 digital images altogether. ian has 1 __ 5 the number of digital images as max. let represent the number of images ian has. a. complete the picture to represent the situation. lan: max: 528 b. determine the number of digital images they each have. explain your reasoning. i divided 528 by 6 to get 88. each box represents 88. so, ian has 88 digital images and max has 440 digital images. c. write an equation to represent the situation. let d represent the number of digital images ian has. d 5d 5286 142 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 198 mid-chapter test page 2 3. falyn and rit collected 72 state quarters between them. falyn has twice as many quarters as rit. let represent the number of quarters rit collected. a. complete the picture to represent the situation. rit: falyn: 72 b. determine the number of quarters collected by each person. explain your reasoning. i divided 72 by 3 to get 24. each box represents 24 quarters. rit collected 24 quarters and falyn collected 48 quarters. c. write an equation to represent the situation. let q represent the number of quarters in rit’s collection. q 2q 72 4. state the inverse operation needed to isolate the variable. then solve each equation. show your work. then check to see if the value of your solution balances the original equation. a. x 3 9 add 3 to both sides. x 3 9 check: x 3 3 9 3 12 3 9 x 12 9 96 chapter 6 assessments 143 2010 carnegie learning 2010 georgia middle school sampler 199 name _____________________________________________ date ____________________ mid-chapter test page 3 b. 5x 85 divide both sides by 5. 5x 85 check: 5x ___ 5 85 ___ 5 5 17 85 x 17 85 85 c. x __ 7 14 multiply both sides by 7. x __ 7 14 check: x __ 7 7 14 7 98 ___ 7 14 x 98 14 14 5. adam and john read 245 pages between the two of them. adam read 39 fewer pages than john. let represent the number of pages john read. a. complete the picture to represent the situation. john: adam: 245 39 b. determine the number of pages read by each person using the picture. explain your reasoning. i added 39 to 245 to get 284. then i divided 284 by 2 to get 142. john read 142 pages. adam read 142 39 103 pages. c. write an equation to represent the situation. let r represent the number of pages john read. r r 39 2456 144 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 200 mid-chapter test page 4 6. use the pan balance to complete parts (a) and (b). a. what will balance 1 rectangle? explain your strategy. two squares will balance 1 rectangle. i subtracted 1 rectangle and 4 squares from each side, which left 3 rectangles on one side and 6 squares on the other side. then, i divided 6 squares into 3 equal groups of 2. b. use symbols to rewrite the representation shown on the balance. let x and 1 unit. then determine what balances x. tell your strategies for solving. 4x 4 x 10 4x x 6 subtract 4 from each side. 3x 6 subtract x from each side. x 2 divide each side by 3. 7. how many circles will balance one square? explain your strategy. in the first balance, i know that 3 rectangles balance 1 square. so, in the second balance i can replace the 3 rectangles with 1 square. i now have 5 squares balancing 15 circles. i can divide 15 into 5 groups of 3, so 1 square balances 3 circles.6 chapter 6 assessments 145 2010 carnegie learning 2010 georgia middle school sampler 201 name _____________________________________________ date ____________________ 6 mid-chapter test page 5 8. write a sentence to describe how to apply inverse operations to solve each equation. then, solve each equation and verify your solution. a. 3x 4 22 i will subtract first, and then divide. 3x 4 22 check: 3x 4 4 22 4 3(6) 4 22 3x 18 18 4 22 x 6 22 22 b. x __ 2 9 4 i will add first, and then multiply. x __ 2 9 4 check: x __ 2 9 9 4 9 (26) ____ 2 9 4 x __ 2 13 13 9 4 x 26 4 4 9. what is a number that when you multiply it by 4 and subtract 7 from the product, you get 29? show your work. 4x 7 29 4x 36 x 9 the number is 9.6 146 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 2026 chapter 6 assessments 147 2010 carnegie learning 2010 georgia middle school sampler 203 end of chapter test name _____________________________________________ date ____________________ 1. ellyx and sue planted 144 seedlings between them. sue planted 8 fewer seedlings than ellyx. let represent the number of seedlings ellyx planted. a. complete the picture to represent the situation. ellyx: sue: 144 8 b. determine the number collected by each person using the picture. explain your reasoning. i added 8 to 144 to get 152. then i divided 152 by 2 to get 76. ellyx planted 76 seedlings. sue planted 76 8 68 seedlings. c. write an equation to represent the situation. let p represent the number of seedlings ellyx planted. p p 8 1446 148 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 204 end of chapter test page 2 2. use the pan balance to answer parts (a) and (b). a. what will balance 1 rectangle? explain your strategy. 2 squares will balance 1 rectangle. i subtracted 2 rectangles and 1 square from each side, which left 2 rectangles on one side and 4 squares on the other side. then, i divided 4 squares into 2 equal groups of 2. b. use symbols to rewrite the representation shown on the balance. let x and 1 unit. then determine what balances x. tell your strategies for solving. 4x 1 2x 5 4x 2x 4 subtract 1 from each side. 2x 4 subtract 2x from each side. x 2 divide each side by 2.6 chapter 6 assessments 149 2010 carnegie learning 2010 georgia middle school sampler 205 name _____________________________________________ date ____________________ 6 end of chapter test page 3 3. sadiq makes tables with a mosaic top. he earns 125 per table sold. a. name the quantity that is constant. the amount sadiq earns per table, 125, is constant. b. which quantity depends on the other? the amount sadiq earns depends on the number of tables he sells. c. complete the table that represents various numbers of tables that sadiq has sold. number of tables 0 1 2 5 7 10 13 earnings (in dollars) 0 125 250 625 875 1250 1625 d. create a graph of the data from the table. 2 0 4 6 8 10 12 14 16 number of tables sadiq's earnings earnings (in dollars) 400 200 600 800 1000 1200 1400 1600 1800 tables e. write an algebraic equation to represent this situation. define your variables. use your equation to determine how much sadiq will earn if he sells 22 tables. d 125t, where d represents total earnings, in dollars, and t represents the number of tables sold; d 125(22); d 2750; sadiq will earn 2750 for selling 22 tables.6 150 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 206 end of chapter test page 4 4. what is a number that when you divide it by 3 and subtract 8 from the quotient, you get 12? show your work. x __ 3 8 12 x __ 3 20 x 60 the number is 60. 5. carla runs a website design business. she charges 60 per hour plus a 125 initial consultation fee. a. complete the table that represents various web sites that carla has designed recently. label the units of measure. time spent hours 1 2 4 5 10 16 25 consultation fees dollars 125 125 125 125 125 125 125 total earnings dollars 185 245 365 425 725 1085 1625 b. create a graph of the data from the table. 9 0 18 27 hrs time spent (in hours) web site earnings earnings (in dollars) 400 200 600 800 1000 1200 1400 1600 18006 chapter 6 assessments 151 2010 carnegie learning 2010 georgia middle school sampler 207 name _____________________________________________ date ____________________ 6 end of chapter test page 5 c. write an algebraic equation to represent this situation. define your variables. t 60h 125, where t represents total earnings in dollars, and h represents the number of hours worked d. carla worked 20 hours on a web site. she received a check for 1225. is this amount correct? if not, state the correct amount she should have received. 60(20) 125 ? 1225 1325 1225 carla should have received a check for 1325, so the check amount is incorrect. e. carla worked 11 hours on a web site. she received a check for 785. is this amount correct? if not, state the correct amount she should have received. 60(11) 125 ? 785 785 785 carla should have been paid 785, so the check amount is correct.6 152 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 208 end of chapter test page 6 6. a company stacks lampshades in packing boxes so that they can be shipped. the table shows the total stack heights of various numbers of medium and large lampshades. a. complete the table. number of lampshades stack height (in inches) medium large 1 8 9.5 2 10.5 11.5 3 13 13.5 4 15.5 15.5 5 18 17.5 6 20.5 19.5 7 23 21.5 8 25.5 23.5 b. what quantity depends on the other? the stack height depends on the number of lampshades. c. what is the difference in heights between a stack of 4 medium lampshades and 3 medium lampshades? between a stack of 6 medium lampshades and 5 medium lampshades? between 7 medium lampshades and 6 medium lampshades? the difference in heights for each pair of medium lampshades is 2.5 inches. 15.5 13 2.5; 20.5 18 2.5; 23 20.5 2.5 d. what is the difference in heights between a stack of 3 large lampshades and 2 large lampshades? between a stack of 6 large lampshades and 5 large lampshades? between 8 large lampshades and 7 large lampshades? the difference in heights for each pair of large lampshades is 2 inches. 13.5 11.5 2; 19.5 17.5 2; 23.5 21.5 26 chapter 6 assessments 153 2010 carnegie learning 2010 georgia middle school sampler 209 name _____________________________________________ date ____________________ 6 end of chapter test page 7 e. determine the height of a stack of 12 medium lampshades and the height of 12 large lampshades. medium: 8 2.5(11) 35.5 inches large: 9.5 2(11) 31.5 inches f. let h represent the stack height. write an equation that represents the height of a stack of medium lampshades in terms of the number of lampshades, s, in the stack. h 8 2.5(s 1) g. let h represent the stack height. write an equation that represents the height of a stack of large lampshades in terms of the number of lampshades, t, in the stack. h 9.5 2(t 1)6 154 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 2106 chapter 6 assessments 155 2010 carnegie learning 2010 georgia middle school sampler 211 standardized test practice name _____________________________________________ date ____________________ 1. what is the solution of the equation? 4x 12 48 a. 9 b. 15 c. 144 d. 240 2. dillon is fencing a rectangular pasture. the length and width of the pasture are related according to a specific formula. he developed this table to see how many total feet of fencing he will need for pastures with different dimensions. which equation could dillon use to find the total number of feet of fencing, t, in terms of different pasture widths, w? width (in feet) 15 16 17 18 20 24 length (in feet) 23 25 27 29 33 41 total fencing needed (in feet) 76 82 88 94 106 130 a. t 2(w w 8) b. t w 2w 7 c. t 2w 2w 7 d. t 2(w 2w 7)6 156 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 212 standardized test practice page 2 3. molly and george earned 352 points together. molly has half the number of points as george. which equation could represent this situation? let g represent the number of points george earned. a. g 2g 352 b. g 1 __ 2 g 352 c. g g ______ 2 352 d. 1 __ 2 g 352 4. what is the solution to the equation? 28 6x 9 a. 19 ___ 6 b. 37 ___ 6 c. 114 d. 222 5. which sentence describes how to apply inverse operations to solve the equation? 1 __ 2 x 3 9 a. add 3, then divide by 2. b. add 3, then multiply by 2. c. subtract 3, and then divide by 2. d. subtract 3, and then multiply by 2.6 chapter 6 assessments 157 2010 carnegie learning 2010 georgia middle school sampler 213 name _____________________________________________ date ____________________ standardized test practice page 3 6. cynthia and amy are customer service representatives. cynthia earns 13 per hour and amy earns 1.50 more per hour than cynthia. amy worked 43.5 hours in the last two weeks. which equation could she use to find the amount of money she should receive? a. m 13(43.5 1.5) b. m 13(43.5 1.5) c. m 43.5(13 1.5) d. m 43.5(13 1.5) 7. to print postcards, a printing company charges a one-time image-upload fee of 5.89 plus 0.67 per postcard. which algebraic equation represents this situation, if t represents the total cost and p represents the number of postcards printed? a. t 5.89p 0.67 b. t 5.89 _____ p 0.67 c. t p(5.89 0.67) d. t 5.89 0.67p 8. jim is an electrician. he charges a 150 home-visit fee and 45 per hour, with the first two hours free. which equation could elyse use to find the cost, c, to have jim work at her house for 5 hours? a. c 150 45(5) b. c 150 45(5 2) c. c 150 45(5 2) d. c (150 45)(5 2)6 158 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 214 standardized test practice page 4 9. what is a number that when you multiply it by 6 and you subtract 4 from the product, you get 50? a. 7 2 __ 3 b. 9 c. 14 d. 324 10. angel and dj assembled 45 fence sections between them. angel assembled twice as many fence sections as dj. let represent the number of fence sections dj assembled. which picture best represents the situation? a. angel: dj: 45 b. angel: dj: 45 c. angel: 45 dj: d. angel: 45 dj: 11. corinne is taking a train into the city. the train travels at a rate of 45 miles per hour. which quantity depends on the other? a. the distance traveled depends on the number of hours traveled. b. the number of hours traveled depends on the speed of the train. c. the distance traveled depends on the speed of the train. d. the speed of the train depends on the number of hours traveled.6 chapter 6 assessments 159 2010 carnegie learning 2010 georgia middle school sampler 215 name _____________________________________________ date ____________________ standardized test practice page 5 12. what is the solution to the equation? 3.5x 6.1 18.4 a. 3.51 b. 7.0 c. 24.5 d. 85.75 13. deirdre is buying plastic mugs to send to her cousin. she made a table of the various heights of stacks of different numbers of mugs. number of plastic mugs stack height (in inches) 1 6 2 6.75 3 7.5 4 8.25 5 9 6 9.75 which equation represents the height, h, of a stack of mugs in terms of the number of mugs, t, in the stack? a. h 6 0.75(t 1) b. h 6 0.75t c. h 0.75 t(t 1) d. h 0.75t 66 2010 carnegie learning 2010 georgia middle school sampler 216 160 chapter 6 assessments standardized test practice page 6 14. use the pan balance to determine the number of squares that will balance 1 rectangle. a. 1 b. 2 c. 5 d. 13 15. kelly and rachel collected 193 pounds of newspaper between them. kelly collected 35 pounds more than rachel. how many pounds of newspaper did each girl collect? a. kelly: 79 pounds; rachel: 114 pounds b. kelly: 114 pounds; rachel: 149 pounds c. kelly: 114 pounds; rachel: 79 pounds d. kelly: 149 pounds; rachel: 114 pounds6 chapter 6 assessments 161 2010 carnegie learning 2010 georgia middle school sampler 217 name _____________________________________________ date ____________________ 6 standardized test practice page 7 16. delia is a graphic designer who charges an hourly rate. she made the graph to show her total earnings for various numbers of hours worked. approximately how many hours would it take delia to earn 250? 1 0 2 3 4 5 6 7 8 9 hrs time worked (in hours) delia's earnings earnings () 100 50 150 200 250 300 350 400 450 a. 6 hours b. 7 hours c. 8 hours d. 9 hours 17. use the pan balances to determine what will balance two circles. a. 1 square b. 2 squares c. 1 rectangle d. 2 rectangles6 162 chapter 6 assessments 2010 carnegie learning 2010 georgia middle school sampler 218 standardized test practice page 8 18. mr. allen orders ceramic kits from an art supplier. each kit costs 35 and comes with tools for 6 students. if mr. allen has 175 to spend on ceramic kits, how many students will be able to have their own set of tools? a. 5 b. 11 c. 30 d. 35 19. jerry repairs computers. he charges 65 per hour. if t represents his total earnings and h represents the number of hours spent on a repair, which equation represents this situation? a. t 65 h b. t 65 h c. t 65h d. t 65 ___ h 20. analee is a sales associate. she earns a weekly wage of 300 plus a 5% commission on every item she sells. if analee earned 675 last week, what were her total sales for the week? a. 750 b. 1950 c. 7500 d. 13,500georgia middle school math series level 3 2010 carnegie learning 2010 georgia middle school sampler 2192010 carnegie learning 2010 georgia middle school sampler 2202.4 investigate magnitude through theoretical probability and experimental probability 73 2010 carnegie learning 2010 georgia middle school sampler 221 2.4 do you have a better chance of winning the lottery or getting struck by lightning? investigate magnitude through theoretical probability and experimental probability learning goals in this lesson, you will l use the multiplication rule of probability for compound independent events. l simulate events using the random number generator on a graphing calculator. l compare experimental and theoretical probability. key term l simulation state lotteries have received mixed reviews ever since the ﬁrst u.s. state lottery was established in new hampshire in 1964. some people feel that state lotteries promote gambling and irresponsible game playing. others feel that state lotteries are beneﬁcial to schools and other public services. can you think of some ideas that would beneﬁt public services, but not be associated with gambling? discuss your ideas with your partner. 2chapter 2 likelihoods 74 2010 carnegie learning 2 2010 georgia middle school sampler 222 problem 1 select a 3-digit number and win a prize! in many states, a common lottery game requires a player to select a 3-digit number; each digit in the number ranges from 0 through 9 and repetition of numbers is allowed. then, a number is randomly chosen from three bins, each bin containing numbers 0 through 9. if a player’s 3-digit number matches the number randomly selected from the three bins, the player wins a cash lottery prize. 8 7 6 4 2 9 0 1 3 5 0 0 2 4 6 7 5 3 1 9 8 5 6 7 8 9 1 2 3 4 1. how many 3-digit lottery numbers are possible? explain your reasoning. 2. mrs. mason chooses the number 2-7-9. what is the probability that she will win the lottery?2.4 investigate magnitude through theoretical probability and experimental probability 75 2010 carnegie learning 2 2010 georgia middle school sampler 223 3. mrs. mason’s book club occasionally plays the lottery as a group. there are 12 members in mrs. mason’s group. if each book club member chooses a different 3-digit number, what is the chance the book club will win a lottery cash prize? 4. at one book club meeting, mr. kaplan said, “if i play the lottery game that has players choose a 4-digit number by myself, the chance of winning will decrease.” is mr. kaplan correct? explain. 5. a similar lottery game claims that there is a “one in a million” chance to win. assuming that the slogan is true, how many numbers would a player need to choose to participate in this lottery game? explain your reasoning.chapter 2 likelihoods 76 2010 carnegie learning 2 2010 georgia middle school sampler 224 problem 2 a good chance of winning? think again! sometimes, lottery companies encourage players to play for greater prizes, but fail to mention that the chances of winning are less than the lotteries in problem 1. many times, lottery companies create different games where the chances of winning are not as obvious. to participate in the supersonic lottery, players choose 3 numbers, each number ranging from 0 through 49 and repetition of numbers is allowed. then, the winning numbers are randomly selected from 3 bins, each bin containing numbers 0 through 49. the player’s three numbers need to match the randomly selected three numbers in the same order to win the cash lottery prize. in a similar fashion, players participate in the wishes come true lottery by choosing 6 numbers, each number ranging from 0 through 24. then, the winning numbers are randomly selected from six bins, each bin containing numbers 0 through 24. 1. use the information given to complete the table. lottery name number of independent events number of choices chance winning supersonic lottery 3 wishes come true lottery 25 2. explain how you calculated the probability of winning each lottery.2.4 investigate magnitude through theoretical probability and experimental probability 77 2010 carnegie learning 2 2010 georgia middle school sampler 225 3. does the number of independent events or the number of choices have a bigger impact on the probability? how can you tell? problem 3 lottery vs. lightning use the information provided to answer questions 1 and 2. l the special 5 lottery has players choose 5 numbers, each number ranges from 0 through 18 and repetition of numbers is allowed. the winning numbers are randomly selected from 5 bins, each bin containing numbers 0 through 18. the player’s ﬁve numbers need to match the randomly selected ﬁve numbers in the same order to win the cash lottery prize. l in 2008, the national oceanic and atmospheric administration (noaa) estimated that there were 600 victims of lightning strikes in the united states for the year. in 2008, the approximate population of the united states was 300,000,000. 1. do you have a better chance of winning the super 5 lottery or getting struck by lightning? explain your reasoning.chapter 2 likelihoods 78 2010 carnegie learning 2 2010 georgia middle school sampler 226 2. based on your answer to question 1, how many times greater is the probability of the more likely event to occur than the less likely event? problem 4 a simulation of a rafﬂe some rafﬂes operate in a similar fashion to lotteries. in this rafﬂe, players choose a two-digit number, between 00 and 99. a number is randomly chosen from 2 bins, each bin containing the numbers 0 through 9. 8 7 6 4 2 9 0 1 3 5 0 5 6 7 8 9 1 2 3 4 1. as a class, select a 2-digit rafﬂe number. 2. calculate the chance of winning this rafﬂe. use the theoretical probability formula to determine the probability of winning the rafﬂe. show your work.2.4 investigate magnitude through theoretical probability and experimental probability 79 2010 carnegie learning 2 2010 georgia middle school sampler 227 a simulation is an experiment that models a real-life situation. you can simulate the selection of rafﬂe numbers by using the random number generator on a graphing calculator. because you are completing an experiment, you will use the experimental probability formula. experimental probability number of times an outcome occurs _________________________________ total amount of trials performed use your graphing calculator to generate random numbers by following these key strokes. step 1: press math. 2:›dec 3: 4:( 6:fmin( 7fmax( 5: math num cpx prb 1:›frac step 2: scroll across to prb using the arrows. 2:npr 3:ncr 4:! 6:randnorm( 7:randbin( 5:randint( math num cpx prb 1:rand step 3: scroll down to randint( using the arrows. this command is to generate random integers. 2:npr 3:ncr 4:! 6:randnorm( 7:randbin( math num cpx prb 1:rand 5:randint(chapter 2 likelihoods 80 2010 carnegie learning 2 2010 georgia middle school sampler 228 step 4: press enter. your screen should show: randint( randint( step 5: following the parenthesis, enter 0,99,5) your screen should show: randint(0,99,5). randint(0,99,5) this means random numbers between and including 0 and 99 will be generated. the numbers will be generated 5 at a time. the reason why 5 numbers will be generated at a time is that you will want 100 numbers randomly generated, and it is easier to read 20 lines with 5 numbers in each line. step 6: press enter. your screen should show a series of 5 numbers. randint(0,99,5) {11 10 68 70 50}2.4 investigate magnitude through theoretical probability and experimental probability 81 2010 carnegie learning 2 2010 georgia middle school sampler 229 3. work with a partner. complete this activity twice, once for each of you. l keep a record of how many times the class’s rafﬂe number appears. l press the enter key a total of 20 times, including the ﬁrst time with the directions. this will provide 100 trials of drawing rafﬂe numbers. l your partner will keep count so that you press the enter key exactly 20 times. the class’s rafﬂe number appeared ___________ times out of 100 trials. 4. determine the experimental probability for the rafﬂe simulation. 5. are the values for the theoretical probability and the experimental probability equivalent? 6. collect the data for the entire class. determine the experimental probability using the results from the entire class. write the experimental probability as a fraction with a denominator of 100, and a numerator that may be a decimal number rounded to one decimal place. in mathematical simulations, the value of the experimental probability approaches the value of the theoretical probability when the number of trials increases. take note record your information as the numbers are generated, you can not scroll up the list.chapter 2 likelihoods 82 2010 carnegie learning 2 2010 georgia middle school sampler 230 problem 5 summary 1. how are experimental probability and theoretical probability alike? 2. how are experimental probability and theoretical probability different? be prepared to show your methods and solutions.4.3 operations with square roots 209 2010 carnegie learning 2010 georgia middle school sampler 231 4 4.3 be radical! operations with square roots learning goal in this lesson, you will: l perform operations involving square roots. key terms l simplest radical form l product property of radicals l quotient property of radicals l rationalizing the denominator the lead in your pencil is an element—the simplest form of matter. but this lead is made up of very tiny particles called atoms. and these atoms are made up of protons, neutrons, and electrons. are there still tinier particles? yes! protons and neutrons are made up of particles called quarks and gluons. if elements are composed of so many parts, why are they called “the simplest forms of matter”? how do you think about simplifying in mathematics? discuss your answers with your group. problem 1 simplify that please 1. circle the expressions that are already in simplest form. simplify the remaining expressions. ( x 2 )( x 3 ) 6 __ 8 106 ____ 106 x 5 __ x 3 3x 4x 4 6 pens and 3 pencils __ 5 4m 3n ( x 2 )( y 2 ) (4)(5) x 4 __ y 3 ___ 16 2 __ 5 4.25 ( 4 __ 5 )( 1 __ 3 ) 4 oranges and 2 oranges 4chapter 4 squares and the pythagorean theorem 210 2010 carnegie learning 4 2010 georgia middle school sampler 232 4 2. describe how you know an expression is simpliﬁed. it is possible to simplify radical expressions. because ___ 36 ______ (4 9) , you can also write this as: ___ 36 ______ (4 9) ___________ 2 2 3 3 ______ 2 2 3 2 2 3 6. in words, the square root of a product equals the product of the square roots of its factors. for some radicals, it is better not to use a calculator to simplify. though a radicand may not appear to be a perfect square, one of the factors of that radicand may be a perfect square. a radical is in simplest radical form when the radicand contains no perfect squares. for example: ___ 40 _______ (4 10) or ___ 40 _________ 2 2 10 __ 4 ___ 10 _______ 2 2 10 2 ___ 10 2 ___ 10 the radical 2 ___ 10 is simpliﬁed because there are no additional perfect square factors in the radicand. 3. rewrite each radical in simplest radical form by writing the radicand as a product of a perfect square and another number. a. ___ 90 b. __ 8 c. ___ 24 d. ___ 45 e. ____ 160 f. ____ 100 take note perfect squares are 1, 4, 9, 16 etc.4.3 operations with square roots 211 2010 carnegie learning 4 2010 georgia middle school sampler 233 4 g. ___ 12 h. ___ 47 i. ___ 72 problem 2 multiplying radicals 1. analyze each mathematical sentence. then, state why it is correct. a. x 2 y 2 ( xy) 2 b. m 3 n 3 p 2 (mn) 3 p 2 c. (4 2 )(3 2 ) (4 3) 2 it is possible to simplify radical expressions that contain multiplication by using the product property of radicals. the product property of radicals states: the square root of a product equals the product of the square roots of the factors. when a and b are positive numbers, ___ ab __ a __ b. look at the three examples shown. example 1: _____ 3 2 __ 3 __ 2 __ 5 __ 3 _____ 5 3 __ 6 ___ 15 1. are __ 6 and ___ 15 in simplest radical form? explain your reasoning.chapter 4 squares and the pythagorean theorem 212 2010 carnegie learning 4 2010 georgia middle school sampler 234 4 example 2: __ 5 ___ 10 ______ 5 10 ___ 50 the expression ___ 50 is not in simplest radical form because the radicand contains a perfect square, which is 25. therefore, ___ 50 ______ 25 2 5 __ 2. another way to think about simplifying ___ 10 __ 5 is to rewrite the expression as ______ 10 5 ________ 2 5 5 ______ 2 5 2 5 __ 2. instead of initially multiplying the radicands, this example demonstrates how to work with the factors of the radicand. example 3: 4 __ 3 2 __ 5 (4 2) _____ 3 5 8 ___ 15 in this example, you would multiply the coefﬁcients, and then multiply the radicands. 2. simplify each product. a. __ 3 __ 5 b. __ 6 ___ 10 c. ___ 12 __ 3 d. __ 5 ___ 154.3 operations with square roots 213 2010 carnegie learning 4 2010 georgia middle school sampler 235 4 e. ___ 21 __ 7 f. ___ 22 ___ 11 g. 3 __ 6 5 __ 8 h. 4 __ 3 2 ___ 12 3. summarize how to multiply two radicals. problem 3 dividing radicals 1. use a calculator to rewrite the radical expression ___ 10 ____ __ 3 as a decimal with six signiﬁcant digits. 2. use a calculator to rewrite the radical expression ___ 10 ___ 3 as a decimal with six signiﬁcant digits. 3. what can you conclude about ___ 10 ____ __ 3 and ___ 10 ___ 3 ? the quotient property of radicals states: “the square root of a quotient equals the quotient of the square roots of the numerator and denominator.” when a and b are positive numbers, __ a __ b __ a ___ __ b .chapter 4 squares and the pythagorean theorem 214 2010 carnegie learning 4 2010 georgia middle school sampler 236 4 in order for a radical expression in fractional form to be completely simpliﬁed, no radical should appear in the denominator. rationalizing the denominator means to rewrite the fraction so that the new fraction is equivalent to the original, but has a rational denominator. recall that a rational number is a number that either terminates or repeats. to rewrite a fraction with a radical in the denominator into an equivalent fraction with a speciﬁed denominator, multiply the original fraction by a form of 1. to rationalize the denominator, multiply the original fraction by a form of 1 that results in a perfect square in the radicand in the denominator. for example, consider the expression __ 2 ___ __ 3 . to rationalize the denominator, multiply by a form of 1. in this example, the expression __ 3 appears in the denominator, so the form of 1 would be __ 3 ___ __ 3 . __ 2 ___ __ 3 __ 3 ___ __ 3 multiply the original expression by a form of 1. _____ 2 3 ______ _____ 3 3 apply the product property of radicals. __ 6 ___ 3 multiply the numerators and denominators. 4. use a calculator to verify that __ 2 ___ __ 3 and __ 6 ___ 3 are equivalent fractions. 5. what form of 1 would you multiply by to rationalize each denominator? a. 5 ___ __ 2 b. __ 6 ___ __ 74.3 operations with square roots 215 2010 carnegie learning 4 2010 georgia middle school sampler 237 c. __ 1 __ 3 d. ___ 12 ____ ___ 10 e. __ 3 ___ __ 4 problem 4 simplifying using addition or subtraction it is possible to simplify radical expressions using addition and subtraction. when adding and subtracting square roots, only roots that have the same radicand can be added or subtracted. can be simpliﬁed already simpliﬁed 4 __ 3 7 __ 3 3 ___ 10 5 ___ 11 8 __ 7 2 __ 7 __ 3 2 1. describe why the two expressions on the left can be simpliﬁed and the two expressions on the right are already simpliﬁed. 2. simplify each expression, if possible. a. 3 __ 2 5 __ 2 b. 9 ___ 11 8 ___ 11 c. ___ 23 4 ___ 21 d. 7 __ 2 __ 8chapter 4 squares and the pythagorean theorem 216 2010 carnegie learning 4 2010 georgia middle school sampler 238 4 e. __ 9 ___ 27 f. 3 __ 6 ___ 24 __ 8 g. ___ 11 ___ 10 h. __ 6 4 __ 6 3. explain how to simplify radical expressions using addition and subtraction. problem 5 problem summary properties of radicals words rule product property of radicals the square root of a product equals the product of the square roots of the factors ___ ab __ a __ b when a and b are positive numbers. quotient property of radicals the square root of a quotient equals the quotient of the square roots of the numerator and denominator. __ a __ b __ a ___ __ b when a and b are positive numbers. 1. simplify each expression. write the expression in simplest radical form. show your work. a. ___ 72 b. ___ 7 ___ 814.3 operations with square roots 217 2010 carnegie learning 4 2010 georgia middle school sampler 239 4 c. 3 __ 3 5 __ 3 d. 8 __ 6 2 ___ 54 e. ____ 150 f. ___ 12 5 __ 3 9 __ 6 g. ___ 15 ___ 12 h. 24 ____ ___ 12 i. 15 __ 3 __ 5 j. 5 ___ __ 3 10 ___ __ 3 be prepared to share your solutions and methods.chapter 4 squares and the pythagorean theorem 218 2010 carnegie learning 4 2010 georgia middle school sampler 2404.4 the pythagorean theorem 219 2010 carnegie learning 2010 georgia middle school sampler 241 4 4.4 soon you will determine the right triangle connection learning goals in this lesson, you will: l use mathematical properties to discover the pythagorean theorem. l solve problems involving right triangles. key terms l right triangle l pythagorean theorem l right angle l theorem l leg l postulate l hypotenuse l proof what do ﬁreﬁghters and roofers have in common? if you said they both use ladders, you would be correct! many people who use ladders as part of their job must also take a class in ladder safety. what type of safety tips would you recommend? do you think the angle of the ladder is important to safety? discuss your ideas with your partner. problem 1 side lengths of right triangles a right triangle is a triangle with exactly one right angle. a right angle has a measure of 90 and is indicated by the black square in a right triangle. the leg of a right triangle is either of the two shorter sides. together, the two legs form the right angle of a right triangle. the hypotenuse of a right triangle is the side opposite the right angle. hypotenuse leg leg right angle symbol the pythagorean theorem 4chapter 4 squares and the pythagorean theorem 220 2010 carnegie learning 4 2010 georgia middle school sampler 242 4 1. the 3 side lengths of right triangles are given. determine which length represents the hypotenuse. a. 5, 12, 13 b. 1, 1, __ 2 c. 2.4, 5.1, 4.5 d. 75, 21, 72 e. 15, 39, 36 f. 7, 24, 25 2. how did you decide which length represented the hypotenuse? yumi, lynn, and fernando must determine all side lengths of a right triangle shown. determining the lengths of the legs appeared simple, but determining the length of the hypotenuse posed a more difﬁcult challenge. yumi suggests that they count the spaces between the dots. fernando did not think that would work if the lines were not horizontal or vertical. lynn remembered that in a previous lesson, she drew squares on slanted lines to determine the length of the side. lynn said, “remember, the length of the side of a square is the square root of its area.” 3. determine the side lengths of the right triangle shown. explain your reasoning. c b a4.4 the pythagorean theorem 221 2010 carnegie learning 4 2010 georgia middle school sampler 243 4 problem 2 explore right triangles let’s explore the side lengths of more right triangles. 1. an isosceles right triangle is drawn on the grid shown. a. use a straightedge to draw squares on each side of the triangle. b. draw two diagonals in each of the two smaller squares. c. cut out the two smaller squares along the legs. then, cut those squares into fourths along the diagonals you drew. d. arrange the pieces on top of the larger square along the hypotenuse. tape the triangles on top of the square you created from the hypotenuse.chapter 4 squares and the pythagorean theorem 222 2010 carnegie learning 4 2010 georgia middle school sampler 2444.4 the pythagorean theorem 223 2010 carnegie learning 4 2010 georgia middle school sampler 245 4 e. what do you notice? f. write a sentence that describes the relationship between the areas of each of the squares. 2. a right triangle is shown with one leg that is 4 units in length and the other leg is 3 units in length. a. use a straightedge to draw squares on each side of the triangle. b. cut out the two smaller squares along the legs. c. cut the two squares into strips that are either 4 units by 1 unit or 3 units by 1 unit. d. arrange the strips and squares on top of the square along the hypotenuse. you may need to make additional cuts to the strips to create individual squares that are 1 unit by 1 unit. tape the strips on top of the square you drew made from the hypotenuse.chapter 4 squares and the pythagorean theorem 224 2010 carnegie learning 4 2010 georgia middle school sampler 2464.4 the pythagorean theorem 225 2010 carnegie learning 4 2010 georgia middle school sampler 247 4 e. what do you notice? f. write a sentence that describes the relationship between the areas of each of the squares. 3. a right triangle is shown with one leg that is 2 units in length and the other leg is 4 units in length. a. use a straightedge to draw squares on each side of the triangle. b. cut out the two smaller squares. c. draw four congruent right triangles on the square with side lengths of 4 units. then, cut the four congruent right triangles you drew. d. arrange the small square and the 4 congruent triangles over the square that has one of its sides as the hypotenuse. e. what do you notice? f. write a sentence that describes the relationship between the areas of each of the squares.chapter 4 squares and the pythagorean theorem 226 2010 carnegie learning 4 2010 georgia middle school sampler 2484.4 the pythagorean theorem 227 2010 carnegie learning 4 2010 georgia middle school sampler 249 4 4. compare the sentences you wrote for part (f) in problem 2, questions 1 through 3. what do you notice? 5. write an equation that represents the relationship between the areas of the squares if the length of the shortest leg of the right triangle is “a,” the length of the longest leg of the right triangle is “b,” and the length of the hypotenuse is “c.” a c b this special relationship that exists between the squares of the lengths of the sides of a right triangle is known as the pythagorean theorem. the sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. the pythagorean theorem states: “if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a 2 b 2 c 2 .” a c bchapter 4 squares and the pythagorean theorem 228 2010 carnegie learning 4 2010 georgia middle school sampler 250 4 a theorem is a mathematical statement that can be proven using deﬁnitions, postulates, and other theorems. a postulate is a mathematical statement that cannot be proved, but is considered true. the pythagorean theorem is one of the earliest known to ancient civilization and one of the most famous. this theorem was named after pythagoras (580 to 496 b.c.), a greek mathematician and philosopher who was the ﬁrst to prove the theorem. a proof is a series of steps used to prove the validity of an if–then statement. while it is called the pythagorean theorem, the mathematical knowledge was used by the babylonians 1000 years before pythagoras. many proofs followed that of pythagoras, including euclid, socrates, and even the twentieth president of the united states, president james a. garﬁeld. 6. use the pythagorean theorem to verify the length of the hypotenuse you calculated in problem 1, question 3. 7. use the pythagorean theorem to determine the length of the hypotenuse in problem 2, question 1. 8. use the pythagorean theorem to determine the length of the hypotenuse in problem 2, question 3.4.4 the pythagorean theorem 229 2010 carnegie learning 4 2010 georgia middle school sampler 251 4 problem 3 maintaining school grounds mitch maintains the magnolia middle school campus. use the pythagorean theorem to help mitch with some of his jobs. 1. mitch needs to wash the windows on the second ﬂoor of a building. he knows the windows are 12 feet above the ground. because of dense shrubbery, he has to put the base of the ladder 5 feet from the building. what ladder length does he need? 12' 5' 2. the gym teacher, mrs. fisher, asked mitch to put up the badminton net. mrs. fisher speciﬁed that the top of the net must be 5 feet above the ground. she knows that mitch will need to put stakes in the ground for rope supports. she speciﬁed that the stakes be placed 6 feet from the base of the poles. mitch has two pieces of rope, one that is 7 feet long and a second that is 8 feet long. will these two pieces of rope be enough to secure the badminton poles? explain your reasoning. 5' 6'chapter 4 squares and the pythagorean theorem 230 2010 carnegie learning 4 2010 georgia middle school sampler 252 4 3. mitch stopped by the baseball ﬁeld to watch the team practice. the ﬁrst baseman caught a line drive right on the base. he touched ﬁrst base for one out and quickly threw the ball to third base to get the last out of the scrimmage. how far did he throw the ball? 90 feet 90 feet home pitcher’s mound 1st 3rd 90 feet 90 feet 2nd 4. the skate ramp on the playground of a neighboring park is going to be replaced. mitch needed to determine how long the ramp is to get estimates on the cost of a new skate ramp. he knows the measurements shown in the ﬁgure. how long is the existing skate ramp? 15 feet 8 feet 5. to meet the american disabilities act (ada) guidelines, mitch learned that a wheelchair ramp that is constructed to rise 1 foot off the ground must extend 12 feet along the ground. how long will the wheelchair ramp be? 1 foot 12 feet4.4 the pythagorean theorem 231 2010 carnegie learning 4 2010 georgia middle school sampler 253 4 6. the eighth grade math class maintains a ﬂower garden in the front of the building. the garden is in the shape of a right triangle, and its dimensions are shown. 9' 12' a. what is the length of the hypotenuse? b. the class wants to install a 3-foot high picket fence around the garden to keep students from stepping on the ﬂowers. how much fencing do the students need? c. the picket fence costs 5 a linear foot. how much will the fence cost? do not calculate sales tax.chapter 4 squares and the pythagorean theorem 232 2010 carnegie learning 4 2010 georgia middle school sampler 254 4 problem 4 solve for the missing side 1. write an equation to determine the unknown length. then, solve the equation. make sure your answer is simpliﬁed. a. 5 12 b b. 9 11 a c. 10 5.1 x d. 2 15 x be prepared to share your solutions and methods.9.5 writing explicit formulas for arithmetic sequences 261 2010 carnegie learning 2010 georgia middle school sampler 255 9 9.5 the nth term of a sequence writing explicit formulas for arithmetic sequences learning goals in this lesson, you will: state the difference between a recursive formula and an explicit formula. write explicit formulas for arithmetic sequences. calculate the values of terms in an arithmetic sequence using an explicit formula. graph arithmetic sequences deﬁned by explicit formulas. write explicit formulas from graphs of arithmetic sequences. explain how the pattern of growth by a constant value is represented by an arithmetic sequence, a recursive formula, a graph, and an explicit formula. key terms nth term explicit formula e mc 2 is one of the most famous mathematical formulas of all time, discovered by albert einstein. what einstein realized is that, given the right conditions, it is possible to turn energy into mass and mass into energy. his famous formula shows that if it were possible to turn a certain mass into energy, then the energy (e ) would be equal to the mass (m) multiplied by the speed of light squared (c 2 ), which is about 449,704,360,000,000,000 miles per hour. how are formulas important in mathematics? what formulas do you use often? discuss your answers with your group. 9chapter 9 arithmetic sequences, relations, and functions 262 2010 carnegie learning 9 2010 georgia middle school sampler 256 9 problem 1 calculating the hundredth term mr. lewis promised a prize to the ﬁrst student who could determine the hundredth term of this sequence without using a calculator. sequence a: 8, 14, 20, 26, 32,… 1. can you develop a faster way to determine the hundredth term without calculating the 99 terms before it? use what you know about sequences and give it a try. show your work.9.5 writing explicit formulas for arithmetic sequences 263 2010 carnegie learning 9 2010 georgia middle school sampler 257 9 several groups in mr. lewis’s class developed methods to calculate the hundredth term. read the methods used in riley’s group and in michael’s group. riley’s group 8, 14, 20, 26, 32,... 6 6 we made it a smaller problem and tried to get the 10th term by a faster method. we wrote the ﬁrst 10 terms so we could check our work. term number 1 2 3 4 5 6 7 8 9 10 100 term 8 14 20 26 32 38 44 50 56 62 ? multiplying would be faster. so, to get the 10th term, 10 6 made sense to us instead of adding 6’s. 10 6 60. according to the table, we were close, but we needed to get 62. we decided to check this method with the other table values to see if we were close on those. 6 6 6 12 6 18 6 24 6 30 6 36 6 42 6 48 6 54 6 60 term number 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 100 term 8 14 20 26 32 38 44 50 56 62 ? we found a pattern! every time, we get an answer 2 less than the term in the sequence. to ﬁx it we need to multiply by 6, then add 2. to get the 10th term: 10 6 60 60 2 62 to get the 100th term: 100 6 600 600 2 602chapter 9 arithmetic sequences, relations, and functions 264 2010 carnegie learning 9 2010 georgia middle school sampler 258 9 michael’s group 8, 14, 20, 26, 32 6 6 6 6 8 6 6 6 6 5th term (there are four 6’s start because the ﬁrst term was given.) 8 6 6 6 6 6 6th term has ﬁve 6’s 8 6 6 6 6 6 … 100th term there would be 99 of them. 8 99 6 8 594 602 the 100th term is 602. 2. was your method similar to either method shown? explain. 3. how were the methods similar?9.5 writing explicit formulas for arithmetic sequences 265 2010 carnegie learning 9 2010 georgia middle school sampler 259 9 4. how were the methods different? 5. calculate the hundredth term of this sequence using any non-calculator method. explain. sequence b: 12, 16, 20, 24, 28,… 6. calculate the ﬁftieth term of this sequence using any non-calculator method. explain. sequence c: 300, 295, 290, 285,…chapter 9 arithmetic sequences, relations, and functions 266 2010 carnegie learning 9 2010 georgia middle school sampler 260 9 problem 2 determining the nth term of a sequence the nth term of a sequence represents the term with n as its term number. example sequence t: 9, 13, 17, 21,… if sequence t was written in a table, it would appear as shown. term number, n 1 2 3 4 n term, t n 9 13 17 21 ? you do not know the nth term of a sequence right away because the term’s position in the sequence is not known. instead, the nth term of a sequence can be represented by an explicit formula. an explicit formula is an algebraic expression that deﬁnes the nth term of a sequence in relation to n. it is different from a recursive formula, which deﬁnes the nth term of a sequence in relation to the previous term. let’s practice writing an explicit formula by expanding riley’s and michael’s methods. riley’s method 1. multiply the common difference by each term number. term number, n 1 2 3 4 n term, t n 9 13 17 21 2. according to the pattern, what value must be added to (or subtracted from) each product to equal each term? show your calculation. term number, n 1 2 3 4 n term, t n 9 13 17 219.5 writing explicit formulas for arithmetic sequences 267 2010 carnegie learning 9 2010 georgia middle school sampler 261 9 3. write the explicit formula for the nth term. michael’s method 9, 13, 17, 21,… 1. identify the initial term. 2. identify the common difference. 3. how many times is the common difference repeatedly added to get each term? express your answer in terms of n. 4. write the explicit formula for the nth term. the general explicit formula for an arithmetic sequence is t n d(n 1) t 1 , where t n represents the value of the nth term, t 1 represents the initial term, d represents the common difference, and n represents the term number. 5. did riley’s method and michael’s method yield the same formula? explain. explicit formulas can be used to determine the value of any term without having to calculate the values of all the terms before it in the sequence. 6. calculate the value of the hundredth term in the sequence t.chapter 9 arithmetic sequences, relations, and functions 268 2010 carnegie learning 9 2010 georgia middle school sampler 262 9 problem 3 writing and using explicit formulas write an explicit formula for each arithmetic sequence and calculate the value of the speciﬁed term. 1. 30, 42, 54, 66, 78,…, t 50 ? 2. 91, 88, 85, 82, 79,…, t 29 ? 3. 2, 12, 22, 32, 42,…, t 77 ? 4. 6, 0, 6, 12, 18,…, t 15 ? 5. 14, 11, 8, 5, 2,…, t 40 ? 6. 8, 16, 24, 32, 40,…, t 11 ? problem 4 revisiting sequences from lesson 9.1 1. robert is crafting toothpick houses for the background of a diorama. he creates one house, and then adds additional houses by adjoining them as shown. a. write and use an explicit formula to calculate the number of toothpicks needed for 13 houses.9.5 writing explicit formulas for arithmetic sequences 269 2010 carnegie learning 9 2010 georgia middle school sampler 263 9 b. explain the meaning of each value in the explicit formula. 2. the ﬁrst diagram shows the number of students that can be seated around a trapezoid table. the second diagram shows how the tables can be joined at the sides to make one longer table. 2 5 1 4 3 a. write and use an explicit formula to calculate the number of students that can be seated around a large table constructed of eight trapezoid tables.chapter 9 arithmetic sequences, relations, and functions 270 2010 carnegie learning 9 2010 georgia middle school sampler 264 b. explain the meaning of each value in the explicit formula. problem 5 connecting explicit formulas and the graphs of sequences complete the table and graph the ﬁrst six values generated by each explicit formula. 1. t n 2(n 1) 3 x 16 12 16 18 8 8 12 4 4 14 6 10 2 y 18 10 14 6 2 term number term value9.5 writing explicit formulas for arithmetic sequences 271 2010 carnegie learning 9 2010 georgia middle school sampler 265 9 2. t n 1(n 1) 14 x 16 12 16 18 8 8 12 4 4 14 6 10 2 y 18 10 14 6 2 write the explicit formula represented by the graph of each arithmetic sequence. 3. x 8 6 8 9 4 4 6 2 2 7 3 5 1 y 9 5 7 3 1 term number term valuechapter 9 arithmetic sequences, relations, and functions 272 2010 carnegie learning 9 2010 georgia middle school sampler 266 9 4. x 8 4 8 10 4 –4 –4 6 –2 2 –6 y 10 12 2 6 –2 –6 problem 6 graphic organizer describe how each representation demonstrates growth by a constant value. provide a general description and speciﬁc examples. be prepared to share your solutions and methods.9.5 writing explicit formulas for arithmetic sequences 273 2010 georgia middle school sampler 267 graph explicit formula arithmetic sequence recursive formula a pattern representing growth by a constant value 2010 carnegie learning 92010 georgia middle school sampler 268 sides of an angle the sides of an angle are the two rays that form the angle. sketch when you sketch a geometric figure, you create it without the use of tools. skew lines skew lines, or non-coplanar lines, are lines that are not located in the same plane. example straight angle a straight angle is an angle whose measure is equal to 180. example a b d adb is a straight angle. straightedge a straightedge is a ruler with no numbers. supplementary angles two angles are supplementary angles if the sum of their angle measures is equal to 180. example 1 2 1 and 2 are supplementary angles. t transformation a transformation is the mapping, or movement, of all the points of a figure in a plane according to a common operation. examples translations, reflections, and rotations are examples of transformations. translation a translation is a transformation that “slides” each point of a figure the same distance and direction. example x y –5 –5 5 5 0 g-10 glossary g 2010 carnegie learninggeorgia middle school math series sample content from texts georgia middle school math series sample content from texts about carnegie learning, inc. carnegie learning is focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standard • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com georgia middle school math series sample content from texts carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-022-3 georgia middle school math level 1 student edition georgia middle school math level 1 student edition volume 1 georgia middle school math level 1 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-025-4 georgia middle school math level 2 student edition georgia middle school math level 2 student edition volume 1 georgia middle school math level 2 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available carnegie learning, inc. 437 grant st., suite 2000 pittsburgh, pa 15219 phone 412-690-2442 customer service phone 877-401-2527 www.carnegielearning.com volume 1 vol 1 isbn 978-1-60972-028-5 georgia middle school math level 3 student edition georgia middle school math level 3 student edition volume 1 georgia middle school math level 3 student edition about carnegie learning, inc. carnegie learning is the only publisher in the georgia adoption focused exclusively on mathematics. we are dedicated to helping every single student in georgia succeed in math as a critical step toward high school graduation. carnegie learning’s georgia middle school math instructional materials are customized to georgia’s mathematics performance standards: • designed for georgia’s performance-based collaborative classrooms • fully compliant with georgia’s mathematics performance standards • sequenced to georgia frameworks • research-based pedagogy • proven effective in third party studies • supplemental software available