ample text 2008 carnegie learning, inc.ittsburgh, pa phone 888.851.7094 fax 412.690.2444 www.carnegielearning.com copyright 2008 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. 2008 carnegie learning, inc. isbn: 978-1-934800-98-0 fall 2008 sample text printed in the united states of americaable of contents introduction…………………………………………….p. 1 sampler table of contents • p. 1 introduction to carnegie learning tm math curricula • p. 3 mathematical representations • p. 6 what is included in a carnegie learning text set? • p. 7 collaborative classroom • p. 8 bridge to algebra …………………………………….p. 18 course description • p. 18 table of contents • p. 20 student text lesson 2.1 • p. 27 look ahead • p. 28 lesson • p. 31 teacher’s implementation guide lesson 2.1 • p. 37 lesson map • p. 41 lesson with answers and teacher notes • p. 43 wrap-up • p. 48 teacher’s resources and assessments (tra) • p. 51 assignment with answers lesson 2.1 • p. 53 skills practice with answers lesson 2.1 • p. 54 teacher notes on chapter 2 assessments • p. 55 pre-test chapter 2 • p.57 post test chapter 2 • p. 61 mid chapter test chapter 2 • p. 65 end of chapter test chapter 2 • p. 67 standardized practice test chapter 2 • p. 69 homework helper • p. 75 lesson 2.1 • p. 77 algebra i …………………………………………........p. 78 course description • p. 78 table of contents • p. 80 student text lesson 1.8 • p. 87 look ahead • p. 88 lesson • p. 91 teacher’s implementation guide lesson 1.8 • p. 95 lesson map • p. 99 lesson with answers and teacher notes • p. 101 wrap up • p. 105 teacher’s resources and assessments (tra) • p. 107 assignment with answers lesson 1.8 • p. 110 pre-test chapter 1 • p.113 post test chapter 1 • p. 119 2008 text sampler page 1mid chapter test chapter 1 • p. 125 end of chapter test chapter 1 • p. 129 standardized practice test chapter 1 • p. 135 homework helper • p. 142 lesson 1.8 • p. 143 geometry …………………………………………….p. 144 course description • p. 144 table of contents • p. 146 student text lesson 5.4 • p. 151 look ahead • p. 152 lesson • p. 155 teacher’s implementation guide lesson 5.4 • p. 165 lesson map • p. 169 lesson with answers and teacher notes • p. 171 wrap-up • p. 181 teacher’s resources and assessments (tra) • p. 183 assignment with answers lesson 5.4 • p. 185 pre-test chapter 5 • p.187 homework helper • p. 193 lesson 5.4 • p. 294 algebra ii …………………………………………….p. 196 course description • p. 196 table of contents • p. 198 student text lesson 1.5 • p. 204 lesson • p. 207 teacher’s implementation guide lesson 1.5 • p. 216 lesson map • p. 217 lesson with answers and teacher notes • p. 219 wrap-up • p. 228 teacher’s resources and assessments (tra) • p. 229 assignment with answers lesson 1.5 • p. 231 pre-test chapter 1.5 • p.235 2008 text sampler page 2ntroduction to carnegie learning tm math curricula we are excited that you are interested in exploring our unique approach to mathematics instruction. what does carnegie learning offer? x over 20 years of research into how students think and learn x blended solutions offering instructional materials in software and text x extensive use of real-world scenarios to reinforce a conceptual understanding of mathematics x instruction tailored to the individual needs of each student a research-based approach to mathematics for your student carnegie learning provides some of the only truly research-based math curricula in the country. controlled field studies have validated that carnegie learning’s approach helps students improve their course grades and overall achievement. whether your student excels or struggles with mathematics, carnegie learning curricula will help strengthen their skills, content knowledge, and confidence. the curriculum uses a scientifically-researched approach incorporating the latest discoveries into how students think, learn, and apply new knowledge in mathematics. building on over 20 years of research and design at carnegie mellon university and field tests by leading mathematics educators, our approach uses students’ intuitive problem solving abilities as a powerful bridge to a more formal understanding of mathematics. independent studies of carnegie learning implementations in miami-dade, fl; pittsburgh, pa; moore, ok, and kent, wa, and elsewhere demonstrate that carnegie learning curricula deliver a positive shift in standardized test scores, student attitudes toward math and problem solving, and critical thinking skills. research also indicates strong results with title i and special-needs populations, including exceptional student education students, those with limited english proficiency and students receiving free or reduced lunch. research shows that students using the carnegie learning tm algebra i curriculum: • perform 30% better on questions from the timss assessment • demonstrate 85% better performance on assessments of complex mathematical problem solving and thinking • have a 70% greater likelihood of completing subsequent geometry and algebra ii courses • achieve 15-25% better scores on the sat and iowa algebra aptitude test • experience equivalent results for both minority and nonminority students carnegie learning tm blended math curricula the carnegie learning blended solutions contain software and text components that complement one another. your students will spend about 40% of instructional time using computer-based tutorials, and 60% using a student text to collaborate with peers in the classroom. 2008 text sampler page 3the skills learned in the text are enhanced by the cognitive tutor software. the cognitive tutor software elaborates on the lessons introduced in the text. students work at their own pace in the cognitive tutor software component of the curriculum. the learning system is built on cognitive models, which represent the knowledge that a student might possess about the mathematics that they are studying. the software assesses students’ prior mathematical knowledge on a step-by-step basis and presents problems tailored to their individual skill levels. using the cognitive tutor software, your student will receive the benefits of individualized instruction, ample practice, immediate feedback, and coaching. just-in-time hints, on-demand hints, and positive reinforcement will put your student in control of his or her own learning. the student text provides an opportunity for analysis, extended investigation, and the exploration of alternate solution paths. students engage in problem solving and reasoning, and communicate using multiple representations of math concepts. real-world situations are used in problems designed to emphasize conceptual understanding. the goal of the student text is to be engaging and effective so your student will have fun while learning by doing. a typical week in a carnegie learning classroom the classroom is a dynamic, adaptive environment. while no two weeks will be exactly the same, most weeks will be split between classroom activities and work in the computer lab. the number of each type of session that the teacher schedules depends on the teacher’s preference and the availability of lab time. carnegie learning suggests that students spend 40% of their class time in the computer lab working with the computer and 60% with student text investigations. below is an itinerary outlining a typical mid-semester week. monday x students complete the student text investigation started on friday with group presentations. x teacher solicits questions on the completed investigation and wraps up the investigation by asking questions that lead students to reflect on the material covered. x students begin a new student text investigation. tuesday x students complete about half of the investigation started on monday. x teacher has students respond to a writing prompt to summarize their work. 2008 text sampler page 4ednesday x students work with the software in the computer lab. thursday x students complete the investigation started on tuesday. x partners present their findings of tuesday’s investigation using a written format. x teacher solicits questions and comments on the completed investigation, wraps up the investigation by asking questions that lead students to apply their knowledge of the material covered. friday x students work with the software in the computer lab. carnegie learning’s ongoing support teacher training to ensure that every implementation yields successful results, carnegie learning offers a comprehensive professional development program, delivered by certified implementation specialists (cis) using best practices knowledge of our curricula and years of classroom mathematics experience. our professional development programs provide teachers with the experience, insight, and support needed to grow as more reflective practitioners. our emphasis is on aligning teaching to learning using standards-based curriculum, student-centered instruction, and the integration of technology. throughout the sessions teachers learn best practices based on the latest research in the science of learning, and are provided with the opportunity to experience the student-centered classroom from the perspective of both student and teacher. the workshops emphasize our learning by doing approach, where the role of the teacher is to facilitate student interaction, communication, and problem solving. for more information on professional development, please contact your carnegie learning sales representative. carnegie learning tm resource center our online resource center provides trained teachers with access to assessments, discussion forums, technical support, and other educational materials. in addition, teachers and administrators can access software updates, installer files, and serial numbers for all of their carnegie learning products. 2008 text sampler page 5chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 mathematical representations introduction mathematics is a human invention, developed as people encountered problems that they could not solve. for instance, when people first began to accumulate possessions, they needed to answer questions such as: how many? how many more? how many less? people responded by developing the concepts of numbers and counting. mathematics made a huge leap when people began using symbols to represent numbers. the first “numerals” were probably tally marks used to count weapons, livestock, or food. as society grew more complex, people needed to answer questions such as: who has more? how much does each person get? if there are 5 members in my family, 6 in your family, and 10 in another family, how can each person receive the same amount? during this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. the following processes can help you solve problems. discuss to understand • read the problem carefully. • what is the context of the problem? do you understand it? • what is the question that you are being asked? does it make sense? think for yourself • do i need any additional information to answer the question? • is this problem similar to some other problem that i know? • how can i represent the problem using a picture, diagram, symbols, or some other representation? work with your partner • how did you solve the problem? • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? work with your group • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? • how can we explain our solution to one another? to the class? share with the class • here is our solution and how we solved it. • we could only get this far with our solution. how can we finish? • could we have used a different strategy to solve the problem? a1s10101.qxd 4/11/08 7:56 am page 4 2008 text sampler page 6 the icons on this page help guide teachers in a collaborative classroom. these icons appear in the student texts and teacher's implementation guides of all carnegie learning textbooks.hat is included in a carnegie learning text set? student text the student text is a consumable textbook with room for students to take notes and work problems directly on the lesson page. each lesson contains objectives, key terms, and problems that help the students to discover and master mathematical concepts. this sample text contains selected student text pages from each of our curricula. student assignments the student assignments book contains one assignment worksheet per lesson. the student assignment book is designed to move with the student from classroom to home to computer lab, so that students can continually practice the skills taught in the lesson. this sample text contains matching assignments for the selected sample of student text lessons. homework helper the homework helper book is designed to help parents and caregivers become more informed about the concepts being covered in the student’s math course. students are encouraged to keep the homework helper at home. the homework helper includes a practice page for each lesson in the student text. the page includes a worked example of the skills covered in the lesson. each page of the homework helper also has practice exercises that the student can try. the answers to the exercises are included at the back of the homework helper. this sample text contains matching homework helper pages for the selected sample of student texts lessons. teacher’s implementation guide the teacher’s implementation guide contains a lesson map for each student text lesson. the lesson map includes each lesson’s objectives, key terms, nctm standards, essential questions, warm-up questions, open-ended questions, and closing activities. an image of each student text page, including answers, is provided in the teacher’s implementation guide, along with teacher notes intended to assist with classroom practice. this sample text contains the teacher’s implementation guide pages for the selected sample of student text lessons. teacher’s resources and assessments the teacher’s resources and assessments (tra) contains five tests per chapter of the student text. the tests are a pre-test, a post-test, a mid-chapter test, an end-of-chapter test, and a standardized test practice. the tra contains the answer keys to all assessments. editable black-line masters of the assessments are provided digitally in the carnegie learning resource center. the tra also contains answer keys for the student assignments. this sample text contains assignment answer keys for the selected sample of lessons. this sample text contains the full set of assessment answer keys for bridge to algebra chapter 2 and algebra i chapter 1. a sample pre-test is included for geometry chapter 3 and algebra ii chapter 1. finally, this sampler is printed in black and white. actual carnegie learning texts are more colorful. 2008 text sampler page 7iii collaborative classroom 2008 carnegie learning, inc. collaborative classroom collaborative classroom as you begin the process of planning for the school year , you will want to give serious consideration to how your classroom is structured. early research on teaching and learning has revealed that what happens in the classroom in the first three days determines the environment for the entire year . this insight is important as you begin to think about your classroom and the cognitive tutor algebra i curriculum. an effective implementation of the curriculum is most likely to occur in the collaborative classroom, a classroom in which knowledge is shared. carnegie learning’s philosophy—learning by doing—captures the belief that students develop understanding and skill by taking an active role in their environment. furthermore, it is carnegie learning’s belief that effective communication and collaboration are essential skills for the successful learner . it is through dialogue and discussion of different strategies and perspectives that students become knowledgeable independent learners. these beliefs can be realized in the collaborative classroom. defining a collaborative classroom a collaborative classroom is an environment in which knowledge and authority are shared between the teacher and the students. in a collaborative classroom, teachers are facilitators and students are active participants. all students, not segregated by ability level, interest, or achievement, benefit from the environment created in the collaborative classroom. teachers in the collaborative classroom combine their extensive knowledge about teaching and learning, content, and skills with the informal and formal knowledge, strategies, and individual experiences of their students. the collaborative classroom differs from the traditional classroom in which the teacher is seen as an information giver (tinzmann, m.b.; jones, b.f .; fennimore, t .f .; bakker , j; fine, c.; and pierce, j., 1990, www.ncrel.org/sdrs/areas/ rpl_esys/collab.htm). characteristics of the collaborative classroom the collaborative classroom is identified by discussion, with in-depth accountable talk and two-way interactions, whether among members of the whole class or small groups. it is a well-structured environment in which questioning and dialogue are valued and appropriate parameters are set so that active learning can occur . careful planning by the teacher ensures that students can work together to attain individual and collective goals and to develop learning strategies. in the collaborative classroom, students are encouraged to take responsibility for their learning through monitoring and reflective self-evaluation. the collaborative classroom is one in which teachers spend more time in true academic interactions as they guide students to search for information and help students to share what they know. as facilitators, teachers have the opportunity to provide the correct amount of help to individual students by providing appropriate hints, probing questions, feedback, and help in clarifying thinking or the use of a particular strategy (tinzmann, m.b.; jones, b.f .; fennimore, t .f .; bakker , j; fine, c.; and pierce, j., 1990, www.ncrel.org/sdrs/areas/rpl_esys/collab.htm). a1t1_fm_v1.qxd 4/11/08 10:30 am page viii 2008 text sampler page 8ollaborative classroom ix 2008 carnegie learning, inc. collaborative classroom collaborative learning versus cooperative learning two types of learning occur in the collaborative classroom; collaborative learning, which focuses on interaction, and cooperative learning, which is a structure of interaction that helps students to accomplish a goal or end product. while these two forms of learning are often described and used interchangeably, differences do exist. the significant difference between collaborative and cooperative learning environments is the amount of control that the teacher exercises in setting goals and providing choice. for instance, in a collaborative classroom, students are positioned to set their own goals and choose activities, whereas in the cooperative learning environment, the teacher directs these activities. (ten panitz, 1996, http://www.city.londonmet.ac.ul/deliberations/collab.learning/panitz2.html) learning in the collaborative classroom critical to teaching and learning in the collaborative-cooperative environment is the ability to define the responsibilities of the teacher and students. for effective collaboration and cooperative teamwork, teachers and students must agree to certain responsibilities that support the learning process. the table below reflects the parallel responsibilities of teachers and students. effective collaboration and cooperative t eamwork teacher responsibilities student responsibilities monitor student behavior . develop the skills to work cooperatively. provide assistance when needed. learn to talk and discuss problems with each other in order to accomplish the group goal. answer questions only when they are group questions. ask for help only after each person in the group has considered the problem and the group has a question for the teacher . interrupt the process to reinforce cooperative skill or to provide direct instructions to all students. believe that all members of the group work together toward a common goal. understand that the success or failure of the group is to be shared by all members. provide closure for the lesson. reflect on the work of the group. evaluate the group process by discussing the actions of the group members. appreciate that working together is a process and encourage each group member to interact and relate to the rest of the group members. help students to become individually accountable for learning and reinforce this understanding regularly. realize that each member must contribute as much as he or she can to the group goal. understand that the success of the group is dependent on the individual work of each member of the group. understand that group members are individually accountable for their own learning. a1t1_fm_v1.qxd 4/11/08 10:30 am page ix 2008 text sampler page 9collaborative classroom 2008 carnegie learning, inc. collaborative classroom what the collaborative classroom is not it must be agreed upon by the teacher and the students that the collaborative classroom is not one in which students: work in small groups on a problem or group of problems without direction or individual responsibility. work individually while sitting in a group working on problems. work without conversation or interaction regarding the method or process being used to solve the problem. allow one member of the group to do all of the work while others sit passively. shaping the collaborative classroom to ensure that the spirit and purpose of the collaborative classroom is clear from the onset of school, you will want to engage your students in a collaborative activity on the first day. in doing so, you can accomplish two important goals. first, students immediately understand the importance and value of working together , and secondly, students quickly move into their role as active participants. the activity “facts in five,” as you may have experienced in training, is an activity designed to meet these goals. another popular activity with students is known as “broken squares” (spencer kagan: cooperative learning). in this activity, members of the team are each given several pieces of a broken square. the pieces belong to different squares. students must create the whole square by taking turns giving each other one piece. no one may speak during the activity, that is, no one can ask for what he or she needs. this activity is perfect for teaching sensitivity and the importance of communication. during the first few days of class, it is extremely important that expectations and the “rules of the game” be defined. the best approach is to have the students work together in small groups to generate the guidelines for teamwork (see lesson: creating collaborative classroom guidelines on page xvi). as a guidepost for identifying the elements for successful group interactions, we suggest reviewing the “ten guidelines for students doing group work in mathematics” written by anne e. brown for the clume project (http://www.uwplatt.edu/clume/tenguide.htm). brown developed these guidelines after viewing the video and audio tapes of more than a dozen group sessions of her students. this list reflects the apparent actions critical to the success or failure of the group. in summary, the guidelines state the following: 1. groups should be formed quickly and members of the group should sit together , facing each other , and get to work quickly. members should call each other by first name. members should not engage in “off-task” discussion. everyone should be encouraged to participate. 2. all instructions should be read aloud so that everyone is aware of the expectations of the assignment. a1t1_fm_v1.qxd 4/11/08 10:30 am page x 2008 text sampler page 10ollaborative classroom xi 2008 carnegie learning, inc. collaborative classroom 3. members of the group should listen to each other and not interrupt. comments or questions should be acknowledged and responded to by other group members. 4. members of the group should not accept being confused. if a member of the group does not understand the information that is presented, this person should ask someone to paraphrase or re-phrase what was said. 5. members of the groups should ask for clarification if a word is used in a way that is confusing. 6. the members of the group should work together on the same problem and check for agreement frequently. 7. members of the group should explain their reasoning by “thinking out loud” and ask others to do the same. this helps everyone to relate the information being presented to what they already know. 8. members of the group should monitor the group’s progress and be aware of time constraints so that all members of the group meet the goals of the assignment. 9. if the group gets stuck, the members of the group should review and summarize what they have done so far . the group can then ask for questions to find errors or missing connections to help the group’s work to proceed. 10. members of the group should engage in questioning, the engine that drives mathematical investigation. group work in the collaborative classroom if we expect students to work well in groups, they will need to understand what it means to learn collaboratively and how it will benefit them. a good description of collaborative learning used by many of our teachers is: collaborative learning is a process in which each individual contributes personal knowledge and skill with the intent of improving his or her learning accomplishments along with those of others. students should be aware that one of the most important goals of collaborative learning is to create a “community of learners.” they should understand that the community will grow and thrive only if all members of the group are active participants. students must also understand that their role in the classroom will be different than what they may have experienced in other classes and so will the teacher’s role! you will want to introduce the features of a collaborative classroom to your students. important characteristics of the collaborative classroom include: shared responsibility choice discussion about how we learn from what is right as well as what is wrong working in groups, whether as an entire class or as several small groups a1t1_fm_v1.qxd 4/11/08 10:30 am page xi 2008 text sampler page 11ii collaborative classroom 2008 carnegie learning, inc. collaborative classroom finally, you will want students to understand the goals and expectations of a collaborative classroom: students learn collaboratively to gain greater individual proficiency. groups “sink or swim” together . everyone suggests, questions, and encourages. group members are responsible for each other’s learning. all group members bring valued talents and information to the task at hand. getting started in the collaborative classroom when problems and investigations in the text require that students work in groups, you will want to structure the groups. when problems and investigations in the text require that students work individually, it is possible to maintain a collaborative classroom where students are free to communicate with each other and to share information. to form groups initially, you may want to set arbitrary groups and make changes as you observe students. one suggestion for structuring groups is to think about having two types of groups, long-term groups and short-term groups. the long-term groups, or home groups, stay together for the entire school year and sit together in class. long-term groups enable students to build trust and confidence and to learn how to negotiate with each other to derive success. on the other hand, the short-term groups are randomly assigned for specific tasks. short-term groups allow students to develop the ability to work with many different people. clearly, how you arrange the groups will depend on how to best meet your students’ needs. most importantly, you want to make sure that students are respectful of one another at all times. the success of the group depends on cooperation, which can be achieved only if students accept one another and value the contributions of others. if you have students who do not want to work in groups, do not force the issue. allow those students to work alone. it is important that the student who is working alone understands that the teacher is not a member of his or her group. after these students find that they cannot talk with others and that those who are sharing information are progressing more easily, they will naturally gravitate to a group. you want to structure the success of the group experience, so it is important to use guidelines and timelines. although you will want the students to come up with the operating guidelines, timelines are probably better left to you to determine. after the groups are formed, you may want to have one person from each group be designated as a facilitator . some responsibilities of the facilitator include: obtaining and returning all materials communicating information from the teacher to the group handing in the completed assignments for the group a1t1_fm_v1.qxd 4/11/08 10:30 am page xii 2008 text sampler page 12ollaborative classroom xiii 2008 carnegie learning, inc. collaborative classroom success while working collaboratively depends upon every group member working on every part of the problem, so you may find that you do not want to assign roles such as recorder or reader to group members. students working together should generate noise and movement in the room. some have defined this attribute as “controlled chaos.” to ensure that the group work remains in control, you will want to monitor group interactions and check for understanding of the task at hand. you may also want to ask students to complete parts of the problem or investigation, stop and discuss the work done, summarize the main points of the task, and then continue. this works well when the problem or investigation is lengthy. because groups will work at different paces, you might want to prepare some additional tasks or extensions of the problem or investigation for those groups who finish quickly. facilitating groups in the collaborative classroom facilitating the group process is critical. as noted earlier , you should only answer a question posed by the group rather than by individual students. you may also restrict the number of questions that a group can ask, being generous the first few times that students work in groups. when a group asks questions, answer by redirecting with guiding questions such as: what does your group think? how did you arrive at that answer? how does this relate to past activities? what work have you done so far? what do you know about the problem? what do you need to figure out? what materials might help you to figure this out? are there other parts of the problem that you can do first? other tips to consider as you manage your collaborative classroom include: provide additional instruction to those struggling with a task. listen carefully and value diversity of thought that often provides instructional opportunities. balance learning with working effectively. remember that no one is on task 100% of the time. deal with conflict constructively. ask students to sign-off on other group members’ papers to acknowledge that everyone understands the group’s results. a1t1_fm_v1.qxd 4/11/08 10:30 am page xiii 2008 text sampler page 13iv collaborative classroom 2008 carnegie learning, inc. collaborative classroom holding the groups accountable for an end product, such as a presentation, will add further value to the learning activity. as you have surely discovered, when you truly understand a concept or idea yourself, then you are able to explain that concept or idea to someone else. presentations and discussions in the collaborative classroom to successfully close or wrap-up a problem or investigation with a presentation and discussion, students must know exactly what you expect from them. you should also make sure that students know that you will hold the entire group accountable for the presentation. (this helps to ensure that students will hold each other accountable.) some suggestions for facilitating the presentation process include: choose presenters in a group to ensure that all students have the opportunity to present. require that students defend and talk about their solutions. hold all students accountable by asking questions of group members who are not presenting. ask presenters to make connections and generalizations and extend concepts. allow groups time to process feedback and to celebrate their achievements. to bring closure to the group work and presentation process, engage students in discussion or have them keep learning journals. some suggestions for summary wrap-up questions include: what was something that you learned from this problem? what were the mathematical concepts that you applied in solving this problem? about what concepts do you still have questions? what are three things that your group did well? what is at least one thing that your group could do even better the next time? a1t1_fm_v1.qxd 4/11/08 10:30 am page xiv 2008 text sampler page 14ollaborative classroom xv 2008 carnegie learning, inc. collaborative classroom checklist of t eacher-directed and learner-centered classrooms to understand where you are in the transition process from creating a teacher-directed classroom to creating a learner-centered classroom, you may use the criteria below to evaluate your classroom (courtesy of jacquelyn snyder , jan sinopoli, and vince vernachhio, pittsburgh public schools). use your initial evaluation as a baseline measure and check yourself at regular intervals throughout the school year . teacher-directed classroom learner-centered classroom the teacher directs all classroom activity. the teacher facilitates classroom activity. each activity is dependent on the teacher . most activities require only guidance by the teacher . the teacher is in the front of the room instructing the entire class using the blackboard or over-head most of the time. the teacher walks around the classroom during all activities, watching and listening to student-to-student discourse. the teacher models examples of the lesson objective and directs students to practice similar problems found in the text or on handouts designed by the teacher . the teacher monitors the students to keep them on task, while the students actively work together on an activity. students are seated in rows, working as a class with the teacher at the front of the class or working independently. the students are typically paired or grouped to work together while the teacher facilitates the process. the teacher presents the material while students watch and take notes. the teacher systematically brings the class together on several occasions, assuring that the mathematics of the lesson is understood. the students work independently as the teacher tries to help each student individually. students are required to make presentations, explaining their progress within the activity. the teacher completely answers the problem for the student when he or she is having difficulty. if a student is having difficulty understanding something, even after consulting with his or her group members, the teacher asks the group leading questions to guide them to the desired outcome. the teacher does the thinking and the work. the students do the thinking and the work. the teacher asks low-level or fill-in-the-blank types of questions that can be answered with a single number or in a word or two. the teacher asks thought-provoking questions that require students to explain their thinking and processes. the majority of classroom discourse is teacher-to-student discourse. the majority of discourse is student-to-student discourse. the teacher encourages students to memorize rules, procedures, and formulas. the teacher encourages students to construct knowledge. prior knowledge is assessed as new concepts emerge. a1t1_fm_v1.qxd 4/11/08 10:30 am page xv 2008 text sampler page 15vi collaborative classroom 2008 carnegie learning, inc. collaborative classroom lesson: creating collaborative classroom guidelines preparing for the lesson arrange the class seats in groups of three or four such that students face each other . position the desks in such a way that students need only do a half turn of their heads if you call their attention to the front of the room. give poster boards to each group. give colorful markers to each group. expected student growth students will gain experience in working cooperatively, listening and respecting the ideas of others, and coming to a consensus regarding the final product. students will learn how to share power with the teacher . initiating the activity ask students if they have ever worked in groups. ask students to think about good and not-so-good group experiences. have students make lists of things that happen in groups or things that they think should happen in groups to have a group work more productively to complete a task. direct students to develop social guidelines for group work in class. the guidelines should be phrased positively and refer to observable behavior . lists of guidelines should not be too long. facilitating the activity monitor student behavior . offer assistance only if necessary. for instance, students may be making their lists too long. interrupt the process to reinforce cooperative skills or to provide directions. student presentations have all groups present their guidelines. have students determine which guidelines are similar and record those. have students look at the remainder of the guidelines and determine which should be included in the list of guidelines. students should be able to justify their choices. as all students will be using these guidelines, there should be consensus on the final list. lesson closure indicate that the final list will be generated and every member in the class must agree by signing off on the list. by doing so, students have agreed to honor the list of guidelines and will be held accountable. indicate that groups not adhering to the guidelines may have their group grades reduced. a1t1_fm_v1.qxd 4/11/08 10:30 am page xvi 2008 text sampler page 16ollaborative classroom 1 2008 carnegie learning, inc. collaborative classroom presentation rubric the rubric below can be used to help you score group presentations to the entire class. the presentation scores, which range from 1 to 5, are detailed in the rubric. it is a good idea to copy the presentation rubric and distribute it to the class so that students understand how they are scored. 5 you earn a 5 for your presentation if your presentation is nearly perfect. your mathematics must be correct with only a very minor flaw (not having to do with the main idea of the problem). your public speaking skills must also be perfect or quite close to perfect. you must look at your audience. your must present yourself well and not make distracting gestures or hand motions during the presentation. your rate of speech must be neither too fast nor too slow. 4 you earn a 4 for your presentation if you miss one thing within the mathematical content of your presentation. or you earn a 4 for your presentation if there is one thing that you do not do very well within the public speaking part of the presentation. 3 you earn a 3 for your presentation if you can complete the problem, but your public speaking skills are poor . this score means that you do not make eye contact, you speak inaudibly, your mumble your words, etc. or you earn a 3 for your presentation if you have some content knowledge and make one major error , as well as omit one of the important aspects of good public speaking. 2 you earn a 2 for your presentation if you stand up for your presentation but really have very little content knowledge. this score means that you are unable to complete the problem and your speaking skills are poor . 1 you earn a 1 for your presentation for being willing to stand up and try to present. 0 you earn a 0 for your presentation if you refuse to stand up and try to present. a1t1_fm_v1.qxd 4/11/08 10:30 am page 1 2008 text sampler page 17ourse description – bridge to algebra carnegie learning tm bridge to algebra is designed as the course taken immediately prior to an algebra i course. it can be implemented with students who lack the prerequisites necessary for success in algebra i as well as advanced middle school students. the first part of bridge to algebra focuses heavily on numeracy. students work with multiple representations such as models and number lines to develop a strong conceptual understanding of fractions, decimals, and percents. students use that conceptual knowledge to develop an understanding of algorithms used to operate on and convert between various numbers. students are also introduced to ratios and proportions, signed numbers, exponents, roots, and absolute value. the second part of bridge to algebra focuses on algebra. students use their intuitive understanding of linear relationships to detect and describe linear patterns using graphs, tables, and equations. students solve simple one- and two-step linear equations and begin to develop an understanding of slope as a rate of change. the third part of bridge to algebra focuses on select topics in geometry, probability, and statistics. students are introduced to geometric topics including angle relationships, similarity, area and perimeter, volume and surface area, and the pythagorean theorem. students find simple and compound probabilities. students explore measures of central tendency and ways of representing data visually. 2008 text sampler page 18ridge to algebra text setv contents 2008 carnegie learning, inc. contents contents 1 number sense and algebraic thinking • p. 2 1.1 money, money, who gets the money? introduction to picture algebra • p. 5 1.2 collection connection factors and multiples • p. 11 1.3 dogs and buns least common multiple • p. 15 1.4 kings and mathematicians prime and composite numbers • p. 19 1.5 i scream for ice cream prime factorization • p. 23 1.6 powers that be powers and exponents • p. 27 1.7 beads and baubles greatest common factor • p. 29 fractions • p. 36 2.1 comic strips dividing a whole into fractional parts • p. 39 2.2 dividing quesadillas dividing more than one whole into parts • p. 45 2.3 no “i” in team dividing groups into fractional parts • p. 49 2.4 fair share of pizza equivalent fractions • p. 53 2.5 when twelfths are eighths simplifying fractions • p. 57 2.6 when bigger means smaller comparing and ordering fractions • p. 63 2 bas1fm00.qxd 8/31/07 3:11 pm page iv 2008 text sampler page 20ontents v 2008 carnegie learning, inc. operations with fractions and mixed numbers • p. 70 3.1 who gets what? adding and subtracting fractions with like denominators • p. 73 3.2 old-fashioned goodies adding and subtracting fractions with unlike denominators • p. 77 3.3 fun and games improper fractions and mixed numbers • p. 81 3.4 parts of parts multiplying fractions • p. 85 3.5 parts in a part dividing fractions • p. 89 3.6 all that glitters adding and subtracting mixed numbers • p. 93 3.7 project display multiplying and dividing mixed numbers • p. 97 3.8 carpenter , baker , mechanic, chef working with customary units • p. 101 decimals • p. 108 4.1 cents sense decimals as special fractions • p. 111 4.2 what’s in a place? place value and expanded form • p. 115 4.3 my dog is bigger than your dog decimals as fractions: comparing and rounding decimals • p. 119 4.4 making change and changing hours adding and subtracting decimals • p. 123 4.5 rules make the world go round multiplying decimals • p. 127 4.6 the better buy dividing decimals • p. 129 4.7 bonjour! working with metric units • p. 133 3 4 contents bas1fm00.qxd 8/31/07 3:11 pm page v 2008 text sampler page 21i contents 2008 carnegie learning, inc. contents ratio and proportion • p. 142 5.1 heard it and read it ratios and fractions • p. 145 5.2 equal or not, that is the question writing and solving proportions • p. 149 5.3 the survey says using ratios and rates • p. 155 5.4 who’s got game? using proportions to solve problems • p. 159 percents • p. 166 6.1 one in a hundred percents • p. 169 6.2 brain waves making sense of percents • p. 173 6.3 commissions, taxes, and tips finding the percent of a number • p. 177 6.4 find it on the fifth floor finding one whole, or 100% • p. 181 6.5 it’s your money finding percents given two numbers • p. 185 6.6 so you want to buy a car percent increase and percent decrease • p. 189 integers • p. 196 7.1 i love new york negative numbers in the real world • p. 199 7.2 going up? adding integers • p. 203 7.3 test scores, grades, and more subtracting integers • p. 207 7.4 checks and balances multiplying and dividing integers • p. 211 7.5 weight of a penny absolute value and additive inverse • p. 215 7.6 exploring the moon powers of 10 • p. 219 7.7 expanding our perspective scientific notation • p. 223 5 6 7 bas1fm00.qxd 8/31/07 3:11 pm page vi 2008 text sampler page 22ontents vii 2008 carnegie learning, inc. contents algebraic problem solving • p. 228 8.1 life in a small town picture algebra • p. 231 8.2 computer games, cds, and dvds writing, evaluating, and simplifying expressions • p. 237 8.3 selling cars solving one-step equations • p. 241 8.4 a park ranger’s work is never done solving two-step equations • p. 245 8.5 where’s the point? plotting points in the coordinate plane • p. 251 8.6 get growing! using tables and graphs • p. 255 8.7 saving energy solving problems using multiple representations • p. 261 geometric figures and their properties • p. 270 9.1 figuring all of the angles angles and angle pairs • p. 273 9.2 a collection of triangles classifying triangles • p. 279 9.3 the signs are everywhere quadrilaterals and other polygons • p. 283 9.4 how does your garden grow? similar polygons • p. 287 9.5 shadows and mirrors indirect measurement • p. 291 9.6 a geometry game congruent polygons • p. 295 8 9 bas1fm00.qxd 8/31/07 3:11 pm page vii 2008 text sampler page 23iii contents 2008 carnegie learning, inc. contents area and the pythagorean theorem • p. 302 10.1 all skate! perimeter and area • p. 305 10.2 round food around the world circumference and area of a circle • p. 311 10.3 city planning areas of parallelograms, triangles, trapezoids, and composite figures • p. 315 10.4 sports fair and square squares and square roots • p. 321 10.5 are you sure it’s square? the pythagorean theorem • p. 325 10.6 a week at summer camp using the pythagorean theorem • p. 329 probability and statistics • p. 338 11.1 sometimes you’re just rained out finding simple probabilities • p. 341 11.2 socks and marbles finding probabilities of compound events • p. 345 11.3 what do you want to be? mean, median, mode, and range • p. 351 11.4 get the message? histograms • p. 357 11.5 go for the gold! stem-and-leaf plots • p. 363 11.6 all about roller coasters box-and-whisker plots • p. 367 11.7 what’s your favorite flavor? circle graphs • p. 371 10 11 bas1fm00.qxd 8/31/07 3:11 pm page viii 2008 text sampler page 24ontents ix 2008 carnegie learning, inc. contents volume and surface area • p. 378 12.1 your friendly neighborhood grocer three-dimensional figures • p. 381 12.2 carnegie candy company volumes and surface areas of prisms • p. 385 12.3 the playground olympics volumes and surface areas of cylinders • p. 391 12.4 the rainforest pyramid volumes of pyramids and cones • p. 395 12.5 what on earth? volumes and surface areas of spheres • p. 401 12.6 engineers and architects nets and views • p. 405 12.7 double take similar solids • p. 409 linear functions • p. 416 13.1 running a tree farm relations and functions • p. 419 13.2 scaling a cliff linear functions • p. 423 13.3 biking along slope and rates of change • p. 427 13.4 let’s have a pool party! finding slope and y-intercepts • p. 435 13.5 what’s for lunch? using slope and intercepts to graph lines • p. 441 13.6 healthy relationships finding lines of best fit • p. 449 12 13 bas1fm00.qxd 8/31/07 3:11 pm page ix 2008 text sampler page 25contents 200 carnegie learning, inc. contents number systems • p. 458 14.1 is it a bird or a plane? rational numbers • p. 461 14.2 how many times? powers of rational numbers • p. 467 14.3 sew what? irrational numbers • p. 471 14.4 worth 1000 words real numbers and their properties • p. 475 14.5 the house that math built the distributive property • p. 481 transformations • p. 488 15.1 worms and ants graphing in four quadrants • p. 491 15.2 maps and models scale drawings and scale models • p. 497 15.3 designer mathematics sliding and spinning • p. 503 15.4 secret codes flipping, stretching, and shrinking • p. 509 15.5 a stitch in time multiple transformations • p. 515 glossary • p. g1 index • p. i1 14 15 bas1fm00.qxd 8/31/07 3:11 pm page x 8 2008 text sampler page 262008 carnegie learning, inc. bridge to algebra student text bas1fm00.qxd 8/31/07 3:11 pm page i 2008 text sampler page 276 chapter 2 • fractions 2008 carnegie learning, inc. 2 fraction • p. 40 numerator • p. 40 denominator • p. 40 reasonable solution • p. 48 equivalent fractions • p. 54 equation • p. 54 simplest form • p. 58 simplest terms • p. 58 completely simplified • p. 58 least common denominator • p. 66 less than • p. 66 greater than • p. 66 key terms looking ahead to chapter 2 focus in chapter 2, you will work with fractions. you will use fractions to represent portions of a whole and divide more than one whole into parts. you will also find equivalent fractions, simplify fractions, and compare and order fractions. chapter warm-up answer these questions to help you review skills that you will need in chapter 2. find the product or quotient. 1. 5 21 2. 124 31 3. 87 29 find the least common multiple of the pair of numbers. 4. 6, 30 5. 9, 15 6. 18, 24 find the greatest common factor of the pair of numbers. 7. 51, 27 8. 48, 64 9. 78, 90 read the problem scenario below. sitara, cecilia, and kym have a total of 80 quarters. sitara has 20 more quarters than cecilia. kym has 10 more quarters than sitara. 10. how many quarters does sitara have? 11. how many quarters does cecilia have? bas10200.qxd 8/31/07 3:37 pm page 36 2008 text sampler page 28hapter 2 • fractions 37 2008 carnegie learning, inc. 2 fractions 2.1 comic strips dividing a whole into fractional parts • p. 39 2.2 dividing quesadillas dividing more than one whole into parts • p. 45 2.3 no “i” in team dividing groups into fractional parts • p. 49 2.4 fair share of pizza equivalent fractions • p. 53 2.5 when twelfths are eighths simplifying fractions • p. 57 2.6 when bigger means smaller comparing and ordering fractions • p. 63 2 soccer , played with two teams of 11 players, is a popular sport in many parts of the world, particularly in europe, latin america, and africa. in lesson 2.3, you will answer questions about soccer teams in a local neighborhood sports organization. bas10200.qxd 8/31/07 3:37 pm page 37 2008 text sampler page 298 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 38 2008 text sampler page 302008 carnegie learning, inc. 2 lesson 2.1 • dividing a whole into fractional parts 39 comic strips dividing a whole into fractional parts objectives in this lesson, you will: • use fractions to represent parts of a whole. key terms • fraction • numerator • denominator problem 1 comic strip preparation you decide to create a comic strip for your school’s newspaper . to do this, you cut a strip of paper that is a little narrower than the width of a newspaper page. the strip represents one whole comic. a. for your first comic, you want to have two frames. your teacher will provide you with a strip of paper that represents one whole comic. work with your partner to divide the strip into two parts of equal size by folding the strip like the one shown below. do not measure the strip. note: throughout this lesson, always fold the strip as shown above so that the fold decreases the length of the longest side. b. write a complete sentence that describes how you divided the strip into two equal parts. c. how can you be sure that you have two parts that are exactly the same size? use complete sentences to explain. 2.1 bas10200.qxd 8/31/07 3:37 pm page 39 2008 text sampler page 310 chapter 2 • fractions 2008 carnegie learning, inc. 2 investigate problem 1 1. math path: fractions we can use a fraction to represent one or more parts of a whole. a fraction is a number of the form where a is the numerator and b is the denominator . the denominator tells us how many equal parts the whole is divided into and the numerator tells us how many of these parts we have. the denominator of a fraction cannot be 0. write a fraction that represents one frame of the strip. 2. what is the denominator of the fraction from question 1? use complete sentences to explain what the denominator represents. 3. what is the numerator of the fraction from question 1? use complete sentences to explain what the numerator represents. 4. label each part of the strip with this fraction. a b problem 2 comic strips with equal parts a. fold a new strip of paper . without opening up the strip, fold the strip again. how many frames would be in your comic if you used this strip? how many equal parts do you have? label each of the parts with the appropriate fraction. b. repeat the process in part (a) with a new strip of paper , but this time fold the strip a total of three times. how many frames would be in your comic if you used this strip? how many equal parts do you have? label each of the parts with the appropriate fraction. c. repeat the process in part (b) with a new strip of paper , but this time fold the strip a total of four times. how many equal parts has this strip been divided into? could you use this strip for frames of a comic? use complete sentences to explain why or why not. label each of the parts with the appropriate fraction. bas10200.qxd 8/31/07 3:37 pm page 40 2008 text sampler page 32esson 2.1 • dividing a whole into fractional parts 41 2008 carnegie learning, inc. 2 problem 3 more strips with equal parts a. divide a strip into exactly three equal parts by folding. is this more or less difficult than dividing a strip into two equal parts? explain your answer using complete sentences. label each part of the strip with a fraction. b. divide another strip into three equally sized parts by folding. then divide each of these parts into two equal parts by folding. label each part of the strip with a fraction. c. take a third strip and repeat the procedure you used in part (b). then divide each of the parts of the strip into two equal parts by folding. label each part of the strip with an appropriate fraction. investigate problem 2 1. how difficult was the process in problem 2? use complete sentences to describe the fractions that you can find using this process. 2. arrange your strips in a column so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. if you folded carefully, you will notice that some of the folds line up with each other . use your fraction strips to complete each statement below. it takes _____ of the parts labeled as to make up one of the parts labeled as . it takes _____ of the parts labeled as to make up two of the parts labeled as . it takes _____ of the parts labeled as to make up three of the parts labeled as . 3. write two other sentences similar to those in question 2 relating the parts of your fraction strips. 1 8 1 16 1 4 1 8 1 2 1 4 bas10200.qxd 8/31/07 3:37 pm page 41 2008 text sampler page 332 chapter 2 • fractions 2008 carnegie learning, inc. 2 investigate problem 3 1. you have created fraction strips for many common fractions. you can create three additional strips that are useful. divide a strip into exactly five equal parts by folding. label each part of the strip with a fraction. divide another strip into exactly five equal parts by folding. then divide each of these parts into two equal parts by folding. label each part of the strip with a fraction. divide a third strip and repeat the procedure you used in the previous step. then divide each of the parts of the strip into two equal parts by folding. label each part of the strip with a fraction. 2. arrange your strips from question 1 in a column so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. write as many sentences as you can that relate the sizes of your fraction pieces. we will be using these fraction strips throughout this chapter and throughout the course, so be sure to keep them. 3. circles, squares, and rectangles can also be used to represent fractions. represent each fraction as indicated. use a circle to represent . use a rectangle to represent . use a square to represent . use a circle to represent . 5 8 1 3 6 7 3 4 bas10200.qxd 8/31/07 3:37 pm page 42 2008 text sampler page 342008 carnegie learning, inc. 2 investigate problem 3 use a square to represent . use a square to represent . use a rectangle to represent . use a square to represent . use a circle to represent . use a circle to represent . 4. which fractions in question 3 were more difficult to represent accurately? which were easier? use complete sentences to explain your reasoning. 11 12 5 7 4 9 7 8 10 11 5 12 lesson 2.1 • dividing a whole into fractional parts 43 bas10200.qxd 8/31/07 3:37 pm page 43 2008 text sampler page 354 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 44 2008 text sampler page 36ridge to algebra teacher’s implementation guide volume 1 cl_te_fm_vol1_i_vi_5.qxd 8/31/07 12:18 pm page i 2008 carnegie learning, inc. 2008 text sampler page 3736 chapter 2 • fractions fraction • p. 40 numerator • p. 40 denominator • p. 40 reasonable solution • p. 48 equivalent fractions • p. 54 equation • p. 54 simplest form • p. 58 simplest terms • p. 58 completely simplified • p. 58 least common denominator • p. 66 less than • p. 66 greater than • p. 66 key terms looking ahead to chapter 2 focus in chapter 2, you will work with fractions. you will use fractions to represent portions of a whole and divide more than one whole into parts. you will also find equivalent fractions, simplify fractions, and compare and order fractions. chapter warm-up answer these questions to help you review skills that you will need in chapter 2. find the product or quotient. 1. 5 21 2. 124 31 3. 87 29 find the least common multiple of the pair of numbers. 4. 6, 30 5. 9, 15 6. 18, 24 find the greatest common factor of the pair of numbers. 7. 51, 27 8. 48, 64 9. 78, 90 read the problem scenario below. sitara, cecilia, and kym have a total of 80 quarters. sitara has 20 more quarters than cecilia. kym has 10 more quarters than sitara. 10. how many quarters does sitara have? 11. how many quarters does cecilia have? 105 4 3 30 45 72 3 16 6 30 quarters 10 quarters cl_te_ch02.qxd 8/31/07 12:50 pm page 36 2008 carnegie learning, inc. 2008 text sampler page 38fractions 37 fractions 2.1 comic strips dividing a whole into fractional parts • p. 39 2.2 dividing quesadillas dividing more than one whole into parts • p. 45 2.3 no “i” in team dividing groups into fractional parts • p. 49 2.4 fair share of pizza equivalent fractions • p. 53 2.5 when twelfths are eighths simplifying fractions • p. 57 2.6 when bigger means smaller comparing and ordering fractions • p. 63 2 soccer , played with two teams of 11 players, is a popular sport in many parts of the world, particularly in europe, latin america, and africa. in lesson 2.3, you will answer questions about soccer teams in a local neighborhood sports organization. cl_te_ch02.qxd 8/31/07 12:50 pm page 37 2008 carnegie learning, inc. 2008 text sampler page 3938 chapter 2 • fractions cl_te_ch02.qxd 8/31/07 12:50 pm page 38 2008 carnegie learning, inc. 2008 text sampler page 40esson 2.1 • comic strips: dividing a whole into fractional parts 39a 2 comic strips dividing a whole into fractional parts 2.1 objectives in this lesson, students will • use fractions to represent parts of a whole. key terms • fraction • numerator • denominator materials • strips of paper to represent a newspaper comic strip, 9 per student (note: prepare strips before class.) nctm content standards number and operations standards grades 6-8 expectations • work flexibly with fractions, decimals, and percents to solve problems • understand the meaning and effects of arith-metic operations with fractions, decimals, and integers lesson overview within the context of this lesson, students will be asked to • use fractions to represent parts of a whole. • label parts of a whole with fractions. • use diagrams to represent fractions. students will physically create accurate representations of fractions. the focus will be on types of fractions that may be easier to represent in this form. students review definition of and parts of a frac-tional number . students will also begin to explore equivalent fractions, which will be covered more in-depth later in this unit. also later in the unit, students will work with fractions in physical as well as abstract representations. students will need to keep the fraction strips they create during this lesson for future lessons, or you may wish to store the strips for the students. essential questions the following key questions are addressed in this section: 1. how can you use fractions to represent parts of a whole? 2. what does each digit in a fraction represent? learning by doing lesson map get ready cl_te_ch02.qxd 8/31/07 12:50 pm page 39 2008 carnegie learning, inc. 2008 text sampler page 419b chapter 2 • fractions warm-up place the following questions or an applicable subset of these questions on the board or project on an overhead projector before students enter class. students should begin working as soon as they are seated. while students are working on the warm-up exercises, you can attend to clerical tasks like tak-ing attendance or returning student work. 1. write the fraction that is represented by the shaded part in each figure. motivator start the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson concerns dividing things evenly to represent fractions. the motivating questions ask about students' experience relating physical objects using the vocabulary of fractions. students briefly dis-cuss the vocabulary and definition of fractions. we all have eaten half of a sandwich before. how did you know that you had eaten "one half"? how else could it have been cut to equal one half? what other things that you've seen can you quickly rec-ognize as being one half? what about one quarter or one third? when introducing this lesson, discuss different ways to write fractions: using words (one half), numbers ( ), pictures, etc. have students identify the different parts of the representations, such as "one" in "one half" being just part of the whole. 1 2 2 2 or 1 6 9 or 2 3 6 7 2 4 or 1 2 3 10 1 3 2 show the way cl_te_ch02.qxd 8/31/07 12:50 pm page 40 2008 carnegie learning, inc. 2008 text sampler page 42roblem 1 grouping have students work in pairs. each stu-dent should work together , but fold and label his or her own strips. ask for a volunteer to read problem 1 aloud. after a student has read the problem, ask students guiding ques-tions to make sure they understand the main task as presented in part a. have a student restate the problem in his or her own words. guiding questions • how can you divide something into equal parts without measuring it? • what do you think is the most accurate way of dividing something? • why should the parts be equal in size? • how would you represent parts that are not equal in size? common student errors parts b or c, tell students that they may not have to use normal math vocabulary in their sentences, but that their answer for b may still include number words. if they are still having trouble expressing this simplistic task, help them focus on the normal word to describe two equal parts (half). explore together problem 1 comic strip preparation you decide to create a comic strip for your school’s newspaper . to do this, you cut a strip of paper that is a little narrower than the width of a newspaper page. the strip represents one whole comic. a. for your first comic, you want to have two frames. your teacher will provide you with a strip of paper that represents one whole comic. work with your partner to divide the strip into two parts of equal size by folding the strip like the one shown below. do not measure the strip. note: throughout this lesson, always fold the strip as shown above so that the fold decreases the length of the longest side. b. write a complete sentence that describes how you divided the strip into two equal parts. c. how can you be sure that you have two parts that are the same size? use complete sentences to explain. 1 2 i put the ends together , then folded the paper in half to get two equal parts. the halves match. when i look at either side of the folded paper , i can’t see any part of the other half behind the front half. i could measure each half with a ruler , but that will prove what i already can tell by just looking at the paper . 1 2 lesson 2.1 • comic strips: dividing a whole into fractional parts 39 2 cl_te_ch02.qxd 1/14/08 1:34 pm page 39 if students are having trouble with 2008 carnegie learning, inc. 2008 text sampler page 43nvestigate problem 1 math path ask a volunteer to read math path and the information about fractions below it. the information is a continuation of the warm-up, which is a review of basic fraction facts. this activity gets students started on creating their own fractions strips, which they will use throughout the unit. have students work in pairs to complete questions 1-4. problem 2 as in the math path, students continue to create fraction strips. common student errors students may label strips consecutively instead of individually. for example, stu-dents may write , , , instead of labeling each section . help them focus on each section on its own. 1 4 4 4 3 4 2 4 1 4 explore together investigate problem 1 1. math path: fractions we can use a fraction to represent one or more parts of a whole. a fraction is a number of the form where a is the numerator and b is the denominator . the denominator tells us how many equal parts the whole is divided into and the numerator tells us how many of these parts we have. the denominator of a fraction cannot be 0. write a fraction that represents one frame of the strip. 2. what is the denominator of the fraction from question 1? use complete sentences to explain what the denominator represents. 3. what is the numerator of the fraction from question 1? use complete sentences to explain what the numerator represents. 4. label each part of the strip with this fraction. a b problem 2 comic strips with equal parts a. fold a new strip of paper . without opening up the strip, fold the strip again. how many frames would be in your comic if you used this strip? how many equal parts do you have? label each of the parts with the appropriate fraction. b. repeat the process in part (a) with a new strip of paper , but this time fold the strip a total of three times. how many frames would be in your comic if you used this strip? how many equal parts do you have? label each of the parts with the appropriate fraction. c. repeat the process in part (b) with a new strip of paper , but this time fold the strip a total of four times. how many equal parts has this strip been divided into? could you use this strip for frames of a comic? use complete sentences to explain why or why not. label each of the parts with the appropriate fraction. 1 2 the denominator is two to represent two equal parts. the numerator is one to represent a single frame. there are 4 equal parts. each part should be labeled . 1 4 there are 8 equal parts. each part should be labeled . 1 8 there are 16 equal parts. each part should be labeled . answers will vary. 1 16 see diagram in part (a). 40 chapter 2 • fractions 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 40 2008 carnegie learning, inc. 2008 text sampler page 44nvestigate problem 2 key formative assessments • which fraction strips are easiest to compare? • what do you notice about the sec-tions of equal size? • how can you predict how many sections of one strip will equal another? problem 3 have a volunteer read problem 3. you may wish to have students suggest ways to divide strips evenly into thirds before allowing students to work in pairs to do so. common student errors students may try to fold the strips into thirds by folding each outward edge to meet in the middle. this only creates a strip with the outer sections equaling one quarter each and the inner section equaling one half. explore together problem 3 more strips with equal parts a. divide a strip into exactly three equal parts by folding. is this more or less difficult than dividing a strip into two equal parts? explain your answer using complete sentences. label each part of the strip with a fraction. b. divide another strip into three equally sized parts by folding. then divide each of these parts into two equal parts by folding. label each part of the strip with a fraction. c. take a third strip and repeat the procedure you used in part (b). then divide each of the parts of the strip into two equal parts by folding. label each part of the strip with an appropriate fraction. investigate problem 2 1. how difficult was the process in problem 2? use complete sentences to describe the fractions that you can find using this process. 2. arrange your strips in a column so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. if you folded carefully, you will notice that some of the folds line up with each other . use your fraction strips to complete each statement below. it takes _____ of the parts labeled as to make up one of the parts labeled as . it takes _____ of the parts labeled as to make up two of the parts labeled as . it takes _____ of the parts labeled as to make up three of the parts labeled as . 3. write two other sentences similar to those in question 2 relating the parts of your fraction strips. 1 8 1 16 1 4 1 8 1 2 1 4 it got harder with each fold. you could find fractions with the denominators: 2, 4, 8, 16, 32, 64, 128, etc. if you had a large enough piece of paper to fold into that many smaller sections. 2 4 6 it takes 6 of the parts labeled as to make up three of the parts labeled as . it takes 1 of the parts labeled as to make up eight of the parts labeled as . 1 2 1 8 it is more difficult to fold a strip into three parts than to fold it in two parts. 1 6 1 12 1 16 1 16 lesson 2.1 • comic strips: dividing a whole into fractional parts 41 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 41 2008 carnegie learning, inc. 2008 text sampler page 45nvestigate problem 3 grouping depending on how well the students are doing with the concepts and how well they are working together , allow pairs or individuals to complete questions 1-4. while students are working, circulate around the classroom to observe their progress. if students are having trouble with question 2, have them compare only two strips at a time. this should make it easier for them to focus on finding the largest parts. more than two strips may be too much input, making it easy to forget what they should be looking for on each. for question 3, you may want to model ways to divide each figure, such as a crosshatch over the circle for . you may also wish to have students find a template for each shape, such as a penny for the circle. tell students that the divisions do not need to be exact, but as close as they can get. don't let students spend too much time agoniz-ing over making exact divisions. 3 4 explore together investigate problem 3 1. you have created fraction strips for many common fractions. you can create three additional strips that are useful. divide a strip into exactly five equal parts by folding. label each part of the strip with a fraction. divide another strip into exactly five equal parts by folding. then divide each of these parts into two equal parts by folding. label each part of the strip with a fraction. divide a third strip and repeat the procedure you used in the previous step. then divide each of the parts of the strip into two equal parts by folding. label each part of the strip with a fraction. 2. arrange your strips from question 1 in a column so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. write as many sentences as you can that relate the sizes of your fraction pieces. we will be using these fraction strips throughout this chapter and throughout the course, so be sure to keep them. 3. circles, squares, and rectangles can also be used to represent fractions. represent each fraction as indicated. use a circle to represent . use a rectangle to represent . use a square to represent . use a circle to represent . 5 8 1 3 6 7 3 4 1 20 1 10 1 5 it takes 2 of the parts labeled as to make up 1 of the parts labeled as . it takes 2 of the parts labeled as to make up four of the parts labeled as . 1 10 1 10 1 5 1 5 42 chapter 2 • fractions 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 42 2008 carnegie learning, inc. 2008 text sampler page 46nvestigate problem 3 after pairs or individuals finish, you may wish to have students compare their answers in pairs, groups, or as a class. ask students why their answers in question 3 look alike or unlike. students will note that they may or may not have divided certain shapes the same way. ask students to express whether there's a right way or a wrong way to divide shapes. students should decide that as long as the parts are all equal, it is fine to divide shapes up different ways (such as horizontally, vertically, etc.). explore together investigate problem 3 use a square to represent . use a square to represent . use a rectangle to represent . use a square to represent . use a circle to represent . use a circle to represent . 4. which fractions in question 3 were more difficult to represent accurately? which were easier? use complete sentences to explain your reasoning. 11 12 5 7 4 9 7 8 10 11 5 12 shapes with odd-numbered divisions were generally the hardest because it was harder to make each part equal. lesson 2.1 • comic strips: dividing a whole into fractional parts 43 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 43 2008 carnegie learning, inc. 2008 text sampler page 473a chapter 2 • fractions 2 wrap-up close to close your lesson, ask students to answer the following questions. their answers should help summarize the lesson and the learning for the day. you may want to review the essential questions from the get ready and/or the key formative and guiding questions found throughout the lesson map. how can you divide something into equal parts? why should the parts be equal in size? what does the numerator in the fraction represent? what does the denominator in the fraction represent? how do you write a fraction after you've folded a strip of paper? how do you write a fraction after drawing a diagram? another alternative for closing your lesson is using the open-ended writing task provided on the next page. t ies to the cognitive tutor software students will be working in a variety of software units when they encounter this les-son. ties between text and software can be made because both software and text include modeling with algebraic expressions, graphing, and examining the relation-ship between written, numeric, graphic, and algebraic representations. for more specific information on correlations between the software and text, con-sult the software implementation guide. cl_te_ch02.qxd 8/31/07 12:50 pm page 44 2008 carnegie learning, inc. 2008 text sampler page 48esson 2.1 • comic strips: dividing a whole into fractional parts 43b 2 follow-up assignment use assignment 2.1 in the student assignments book. see the teacher’s resources and assessments book for answers. assessment see the assessments provided in the teacher’s resources and assessments book for this chapter . open-ended writing task your next comic strip in the newspaper is two lines or two strips long. how will you divide each line or strip? how much of your strip will each frame be? which would you use to have the largest frames to draw in? which would you use to have the greatest number of frames? reflections insert your reflections on the lesson as it played out in class today. what went well? what did not go as well as you would have liked? how would you like to change the lesson in order to improve the things that did not go well? how would you like to change the lesson in order to capitalize on the things that did go well? are there ways in which you feel the lesson could have been enriched for students? cl_te_ch02.qxd 8/31/07 12:50 pm page 45 2008 carnegie learning, inc. 2008 text sampler page 492008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 50 2008 text sampler page 502008 carnegie learning, inc. bridge to algebra teacher’ s resources and assessments 1_bam1_fm.qxd 3/24/08 9:01 am page i 2008 text sampler page 51ontents iii 2008 carnegie learning, inc. contents contents section 1 assessments with answers section 2 assignments with answers section 3 skills practice with answers 1_bam1_fm.qxd 3/24/08 9:01 am page iii 2008 text sampler page 522008 carnegie learning, inc. 2 chapter 2 assignments 15 assignment name ___________________________________________________ date _____________________ assignment name ___________________________________________________ date _____________________ assignment for lesson 2.1 comic strips dividing a whole into fractional parts write the fraction that is represented by the fraction model. 1. 2. 3. divide and shade each rectangle to represent the fraction. 4. 5. 6. 7. 8. divide and shade the circle to represent 3 8 . 7 12 1 8 3 5 2 3 assignment name ___________________________________________________ date _____________________ assignment name ___________________________________________________ date _____________________ assignment for lesson 2.1 1 4 2 5 5 9 bag1_te_v1.qxd 1/7/08 10:34 am page 8 2008 text sampler page 532008 carnegie learning, inc. 2 chapter 2 skills practice 203 skills practice name ___________________________________________________ date _____________________ lesson 2.1 reflect & review 1. you and your friends have 18 for lunch. you buy three sandwiches for 4 each and three drinks for 1 each. assuming that the tax has already been included, do you have enough to pay for your lunch? show all your work. 2. this summer , you earned 1920 in 12 weeks. how much money did you earn each week? 3. what are the factors of 64? 4. use mental math to find the quotient practice 5. identify the numerator of the fraction 6. identify the denominator of the fraction 7. write the fraction that has a numerator of 10 and a denominator of 19. 8. write the fraction that has a denominator of 11 and a numerator of 5. complete each statement. 9. is the same as ___ s. 10. is the same as ___ s. 11. is the same as ___ s. represent each fraction by drawing the specified figure. 12. of a circle 13. of a rectangle 14. of a square 7 9 1 4 4 5 1 25 1 5 1 18 4 9 1 12 1 6 15 44 . 3 8 . 1200 60. 10 19 1, 2, 4, 8, 16, 32, 64 amount earned each week 1920 12 160 total cost yes. 3(4) 3(1) 15 18; 3 44 5 11 2 8 20 bak1_te_v1.qxd 1/9/08 4:25 pm page 8 5 2008 text sampler page 54hapter 2 assessments 15a 2 teacher notes pre- and post-tests this pre-assessment is designed to be completed by students before they start work on chapter 2. students should be given about 20–30 minutes to complete this pre-assessment. the teacher can use this assessment formatively, to understand what material students have mastered going into the start of chapter 2. this post-assessment is designed to be completed at the culmination of chapter 2. this test is a parallel form to the pre-assessment. in accordance, students should be allowed 20–30 minutes to complete this post-assessment. student performance on the post- and pre-assessment can be compared in order to measure gains in student learning from chapter 2. mid-chapter test this mid-chapter assessment is designed to be completed by students when they have completed the first four lessons in chapter 2. students should be given about 20–30 minutes to complete this mid-chapter assessment. students should be encouraged to spend about 5–10 minutes on each question in this mid-chapter test. test item 1 this test item aims to get students thinking about what happens when you are asked to compare two fractions without knowing what the whole is that the fractions refer to. if students have a hard time with this problem, it may be helpful to suggest they use different amounts for the magician and the ringleader , to see what the effect is on the comparison. test item 2 this test item asks students to sort fractions into two piles. if students have a hard time with these problems, it may mean that they do not have a firm grasp on the syntax associated with fractions. for instance, you may find that students want to group statements such as and together , when in fact, these two statements are not the same. 6 4 6 4 bam102.qxd 9/5/07 12:57 pm page 13 2008 carnegie learning, inc. 2008 text sampler page 555b chapter 2 assessments 2 teacher notes page 2 teacher notes page 2 end of chapter test this is an end of chapter assessment. give students one class period to complete this assessment. test item 1 for this test item students must determine which fractions are equivalent. students must be able to reduce fractions successfully. this problem also requires students to be organized in their approach, and this should be noted to students. test item 2 students must be able to compare fractions. students may complete this task by dividing the numerator by the denominator of each fraction and then ordering the decimals on a number line. or , students may write all of the fractions with a common denominator , in which case 32 is the least common denominator . test item 3 this test item requires students to estimate the height of each bar in the graph. student answers may vary slightly because of this estimation. test item 5 for this item, students must understand that there are values between and one way to show this is to write equivalent fractions. for instance, and you can see that is between and standardized test practice this test will provide practice for standardized tests that students may take during the school year . content from all previous chapters will be included on the practice exams. to prepare students for standardized testing, allow students 15 to 20 minutes of time to complete the exam. emphasize that students should work quickly but carefully to perform well. 5 6 . 4 6 9 12 5 6 2 2 10 12 . 4 6 2 2 8 12 5 6 . 4 6 teacher notes page 2 teacher notes page 2 bam102.qxd 9/5/07 12:57 pm page 14 2008 carnegie learning, inc. 2008 text sampler page 56hapter 2 assessments 15 2 pre-test name ___________________________________________________ date _____________________ 1. what fraction of the figure is shaded? 2. write the fraction that is represented by the point on each number line. 3. use a number line to represent . 4. use a rectangle to represent 5 8 . 2 3 0 1 2 3 4 0 1 2 3 4 the shaded region represents of the figure. both points are at on the number lines. 0 1 2 sample answer: bam102.qxd 9/5/07 12:57 pm page 15 2008 carnegie learning, inc. 2008 text sampler page 576 chapter 2 assessments 2 pre-test page 2 5. you share 5 pizzas equally among 4 people. how much pizza does each person get? 6. the fractions are examples of equivalent fractions. explain what is meant by the term equivalent fraction. 7. the fractions are written in simplest form. explain what is meant by the term simplest form. 8. kenya and jada started with the same amount of colored paper . kenya used of her colored paper . jada used of hers. who has more paper now, kenya or jada? explain your reasoning. 9. maya’s book has 168 pages. she reads 24 pages each night before she goes to bed. what fraction of her book does she read each night? write your answer in simplest form. how long will it take maya to finish the book? 1 3 1 3 , 2 3 , and 3 5 3 4 , 6 8 , and 9 12 1 4 kenya used more paper than jada because one third is more than one fourth. so, jada has more paper left than kenya. maya reads of her book nightly. it will take her 7 days, or 1 week, to finish the book. 168 1 7 each person receives or pizzas. 1 1 4 5 4 two fractions that are equal when reduced to simplest form are equivalent fractions. fractions are in simplest form when the numerator and denominator have no common factors other than 1. bam102.qxd 9/5/07 12:57 pm page 16 2008 carnegie learning, inc. 2008 text sampler page 58hapter 2 assessments 17 2 10. find the missing numbers in the equivalent fractions below. 11. order the fractions from least to greatest. 1 2 , 2 3 , and 3 5 2 5 24 ? 3 8 15 ? pre-test page 3 name ___________________________________________________ date _____________________ in order from least to greatest, the fractions are 5 and the missing value is 40. the missing value is 60. 5 6 ; 8 5 5 5 ; bam102.qxd 9/5/07 12:57 pm page 17 2008 carnegie learning, inc. 2008 text sampler page 598 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 18 2008 carnegie learning, inc. 2008 text sampler page 60hapter 2 assessments 19 2 post-test name ___________________________________________________ date _____________________ 1. what fraction of the figure is shaded? 2. write the fraction that is represented by the point on the number line. 3. use a number line to represent . 4. use a rectangle to represent 2 5 . 4 5 0 1 2 the shaded region represents of the figure. 7 the point on the number line is . 5 sample answer: sample answer: 0 1 2 4 5 bam102.qxd 9/5/07 12:57 pm page 19 2008 carnegie learning, inc. 2008 text sampler page 610 chapter 2 assessments 2 5. you share 3 pizzas equally among 4 people. how much pizza does each person get? 6. the fractions are examples of equivalent fractions. explain what is meant by the term equivalent fraction. 7. the fractions are written in simplest form. explain what is meant by the term simplest form. 8. on thursday afternoon of the library computers were being used by students. what fraction of the computers were not being used? 9. sumi collects mugs. she has 5 yellow, 8 blue, and some red mugs. the 5 yellow mugs represent of her total mugs. how many mugs does sumi have? how many red mugs does she have? 1 5 6 9 2 5 , 3 7 , and 3 8 2 3 , 6 9 , and 8 12 post-test page 2 if of the computers were being used, then were not being used. 6 9 9 6 9 if 5 yellow mugs represent of the total mugs, then sumi has 25 mugs in all. because 5 are yellow and 8 are blue, then of the mugs are red. 5 5 8 5 each person receives pizza. two fractions that are equal when reduced to simplest form are equivalent fractions. fractions are in simplest form when the numerator and denominator have no common factors other than 1. bam102.qxd 9/5/07 12:57 pm page 20 2008 carnegie learning, inc. 2008 text sampler page 62hapter 2 assessments 21 2 10. find the missing numbers in the equivalent fractions below. 11. order the fractions from least to greatest. 5 8 , 1 4 , 1 3 , 1 2 , 3 8 , 7 16 , and 3 4 4 ? 16 24 2 3 ? 18 post-test page 3 name ___________________________________________________ date _____________________ to solve this problem, students can place each fraction on a number line. or , they can write the fractions with the common denominator of 48 and then place them in order . in order from least to greatest, the fractions are 4 , 1 3 , 3 8 , 7 16 , 1 2 , 5 8 , and 3 4 . the missing value is 12. the missing value is 6. 4 6 ( 4 4 ) 16 24 ; 2 3 ( 6 6 ) 12 18 ; bam102.qxd 9/5/07 12:57 pm page 21 2008 carnegie learning, inc. 2008 text sampler page 632 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 22 2008 carnegie learning, inc. 2008 text sampler page 64hapter 2 assessments 23 2 mid-chapter test name ___________________________________________________ date _____________________ 1. what is the wrong with the reasoning below? 2. sort these statements into two piles. each pile must contain matching statements. you can win half of my money at my booth. you can win more at my booth because i give away one quarter of my money to winners. 6 4 4 6 6 4 4 6 4 6 6 4 if i share 6 pizzas equally among 4 people, how much pizza will each person get? if i share 4 pizzas equally among 6 people, how much pizza will each person get? pile 1: “if i share 6 pizzas equally among 4 people ...” pile 2: “if i share 4 pizzas equally among 6 people ...” 6 6 6 6 6 6 the reasoning is wrong because it is impossible to compare two fractions without knowing the total amounts that they refer to. for example, suppose that the magician has 100. then players in his game can win 50. then suppose that the ringleader has 400. then players in his game can win 100. so, the ringleader’s statement is true. however , if both the magician and the ringleader have the same amount of money, then the players at the magician’s booth will win more money. bam102.qxd 9/5/07 12:57 pm page 23 2008 carnegie learning, inc. 2008 text sampler page 654 chapter 2 assessments 2 mid-chapter test page 2 3. the bar graph shows the heights of students in a class. use the graph to answer the following questions. what fraction of the students are 150 centimeters tall? what fraction of the students are taller than 150 centimeters? what fraction of students are in the tallest group? 148 10 8 6 4 2 0 height (centimeters) number of students 149 150 151 152 you can find the total number of students in the class by adding the number of students from each column. so, there are students in the class. there are 8 students who are 150 centimeters tall, so of the students are 150 centimeters tall. 8 5 8 9 7 there are 9 students who are 151 centimeters tall and 7 students who are 152 centimeters tall, so of the students are taller than 150 centimeters. 9 7 6 there are 7 students in the tallest group, so of the students are in the tallest group. 7 bam102.qxd 9/5/07 12:57 pm page 24 2008 carnegie learning, inc. 2008 text sampler page 66hapter 2 assessments 25 2 end of chapter test name ___________________________________________________ date _____________________ 1. nine fractions are shown below. which fractions are equivalent? 2. in the united states, hardwood flooring comes in different thicknesses. maple, beech, or birch is inch thick. both oak and pecan flooring can be inch, inch, or inch thick. put these thicknesses in order , thickest first. 1 2 25 32 3 8 25 32 16 20 4 5 8 10 15 18 20 24 7 8 10 12 12 15 5 6 5 inch inch 8 inch all of the fractions reduce to one of three values in simplest form. students should group the fractions as follows: are equivalent fractions. are equivalent fractions. is not equivalent to any of the given fractions. 7 8 5 6 5 8 and 5 8 5 and 6 bam102.qxd 9/5/07 12:57 pm page 25 2008 carnegie learning, inc. 2008 text sampler page 676 chapter 2 assessments 2 end of chapter test page 2 3. the bar graph shows the number of people who visited a musum each day during one week. about how many people visited each day? about how many people visited in total during the week? about what fraction of people visited on wednesday? about what fraction of people visited before wednesday? 4. explain why the fractions are equivalent fractions. 5. find a number that lies between 6. the fractions are in simplest form. explain what is meant by simplest form. 3 5 , 3 10 , 2 15 , and 7 20 4 6 and 5 6 . 3 5 , 6 10 , 9 15 , and 12 20 monday tuesday wednesday thursday friday 500 450 400 350 300 250 200 150 100 50 0 students must estimate for this problem. the graph shows that about 250 people visited on monday, 310 people visited on tuesday, 150 people visited on wednesday, 250 people visited on thursday, and 440 people visited on friday. about people visited during the week. 5 5 5 about of the visitors came to the museum on wednesday. 5 8 about of the visitors came to the museum before wednesday. 5 56 5 the fractions all reduce to 5 is between and 5 6 6 9 fractions are in simplest form when the numerator and denominator have no common factors other than 1. bam102.qxd 9/5/07 12:57 pm page 26 2008 carnegie learning, inc. 2008 text sampler page 68hapter 2 assessments 27 2 1. what is the value of the expression , when a. 10 b. 13 c. 15 d. 21 2. what is the prime factorization of 96? a. b. c. d. 3. which number is exactly divisible by 7? a. 27 b. 34 c. 57 d. 91 4. what is the greatest common factor (gcf) of 18 and 54? a. 3 b. 6 c. 9 d. 18 12 8 9 6 3 3 2 3 2 5 3 3? 4 3 standardized test practice name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 27 2008 carnegie learning, inc. 2008 text sampler page 698 chapter 2 assessments 2 5. simplify the expression a. b. c. 1 d. 6. the figure below represents one whole divided into equal parts. what fractional part of the figure is shaded? a. b. c. d. 7. three circles are divided into 4 equal pieces. how many pieces are there? a. 4 b. 7 c. 10 d. 12 4 7 1 4 3 4 1 3 1 8 10 9 10 6 10 4 2 3 5 2 . standardized test practice page 2 bam102.qxd 9/5/07 12:57 pm page 28 2008 carnegie learning, inc. 2008 text sampler page 70hapter 2 assessments 29 2 8. you want to divide 2 circles so that you have a total of 12 equal pieces. into how many pieces should each circle be divided? a. 6 b. 10 c. 14 d. 24 9. which fraction is equivalent to ? a. b. c. d. 10. which fraction is the simplest form of ? a. b. c. d. 7 18 8 12 4 9 2 3 72 108 12 20 6 8 6 7 2 3 3 4 standardized test practice page 3 name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 29 2008 carnegie learning, inc. 2008 text sampler page 710 chapter 2 assessments 2 11. which fraction lies between and ? a. b. c. d. 12. a strip is divided into 5 equal parts. what fraction represents 2 parts of the strip? a. b. c. d. 13. which fraction is equivalent to ? a. b. c. d. 3 7 3 5 2 3 1 2 15 35 2 10 2 7 2 5 2 3 2 8 4 5 1 4 1 2 5 8 3 8 standardized test practice page 4 bam102.qxd 9/5/07 12:57 pm page 30 2008 carnegie learning, inc. 2008 text sampler page 72hapter 2 assessments 31 2 14. which fraction is written in simplest form? a. b. c. d. 15. which set of fractions is written in order from least to greatest? a. b. c. d. 1 2 , 1 3 , 1 4 5 16 , 2 8 , 2 4 1 3 , 2 9 , 5 12 1 3 , 2 4 , 4 6 9 18 4 16 3 9 2 5 standardized test practice page 5 name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 31 2008 carnegie learning, inc. 2008 text sampler page 732 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 32 2008 carnegie learning, inc. 2008 text sampler page 74ridge to algebra homework helper 2008 carnegie learning, inc. bah1fm00.qxd 1/11/08 11:24 am page i 2008 text sampler page 752008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 76 2008 text sampler page 76chapter 2 • fractions 2008 carnegie learning, inc. 2 comic strips dividing a whole into fractional parts 2.1 students should be able to answer these questions after lesson 2.1: • how can you use fractions to represent parts of a whole? • what does each digit in a fraction represent? • how can you use pictures to represent a fraction? directions read question 1 and its solution. then, for questions 2 and 3 draw a picture to represent the fraction. finally, write the fraction to answer the question. 1. the school newspaper will print 6 comic strips on a page. you have created 5 comic strips. how much of the page is going to have your work? 2. the newspaper will print 3 different stories on each page. you wrote 1 story. how much of the page will contain your work? 3. the newspaper will have 8 ads on a page. you put in 3 ads—one ad for your tutoring business, one ad to sell candy for the spanish club, and one ad to sell your bicycle. how much of the page shows your ads? draw a rectangle to represent a page. divide the rectangle into 6 equal parts. step 2 step 1 shade the amount of the page that will contain your work. write the fraction to answer the question. five out of the 6 parts, or , of the page will be your work. 5 6 step 4 step 3 bah10200.qxd 12/17/07 3:21 pm page 8 2008 text sampler page 77ourse description – algebra i carnegie learning tm algebra i is designed as a first-year algebra course. it can be implemented with students at a variety of ability and grade levels. algebra i focuses heavily on linear functions. students use their intuitive understanding of linear relationships to detect and describe linear patterns. students are introduced to multiple representations of functions including verbal, numeric, graphical, and algebraic. students develop an understanding of the equivalence of relationships and the ability to convert between representations. students explore the graphs of linear functions and develop an understanding of slope as a rate of change. students model data with a linear function and use the regression equation to make predictions. students solve linear equations and inequalities, including absolute value equations and inequalities. students solve systems of linear equations and inequalities graphically and algebraically. algebra i also includes select topics in non-linear algebra, probability, and statistics. students are introduced to quadratic, polynomial, and exponential functions. these functions are covered in more depth in algebra ii. students find simple and compound probabilities. students explore measures of central tendency and ways of representing data visually. 2008 text sampler page 78lgebra i text setv contents 2008 carnegie learning, inc. contents contents patterns and multiple representations p. 2 1.1 designing a patio patterns and sequences p. 5 1.2 lemonade, anyone? finding the 10th term of a sequence p. 9 1.3 dinner with the stars finding the nth term of a sequence p. 13 1.4 working for the cia using a sequence to represent a problem situation p. 17 1.5 gauss’s formula finding the sum of a finite sequence p. 23 1.6 8 an hour problem using multiple representations, part 1 p. 25 1.7 the consultant problem using multiple representations, part 2 p. 31 1.8 u.s. shirts using tables, graphs, and equations, part 1 p. 37 1.9 hot shirts using tables, graphs, and equations, part 2 p. 41 1.10 comparing u.s. shirts and hot shirts comparing problem situations algebraically and graphically p. 45 proportional reasoning, percents, and direct variation p. 48 2.1 left-handed learners using samples, ratios, and proportions to make predictions p. 51 2.2 making punch ratios, rates, and mixture problems p. 57 2.3 shadows and proportions proportions and indirect measurement p. 63 2.4 tv news ratings ratios and part-to-whole relationships p. 69 2.5 women at a university ratios, part-to-part relationships, and direct variation p. 73 2.6 tipping in a restaurant using percents p. 81 2.7 taxes deducted from your paycheck percents and taxes p. 87 1 2 a1s1_fm.qxd 4/11/08 7:48 am page iv 2008 text sampler page 80ontents v 2008 carnegie learning, inc. contents solving linear equations p. 92 3.1 collecting road tolls solving one-step equations p. 95 3.2 decorating the math lab solving two-step equations p. 101 3.3 earning sales commissions using the percent equation p. 107 3.4 rent a car from go-go car rentals, wreckem rentals, and good rents rentals using two-step equations, part 1 p. 113 3.5 plastic containers using two-step equations, part 2 p. 125 3.6 brrr! it’s cold out there! integers and integer operations p. 131 3.7 shipwreck at the bottom of the sea the coordinate plane p. 139 3.8 engineering a highway using a graph of an equation p. 143 linear functions and inequalities p. 150 4.1 up, up, and away! solving and graphing inequalities in one variable p. 153 4.2 moving a sand pile relations and functions p. 159 4.3 let’s bowl! evaluating functions, function notation, domain, and range p. 165 4.4 math magic the distributive property p. 169 4.5 numbers in your everyday life real numbers and their properties p. 175 4.6 technology reporter solving more complicated equations p. 183 4.7 rules of sports solving absolute value equations and inequalities p. 187 3 4 a1s1_fm.qxd 4/11/08 7:48 am page v 2008 text sampler page 81i contents 2008 carnegie learning, inc. contents writing and graphing linear equations p. 194 5.1 widgets, dumbbells, and dumpsters multiple representations of linear functions p. 197 5.2 selling balloons finding intercepts of a graph p. 205 5.3 recycling and saving finding the slope of a line p. 211 5.4 running in a marathon slope-intercept form p. 219 5.5 saving money writing equations of lines p. 227 5.6 spending money linear and piecewise functions p. 233 5.7 the school play standard form of a linear equation p. 239 5.8 earning interest solving literal equations p. 245 lines of best fit p. 248 6.1 mia’s growing like a weed drawing the line of best fit p. 251 6.2 where do you buy your music? using lines of best fit p. 259 6.3 stroop test performing an experiment p. 267 6.4 jumping correlation p. 273 6.5 human chain: wrist experiment using technology to find a linear regression equation, part 1 p. 279 6.6 human chain: shoulder experiment using technology to find a linear regression equation, part 2 p. 285 6.7 making a quilt scatter plots and non-linear data p. 291 5 6 a1s1_fm.qxd 4/11/08 7:48 am page vi 2008 text sampler page 82ontents vii 2008 carnegie learning, inc. contents systems of equations and inequalities p. 296 7.1 making and selling markers and t -shirts using a graph to solve a linear system p. 299 7.2 time study graphs and solutions of linear systems p. 307 7.3 hiking trip using substitution to solve a linear system p. 315 7.4 basketball tournament using linear combinations to solve a linear system p. 323 7.5 finding the better paying job using the best method to solve a linear system, part 1 p. 329 7.6 world oil: supply and demand using the best method to solve a linear system, part 2 p. 333 7.7 picking the better option solving linear systems p. 339 7.8 video arcade writing and graphing an inequality in two variables p. 345 7.9 making a mosaic solving systems of linear inequalities p. 351 quadratic functions p. 356 8.1 website design introduction to quadratic functions p. 359 8.2 satellite dish parabolas p. 369 8.3 dog run comparing linear and quadratic functions p. 377 8.4 guitar strings and other things square roots and radicals p. 385 8.5 tent designing competition solving by factoring and extracting square roots p. 389 8.6 kicking a soccer ball using the quadratic formula to solve quadratic equations p. 397 8.7 pumpkin catapult using a vertical motion model p. 403 8.8 viewing the night sky using quadratic functions p. 411 7 8 a1s1_fm.qxd 4/11/08 7:48 am page vii 2008 text sampler page 83iii contents 2008 carnegie learning, inc. contents properties of exponents p. 418 9.1 the museum of natural history powers and prime factorization p. 421 9.2 bits and bytes multiplying and dividing powers p. 425 9.3 as time goes by zero and negative exponents p. 429 9.4 large and small measurements scientific notation p. 433 9.5 the beat goes on properties of powers p. 437 9.6 sailing away radicals and rational exponents p. 443 polynomial functions and rational expressions p. 450 10.1 water balloons polynomials and polynomial functions p. 453 10.2 play ball! adding and subtracting polynomials p. 459 10.3 se habla español multiplying and dividing polynomials p. 463 10.4 making stained glass multiplying binomials p. 471 10.5 suspension bridges factoring polynomials p. 477 10.6 swimming pools rational expressions p. 483 probability p. 490 11.1 your best guess introduction to probability p. 493 11.2 what’s in the bag theoretical and experimental probabilities p. 499 11.3 a brand new bag using probabilities to make predictions p. 503 11.4 fun with number cubes graphing frequencies of outcomes p. 507 11.5 going to the movies counting and permutations p. 513 9 1 0 1 1 a1s1_fm.qxd 4/11/08 7:48 am page viii 2008 text sampler page 84ontents 1 2008 carnegie learning, inc. contents 11.6 going out for pizza permutations and combinations p. 521 11.7 picking out socks independent and dependent events p. 527 11.8 probability on the shuffleboard court geometric probabilities p. 533 11.9 game design geometric probabilities and fair games p. 537 statistical analysis p. 544 12.1 taking the psat measures of central tendency p. 547 12.2 compact discs collecting and analyzing data p. 553 12.3 breakfast cereals quartiles and box-and-whisker plots p. 557 12.4 home team advantage? sample variance and standard deviation p. 563 quadratic and exponential functions and logic p. 568 13.1 solid carpentry the pythagorean theorem and its converse p. 571 13.2 location, location, location the distance and midpoint formulas p. 575 13.3 “old” mathematics completing the square and deriving the quadratic formula p. 583 13.4 learning to be a teacher vertex form of a quadratic equation p. 587 13.5 screen saver graphing by using parent functions p. 593 13.6 science fair introduction to exponential functions p. 601 13.7 money comes and money goes exponential growth and decay p. 607 13.8 camping special topic: logic p. 615 glossary p. g-1 index p. i-1 1 2 1 3 a1s1_fm.qxd 4/11/08 7:48 am page 1 2008 text sampler page 852008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 86 2008 text sampler page 862008 carnegie learning, inc. algebra i student t ext a1s1_fm.qxd 4/11/08 7:48 am page i 2008 text sampler page 87ooking ahead to chapter 1 focus in chapter 1, you will work with patterns and sequences, find terms in sequences, and use sequences and patterns to solve problems. you will also learn how to represent problem situations in different ways using tables, graphs, and algebraic equations. chapter warmup answer these questions to help you review the skills that you will need in chapter 1. find each sum, difference, product, or quotient. 1. 2. 3. 4. 5. 6. 7. 8. 9. compare the two numbers using the symbol or . 10. 951 876 11. 48.2 48.7 12. read the problem scenario below. you grow your own vegetables. to keep the rabbits from eating all your carrots, you decide to build a rectangular wire fence around your garden. how much fencing will you need to enclose each garden described below? what is the area of each garden described below? be sure to include the correct units in your answers. write your answers using a complete sentence. 13. the length of the garden is 7 feet and the width is 9 feet. 14. the length of the garden is 21 meters and the width is 14 meters. 15. the length of the garden is 65 inches and the width is 140 inches. 2 5 2 3 139.05 12.4 13(12.45) 16.95 0.5 0.78 6.05 8 8 8 342 6 54 29 124 18 7(31) 2 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 pattern p. 5 sequence p. 6 term p. 6 area p. 7 profit p. 10 power p. 11 base p. 11 exponent p. 11 order of operations p. 12 numerical expression p. 13 algebraic expression p. 13 variable p. 13 coefficient p. 14 evaluate p. 14 value p. 14 nth term p. 16 sum p. 23 labels p. 26 units p. 26 bar graph p. 27 bounds p. 28 graph p. 28 algebraic equation p. 30 solution p. 30 dependent variable p. 35 independent variable p. 35 estimation p. 42 point of intersection p. 47 key t erms a1s10100.qxd 4/11/08 7:55 am page 2 2008 text sampler page 88hapter 1 patterns and multiple representations 3 2008 carnegie learning, inc. 1 patterns and multiple representations c h a p t e r 1 tiled sidewalks can be seen around public parks, swimming pools, monuments, and other outdoor sites. the tiles provide a decorative backdrop on which visitors can walk without harming the natural surroundings. in lesson 1.1, you will continue a pattern to create a tiled sidewalk. 1.1 designing a patio patterns and sequences p. 5 1.2 lemonade, anyone? finding the 10th term of a sequence p. 9 1.3 dinner with the stars finding the nth term of a sequence p. 13 1.4 working for the cia using a sequence to represent a problem situation p. 17 1.5 gauss’s formula finding the sum of a finite sequence p. 23 1.6 8 an hour problem using multiple representations, part 1 p. 25 1.7 the consultant problem using multiple representations, part 2 p. 31 1.8 u.s. shirts using tables, graphs, and equations, part 1 p. 37 1.9 hot shirts using tables, graphs, and equations, part 2 p. 41 1.10 comparing u.s. shirts and hot shirts comparing problem situations algebraically and graphically p. 45 a1s10100.qxd 4/11/08 7:55 am page 3 2008 text sampler page 89chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 mathematical representations introduction mathematics is a human invention, developed as people encountered problems that they could not solve. for instance, when people first began to accumulate possessions, they needed to answer questions such as: how many? how many more? how many less? people responded by developing the concepts of numbers and counting. mathematics made a huge leap when people began using symbols to represent numbers. the first “numerals” were probably tally marks used to count weapons, livestock, or food. as society grew more complex, people needed to answer questions such as: who has more? how much does each person get? if there are 5 members in my family, 6 in your family, and 10 in another family, how can each person receive the same amount? during this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. the following processes can help you solve problems. discuss to understand • read the problem carefully. • what is the context of the problem? do you understand it? • what is the question that you are being asked? does it make sense? think for yourself • do i need any additional information to answer the question? • is this problem similar to some other problem that i know? • how can i represent the problem using a picture, diagram, symbols, or some other representation? work with your partner • how did you solve the problem? • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? work with your group • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? • how can we explain our solution to one another? to the class? share with the class • here is our solution and how we solved it. • we could only get this far with our solution. how can we finish? • could we have used a different strategy to solve the problem? a1s10101.qxd 4/11/08 7:56 am page 4 2008 text sampler page 90esson 1.8 using tables, graphs, and equations, part 1 37 2008 carnegie learning, inc. 1 u.s. shirts using t ables, graphs, and equations, part 1 objectives in this lesson, you will: use different methods to represent a problem situation. determine an initial value when given a final result. identify the advantages and disadvantages of using a particular representation. key t erms independent variable dependent variable scenario this past summer you were hired to work at a custom t -shirt shop, u.s. shirts. problem 1 cost analysis a. one of your responsibilities is to find the total cost of customers’ orders. the shop charges 8 per shirt plus a one-time charge of 15 to set up the t -shirt design. use complete sentences to describe the problem situation in your own words. b. how does this problem situation differ from the problem situations in lesson 1.6 and lesson 1.7? use a complete sentence in your answer . c. what is the total cost of an order for 3 shirts? use a complete sentence in your answer . what is the total cost of an order for 10 shirts? use a complete sentence in your answer . what is the total cost of an order for 100 shirts? use a complete sentence in your answer . d. use complete sentences to explain how you found the total costs. 1 .8 a1s10108.qxd 4/11/08 8:02 am page 37 2008 text sampler page 918 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 investigate problem 1 1. a customer has 50 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . a customer has 60 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . a customer has 220 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . use complete sentences to explain how you found the number of shirts that can be ordered. 2. complete the table of values for the problem situation. quantity name unit number of shirts ordered total cost shirts dollars a1s10108.qxd 4/11/08 8:02 am page 38 2008 text sampler page 92nvestigate problem 1 3. what are the variable quantities in this problem situation? assign letters to represent these quantities including each quantity’s units. use a complete sentence in your answer . 4. what are the constant quantities in this problem situation? include the units that are used to measure these quantities. use a complete sentence in your answer . 5. which variable quantity depends on the other variable quantity? 6. which of the variables from question 3 is the independent variable and which is the dependent variable? 7. use the grid below to create a graph of the data from the table in question 2. first, choose your bounds and intervals. lesson 1.8 using tables, graphs, and equations, part 1 39 2008 carnegie learning, inc. 1 variable quantity lower bound upper bound interval number of shirts total cost t ake note remember to label your graph clearly and add a title to the graph. (label) (units) (label) (units) a1s10108.qxd 4/11/08 8:02 am page 39 2008 text sampler page 930 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 investigate problem 1 8. write an algebraic equation for the problem situation. use a complete sentence in your answer . 9. in this lesson, you have represented the problem situation 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 by writing a paragraph. use complete sentences. be prepared to share your answers with the class. t ake note whenever you see the share with the class icon, your group should prepare a short presentation to share with the class that describes how you solved the problem. be prepared to ask questions during other groups’ presentations and to answer questions during your presentation. a1s10108.qxd 4/11/08 8:02 am page 40 2008 text sampler page 942008 carnegie learning, inc. algebra i t eacher’s implementation guide volume 1 a1t1_fm_v1.qxd 4/11/08 10:30 am page i 2008 text sampler page 95chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 looking ahead to chapter 1 focus in chapter 1, you will work with patterns and sequences, find terms in sequences, and use sequences and patterns to solve problems. you will also learn how to represent problem situations in different ways using tables, graphs, and algebraic equations. chapter warmup answer these questions to help you review the skills that you will need in chapter 1. find each sum, difference, product, or quotient. 1. 2. 3. 4. 5. 6. 7. 8. 9. compare the two numbers using the symbol or . 10. 951 876 11. 48.2 48.7 12. read the problem scenario below. you grow your own vegetables. to keep the rabbits from eating all your carrots, you decide to build a rectangular wire fence around your garden. how much fencing will you need to enclose each garden described below? what is the area of each garden described below? be sure to include the correct units in your answers. write your answers using a complete sentence. 13. the length of the garden is 7 feet and the width is 9 feet. 14. the length of the garden is 21 meters and the width is 14 meters. 15. the length of the garden is 65 inches and the width is 140 inches. 2 5 2 3 139.05 12.4 13(12.45) 16.95 0.5 0.78 6.05 8 8 8 342 6 54 29 124 18 7(31) pattern p. 5 sequence p. 6 term p. 6 area p. 7 profit p. 10 power p. 11 base p. 11 exponent p. 11 order of operations p. 12 numerical expression p. 13 algebraic expression p. 13 variable p. 13 coefficient p. 14 evaluate p. 14 value p. 14 nth term p. 16 sum p. 23 labels p. 26 units p. 26 bar graph p. 27 bounds p. 28 graph p. 28 algebraic equation p. 30 solution p. 30 dependent variable p. 35 independent variable p. 35 estimation p. 42 point of intersection p. 47 key t erms 217 142 25 57 512 6.83 33.9 161.85 126.65 the amount of fencing needed is 32 feet and the area is 63 square feet. the amount of fencing needed is 70 meters, and the area is 294 square meters. the amount of fencing needed is 410 inches, and the area is 9100 square inches. a1t10100.qxd 4/1/08 9:41 am page 2 2008 text sampler page 96hapter 1 patterns and multiple representations 3 2008 carnegie learning, inc. 1 patterns and multiple representations c h a p t e r 1 tiled sidewalks can be seen around public parks, swimming pools, monuments, and other outdoor sites. the tiles provide a decorative backdrop on which visitors can walk without harming the natural surroundings. in lesson 1.1, you will continue a pattern to create a tiled sidewalk. 1.1 designing a patio patterns and sequences p. 5 1.2 lemonade, anyone? finding the 10th term of a sequence p. 9 1.3 dinner with the stars finding the nth term of a sequence p. 13 1.4 working for the cia using a sequence to represent a problem situation p. 17 1.5 gauss’s formula finding the sum of a finite sequence p. 23 1.6 8 an hour problem using multiple representations, part 1 p. 25 1.7 the consultant problem using multiple representations, part 2 p. 31 1.8 u.s. shirts using tables, graphs, and equations, part 1 p. 37 1.9 hot shirts using tables, graphs, and equations, part 2 p. 41 1.10 comparing u.s. shirts and hot shirts comparing problem situations algebraically and graphically p. 45 a1t10100.qxd 4/1/08 9:41 am page 3 2008 text sampler page 97chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 mathematical representations introduction mathematics is a human invention, developed as people encountered problems that they could not solve. for instance, when people first began to accumulate possessions, they needed to answer questions such as: how many? how many more? how many less? people responded by developing the concepts of numbers and counting. mathematics made a huge leap when people began using symbols to represent numbers. the first “numerals” were probably tally marks used to count weapons, livestock, or food. as society grew more complex, people needed to answer questions such as: who has more? how much does each person get? if there are 5 members in my family, 6 in your family, and 10 in another family, how can each person receive the same amount? during this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. the following processes can help you solve problems. discuss to understand • read the problem carefully. • what is the context of the problem? do you understand it? • what is the question that you are being asked? does it make sense? think for yourself • do i need any additional information to answer the question? • is this problem similar to some other problem that i know? • how can i represent the problem using a picture, diagram, symbols, or some other representation? work with your partner • how did you solve the problem? • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? work with your group • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? • how can we explain our solution to one another? to the class? share with the class • here is our solution and how we solved it. • we could only get this far with our solution. how can we finish? • could we have used a different strategy to solve the problem? a1t10100.qxd 4/1/08 9:41 am page 4 2008 text sampler page 98esson 1.8 using tables, graphs, and equations, part 1 37a 2008 carnegie learning, inc. 1 u.s. shirts using t ables, graphs, and equations, part 1 objectives in this lesson, you will: use different methods to represent a problem situation. determine an initial value when given a final result. identify the advantages and disadvantages of using a particular representation. key t erms independent variable dependent variable nctm content standards grades 9–12 expectations algebra standards generalize patterns using explicitly defined and recursively defined functions. interpret representations of functions of two variables. use symbolic algebra to represent and explain mathematical relationships. draw reasonable conclusions about a situation being modeled. approximate and interpret rates of change from graphical and numerical data. measurement standards make decisions about units and scales that are appropriate for problem situations involving measurement. 1 .8 lesson overview within the context of this lesson, students will be asked to: represent a problem situation in a sentence, using a table of values, using an equation in two variables, and using a graph. determine an initial value for the number of shirts ordered if given the final result of the total cost of the order . write an equation with two variables. classify variables as dependent or independent. identify the advantages and disadvantages of each form of representing the problem situation: a sentence, an equation, a table of values, and a graph. essential questions the following key questions are addressed in this lesson: 1. what is a set-up fee? 2. how can you determine the initial number of shirts ordered if given the total cost of an order of shirts? 3. what is an independent variable? 4. what is a dependent variable? 5. what different representations can you use to represent a problem situation? get ready learning by doing lesson map a1t10108.qxd 4/1/08 9:49 am page 35 2008 text sampler page 997b chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 warm up place the following questions or an applicable subset of these questions on the board before students enter class. students should begin working as soon as they are seated. evaluate each expression. 1. 32 2. 27 3. 4 4. 9 5. 39 6. 364 7. 3 8. 32 9. 0 motivator begin the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson is about making custom t -shirts. the motivating questions are about working for a custom t -shirt shop. ask the students the following questions to get them interested in the lesson. suppose your club wanted to have t -shirts printed with a design that you drew. what work would have to be done by the t -shirt shop to set up your design to put onto t -shirts? would the cost for the work by the t -shirt shop to scan the design into their computer system be different if the shop was going to print 1 shirt, 10 shirts, or 100 shirts? why might a t -shirt shop charge for that service as a fee that is not based on the number of shirts made? how much money do you think a t -shirt shop should charge for the set-up service? imagine the following situation. a club decides to have t -shirts made and goes to the t -shirt shop to set up the design and to get a price for printing the shirts. before they have any shirts printed, they advertise the shirts at school. the principal and teachers feel that the statement on the shirts is offensive, so the group is not allowed to use that design. in that situation, why would the group still have to pay the set-up fee? (8 4 2)45 5(7) 3 9 6 ( 12 6 ) 3.6(100) 4 54 15 6(14) 21 7 64 (2 8) 16(3 2) 11 3(12) 4 show the way a1t10108.qxd 4/1/08 9:49 am page 36 2008 text sampler page 100esson 1.8 using tables, graphs, and equations, part 1 37 2008 carnegie learning, inc. 1 problem 1 students will calculate the cost for orders of t -shirts for various given values. this problem is intentionally mathematically related to the 8 an hour problem in lesson 1.6. the rate of change in both lessons is a constant 8, but the y-intercept was zero in lesson 1.6 and the y-intercept is 15 in this lesson. grouping ask for a student volunteer to read the scenario and problem 1 aloud. have a student restate the problem. pose the guiding questions below to verify student understanding. have students work together in small groups to complete parts (a) through (d) of problem 1. guiding questions what information is given in this problem? what is a custom t -shirt? what does the number 8 represent in this situation? what does the number 15 represent in this situation? what is a set-up fee? why would the t -shirt shop charge for the set-up? common student errors students often will incorrectly add the cost per shirt of 8 to the set-up cost of 15 to get 23 and then multiply the number of shirts by 23. if this happens, refocus the students by asking them what the set-up fee is for , and how many times they would have to pay the set-up fee. grouping call the class back together to have the students discuss and present their work for parts (a) through (d) of problem 1. scenario this past summer you were hired to work at a custom t -shirt shop, u.s. shirts. problem 1 cost analysis a. one of your responsibilities is to find the total cost of customers’ orders. the shop charges 8 per shirt plus a one-time charge of 15 to set up the t -shirt design. use complete sentences to describe the problem situation in your own words. b. how does this problem situation differ from the problem situations in lesson 1.6 and lesson 1.7? use a complete sentence in your answer . c. what is the total cost of an order for 3 shirts? use a complete sentence in your answer . what is the total cost of an order for 10 shirts? use a complete sentence in your answer . what is the total cost of an order for 100 shirts? use a complete sentence in your answer . d. use complete sentences to explain how you found the total costs. sample answer: i will calculate the total cost of orders. the total cost of an order is the cost of each shirt ordered plus a set-up fee. the cost of one shirt is 8 and the set-up fee is 15. sample answer: there is a set-up fee in this problem situation. total cost in dollars: 100(8) 15 815 an order of 100 shirts will cost 815. the total costs were found by first multiplying the number of shirts by the cost of one shirt and then adding the set-up fee to the result. total cost in dollars: 3(8) 15 39 an order of 3 shirts will cost 39. total cost in dollars: 10(8) 15 95 an order of 10 shirts will cost 95. explore together a1t10108.qxd 4/1/08 9:49 am page 37 2008 text sampler page 1018 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 investigate problem 1 1. a customer has 50 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . a customer has 60 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . a customer has 220 to spend on t -shirts. how many shirts can the customer buy? use a complete sentence in your answer . use complete sentences to explain how you found the number of shirts that can be ordered. 2. complete the table of values for the problem situation. quantity name unit number of shirts ordered total cost shirts dollars the numbers of shirts were found by first subtracting the set-up fee from the amount of money available and then dividing the result by the cost for one shirt. the decimal portion of each result is dropped, because a customer cannot receive a partial shirt. number of shirts: because the customer cannot receive a partial shirt, the customer receives 4 shirts. 50 15 8 4.375 39 47 95 215 815 975 1215 number of shirts: because the customer cannot receive a partial shirt, the customer receives 5 shirts. 60 15 8 5.625 number of shirts: because the customer cannot receive a partial shirt, the customer receives 25 shirts. 220 15 8 25.625 3 4 10 25 100 120 150 the table shows sample answers. explore together investigate problem1 students will calculate the number of shirts that can be purchased for various given amounts of money. students will create a table of values for the problem situation. grouping ask for a student volunteer to read question 1 aloud. have a student restate the problem. pose the guiding questions below to verify student understanding. have students work together in small groups to complete questions 1 through 6. guiding questions how did you find the cost for an order of t -shirts in parts (a) through (d) of problem 1? what are you asked to find in question 1? how is question 1 different from parts (a) through (d)? how can you find the number of shirts that can be purchased for a given amount of money? what is the smallest possible number of shirts that could be ordered? why might a group pay the set-up fee for a shirt design, but not buy any shirts? do you think that it is very common for a group to pay the set-up fee, but order no shirts? what is the largest reasonable number of shirts of one design that you think may be ordered? when might a person or group order such a large number of shirts of the same design? notes in question 2, students are asked to create a table of values for this problem situation. it is important that they be required to create a table of values that is reasonable for the situation and that also models examples of different-sized orders. many students will use the values of 1 through 5 for the number of t -shirts, if allowed to do so. such a limited table makes graphing and fully understanding the situation very difficult. always require that students’ tables and graphs encompass the entire situation. a1t10108.qxd 4/1/08 9:49 am page 38 2008 text sampler page 102esson 1.8 using tables, graphs, and equations, part 1 39 2008 carnegie learning, inc. 1 investigate problem 1 3. what are the variable quantities in this problem situation? assign letters to represent these quantities including each quantity’s units. use a complete sentence in your answer . 4. what are the constant quantities in this problem situation? include the units that are used to measure these quantities. use a complete sentence in your answer . 5. which variable quantity depends on the other variable quantity? 6. which of the variables from question 3 is the independent variable and which is the dependent variable? 7. use the grid below to create a graph of the data from the table in question 2. first, choose your bounds and intervals. variable quantity lower bound upper bound interval number of shirts total cost the variable quantities are the number of shirts ordered s and the total cost c in dollars. the constant quantities are the cost per shirt in dollars and the set-up fee in dollars. 0 150 10 0 1500 100 c s 10 20 30 40 50 0 60 70 80 90 100 110 150 120 130 140 100 200 300 400 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 u.s. shirts number of shirts total cost (dollars) c 8s 15 the total cost depends on the number of shirts ordered. the variable s is the independent variable and the variable c is the dependent variable. explore together investigate problem 1 students will classify the variables in this situation as dependent or independent and create a graph. grouping call the class back together to have the students discuss and present their work for questions 1 through 6. notes when discussing question 3, talk about the benefits of using meaningful variables such as s or n for the number of shirts and c or d for the cost of the shirts in dollars. some students are exposed so frequently to the variables x and y that it is more difficult for them to solve equations in geometry and physics courses with meaningful variables. later in this course, the students will have considerable practice writing, solving, and graphing equations in terms of the variables x and y. common student errors students may still struggle to classify each variable as a dependent or an independent variable. be ready to refocus students when debriefing questions 4 through 6. grouping have students work in small groups to complete questions 7 through 9. pose the guiding questions below to verify student understanding. guiding questions what bounds would be appropriate for the number of shirts? explain. what bounds would be appropriate for the total cost of the order? explain. what interval is appropriate for the number of shirts? explain. what interval is appropriate for the total cost of the order? explain. t ake note remember to label your graph clearly and to add a title to the graph. a1t10108.qxd 4/1/08 9:49 am page 39 2008 text sampler page 1030 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 investigate problem 1 8. write an algebraic equation for the problem situation. use a complete sentence in your answer . 9. in this lesson, you have represented the problem situation 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 by writing a paragraph. use complete sentences. be prepared to share your answers with the class. the equation is c 8s 15, where c represents the total cost in dollars and s represents the number of shirts ordered. sample answer: sentences allow you to understand what information you need to find. however , sentences don’t give a visual representation of a problem situation. a table gives you specific values for the problem situation. however , it does not show values between those given. a graph allows you to find different values for the problem situation and to visually see how the data in the problem situation are related. however , the values may not be exact. an equation allows you to generalize the problem situation and find any value exactly. however , an equation doesn’t give a visual representation of the problem situation. explore together investigate problem 1 students will analyze the various representations that they used to model the problem situation. grouping call the class back together to have the students discuss and present their work for questions 6 through 9. common student errors some students may not have noticed the various representations of the problem situation. you may need to call the students together and discuss question 9 if several of the groups are unable to complete the question on their own. guiding questions where in this lesson was the problem situation represented in a sentence? were there any other parts of this lesson in which the problem situation was represented in a sentence? where was this problem situation represented using a table in this lesson? where was this problem modeled using a graph in this lesson? where was this problem modeled using an equation? key formative assessments compare your graph in lesson 1.6 for the 8 an hour problem with your graph for this situation. how are the graphs similar? how are they different? what was different in the scenario of the 8 an hour problem and the u.s. shirts problem? how do the differences in the scenarios and graphs relate to each other? summarize the advantages and disadvantages for each representation used in this lesson. t ake note whenever you see the share with the class icon, your group should prepare a short presentation to share with the class that describes how you solved the problem. be prepared to ask questions during other groups’ presentations and to answer questions during your presentation. a1t10108.qxd 4/1/08 9:49 am page 40 2008 text sampler page 104esson 1.8 using tables, graphs, and equations, part 1 40a 2008 carnegie learning, inc. 1 close review all key terms and their definitions. include the terms independent variable and dependent variable. you may also want to review any other vocabulary terms that were discussed during the lesson, which may include representations and models, and for more advanced students possibly slope and y-intercept. remind the students to write the key terms and their definitions in the notes section of their notebooks. you may also want the students to include examples. ask the students to summarize the mathematics from this lesson. ask the students to construct another scenario that could be modeled by the equation of . then ask how the graph of such an equation would compare to the graph of the problem situation for the problem they just finished. finally, verify their thoughts by graphing this equation as a whole class. discuss with the students how the values for c will compare from the equation in the u.s. shirts problem and the new equation . ties to the cognitive tutor software in some cognitive tutor software units, students graph the relationship between quantities by plotting points. this helps students understand the correspondence between rows in a table and points on a graph. students should reflect on the fact that the graph of the relationship is a straight line. this is no coincidence. in these problems, the dependent quantity changes at a constant rate, and the graph of a relationship with a constant rate of change will always be a line. c 8s 12 c 8s 15 c 8s 12 wrap up assignment use the assignment for lesson 1.8 in the student assignments book. see the teacher’s resources and assessments book for answers. assessment see the assessments provided in the teacher’s resources and assessments book for chapter 1. open-ended writing t ask ask the students to compare the problem situation in the consultant’s problem in the previous lesson with the problem situation in the u.s. shirts problem. how are they similar and how are they different? how did the similarities and differences affect the different representations of the problem situations? follow up a1t10108.qxd 4/1/08 9:49 am page 41 2008 text sampler page 1050b chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 reflections insert your reflections on the lesson as it played out in class today. what went well? _______________________________________________________________________________________________ _______________________________________________________________________________________________ what did not go as well as you would have liked? _______________________________________________________________________________________________ _______________________________________________________________________________________________ how would you like to change the lesson in order to improve the things that did not go well and capitalize on the things that did go well? _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ notes a1t10108.qxd 4/1/08 9:49 am page 42 2008 text sampler page 1062008 carnegie learning, inc. algebra i t eacher’s resources and assessments a1tra_fm.qxd 5/6/08 9:42 am page i 2008 text sampler page 1072008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 108 2008 text sampler page 108ontents iii 2008 carnegie learning, inc. contents contents section 1 assessments with answers section 2 assignments with answers assessment answer keys and master copies the answer keys and master copies of the student assessments are available online in the carnegie leaning k-12 community. to access the answer keys and master copies of the student assessments, go to http://k12.carnegielearning.com and login with your k-12 password. if you are a first time user or unsure of your password, choose the new users / account help link in the login box for information on registering and/or setting up your password. contact carnegie learning customer support at 1-888-851-7094, option 3, or via email at help@carnegielearning.com for additional assistance. a1tra_fm.qxd 5/6/08 9:42 am page iii 2008 text sampler page 1092008 carnegie learning, inc. 1 chapter 1 assignments 15 u.s. shirts using tables, graphs, and equations, part 1 define each term in your own words. 1. variable quantity 2. constant quantity evaluate each algebraic expression for the value given. show your work. 3. when 4. when 5. when you want to save money for college. you have already saved 500, and you are able to save 75 each week. 6. if you continue to save money at this rate, what will your total savings be in 3 weeks? what will your total savings be in 10 weeks? what will your total savings be in 6 months? (hint: there are four weeks in one month.) 7. use a complete sentence to explain how you found the total savings in question 6. 8. if you continue to save money at this rate, how long will it take you to save 2000? how long will it take you to save 8000? how long will it take you to save 11,750? 9. use a complete sentence to explain how you found the answers to the number of weeks in question 8. r 10 1 2 r 30 m 4 10 2m s 20 8s 15 assignment name ___________________________________________________ date _____________________ assignment for lesson 1.8 sample answer: a variable quantity is a quantity that does not have a fixed value. sample answer: a constant quantity is a quantity that has a fixed value. 8(20) 15 175 10 – 2(4) 2 (10) 30 35 1 2 725; 1250; 2300 sample answer: the total savings were found by first multiplying the weekly savings rate by the number of weeks and then adding the amount already saved. 20 weeks; 100 weeks; 150 weeks sample answer: the numbers of weeks were found by first subtracting the amount already saved from the amount saved and then dividing the result by the weekly savings rate. a1g1_1_te.qxd 5/6/08 6:56 am page 10 2008 text sampler page 1102008 carnegie learning, inc. 1 16 chapter 1 assignments variable quantity lower bound upper bound interval time total savings 10. complete the table using the data from questions 6 and 8. be sure to fill in your labels and units. quantity name unit 11. use the grid below to create a line graph of the data from the table in question 10. first, choose your bounds and intervals. be sure to label your graph clearly. time total savings weeks dollars 3 725 10 1250 20 2000 24 2300 100 8000 150 11,750 0 150 10 0 15,000 1000 s t 10 20 30 40 50 0 60 70 80 90 100 110 150 120 130 140 1000 2000 3000 4000 0 5000 6000 7000 8000 9000 10,000 11,000 12,000 13,000 14,000 15,000 college savings time (weeks) total savings (dollars) a1g1_1_te.qxd 5/6/08 6:56 am page 11 2008 text sampler page 1112008 carnegie learning, inc. 1 chapter 1 assignments 17 name ___________________________________________________ date _____________________ 12. write an algebraic equation for the problem situation. use a complete sentence in your answer . sample answer: the equation s 75t 500, where s represents the total savings in dollars and t represents the time in weeks. a1g1_1_te.qxd 5/6/08 6:56 am page 12 2008 text sampler page 112hapter 1 assessments 1 2008 carnegie learning, inc. 1 pre-t est name ___________________________________________________ date _____________________ find the next two terms in each sequence. 1. 2. 2, 4, 8, 16, 32, ______ , ______ 3. use the nth term to list the first five terms of the sequence. show your work. use the sequence below to answer questions 4 through 6. 4. complete the table by filling in the number of hexagons in each term of the sequence. 5. write an expression showing the relationship between the term and the number of hexagons in that term. let n represent the term. 6. use the expression from question 5 to find the 10th term of the sequence. show your work. 7. write the power as a product. 3 4 a 5 ____________ a 4 ____________ a 3 ____________ a 2 ____________ a 1 ____________ a n 2n 4 term (n) 1 2 3 4 5 number of hexagons 64 128 2(1) 4 6 2(2) 4 8 2(3) 4 10 2(4) 4 12 2(5) 4 14 3 6 9 12 15 3n 3(10) 30 (3)(3)(3)(3) a1m101.qxd 5/2/08 3:03 pm page 1 2008 text sampler page 113chapter 1 assessments 2008 carnegie learning, inc. 1 pre-t est page 2 8. write the product as a power . (7)(7)(7)(7)(7)(7)(7)(7) 9. perform the indicated operations. show your work. 10. you and your classmates have set up a phone chain to call each other if school is can-celled due to bad weather . you call two classmates, then each of them calls two class-mates, and so on until everyone in your class has been notified. there are 31 students in your class. draw a diagram to show how each student would be reached to be notified of school cancellations. 11. find the sum of the numbers from 1 to 10. show your work. 12. write an expression for the sum of the numbers from 1 to n. 4 2 (8 3)6 7 8 46 16 (5)6 16 30 1 2 3 4 5 6 7 8 9 10 55 or 55 10(11) 2 n(n 1) 2 a1m101.qxd 5/2/08 3:03 pm page 2 2008 text sampler page 114hapter 1 assessments 3 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 13 through 16. the spanish club at your school is selling animal piñatas to raise money for a trip to mexico city. the club earns a profit of 3 on each piñata sold. the sale runs for 5 weeks. the number of piñatas sold each week are 15, 22, 8, 35, and 42. 13. make a table to show the number of piñatas sold and the profit made for each week of the sale. 14. create a bar graph to display the profit for each week of the sale. pre-t est page 3 name ___________________________________________________ date _____________________ 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 spanish club piñata sale profts proft (dollars) week 1 week 2 week 3 week 4 week 5 week number of piñatas profit piñatas dollars week 1 15 45 week 2 22 66 week 3 8 24 week 4 35 105 week 5 42 126 a1m101.qxd 5/2/08 3:03 pm page 3 2008 text sampler page 1155. create a graph to display the relationship between the number of piñatas sold and the profit. first, choose your bounds and intervals. be sure to label your graph clearly. 16. write an algebraic equation that you could use to show the profit for any number of piñatas sold. 4 chapter 1 assessments 2008 carnegie learning, inc. 1 pre-t est page 4 variable quantity lower bound upper bound interval number of piñatas profit sample answer: p 3n, where p is profit and n is the number of piñatas sold. p n 3 6 9 12 15 0 18 21 24 27 30 33 45 36 39 42 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 spanish club piñata sale profits number of piñatas profit (dollars) 0 45 3 0 150 10 a1m101.qxd 5/2/08 3:03 pm page 4 2008 text sampler page 116hapter 1 assessments 5 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 17 and 18. two airlines offer special group rates to your school’s spanish club for the trip to mexico city. the mexican air airline offers a roundtrip airfare of 250 per person. the fiesta airline offers a roundtrip airfare of 150 per person if the club agrees to pay a one-time group rate processing fee of 1000. 17. which airline offers the better deal if only nine students from the spanish club are able to fly to mexico city? show all your work and use complete sentences in your answer . 18. which airline offers the better deal if 20 students are able to fly to mexico city? show all your work and use complete sentences in your answer . pre-t est page 5 name ___________________________________________________ date _____________________ mexican air: fiesta: total cost in dollars: 250(9) 2250 150(9) 1000 2350 the total cost for nine students on mexican air is 2250, and the total cost for nine students on fiesta is 2350. therefore, mexican air offers a better deal if only nine students are able to fly to mexico city. mexican air: fiesta: total cost in dollars: 250(20) 5000 150(20) 1000 4000 the total cost for 20 students on mexican air is 5000, and the total cost for 20 students on fiesta is 4000. therefore, fiesta offers a better deal if 20 students are able to fly to mexico city. a1m101.qxd 5/2/08 3:03 pm page 5 2008 text sampler page 117chapter 1 assessments 2008 carnegie learning, inc. 1 a1m101.qxd 5/2/08 3:03 pm page 6 2008 text sampler page 118hapter 1 assessments 7 2008 carnegie learning, inc. 1 find the next two terms in each sequence. 1. 2. 4, 7, 10, 13, ______ , ______ 3. use the nth term to list the first five terms of the sequence. show your work. use the sequence below to answer questions 4 through 6. 4. complete the table by filling in the number of triangles in each term of the sequence. 5. write an expression showing the relationship between the term and the number of triangles in that term. let n represent the term. 6. use the expression from question 5 to find the 10th term of the sequence. show your work. 7. write the power as a product. 5 6 a 5 _______________ a 4 _______________ a 3 _______________ a 2 _______________ a 1 _______________ a n 3n 5 post-t est name ___________________________________________________ date _____________________ term 1 2 3 4 5 number of triangles 2 3 4 5 6 16 19 3(1) 5 8 3(2) 5 11 3(3) 5 14 3(4) 5 17 3(5) 5 20 n 1 10 1 11 (5)(5)(5)(5)(5)(5) a1m101.qxd 5/2/08 3:03 pm page 7 2008 text sampler page 119chapter 1 assessments 2008 carnegie learning, inc. 1 post-t est page 2 8. write the product as a power . (3)(3)(3)(3)(3)(3)(3)(3)(3) 9. perform the indicated operations. show your work. 10. you and your classmates have set up an email chain to notify each other if school is cancelled due to bad weather . you email three classmates, then each of them emails three classmates, and so on until everyone in your class has been notified. there are 40 students in your class. draw a diagram to show how each student would be reached to be notified of school cancellations. 11. find the sum of the numbers from 1 to 200. show your work. 12. write an expression for the sum of the numbers from 1 to n. (2 1) 3 5(2) 3 9 17 3 3 10 27 10 200(201) 2 20,100 n(n 1) 2 a1m101.qxd 5/2/08 3:03 pm page 8 2008 text sampler page 120hapter 1 assessments 9 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 13 through 16. a local ballet company is selling tickets for their upcoming performances of swan lake. the company earns a profit of 8 on each ticket they sell. the first week that tickets are on sale, they sold 30 tickets on monday, 27 on tuesday, 18 on wednesday, 6 on thursday, and 41 tickets on friday. 13. make a table to show the number of tickets sold each day during the first week and the profit made on each of those days. 14. create a bar graph to display the profit for each day of ticket sales in the first week. post-t est page 3 name ___________________________________________________ date _____________________ 25 50 75 100 0 125 150 175 200 225 250 275 300 325 350 375 profts from ticket sales for swan lake proft (dollars) monday tuesday wednesday thursday friday day day number of tickets profit tickets dollars monday 30 240 tuesday 27 216 wednesday 18 144 thursday 6 48 friday 41 328 a1m101.qxd 5/2/08 3:03 pm page 9 2008 text sampler page 1210 chapter 1 assessments 2008 carnegie learning, inc. 1 post-t est page 4 variable quantity lower bound upper bound interval number of tickets profit (dollars) 15. create a graph to display the relationship between the number of tickets sold and the profit. first, choose your bounds and intervals. be sure to label your graph clearly. 16. write an algebraic equation that you could use to show the profit for any number of tickets sold. sample answer: p 8n, where p is profit and n is the number of tickets sold. p n 3 6 9 12 15 0 18 21 24 27 30 33 45 36 39 42 25 50 75 100 0 125 150 175 200 225 250 275 300 325 350 375 profits from ticket sales for swan lake number of tickets sold profit (dollars) 0 45 3 0 375 25 a1m101.qxd 5/2/08 3:03 pm page 10 2008 text sampler page 122hapter 1 assessments 11 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 17 and 18. the director of the local ballet company needs to print the programs for swan lake. janet’s print shop charges .25 a program plus a 35 set-up fee. the printing press charges .18 a program plus a 50 set-up fee. 17. which printing company offers the better deal if 200 programs are printed? show your work and use complete sentences in your answer . 18. which printing company offers the better deal if 300 programs are printed? show your work and use complete sentences in your answer . post-t est page 5 name ___________________________________________________ date _____________________ janet’s print shop: the printing press: total cost in dollars: 0.25(200) 35 85 0.18(200) 50 86 the total cost for printing 200 programs at janet’s print shop is 85, and the total cost for printing 200 programs at the printing press is 86. therefore, janet’s print shop offers a better deal if 200 programs are printed. janet’s print shop: the printing press: total cost in dollars: 0.25(300) 35 110 0.18(300) 50 104 the total cost for printing 300 programs at janet’s print shop is 110, and the total cost for printing 300 programs at the printing press is 104. therefore, the printing press offers a better deal if 300 programs are printed. a1m101.qxd 5/2/08 3:03 pm page 11 2008 text sampler page 1232 chapter 1 assessments 2008 carnegie learning, inc. 1 a1m101.qxd 5/2/08 3:03 pm page 12 2008 text sampler page 124hapter 1 assessments 13 2008 carnegie learning, inc. 1 mid-chapter t est name ___________________________________________________ date _____________________ read the scenario below. use the scenario to complete questions 1 through 8. a local concrete company has hired you for the summer . your first project is to help pour a driveway. before you can pour the driveway, you must put wood forms in place to hold the concrete until it is dry. the first form in a driveway is made by constructing a square out of four 2 4’s that are each ten feet long. the second form is made by using one side of the first form and three other 2 4’s to make a second square. this process is continued to the end of the driveway. the diagrams show the first two steps in completing the form. 1. draw a diagram that shows steps 3, 4, and 5 in the form construction. step 3 step 4 step 5 2. the completed driveway will be 80 feet long. the company would refer to this as an eight-form driveway. draw a diagram that shows what the completed form will look like. (hint: draw the eighth step in the form construction.) step 1 step 2 a1m101.qxd 5/2/08 3:03 pm page 13 2008 text sampler page 1254 chapter 1 assessments 2008 carnegie learning, inc. 1 3. the company needs to know the number of 2 4’s needed to complete the driveway. they will also need to know the area of the driveway in order to properly bill their clients. complete the table to help organize this information. 4. write the sequence of numbers formed by the area of one form, two forms, three forms, and so on. use a complete sentence to describe the pattern produced by the area. 5. use a complete sentence to describe the relationship between the number of forms and the number of 2 4’s required to complete the forms. 6. write an expression that gives the number of 2 4’s needed to complete a driveway with n forms. 7. use your expression from question 6 to determine the number of 2 4’s needed for a 12-form driveway. show your work. 8. a. write the sequence of numbers formed by the number of 2 4’s. b. find the 10th term of this sequence. show your work. c. use a complete sentence to explain what the 10th term represents. mid-chapter t est page 2 number of forms 1 2 3 4 5 6 number of 2 4’s area (square feet) 4 7 10 13 16 19 100 200 300 400 500 600 100, 200, 300, 400, 500, 600, … sample answer: add 100 to the previous term to get the next term. sample answer: multiply the number of forms by 3 and then add 1 to get the number of 2 4’s needed. sample answer: 3n 1 3(12) 1 37 4, 7, 10, 13, 16, 19, … 3(10) 1 31 the 10th term represents the number of 2 x 4’s it would take to make the forms for a 10-form driveway. a1m101.qxd 5/2/08 3:03 pm page 14 2008 text sampler page 126hapter 1 assessments 15 2008 carnegie learning, inc. 1 write each power as a product. 9. 7 4 10. 12 6 write each product as a power . 11. (15)(15)(15)(15)(15) 12. (6)(6)(6)(6)(6)(6)(6)(6)(6) perform the indicated operations. show your work. 13. 14. evaluate each expression for the given value of the variable. show your work. 15. evaluate when r is 12. 16. evaluate when t is 36. 17. use the nth term to list the first five terms of the sequence. show your work. a 5 _______________ a 4 _______________ a 3 _______________ a 2 _______________ a 1 _______________ a n 20 2n t 4 2r 8 25 (3 5) 2 4 (6 3) 3 2(1 4) mid-chapter t est page 3 name ___________________________________________________ date _____________________ 20 – 2(1) 18 20 – 2(2) 16 20 – 2(3) 14 20 – 2(4) 12 20 – 2(5) 10 (7)(7)(7)(7) (12)(12)(12)(12)(12)(12) 15 5 6 9 37 3 3 2(5) 27 10 33 25 8 16 17 16 32 2(12) 8 24 8 36 4 9 a1m101.qxd 5/2/08 3:03 pm page 15 2008 text sampler page 1276 chapter 1 assessments 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 18 through 20. a local college has decided to build new sidewalks to connect the main administration building to the other buildings on campus. they can only build two new sidewalks a month. it will take 6 months to connect the administration building to all of the other buildings on campus. the diagrams show the number of sidewalks that have been built after 1, 2, and 3 months. 18. draw diagrams that would represent the number of sidewalks after 4, 5, and 6 months. 19. complete the table below to show the number of sidewalks built after 1, 2, 3, 4, 5, and 6 months. 20. the college wants to put in sidewalks connecting the library to the other buildings on campus. there is already a sidewalk connecting the library to the main administration building. how many more sidewalks will need to be built in order to connect the library to the remaining buildings on campus? 21. write an algebraic expression to find the sum of the numbers from 1 to n. 22. use your answer to question 21 to find the sum of the numbers from 1 to 85. show your work. 2 months 3 months 1 month mid-chapter t est page 4 number of months 1 2 3 4 5 6 number of sidewalks 2 4 6 8 10 12 6 months 5 months 4 months 11 n(n 1) 2 3655 7310 2 85(85 1) 1 85(86) 2 a1m101.qxd 5/6/08 9:52 am page 16 2008 text sampler page 128hapter 1 assessments 17 2008 carnegie learning, inc. 1 term (n) 1 2 3 4 5 sequence 5 25 125 625 3125 term (n) 1 2 3 4 5 sequence 5 15 25 35 45 end of chapter t est name ___________________________________________________ date _____________________ for each sequence, find the next two terms and describe the pattern. 1. 3, 8, 13, 18, 23, ______ , ______ 2. 3, 9, 27, 81, ______ , ______ 3. 4. for each sequence, find the expression for the nth term and describe the pattern. 5. 6. 28 33 add 5 to the previous term to find the next term. 243 729 multiply the previous term by 3 to find the next term. start with the previous term. draw a concentric circle around the other circles to find the next term. start with the previous term. add a row of triangles below the figure to find the next term. the expression is 10n – 5, so multiply the term number (n) by 10 and then subtract 5. the expression is 5 n , so raise 5 to the power of the term number (n). a1m101.qxd 5/2/08 3:03 pm page 17 2008 text sampler page 1298 chapter 1 assessments 2008 carnegie learning, inc. 1 end of chapter t est page 2 7. write the power as a product. 8. write the product as a power . 5 4 (8)(8)(8) use the nth term to list the first five terms of each sequence. show your work. 9. 10. ________________ ________________ ________________ ________________ ________________ read the scenario below. use the scenario to answer questions 11 through 23. you are a volunteer for the school store. one of the most popular items is strawberry-banana-orange juice. there are two local vendors that will deliver the juice to the school store at the beginning of every month. healthy drinks, inc. charges .39 per bottle with a delivery fee of 25. the squeeze charges .18 per bottle with a delivery fee of 65. 11. you want to stock the store with juice at the beginning of the school year . how much will it cost to purchase 300 bottles of strawberry-banana-orange juice from healthy drinks, inc.? show your work and use a complete sentence in your answer . 12. how much will it cost to purchase 300 bottles of strawberry-banana-orange juice from the squeeze? show your work and use a complete sentence in your answer . a 5 a 4 a 3 a 2 a 1 a n (n 1) 2 3 a n 3(n 1) 2 ________________ ________________ ________________ ________________ ________________ a 5 a 4 a 3 a 2 a 1 (5)(5)(5)(5) 8 3 cost for 300 bottles of strawberry-banana-orange juice: 0.39(300) 25 142 it will cost 142 to purchase 300 bottles of strawberry-banana-orange juice from healthy drinks, inc. cost for 300 bottles of strawberry-banana-orange juice: 0.18(300) 65 119 it will cost 119 to purchase 300 bottles of strawberry-banana-orange juice from the squeeze. (1 – 1) 2 3 3 (2 – 1) 2 3 4 (3 – 1) 2 3 7 (4 – 1) 2 3 12 (5 – 1) 2 3 19 3(5 1) 2 9 3(4 1) 2 15 2 3(3 1) 2 6 3(2 1) 2 9 2 3(1 1) 2 3 a1m101.qxd 5/2/08 3:03 pm page 18 2008 text sampler page 130hapter 1 assessments 19 2008 carnegie learning, inc. 1 13. complete the table summarizing the cost of purchasing strawberry-banana-orange juice from each vendor based on last year’s actual monthly sales. remember to label units. 14. let c represent the cost of purchasing bottles of juice from healthy drinks, inc. and b represent the number of bottles. write an equation that relates c and b for this problem situation. 15. let c represent the cost of purchasing bottles of juice from the squeeze and b represent the number of bottles. write an equation that relates c and b for this problem situation. 16. what was the average number of bottles of juice sold in a month last year? show your work and use a complete sentence in your answer . end of chapter t est page 3 name ___________________________________________________ date _____________________ month number of bottles of juice purchased cost of purchasing from healthy drinks, inc. cost of purchasing from the squeeze bottles dollars dollars september 187 october 229 november 162 december 137 january 171 february 201 march 192 april 258 may 214 june 79 97.93 98.66 114.31 106.22 88.18 94.16 78.43 89.66 91.69 95.78 103.39 101.18 99.88 99.56 125.62 111.44 108.46 103.52 55.81 79.22 average number of bottles of juice sold in a month: 1830 10 183 the average number of bottles of juice sold in a month was 183 bottles. c 0.39b 25 c 0.18b 65 a1m101.qxd 5/2/08 3:03 pm page 19 2008 text sampler page 1310 chapter 1 assessments 2008 carnegie learning, inc. 1 17. create a bar graph to display the costs of purchasing strawberry-banana-orange juice from healthy drinks, inc. each month. 18. create a graph displaying the cost of purchasing strawberry-banana-orange juice from both healthy drinks, inc. and the squeeze. first, choose your bounds and intervals. be sure to label your graph clearly. end of chapter t est page 4 variable quantity lower bound upper bound interval bottles cost 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 cost of purchase from healthy drinks, inc. cost (dollars) sep oct nov dec jan feb mar apr may jun 0 300 20 0 150 10 a1m101.qxd 5/2/08 3:03 pm page 20 2008 text sampler page 132hapter 1 assessments 21 2008 carnegie learning, inc. 1 19. estimate the number of bottles of juice for which the total costs for each company are the same and explain how you found your answer . use complete sentences in your answer . 20. for how many bottles of juice is healthy drinks, inc. more expensive to order from? use a complete sentence in your answer . end of chapter t est page 5 name ___________________________________________________ date _____________________ c b 20 40 60 80 100 0 120 140 160 180 200 220 300 240 260 280 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 cost of juice from healthy drinks, inc. and the squeeze number of bottles of juice cost (dollars) c 0.18b 65 c 0.39b 25 sample answer: the total costs are about the same when about 190 bottles of juice are purchased. the total costs are the same where the lines intersect each other on the graph. to find this number , start at the point of intersection and move straight down to the horizontal axis to read the number of bottles of juice for this total cost. healthy drinks, inc. is more expensive to order from when you order more than 190 bottles of juice. a1m101.qxd 5/2/08 3:03 pm page 21 2008 text sampler page 1332 chapter 1 assessments 2008 carnegie learning, inc. 1 21. for how many bottles of juice is the squeeze more expensive to order from? use a complete sentence in your answer . 22. the faculty sponsor who is responsible for the school store has asked you to write a report that compares the costs of ordering from each vendor . she would also like you to make a recommendation about which vendor you would choose if you had to order from the same vendor for the entire school year . use complete sentences in your answer . 23. if you were able to choose a different vendor each month, would you? use complete sentences in your answer . end of chapter t est page 6 the squeeze is more expensive to order from when you order 190 or fewer bottles of juice. sample answer: the total cost is the same when approximately 190 bottles of juice are purchased. if you purchase less than 190 bottles, healthy drinks, inc. is cheaper . if you purchase more than 190 bottles, the squeeze is cheaper . because the average number of bottles sold each month last year is 183, i would recommend healthy drinks, inc. as the vendor for the entire school year . answers will vary. a1m101.qxd 5/2/08 3:03 pm page 22 2008 text sampler page 134hapter 1 assessments 23 2008 carnegie learning, inc. 1 1. which choice shows the next two terms in the sequence? 1, 101, 2, 102, 3, 103, … a. 4 and 140 b. 4 and 401 c. 4 and 104 d. 4 and 114 2. which choice shows the next two items in the sequence? a. b. c. d. 3. which statement describes the pattern? –1, 10, –100, 1000, … a. start with the previous term, and multiply by 10 to get the next term. b. start with the previous term, and add a zero to get the next term. c. start with the previous term, and multiply by 100 to get the next term. d. start with the previous term, and multiply by –10 to get the next term. 4. simplify a. –1 b. 3 c. 14 d. 18 5 2 (3 6) 14 7 . standardized t est practice name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 23 2008 text sampler page 1354 chapter 1 assessments 2008 carnegie learning, inc. 1 5. which expression is equivalent to 4 6 ? a. (4)(6) b. (4)(4)(4)(4)(4)(4) c. (6)(6)(6)(6) d. 46 6. how many cups of flour are needed for 10 loaves of bread? a. 12 b. 16 c. 20 d. 24 7. evaluate when . a. 14 b. 18 c. 23 d. 41 8. which expression represents the nth term of the sequence? a. 7n 1 b. 7n c. 6n d. n 7 p 6 3p 5 standardized t est practice page 2 loaves of bread 1 2 3 4 5 cups of flour 2 4 6 8 10 term (n) 1 2 3 4 5 sequence 6 13 20 27 34 a1m101.qxd 5/2/08 3:03 pm page 24 2008 text sampler page 136hapter 1 assessments 25 2008 carnegie learning, inc. 1 9. which number represents a 4 ? a. 6 b. 8 c. 10 d. 12 10. a family of 8 has just signed a contract for a new cellular phone service so that they can call each other for free. to try it out, each person in the family calls every other person once unless that person has already called them. how many calls does the family make? a. 10 b. 15 c. 21 d. 28 11. what is the sum of the numbers from 1 to 300? a. 301 b. 4515 c. 45,150 d. 300,001 12. the average speed of an airplane is 325 miles per hour . which expression shows the distance in miles an airplane could travel in n hours? a. b. 325n c. d. n 325 325 n 325 n a n 2n 4 standardized t est practice page 3 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 25 2008 text sampler page 1376 chapter 1 assessments 2008 carnegie learning, inc. 1 13. james earns 6.25 an hour at work. the table shows his hours and earnings for each week in one month. which graph correctly displays the relationship between hours worked and earnings? a. b. c. d. 0 15 30 45 60 75 90 105 120 135 150 0 2 4 6 8 10 12 14 16 18 20 h e time worked (hours) earnings (dollars) james’ earnings 0 15 30 45 60 75 90 105 120 135 150 0 2 4 6 8 10 12 14 16 18 20 e h earnings (dollars) time worked (hours) james’ earnings 0 2 4 6 8 10 12 14 16 18 20 0 15 30 45 60 75 90 105 120 135 150 e h earnings (dollars) time worked (hours) james’ earnings 0 2 4 6 8 10 12 14 16 18 20 0 15 30 45 60 75 90 105 120 135 150 h e time worked (hours) earnings (dollars) james’ earnings standardized t est practice page 4 week time worked earnings hours dollars week 1 15 93.75 week 2 18 112.50 week 3 12 75 week 4 13 81.25 a1m101.qxd 5/2/08 3:03 pm page 26 2008 text sampler page 138hapter 1 assessments 27 2008 carnegie learning, inc. 1 14. james earns 6.25 an hour at work. the table in question 13 shows his hours and earnings for each week in one month. which bar graph correctly displays the relationship between the week and the time worked? a. b. c. d. 0 15 30 45 60 75 90 105 120 135 150 week time worked per week 1 2 3 4 time worked (hours) 0 15 30 45 60 75 90 105 120 135 150 week time worked per week 1 2 3 4 earnings (dollars) 0 2 4 6 8 10 12 14 16 18 20 week time worked per week 1 2 3 4 time worked (hours) 0 2 4 6 8 10 12 14 16 18 20 time worked (hours) time worked per week 1 2 3 4 week standardized t est practice page 5 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 27 2008 text sampler page 1398 chapter 1 assessments 2008 carnegie learning, inc. 1 15. james earns 6.25 an hour at work. which algebraic equation shows the amount of money e that james earns in n hours? a. b. c. d. 16. angelica earns 7.50 each hour she works. how many hours will she have to work to buy a bicycle that costs 90? a. 6 b. 9 c. 12 d. 15 17. angelica earns 7.50 each hour she works. how much money will angelica earn if she works for 6 hours and 12 minutes? a. 45 b. 46.50 c. 46.75 d. 135 18. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. the total cost of an order was 562. how many frisbees were printed? a. 47 b. 275 c. 281 d. 550 e 6.25n n 6.25e e 6.25 n e 6.25 n standardized t est practice page 6 a1m101.qxd 5/2/08 3:03 pm page 28 2008 text sampler page 140hapter 1 assessments 29 2008 carnegie learning, inc. 1 19. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. which algebraic equation shows the cost c of printing f frisbees? a. b. c. d. 20. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. you say it, we print it will print any image on a frisbee for a cost of 5 per frisbee and no set-up fee. which statement is true? a. you say it, we print it is a better buy if you purchase more than four frisbees. b. t -shirts & more print shop is always the better buy. c. you say it, we print it is always the better buy. d. t -shirts & more print shop is the better buy if you purchase more than four frisbees. c 2f 12 f 2c 12 c 12f 2 c 2f standardized t est practice page 7 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 29 2008 text sampler page 1412008 carnegie learning, inc. algebra i homework helper a1h1_fm.qxd 5/7/08 1:34 pm page i 2008 text sampler page 1422 chapter 1 homework helper 2008 carnegie learning, inc. 1 students should be able to answer these questions after lesson 1.8: what are the four different ways you have learned to represent a problem situation? what are the advantages and disadvantages of each method? read question 1 and its solution. then, use similar steps to complete questions 2 and 3. 1. you have 183 in your savings account. you want to add 30 to your account each week. how much will you have in your account after 6 weeks? step 1 describe the problem situation. i will calculate the total amount in my savings. this amount is amount saved each week (30) plus the starting value (183). step 2 total in dollars: after 6 weeks, i will have 363 in my savings account. 2. a storage company charges 100 to store your furniture in their warehouse plus an additional 35 per month. how much will they charge to rent their space for 3 months? use a complete sentence in your answer . 3. a car-rental company charges 30 to rent a car plus .25 per mile. how much will the rental company charge a customer who drives 200 miles? use a complete sentence in your answer . use the scenario in question 3 to answer questions 4 through 8. 4. how much will the rental company charge a customer who drives 364 miles? 5. the rental company charges you 135.25 to rent a car . how many miles did you travel? 6. what are the two variable quantities in this problem situation? assign letters to represent these quantities and include the units that are used to measure these quantities. use a complete sentence in your answer . 7. which of the variables from question 6 is the independent variable and which is the dependent variable? use a complete sentence in your answer . 8. write an algebraic equation for the problem situation. use a complete sentence in your answer . 6(30) 183 363 u.s. shirts using t ables, graphs, and equations, part 1 1 .8 directions a1h101.qxd 5/7/08 1:34 pm page 12 2008 text sampler page 143ourse description – geometry carnegie learning tm geometry is designed to be taken after an algebra course. it can be implemented with students at a variety of ability and grade levels. the course assumes number fluency and basic algebra skills such as equation solving. geometry focuses heavily on developing spatial relationships, measurement, and reasoning. students develop properties of figures in two and three dimensions and use these properties to prove statements and calculate measurements. students are introduced to the basic building blocks of geometry: points, lines, and angles. students develop properties of angles and angle pairs, including angles formed by parallel lines. students explore triangles using the pythagorean theorem, special right triangles, the triangle inequality, and trigonometric ratios. students explore quadrilaterals and understand the relationship between squares, rectangles, parallelograms, trapezoids, and rhombi. students explore circles including angles, arcs, chords, tangents, and sectors. students explore polygons including area and perimeter, similarity, congruence, and angle sums. students use reflections, rotations, translations, dilations, and symmetry to transform shapes in the coordinate plane. students calculate slope, distance, and midpoint and use these measures to explore shapes in the coordinate plane. students calculate the volume and surface area of three dimensional figures. students explore ways to represent three dimensional figures including nets and cross sections. 2008 text sampler page 144eometry text setv contents 2008 carnegie learning, inc. contents contents perimeter and area p. 2 1.1 building a deck introduction to polygons, perimeter , and area p. 5 1.2 weaving a rug area and perimeter of a rectangle and area of a parallelogram p. 13 1.3 sailboat racing area of a triangle p. 21 1.4 the keystone effect area of a trapezoid p. 27 1.5 traffic signs area of a regular polygon p. 33 1.6 photography circumference and area of a circle p. 39 1.7 installing carpeting and tile composite figures p. 49 volume and surface area p. 56 2.1 backyard barbecue introduction to volume and surface area p. 59 2.2 turn up the volume volume of a prism p. 65 2.3 bending light beams surface area of a prism p. 73 2.4 modern day pyramids volume of a pyramid p. 79 2.5 soundproofing surface area of a pyramid p. 85 2.6 making concrete stronger volume and surface area of a cylinder p. 91 2.7 sand piles volume and surface area of a cone p. 97 2.8 ball bearings and motion volume and surface area of a sphere p. 103 1 2 ges1_fm.qxd 4/25/08 9:28 am page iv 2008 text sampler page 146ontents v 2008 carnegie learning, inc. contents introduction to angles and t riangles p. 110 3.1 constellations naming, measuring, and classifying angles p. 113 3.2 cable-stayed bridges special angles p. 121 3.3 designing a kitchen angles of a triangle p. 129 3.4 origami classifying triangles p. 137 3.5 building a shed the triangle inequality p. 145 right t riangle geometry p. 150 4.1 tiling a bathroom wall simplifying square root expressions p. 153 4.2 installing a satellite dish the pythagorean theorem p. 157 4.3 drafting equipment properties of 45º–45º–90º triangles p. 163 4.4 finishing concrete properties of 30º–60º–90º triangles p. 167 4.5 meeting friends the distance formula p. 175 4.6 treasure hunt the midpoint formula p. 183 parallel and perpendicular lines p. 188 5.1 visiting washington, d.c. transversals and parallel lines p. 191 5.2 going up? introduction to proofs p. 199 5.3 working with iron parallel lines and proofs p. 205 5.4 parking lot design parallel and perpendicular lines in the coordinate plane p. 213 5.5 building a henge exploring triangles in the coordinate plane p. 223 5.6 building a roof truss angle and line segment bisectors p. 229 5.7 warehouse space points of concurrency in triangles p. 235 3 4 5 ges1_fm.qxd 4/25/08 9:28 am page v 2008 text sampler page 147i contents 2008 carnegie learning, inc. contents simple t ransformations p. 240 6.1 paper snowflakes reflections p. 243 6.2 good lighting rotations p. 253 6.3 web page design translations p. 261 6.4 shadow puppets dilations p. 269 6.5 cookie cutters symmetry p. 277 similarity p. 280 7.1 ace reporter ratio and proportion p. 283 7.2 framing a picture similar and congruent polygons p. 289 7.3 using an art projector proving triangles similar: aa, sss, and sas p. 297 7.4 modeling a park indirect measurement p. 305 7.5 making plastic containers similar solids p. 311 congruence p. 316 8.1 glass lanterns introduction to congruence p. 319 8.2 computer graphics proving triangles congruent by using sss and sas p. 323 8.3 wind triangles proving triangles congruent by using asa and aas p. 329 8.4 planting graph vines proving triangles congruent by using hl p. 337 8.5 koch snowflake fractals p. 341 6 7 8 ges1_fm.qxd 4/25/08 9:28 am page vi 2008 text sampler page 148ontents vii 2008 carnegie learning, inc. contents quadrilaterals p. 348 9.1 quilting and tessellations introduction to quadrilaterals p. 351 9.2 when trapezoids are kites kites and trapezoids p. 357 9.3 binocular stand design parallelograms and rhombi p. 363 9.4 positive reinforcement rectangles and squares p. 369 9.5 stained glass sum of the interior angle measures in a polygon p. 373 9.6 pinwheels sum of the exterior angle measures in a polygon p. 377 9.7 planning a subdivision rectangles and parallelograms in the coordinate plane p. 383 circles p. 388 10.1 riding a ferris wheel introduction to circles p. 391 10.2 holding the wheel central angles, inscribed angles, and intercepted arcs p. 397 10.3 manhole covers measuring angles inside and outside of circles p. 401 10.4 color theory chords and circles p. 407 10.5 solar eclipses tangents and circles p. 413 10.6 gears arc length p. 417 10.7 playing darts areas of parts of circles p. 421 9 1 0 ges1_fm.qxd 4/25/08 9:28 am page vii 2008 text sampler page 149iii contents 2008 carnegie learning, inc. contents right t riangle t rigonometry p. 426 11.1 wheelchair ramps the tangent ratio p. 429 11.2 golf club design the sine ratio p. 435 11.3 attaching a guy wire the cosine ratio p. 439 11.4 using a clinometer angles of elevation and depression p. 443 extensions in area and volume p. 446 12.1 replacement for a carpenter’s square inscribed polygons p. 449 12.2 box it up nets p. 455 12.3 tree rings cross sections p. 459 12.4 minerals and crystals polyhedra and euler’s formula p. 463 12.5 isometric drawings compositions p. 469 glossary p. g-1 index p. i-1 1 2 1 1 ges1_fm.qxd 4/25/08 9:28 am page viii 2008 text sampler page 1502008 carnegie learning, inc. geometry student t ext ges1_fm.qxd 4/25/08 9:28 am page i 2008 text sampler page 15188 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 looking ahead to chapter 5 focus in chapter 5, you will learn about properties of parallel and perpendicular lines, angle and line segment bisectors, and points of concurrency in triangles. you will also learn how to write proofs. chapter warmup answer these questions to help you review skills that you will need in chapter 5. find the slope of the line that passes through the given points. 1. (3, 2) and (5, 6) 2. (–1, 0) and (11, –8) 3. (7, 9) and (3, 4) read the problem scenario below. each square on the grid represents a square that is 1 mile long and 1 mile wide. 4. find the distance between the museum and the school. 5. the library is located halfway between the museum and the school. find the coordinates of the point that represents the library. plot and label this point on the grid. 6. find the distance between the school and the library. plane p. 191 coplanar p. 191 parallel lines p. 192 skew lines p. 192 transversal p. 193 interior angle p. 193 exterior angle p. 193 alternate interior angles p. 193 alternate exterior angles p. 194 same-side interior angles p. 194 same-side exterior angles p. 194 corresponding angles p. 194 congruent p. 199 conditional statement p. 201 hypothesis p. 201 conclusion p. 201 if-then form p. 201 proof p. 201 postulate p. 201 two-column proof p. 202 slope p. 214 y-intercept p. 214 point-slope form p. 214 slope-intercept form p. 214 perpendicular lines p. 216 reciprocal p. 218 negative reciprocal p. 218 horizontal line p. 221 vertical line p. 221 inscribed triangle p. 224 midsegment p. 227 angle bisector p. 230 line segment bisector p. 233 perpendicular bisector p. 233 key t erms y 1 2 3 4 5 6 1 2 3 4 5 6 x school museum ges10500.qxd 4/24/08 9:10 am page 188 2008 text sampler page 152hapter 5 parallel and perpendicular lines 189 2008 carnegie learning, inc. 5 parallel and perpendicular lines c h a p t e r 5 the lines drawn in parking lots are typically designed to allow the most cars to fit in the parking area. sometimes, the cars in a parking lot will be parked perpendicular to the curb. sometimes, the cars will be parked at other angles with the curb. in lesson 5.4, you will explore different designs that are used in parking lots 5.1 visiting washington, d.c. transversals and parallel lines p. 191 5.2 going up? introduction to proofs p. 199 5.3 working with iron parallel lines and proofs p. 205 5.4 parking lot design parallel and perpendicular lines in the coordinate plane p. 213 5.5 building a henge exploring triangles in the coordinate plane p. 223 5.6 building a roof truss angle and line segment bisectors p. 229 5.7 warehouse space points of concurrency in triangles p. 235 ges10500.qxd 4/24/08 9:10 am page 189 2008 text sampler page 15390 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 ges10501.qxd 4/24/08 9:11 am page 190 2008 text sampler page 154esson 5.4 parallel and perpendicular lines in the coordinate plane 213 2008 carnegie learning, inc. 5 parking lot design parallel and perpendicular lines in the coordinate plane objectives in this lesson, you will: determine whether lines are parallel. find the equations of lines parallel to given lines. determine whether lines are perpendicular . find the equations of lines perpendicular to given lines. determine equations of horizontal and vertical lines. key t erms slope y-intercept point-slope form slope-intercept form parallel lines perpendicular lines reciprocal negative reciprocal horizontal line vertical line scenario large parking lots, such as those located in a shopping center or at a mall, have line segments painted to mark the locations where vehicles are supposed to park. the layout of these line segments must be considered carefully so that there is enough room for the vehicles to move and park in the lot without the vehicles being damaged. problem 1 parking spaces some line segments that form parking spaces in a parking lot are shown on the grid below. one grid square represents a square that is one meter long and one meter wide. a. what do you notice about the line segments that form the parking spaces? use a complete sentence to explain your reasoning. b. what is the vertical distance between and and between and ? use a complete sentence in your answer . ef cd cd ab y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 e f d c a b x 5.4 ges10504.qxd 4/24/08 9:12 am page 213 2008 text sampler page 15514 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 1 parking spaces c. carefully extend into line p, extend into line q, and extend into line r. d. use the graph to identify the slope of each line. what do you notice? use a complete sentence in your answer . e. use the point-slope form to write the equations of lines p, q, and r. then write the equations in slope-intercept form. what do you notice about the y-intercepts of these lines? use a complete sentence in your answer . what do the y-intercepts tell you about the relationship between these lines? use a complete sentence in your answer . f . if you were to draw a line segment above to form another parking space, what would be the equation of the line that coincides with this line segment? determine your answer without graphing the line. use complete sentences to explain how you found your answer . ef ef cd ab t ake note remember that the slope of a line is the ratio of the rise to the run: rise run slope rise run . t ake note remember that the point-slope form of the equation of the line that passes through and has slope m is the slope-intercept form of the equation of the line that has slope m and y-intercept b is . y mx b y y 1 m(x x 1 ). (x 1 , y 1 ) ges10504.qxd 4/24/08 9:12 am page 214 2008 text sampler page 156esson 5.4 parallel and perpendicular lines in the coordinate plane 215 2008 carnegie learning, inc. 5 investigate problem 1 1. what can you conclude about the slopes of parallel lines in the coordinate plane? use a complete sentence in your answer . what can you conclude about the y-intercepts of parallel lines in the coordinate plane? use a complete sentence in your answer . 2. write equations for three lines that are parallel to the line given by . use complete sentences to explain how you found your answers. 3. write an equation for the line that is parallel to the line given by and passes through the point (4, 0). show all your work. use complete sentences to explain how you found your answer . 4. without graphing the equations, determine whether the lines given by and are parallel. show all your work and use a complete sentence in your answer . 2x y 4 y 2x 5 y 5x 3 y 2x 4 ges10504.qxd 4/24/08 9:12 am page 215 2008 text sampler page 15716 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 2 more parking spaces another arrangement of line segments that form parking spaces in a truck parking lot is shown on the grid below. one grid square represents a square that is one meter long and one meter wide. a. use a protractor to find the measures of and what do you notice about the angles? use a complete sentence in your answer . when lines or line segments intersect at right angles, we say that the lines or line segments are perpendicular . for instance, is perpendicular to . in symbols, we can write this as where means "is perpendicular to." b. carefully extend into line p, extend into line q, extend into line r, and extend into line s. c. how do these lines relate to each other? use complete sentences in your answer . yz wx uv uy uv uw uw uv zyw. xwy, vuw, y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x v x z u w y ges10504.qxd 4/24/08 9:12 am page 216 2008 text sampler page 158esson 5.4 parallel and perpendicular lines in the coordinate plane 217 2008 carnegie learning, inc. 5 problem 2 more parking spaces complete each statement by using or . line p ____ line r line q ____ line s d. without actually determining the slopes, how will the slopes of the lines compare? explain your reasoning. use complete sentences in your answer . e. what do you think must be true about the signs of the slopes of two lines that are perpendicular? use complete sentences in your answer . f . use the graph to find the slopes of lines p, q, r, and s. g. how does the slope of line p compare to the slopes of lines q, r, and s? use a complete sentence in your answer . h. what is the product of the slopes of two of your perpendicular lines? use a complete sentence in your answer . investigate problem 2 1. what can you conclude about the product of the slopes of perpendicular lines in the coordinate plane? use a complete sentence in your answer . ges10504.qxd 4/24/08 9:12 am page 217 2008 text sampler page 15918 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 investigate problem 2 when the product of two numbers is 1, the numbers are reciprocals of one another . when the product of two numbers is –1, the numbers are negative reciprocals of one another . so the slopes of perpendicular lines are negative reciprocals of each other . 2. find the negative reciprocal of each number . 5 –2 3. do you think that the y-intercepts of perpendicular lines tell you anything about the relationship between the perpendicular lines? use a complete sentence to explain your reasoning. 4. write equations for three lines that are perpendicular to the line given by . use complete sentences to explain how you found your answers. 5. write an equation for the line that is perpendicular to the line given by and passes through the point (4, 0). show all your work. use complete sentences to explain how you found your answer . y 5x 3 y 2x 4 1 3 ges10504.qxd 4/24/08 9:12 am page 218 2008 text sampler page 160esson 5.4 parallel and perpendicular lines in the coordinate plane 219 2008 carnegie learning, inc. 5 investigate problem 2 6. without graphing the equations, determine whether the lines given by and are perpendicular . show all your work and use a complete sentence in your answer . 7. complete each statement. when two lines are parallel, their slopes are _______________. when two lines are perpendicular , their slopes are _____________ ___________________________________. 8. suppose that you have a line and you choose one point on the line. how many lines perpendicular to the given line can you draw through the given point? use a complete sentence in your answer . 9. suppose that you have a line and you choose one point that is not on the line. how many lines can you draw through the given point that are perpendicular to the given line? how many lines can you draw through the given point that are parallel to the given line? use complete sentences in your answer . 2x y 4 y 2x 5 ges10504.qxd 4/24/08 9:12 am page 219 2008 text sampler page 16120 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 3 a very simple parking lot one final truck parking lot is shown below. one grid square represents a square that is one meter long and one meter wide. a. what angles are formed by the intersection of the parking lot line segments? how do you know? use complete sentences in your answer . b. carefully extend into line p, extend into line q, extend into line r, and extend into line s. c. choose any three points on line q and list their coordinates. choose any three points on line r and list their coordinates. choose any three points on line s and list their coordinates. what do you notice about the x- and y-coordinates of these points? use complete sentences in your answer . kl ij gh gk y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x k l i j h g ges10504.qxd 4/24/08 9:12 am page 220 2008 text sampler page 162esson 5.4 parallel and perpendicular lines in the coordinate plane 221 2008 carnegie learning, inc. 5 problem 3 a very simple parking lot what do you think should be the equations of lines q, r, and s? use complete sentences to explain your reasoning. d. choose any three points on line p and list their coordinates. what do you notice about the x- and y-coordinates of these points? use complete sentences in your answer . what do you think should be the equation of line p? use complete sentences to explain your reasoning. investigate problem 3 1. just the math: horizontal and vertical lines in problem 3, you wrote the equations of horizontal and vertical lines. a horizontal line has an equation of the form where a is any real number . a vertical line has an equation of the form where b is any real number . consider your horizontal lines in problem 3. for any horizontal line, if x increases by 1 unit, by how many units does y change? use a complete sentence in your answer . what is the slope of any horizontal line? use complete sentences to explain your reasoning. consider your vertical line in problem 3. suppose that y increases by one unit. by how many units does x change? use a complete sentence in your answer . x b y a ges10504.qxd 4/24/08 9:12 am page 221 2008 text sampler page 16322 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 investigate problem 3 what is the rise divided by the run? does this make any sense? use a complete sentence to explain. because division by zero is undefined, we say that a vertical line has an undefined slope. 2. consider the statements about parallel and perpendicular lines in question 7 of problem 2. are these statements true for horizontal and vertical lines? use complete sentences to explain your reasoning. complete the following statements. ________ vertical lines are parallel. ________ horizontal lines are parallel. write a statement that describes the relationship between a vertical line and a horizontal line. use a complete sentence in your answer . 3. write equations for a horizontal line and a vertical line that pass through the point (2, –1). 4. write an equation of the line that is perpendicular to the line given by and passes through the point (1, 0). write an equation of the line that is perpendicular to the line given by and passes through the point (5, 6). y 2 x 5 ges10504.qxd 4/24/08 9:12 am page 222 2008 text sampler page 1642008 carnegie learning, inc. geometry t eacher’s implementation guide volume 1 get1_fm_v1.qxd 4/29/08 9:05 am page i 2008 text sampler page 16588 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 looking ahead to chapter 5 focus in chapter 5, you will learn about properties of parallel and perpendicular lines, angle and line segment bisectors, and points of concurrency in triangles. you will also learn how to write proofs. chapter warmup answer these questions to help you review skills that you will need in chapter 5. find the slope of the line that passes through the given points. 1. (3, 2) and (5, 6) 2. (–1, 0) and (11, –8) 3. (7, 9) and (3, 4) read the problem scenario below. each square on the grid represents a square that is 1 mile long and 1 mile wide. 4. find the distance between the museum and the school. 5. the library is located halfway between the museum and the school. find the coordinates of the point that represents the library. plot and label this point on the grid. 6. find the distance between the school and the library. plane p. 191 coplanar p. 191 parallel lines p. 192 skew lines p. 192 transversal p. 193 interior angle p. 193 exterior angle p. 193 alternate interior angles p. 193 alternate exterior angles p. 194 same-side interior angles p. 194 same-side exterior angles p. 194 corresponding angles p. 194 congruent p. 199 conditional statement p. 201 hypothesis p. 201 conclusion p. 201 if-then form p. 201 proof p. 201 postulate p. 201 two-column proof p. 202 slope p. 214 y-intercept p. 214 point-slope form p. 214 slope-intercept form p. 214 perpendicular lines p. 216 reciprocal p. 218 negative reciprocal p. 218 horizontal line p. 221 vertical line p. 221 inscribed triangle p. 224 midsegment p. 227 angle bisector p. 230 line segment bisector p. 233 perpendicular bisector p. 233 key t erms y 1 2 3 4 5 6 1 2 3 4 5 6 x school museum 2 5 4 2 3 2.5 miles ( 3, 3 1 2 ) 5 miles library get10500.qxd 4/18/08 12:26 pm page 188 2008 text sampler page 166hapter 5 parallel and perpendicular lines 189 2008 carnegie learning, inc. 5 parallel and perpendicular lines c h a p t e r 5 the lines drawn in parking lots are typically designed to allow the most cars to fit in the parking area. sometimes, the cars in a parking lot will be parked perpendicular to the curb. sometimes, the cars will be parked at other angles with the curb. in lesson 5.4, you will explore different designs that are used in parking lots 5.1 visiting washington, d.c. transversals and parallel lines p. 191 5.2 going up? introduction to proofs p. 199 5.3 working with iron parallel lines and proofs p. 205 5.4 parking lot design parallel and perpendicular lines in the coordinate plane p. 213 5.5 building a henge exploring triangles in the coordinate plane p. 223 5.6 building a roof truss angle and line segment bisectors p. 229 5.7 warehouse space points of concurrency in triangles p. 235 get10500.qxd 4/18/08 12:26 pm page 189 2008 text sampler page 16790 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 get10500.qxd 4/18/08 12:26 pm page 190 2008 text sampler page 168esson 5.4 parallel and perpendicular lines in the coordinate plane 213a 2008 carnegie learning, inc. 5 parking lot design parallel and perpendicular lines in the coordinate plane objectives in this lesson, you will: determine whether lines are parallel. find the equations of lines parallel to given lines. determine whether lines are perpendicular . find the equations of lines perpendicular to given lines. determine equations of horizontal and vertical lines. key t erms slope perpendicular lines point-slope form reciprocal slope-intercept form negative reciprocal y-intercept horizontal line parallel lines vertical line materials graph paper rulers protractors nctm content standards grades 9–12 expectations algebra standards analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior . use symbolic algebra to represent and explain mathematical relationships. draw reasonable conclusions about a situation being modeled. approximate and interpret rates of change from graphical and numerical data. 5.4 geometry standards use cartesian coordinates and other coordinate systems, such as navigational, polar , or spherical systems, to analyze geometric situations. lesson overview within the context of this lesson, students will be asked to: determine whether lines are parallel. find the equations of lines parallel to given lines. determine whether lines are perpendicular . find the equations of lines perpendicular to given lines. determine equations of horizontal and vertical lines. essential questions the following key questions are addressed in this lesson: 1. how can you determine whether lines are parallel given the equations? 2. how can you determine whether lines are parallel given the graphs of the lines? 3. what formula is used to find the equation of a line given two points? 4. how can you determine whether lines are perpendicular given the equations? 5. how can you determine whether lines are perpendicular given the graph? 6. how do you determine equations of horizontal and vertical lines? 7. what is the slope of a horizontal line? 8. what is the slope of a vertical line? get ready learning by doing lesson map get10504.qxd 4/23/08 12:31 pm page 211 2008 text sampler page 16913b chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 warm up place the following questions or an applicable subset of these questions on the board before students enter class. students should begin working as soon as they are seated. graph the following equations on your graph paper . 1. 2. 3. 4. motivator begin the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson is about designing a parking lot. the motivating questions are about parking lots. ask the students the following questions to get them interested in the lesson. who has his or her driver’s license? are you good at parallel parking? do you prefer parking in a parking lot? at what angles are cars normally parked in a parking lot? is there a reason why lines in a parking lot are on a diagonal? y 1 1 3 4 5 2 3 4 5 1 2 3 4 5 5 4 3 2 1 x y 1 1 2 3 4 5 2 3 4 5 1 2 4 5 5 4 3 2 1 x 9x 4y 12 3x 2y 8 y 1 1 2 3 4 5 3 4 5 1 2 3 4 5 5 4 3 2 1 x y 1 1 2 2 3 5 6 7 8 1 2 3 4 5 5 4 2 1 x y 2x 1 y 2x 5 show the way get10504.qxd 4/18/08 1:02 pm page 212 2008 text sampler page 170esson 5.4 parallel and perpendicular lines in the coordinate plane 213 2008 carnegie learning, inc. 5 problem 1 students will determine that parallel lines have the same slopes and different y-intercepts. grouping ask for a student volunteer to read the scenario and problem 1 aloud. pose the guiding questions below to verify student understanding. have students work together as a whole class to complete part (a) through part (f) of problem 1. guiding questions what information is given in this problem? how much does each square on the coordinate plane represent? why do the lines need to be slanted in some parking lots instead of horizontal or vertical lines? common student errors some students will have forgotten how to read a graph. when the students complete the warm up questions discuss graphing by using slope-intercept form and by using a table of values (or a graphing calculator). scenario large parking lots, such as those located in a shopping center or at a mall, have line segments painted to mark the locations where vehicles are supposed to park. the layout of these line segments must be considered carefully so that there is enough room for the vehicles to move and park in the lot without the vehicles being damaged. problem 1 parking spaces some line segments that form parking spaces in a parking lot are shown on the grid below. one grid square represents a square that is one meter long and one meter wide. a. what do you notice about the line segments that form the parking spaces? use a complete sentence to explain your reasoning. b. what is the vertical distance between and and between and ? use a complete sentence in your answer . ef cd cd ab y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 e f d c a b x r q p sample answer: the segments are all the same distance apart, which means they are parallel. the vertical distance between and and between and is 6 meters. ef cd cd ab explore together get10504.qxd 4/23/08 12:39 pm page 213 2008 text sampler page 17114 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 1 parking spaces c. carefully extend into line p, extend into line q, and extend into line r. d. use the graph to identify the slope of each line. what do you notice? use a complete sentence in your answer . e. use the point-slope form to write the equations of lines p, q, and r. then write the equations in slope-intercept form. what do you notice about the y-intercepts of these lines? use a complete sentence in your answer . what do the y-intercepts tell you about the relationship between these lines? use a complete sentence in your answer . f . if you were to draw a line segment above to form another parking space, what would be the equation of the line that coincides with this line segment? determine your answer without graphing the line. use complete sentences to explain how you found your answer . ef ef cd ab sample answer: line p: line q: line r: y 12 3 2 (x 0); y 3 2 x 12 y 6 3 2 (x 0); y 3 2 x 6 y 0 3 2 (x 0); y 3 2 x sample answer: the y-intercepts are all multiples of 6. ; sample answer: the slope would be the same as the other lines, and the y-intercept would be 6 units above the y-intercept for line r. y 3 2 x 18 the slope is the same for all three lines. this slope is . 3 2 sample answer: the y-intercepts tell you how many units one line is above (or below) the other . explore together problem 1 grouping students will be working on part (a) through part (f) as a whole class. after completing part (d), have a student read the take note boxes aloud. guiding questions what does slope mean? how do we look at the graph and determine the slope? what happens to the value of rise if we count down and then over? what happens to the value of the run when we count left instead of right? what do x 1 and y 1 represent? what does m represent? what does b represent? what does y-intercept mean? t ake note remember that the slope of a line is the ratio of the rise to the run: rise run slope rise run . t ake note remember that the point-slope form of the equation of the line that passes through and has slope m is the slope-intercept form of the equation of the line that has slope m and y-intercept b is . y mx b y y 1 m(x x 1 ). (x 1 , y 1 ) get10504.qxd 4/23/08 12:40 pm page 214 2008 text sampler page 172esson 5.4 parallel and perpendicular lines in the coordinate plane 215 2008 carnegie learning, inc. 5 investigate problem 1 students will make conclusions about the slopes of parallel lines. students will examine equations to determine whether the lines are parallel. grouping ask a student volunteer to read question 1 aloud. have students work together in small groups to complete questions 1 through 4. common student errors students might need direction in question 3 that they need to use point-slope form. in question 4, students who are weak equation solvers will struggle with isolating the variable y. some students will not recognize that they need to change the equations to slope-intercept form in order to compare the slopes. guiding questions will the slopes of parallel lines always be the same? explain. will the y-intercepts of parallel lines ever be the same? explain. is it easier to pick out the value of slope from an equation in standard form or an equation in slope-intercept form? how do you determine whether two lines are parallel without graphing the lines? call the class back together to have the students discuss and present their work for questions 1 through 4. investigate problem 1 1. what can you conclude about the slopes of parallel lines in the coordinate plane? use a complete sentence in your answer . what can you conclude about the y-intercepts of parallel lines in the coordinate plane? use a complete sentence in your answer . 2. write equations for three lines that are parallel to the line given by . use complete sentences to explain how you found your answers. 3. write an equation for the line that is parallel to the line given by and passes through the point (4, 0). show all your work. use complete sentences to explain how you found your answer . 4. without graphing the equations, determine whether the lines given by and are parallel. show all your work and use a complete sentence in your answer . 2x y 4 y 2x 5 y 5x 3 y 2x 4 parallel lines in the coordinate plane have the same slope. parallel lines in the plane have different y-intercepts. sample answer: any line that is parallel to the line given by will have a slope of and a y-intercept that is not 4. 2 y 2x 4 y 2x; y 2x 5, y 2x 3 a line parallel to the line given by must have a slope of 5. because you know that the line must pass through the point (4, 0), you can use the point-slope form to write the equation. y 0 5(x 4); y 5x 20 y 5x 3 the slopes are the same and the y-intercepts are different, so the lines are parallel. y 2x 4 y 2x 4 y 2x 5 2x y 4 y 2x 5 explore together get10504.qxd 4/23/08 12:41 pm page 215 2008 text sampler page 17316 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 2 more parking spaces another arrangement of line segments that form parking spaces in a truck parking lot is shown on the grid below. one grid square represents a square that is one meter long and one meter wide. a. use a protractor to find the measures of and what do you notice about the angles? use a complete sentence in your answer . when lines or line segments intersect at right angles, we say that the lines or line segments are perpendicular . for instance, is perpendicular to . in symbols, we can write this as where means "is perpendicular to." b. carefully extend into line p, extend into line q, extend into line r, and extend into line s. c. how do these lines relate to each other? use complete sentences in your answer . yz wx uv uy uv uw uw uv zyw. xwy, vuw, y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x v x z u w y q r s p all of the angles are right angles. sample answer: lines q, r, and s are parallel, lines p and q are perpendicular , lines p and r are perpendicular , and lines p and s are perpendicular . explore together problem 2 students will discover that perpendi-cular lines have negative reciprocal slopes. grouping ask for a student volunteer to read the scenario for problem 2 aloud. pose the guiding questions below to verify student understanding. have students work together in pairs to complete part (a) through part (h). guiding questions what information is given in this problem? what does each square represent? what would you guess about the relationship of the lines? which lines look to be parallel? what do you know about the slopes of each line? which slopes are positive? which slope is negative? common student errors remind students that they should not assume that lines are parallel or perpendi-cular based on a diagram. there should be marks on the diagram indicating parallel or perpendicular lines, or students should measure the angles. get10504.qxd 4/23/08 12:41 pm page 216 2008 text sampler page 174esson 5.4 parallel and perpendicular lines in the coordinate plane 217 2008 carnegie learning, inc. 5 problem 2 students will continue to investigate the slope of perpendicular lines. grouping students will be working together in pairs to complete part (d) through part (f). notes students may be unfamiliar with the symbols , which indicates that lines are perpendicular , and , which indicates that lines are parallel. call the class back together to discuss and present their work for part (a) through part (h). investigate problem 2 students will be formally introduced to the terms reciprocal and negative reciprocal in question 1. grouping ask a student volunteer to read question 1 aloud. have the students complete question 1 individually. note that question 1 is continued on the next page. problem 2 more parking spaces complete each statement by using or . line p ____ line r line q ____ line s d. without actually determining the slopes, how will the slopes of the lines compare? explain your reasoning. use complete sentences in your answer . e. what do you think must be true about the signs of the slopes of two lines that are perpendicular? use complete sentences in your answer . f . use the graph to find the slopes of lines p, q, r, and s. g. how does the slope of line p compare to the slopes of lines q, r, and s? use a complete sentence in your answer . h. what is the product of the slopes of two of your perpendicular lines? use a complete sentence in your answer . investigate problem 2 1. what can you conclude about the product of the slopes of perpendicular lines in the coordinate plane? use a complete sentence in your answer . sample answer: the slopes of perpendicular lines have a product of –1. sample answer: the slopes must have opposite signs. line p: ; lines q, r, and s: 2 1 1 2 sample answer: the slopes have opposite signs. the absolute value of the slope of line p is the reciprocal of the slopes of lines q, r, and s. sample answer: the slopes of lines q, r, and s will be the same because the lines are parallel, and the slopes of lines q, r, and s will be different from the slope of line p because the lines are not parallel. the product of the slopes is –1. explore together get10504.qxd 4/23/08 12:42 pm page 217 2008 text sampler page 17518 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 investigate problem 2 when the product of two numbers is 1, the numbers are reciprocals of one another . when the product of two numbers is –1, the numbers are negative reciprocals of one another . so the slopes of perpendicular lines are negative reciprocals of each other . 2. find the negative reciprocal of each number . 5 –2 3. do you think that the y-intercepts of perpendicular lines tell you anything about the relationship between the perpendicular lines? use a complete sentence to explain your reasoning. 4. write equations for three lines that are perpendicular to the line given by . use complete sentences to explain how you found your answers. 5. write an equation for the line that is perpendicular to the line given by and passes through the point (4, 0). show all your work. use complete sentences to explain how you found your answer . y 5x 3 y 2x 4 1 3 3 1 2 1 5 sample answer: yes, taken together with the slopes, the y-intercepts can give you a general idea of where the lines intersect in the plane. sample answer: any line that is perpendicular to the line given by will have a slope of . 1 2 y 2x 4 y 1 2 x; y 1 2 x 5; y 1 2 x 3 a line perpendicular to the line given by must have a slope of . because you know that the line must pass through the point (4, 0), you can use the point-slope form to write the equation. y 0 1 5 (x 4); y 1 5 x 4 5 1 5 y 5x 3 explore together investigate problem 2 students will make conclusions about the slopes of perpendicular lines. students will examine equations to determine whether the lines are perpendicular . grouping question 1 is continued from the previous page. call the class back together to discuss and present their answer for question 1. ask a student volunteer to read the paragraph at the top of the page aloud. pose the guiding questions below to verify student understanding. guiding questions what does reciprocal mean? what does negative reciprocal mean? what is the product of negative reciprocals? grouping have the students work in small groups to complete questions 2 through 6. pose the guiding questions to verify student understanding. common student errors students might need direction in question 5 that they need to use point-slope form. in questions 6, students who are weak equation solvers will struggle with isolating the variable y. some students will not recognize that they need to change the equations to slope-intercept form in order to compare the slopes. guiding questions will the slopes of perpendicular lines ever be the same? explain. will the y-intercepts of perpendicular lines ever be the same? explain. is it easier to pick out the value of slope from an equation in standard form or an equation in slope-intercept form? how do you determine whether two lines are parallel without graphing the lines? get10504.qxd 4/23/08 12:43 pm page 218 2008 text sampler page 176esson 5.4 parallel and perpendicular lines in the coordinate plane 219 2008 carnegie learning, inc. 5 investigate problem 2 students will summarize their findings from investigate problem 2. students will be working in small groups to complete question 6. call the class back together to discuss and present their work for questions 2 through 6. grouping ask a student volunteer to read question 7 aloud. complete question 7 as a whole class. then have students work in groups to complete questions 8 and 9. call the class back together to discuss and present their answers for questions 8 and 9. investigate problem 2 6. without graphing the equations, determine whether the lines given by and are perpendicular . show all your work and use a complete sentence in your answer . 7. complete each statement. when two lines are parallel, their slopes are _______________. when two lines are perpendicular , their slopes are _____________ ___________________________________. 8. suppose that you have a line and you choose one point on the line. how many lines perpendicular to the given line can you draw through the given point? use a complete sentence in your answer . 9. suppose that you have a line and you choose one point that is not on the line. how many lines can you draw through the given point that are perpendicular to the given line? how many lines can you draw through the given point that are parallel to the given line? use complete sentences in your answer . 2x y 4 y 2x 5 the same negative reciprocals of each other you can draw one line through the given point that is perpendicular to the given line. you can draw one line through the given point that is perpendicular to the given line. you can draw one line through the given point that is parallel to the given line. the slopes are not negative reciprocals because . so, the lines are not perpendicular . 2(2) 4 y 2x 4 y 2x 4 y 2x 5 2x y 4 y 2x 5 explore together get10504.qxd 4/23/08 12:47 pm page 219 2008 text sampler page 17720 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 problem 3 a very simple parking lot one final truck parking lot is shown below. one grid square represents a square that is one meter long and one meter wide. a. what angles are formed by the intersection of the parking lot line segments? how do you know? use complete sentences in your answer . b. carefully extend into line p, extend into line q, extend into line r, and extend into line s. c. choose any three points on line q and list their coordinates. choose any three points on line r and list their coordinates. choose any three points on line s and list their coordinates. what do you notice about the x- and y-coordinates of these points? use complete sentences in your answer . kl ij gh gk y 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 x k l i j h g p s r q sample answer: the angles formed are right angles because they meet at the intersection of grid lines. sample answer: (0, 4), (1, 4), (2, 4) sample answer: (0, 8), (1, 8), (2, 8) sample answer: (0, 12), (1, 12), (2, 12) sample answer: for each line, the x-coordinates are different, but the y-coordinates are the same. explore together problem 3 students will investigate vertical and horizontal lines that are also parallel and perpendicular lines. grouping ask for a student volunteer to read the scenario of problem 3 aloud. pose guiding questions to verify student understanding. complete part (a) through part (d) as a whole class. guiding questions what information is given in this problem? what does each square represent? what would you guess about the relationship of the lines? which lines look parallel? what do you know about the slopes of each line? get10504.qxd 4/18/08 1:02 pm page 220 2008 text sampler page 178esson 5.4 parallel and perpendicular lines in the coordinate plane 221 2008 carnegie learning, inc. 5 problem 3 students will continue to investigate horizontal and vertical lines. students will be working as a whole class to complete part (c) and part (d). investigate problem 3 students will formalize their findings about horizontal and vertical lines. solve and discuss question 1 together as a whole class. just the math the terminology for horizontal and vertical lines is formally presented in question 1. grouping ask a student volunteer to read question 1 aloud. have a student restate the problem. then, have students work in pairs to complete question 1. when the students are finished, call the class back together to discuss and present their work for question 1. problem 3 a very simple parking lot what do you think should be the equations of lines q, r, and s? use complete sentences to explain your reasoning. d. choose any three points on line p and list their coordinates. what do you notice about the x- and y-coordinates of these points? use complete sentences in your answer . what do you think should be the equation of line p? use complete sentences to explain your reasoning. investigate problem 3 1. just the math: horizontal and vertical lines in problem 3, you wrote the equations of horizontal and vertical lines. a horizontal line has an equation of the form where a is any real number . a vertical line has an equation of the form where b is any real number . consider your horizontal lines in problem 3. for any horizontal line, if x increases by 1 unit, by how many units does y change? use a complete sentence in your answer . what is the slope of any horizontal line? use complete sentences to explain your reasoning. consider your vertical line in problem 3. suppose that y increases by one unit. by how many units does x change? use a complete sentence in your answer . x b y a the equations should be and because no matter what the x-coordinates are, the y-coordinates are constant. y 12 y 8, y 4, sample answer: (4, 0), (4, 1), (4, 2) the x-coordinates are the same and y-coordinates are different. the equation should be because no matter what the y-coordinates are, the x-coordinates are constant. x 4 the value of y does not change at all. the slope of any horizontal line is zero because the rise is always zero and zero divided by any number is zero. the value of x does not change at all. explore together get10504.qxd 4/23/08 12:47 pm page 221 2008 text sampler page 17922 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 investigate problem 3 what is the rise divided by the run? does this make any sense? use a complete sentence to explain. because division by zero is undefined, we say that a vertical line has an undefined slope. 2. consider the statements about parallel and perpendicular lines in question 7 of problem 2. are these statements true for horizontal and vertical lines? use complete sentences to explain your reasoning. complete the following statements. ________ vertical lines are parallel. ________ horizontal lines are parallel. write a statement that describes the relationship between a vertical line and a horizontal line. use a complete sentence in your answer . 3. write equations for a horizontal line and a vertical line that pass through the point (2, –1). 4. write an equation of the line that is perpendicular to the line given by and passes through the point (1, 0). write an equation of the line that is perpendicular to the line given by and passes through the point (5, 6). y 2 x 5 ; no, because you cannot divide by zero. 1 0 sample answer: the statement about parallel lines is true for horizontal lines because you can compare the slopes, but this statement is not true for vertical lines because you cannot compare their slopes. the statement about perpendicular lines is not true for horizontal and vertical lines because vertical lines have no slope. sample answer: any vertical line and any horizontal line are perpendicular . all all y 1; x 2 y 0 x 5 explore together investigate problem 3 students will write equations of horizontal and vertical lines. grouping ask for a student volunteer to read question 2 aloud. then, have students work with a partner to complete questions 2 through 4. call the class back together to discuss and present their answers for questions 2 through 4. key formative assessments how can you determine whether lines are parallel given the equations of the lines? how can you determine whether lines are parallel given the graph of the lines? what formula is used to find the equation of a line given two points? how can you determine whether lines are perpendicular given the equations of the lines? how can you determine whether lines are perpendicular given the graph of the lines? how do you determine equations of horizontal and vertical lines? what is the slope of a horizontal line? what is the slope of a vertical line? get10504.qxd 4/24/08 8:39 am page 222 2008 text sampler page 180esson 5.4 parallel and perpendicular lines in the coordinate plane 222a 2008 carnegie learning, inc. 5 close review all key terms and their definitions. include the terms slope, y-intercept, point-slope form, slope-intercept form, parallel lines, perpendicular lines, reciprocal, negative reciprocal, horizontal line, vertical line, zero slope, and undefined slope. remind the students to write the key terms and their definitions in the notes section of their notebooks. you may also want the students to include examples. ask the students to explain the meaning of the term slope. write several equations on the board in slope-intercept form. have the students determine which lines are parallel, perpendicular , or intersecting. ask the students to compare the similarities and contrast the differences between parallel and perpendicular lines. wrap up assignment use the assignment for lesson 5.4 in the student assignments book. see the teacher’s resources and assessments book for answers. assessment see the assessments provided in the teacher’s resources and assessments book for chapter 5. open-ended writing t ask ask the students to design a parking lot that is 150 feet by 400 feet. they must maximize the number of cars in the parking lot. how will they paint the lines in the parking lot? explain and show a diagram. follow up get10504.qxd 4/18/08 1:02 pm page 223 2008 text sampler page 18122b chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 reflections insert your reflections on the lesson as it played out in class today. what went well? _______________________________________________________________________________________________ _______________________________________________________________________________________________ what did not go as well as you would have liked? _______________________________________________________________________________________________ _______________________________________________________________________________________________ how would you like to change the lesson in order to improve the things that did not go well and capitalize on the things that did go well? _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ notes get10504.qxd 4/18/08 1:02 pm page 224 2008 text sampler page 1822008 carnegie learning, inc. geometry t eacher’s resources and assessments 1_gem1_fm_v1.qxd 4/25/08 8:03 am page i 2008 text sampler page 183ontents iii 2008 carnegie learning, inc. contents contents section 1 assessments with answers section 2 assignments with answers assessment answer keys and master copies the answer keys and master copies of the student assessments are available online in the carnegie leaning k-12 community. to access the answer keys and master copies of the student assessments, go to http://k12.carnegielearning.com and login with your k-12 password. if you are a first time user or unsure of your password, choose the new users / account help link in the login box for information on registering and/or setting up your password. contact carnegie learning customer support at 1-888-851-7094, option 3, or via email at help@carnegielearning.com for additional assistance. 1_gem1_fm.qxd 5/1/08 8:44 am page iii 2008 text sampler page 184hapter 5 assignments 67 2008 carnegie learning, inc. 5 assignment name ___________________________________________________ date _____________________ assignment for lesson 5.4 parking lot design parallel and perpendicular lines in the coordinate plane state whether each pair of lines is parallel, perpendicular , or neither . explain your answer using a complete sentence. 1. 2. 3. 4. 5. 6. y x y x y 7x 7x 2 y y 1 2 x 2y 2x 10 y 1 6 x 5 y 6x y 1 2 x 9 y 2x 6 y 4x y 4x 18 the lines are neither parallel nor perpendicular because their slopes are not equal and the product of their slopes is not –1. the lines are perpendicular because the product of their slopes is –1. the lines are neither parallel nor perpendicular because their slopes are not equal and the product of their slopes is not –1. the lines are parallel because their slopes are equal. the lines are parallel because their slopes are equal. the lines are perpendicular because the product of their slopes is –1. geg105.qxd 4/22/08 10:30 am page 67 2008 text sampler page 1858 chapter 5 assignments 2008 carnegie learning, inc. 5 7. 8. write the equations of 3 lines that are parallel to 9. write the equations of 3 lines that are perpendicular to 10. write the equation of a line that is perpendicular to 11. write the equation of a line that is perpendicular to the line in your answer to question 10. 12. is the line from your answer in question 11 parallel, perpendicular or neither to the original line in number 10? explain. y 1 3 x 2. y 4x 2. y 2 3 x 7. y 2 x 5 the line is parallel to the line given in question 10 because it has the same slope. sample answer: y 2 3 x 3, y 2 3 x 25, y 2 3 x 1.75 the lines are perpendicular because x 5 is a vertical line and y 2 is a horizontal line. sample answer: y 1 4 x, y 1 4 x 5, y 1 4 x 2 3 sample answer: y 3x 2 sample answer: y 1 3 x 7 geg105.qxd 4/22/08 10:30 am page 68 2008 text sampler page 186hapter 5 assessments 77 2008 carnegie learning, inc. 5 pre-t est name ___________________________________________________ date _____________________ is each pair of lines parallel or skew? use a complete sentence to explain your reasoning. 1. 2. use the figure shown below to answer questions 3 through 6. 3. which line given in the figure is a transversal? use a complete sentence to explain your reasoning. 4. name a pair of alternate interior angles in the figure. use a complete sentence in your answer . a b c 1 2 3 4 5 6 7 8 a b p m sample answer: the lines are skew, because they are not coplanar and they do not intersect. angle 4 and or are alternate interior angles. 3 and 6 5 sample answer: line c is a transversal, because it intersects two lines at different points. sample answer: the lines are parallel, because they are coplanar and they do not intersect. gem105.qxd 4/22/08 12:43 pm page 77 2008 text sampler page 1878 chapter 5 assessments 2008 carnegie learning, inc. 5 5. name a pair of alternate exterior angles in the figure. use a complete sentence in your answer . 6. name a pair of corresponding angles in the figure. use a complete sentence in your answer . 7. in the figure shown below, line x is parallel to line y and the measure of 1 is 64º. find the missing angle measures without using a protractor . use complete sentences to explain how you found your answers. z x y 1 2 3 4 5 6 7 8 pre-t est page 2 angle 1 and 5 or 2 and 6 or 3 and 7 or 4 and 8 are corresponding angles. angle 2 and 7 or 1 and 8 are alternate exterior angles. m 2 is 116 , because 1 and 2 are supplementary angles. m 3 is 116 , because 2 and 3 are vertical angles. m 4 is 64 , because 1 and 4 are vertical angles. m 5 is 64 , because 4 and 5 are alternate interior angles. m 6 is 116 , because 2 and 6 are corresponding angles. m 7 is 116 , because 2 and 7 are alternate exterior angles. m 8 is 64 , because 4 and 8 are corresponding angles. º º º º º º º gem105.qxd 4/22/08 12:43 pm page 78 2008 text sampler page 188hapter 5 assessments 79 2008 carnegie learning, inc. 5 pre-t est page 3 name ___________________________________________________ date _____________________ pre-t est page 3 8. write the equation of a line that is parallel to the line and passes through the point (–3, 1). show all your work and use complete sentences to explain how you found your answer . 9. write the equation of a line that is perpendicular to the line and passes through the point (–3, 1). show your work and use complete sentences to explain how you found your answer . 10. write the equations for a horizontal line and a vertical line that pass through the point (–6, 4). horizontal line: ________ vertical line: ________ y 3x 4 y 3x 4 sample answer: a line parallel to the line must have a slope of 3. because you know the line must pass through the point (–3, 1), use the point-slope form to write the equation. y 1 3(x (3) ); y 3x 10 y 3x 4 sample answer: a line perpendicular to the line must have a slope of . because you know that the line must pass through (–3, 1), use the point-slope form to write the equation. y 1 1 3 (x (3) ); y 1 3 x 1 3 y 3x 4 y 4 x –6 gem105.qxd 4/22/08 12:43 pm page 79 2008 text sampler page 1890 chapter 5 assessments 2008 carnegie learning, inc. 5 11. what is the length of ? use a complete sentence to explain your reasoning. 12. in the figure shown below, bisects a, which has a measure of 84 . what is the value of x? use a complete sentence in your answer . a b c x d º ad y 1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 x a b c x y xy pre-t est page 4 sample answer: is a midsegment. so it is parallel to and half the length of the side that it does not intersect, . therefore, the length of is 7 units. xy ab xy the value of x is 42 . º gem105.qxd 4/22/08 12:43 pm page 80 2008 text sampler page 190hapter 5 assessments 81 2008 carnegie learning, inc. 5 13. sketch the perpendicular bisector of segment . match each point of concurrency with its definition. 14. incenter a. the point at which the altitudes of the triangle intersect. 15. circumcenter b. the point at which the angle bisectors of the triangle intersect. 16. centroid c. the point at which the medians of the triangle intersect. 17. orthocenter d. the point at which the perpendicular bisectors of the triangle intersect. y x xy pre-t est page 5 name ___________________________________________________ date _____________________ a c d b gem105.qxd 4/22/08 12:43 pm page 81 2008 text sampler page 1912 chapter 5 assessments 2008 carnegie learning, inc. 5 gem105.qxd 4/22/08 12:43 pm page 82 2008 text sampler page 1922008 carnegie learning, inc. geometry homework helper geh1_fm.qxd 4/28/08 10:13 am page i 2008 text sampler page 1936 chapter 5 homework helper 2008 carnegie learning, inc. 5 students should be able to answer these questions after lesson 5.4: how are the slopes of parallel and perpendicular lines determined? how are the equations of parallel and perpendicular lines determined? how are the equations of horizontal and vertical lines determined? read question 1 and its solution. then, write the equations of one line parallel to and one line perpendicular to the given line in questions 2 and 3. 1. write the equation of one line that is parallel and one that is perpendicular to the line represented by the equation . step 1 identify the slope of the equation. the slope is the coefficient of the x-term. in this example, the slope is 2. step 2 determine the slopes of the parallel and perpendicular lines. parallel lines have the same slope. lines parallel to have a slope of 2. perpendicular lines have slopes that are negative reciprocals of each other . lines perpendicular to have a slope of . step 3 write the equations of the lines. the value of the constant in the equation does not affect whether the line is parallel or perpendicular . parallel line: perpendicular line: 2. 3. parallel line: ____________________ parallel line: ____________________ perpendicular line: ____________________ perpendicular line: ____________________ read question 4 and its solution. then complete question 5. 4. write the equation of a horizontal line and a vertical line that passes through the point (–1, 2). step 1 a horizontal line will pass through the y-coordinate. the horizontal line has the equation step 2 a vertical line will pass through the x-coordinate. the vertical line has the equation 5. write the equation of a horizontal line and a vertical line that passes through (3, –6). x 1. y 2. y 2 3 x 1 y 3x 5 y 1 2 x 4 y 2x 7 1 2 y 2x 1 y 2x 1 y 2x 1 parking lot design parallel and perpendicular lines in the coordinate plane 5.4 directions geh105.qxd 4/28/08 10:17 am page 36 2008 text sampler page 1942008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 195 2008 text sampler page 195ourse description – algebra ii carnegie learning tm algebra ii is designed as a second-year algebra course. it can be implemented with students at a variety of ability and grade levels. algebra ii focuses heavily on functions. throughout the course, students understand and compare the properties of classes of functions including quadratic, polynomial, exponential, logarithmic, rational, radical, and circular. students extend their understanding of linear functions to include a wide range of function types. for each family of function, students explore graphical behavior and characteristics including general shape, x- and y-intercepts, rate of change, extrema, intervals of increase and decrease, domain, and range. students simplify expressions using techniques including factoring and properties of exponents and radicals. students develop ability to solve equations for each function and understand the relationship between solutions algebraically and graphically. students develop an understanding of arithmetic and geometric sequences as linear and exponential functions with whole number domains. students determine arithmetic and geometric series, including infinite series. students explore conic sections algebraically and graphically. algebra ii also includes select topics in probability and statistics. students find simple and compound probabilities, including experimental probabilities, and are introduced to combinations and permutations. students explore variation, standard deviation, and variance. 2008 text sampler page 196lgebra ii text set2008 carnegie learning, inc. contents linear functions, equations, and inequalities p. 1 1.1 tanks a lot introduction to linear functions p. 3 1.2 calculating answers solving linear equations and linear inequalities in one variable p. 11 1.3 running a 10k slope–intercept form of linear functions p. 21 1.4 pump it up standard form of linear functions p. 29 1.5 shifts and flips basic functions and linear transformations p. 37 1.6 inventory and sand determining the equations of linear functions p. 47 1.7 absolutely! absolute value in equations and inequalities in one and two variables p. 55 1.8 inverses and pieces functional notation, inverses, and piecewise functions p. 67 systems of linear equations and inequalities p. 75 2.1 finding a job introduction to systems of linear equations p. 77 2.2 pens-r-us solving systems of linear equations: graphing and substitution p. 83 2.3 tickets solving systems of linear equations: linear combinations p. 91 2.4 cramer’s rule solving systems of linear equations: cramer’s rule p. 99 2.5 consistent and independent systems of linear equations: consistent and independent p. 105 2.6 inequalities–infinite solutions solving linear inequalities and systems of linear inequalities in two variables p. 113 2.7 three in three or more solving systems of three or more linear equations in three or more p. 121 contents 1 2 iv contents a2s10100.qxd 7/23/08 1:01 pm page iv 2008 text sampler page 1982008 carnegie learning, inc. quadratic functions p. 129 3.1 lots and projectiles introduction to quadratic functions p. 131 3.2 intercepts, vertices, and roots quadratic equations and functions p. 137 3.3 quadratic expressions multiplying and factoring p. 143 3.4 more factoring special products and completing the square p. 155 3.5 quadratic formula solving quadratic equations using the quadratic formula p. 165 3.6 graphing quadratic functions properties of parabolas p. 173 3.7 graphing quadratic functions basic functions and transformations p. 181 3.8 three points determine a parabola deriving quadratic functions p. 193 3.9 the discriminant the discriminant and the nature of roots/vertex form p. 199 the real number system p. 209 4.1 thinking about numbers counting numbers, whole numbers, integers, rational and irrational numbers p. 211 4.2 real numbers properties of the real number system p. 217 4.3 man-made numbers imaginary numbers and complex numbers p. 223 4.4 the complete number system operations with complex p. 229 polynomial functions p. 235 5.1 many terms introduction to polynomial expressions, equations, and functions p. 237 5.2 roots and zeros solving polynomial equations and inequalities: factoring p. 245 3 contents 4 5 contents v a2s10100.qxd 7/23/08 1:01 pm page v 2008 text sampler page 1992008 carnegie learning, inc. 5.3 successive approximations, tabling, zooming/tracing, and calculating solving polynomial equations: approximations and graphing p. 253 5.4 it’s fundamental the fundamental theorem of algebra p. 259 5.5 when division is synthetic polynomial and synthetic division p. 263 5.6 remains of a polynomial the remainder and factor theorems p. 273 5.7 out there and in between extrema and end behavior p. 279 exponential and logarithmic functions p. 289 6.1 the wizard and the king introduction to exponential functions p. 291 6.2 a review properties of whole number exponents p. 297 6.3 exponents, reciprocals, and roots integral and rational exponents p. 307 6.4 the hockey stick graph applications of exponential functions p. 313 6.5 log a what? inverses of exponential functions: logarithmic functions p. 325 6.6 properties of logarithms the remainder and factor theorems p. 333 6.7 continuous growth, decay, and interest solving exponential and logarithmic equations p. 339 rational equations and functions p. 351 7.1 cars and growing old introduction to rational functions p. 353 7.2 rational expressions, part i simplifying, adding, and subtracting rational expressions p. 361 7.3 rational expressions, part ii multiplying and dividing rational expressions p. 365 7.4 solutions solving rational equations and inequalities p. 369 7.5 holes and breaks graphing rational functions and discontinuities p. 383 contents 7 vi contents 6 a2s10100.qxd 7/23/08 1:01 pm page vi 2008 text sampler page 2002008 carnegie learning, inc. 7.6 work, mixture, and more applications of rational equations and functions p. 399 radical equations and functions p. 411 8.1 inverses of inverses introduction to radical functions and expressions p. 413 8.2 radical expressions simplifying, adding, and subtracting radical expressions p. 423 8.3 solutions solving radical equations p. 429 8.4 graphs graphing radical functions p. 437 conic sections p. 449 9.1 conics? conic sections p. 451 9.2 round and round circles p. 457 9.3 it’s a stretch ellipses p. 465 9.4 more asymptotes hyperbolas p. 477 9.5 ultimate focus parabolas p. 493 9.6 antennas, whispering rooms, and more applications of conics p. 507 t rigonometric ratios and circular functions p. 527 10.1 the unit circle angle measures p. 529 10.2 circular functions, part i sine and cosine functions p. 537 10.3 circular functions, part ii tangent function p. 543 10.4 you mean there are more? other circular functions p. 551 10.5 arc functions inverses of circular functions p. 557 8 9 1 0 contents contents vii a2s10100.qxd 7/23/08 1:01 pm page vii 2008 text sampler page 2012008 carnegie learning, inc. t rigonometric graphs, identities, and equations p. 565 11.1 ups and downs graphs of circular functions p. 567 11.2 transformations amplitude, period, phase shift p. 575 11.3 identical? trigonometric identities p. 587 11.4 solutions solving trigonometric equations p. 595 11.5 rabbits and seasonal affective disorder applications of circular functions p. 601 11.6 angle-angle-side and angle-side-angle law of sines p. 613 11.7 side-angle-side and side-side-side law of cosines p. 621 sequences and series p. 627 12.1 college tutoring introduction to arithmetic sequences p. 629 12.2 too much homework! introduction to geometric sequences p. 637 12.3 sums a lot arithmetic and geometric series p. 641 12.4 summing forever sum of infinite geometric series p. 647 counting methods and probability p. 655 13.1 rolling, flipping, and pulling probability and sample spaces p. 657 13.2 multiple trials compound probability p. 665 13.3 counting permutations and combinations p. 673 13.4 pascal and independent events pascal’s triangle and the binomial theorem p. 681 13.5 the theoretical and the actual experimental versus theoretical probability p. 689 contents 1 3 viii contents 1 1 1 2 a2s10100.qxd 7/23/08 1:01 pm page viii 2008 text sampler page 2022008 carnegie learning, inc. statistics p. 695 14.1 averages measures of central tendency, quartiles, and percentiles p. 697 14.2 spread variation and standard deviation p. 703 14.3 normal? distribution p. 709 14.4 line of best fit linear regressions p. 719 14.5 not all data are linear other regressions p. 727 matrices p. 735 15.1 arrays, arrays! introduction to matrices and matrix operations p. 737 15.2 rows times columns matrix multiplication p. 743 15.3 solving systems of linear equations matrices p. 747 15.4 multiplicative inverses solving matrix equations p. 753 15.5 calories and lunch applications of matrices p. 759 glossary g-1 index i-1 1 4 1 5 contents ix contents a2s10100.qxd 7/23/08 1:01 pm page ix 2008 text sampler page 2032008 carnegie learning, inc. algebra ii student t ext a2s10100.qxd 7/23/08 1:01 pm page i 2008 text sampler page 2042008 carnegie learning, inc. 1.1 tanks a lot introduction to linear functions p. 3 1.2 calculating answers solving linear equations and linear inequalities in one variable p. 11 1.3 running a 10k slope–intercept form of linear functions p. 21 1.4 pump it up standard form of linear functions p. 29 1.5 shifts and flips basic functions and linear transformations p. 37 1.6 inventory and sand determining the equations of linear functions p. 47 1.7 absolutely! absolute value equations and inequalities p. 55 1.8 inverses and pieces functional notation, inverses, and piecewise functions p. 67 inventory is the list of items that businesses stock in stores and warehouses to supply customers. businesses in the united states keep about 1.5 trillion dollars worth of goods in inventory. you will use linear functions to manage the inventory levels of a business. 1 chapt er linear functions, equations, and inequalities chapter 1 linear functions, equations, and inequalities 1 1 a2s10101.qxd 7/10/08 12:46 pm page 1 2008 text sampler page 205chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. mathematical representations introduction mathematics is a human invention, developed as people encountered problems that they could not solve. for instance, when people first began to accumulate possessions, they needed to answer questions such as: how many? how many more? how many less? people responded by developing the concepts of numbers and counting. mathematics made a huge leap when people began using symbols to represent numbers. the first “numerals” were probably tally marks used to count weapons, livestock, or food. as society grew more complex, people needed to answer questions such as: who has more? how much does each person get? if there are 5 members in my family, 6 in your family, and 10 in another family, how can each person receive the same amount? during this course, we will solve problems and work with many different representations of mathematical concepts, ideas, and processes to better understand our world. the following processes can help you solve problems. discuss to understand • read the problem carefully. • what is the context of the problem? do you understand it? • what is the question that you are being asked? does it make sense? think for yourself • do i need any additional information to answer the question? • is this problem similar to some other problem that i know? • how can i represent the problem using a picture, a diagram, symbols, or some other representation? work with your partner • how did you do the problem? • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? work with your group • show me your representation. • this is the way i thought about the problem—how did you think about it? • what else do we need to solve the problem? • does our reasoning and our answer make sense to one another? • how can we explain our solution to one another? to the class? share with the class • here is our solution and how we solved it. • we could only get this far with our solution. how can we finish? • could we have used a different strategy to solve the problem? 1 a2s10101.qxd 7/10/08 12:46 pm page 2 2008 text sampler page 206se the table to graph the functions, and indicate the transformations, both in terms of transforming the equation and the graph, which were performed on the basic function to arrive at the transformed function. 2008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 37 problem 1 we have now worked with two different forms of linear functions, slope–intercept form and standard form. you should remember that a function is defined as a relation for which every input value has one and only one output value. we are going to look at linear functions from the view of a family of functions. the most basic form for a linear function is y x which is called the basic function. any linear function can be constructed through a series of transformations to the basic function. objectives in this lesson, you will define basic functions. use translations, dilations, and reflections to transform linear functions. graph parallel lines. graph perpendicular lines. key t erms basic function dilation reflection line of reflection 1.5 shifts and flips basic functions and linear t ransformations remember a dilation is a transformation of a figure in which the figure stretches or shrinks with respect to a fixed point. remember a reflection is a transformation in which a figure is reflected, or flipped, in a given line called the line of reflection. 1 algebraic graphical transformations transformations add a constant shift up subtract a constant shift down multiply or divide by a positive constant dilation multiply by 1 reflection 1 a2s10101.qxd 7/22/08 4:46 pm page 37 2008 text sampler page 207lgebraic transformation: graphical transformation: 38 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1. basic function y x algebraic transformation: graphical transformation: 2. y x 3 1 a2s10101.qxd 7/10/08 1:14 pm page 38 2008 text sampler page 2082008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 39 3. y x 4 algebraic transformation: graphical transformation: 4. y 2x algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 39 2008 text sampler page 2090 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 5. y 2x 1 algebraic transformation: graphical transformation: 6. y 3x algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 40 2008 text sampler page 2102008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 41 7. y 4x 1 algebraic transformation: graphical transformation: 8. y 3x 5 algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 41 2008 text sampler page 2112 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 9. y 2 3 x 1 algebraic transformation: graphical transformation: 10. y 1 2 x 3 algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 42 2008 text sampler page 2122008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 43 for each of the following equations of linear functions, describe the transformations you would need to perform to the graph of the basic function in order to transform it into the given function. 11. y 4x 12. y x 7 13. y 2x 7 14. y 7x 11 problem 2 graph the following equations on the same grid. 1. y 2x and y 2x 5 2. describe how the graphs are related geometrically. 1 a2s10101.qxd 7/10/08 1:14 pm page 43 2008 text sampler page 2134 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 3. graph the following equations on the same grid. y 3x and y 3x 5 4. describe how the graphs are related geometrically. 5. what conclusion might you make about equations with the same slope? 6. the graphs of y 2x and are shown on the graph. y 1 2 x from the x-axis, draw a line segment vertically from (2, 0) to the line y 2x to form a right triangle. from the x-axis, draw a line segment vertically from (4, 0) to the line y to form a second right triangle. 1 2 x 1 4 2 6 8 –4 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x – 1 2 a2s10101.qxd 7/10/08 1:14 pm page 44 2008 text sampler page 2142008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 45 7. using what you know from geometry, why are the two triangles congruent? what can you conclude about the angles formed by the intersecting lines y 2x and y ? 1 2 x 8. the graphs of y 3x and are on the grid. y 1 3 x from the x-axis, draw a line segment vertically from (2, 0) to the line y 3x to form a right triangle. from the x-axis, draw a line segment vertically from (4, 0) to the line y to form a second right triangle. 9. using what you know from geometry, why are the two right triangles congruent? what can you conclude about the angles formed by the intersecting lines y 3x and y ? 10. what conclusion can you draw about linear functions with related slopes? be prepared to share your work with another pair , group, or the entire class. 1 3 x 1 3 x 1 4 2 6 8 6 8 4 –6 –4 –8 –2 y x –8 –6 y 3x y x – 1 3 a2s10101.qxd 7/10/08 1:14 pm page 45 2008 text sampler page 215algebraii teacher’s implementationguide 2008 text sampler page 216 2008 text sampler page 216esson 1.5 basic functions and linear transformations 37a 2008 carnegie learning, inc. 1 1.5 shifts and flips basic functions and linear t ransformations objectives define basic functions. use translations, dilations and reflections to transform linear functions. graph parallel lines. graph perpendicular lines. key t erms basic function dilation reflection line of reflection nctm content standards grades 9–12 expectations algebra standards understand relations and functions and select, convert flexibly among, and use various representations for them. use symbolic algebra to represent and explain mathematical relationships. geometry standards understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices. use various representations to help understand the effects of simple transformations and their compositions. essential ideas algebraic and geometric transformations of linear functions are distinct. vertical shifts change the y-intercept. dilations are changes in the coefficient of the slope. reflections are a change in the sign of the slope. parallel lines have the same slope. perpendicular lines have slopes that are negative reciprocals. essential questions 1. describe the similarities and differences between the graph of y x and the graph of y x 5. 2. how can you tell the graph of a linear function has been shifted down when you look at its equation? 3. how can you tell the graph of a linear function has been dilated when you look at its equation? 4. describe the similarities and differences between the graph of y x and the graph of y 3x. 5. describe the similarities and differences between the graph of y 5x and the graph of y 5x. 6. can you look at two equations and tell if the lines are parallel? explain. 7. can you look at two equations and tell if the lines are perpendicular? explain.7b chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 show the way warm up provide students with a graph of y 2x and ask them to write a different equations that would be parallel to the equation y 2x the graph should be of any equation with a slope of 2. perpendicular to the equation y 2x the graph should be of any equation with a slope of . the same as the equation y 2x the original graph. motivator 1. describe the difference between the graph of y 2x and the graph of y 2x 8 without graphing the linear functions. the second graph is the first shifted down 8 units. 2. describe the difference between the graph of y 2x and the graph of y 2x 8 without graphing the linear functions. the second graph is the first shifted up 8 units. 3. identify the x- and y-intercept in the equation y 2x 8. the x-intercept is 4 and the y-intercept is 8. 4. identify the slope in the equation y 2x 8. the slope is 2. 1 2esson 1.5 basic functions and linear transformations 37 2008 carnegie learning, inc. 1 problem 1 this activity is both a review of geometric transformations and an introduction to transformations of linear functions from a basic function and through algebraic transformations. students will also find how the slopes of parallel and perpendicular lines are related. grouping have a student read the first paragraph of problem 1. you may want to ask some or all of the following questions: have we worked with other families of functions? if so, what were they? what is functional notation? use the table to graph the functions and indicate the transformations, both in terms of transforming the equation and the graph, which were performed on the basic function to arrive at the transformed function. problem 1 we have now worked with two different forms of linear functions, slope–intercept form and standard form. you should remember that a function is defined as a relation for which every input value has one and only one output value. we are going to look at linear functions from the view of a family of functions. the most basic form for a linear function is y x which is called the basic function. any linear function can be constructed through a series of transformations to the basic function. objectives in this lesson, you will define basic functions. use translations, dilations, and reflections to transform linear functions. graph parallel lines. graph perpendicular lines. key t erms basic function dilation reflection line of reflection 1.5 shifts and flips basic functions and linear t ransformations remember a dilation is a transformation of a figure in which the figure stretches or shrinks with respect to a fixed point. remember a reflection is a transformation in which a figure is reflected, or flipped, in a given line called the line of reflection. algebraic graphical transformations transformations add a constant shift up subtract a constant shift down multiply or divide by a positive constant dilation multiply by 1 reflection remember a dilation is a transformation of a figure in which the figure stretches or shrinks with respect to a fixed point. remember a reflection is a transformation in which a figure is reflected, or flipped, in a given line called the line of reflection.8 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 guiding questions have another student read the rest of problem 1. you may want to ask some or all of the following questions: where have you used these algebraic transformations? where have you used these geometric transformations? how do the algebraic transformations relate to the geometric transformations? note make sure that students know how to graph the basic function and the transformed function or image of the transformation. grouping have students work in pairs or groups to complete problem 1. this should take about 15 minutes. be sure to instruct the students to graph the basic function on each graph first so the transformation is visual. algebraic transformation: graphical transformation: 1. basic function y x algebraic transformation: graphical transformation: 2. y x 3 there were no transformations. there were no transformations. add 3. the line shifts up 3 units. 2 4 6 8 –2 –4 2 o 6 8 4 –6 –8 –4 –2 y x –8 –6 y x 4 6 8 –2 –4 2 o 6 8 4 –6 –8 –2 y x –8 –6 y x y x 3esson 1.5 basic functions and linear transformations 39 2008 carnegie learning, inc. 1 note make sure to circulate throughout the class to monitor student progress and facilitate student learning. most students will have little difficulty with vertical shifts. 3. y x 4 algebraic transformation: graphical transformation: 4. y 2x algebraic transformation: graphical transformation: 4 2 6 8 –2 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y x – 4 subtract 4. the line shifts down 4 units. double the slope. there is a dilation of 2. 4 2 6 8 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 2x0 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 common student errors students may struggle with the change in slope being a dilation. unlike in transformations of geometric figures where a dilation enlarges or shrinks the figure, with linear functions, a dilation just changes the slope—the unit rate of change. students should be encouraged to apply multiple transformations in different orders to determine if order matters. as you circulate, if you see “different” student solutions, make sure that these are reported out when the groups share their answers. in the event of different visuals for the same problem, usually due to scaling, ask students to explain their thinking, and ask the class to accept or deny their explanations with a supporting counter example where appropriate. note the change in sign of the slope produces a reflection. 5. y 2x 1 algebraic transformation: graphical transformation: 6. y 3x algebraic transformation: graphical transformation: multiply by 3. there is a dilation of 3 followed by a reflection around the x-axis. double slope and then subtract 1. there is a dilation of 2 followed by a shift down of 1 unit. 4 2 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x y 2x – 1 4 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 3x y –3xesson 1.5 basic functions and linear transformations 41 2008 carnegie learning, inc. 1 7. y 4x 1 algebraic transformation: graphical transformation: 8. y 3x 5 algebraic transformation: graphical transformation: 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 4x y –4x y –4x 1 multiply by 4 and then add 1. there is a dilation of 4, followed by reflection around x-axis, and then shift up 1 unit. 4 2 6 8 2 o 6 8 4 –6 –4 –8 –2 y x –6 y x y 3x y 3x – 5 multiply by 3 and then subtract 5. there is a dilation by a factor of 3 and then shift down 5 units.2 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 9. y 2 3 x 1 algebraic transformation: graphical transformation: 10. y 1 2 x 3 algebraic transformation: graphical transformation: 4 2 6 8 2 o 6 8 4 –6 –4 –8 y x –6 –8 –2 –4 y x y 3x y x 2 3 y x 1 2 3 multiply by and then subtract 3. 1 2 there is a dilation by a factor of , reflection around the x-axis and shift down 3 units. 1 2 multiply by and then add 1. 2 3 there is a dilation by a factor of and then shift up 1 unit. 2 3 4 2 6 8 6 8 4 –6 –8 y x –6 –8 –2 –4 y x y x – 3 – 1 2 y x – 1 2 y x 1 2esson 1.5 basic functions and linear transformations 43 2008 carnegie learning, inc. 1 grouping after an appropriate amount of time, pull the class back together to share their solutions. problem 2 problem 2 asks students to determine how the slopes of parallel and perpendicular lines are related. the relationship for parallel lines is more obvious in that vertical shifts produce parallel lines. grouping have students work though problem 2 in pairs or groups for about 15 minutes. for each of the following equations of linear functions, describe the transformations you would need to perform to the graph of the basic function in order to transform it into the given function. 11. y 4x 12. y x 7 13. y 2x 7 14. y 7x 11 problem 2 graph the following equations on the same grid. 1. y 2x and y 2x 5 2. describe how the graphs are related geometrically. perform a dilation by a factor of 4. perform a shift down 7 units. perform a dilation by a factor of 2, a reflection around the x-axis, and a shift up 7 units. perform a dilation by a factor of 7, a reflection around the x-axis, and a shift down 11 units. 2 6 8 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y 2x 5 the lines are parallel.4 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 3. graph the following equations on the same grid. y 3x and y 3x 5 4. describe how the graphs are related geometrically. 5. what conclusion might you make about equations with the same slope? 6. the graphs of y 2x and are shown on the graph. y 1 2 x from the x-axis, draw a line segment vertically from (2, 0) to the line y 2x to form a right triangle. from the x-axis, draw a line segment vertically from (4, 0) to the line y to form a second right triangle. 1 2 x 4 2 6 8 –4 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x – 1 2 the lines are parallel. equations with the same slope have graphs that are parallel. 4 6 8 –4 2 o 6 8 4 –6 –4 –8 y x –8 –6 y –3x y –3x – 5esson 1.5 basic functions and linear transformations 45 2008 carnegie learning, inc. 1 math note the relationship for slopes of perpendicular lines may be shown by drawing the appropriate right triangles and proving the triangles congruent. the angle formed by the lines is the sum of the two acute complimentary angles and therefore forms a right angle, thereby making the lines perpendicular . although this is not a “proof” this is a fairly straightforward illustration of this relationship. grouping after an appropriate amount of time, pull the class back together to share their solutions. essential ideas algebraic and geometric transfor-mations of linear functions are distinct. vertical shifts change the y-intercept. dilations are changes in the coefficient of the slope. reflections are a change in the sign of the slope. parallel lines have the same slope. perpendicular lines have slopes that are negative reciprocals. 7. using what you know from geometry, why are the two triangles congruent? what can you conclude about the angles formed by the intersecting lines y 2x and y ? 1 2 x 8. the graphs of y 3x and are on the grid. y 1 3 x from the x-axis, draw a line segment vertically from (2, 0) to the line y 3x to form a right triangle. from the x-axis, draw a line segment vertically from (4, 0) to the line y to form a second right triangle. 9. using what you know from geometry, why are the two right triangles congruent? what can you conclude about the angles formed by the intersecting lines y 3x and y ? 10. what conclusion can you draw about linear functions with related slopes? be prepared to share your work with another pair , group, or the entire class. 1 3 x 1 3 x 4 2 6 8 6 8 4 –6 –4 –8 –2 y x –8 –6 y 3x y x – 1 3 the right triangles are congruent because their angles are congruent. the angles formed by the lines are right angles and the lines are perpendicular . the right triangles are congruent because their angles are congruent. the angles formed by the lines are right angles and the lines are perpendicular . linear functions with slopes that are negative reciprocals have graphs that are perpendicular . 4 2 6 8 4 –6 –4 –8 –2 y 6 y y x – 1 36 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 follow up assignment use the assignment for lesson 1.5 in the student assignments book. see the teacher’s resources and assessments book for answers. assessment see the assessments provided in the teacher’s resources and assessments book for chapter 1. check students’ understanding using y 3x, ask the students to write equations and sketch graphs that would model the following trans-formations: a vertical shift of 7 a horizontal shift of 3 a dilation of 4 a reflection and dilation of .5 a reflection and vertical shift of 3 notesalgebraii teacher’s resourcesandassessments 2008 text sampler page 229 2008 text sampler page 2292008 carnegie learning, inc. contents section 1 assessments with answers section 2 assignments with answers assessment answer keys and master copies the answer keys and master copies of the student assessments are available online in the carnegie leaning k-12 community. to access the answer keys and master copies of the student assessments, go to http://k12.carnegielearning.com and login with your k-12 password. if you are a first time user or unsure of your password, choose the new users / account help link in the login box for information on registering and/or setting up your password. contact carnegie learning customer support at 1-888-851-7094, option 3, or via email at help@carnegielearning.com for additional assistance. contents 2008 text sampler page 2302008 carnegie learning, inc. 1 shifts and flips basic functions and linear transformations graph the basic function y x on each grid. then graph the given function and describe the transformation that was performed on the basic function to result in the given function. describe the transformation both algebraically and graphically. 1. 2. y 3x y x 5 assignment name _____________________________________________ date _____________________ assignment for lesson 1.5 y x 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 –5 –4 –3 –2 –1 5 6 4 3 2 1 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 0 algebraically: subtract 5. algebraically: multiply by 3. graphically: shift down 5 units. graphically: dilate by 3. chapter 1 assignments 13 2008 text sampler page 2312008 carnegie learning, inc. 1 3. 4. graph the basic function y x on each grid. then graph the function that results after performing the given transformation on the parent function. then write an equation for the new function. 1. shift up 2 units. 2. reflect in the x -axis. y 2x 3 y 1 2 x 4 algebraically: multiply by then add 4. algebraically: multiply by 2, then subtract 3. graphically: dilate by and shift up graphically: dilate by 2, reflect in 4 units. x-axis, and shift down 3 units. 1 2 1 2 , y x 12 10 8 6 4 2 –2 –4 –6 –8 –10 –10 –8 –6 –4 –2 10 12 8 6 4 2 0 y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 y x y x 2 14 chapter 1 assignments 2008 text sampler page 2322008 carnegie learning, inc. 1 name______________________________________________ date _____________________ 3. dilate by 4. then shift down 3 units. 4. dilate by then reflect in x -axis. 1 3 . y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y 1 3 x y 4x 3 chapter 1 assignments 15 2008 text sampler page 23316 chapter 1 assignments 2008 carnegie learning, inc. 2008 text sampler page 234hapter 1 assessments 1 1. tommy’s farm has a water tank to supply all the cattle with fresh water . the tank contains 400 gallons of water . water is pumped out to the animals at a rate of 15 gallons per minute. a. write a linear equation that represents the volume of water in the tank at any given time since it was full. b. to the nearest gallon, how much water will be left in the tank after 10 minutes? c. to the nearest minute, how long will it take the entire tank to empty? d. for what time period will the tank contain more than 200 gallons? show your answer as an inequality and to the nearest minute. pre-t est name _________________________________________________________ date _________________________ 2008 carnegie learning, inc. 1 400 15t x gallons after 10 minutes, 250 gallons will be left in the tank. x 250 400 150 x 400 15(10) x 400 15t x minutes it will take the tank about 27 minutes to empty. t 27 t 400 15 400 15t 400 15t 0 minutes t 13 15t 200 400 15t 200 2008 text sampler page 235chapter 1 assessments 2. a. graph the equation . b. what are the x-intercept, y-intercept, and slope of the line? c. if the graph is shifted up 4 units, write the equation of the new graph. 3. determine the equation of the line in slope-intercept form that passes through the points (4, 6) and (1, 3). y 2x 2 pre-t est page 2 2008 carnegie learning, inc. 1 x-intercept 1 y-intercept 2 m 2 (4, 6) and (1, 3) substitute (4, 6). y x 2 b 2 6 (4) b y x b m ( y 2 y 1 ) (x 2 x 1 ) (3 6) (1 4) 3 3 1 y 1 6, y 2 3 x 1 4, x 2 1 y mx b y 2x 6 y x –8 –6 –4 –2 8 6 4 2 6 8 4 2 –2 –4 –6 –8 0 2008 text sampler page 236hapter 1 assessments 3 4. determine the equation of the line that is perpendicular to the line y 2x 5 and passes through the point (8, 4). 5. determine the inverse of the function . f(x) 5x pre-t est page 3 name _________________________________________________________ date _________________________ 2008 carnegie learning, inc. 1 f 1 (x) x 5 f(x) 5x perpendicular line: substitute (8, 4). y 1 2 x 8 b 8 4 1 2 (8) b y 1 2 x b y mx b m 1 2 m 2 y 2x 5 2008 text sampler page 237chapter 1 assessments 2008 carnegie learning, inc. 1 2008 text sampler page 238