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Sullivan Algebra & Trigonometry 9th txtbk

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Welcome to the Math Online Wikidot page! Here you are able to browse through a wide range of mathematics topics and enhance your understanding of mathematics! The pages on Math Online provide many examples and visuals to enhance learning while still delivering the important key concepts to understanding undergraduate mathematics. Be careful though! All of these notes have been written by me throughout my undergraduate and graduate degree, so many pages contain typos, errors, or are not written in the most suitable manner, so use these notes at your own risk!

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MAY 1ST, 2019: Sorry for being away for over half a year! I have been quite busy with my own personal studies. I plan to continue to add pages on a slightly more consistent basis from now on, but not as frequently as I used to due to time constraints. I have received a few very nice comments from you guys saying that you appreciate the site, and I'm really glad for it! I created this site 6 years ago when I was starting my undergraduate degree as a way for me to keep my notes and solved problems in case I needed or wanted to relearn them down the road. I had no idea that so many people would enjoy reading my notes and problems (although a lot of them are error-riden!) Thank you all for your kind words and I'm glad you're enjoying the site. Also, I have corrected a lot of the errors posted on the errors page. Many reported errors haven't been fixed yet (and I probably won't be able to correct them at this point in time unless I relearn the material from scratch, which isn't an option for me at the present time unfortunately). Regardless, thank you all who have reported errors, and I'm sorry if any reported error hasn't been fixed by me yet. I hope to get around to it... some day. Keep your eyes keen though, as a large part of math is questioning things that don't look quite right =)!

SEPTEMBER 2ND, 2015: Just changed the Math Online favicon to a modern and sleek $*$ operation symbol!

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Welcome to the Math Online Wikidot page! Here you are able to browse through a wide range of mathematics topics and enhance your understanding of mathematics! The pages on Math Online provide many examples and visuals to enhance learning while still delivering the important key concepts to understanding undergraduate mathematics.

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AcademyScope is best experienced on a desktop or tablet display. Unfortunately, your device isn't wide enough to support browsing on AcademyScope. Come back and explore our books when you have access to a larger screen! While you're on the go, you can still browse our publications by topic.

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AcademyScope | The National Academies Press

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AcademyScope is best experienced on a desktop or tablet display. Unfortunately, your device isn't wide enough to support browsing on AcademyScope. Come back and explore our books when you have access to a larger screen! While you're on the go, you can still browse our publications by topic.

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Data science is emerging as a field that is revolutionizing science and industries alike. Work across nearly all domains is becoming more data driven, affecting both the jobs that are available and the skills that are required. As more data and ways of analyzing them become available, more aspects of the economy, society, and daily life will become dependent on data. It is imperative that educators, administrators, and students begin today to consider how to best prepare for and keep pace with this data-driven era of tomorrow. Undergraduate teaching, in particular, offers a critical link in offering more data science exposure to students and expanding the supply of data science talent.

Data Science for Undergraduates: Opportunities and Options offers a vision for the emerging discipline of data science at the undergraduate level. This report outlines some considerations and approaches for academic institutions and others in the broader data science communities to help guide the ongoing transformation of this field.

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Read "Data Science for Undergraduates: Opportunities and Options" at NAP.edu

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Data science is emerging as a field that is revolutionizing science and industries alike. Work across nearly all domains is becoming more data driven, affecting both the jobs that are available and the skills that are required. As more data and ways of analyzing them become available, more aspects of the economy, society, and daily life will become dependent on data.

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Mentorship is a catalyst capable of unleashing one's potential for discovery, curiosity, and participation in STEMM and subsequently improving the training environment in which that STEMM potential is fostered. Mentoring relationships provide developmental spaces in which students' STEMM skills are honed and pathways into STEMM fields can be discovered. Because mentorship can be so influential in shaping the future STEMM workforce, its occurrence should not be left to chance or idiosyncratic implementation. There is a gap between what we know about effective mentoring and how it is practiced in higher education.

The Science of Effective Mentorship in STEMM studies mentoring programs and practices at the undergraduate and graduate levels. It explores the importance of mentorship, the science of mentoring relationships, mentorship of underrepresented students in STEMM, mentorship structures and behaviors, and institutional cultures that support mentorship. This report and its complementary interactive guide present insights on effective programs and practices that can be adopted and adapted by institutions, departments, and individual faculty members.

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Mentorship is a catalyst capable of unleashing one's potential for discovery, curiosity, and participation in STEMM and subsequently improving the training environment in which that STEMM potential is fostered. Mentoring relationships provide developmental spaces in which students' STEMM skills are honed and pathways into STEMM fields can be discovered.

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March 14, 2020

Kelvin Droegemeier, Ph.D.

Office of Science and Technology Policy

Executive Office of the President

Eisenhower Executive Office Building

1650 Pennsylvania Avenue, NW

Washington, DC 20504

Robert Kadlec, M.D.

Assistant Secretary for Preparedness and Response

200 Independence Avenue, SW

Washington, DC 20201

Dear Drs. Droegemeier and Kadlec:

Attached is a brief response to your question on whether reports of severe illness in younger adults in Italy may represent a genetic change to the virus. As explained in the note, the reports from Italy of severe illness in young adults may not represent a change in the pattern of susceptibility, as even the earliest reports from China indicated severe illness among young adults, though at a lower frequency than among older persons. At the present time, the genetic make up of the virus circulating in Italy appears to be the same as that found in other countries of Europe.

The enclosed document was prepared by staff of the National Academies of Sciences, Engineering, and Medicine based on input from Trevor Bedford, David Walt, and me.

My colleagues and I hope this input is helpful to you as you continue to guide the nation's response in this ongoing public health crisis.

Respectfully,

Harvey V. Fineberg, M.D., Ph.D.

Chair

Standing Committee on Emerging Infectious Diseases and 21st Century Health Threats

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Read "Rapid Expert Consultation on Severe Illness in Young Adults for the COVID-19 Pandemic (March 14, 2020)" at NAP.edu

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March 14, 2020 Kelvin Droegemeier, Ph.D. Office of Science and Technology Policy Executive Office of the President Eisenhower Executive Office Building 1650 Pennsylvania Avenue, NW Washington, DC 20504 Robert Kadlec, M.D. Assistant Secretary for Preparedness and Response 200 Independence Avenue, SW Washington, DC 20201 Dear Drs.

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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

1 Introduction N ot since the launch of Sputnik has the United States seen such tre- mendous attention to the value and importance of science, technol- ogy, engineering, and mathematics (STEM) education. In a world with rapidly expanding technologies and increasingly complex political and social challenges, Americans are expected to able to put STEM knowledge and skills to work in their daily lives. Recognizing this, major stakeholders in Americaâ€™s STEM landscapeâ€"from federal agencies to private enterprise to local communitiesâ€"are investing in efforts to enhance STEM education and stimulate public engagement in STEM issues. As one of the countryâ€™s leading STEM agencies with a brand that resonates deeply with the American public, the National Aeronautics and Space Administration (NASA) has a particular role to play in helping America realize the promise of STEM education. In an effort to harness the potential of NASAâ€™s resources, NASAâ€™s Sci- ence Mission Directorate (SMD) underwent a significant restructuring of STEM education-related activities in 2013. The new portfolio of competi- tive award investments, NASA SMDâ€™s Science Activation (SciAct) Program, endeavors to increase the overall coherence of SMDâ€™s efforts in support of more effective, sustainable, and efficient use of SMD science discoveries and learning experiences. One of SciActâ€™s stated goals is to enable NASA scientists and engineers to engage more effectively and efficiently in STEM learning environments with learners of all ages. In 2019, SciAct is nearing the end of its first funding cycle. As SciAct transitions into its second round of funding, it is important that SciAct stake- holders take stock of their work, address challenges, and plan for the future. A few critical questions emerge: How is SciAct designed, structured, and 11

12 NASAâ€™S SCIENCE ACTIVATION PROGRAM BOX 1-1 Charge to the Committee An ad hoc committee will assess the SMDâ€™s Science Activation (SciAct) Pro- gramâ€™s efforts toward meeting the following objectives: (1) enable STEM educa- tion, (2) improve U.S. scientific literacy, (3) advance national education goals, and (4) leverage efforts through partnerships. The review will provide an independent, authoritative forum for identifying and discussing SciAct issues in earth and space science related to NASA SMDâ€™s SciAct Program and will include the following: [A] concise written assessment of the status of the SMD Science Activation Program including feedback on improving the program. The assessment will be based on evidence gathered by the committee at its in-person and virtual meetings and on established principles for evidence-based science education as summarized in previ- ous reports from the Board on Science Education. The committeeâ€™s assessment final report may include findings and conclusions related to management of and priorities for the next phase of the program, including the identification of any gaps in the SciAct approach, given new advances in science education pedagogy and recent Decadal recommendations. The assessment will be subject to review in accordance with the National Academiesâ€™ institutional policies. implemented to address its stated goals? What modifications can be made to current program elements to improve its efficacy? What should SciAct stakeholders consider when making changes to the existing program? Finally, what is NASAâ€™s role in supporting STEM education in the United States, and how well is SciAct positioned to fulfill that role? In order to address these questions, NASA tasked the Board on Science Education (BOSE) of the National Academies of Sciences, Engineering, and Medicine with conducting a review of the SciAct portfolio. In response to NASAâ€™s request, BOSE convened the Committee to Assess Science Activa- tion (see the committeeâ€™s charge in Box 1-1). The 14-member committee included individuals with expertise in earth science, space science, plan- etary science, collaborative models and partnerships, collective impact, education policy, and learning and teaching in science and engineering (see Appendix B for biographical sketches). In this report, the committee provides a comprehensive review of the SciAct portfolio and offers recom- mendations for improving the program. CONTEXT OF THIS REPORT NASA is one of many entities (i.e., agencies, institutions, and organi- zations) working to improve educational experiences and opportunities in STEM for learners of all ages. To address its charge, it was important for

INTRODUCTION 13 the committee to understand the complex education landscape in which NASA SciAct operates, and the contextual factors that NASA should con- sider in order to define SciActâ€™s appropriate role in STEM education nation- ally. This section briefly describes the U.S. education system as it relates to the role of the federal government. Additionally, it reviews the current STEM education context in the nation, and why it is appropriate for NASA to have a specific role in this domain. Federal Involvement in STEM Education In the United States, state, local, and tribal governments are primarily responsible for public preKâ€"12 education. The federal government does not set a national curriculum and it cannot mandate state or local partici- pation in federal programs. Yet, even with limited authority over public education, the federal government is uniquely positioned to augment ongo- ing efforts in the formal education system through legislation and funding. The federal government also plays an important role in informal educa- tion, providing management and financial support for many museums, aquariums, planetariums, zoos, science centers, libraries, and after-school programs. Moreover, the government can shape how different parts of the education system interact, for example, by calling for the blending of best practices across different education communities (e.g., informal and formal) to ensure that consistent, high-quality education experiences are uniformly afforded across the entire system (National Science and Technol- ogy Council, 2018). Several federal agencies receive funding through Congress in support of their STEM education-related programs, research, and activities, including the Department of Education, the National Science Foundation (NSF), the Department of Defense, the Department of Health and Human Services (i.e., National Institutes of Health), the Department of Commerce (i.e., Â ational Oceanic and Atmospheric Administration [NOAA]), the Depart- N ment of Energy, the Department of Transportation, the Department of Agri- culture, and the Environmental Protection Agency, as well as NASA. These agencies support STEM education in a number of ways, including but not limited to sharing expertise in science, developing programs and curricular materials that foster engaging opportunities for learners to understand the nature of science, offering in-service training and instructional resources to teachers, and funding university-based research and development and research on STEM education (American Association for the Advancement of Science, 2018; National Science and Technology Council, 2015). In 2016, the most current year with available data, 163 STEM edu- cation programs and activities were in operation across all federal agen- cies. The National Science and Technology Councilâ€™s Committee on STEM

14 NASAâ€™S SCIENCE ACTIVATION PROGRAM Education (CoSTEM)1 is an interagency body responsible for coordinating STEM education programs, investments, and activities across federal agen- cies, and developing and implementing a STEM education strategic plan, to be updated every 5 years. The Subcommittee on Federal Coordination in STEM Education (FC-STEM)2 advises and supports the work of CoSTEM in developing strategic investments in STEM education and in formulating and implementing the strategic plan. Historically, the primary objectives for federal investments in STEM education have been to (1) increase the number of STEM degrees, (2) prepare people to enter the STEM workforce, and (3) support STEM education research and development (Granovskiy, 2018). Indeed, the latest 5-year strategic plan for the federal governmentâ€™s efforts in STEM education emphasized STEM workforce development and STEM literacy as top national priorities (National Science and Technology Council, 2018). In the last decade, annual federal investments in STEM education have remained relatively stable ranging from $2.8 billion to $3.4 billion (Granovskiy, 2018). NSF, the Department of Health and Human Services, and the Department of Education are the major federal sponsors in STEM education both in terms of number of programs and total investment, but other agencies, such as NASA and NOAA, have education agendas as part of their missions. In fiscal 2019, NASA received $155 million for its STEM education and public outreach activities. Of this amount, $110 million was directed toward the Office of STEM Engagement (formerly the Office of Education) and $45 million was directed toward the SMD, which houses the SciAct Program (American Institute of Physics, 2018). The National Landscape of STEM Education Understanding the potential contributions of the SciAct portfolio re- quires an understanding of the current landscape of STEM education. Re- cent data indicate that while Americansâ€™ basic STEM skills have modestly improved over the past two decades, other countries are doing a better job of preparing their students in science, mathematics, and engineering fields, indicating the growing need for improvements in STEM education in the United States (National Science Board, 2018; National Science and Technology Council, 2018). To help meet this need, national educa- tion goals have evolved over the past two decades to reflect new science, 1â€‰See https://obamawhitehouse.archives.gov/sites/default/files/microsites/ostp/costem_Âcharter_ signed_01-31-11.pdf. 2â€‰FC-STEM, chartered in 2014, is a committee of members from 11 federal agencies and the Smithsonian Institution: https://obamawhitehouse.archives.gov/sites/default/files/microsites/ ostp/NSTC/CoSTEM-FCSTEM-%20Charter-0114-SIGNED.pdf.

INTRODUCTION 15 matheÂ atics, and engineering education standards and new evidence on m how people learn. New research on the nature of learning across the life span has led to greater understanding of the factors that shape learning and what this means for the design of learning experiences. Learning is influenced by the knowledge, motivation, and cultural experiences of the individual, but also by the cultural and social characteristics of the learning environment (National Academies of Sciences, Engineering, and Medicine [NASEM], 2018a). Learning is a dynamic, ongoing process, during which the learner actively constructs their own under- standing through social interactions with other learners and with teachers or facilitators. Research also confirms that learning occurs across multiple spaces over the life span, not solely in formal school settings (National Research Council [NRC], 2009). Indeed, much of NASAâ€™s programming occurs in these informal settings. For more information on informal learning environments, see Chapter 4 of this report. These insights about learning have led to new standards for STEM education that call for learners not just to accumulate factual knowledge, but to be able to deploy their knowledge in meaningful ways. For example, in 2012, the National Academies released A Framework for Kâ€"12 Science EduÂcation, which articulated a new vision for science education in which leaners actively engage in science and engineering practices over multiple years of school to deepen their understanding of the core ideas inherent in these disciplines and make connections to cross-cutting concepts (NRC, 2012a). The Framework outlines core ideas in the physical sciences, life s Âciences, earth and space sciences, and engineering, as well as the cross- cutting concepts that are relevant across these disciplines (i.e., structure and function or cause and effect). It also describes the science and engineering practices that scientists and engineers employ in their work and that Âearners l should engage in as they learn science and engineering. The Framework was the impetus for the development of new Kâ€"12 science standards, the Next Generation Science Standards (NGSS), which have been adopted or adapted by 44 states, reaching about 71 percent of U.S. students (National Science Teaching Association, 2019). The NGSS and similar standards outline a set of performance expectations that integrate the vision of the Framework into statements that articulate the ways students will demonstrate competency. It is important for federal science agencies, including NASA, to be aware of advances in the understanding of learning and of current efforts to improve science education through new standards. STEM Education at NASA NASA has been a long-standing stakeholder in science and engineering education with programs dating back to its authorizing legislation in 1958.

16 NASAâ€™S SCIENCE ACTIVATION PROGRAM This legislation established the agencyâ€™s responsibility to ensure that the United States remains a leader in space exploration, scientific discovery, and aeronautics research. Additionally, this legislation codified NASAâ€™s obliga- tion to effectively share knowledge of the atmosphere and space with the public, thus providing a return on the publicâ€™s investment in NASA science and engineering. NASAâ€™s involvement in STEM education is supported by the science, engineering, and exploration missions of the agency, and NASAâ€™s notable assets offer a rare opportunity to contribute to the STEM education land- scape. These assets include state-of-the-art facilities; cutting-edge technol- ogy; awe-inspiring missions; expert astronauts, scientists, and engineers; and a wealth of images, data, scientific findings, and compelling narratives from more than five decades of spaceflight missions. These exceptional re- sources provide opportunities for students, teachers, and the general public to engage meaningfully with the latest science and engineering advance- ments, as well as to learn about new science discoveries and to acquire new skills in STEM. Additionally, NASAâ€™s resources are especially well suited to inspiring and motivating learners. Missions involving human spaceflight, as well as missions such as the James Webb Space Telescope and Mars Exploration Rovers, are powerful ways to naturally pique learnersâ€™ inter- est. The search for life beyond Earth and the exploration of the moon and Mars are intrinsically attractive to learners, with the potential to motivate them to pursue eventual careers in STEM areas. Moreover, NASA is more widely known to the U.S. public than any other federal science agency, and has the convening power to facilitate the creation of partnerships among mission scientists, educators, and communications specialists that can serve as one avenue for bringing the expertise and enthusiasm of scientists into the public domain. NASAâ€™s many assets uniquely position it among the federal agencies involved in STEM education and strongly support its role as a key resource for the motivational and content aspects of learning in STEM subjects. The bulk of STEM education activities at NASA are in the Office of STEM Engagement and the SMD. Traditionally, these two units have employed different approaches to developing and implementing STEM education projects (NRC, 2008). In the case of SMD, a percentage of funds from the major science mission budgets was originally devoted to support- ing education activities related to each mission. However, in an effort to improve coordination and maximize collaboration across education and communications activities at NASA, SMD undertook significant restruc- turing of its STEM education related activities and established the SciAct Program in 2015. This program was established as a collective comprised of 27 competitively selected awardees to further enable NASA scientists and engineers to engage more effectively and efficiently with learners of

INTRODUCTION 17 all ages, and to foster more effective, sustainable, and efficient use of SMD science discoveries. Prior to the establishment of the SciAct Program in 2015, the SMD implemented its education and public outreach (E/PO) activities primar- ily through two means3 (National Aeronautics and Space Administration, 2010, p. 70). First, an SMD directive issued in 1993 called for each flight mission to allocate at least 1 percent of its total mission budget, exclud- ing the cost of launch vehicles, to E/PO activities (1% model). The sec- ond method supported E/PO projects through awards from solicited (e.g., GLOBE Implementation Office) or unsolicited proposals, or as elements of a major research-enabling program (e.g., suborbital science). The 1 percent model implemented in individual missions mobilized earth and space scientists in a wide range of activities, from speaking at public forums or in classrooms to publishing web content to providing professional development. For example, in early 2015, the New Horizons mission focused on setting the stage for the upcoming first-ever flyby of Pluto through a variety of events called â€œPlutopaloozas.â€ These public events were staged at museums and science centers throughout the country, providing educational activities for children and adults. Each event featured several mission scientists and engineers who provided first-person accounts of mission planning and its scientific foundations. In the past 25 years at NASA, more than 100 missions, on aver- age, have been in development, in operation, or in extension, with equal numbers of education teams funded in those missions under the 1 percent model. Such a large number of teams made coordination across mission activities difficult, leading to duplication and fragmentation of efforts, and potentially inequitable distribution of educational resources. Additionally, under this model, it was difficult to demonstrate impact of the educational efforts related to highly valued goals in education (e.g., systemic improve- ments and activities that are closely aligned with learnersâ€™ needs). Given that the mission education activities did not always leverage education resources within and outside NASA, their reach and potential to go to scale were presumably limited. In 2013, two separate events occurred that would precede the de- velopment of the current SciAct portfolio. First, the release of the 2015 Presidentâ€™s Budget showed a dramatic loss of funding for these activities. Additionally, the release of the 2013 CoSTEM report4 prompted SMD to develop a more efficient internal structure for its program in an effort to 3â€‰See https://smd-prod.s3.amazonaws.com/science-pink/s3fs-public/atoms/files/2010Science Plan_TAGGED_0.pdf. 4â€‰See https://obamawhitehouse.archives.gov/sites/default/files/microsites/ostp/stem_stratplan_ 2013.pdf.

18 NASAâ€™S SCIENCE ACTIVATION PROGRAM strengthen its contribution to the national efforts to make progress on improving STEM education. This coincided with an internal clarification on the distinction between education and communications at NASA.5 With the approval of the SMD Management Council and subsequent approval of the Office of Management and Budget (OMB), SMD restruc- tured its approach to E/PO and established one desired outcome and four overarching objectives for its education-related endeavor: (1) enable STEM education, (2) improve U.S. scientific literacy, (3) advance national educa- tion goals, and (4) leverage efforts through partnerships (for more informa- tion on these goals and the vision of SciAct, see Chapter 2 of this report). Additionally, SMD appointed Kristen Erickson as the director for science engagement and partnerships to oversee and provide strategic direction for the SciAct Program. Under the leadership of Dr. Thomas Zurbuchen, associate administrator for the SMD, the current iteration of SciAct was developed and implemented. SciAct aims to further enable scientists and engineers to engage more effectively with learners of all ages. In addition to recognizing the essential role of the agencyâ€™s subject matter experts (SMEs), it places emphasis on the needs of learners and educators; independent evaluation as practiced in the field of STEM education; and collaboration with external partners who deliver learning and instruction more systemically or directly to leverage their expertise for greater and more extensive impact. After issuing an open call through a NASA Cooperative Agreement No- tice (CAN, or a solicitation for proposals), and carrying out a subsequent competitive process of selection, SMD awarded 27 multiyear cooperative agreements in 2016 for a maximum of 10 years: projects are awarded a 5-year base award, and have one 5-year option to extend their agreements. Currently, there are 24 awards (or projects) still in progress that constitute the SciAct Phase 16 portfolio (see Appendix A for more information on the 24 projects). These projects leverage SMDâ€™s astrophysics, earth, helioÂ physics, and planetary science content and experts to engage learners of 5â€‰Education comprises activities designed to enhance learning in science, technology, en- gineering, and mathematics (STEM) content areas using NASAâ€™s unique capabilities, while communications comprises the comprehensive set of functions necessary to effectively convey and provide an understanding of the program, its objectives and benefits to target audiences, the public, and other stakeholders. This includes a diverse, broad, and integrated set of efforts intended to promote interest and foster participation in NASAâ€™s endeavors and to develop exposure to and appreciation for STEM (e.g., media services, multimedia products and ser- vices (including Web, social media, and nontechnical publications), and public engagement (outreach) activities and events). 6â€‰The committee notes that in order to delineate timeframe, it refers to SciActâ€™s â€œPhase 1â€ as the first 5 years of SciAct programming. The committee refers to â€œPhase 2â€ to signify the next 5 years of programming, to commence in 2020.

INTRODUCTION 19 all ages in STEM, in both formal and informal settings (see Chapter 3 of this report for more discussion of the portfolio). Additionally, internal collaboration and coordination among the funded projects, as well as in- frastructure support, have been put in place to promote overall coherence. Decisions on which projects will continue during Phase 2 of the program (potentially at different levels) will be made in 2020, and opportunities for new projects will be afforded through SMDâ€™s Research Opportunities in Space and Earth Sciences (ROSES) grant solicitation, an annual omnibus solicitation for proposals across all disciplinary areas in SMD. The establishment of the SciAct Program has enabled SMD to make a more concerted effort in contributing to the national STEM education agenda and enhancing operational effectiveness and efficiency, which is the focus of this report. However, a recent midterm review of NASAâ€™s planetary science invest- ments revealed that some of NASAâ€™s assets are not optimized as much as they might be under this new model (National Academies of Sciences, Engineering, and Medicine, 2018d, pp. 77â€"78). These assets include NASA missionsâ€™ results and mission science experts when it comes to defining and providing science content within the E/PO activities, as described earlier in this chapter. This report considers this finding in the context of how SciAct can best leverage its considerable assets while also meeting the needs of the communities it serves. Summary Based on current national priority areas, federal science agencies have been encouraged to incorporate STEM education more explicitly into their programs, to prioritize policies and practices that place an emphasis on ex- panding the STEM workforce, and to develop methods and metrics to collect data and track the effectiveness of the STEM programs (National Science and Technology Council, 2018; Office of Science and Technology Policy, 2017). Indeed, these priorities are relevant to SMDâ€™s education mission; therefore, this request for an expert assessment of the SciAct Program to establish priorities for the next phase of the program is timely. National agendas will continue to shape the involvement of federal agencies in STEM education, highlighting the importance of demonstrated efforts (e.g., periodic expert portfolio reviews) to provide compelling justifications for budgeted STEM activities at these agencies going forward. Lastly, given that NASA is one of many federal stakeÂ holders in the STEM education landscape, it is critical for the agency to con- sider its goals, objectives, and strategy for uniquely contributing to this space. PREVIOUS REVIEWS OF NASA EDUCATION ACTIVITIES The National Academies has a long history of providing strategic advice to NASA and other U.S. government agencies on scientific and technical

20 NASAâ€™S SCIENCE ACTIVATION PROGRAM matters. A snapshot of some of the reports produced in the last decade as a result of these relationships is shown in Box 1-2. At the request of NASA, NOAA, and the U.S. Geological Survey (USGS), the first decadal survey in earth science was conducted by the Space Studies Board (NRC, 2007). Since the 1960s, the National Academies has been asked to conduct these types of studies for federal agencies, and in particular, they have been instrumental in helping NASA to strategically plan for 10 or more years into the future, as it relates to identifying and prioritizing research areas, leading-edge sci- entific questions, the observations required to answer these questions, and the missions required to facilitate those observations. In recent history, a number of decadal surveys have been conducted by the National Academies in the areas of astronomy and astrophysics (NRC, 2010), life and physical sciences (NRC, 2011a), planetary sciences (NRC, 2011b), and solar and space physics (NRC, 2013b), and again in earth science (NASEM, 2017). In the area of STEM education, the National Academies has carried out several previous efforts to review and evaluate both NASAâ€™s and NOAAâ€™s education activities. Beginning in 2008, the Board on Science Education of the National Academies was asked to conduct a review and evaluation of NASAâ€™s precollege science, technology, and mathematics education program (housed in NASA headquartersâ€™ Office of STEM Engagement, formerly the Office of Education) as mandated by the NASA Authorization Act of 2005 (P.L. 109-555 Subtitle B-Education, Sec. 614).7 A 12-member expert com- mittee was appointed to carry out this study, which focused on four areas: (1) the effectiveness of the overall program in meeting its defined goals and objectives, (2) the quality and educational effectiveness of the major compo- nents of the program, (3) the funding priorities in the program, and (4) the extent and effectiveness of coordination and collaboration between NASA and other federal agencies invested in science, technology and mathematics education (NRC, 2008). The committee concluded that NASAâ€™s strengths were its efforts to reach underrepresented groups, its effectiveness in raising awareness of the science and engineering of its missions among Kâ€"12 students and teach- ers, and its demonstrated commitment to funding Kâ€"12 STEM education activities. However, the report also highlighted the following barriers to the overall effectiveness of the program: program instability, lack of rigor- ous evaluation and funding for it, lack of a strategic plan, lack of in-depth experiences to support learning of STEM content, and the nature of science and engineering, budget fluctuations, and lack of systemic coordination with other federal agencies to draw on needed expertise in STEM educa- tion project design. The committee also identified a set of recommendations across four broad areas deemed important for improving NASAâ€™s efforts 7â€‰See https://www.congress.gov/109/plaws/publ155/PLAW-109publ155.pdf.

INTRODUCTION 21 BOX 1-2 National Academies Reports on STEM Activities at Federal Agencies: 2007â€"2017 2007: arth Science and Applications from Space: National Imperatives for E the Next Decade and Beyond (study conducted by the Space Studies Board) 2008: ASAâ€™s Elementary and Secondary Education Program: Review and N Critique (study conducted by the Board on Science Education) 2010: OAAâ€™s Education Program: Review and Critique (study conducted N by the Board on Science Education) 2010: ew Worlds, New Horizons in Astronomy and Astrophysics (study N conducted by the Board on Physics and Astronomy and the Space Studies Board) 2011: ecapturing a Future for Space Exploration: Life and Physical Sci- R ences Research for a New Era (study conducted by the Space Stud- ies Board and the Aeronautics and Space Engineering Board) 2011: ision and Voyages for Planetary Science in the Decade 2013-2022 V (study conducted by the Space Studies Board) 2013: olar and Space Physics: A Science for a Technological Society S (study conducted by the Space Studies Board and the Aeronautics and Space Engineering Board) 2013: eview of the Draft 2014 Science Mission Directorate Science Plan R (study conducted by the Space Studies Board) 2013: reparing the Next Generation of Earth Scientists: An Examination P of Federal Education and Training Programs (study conducted by the Board on Earth Sciences and Resources) 2014: haring the Adventure with the Student: Exploring the Intersections S of NASA Space Science and Education: A Workshop (workshop conducted by the Space Studies Board and the Board on Science Education) 2017: ASA Science Mission Directorate Expert Meetings (convenings con- N ducted by the Board on Science Education and the Space Studies Board) 2017: hriving on Our Changing Planet: A Decadal Strategy for Earth Ob- T servations from Space (study conducted by the Space Studies Board) in elementary and secondary STEM education: (1) defining the nature of NASAâ€™s role in Kâ€"12 STEM education, (2) ensuring continuous improve- ment of projects, (3) establishing partnerships to harness expertise in educa- tion, and (4) leveraging information and communications technology more effectively. It is important to note that the SMDâ€™s work, including SciAct, is largely separate from the STEM education efforts emerging from the Office of STEM Engagement, and therefore the above conclusions and recommen-

22 NASAâ€™S SCIENCE ACTIVATION PROGRAM dations on NASAâ€™s precollege education portfolio do not necessarily reflect the state of the activities within SMDâ€™s education portfolio. More recently, in 2013, the National Academies conducted an assess- ment of NASAâ€™s 2014 SMD Science Plan to gauge the agencyâ€™s responsive- ness to the guidance outlined in the recent NRC decadal surveys (NRC, 2013b). In 2014, the National Academies convened a workshop to explore promising approaches for effectively translating NASAâ€™s missions and sci- entific discoveries into formal and informal science learning experiences for students and teachers in the Kâ€"12 education space (NRC, 2015). In 2017, a series of expert meetings was organized by the National Academies that afforded the space for NASA scientists to hear from education specialists on ways to develop curriculum materials bearing life-relevant content and on ways to leverage subject-matter experts to foster better student engagement in science classrooms (see Box 1-2). The Board on Science Education of the National Academies has been involved in all of the efforts related to federal STEM education programs, and thus was well positioned to undertake this new request to assess the education portfolio of NASAâ€™s SciAct Program. THE COMMITTEEâ€™S APPROACH TO THE CHARGE Unlike traditional consensus studies at the National Academies, the work of this committee revolved primarily around assessing a current pro- gram model by applying committee membersâ€™ expertise to analyze different facets of SciActâ€™s design and implementation. In this report, the â€œevidenceâ€ considered by the committee is not published in peer-reviewed journals, but is the physical and oral documentation of the current program design as provided to the committee. In this section, we provide key notes on our interpretation of the charge, and we detail our processes in order to describe in depth the evidence we considered in our effort to assess SciAct and make recommendations for NASA going forward. We conclude by describing how we organized this report. Interpreting the Charge In the early phases of this committeeâ€™s work, it was necessary to first delineate the task at hand. In consultation with the sponsor, the committee made a series of decisions defining the boundaries of this assignment that have implications for the content of our work. First, this report does not assess the individual efforts of the 24 SciAct projects in meeting the pro- gramâ€™s four objectives, but rather considers the impact of the portfolio as a whole (see Appendix A for descriptive information on all 24 projects). The cooperative agreement process utilized by SciAct affords opportunities for proposals that may have a clearly articulated, narrow focus to be considered

INTRODUCTION 23 for sponsorship without the requirement to address every objective in the CAN. In this way, SciAct seeks to leverage individual project expertise within the collective of the portfolio to yield resources and opportunities that are scientifically accurate, grounded in current education research, reflective of the needs of the education community, and more deeply aligned with science and engineering processes. The committee was charged with conducting an assessment of SciActâ€™s aggregate program efforts, and not of the specific efforts of individual projects. Moreover, the committee took care not to in- advertently endorse or critique individual project efforts; for this reason, we do not comment on the efficacy of individual project approaches to meeting SciAct goals but rather attempt to characterize trends in the portfolio. Additionally, this assessment is not designed to compare the effective- ness of SciAct Phase 1 in relationship to the 1 percent model, or any other NASA programming. As a result, the committee declines to make any summative statements about whether or not this model is â€œbetterâ€ than its predecessor: Rather, we attempt to make sense of the design and imple- mentation of SciAct in its first phase on its own merits. We consider how well SciAct is positioned to meet its stated goals and objectives, as well as whether or not those goals and objectives are appropriate for the scope of this portfolio, among other questions delineated in our charge. GATHERING AND ASSESSING THE EVIDENCE In order to obtain the evidence necessary to complete this review, the committee held a number of open-session conversations with members of the SciAct portfolio, as well as several experts on various aspects of STEM education. In the first meeting of the committee, we heard testimony from Kristen Erickson, director, science engagement and partnerships at NASA SMD, who provided a comprehensive overview of the history of SciAct, its efforts to reorganize, and the current shape and status of the portfolio. For the PowerPoint slides used by Ms. Erickson in her presentation, see AÂ pendix C on The National Academies Press Webpage at http://www.nap. p edu/25569. Ms. Erickson was supported by Dr. Lin Chambers, Science Acti- vation integration manager. Later in that first meeting, the committee heard from three SciAct awardees, who discussed the goals and implementation of their projects: Denise Smith of the Space Telescope Science Institute, Alex Young of NASA Goddard, and Theresa Scherwin of the Institute of Global Environmental Strategies. Throughout the evidence-gathering process, in order to select presenters from the SciAct community of awardees, the committee solicited recommendations from Ms. Erickson in consultation with Dr. Chambers. Following the first in-person committee meeting, the committee held several virtual conversations with other SciAct awardees, who were able to

24 NASAâ€™S SCIENCE ACTIVATION PROGRAM provide further insight into their awards and how they fit into the SciAct port- folio. These conversations were organized around themes within the portfolio identified by Ms. Erickson and Dr. Chambers (see Appendix C at http://www. nap.edu/25569), and as with the in-person presentations, presenters were identified from among the awardees by Ms. Erickson and Dr. Chambers. We held three separate conversations on these topics: 1. On awards that disseminate information to the public, we heard from Rachel Connolly of WGBH and PBS Learning Media; Ariel Anbar of Infiniscope at Arizona State University; and Jon Miller from the University of Michigan, who is responsible for measuring SciActâ€™s progress on its science literacy goal. 2. On engaging diverse audiences in STEM, we heard from Paul Dusenberry, working with libraries from the Space Science Insti- tute; Kevin Czajkowski working with educators at the University of Toledo; Denise Kopecky with the Challenger Centers from the Challenger Center for Space Science Education; and Paul Martin Arizona State University, who works with museums. 3. On the issue of connecting NASA infrastructure, we heard from Kay Ferrari of the Solar Systems Ambassadors Program at the NASA Jet Propulsion Laboratory (NASA JPL), and Emily Law from the Solar Systems Treks Program also of NASA JPL. Finally, in order to understand SciAct awardeeâ€™s efforts to communicate with one another, Andy Shaner of NASA conducted a virtual tour of the SMD E/PO, an online communication space. At our second in-person meeting, we heard from individual project evaluators (and their principal investigator counterparts) who detailed their efforts to evaluate their awardsâ€™ work toward meeting the goals of SciAct. The committee heard from Jackie Delisi of the Education Development Center and project investigator Leigh Peake of the Gulf of Maine Research Institute, Eric Banilower of Horizon Research and Robert Winglee from the University of Washington, and Martin Storksdiek of Oregon State Uni- versity and Rachel Connolly from WGBH. In order to augment committee expertise on critical issues, the committee heard testimony from Carol Oâ€™Donnell of the Smithsonian Institution, who discussed efforts to leverage federal-level funds through local partnerships. In order to center conversa- tions around diversity, equity, inclusion, and access, both Louis Gomez of the University of California, Los Angeles and Julie Johnson of the National Science Foundation offered testimony on best practices and consideration for prioritizing equity in network models. Finally, the committee reviewed SciAct documentation for evidence of the portfolioâ€™s design and implementation. These documents included

INTRODUCTION 25 SciActâ€™s promotional and communication materials, as well as internal documents that describe how the work of awardees is intended to meet specific SciAct objectives. The committee extensively reviewed the CAN, as well as each of the awardeesâ€™ project summaries, ongoing monitoring data, and evaluation reports. The committee also reviewed the SMD E/PO Website in an effort to better understand the support resources available to awardees, as well as the cadence and tenor of awardeesâ€™ online interac- tions across projects. The committee also reviewed the agendas and briefing documents associated with the last few years of SciActâ€™s annual principal investigator meetings so as to understand the nature of those week-long events. Where appropriate, the committee will describe these documents and their contents in greater depth later in this report. As noted, this report varies from traditional National Academiesâ€™ con- sensus studies in that the charge to the committee did not call for a large synthesis of disparate bodies of literature. Instead, the committee was asked to rely on its collective expertise to assess the current SciAct portfolio of an education program in a federal agency, a task that required deep expertise in content areas ranging from federal education policy to STEM engage- ment and learning to the inner workings and functionality of NASA itself. In assessing the portfolio, the committee took care to conduct a full review of all the documents and artifacts described above. When reaching conclu- sions and developing recommendations, the committee drew on multiple streams of evidence: Wherever possible, the committee considered oral testi- mony in conjunction with documents or artifacts in order to make a claim. Where oral testimony alone was used as the basis of a finding, the commit- tee collaborated to ensure collective agreement on how the evidence was interpreted; that is, one individualâ€™s recollection was not sufficient evidence to support a claim. More specifically, the committee took particular care to not offer judgment where evidence was not adequate: In these cases, the committee attempted instead to identify or outline issues and considerations for SciAct to take into account when planning for the future without recom- mending a specific course of action. As a result of these deliberate processes, all conclusions and recommendations outlined in this report reflect the full consensus judgment of the Committee to Assess Science Activation. Throughout these deliberative processes, committee members were asked to formulate assessments of the existing portfolio (as well as sugges- tions for the future) by applying their understanding of the best available research evidence and the current state of their respective fields. Given the nature of the charge, it was not feasible for the committee to do an exten- sive review of all of the research that is relevant to STEM education. For this reason, the report does not provide detailed descriptions of individual studies. Instead, the committee relied on broadly accepted theoretical ap- proaches that are based on large bodies of research across multiple fields

26 NASAâ€™S SCIENCE ACTIVATION PROGRAM that examine learning and teaching. Much of this research is summarized in previous reports from the National Academies. This body of National Academies work includes two reports that examine research on learning, drawing from cognitive science, the learning sciences, developmental psy- chology, social psychology, cultural psychology, educational psychology, sociology, and anthropology: How People Learn: Brain, Mind, Experience, and School (NRC, 2000) and How People Learn II: Culture, Contexts, and Cultures (NASEM, 2018a). It also includes major consensus studies that reviewed evidence on effective approaches to STEM education: Americaâ€™s Lab Report (NRC, 2006), Taking Science to School (NRC, 2007), Learning Science in Informal Environments (NRC, 2009), STEM Integration in Kâ€"12 Education (National Academy of Engineering and NRC, 2014), Science and Engineering in Grades 6â€"12 (NASEM, 2019), and Learning through Citi- zen Science (NASEM, 2018b). The committee notes that these literatures are expanding: As SciAct makes decisions about how to plan for the future, it is critical to note that, as with any science, findings that emerge from the best available and most current evidentiary bases will shift with time. AUDIENCES FOR AND ORGANIZATION OF THE REPORT The committee expects that this report will be of interest to several audiences. First and most immediately, the committee hopes that this report will serve the needs of NASA SMD and the SciAct community so that lead- ership can assess and plan for the future. Second, we expect this report will be of interest to other NASA personnel and other stakeholders invested in NASAâ€™s work in STEM education and engagement. We also expect that this report could serve as a useful template for other federal agencies with their own investments in STEM education and engagement. Finally, this report is intended to be accessible to the taxpaying public: To the extent that NASA is publicly funded, it has an obligation to serve the interests of the American people. This report is an attempt to help SciAct meet that potential. This report is organized into six chapters, with three appendixes. Chapter 2 describes the committeeâ€™s understanding of the vision and objec- tives of SciAct, detailing how it operationalizes its overarching goals and assessing how the portfolio is poised to address those goals. Chapter 3 characterizes the current portfolio, describing the logic undergirding the current organization of the awards and describing the breadth of SciAct programming. Chapter 4 presents our assessment of the SciAct portfolio according to key themes that connect to the overarching program goals and reflects most of the activities across the constituent projects, begin- ning with a discussion of STEM learning and leveraging NASA assets in the portfolio. Particular attention is paid to how these features currently function in the Phase 1 portfolio and considerations for future planning

INTRODUCTION 27 based on the research literature. Chapter 5 continues our assessment of key themes by focusing on broadening participation and collective learn- ing within the portfolio. In Chapter 6, we conclude the report with a set of recommendations for SciAct based on the committeeâ€™s expertise. Given the conclusions drawn by the committee over the course of this report, the committee is confident that careful consideration and implementation of these recommendations can help SciAct continue its good work into the future. Appendix A provides individual descriptions of each of the 24 Â Â SciAct awards. Appendix B contains biographical sketches of committee members and staff. Appendix C includes the PowerPoint slides used by Kristen Erickson in her presentation to the committee and are available on The National Academies Press Webpage at http://www.nap.edu/25569.

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Read "NASA's Science Activation Program: Achievements and Opportunities" at NAP.edu

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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

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There are many reasons to be curious about the way people learn, and the past several decades have seen an explosion of research that has important implications for individual learning, schooling, workforce training, and policy.

In 2000, How People Learn: Brain, Mind, Experience, and School: Expanded Edition was published and its influence has been wide and deep. The report summarized insights on the nature of learning in school-aged children; described principles for the design of effective learning environments; and provided examples of how that could be implemented in the classroom.

Since then, researchers have continued to investigate the nature of learning and have generated new findings related to the neurological processes involved in learning, individual and cultural variability related to learning, and educational technologies. In addition to expanding scientific understanding of the mechanisms of learning and how the brain adapts throughout the lifespan, there have been important discoveries about influences on learning, particularly sociocultural factors and the structure of learning environments.

How People Learn II: Learners, Contexts, and Cultures provides a much-needed update incorporating insights gained from this research over the past decade. The book expands on the foundation laid out in the 2000 report and takes an in-depth look at the constellation of influences that affect individual learning. How People Learn II will become an indispensable resource to understand learning throughout the lifespan for educators of students and adults.

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Read "How People Learn II: Learners, Contexts, and Cultures" at NAP.edu

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There are many reasons to be curious about the way people learn, and the past several decades have seen an explosion of research that has important implications for individual learning, schooling, workforce training, and policy. In 2000, How People Learn: Brain, Mind, Experience, and School: Expanded Edition was published and its influence has been wide and deep.

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Welcome to the Real Analysis page. Here you can browse a large variety of topics for the introduction to real analysis. This hub pages outlines many useful topics and provides a large number of important theorems.

Submit an Error: Do you think that you see an error in any of the pages? Click the link and let us know so that we can fix it as soon as possible! All help is greatly appreciated with there being so many possibilities that can be overlooked.

1. Introduction to Real Analysis (4th Edition) by Robert G. Bartle and Donald R. Sherbert

2. Real Analysis (2nd Edition) by Tom M. Apostol

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Welcome to the Real Analysis page. Here you can browse a large variety of topics for the introduction to real analysis. This hub pages outlines many useful topics and provides a large number of important theorems. Submit an Error : Do you think that you see an error in any of the pages?

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Read "Adding It Up: Helping Children Learn Mathematics" at NAP.edu

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Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years. The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency.

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Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn the subject regardless of whether you are in my classes or not. In other words, they do not assume you've got any prior knowledge other than the standard set of prerequisite material needed for that class. In other words, it is assumed that you know Algebra and Trig prior to reading the Calculus I notes, know Calculus I prior to reading the Calculus II notes, etc. The assumptions about your background that I've made are given with each description below.

I'd like to thank Shane F, Fred J., Mike K. and David A. for all the typos that they've found and sent my way! I've tried to proof read these pages and catch as many typos as I could, however it just isn't possible to catch all of them when you are also the person who wrote the material. Fred, Mike and David have caught quite a few typos that I'd missed and been nice enough to send them my way. Thanks again Fred, Mike and David!

If you are one of my current students and are here looking for homework assignments I've got a set of links that will get you to the right pages listed here.

At present I've gotten the notes/tutorials for my Algebra (Math 1314), Calculus I (Math 2413), Calculus II (Math 2414), Calculus III (Math 2415) and Differential Equations (Math 3301) class online. I've also got a couple of Review/Extras available as well. Among the reviews/extras that I've got are an Algebra/Trig review for my Calculus Students, a Complex Number primer, a set of Common Math Errors, and some tips on How to Study Math.

I've made most of the pages on this site available for download as well. These downloadable versions are in pdf format. Each subject on this site is available as a complete download and in the case of very large documents I've also split them up into smaller portions that mostly correspond to each of the individual topics. To get the downloadable version of any topic navigate to that topic and then under the Download menu you will be presented an option to download the topic.

Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each.

Algebra Cheat Sheets - This is as many common algebra facts, properties, formulas, and functions that I could think of. There is also a page of common algebra errors included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Trig Cheat Sheets - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Calculus Cheat Sheets - These are a series of Calculus Cheat Sheets that covers most of a standard Calculus I course and a few topics from a Calculus II course. There are four different cheat sheets here. One contains all the information, one has just Limits information, one has just Derivatives information and the final one has just Integrals information. Each cheat sheets comes in two versions. One that is full sized and another that has been reduced, with exactly the same information as the full sized version, that prints two pages on the front and/or back of each page of paper.

Common Derivatives and Integrals - Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. here are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper.

Table of Laplace Transforms - Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas. It is currently two pages long with the first page being the Laplace transforms and the second being some information/facts about some of the entries.

All of the classes, with the exception of Differential Equations, have practice problems (with solutions) you can use for practice as well as a set of assignment problems (without solutions/answers) for instructors to use if they wish.

Algebra (Math 1314) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Preliminaries - Exponent Properties, Rational Exponents, Negative Exponents, Radicals, Polynomials, Factoring, Rational Expressions, Complex Numbers

Solving Equations and Inequalities - Linear Equations, Quadratic Equations, Completing the Square, Quadratic Formula, Applications of Linear and Quadratic Equations, Reducible to Quadratic Form, Equations with Radicals, Linear Inequalities, Polynomial & Rational Inequalities, Absolute Value Equations & Inequalities.

Graphing and Functions - Graphing Lines, Circles, and Piecewise Functions, Function Definition, Function Notation, Function Composition, Inverse Functions.

Common Graphs - Parabolas, Ellipses, Hyperbolas, Absolute Value, Square Root, Constant Function, Rational Functions, Shifts, Reflections, Symmetry.

Polynomial Functions - Dividing Polynomials, Zeroes/Roots of Polynomials, Finding Zeroes of Polynomials, Graphing Polynomials, Partial Fractions.

Exponential and Logarithm Functions - Exponential Functions, Logarithm Functions, Solving Exponential Functions, Solving Logarithm Functions, Applications.

Systems of Equations - Substitution Method, Elimination Method, Augmented Matrix, Nonlinear Systems.

The Algebra notes/tutorial assume that you've had some exposure to the basics of Algebra. In particular it is assumed that the exponents and factoring sections will be more of a review for you. Also, it is assumed that you've seen the basics of graphing equations. Graphing particular types of equations is covered extensively in the notes, however, it is assumed that you understand the basic coordinate system and how to plot points.

Calculus I (Math 2413) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Algebra/Trig Review - Trig Functions and Equations, Exponential Functions and Equations, Logarithm Functions and Equations.

Limits - Concepts, Definition, Computing, One-Sided Limits, Continuity, Limits Involving Infinity, L'Hospitals Rule

Derivatives - Definition, Interpretations, Derivative Formulas, Power Rule, Product Rule, Quotient Rule, Chain Rule, Higher Order Derivatives, Implicit Differentiation, Logarithmic Differentiation, Derivatives of Trig Functions, Exponential Functions, Logarithm Functions, Inverse Trig Functions, and Hyperbolic Trig Functions.

Applications of Derivatives - Related Rates, Critical Points, Minimum and Maximum Values, Increasing/Decreasing Functions, Inflection Points, Concavity, Optimization

Integration - Definition, Indefinite Integrals, Definite Integrals, Substitution Rule, Evaluating Definite Integrals, Fundamental Theorem of Calculus

Applications of Integrals - Average Function Value, Area Between Curves, Solids of Revolution, Work.

The Calculus I notes/tutorial assume that you've got a working knowledge of Algebra and Trig. There is some review of a couple of Algebra and Trig topics, but for the most part it is assumed that you do have a decent background in Algebra and Trig. These notes assume no prior knowledge of Calculus.

Calculus II (Math 2414) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Integration Techniques - Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions, Integration using Partial Fractions, Integrals Involving Roots, Integrals Involving Quadratics, Integration Strategy, Improper Integrals, Comparison Test for Improper Integrals, and Approximating Definite Integrals.

Applications of Integrals - Arc Length, Surface Area, Center of Mass/Centroid, Hydrostatic Pressure and Force, Probability.

Parametric Equations and Polar Coordinates - Parametric Equations & Curves, Calculus with Parametric Equations (Tangents, Areas, Arc Length and Surface Area), Polar Coordinates, Calculus with Polar Coordinates (Tangents, Areas, Arc Length and Surface Area).

Sequences and Series - Sequences, Series, Convergence/Divergence of Series, Absolute Series, Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, Ratio Test, Root Test, Estimating the Value of a Series, Power Series, Taylor Series, Binomial Series

Vectors - Basics, Magnitude, Unit Vector, Arithmetic, Dot Product, Cross Product, Projection

Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates

The Calculus II notes/tutorial assume that you've got a working knowledge Calculus I, including Limits, Derivatives, and Integration (up to basic substitution). It is also assumed that you have a fairly good knowledge of Trig. Several topics rely heavily on trig and knowledge of trig functions.

Calculus III (Math 2415) [ Notes] [ Practice Problems] [ Assignment Problems] - Topics included in this set of notes/tutorial are :

Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates

Partial Derivatives - Limits, Partial Derivatives, Higher Order Partial Derivatives, Differentials, Chain Rule, Directional Derivatives, Gradient.

Applications of Partial Derivatives - Tangent Plane, Normal Line, Relative Extrema, Absolute Extrema, Optimization, Lagrange Multipliers.

Multiple Integrals - Iterated Integrals, Double Integrals, Double Integrals in Polar Coordinates, Triple Integrals, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Change of Variables, Surface Area.

Line Integrals - Vector Fields, Line Integrals With Respect to Arc Length, Line Integrals With Respect to x and y, Line Integrals of Vector Fields, Fundamental Theorem of Line Integrals, Conservative Vector Fields, Potential Functions, Green's Theorem, Curl, Divergence.

Surface Integrals - Parametric Surfaces, Surface Integrals, Surface Integrals of Vector Fields, Stokes' Theorem, Divergence Theorem.

The Calculus III notes/tutorial assume that you've got a working knowledge Calculus I, including limits, derivatives and integration. It also assumes that the reader has a good knowledge of several Calculus II topics including some integration techniques, parametric equations, vectors, and knowledge of three dimensional space.

Differential Equations (Math 3301) [ Notes] - Topics included in this set of notes/tutorial are :

First Order Differential Equations - Linear Equations, Separable Equations, Exact Equations, Equilibrium Solutions, Modeling Problems.

Second Order Differential Equations - Homogeneous and Nonhomogeneous Second Order Differential Equations, Fundamental Set of Solutions, Undetermined Coefficients, Variation of Parameters, Mechanical Vibrations

Laplace Transforms - Definition, Inverse Transforms, Step Functions, Heaviside Functions, Dirac-Delta Function, Solving IVP's, Nonhomogeneous IVP, Nonconstant Coefficient IVP, Convolution Integral.

Systems of Differential Equations - Matrix Form, Eigenvalues/Eigenvectors, Phase Plane, Nonhomogeneous Systems, Laplace Transforms.

Series Solutions - Series Solutions, Euler Differential Equations.

Higher Order Differential Equations - n th order differential equations, Undetermined Coefficients, Variation of Parameters, 3 x 3 Systems of Differential Equations.

Boundary Value Problems & Fourier Series - Boundary Value Problems, Eigenvalues and Eigenfunctions, Orthogonal Functions, Fourier Sine Series, Fourier Cosine Series, Fourier Series.

Partial Differential Equations - Heat Equation, Wave Equation, Laplace's Equation, Separation of Variables.

These notes assume no prior knowledge of differential equations. A good grasp of Calculus is required however. This includes a working knowledge of differentiation and integration.

Algebra/Trig Review - This is an Algebra Review and Trig Review that was originally written for my Calculus I students. It it still geared mostly towards Calculus students with occasional comments on how a topic will be used in a Calculus class. However, anyone needing a review of some of the basic algebra, trig, exponential functions and logarithms should find the information of use.

Not all the topics covered in an Algebra or Trig class are covered in this review. I've mostly covered topics that are of particular importance to students in a Calculus class. I have included a couple of topics that are not that important to a Calculus class, but students do seem to have trouble with on occasion. As time permits I will be adding more sections as well.

The review is in the form of a problem set with the first solution containing detailed information on how to work that type of problem. Later solutions are usually not as detailed, but may contain more/new information as required.

Complex Number Primer - This is a brief introduction to some of the basic ideas involved with Complex Numbers. The topics covered are a brief review of arithmetic with complex numbers, the complex conjugate, modulus, polar and exponential form and computing powers and roots of complex numbers.

Note that this primer does assume that you've at least seen some complex numbers prior to reading. The purpose of this document is go a little beyond what most people see when the first are introduced to complex numbers in say a College Algebra class. Also, this document is in no way intended to be a complete picture of complex numbers nor do I cover all the concepts involved (that's a whole class in and of itself).

Common Math Errors - As with the Algebra/Trig review this was originally written for my Calculus I class. However, only one of the five sections that I've given here directly addresses the topic of Calculus. The other four sections are more general errors or cover Algebra and Trig errors. There are a couple of calculus examples in the first four sections, but in all of these cases I've also tried to provide non Calculus examples as well.

This portion of the site should be of interest to anyone looking for common math errors. If you aren't in a Calculus class or haven't taken Calculus you should just ignore the last section.

How To Study Math - This is a short section with some advice on how to best study mathematics.

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Welcome to my math notes site. Contained in this site are the notes (free and downloadable) that I use to teach Algebra, Calculus (I, II and III) as well as Differential Equations at Lamar University. The notes contain the usual topics that are taught in those courses as well as a few extra topics that I decided to include just because I wanted to.

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Preface Features of the Text

Instructors and students alike will find several consistent features in the presentation, including:

Motivating Questions

At the start of each section, we list 2-3 motivating questions that provide motivation for why the following material is of interest to us. One overall goal of each section is to answer each of the motivating questions.

Preview Activities

Each section of the text begins with a short introduction, followed by a preview activity. This brief reading and the preview activity are designed to foreshadow the upcoming ideas in the remainder of the section; both the reading and preview activity are intended to be accessible to students in advance of class, and to be completed by students before the day on which a particular section is to be considered.

Activities

A typical section in the text has at least three activities. These are designed to engage students in an inquiry-based style that encourages them to construct solutions to key examples on their own, working individually or in small groups.

Exercises

There are dozens of calculus texts with (collectively) tens of thousands of exercises. Rather than repeat standard and routine exercises in this text, we recommend the use of WeBWorK with its access to the Open Problem Library and around 20,000 calculus problems. In this text, each section includes a small number of anonymous WeBWorK exercises, as well as 3-4 challenging problems per section. The WeBWorK exercises are best completed in the .html version of the text, as this provides students with immediate feedback without penalty. Almost every non- WeBWorK exercise has multiple parts, requires the student to connect several key ideas, and expects that the student will do at least a modest amount of writing to answer the questions and explain their findings. For instructors interested in a more conventional source of exercises, consider the freely available APEX Calculus text by Greg Hartmann et al., available from www.apexcalculus.com.

Graphics

We strive to demonstrate key fundamental ideas visually, and to encourage students to do the same. Throughout the text, we use full-color graphics to exemplify and magnify key ideas, and to use this graphical perspective alongside both numerical and algebraic representations of calculus.

Links to interactive graphics

Many of the ideas of calculus are best understood dynamically; java applets offer an often ideal format for investigations and demonstrations. Relying primarily on the work of David Austin of Grand Valley State University and Marc Renault of Shippensburg University, each of whom has developed a large library of applets for calculus, we frequently point the reader (through active links in the electronic versions of the text) to applets that are relevant for key ideas under consideration.

Summary of Key Ideas

Each section concludes with a summary of the key ideas encountered in the preceding section; this summary normally reflects responses to the motivating questions that began the section.

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Instructors and students alike will find several consistent features in the presentation, including: Motivating Questions At the start of each section, we list 2-3 motivating questions that provide motivation for why the following material is of interest to us. One overall goal of each section is to answer each of the motivating questions.

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¶
These notes are based on the philosophy that you learn the most about a subject when you are figuring it out directly for yourself, and learn the least when you are trying to figure out what someone else is saying about it. On the other hand, there is a subject called combinatorial mathematics, and that is what we are going to be studying, so we will have to tell you some basic facts. What we are going to try to do is to give you a chance to discover many of the interesting examples that usually appear as textbook examples and discover the principles that appear as textbook theorems. Your main activity will be solving problems designed to lead you to discover the basic principles of combinatorial mathematics. Some of the problems lead you through a new idea, some give you a chance to describe what you have learned in a sequence of problems, and some are quite challenging. When you find a problem challenging, don't give up on it, but don't let it stop you from going on with other problems. Frequently you will find an idea in a later problem that you can take back to the one you skipped over or only partly finished in order to finish it off. With that in mind, let's get started. In the problems that follow, you will see some problems marked on the left with various symbols. The preface gives a full explanation of these symbols and discusses in greater detail why the book is organized as it is! Table 1.1.1, which is repeated from the preface, summarizes the meaning of the symbols.

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¶ These notes are based on the philosophy that you learn the most about a subject when you are figuring it out directly for yourself, and learn the least when you are trying to figure out what someone else is saying about it.

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Read "Adding It Up: Helping Children Learn Mathematics" at NAP.edu

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Read chapter 1 LOOKING AT MATHEMATICS AND LEARNING: Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how tea...

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https://gvsu.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=df8ad125-2934-41d8-8bc6-aa9a00e3af9a

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Some overlap with LC but don't skip the first book

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Active Prelude to Calculus, Active Calculus, and Active Calculus Multivariable are free, open-source texts designed for a more active learning experience for students. Every section of each text has engaging activities for students to complete before and during class, as well as exercises that challenge students to connect and assimilate core concepts. The HTML version of each text offers anonymous interactive WeBWorK exercises, select interactive graphics, and full-color static graphics. Inexpensive print versions are available for purchase from Amazon. You can learn more about the features and ancillary materials for each text by clicking on "learn more" for any text of interest above. You can also request to join the Active Calculus Users Google group for instructors, a low-traffic listserv for discussion among instructors who use one or more of our textbooks.

Active Prelude to Calculus, Active Calculus, and Active Calculus Multivariable are written in PreTeXt, a free and open-source authoring platform for creating the next generation of textbooks. PreTeXt generates beautiful HTML output that is readable on a smartphone and meets most user accessibility requirements.

We also maintain an online archive of past editions from 2017 forward. The most current version of each text will always be found on this page.

© 2019-2022 Matthew Boelkins

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Active Prelude to Calculus, Active Calculus, and Active Calculus Multivariable are free, open-source texts designed for a more active learning experience for students. Every section of each text has engaging activities for students to complete before and during class, as well as exercises that challenge students to connect and assimilate core concepts.

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Download Annual Edition

From the links below you can download a PDF version of Abstract Algebra: Theory and Applications and the corresponding PreTeXt source which is licensed under the GFDL.

Note: The 2019 Annual Edition has been finalized. See the note about the various Editions and changes.

The current edition is for the 2019-20 academic year, with only minor modifications to the content from the previous year's edition. It also verifies and updates substantial material about Sage to Version 8.8, available online, and in a special PDF version (see Sage and AATA). The 2009 PDF version is very close to the original text, as first published by PWS Publishing Company in 1994. Since then, the copyright has transferred back to the author, making this open source version possible. (A full copy of the copyright transfer is distributed with the source code in the local subdirectory).

A print-on-demand version is the most affordable way to get a physical copy (hardcover or paperback) of the entire text. However, a PDF version optimized for print (two-sided, no color, etc.) is made available here for your convenience.

Note: A PDF version that includes all the supplemental material about Sage is available for download from the Sage and AATA page.

Full Text, 2014 edition (PDF, 1.8 MB, August 15, 2014)

Full Text, 2013 edition (PDF, 1.9 MB, August 16, 2013)

Full Text, 2012 edition (PDF, 2.4 MB, August 11, 2012)

Full Text, 2011 edition (PDF, 2.4 MB, August 10, 2011)

Full Text, 2010 edition (PDF, 2.2 MB, August 27, 2010)

Full Text, 2009 edition (PDF, 1.5 MB, February 14, 2009)

Development Versions, Source Code with Revision Control

Source code is available in a git repository, hosted at GitHub, and authored with PreTeXt. Prior to the 2015 edition, the source code was LaTeX and was primarily hosted at bitbucket.org.

Periodically a current snapshot of the book as a PDF is available here. These are not archived, nor tagged in the source code repository. They simply accumulate fixes or additions made to annual official releases. If there is nothing listed here, then use the current Annual Edition.

Updated PDF snapshot (January 1, 2020)

Updated Printable PDF snapshot (January 1, 2020)

The absolute most recent version is available as source code through a git repository at github.com, the specific address is https://github.com/twjudson/aata. Links below to recent source code archives come from github.com, so the current version links below are the most up-to-date version.

If you would like to participate in contributing to the book, you will want to use git. Then you can fork the repository on GitHub, or create a local copy of the source code by doing the following in a terminal window:

1. Set your working directory to a place where a new directory named aata can live.

2. Issue the command: git clone https://github.com/twjudson/aata.git

If you would like to make contributions to the book, sending a pull request via GitHub is welcome, or else a patch file produced by git. If you wish to make substantial contributions, or create an alternate version, please save yourself some trouble and contact the author beforehand.

Source Code Archives

Cover from PWS Publishing Edition, 1994

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Abstract Algebra: Theory and Applications (A Free Textbook)

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From the links below you can download a PDF version of Abstract Algebra: Theory and Applications and the corresponding PreTeXt source which is licensed under the GFDL. Note: The 2019 Annual Edition has been finalized. See the note about the various Editions and changes.

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Is the equation \(x^2 + xy +\tan(y^3)=0\) linear or not? Why or why not?

Find all solutions to the system of two linear equations \(2x+3y=-8\text{,}\) \(x-y=6\text{.}\)

Describe how the production manager might explain the importance of the procedures described in the trail mix application ( Subsection WILA.AA).

In Example TMP the first table lists the cost (per kilogram) to manufacture each of the three varieties of trail mix (bulk, standard, fancy). For example, it costs $3.69 to make one kilogram of the bulk variety. Re-compute each of these three costs and notice that the computations are linear in character.

In Example TMP two different prices were considered for marketing standard mix with the revised recipes (one-third peanuts in each recipe). Selling standard mix at $5.50 resulted in selling the minimum amount of the fancy mix and no bulk mix. At $5.25 it was best for profits to sell the maximum amount of fancy mix and then sell no standard mix. Determine a selling price for standard mix that allows for maximum profits while still selling some of each type of mix.

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Is the equation \(x^2 + xy +\tan(y^3)=0\) linear or not? Why or why not? Find all solutions to the system of two linear equations \(2x+3y=-8\text{,}\) \(x-y=6\text{.}\) Describe how the production manager might explain the importance of the procedures described in the trail mix application ( Subsection WILA.AA).

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Front Matter 1 What is Combinatorics? 2 Applications of Induction and Recursion in Combinatorics and Graph Theory 3 Distribution Problems 4 Generating Functions 5 The Principle of Inclusion and Exclusion Back Matter

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by

David Guichard and friends

Department of Mathematics
Whitman College

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. If you distribute this work or a derivative, include the history of the document.

This text was initially written by David Guichard. The single variable material in chapters 1-9 is a modification and expansion of notes written by Neal Koblitz at the University of Washington, who generously gave permission to use, modify, and distribute his work. New material has been added, and old material has been modified, so some portions now bear little resemblance to the original.

The book includes some exercises and examples from Elementary Calculus: An Approach Using Infinitesimals, by H. Jerome Keisler, available at http://www.math.wisc.edu/~keisler/calc.html under a Creative Commons license. In addition, the chapter on differential equations is largely derived from the corresponding chapter in Keisler's book. Albert Schueller, Barry Balof, and Mike Wills have contributed additional material.

This HTML version was produced by David Farmer, Albert Schueller, and David Guichard. It is generated from the TeX source; the output of TeX should be treated as the definitive version. Please report problems to guichard@whitman.edu.

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by This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA. If you distribute this work or a derivative, include the history of the document.