2.2: New Page - Mathematics

2.2: New Page - Mathematics

2.2: New Page - Mathematics

It started with a simple tweet, before the internet went berserk:

Now, I taught junior high school math for three years, so when I saw this simple equation, I knew I could solve it quickly and I did just that. Before you see my answer, go ahead and solve it for yourself. I dare you!

OK, so what answer did you get?

I used PEMDAS to solve the problem, as follows:

And if you got 16, you're with me, and about 39 percent of the internet, according to this Twitter poll that a whopping 340,000 people took after the math problem went viral on Twitter.

However, that means that a whopping 59 percent of those poll-takers, or about 200,000 people, thought the answer was 1. Were they all wrong? Was I wrong? Or is something else going on here?

As it turns out many people around the world are taught order of operations in math as children using BODMAS rather than PEMDAS. If you were taught math using BODMAS, you probably solved the equation as follows:

So, who is correct, and who is wrong?

The answer is not as simple as the question.

The real answer is, it depends how you were taught. That, and the question isn't written very well: It would be better to using the "/" sign instead of the "÷" sign, which is rarely used after elementary school. (And even better to use more parentheses to indicate the order you want.) So, a better version would be 8/(2(2+2)) = ? or (8/2)(2+2) = ?

I look for life and leadership lessons with every experience I go through. So, what are the lessons here, beyond the math itself, for entrepreneurs and leaders to take away? I thought of three to share:

1. It's important to clearly define the problem.

2. Sometimes, even when you think the answer to a problem is simple, it's really more complicated.

3. Sometimes, there's more than one answer to a problem!

Here's to our better defining problems we face, and always looking carefully for the answers!

Hot Topics

New York City HOLD National Honest Open Logical Decisions on Mathematics Education Reform

    A national nonpartisan advocacy organization that provides parents, K-12 educators, mathematicians and scientists, and others information resources and networking opportunities to support systemic improvements in the quality of mathematics education in our nation's schools.

The performance of American students in mathematics is mediocre at best. Reform efforts over the past two decades have only made matters worse. We are advocates for school mathematics education reforms that support the mathematics interests and abilities of all students. We advocate programs and policies that help all students achieve success in school mathematics courses, that prepare all students for the broadest options in high school math and science courses, and that give them the opportunity to advance into mathematics based college courses and careers.

NYC HOLD National (Honest Open Logical Decisions on Mathematics Education Reform) is a nonpartisan advocacy organization that provides parents, K-12 educators, mathematicians, scientists and other concerned citizens information, resources and networking opportunities to support systemic improvements in the quality of mathematics education in our nation's schools. NYC HOLD was first established in order to address mathematics education in the New York City schools. It is now a national organization

    . One of the advocates for California approved textbooks was Barry Simon, the mathematics chair and IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. Simon asserted that basic skills are essential to mathematics and counselled against redefining algebra via the NCTM aligned programmes in order to increase passage rates. Confronting the social justice arguments of the progressivists, he countered, 'If anyone is racist or sexist, it is those who claim that women and minorities are unable to deal with traditional mathematics'.
    In an open letter dated July 7, 2006, former governors Gray Davis and Pete Wilson expressed unambiguous support for California's K-12 academic content standards, curriculum frameworks, instructional materials, and tests aligned to the standards. They warned that dismantling California's system for K-12 education would be a disastrous step backward.

Save Our Children from Mediocre Math

    Save Our Children from Mediocre Math (SOCMM) is a non partisan organization that advocates ensuring Conejo Valley Unified School District parents have a choice to have students instructed in mathematics using a State Board of Education approved curriculum.
    Great Connected Math News! We know of two Alpine Jr. High Schools that have completely thrown out Connected Mathematics (excepting one teacher) because it failed so badly!
    Johnny Lott's condemnation of the open letter as a form of "stalking" appears to be an attempt to side step legitimate criticisms of the current direction of mathematics education in the United States.

California's Algebra Crisis

    California has had its share of educational crises—such as whole language and fuzzy math. Despite recent improvements, the state is still in the grips of an algebra crisis.
    IMP is a program designed to retain the attention of students who will either not attend college or will major in non math fields. It lacks the depth of study for students who will study math in college. It is not a college prep math curriculum.

    NYC HOLD is a nonpartisan advocacy organization that provides parents, educators, mathematicians and other concerned citizens opportunities to work together to improve the quality of mathematics education in the New York City schools.

    We have followed our children's experience and progress in NCTM Standards-based mathematics programs and have grown increasingly concerned. We have studied the materials and teaching approaches in our children's schools. Some of us have researched the programs and their use in other regions and found that we are not alone in our concerns, rather, our experiences and worries are shared by parents across the country.

    Is LA turning a silk purse into a sow's ear? An Open Letter to the superintendent and school Board suggests LA's Algebra I pacing plan is disastrous and undermines both the California math standards and state approved textbooks.
    In order to raise the level of student achievement in secondary school mathematics, which everyone agrees is urgently necessary, there must be major improvements in the expository procedures employed by teachers.
    The NCEE is satisfied with entry level math. Their "vision" is limited to the needs of everyday life. The concept of prerequisite knowledge is never mentioned. They're not concerned with setting the stage for learning more advanced math. Many of their high school math examples belong at the elementary school level. They claim to emphasize conceptual understanding, but give no evidence that they understand how math ideas are connected. They appear blind to the vertically-structured nature of the math knowledge domain.

Mathematics Education in California

  • Mathematics Framework for California Public Schools: Kindergarten through Grade Twelve [requires Adobe Acrobat reader] [requires Adobe Acrobat reader]
    David E. Joyce provides this review of the Prentice-Hall Geometry text, noting . It's the content that bothers me, in particular, the lack of logical content. The review covers each chapter in a way that is especially informative for those required to use this text.

Does Two Plus Two Still Equal Four? What Should Our Children Know about Math?

    Despite efforts to improve mathematics education in the United States, the August 2001 National Assessment of Educational Progress report found that a majority of children are still unable to perform at a basic level in mathematics and that an achievement gap between white and minority students continues to persist in that subject. The link provides information and transcript of the seminar at AEI.

Stand and Deliver Revisited by Jerry Jesness

    The untold story behind the famous rise -- and shameful fall -- of Jaime Escalante, America’s master math teacher.
    A review of the TERC (Investigations in Number, Data, and Space) mathematics program shows:
    • TERC Omits All Standard Computational Methods
    • TERC Omits Standard Formulas
    • TERC Omits Standard Terminology
      Mathematics education policies and programs for U.S. public schools have never been more contentious than they were during the decade of the 1990s. The immediate cause of the math wars of the 90s was the introduction and widespread distribution of new math textbooks with radically diminished content, and a dearth of basic skills. This led to organized parental rebellions and criticisms of the new math curricula by mathematicians and other professionals.
      In NYC, organized parent and teacher opposition to the new math programs began in District 2, one of the city's best and now extends to District 3, District 10 and District 15. Bronx high school teachers have organized to express opposition to next years' requirement they use only the Interactive Mathematics Project (IMP), an experimental high school math program. The teachers worry the program lacks important mathematical content necessary to prepare large numbers of their students for the Regents A exam and college level coursework.
      The new programs, many without student texts, are based on a "constructivist" teaching philosophy, which discourages teachers from teaching mathematical rules and procedures. Instead, teachers guide students, through group activities, to their own "discovery" of personal solutions. Students are encouraged to seek help from each other, rather than from the teacher.
      A point by point review of CPM Mathematics vis-a-vis the California Standards. Much of Volume 1 actually detracts from developing algebraic competence. Almost all of the mathematical content is at the level of the Grade 7 standards or below . This is not algebra and it is not college preparatory math, no matter what it calls itself. Eventually, Volume 2 starts teaching some algebra but it is too little and too late.
      The list of K-8 programs approved by the State Board of Education on January 10, 2001. These programs were seen as the best fit to the state Mathematics Standards [.pdf file 504k] and the guidelines in the Mathematics Framework [.pdf file 1748k].
      This site provides reviews of textbooks adoped in California for middle school, including texts in algebra 1. The report helps to identify the strengths and weaknesses of these adopted programs.
      This is the press release for the report from Pacific Research Institute for Public Policy. The full text [.pdf file 741k] of the report, Facing the Classroom Challenge Teacher Quality and Teacher Training in California’s Schools of Education is also available.
      What do you say? How do you respond when your school tells you that your child's math program is going to be replaced? What is your reaction when the replacements main advantages are a "Tokyo by Night" layout, fuzzy-headed but politically correct examples, oddball algorithms and methods (or no methods at all), and a big emphasis on writing essays and playing games?
      My disagreement with my father contained the essential elements of the current Math Wars, the debate that is going on today over the way that mathematics should be taught.
      Ralph A. Raimi, professor emeritus of mathematics at the University of Rochester, talks about MSPAP, the examination system Maryland uses to assess student performance. He notes that, The system was of no value for its announced purpose. For the full story see the Washington Times of Sunday, April 1, 2001.

      Opposition to California's mathematics standards from reform leaders continues as of this writing. Former NCTM president Jack Price wrote in a letter published by the Los Angeles Times on May 10, 1998:

    • The NCEE Wants to Demonstrate "How Good is Good Enough"
    • What's Emphasized in the ACPS? What's Missing?
    • A Compact Version of the 38 ACPS Math Examples
      This level of parental outrage and concern is certainly not confined to McFarland. According to Parents Raising Educational Standards in Schools, a Wisconsin-based parent organization, math education has become the number one concern of parents calling for information and assistance. In the last two years it has supplanted the "Reading Wars" and is causing parents across Wisconsin and the nation to organize and rebel.

      Ralph Raimi discusses the new Principles and Standards for School Mathematics (PSSMP) from the NCTM.

      Whole math must lead to a miasma of confusion, boredom, and despair. Rather than encouraging independent, conceptual-level thinking, it is thoroughly anti-conceptual. It dooms children to function on a primitive, perceptual level—i.e., to flounder in a chaotic sea of concretes with no objective principles to guide them. This is cognitive child abuse. Whole-math defenders are shrinking the cognitive capacities of their students to those of infants or even animals.
      "The progressive way of running a school is essentially the opposite of what the 'effective schools' research has taught us." So says Hirsch in the new journal, Education Next
      The U.S. Department of Education issued a statement endorsing some of the worst mathematics programs available. This prompted a rebuttal endorsed by over 200 of the nation's leading mathematicians and has resulted in a congressional hearing.
      Although high-stakes standardized tests are often controversial, Reality Check picks up few signs of public backlash. Neither parents, teachers, nor students themselves voice significant dissatisfaction with testing in their own schools. Large majorities of all groups express strong support for their own district's efforts to raise standards and for using standardized tests to enforce standards .
      Noting the inadequate achievement in school mathematics in the United States, President Clinton proposed a voluntary national test as a way to combat the problem. Examinations can be an effective way to stimulate achievement gains, but the devil is in the details. Clinton's plan got off to a bad start. The first committee working on it was full of fuzzy math supporters, and the initial plans were dismal. The test design was shifted from the initial committee to the National Assessment Governing Board. The progress was often delayed and funding was restricted by congress.
      Similar to the original NCTM Standards, PSSM is vague about the major components of arithmetic mastery:
        1.Memorization of of basic number facts
        2.Mastery of the standard algorithms of arithmetic
        3.Mastery of fractions

      The National Council of Teachers of Mathematics, now led by theoreticians from our Schools of Education, imposes policies that distort the teaching process and heavily impair the learning of school mathematics. Can the NCTM accept the challenge to Save Our Schools?.

        The academic achievement of American students has risen since the 1970s but only at a snail’s pace. Performance in arithmetic remained static or declined slightly. Results for thirteen year olds suggest large numbers of students have not mastered the basic arithmetic skills that are necessary before moving on to algebra.
        In a paper commissioned by the Brookings Institution, David Klein describes characteristics and academic policies of three low income elementary schools in the Los Angeles area whose students are unusually successful in mathematics.
        The Wall Street Journal Editiorial of January 4, 2000 addressed the so-called reform in mathematics education. The editorial concludes that New Math will take its casualties, especially among the poor, adding to the already mounting costs of the decline in national educational standards.
        American education is under intense pressure to produce better results. The increasing importance of education to the economic well-being of individuals and nations will continue feeding this pressure. In the past—and still today—the profession has tended to respond to such pressures by offering untested but appealing nostrums and innovations that do not improve academic achievement.

      Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education [.pdf file 54k], by H. Wu

        The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding.

      Knowing and Teaching Elementary Mathematics [.pdf file 73k], by Richard Askey

      A circular grid like this one can be helpful for performing dilations.

      The radius of the smallest circle is one unit, and the radius of each successive circle is one unit more than the previous one.

      Since the circles on the grid are the same distance apart, segment PA' has twice the length of segment PA , and the same holds for the other points.

      • As of June 25, 2021, Update 2.2 has had the longest wait of any Geometry Dash update, at 4 years 6 months, or about 3 years 8 months if Updates 2.1 and 2.11 are considered separate.
        • As of June 21, 2020, the Geometry Dash community has waited for half the game's existence for 2.2, or the waiting time for all other major updates combined.
        • If not released before roughly February 20, 2022, this will include the waiting time for 2.11, meaning the main Geometry Dash game will have gone half of its entire existence without any updates to it whatsoever.
        • If not released before April 30, 2022, this will also include the release of Geometry Dash SubZero, meaning that Geometry Dash as a franchise will have gone half of its existence without any official content updates whatsoever.
        • Update 2.11 was originally meant to be a small "pre-update", but it has been the latest update for over three years.
        • In these hacked versions, the camera triggers such as Static Camera are confirmed, as well as confirming that there is no object limit in 2.2.
        • However, these versions don't include features such as the new gamemode, the new gauntlets, the multiplayer feature, or any new secrets, shops, or icons.
        • RobTop started to reveal plans for 2.2 before 2.1 was even released, when he teased the new camera controls.
          • He also has done this for Update 2.3 during 2.2's development, teasing another new gamemode.
          • At least one new preview has been revealed for each gamemode, including the new gamemode that is expected to release in 2.2.
          • However, this did not go to plan as expected as of Christmas 2020 (25 December), 2.2 is apparently still yet to be released on application stores. RobTop has not confirmed if a new plan or deadline exists as of this date.
          • A number of famous creators and streamers represented by their icons, such as Juniper, Technical, AmorAltra, AeonAir, Viprin, and more.
          • A sign labelled "PP" in the first results image, a reference to the level PP by AmorAltra.
          • A building covered in yellow hexagons with faces in the first image, referencing C.S.Q.N. by Bluskys.
          • A red circle character in the second results image resembling the character Jeremy from the level Your Brain on Drugs by Jayuff.
          • A blue and white cube (Rafer) fishing a gray and red cube (Alkali) out of the water in the second image, a reference to the animations made for Rafer's levels In Silico and Sky Tower.
          • A computer with a green smiley face in the second image, most likely a reference to Computer Talk by 1374.
          • A large cat having a picnic in the second image, referencing Meow by BlowMyPooh.
          • A volcano in the third image, representing Magma Bound by ScorchVX.
          • The 3D-like wave from the level HOW by Spu7nix in the third image.
          • The large lava monster from Haze N Blaze by Unzor in the third image.
          • Multiple large characters in the background of the fourth image, referencing popular boss levels such as CastleMania, The Mayhem War, Better Than Reality, and Ultra Violence.
          • One of the penguins and a slime from Mastergame by Serponge in the fourth image.
          • An icon contest was hosted in November/December 2020, as people thought 2.2 might be released back then, but the results are currently not out.


            24th June:RobTopreveals plans to add a trigger that will allow players to manipulate the camera angle.

            8th February: Viprin, Etzer, and Michigun announced an icon contest in collaboration with RobTop in which the 10 winners of the respective categories of the contest would get their submissions put into Update 2.2. 1st Place winners got their submissions put into Update 2.11. The rest will be added in future updates.

            8th March: RobTop releases the first teaser image of Update 2.2, showcasing a new game mode.

          • 20th July: RobTop replies to AdvyStyle's comment on Twitter, showing a new comment system.
          • 22nd July: RobTop comments on the respective daily level with the following:
            • "Hmm 2.2 info. Well one thing is it will be out pretty quick after 2.11, that's good."
            • "Secret: In 2.2 I am testing the idea of flymodes without roof or floor, so you have no limits."

              19th and 20th August: RobTop comments on the daily level with the following:
                "@PlatnumGD - I can add a "select All in GroupID" for 2.2."

                1st September: RobTop comments on the daily level with the following: "@Ocharius - More options for blocking players, timeouts etc will be added in 2.2 :)"

              • 7th November: RobTop goes on EricVanWilderman's stream and talks about 2.2. He reveals that he may be adding Unlockable Death Sounds and also says Challenge Mode will not yet be featuring Multiplayer capabilities.
              • 21st November: RobTop uploads a video on YouTube about the "Random Trigger".
              • 29th November: RobTop uploads a video on YouTube showcasing a new desert-like background, camera controls and free-flying.

              • 1st December: RobTop uploads another video, showcasing cube direction change.
              • 12th December: RobTop unveils a preview for a new Geometry Dash project titled "Geometry Dash SubZero". Ώ]
              • 21st December: RobTop releases "Geometry Dash SubZero" on iOS and Android, revealing the 2.2 triggers and more. RobTop also comments on how the flip trigger and percentage will work in 2.2 in this forum post:
              • 22nd December: 2.2 was added on the lite version of Geometry Dash, with a new level, Electroman Adventures.
              • 23rd December: A hacked Android version of "Geometry Dash SubZero" that allows users to use the 2.2 level editor is released, revealing a reverse trigger, new editor features, and the ability to use Free Fly mode for all game modes. Also, the level copy exploits from GD World now has 2.2 features.
              • 31st December: RobTop comments on the level "Wire Frame" stating that he will add new tools to make using camera triggers easier.

                4th January: RobTop comments on the level "Opalescent" saying that he would add new level editor features to make creating easier, since the new camera toggle is hard to control.

              • 2nd February: RobTop comments on the daily level saying that "2.2 is looking good, got nice stuff for everyone :)" and apologizes for being slow. He also confirms that there will be new monsters in 2.2 along with a trigger that stops the player from jumping at any time, also stating that "Challenge Mode > Multiplayer Mode", probably meaning that a Challenge Mode is more likely to come in 2.2 or other future updates than a Multiplayer Mode.
              • 5th February: RobTop again comments on the daily level and gives a small amount of information, saying that "2.2 does not have all 2.2 stuff", suggesting that not all of 2.2's confirmed features have been added yet. He also says that he will probably add a custom art system in the next update after 2.2, also saying that he is 100% planning for 2.3 to be a small update.
              • 6th February: RobTop comments on the daily level for a third time saying that he might add a new level event occurring every 3 days that will only have Insane difficulty levels in a future update.
              • 10th February: RobTop comments on the daily level, Zim by DanZmen, saying "@Robotic24 - [2.2 is] going well. Playing around with code for challenge mode :)"
              • 12th February: RobTop released the fifth teaser which hints the soundtrack for the new level. The teaser depicts a bucket with a Swedish flag in it and the words "I know". On the right, the teaser shows a planet-like symbol along with two other symbols. Based on this, it is assumed that the song is Hinkik - Explorers.
              • 13th February: RobTop comments on the daily level saying that there are numerous features planned for 2.2 that he hasn't mentioned saying "Multiplayer = Challenge Mode. Don't get any ideas" and that there will "hopefully be an alpha version of challenge mode in 2.2. We will see :)" He also mentions that he is currently programming lots of fun stuff.
              • 14th February: RobTop comments on the daily level saying that he is coding the challenge mode system with chat and thinks it will be fun. He says that challenge mode "can be multiple players, random level, x attempts, lots of settings", also saying that there will be more chests to open in the Treasure Room in 2.2. RobTop also says "@DrFox - if I get challenge mode working it will be huge tbh" and that he will probably remove the Force Smooth Fix option. He also says that Challenge Mode is the same as Multiplayer Mode.
              • 16th February: RobTop again comments on the daily level with info that "Challenge mode coding is going well. Now I have a lobby that friends can join etc. :)" He also says that he has been building a new ban system and that in 2.2 elder mods will be able to comment ban users temporarily for an unspecified amount of time. RobTop also mentions that elder mods can pick a custom comment text color in 2.2 and that he needs to set aside some time to fix the global leaderboards. Hours later RobTop comments on the new daily (Cobalt) saying that multiplayer/challenge mode is going well and that he should have an alpha version ready to be used in 2.2.
              • 18th February: RobTop goes onto Knobbleboy's stream announcing that the new collaboration system will be delayed due to it being a pain to make.
              • 23rd February: RobTop comments on the daily level saying that he has added a song trigger which can skip to different times in songs while playing levels.
              • 25th February: RobTop appears briefly on the daily level "Cold Fire" saying "Dont worry 2.2 is coming, all going well :)"

              • 5th March: RobTop comments on the daily level saying "you better not look for my new vault in 2.2. gonna hide it gooooood Kappa", meaning that there will be a new, fourth Vault in 2.2 and its entrance will be harder to find.
              • 6th March: RobTop comments on the daily level saying "@GDGeeDee - Challenge mode has a kind of online/offline indicator for players. Thats something :|"
              • 10th March: RobTop appears on the Demon List stream and says he is considering adding an official FPS (Frames Per Second) option to the game, which would help ease any potential drama over the current FPS bypass.
              • 11th March: RobTop comments on the daily level saying that all is good with 2.2 and that he is still working with some stuff for the challenge mode. RobTop also says that he has added extra steps to stop users from faking account names, and also changed the way the user search system works. Finally, he says "Not sure what I am most excited for. All the new stuff will be big for GD tbh."
              • 13th March: RobTop comments on the daily level with a decent amount of info, saying that he is still planning a reward system for challenge mode and that he is still wondering whether a player will only be able to join games with friends or if the player can join any game - RobTop also says that "Could also make it [challenge mode games] by invitation, so you can invite anyone even if they arent a friend", and that the maximum amount of players in a challenge will probably be 10. He also says "@CircleGMD - New gamemode is 100% done" and "There are more chatmods than SirHadoken, but yea more will be added :) Also added some changes :)" RobTop also confirms that the custom art system will probably be in 2.3.
              • 22nd March: RobTop comments on the new daily level saying that he isn't giving any more 2.2 spoilers yet but he is playing with some more ideas. He also says "Working on lots of stuff, today I was drawing icons. "
              • 24th March: RobTop comments on the daily level with: ". 2.2 is slow, doing all I can to get it ready :) Sorry to keep everyone waiting :/"
              • 26th March: RobTop comments on the daily level saying "You can use challenge mode in 2.2 as a chatroom if you want", hence confirming a chatroom, and "@MarcoDash - Lots of triggers in 2.2, spicy triggers". He also says that the scale trigger will not be in 2.2 but probably in 2.3. RobTop also reconfirmed the usage of the song/sound trigger with "Sound trigger allows you to skip around in a song. So you can skip boring parts and go to the drop!", also saying "Think you are gonna like the new unlockables in 2.2, lots of things setup :)"
              • 28th March: RobTop comments on the daily level stating: "Nah, you can't change song with song trigger. Too much lag" and "With the song trigger you can place it where you want, and enter milliseconds to skip the song to."

              • 7th April: RobTop tweets a link to an unlisted video called: "TimeWarp test" that showcases a new trigger called the "TimeWarp" trigger, which allows the user to manipulate the level speed. This trigger is expected to be used for better sync with the level song.
              • 19th April: The purpose of the 2 triggers of the beta editor of Geometry Dash World/SubZero - whose purpose was unknown before - was leaked by OmegaFalcon.
                • The first trigger turned out to be a 'no jump' trigger, which stops players from jumping when holding unless they stop holding then hold again. ΐ]
                • The second one was a 'don't fade' trigger, which deactivated the object's fade transition effect. Α]

                  27th May: RobTop comments on the daily level that "2.2 has been slow, but it will be pretty big so atleast thats something :)" and "magic section code is secret :) but it seems to work ok."

                  9th June: RobTop replies to a user on Twitter, saying that he can "increase the limit" for the maximum amount of your custom objects in a future update. Β]

                • 13th August: Viprin confirms that a new sneak peek is coming soon. Γ]
                • 16th August: Viprin says that the difficulty of building Challenge Mode, the upcoming multiplayer mode, is the primary reason for the 2.2 delay. Δ]

                • 13th October: RobTop replies to GuitarHeroStyles's video on Future Dash's Adventure Mode saying that he could do the same exact thing in the 2.2 editor. Ε]
                  • RobTop also releases a preview on YouTube, showcasing the arrow system to change the direction of the player, as well as a new monster and cube animation. Ζ]

                  • 7th December: RobTop joins ChrisCredible's Twitch Livestream, in which he mentions that, although he is working to release 2.2 before Christmas, such a goal is difficult for him to achieve. Regardless, he agrees to do his best to release it as soon as possible. Some other features he brings up include a revamped scale system, which consists of a scale trigger and the potential option to make an object's size smaller than the current minimum. Additionally, he brings up the possibility of hiring others to help him work on future updates after 2.2. He also explains why the update is taking as long as it is, saying that "it grew too big" and that "some features take way longer than expected". He agrees that updates after 2.2 will most likely be smaller and will take less time. ⎖]
                  • 21st December: In the daily chat, RobTop confirms that Challenge Mode will contain a chatroom for players to converse with each other before a match. He also reassures that he is "working non stop on the update."

                  • 6th January: RobTop comments on the levels God Eater by Knobbelboy and Beastmode by Pipenashho, bringing up the fact that he plans to have a rating above "epic" in 2.2 that gives the player 4 CP, which he refers to as the "epic+ fire". In addition, he says that the maximum number of groups in 2.2 will be raised to 9999.
                  • 23rd January: RobTop comments on the daily level as well as "Worst LvL Ever Made" with a substantial amount of information. He considers naming the new feature rank as "legendary" as opposed to simply "epic+". He jokingly considers adding a "Why did you do this" rating that takes away 1CP. He also comments that 2.2 is "already too much" and will no longer be adding any new features to it, however he is planning to accept suggestions for 2.3. In addition, he explains that the timewarp trigger can decrease the game's speed down to 0.1x or increase it up to 2.0x. He is planning to implement a trigger that hides the ground as well as a gradient ground. He says that the latter could be made into a plain ground if color 1 and color 2 are chosen to be the same.

                  • 21st February: RobTop confirms in the daily level chat that the chat in Challenge Mode will be "real-time", implying that it will automatically update without needing to refresh it. However, he clarifies that level comments will not do this.
                  • 22nd February: RobTop uploads a new video showcasing the new particle editor for 2.2, which contains a variety of new customization options.
                    • In the video, an FPS counter can be found in the top-left corner, which could be evidence of an FPS option in update 2.2. Also, a new object section and the existence of 27 new objects in the animated objects section were revealed.
                    • RobTop makes on the same day the platformer mode video public and updates its description, officially referring to the mode by that name. He also clarifies it as a separate mode distinct from normal levels. The timewarp trigger demonstration video was also made public for a while before being unlisted again.

                    • 9th March: RobTop comments on the level "shuriken" by Drob3 that 2.2 will have even more gauntlets. He says that they also won't be released at the same time.
                    • 10th March: After joining the official Geometry Dash Discord server on the 8th of March, RobTop posts new information on 2.2, including:
                      • That there could be a second icon contest after 2.2's release. ⎗]
                      • New Map Packs will no longer be released because according to RobTop, gauntlets are "more fun". ⎘]
                      • While there will still be warnings for levels with more than 40000+ objects, the limit for objects in a level will be removed after 80000 objects are used. The reason for the change is because RobTop did not like the idea of using hacks for bypassing the object limit. Infinite object levels will have a warning like the 80000 object warning. He also said that there is a new icon for exceeding 80000 objects in a level. ⎙]
                      • When levels are played, there will be a loading screen before they load. This was made to let players know that the level is loading, not lagging. There is no loading screen for the editor yet.
                      • Teleporting Portals can now be moved on the x-axis in Platformer Mode but not Normal Mode yet.
                      • The FPS bypass will be added to the Windows PC version of the game.
                        • Humorously, he also states that he might only release 2.2 to iOS/Android only for people to brag about others not having phones for 2.2 (which will not happen). ⎚]

                          12th April: RobTop tweets the icons of 30 upcoming gauntlets, all with various themes.

                        • 12th May: RobTop uploads an unlisted video called "Uhm. " that reveals the new scale system. It can change a group of objects in different dimensions so that they can be stretched in a variety of ways. He also shows the scale trigger, which can change the dimensions of an object in-level.
                          • The new system could potentially include rotating, too, which wasn't shown in the preview, and therefore might not be accurate.
                          • There is a new block tab, most likely for all new pixel art blocks.
                          • There's also a "create loop" tool that automatically builds spawn triggers, which causes the selected triggers to loop.
                          • 6th July: RobTop announces on the daily level "Aumetras Wrath" by Cirtrax by that there will be new blocks and art in 2.2. He says he will add "cartoony" and pixel art and it will be more vector-ish.
                            • On the same day he says that new sneak peeks will be coming.
                            • He says that adding a 5x speed portal or beyond is not a good idea in his opinion, but that it wouldn't take too much work and is still possible if the community wants it, liekly in Update 2.3. ⎝] **He spoke how the "Ice Block" option works on objects, explaining that the object will have low friction, making it possible to slide on the object. ⎞]
                            • He said he has planned an option for itemID's to be connected to attempts, explaining: "if attempts are higher than x activate group y." ⎟]
                            • He confirmed that there will be new uses for mana orbs, demon keys and diamonds, stating: "diamonds get much more interesting", as well as stating that there will be 6 new shards.
                            • Lastly, he posted a picture showcasing new pixel art blocks. These were confirmed to be actual new blocks, and not custom pixel art. ⎠]
                            • He also confirms plans for an in-game featured song list that regularly gets updated so unknown NG creators can get noticed.
                            • 19th September: RobTop joined the GD Fanmade ideas server "Collar". There he confirmed that nothing new is being added anymore and that he is cleaning and polishing the update.
                              • He further talked about the idea of 2.3 being an update with mostly ideas made up by the community.

                                6th October: RobTop commented on daily level 'RobTop' confirming that Platformer Mode will be a toggleable setting like 2-player mode, but can't be used normally like dual mode with portals.
                                  Viprin announced on his Twitter that the 2.1 Awards Nominees along with the animation "Prelude" prior to Awards will be premiering on YouTube and Twitch on the 14th of December.
                                    Viprin also hopes that 2.1 Awards Results to come right before New Year, which means we could see an early 2020 release for 2.2, as it only took a little over a month for 2.1 to release after 2.0 Awards.

                                    23rd November: The soundtrack for Prelude premieres on YouTube and Soundcloud. Along the tracks there is one mysteriously named "2.2. is out?". The nature of the track suggests that the release date for 2.2 will be revealed in the 2.1 Awards, even possibly releasing the same time as the 2.1 Awards.

                                  • 5th December: In a tweet on the official RobTop Games Twitter, RobTop states that Geometry Dash Lite and Meltdown have been removed from the Google Play Store, with the Lite version being suspended as well, making him unable to update the Lite version. He also explicitly stated that "2.2 is getting closer, doing my best :)"
                                  • 8th December: Viprin joins Npesta's YouTube stream and tells him to beat Kenos within 5 days. Considering Npesta has explicitly stated his goal of beating Kenos before the release of 2.2 numerous times, there is a possibility that Viprin is implying that 2.2 will release alongside the Prelude animation and the 2.1 Awards nominations. He also says the following: ​"the animation is about 11 minutes, awards nominees is about 20 min, and then we have 1 min of . " The last part may be referring to the 2.2 level sneak peek.
                                    • Additional information is revealed in the GD Discord server:
                                      • The Challenge Mode chat appears to act more like a regular chat, unlike the daily level chat (which happens inside the daily level). It also states that Challenge Mode can "evolve" overtime, although it is unknown how this works.
                                      • There appears to be a new tab in the editor featuring over 500 pixel art objects, which can be scaled to any size without being blurry.
                                      • There appears to be a new option in the editor pause screen that can slow down in-editor level playtesting.
                                      • Also in this build was a sprite sheet with art for new triggers in the update, and a new icon that can be unlocked by following Geometry Dash on Twitch.
                                      • Later on that day, the leak was confirmed to be true by Viprin in this tweet.
                                      • RobTop acknowledges this later in the GD Discord. He also says "Now I can go back to sneak peek" which means that a new sneak peek for the update is in the works.
                                      • RobTop reveals that the new Meltdown leak of "Firebird" by MDK being a new main level is a prank created to fool dataminers with references to files such as "Firebird.mp3".
                                      • RobTop sends 4 images later on this day, they appear to be from "THE MAP" feature.
                                      • RobTop said he won't add Meltdown/World/SubZero songs into the main game due to file sizes. He said they could be special songs downloadable from the GD server.
                                      • He also reveals the idea for a Platformer Mode exclusive gamemode, sharing a concept picture of the character in GD Server general chat. He said that this may be added into 2.3, depending on the popularity of Platformer Mode.
                                      • "Flymode", which is the ship in the current game, was said to have looked weird as a ship in platformer mode. This resulted in it being graphically changed to a jetpack for the purposes of platformer mode. It is not known if this will be customisable like all other vehicles, however.
                                      • He also said that the Platformer Mode excludes the wave, the new gamemode, and dual mode.
                                      • RobTop reveals another gamemode concept on the Geometry Dash Discord server.
                                      • RobTop said Diamonds will have a lot more value in 2.2.
                                      • He says that scale trigger can be performance heavy at times.
                                      • There are blocks that can be walked on but jumped through, for platformer mode.
                                      • RobTop confirms that Everyplay support is being removed in 2.2.
                                      • Physics are fixed to act the same, regardless of the monitor's refresh rate.
                                      • In the GD Discord, RobTop asks the community if a profile link to Discord should be added (similar to the way you can add YouTube and Twitter accounts). He also shares art of a new Discord themed icon, which most likely would be rewarded for viewing the GD Discord server if a link to it were added to the game in the "Follow RobTopGames" section.
                                      • RobTop mentions that an Everyplay alternative is very unlikely to happen soon due to stability issues, but one is still possible in the future.
                                      • RobTop also says that the future of platformer mode after 2.2 will depend on what the community does with it.
                                      • He jokingly confirms that there will be at least one new icon in the update.
                                      • On Discord, Viprin confirms that "swing will appear in Prelude" when asked if the new gamemode will appear, which may mean the gamemode is simply called "Swing" instead of the popular theory of "Swingcopter". It's in the opening of the animation, at exactly 0:56. While this is a simple Easter egg, it gives what possibly was the first look at gameplay for it, yet it went unnoticed.
                                      • He says that there will be no platformer mode exclusive gamemodes in 2.2, but one is a possibility in the next update.
                                      • He says there are "lots of secrets in 2.2."
                                      • He says that scale is unlocked in 2.2, but can run into issues at higher sizes.
                                      • Earlier, Viprin said that the animation would contain "1 min of . ", which was thought to be 2.2 related. This proved the theory to be false.
                                      • Prelude appears to have started some sort of ARG related to Update 2.2, with a link appearing that leads to After reaching a certain point in the game, glitched trees will begin to spawn, these can be touched, which causes the game to crash, giving an error related to a missing or malfunctioning plugin for Unity Engine. Resulting error of touching the glitch tree can be found in this video:
                                      • Later on this day, Viprin confirms that so far, the ARG is completed, and that no further action is required until the 28th of December, the day when 2.1 Awards Results premiere. (The 2.1 Awards Results will now take place on the 29th of December due to a video rendering issue).
                                      • RobTop also explains how the new physics work and how old physics worked for different HZ values.
                                      • He also states that it is trickier to make platformer mode levels, mostly referring to music sync and requirement of loops.
                                      • He later said: "Just gonna fix GD World on Google then back to work on sneaky peeky."
                                      • On the same day, SMJS on his Discord, after a source-code investigation a message from the Keymaster would show up if you clicked on Firebird, hinting it might be an actual in-game level: "The Demon Guardian is the only one that can unlock this level. Maybe he has a special key?"
                                      • RobTop stops by the GD Discord server and reveals that diamonds are a currency that are useable in a new special shop. However, unlocks for diamonds in achievements are based on total amount collected still, so getting diamond based achievements isn't changed. RobTop says that diamonds are split into 2 different types, one type that stays and one that is spendable.
                                      • He says that the joke "Firebird" leak "is or is not going to one day maybe not be a level".
                                      • An update was released for Geometry Dash on Steam. This update consisted of RobTop changing the tags in the Steam page of the game, possibly updating them ready for 2.2.
                                      • RobTop says that when you upload an unlisted level to the servers, you can filter the level so only your friends can see it or you can set it so everyone can see it.
                                      • He confirms that there will be a few icons from one of his previous games "Boomlings", requested by GDColon.
                                      • You can now input numbers with the Alpha Trigger.
                                      • He says that you can see if a level is a platformer level from the menu.
                                      • He talks about new running animations for both Robot and Spider gamemodes.
                                      • Platformer mode has checkpoints you can place, presumably in the editor, that are exclusive to the mode. Normal practice mode is disabled in Platformer Mode.
                                      • RobTop wants to add a Boomlings inspired icon as a secret unlock.
                                      • Hex Codes and RGB for colors.
                                      • A "Disable Optimization" option for the Rotate Trigger, as was mentioned months earlier.
                                      • The name of the monster seen in the Platformer Mode reveal, "Spikeball".
                                      • Move Trigger can now lock to the position of the camera.
                                      • The number inputting with Alpha Triggers was found.
                                      • Collision Blocks let you use the player instead of a dynamic block.
                                      • Previewing the music in the Song Trigger at the selected timestamp.
                                      • A Camera Rotation trigger.
                                      • He says that account registration is now performed on a web page, which means you can no longer make accounts in the game itself.
                                      • RobTop confirms that you get the newly-implemented Discord icon by following him on Discord.
                                      • RobTop confirms that he is not adding anything else to the update once again.
                                      • RobTop mentions hiring a development team again for future updates.
                                      • RobTop talks about possible notifications for future updates. He says that they could be used for when Quests and Daily Chests are ready.
                                      • He mentions a potential "GD Tournament" with prizes.
                                      • He says that you mark blocks "untouchable" which makes the block unable to kill the player, sort of like a "noclip" for specific blocks.
                                      • He says that the Y limit in the editor is increased.
                                      • RobTop says that he wants Update 2.3 to take under 6 months.
                                      • He makes another comment on the possibility of a developer team in the future, saying he should probably have one.
                                      • RobTop mentions that the level editor should be restructured but that he doesn't have time to make changes to it in 2.2.
                                      • He turns down the idea of using the "-" symbol in searches, saying that the symbol shouldn't be usable in level names in the first place.
                                      • When asked which update he thinks had the biggest effect on the game, he says that adding custom songs will likely always be the biggest change. However, he mentions that "2.2 has A LOT of new gameplay stuff."
                                      • Only one song can be used in Platformer Mode at a time, but more may be allowed in the future.
                                      • RobTop says, "No, 2.2 isn't coming out tomorrow. Haven't even released sneaky peaky. )"
                                      • At exactly 8:55, a glitch was shown again which reveals a cryptogram filled with letters and numbers. This is the beginning of the second part of the 2.2 ARG.
                                      • Later on this day, the ARG is confirmed to be completed by Viprin. Another official summary is released, as well as the rewards - a code for the new Vault, and some new icons which would presumably be rewarded for putting in the code.
                                      • RobTop reappears in the GD Discord and confirms that 2.2 will update support so GD works on the latest iPad Pro.
                                      • He says that the update will hopefully be released on all platforms at the same time, but it might release a few days earlier on Steam compared to mobile, so bugs are easier to fix.
                                      • RobTop turns down the idea of GD on Linux.
                                      • He says that he'll rework the Mac version of GD after Update 2.2.
                                      • FPS bypass is a PC exclusive feature and isn't on mobile.
                                      • 3rd January: RobTop talks about the many new animations in the GD Discord. He shares a video showcasing the new animations.
                                        • RobTop gives an update on the sneak peek, saying that it's going well so far.
                                        • He says that animated objects are triggerable in 2.2.
                                        • He says he wants to add a simpler method to browse Newgrounds songs, but it won't be coming yet.
                                        • He says again that multiple songs in a level are not usable yet, but might be in the future, mentioning the usefulness of this feature in Platformer Mode.
                                        • Parts of a song can be looped with the Song trigger.
                                        • RobTop says again that there is a gradient ground that allows for solid ground colors in 2.2, but also mentions that ground is hideable.
                                          • Objects cannot be placed in front of the ground.
                                          • This new block is based on a video by EricVanWilderman that showed a fan gamemode called "Switch mode".
                                            • If you would like to try the block for yourself before it's released, the level ID in the video is 58961357.
                                            • He says that he was working on the sneak peek but ran into annoying level building issues, so he has been working on a new system for it. He says "If I get this thing to work u guys are gonna love it."
                                              • "I liked the new system so much I thought about making a video, but I think people will go berserk if I dont release the sneak peek first."
                                              • He says the new feature helps builders on both PC and mobile.
                                              • A video could come for it in a few days if RobTop finishes it soon.
                                              • He says that since the system is done he's going to go back to work on the sneak peek.
                                              • RobTop says he'll be doing more videos on new features in Update 2.2 to help explain them.
                                              • In the video, 3 new tabs for items in the editor are visible. These 3 tabs are speculated to be one tab specifically for the "monsters", one for the new pixel art blocks, and the last one for the particles.
                                              • Also in the video, the edit button to rotate a selection 45 degrees isn't grayed out when static blocks are selected. This likely means that static blocks are rotatable now.
                                              • RobTop explains his reason for adding the system was because of a new block that needs to be placed in a correct order along with edge variations.
                                              • He also says that templates can be mixed, and different variations of pieces can be created.
                                              • The new block he mentioned is actually an "old stone block".
                                              • 5th February: RobTop goes to the Geometry Dash Discord server once again and reveals some more interesting things.
                                                • RobTop confirms that Map Packs won't be removed.
                                                • If you scale objects too big, it will begin to glitch out. This technically adds a limit to how big you can scale objects. However, shrinking objects don't have a limit.
                                                • Most things in Platformer Mode are finished. He is fixing bugs such as "when you get squeezed between moving objects".
                                                • He says, "Sneak peek going well, just coding a lot".
                                                • He states that "sync for start position is worse in 2.2".
                                                • The saving and loading bug will be fixed in 2.2.
                                                • Account creation in Update 2.2 is moved entirely to the website and is no longer in-game.
                                                • He reconfirms that the sneak peek is going well, and that he has run into a few problems with the preview system.
                                                • 1st March: RobTop goes on Discord and reconfirms that the chicken can be found in-game, mentioning a "prize" received for collecting it.
                                                  • He says that the sneak peek has been going well aside from one issue.
                                                  • RobTop says that mana orbs will have more use in 2.2.
                                                  • He says that 2.2 will be "the buggiest update ever".
                                                  • He mentions that in the future RobTop will address these issues, and that Update 2.2 will have fixes for common issues of frame skips and stuttering.
                                                  • He confirms a few new features in the editor.
                                                    • Dual mode will not be available in Platformer Mode.
                                                    • New glow pieces have been added, specifically for slopes.
                                                    • The custom object limit has been increased.
                                                    • He clarifies that global ranking will be fixed after 2.2's release.
                                                    • He says that there's a lot to explain about Challenge Mode and that he'll explain all of it at a later time.
                                                    • Once again, he talks about a Platformer Mode exclusive gamemode.
                                                    • He talks about how he got an in-game FPS Bypass to work for PC.
                                                    • He says that UHD Textures for mobile are not a thing yet, but he said he'll add them in a future update.
                                                    • You can now spawn particles on command.
                                                    • He states that he is "still fixing a few bugs" before releasing the sneak peek.
                                                      • He also states that 2.2 will fix some lag, particularly on Android.
                                                      • 1st April: RobTop goes on Discord and shows some sketches of 4 new icons.
                                                        • He talks about possibly hosting a creator contest after the release of 2.2, with prizes for winning such as an iPad or a gauntlet entry.
                                                        • When asked about adding a new speed portal, he states that "I don't think 5x speed portal is a good idea."
                                                        • He says he is trying to get the update done, adding that "I understand 100% of people are fed up with waiting."
                                                        • He also says that he will look into adding higher graphical and FPS settings for iPads.
                                                        • He hints at the possibility of adding a trigger that activates after a certain number of global or local attempts.
                                                        • He confirms that Geometry Dash Lite will also receive an update.
                                                          • He says his ideal plan is to release Update 2.2, update Geometry Dash Lite and Geometry Dash World, and then make another free expansion app around Christmastime.

                                                            5th May: Concept art for a new ship that resembles an AK-47 is released in the news channel of the Geometry Dash Discord server by Hyperflame, who is an administrator within the server: The ship art will be used in 2.2 as it was later verified by RobTop on his recent comment on the level "Crazy".

                                                          • 8th July: Viprin showcases art for six new icons on Twitter, which were later reposted on the Geometry Dash Discord server.
                                                          • 31st July: RobTop shows an unlisted video titled "Smooth", which shows two new robot animations. It's also mentioned in a tweet by GuitarHeroStyles.
                                                            • One of the animations shows the "head" of the robot bobbing up and down while running, while the other shows this combined with the robot's legs spinning around its entire body.
                                                            • Additional information is confirmed in the Geometry Dash Discord server:
                                                              • After 2.2 is released, a fix for the global leaderboard will be searched for.
                                                              • 2.2's editor will have many improvements.
                                                              • The in-game story will be expanded on.
                                                              • More chests are on the way.
                                                              • The cube gamemode alone has about 100 new icons.
                                                              • Filters for platformer mode levels will be available.
                                                              • New items will be available for purchase, presumably in the shops.
                                                              • New achievements will be added.

                                                              • 1st August: RobTop returns to Discord and reveals some more information on the update:
                                                                • The corrupt save bug has been fixed in 2.2.
                                                                • The wave trail bug is planned to be fixed.
                                                                • RobTop says that if top lists are fixed, more lists could be added in the future, such as top demons or even top clans.
                                                                  • This possibly means that a sort of "clan" implementation could be planned for a future update.
                                                                  • 2.2 will continue the in-game story.
                                                                  • The notorious robot jump bug is fixed for new levels (it can be enabled on old levels, which are using that bug for gameplay).
                                                                  • Profiles, messaging or the friend system can be updated in 2.3 or 2.21 patch.
                                                                  • You'll no longer have to delete everything in a text box to fix something in the start/mid.
                                                                  • We will see few new editor examples levels.
                                                                  • Teleporters in 2.2 can be disconnected and moved.
                                                                  • There will be another Icon Contest.
                                                                  • 17th September: RobTop says on Discord that Discord Rich Presence will likely be in Update 2.3. He also says he is considering adding some "meme icons" from this list:
                                                                    • Among Us icon
                                                                    • Banana ship
                                                                    • Bunny icon
                                                                    • Troll face something
                                                                    • Amazon box spider
                                                                    • Donut ball
                                                                    • Ahegao Ball
                                                                    • Hot dog something
                                                                    • Paper plane ship
                                                                    • Square wave
                                                                    • Cyclic rabbit
                                                                    • Carrot wave
                                                                      • The ahegao ball was later removed from the list as a result of problems and debate related to the term referring to something inappropriate.
                                                                      • Later that day, RobTop showcased the first effect in a series of effects that distorted the screen when playing a user level.
                                                                      • 3rd October:RobTop reveals an early preview of the unfinished work of what appears to be graphics for GD World, affectionately titled "swamp.png".
                                                                      • 9th October: On Discord, RobTop previews concepts for new level rankings to surpass Epic, titled Legendary and Godlike. Godlike will only be awarded to levels “a few times a year”. The Legendary rating uses a reddish variant of the Epic rating difficulty glow, while the Godlike rating uses a blue variant as well as an entirely new version of the difficulty icon itself.
                                                                        • Presumably, since Legendary awards 4 CP, Godlike will likely give 5 CP.
                                                                        • RobTop also confirms that the previously mentioned Among Us and Cyclic rabbit icons will be added.
                                                                        • We can see at the start that there are some triggers named End, Stop Jump, Options, & Gradient.
                                                                        • RobTop also reveals the icon for the Challenge mode in the menu. It is noted that the new mode will allow you to race others to finish in-game levels and may be renamed to Versus mode in the future. ⎢]
                                                                        • He shows off some artwork for new blocks in-game. ⎣]
                                                                        • RobTop also shows a new icon that resembles a castle. ⎤]
                                                                        • He says that the camera rotation trigger is planned to have an option to add to the current camera rotation as an alternative to setting a degree for it to move to. ⎥]
                                                                        • 21st October: RobTop says that he may do coding streams in 2.3, where he will take requests for new features. ⎦]
                                                                          • He reveals that a bug has been fixed in adventure mode (likely a potential new name for platformer mode). The change causes the player to die when they are squished between blocks. ⎧]
                                                                          • He also showcased gameplay of Jetpack Mode and revealed 5 new shards.
                                                                          • A new background, two new icons, and new level completion messages (shown with the "Just, Don't. " setting enabled, presumably) ⎩]
                                                                          • The ability to move the text cursor when typing comments. ⎪]
                                                                            • This is not known to apply to any other areas where text can be entered, such as level names and descriptions.
                                                                            • 2nd November: RobTop considers making outline colors customisable independently of other icon colors, although is not 100% sure. An accompanying image was shown. ⎱]
                                                                              • Icons are revealed to be reworked in the game code, with each icon being an individual file rather than one sheet. ⎲]
                                                                              • RobTop considers adding inverted blending in Update 2.2, and also says that additive blending is added in the update. ⎳]
                                                                              • The ability to change the music download folder on the PC version is considered for a future update. ⎴]
                                                                              • Custom menu and practice mode songs are revealed to be unlockable in some way. ⎶]
                                                                              • RobTop says that the app size will not increase too much from what it currently is. ⎷]
                                                                              • He says that Geometry Dash on game consoles "would be cool" but that he isn't sure how to do it. ⎸]
                                                                              • RobTop says that levels from the free games such as Geometry Dash Subzero and Meltdown will not be added to the main game, because the advertisement money that they create allows for more free games. ⎹]
                                                                              • He considers adding a "Mega Palette" that would unlock after unlocking all colors. ⎺]
                                                                              • 6th December: RobTop begins revealing winners to the Icon Contest on Twitter. ⏃]
                                                                              • 8th December: More icon contest winners are revealed. ⏄]
                                                                              • 9th December: The third set of contest winners is revealed.
                                                                              • 10th December: The fourth set of contest winners is revealed.
                                                                                • It’s discovered that the four images posted by RobTop can be arranged to form a seamless picture together. This is not the "MAP" that is speculated to be in the game in 2.2, as it was created by a third party.
                                                                                • RobTop considers adding the ability to scale the editor grid in Update 2.3. ⏆]
                                                                                • Contest icons are unlockable "everywhere", spread across several areas of the game. ⏇]
                                                                                • Two new icons are unveiled on Discord. ⏈]⏉]
                                                                                • RobTop says that Discord Rich Presence for the game will be implemented in Update 2.3. ⏊]
                                                                                • He says that a new trigger will allow you to hide only one of the players in dual mode. ⏌]
                                                                                • 3rd January: RobTop previews new icons on Discord. He also claims that the amount of new icons in 2.2 is more than all icons in 2.1 combined, allowing for hundreds of new icons.
                                                                                • 30th January: When asked about how it was impossible to enter more than 6 digits for the Newgrounds song ID (the number of songs on Newgrounds had recently surpassed 1 million), Robtop said that it will be fixed in 2.2.
                                                                                  • He says that account security will be improved in Update 2.2, mentioning HTTPS as something that would possibly be implemented. ⏏]
                                                                                  • 5th February: RobTop mentions that Update 2.2 cannot be released as it currently is right now, considering that some features added are still incomplete. ⏔]
                                                                                  • 18th February: RobTop once again says that he isn't adding anything new to the update, and that requested features will be added after 2.2 is released. ⏕]
                                                                                    • He considers separating like and dislike numbers for online levels in Update 2.3. ⏖]
                                                                                    • RobTop also says that a GD API would be considered after 2.2, noting the creation of editor tools specifically.. ⏗]
                                                                                    • When asked about creating custom level playlists for Update 2.3, RobTop hints at them possibly being in Update 2.2. ⏘]
                                                                                    • When asked what he's working on at the moment, RobTop simply responds with "icons". ⏙]
                                                                                    • RobTop says that an update for Geometry Dash World is not planned to go alongside Update 2.2. ⏚]
                                                                                    • He considers adding a built-in feature for texture packs. ⏛]

                                                                                    • 18th May: Robtop previews new shards, icons and blocks.
                                                                                    • 20th May: Robtop previews a green gravity portal that apparently always flips your gravity. Instead of the usual gravity portals not working if you are the same gravity of the portal.

                                                                                    Right-click on an equation and see what MathPlayer lets you do with it! You can cut-and-paste math into any one of a growing number of MathML-compatible software packages, such as Maple and Mathematica. You can open it in our MathFlow and MathType products for further editing, reuse in your own documents, and much more in our MathPlayer User Manual.

                                                                                    If you publish web pages containing MathML, readers using Internet Explorer will need to install MathPlayer on their computers. Obviously, one way to do this is to have them download MathPlayer from our website like you probably did. To get some tips on how best to communicate this to your readers using our MathPlayer Logo Program, as well as some alternative distribution options, visit Making MathPlayer Available to Your Readers.

                                                                                    8. Look for and make use of repeated reasoning

                                                                                    The seventh and eighth standards are closely related, but it’s important to distinguish between them. Instead of focusing on the repeated structure of an object, the eighth standard encourages students to use past problems as a model for present ones. When students can demonstrate repeated reasoning, it means they’re able to try different solutions for the same problem and adjust as needed. Students can see which elements stay the same and which are variables by testing different methods repeatedly. This process develops both attention to detail and oversight -- controlling the small parts of the problem while making sure that overall, they’re on the right path to a solution.


                                                                                    A great way to promote repeated reasoning is through the use of “fact families.” When students write an equation, challenge them to write two or three more equations that directly relate to the original, like this: Source: Teachers Pay Teachers As students progress and become more advanced, this provides a solid foundation for more complicated equations that include fractions, integers and algebraic elements. Fact families encourage students to maintain a focus on the overall equation, while also manipulating the individual numbers and examining the relationships between them. Working with fact families to express repeated reasoning in early elementary gives students the skills they need for later elementary, high-school level and post-secondary math.

                                                                                    Its founding fathers are dead, its disciples scattered, its millions long spent. Yet countless Americans still carry the revolutionary message of new math in their memories, if not always close to their hearts. Now in their mid-thirties or forties, these “new math kids,” myself among them, were part of a learning crusade that in the 1950s and 1960s marched through schools across the nation. For many of us, new math was a disaster for others, a godsend.

                                                                                    Before the results could even be measured, new math became a near religion, complete with its own high priests and heresies. Chief among the hierophants were the University of Illinois’s Max Beberman and Stanford’s Edward Begle. Together with mathematicians and educators at universities in New York, Indiana, Massachusetts, Minnesota, and Maryland, they took aim at the mindless rigidity of traditional mathematics. They argued that math could be exciting if it showed children the whys of problem solving rather than just the hows. Memorization and rote were wrong. Discovery, deduction, and limited drill were the best routes to arithmetical mastery.

                                                                                    In practice, this meant learning how different number systems worked, that the number 9 in the decimal, or base ten, system would be the number 100 in base three. It meant learning about the set, a grouping of things: a beach as a “set” of grains of sand, for example. It meant learning the difference between a number like 7 and its representation the numeral, which could be expressed many different ways—21 minus 14, 7 times 1, VII. It meant learning to draw rulerlike number lines and divide them into sections to discover fractional multiplication. It meant learning about frames—boxlike symbols used as substitutes for the x, y, z ’s of algebra. It meant learning a new language with terms like open sentence, complementation , and truth set . It meant, in essence, learning to discover the hidden patterns in mathematics before knowing what they were called and reasoning out solutions before knowing rules—all at an earlier age than had ever been attempted before.

                                                                                    Beberman also urged a conceptual overhaul of math education. Mathematics should be taught as a language, he said. And like language, it should be considered a liberal art, a key to clear thinking, and a logic for solving social as well as scientific problems.

                                                                                    No educational proposal, before or since, has won such wide and quick acceptance. PTAs, politicians, and textbook publishers stumbled over one another to endorse the new approach to what was probably the worst-taught subject in American schools. Many high school teachers also were ecstatic, even though new math required that they work harder, perhaps retrain, and drop the drill sergeant’s mask for that of the muse. When the Soviets launched Sputnik, in 1957, the small new-math experiment, previously confined to a few score schools, became a national obsession. Parents went to night school to learn the new approach. The press hailed the reformers as the guiding geniuses of the most important curriculum change since Pythagoras.

                                                                                    By the mid-1960s more than half the nation’s high schools had adopted some form of the new-math curriculum. The figure jumped to an estimated 85 percent of all schools, kindergarten through grade twelve, a decade later. As Robert Davis, founder of the elementary-school program known as the Madison Project and now a professor at Rutgers University, said at the time, “U.S. math literacy can no longer be a matter of God and heredity.”

                                                                                    Now, almost thirty years later, we are hearing the saune lament about our mathematical skills and the same call for educational mobilization. Why? Where did new math go? And what happened to all those schoolchildren ready to experience, as one writer called it, “the wonder of why”?

                                                                                    If they were, like me, high school freshmen in 1964, they may have developed a lifelong aversion to anything associated with new math. Or they may have gone on to become brilliant mathematicians. The differences in experience and outcomes were as varied as the nation’s geography and, most important, only as good as the teachers. But what new math became is not what it was intended to be. In fact, there never was just one new math.

                                                                                    The need for curriculum change was apparent to educators everywhere after World War II. Wartime experience had shown that many high school graduates were too illiterate in math to be trained in radar and navigation. Scientific discoveries and new technology made such illiteracy a threat to America’s future.

                                                                                    Yet everywhere would-be reformers looked, orthodoxy reigned. Population growth and a national shortage of mathematics teachers had forced everyone from coaches to homemaking instructors to the blackboard. Few distinguished themselves. Often inadequately trained, these teachers relied heavily on rules-and-rote textbooks that in many ways had changed little since colonial times. Mathematics was presented not as a human enterprise but as a static subject about which everything was known, including a handy bag of computational tricks—like “carrying” and “borrowing” numbers. (Borrow from where?)

                                                                                    It was not surprising, said the reformers, that students lost interest. By the early 1950s two-thirds of U.S. high school students were ending their math careers after their freshman year. And many of those who went on to “advanced math” were still not prepared for college course work.

                                                                                    The dean of the school of engineering at the University of Illinois, William Everitt, proposed a solution: he would find a young, gifted teacher at the university’s laboratory high school and give him the task of designing a high school math curriculum to help prepare future engineers for college. Had he not found Max Beberman, in 1951, what became the University of Illinois’s Committee on School Mathematics (UICSM) might have remained a prairie dream. But under the guidance of this brilliant, chain-smoking New Yorker, UICSM became a new-math totem.

                                                                                    It is a particularly nice irony that the twenty-five-year-old Beberman was put in charge of this four-member reform committee. A graduate of City College of New York at age nineteen, Beberman had originally been denied a teaching credential because his Bronx accent was so unintelligible. Yet here he was being asked to help shape a new high school curriculum. Beberman loved the role, the limelight, and teaching children. With characteristic sagacity he quickly realized that the dull and lifeless textbooks available would ruin his chances, so with the help of the committee and a logician named Herbert Vaughan, he set out to write his own.

                                                                                    Five years later UICSM had produced loose-leaf texts for all four years of high school and put those texts to experimental use in four locations, including University High on the Urbana-Champaign campus. These early tomes were remarkable for their clarity, their weaving together of algebra and geometry, and their clear, but tacit, criticism of the piece-meal textbooks they were meant to replace. On the opening page of the 1954 edition of the revolutionary High School Mathematics First Course , students were introduced to number lines. By the third page they were negotiating positive and negative numbers and confronting the concept of sets. By the end of Unit 9 they had covered everything from coordinate planes to tangent ratios. A summary of concepts, rules, and principles followed each unit. New math had been born. And as classroom reports from pilot schools flowed in and lessons were revised, the message seemed clear: Children were learning more, staying with math longer, and even saying they enjoyed it.

                                                                                    This was exciting news for the Commission on Mathematics of the College Entrance Examination Board (CEEB), which incorporated many of new math’s guiding tenets in its influential 1959 report “Program for College Preparatory Mathematics.” (The commission’s members included Beberman, Vaughan, and David Page, a UICSM teacher who would soon focus his reform efforts on the elementary school.) CEEB’s goal was both to stimulate change in the high school curriculum and to expound the spirit of reform groups, not only in Illinois but at five other colleges and universities. Missing from the commission’s specific recommendations, however, was any call to modernize teachers.

                                                                                    Beberman was keenly aware of the teacher’s role as gatekeeper for the movement. Beginning in 1958 he used National Science Foundation funds to organize four-week summer teaching institutes on the Illinois campus and elsewhere. Even before Sputnik triggered the funding that broadened the program to nineteen thousand schools, Beberman and Page flew around Illinois, teaching classes and listening to teacher complaints. Such feedback prompted them to add more drill to their courses in 1959. For UICSM, teacher education was as important as its 1,430-page teaching manual.

                                                                                    National fretting over the loss of America’s scientific superiority was unleashing other forces in 1957 and 1958. At Yale the School Mathematics Study Group (SMSG), led by Edward Begle, began gathering teams of schoolteachers and mathematicians for “writing sessions.” Over the next fourteen years, SMSG would produce textbooks for every grade from kindergarten up, including special texts for less able and gifted students. The same federal and foundation grants that fueled SMSG propelled perhaps a dozen other reform programs as well. Together with UICSM, they dissected the traditional math curriculum from all sides. Some, bowing to the recommendations of the Cambridge Conference on School Mathematics, urged a radical acceleration of math curriculum so that calculus could become a high school subject. Others were more in the mainstream. Some aimed at high schools, others the primary grades. Although the public would eventually lump them all together as new math, the work of the ebullient Beberman and the commanding Begle, both of whom appeared in almost every popular periodical of the day, stood apart. Mutual friends and friendly competitors, they became the fathers of new math as well.

                                                                                    Thanks to the diligence of the Dominican nuns in my hometown of Bakersfield, California, I was blissfully unaware of this grand intellectual commotion in junior high school. I was learning math the old way, just as I had learned the multiplication tables, long division, and fractions. Drill, drill, drill.

                                                                                    In some Catholic schools, though, “Illinois math” took root early and flourished. Each summer scores of nuns, most of whom had never had more than a high school geometry course, made a trip to Urbana to “get modern.” In all, several thousand of them went over a thirteen-year period those who couldn’t go could subscribe to one of twenty training films paid for by the U.S. Office of Education. What did these teachers learn? Robert Kanske, a former summer-institute teacher and now the senior program officer for the National Academy of Sciences’ Mathematical Sciences Education Board, remembers that it was supposed to be the Beberman technique. “But in four or six weeks you couldn’t teach Max’s instructional genius. You spent all your time on the math itself.”

                                                                                    While remedial training was undoubtedly useful for the teachers, it denied them initiation in the greater mysteries of discovery learning and nonverbal awareness—Beberman’s twin pillars of pedagogy. Both stemmed from his faith in the mental agility of children to discover an answer. To illustrate, Beberman believed students could “discover” that the order in which numbers were multiplied did not affect the sum, if they could grasp such equations as:

                                                                                    He believed students could “discover” that multiplying parts of an equation together yielded the same result as adding them separately, if they could perform such computations as:

                                                                                    Example Story Problems

                                                                                    In this section, we’ll give you several examples of story (word) problems, starting
                                                                                    with simple problems and working up towards more complex problems.

                                                                                    Nick had 8 toy trucks in his toy box. His friend Nathan brought over 3 more toy
                                                                                    trucks. How many toy trucks did the boys have altogether?

                                                                                    What is the key word in this problem?

                                                                                    If you look back at the list of key words at the top of the page, you’ll find that
                                                                                    altogether listed as a key word.

                                                                                    Altogether is our key word. Now, what operation will we have to perform to get the
                                                                                    answer to this problem?

                                                                                    We know we’ll have to do addition, because altogether is a key word that means adding.

                                                                                    Now, what will our problem look like?

                                                                                    We know we’ll be adding together 8 + 3, because those were the two numbers mentioned
                                                                                    in the problem.

                                                                                    Therefore, our answer is 11 toy trucks altogether.

                                                                                    John had 15 books on his bookshelf. John’s dog, Buster, came in and slobbered all
                                                                                    over four of them. How many books did John have left that were not slobbered on?

                                                                                    What is the key word in this problem?

                                                                                    If you look back at the list of key words at the top of the page, you’ll find that
                                                                                    left is listed as a key word.

                                                                                    Left is our key word. Now, what operation will we have to perform to get the answer
                                                                                    to this problem?

                                                                                    We know we’ll have to do subtraction, because left is a key word that means subtract.

                                                                                    This problem is a subtraction problem. Now, let’s get it set up. How will this problem

                                                                                    We know we’ll be subtracting 15 – 4, because those were the two numbers mentioned
                                                                                    in the problem.

                                                                                    Now, perform the subtraction. What is 15 – 4?

                                                                                    Our final answer is 11 books.

                                                                                    Now, let’s try a couple harder problems.

                                                                                    Dan is getting ready to go to a concert. He wants to figure out how many people
                                                                                    will be there. He knows that there are 250 rows of seats, and each row has 40 seats
                                                                                    in it. How many seats are there in the concert hall in all?

                                                                                    What is the key word in this problem?

                                                                                    If you look back at the list of key words at the top of the page, you’ll find that
                                                                                    in all is listed as a key word.

                                                                                    In all is our key word. Now, what operation will we have to perform to get the answer
                                                                                    to this problem?

                                                                                    We choose multiplication because we see the keyword in all, but also because it
                                                                                    makes sense. Essentially this may be seen as an addition problem, which is why the
                                                                                    keyword is also in the addition section, but since the adding of the rows would
                                                                                    all be the same, we can multiply to make the process faster.

                                                                                    This problem is a multiplication problem. Now, let’s get it set up. How will this
                                                                                    problem look?

                                                                                    We would use 250 x 40 because we decided that this is a multiplication problem.
                                                                                    Since we want to figure out a total number of seats in the hall, we’re going to
                                                                                    multiply the two given numbers together, as if we were calculating area.

                                                                                    Now, perform the multiplication. What is 250 x 40?

                                                                                    Our final answer is that there are 10,000 seats at the concert Dan is attending.

                                                                                    Let’s look at one more example. Three friends go out to dinner. Near the end, they
                                                                                    get the bill and they owe the restaurant $27.89. They want to split the bill evenly
                                                                                    between the three of them. How much will each person pay?

                                                                                    What is the key word in this problem?

                                                                                    If you look back at the list of key words at the top of the page, you’ll find that
                                                                                    split evenly is listed as a key word.

                                                                                    Evenly is our key word. Now, what operation will we have to perform to get the answer
                                                                                    to this problem?

                                                                                    We choose division because we see the keywords “split” and “evenly.” Also, division
                                                                                    makes sense because they want to divide the bill between three people. Because they’re
                                                                                    splitting it up, we would choose division.

                                                                                    This problem is a division problem. Now, let’s get it set up. What would our equation

                                                                                    We choose $27.89 / 3 because we know we have to split up the amount of money, $27.89,
                                                                                    between the three friends, so we know we have to divide by three.

                                                                                    Now, perform the division. What is $27.89 / 3? (Round to the nearest cent)

                                                                                    When we divide, we get an answer of 9.2966666, with a repeating 6 at the end. We
                                                                                    want to round it to the nearest cent, which is the hundredths place after the decimal.
                                                                                    We see that that number is already a 9, and a 6 after it means round up. However,
                                                                                    we can’t make one place value a ten, so we increase the tenths digit by one, turning
                                                                                    the 2 into a 3. If this doesn’t make sense, please read
                                                                                    rounding numbers. Thus, your final answer is $9.30 after rounding.

                                                                                    Now, let’s go through some harder story (word) problems. All of the story problems
                                                                                    we’ve done have had only one step, and we’ve been able to easily decide if they
                                                                                    are addition, subtraction, multiplication, or division. However, some story problems
                                                                                    have more than one step, involving more than one key word and/or operation. We’ll
                                                                                    show you a few of these now.

                                                                                    Carly is making a dress. She needs 1 yard of yellow fabric, 1.5 yards of purple
                                                                                    fabric, and .5 yards of green fabric. Yellow fabric costs $5.95 per yard, purple
                                                                                    fabric costs $3.95 per yard, and green fabric is on sale for $7.00 per yard. How
                                                                                    much will she spend in all if she buys just enough fabric to make her dress? (Ignore
                                                                                    tax in your calculations). Click Next Step for the first part of the solution.

                                                                                    First, we have to figure out how much Carly is spending on each amount of fabric
                                                                                    then, we can use the key word “in all” which tells us that we need to add the amounts
                                                                                    together for a final total. In order to figure out how much each piece of fabric
                                                                                    costs, we need to multiply the price by the amount she needs to get a total.

                                                                                    First, let’s calculate the yellow fabric cost. She needs one yard, and it costs
                                                                                    $5.95 per yard, so she’ll be spending $5.95.

                                                                                    Now, let’s calculate the purple fabric cost. She needs 1.5 yards, and it costs $3.95
                                                                                    per yard therefore we have to multiply 1.5 times $3.95, which comes out to be $5.93
                                                                                    (we round to the nearest cent).

                                                                                    Finally, let’s calculate the green fabric cost. The green fabric is on sale for
                                                                                    $7.00 per yard, and she needs .5 yards of it, so we multiply $7.00 times .5 and
                                                                                    get $3.50.

                                                                                    Now, we have three money amounts (one for each color fabric) that we can now add
                                                                                    together to get a total amount that Carly will spend. We know that we have to add
                                                                                    these amounts together, like this:

                                                                                    Thus, our final answer is that Carly will spend $15.38 on fabric for her dress.

                                                                                    Now, we’ll give you one to practice on. John is planning to carpet three rooms in
                                                                                    his house. One room is 15 by 12 feet, one room is 17 by 14 feet, and the last room
                                                                                    is 10 by 12 feet. John has 130 square feet of carpeting already. How much more carpeting
                                                                                    does he need in order to carpet all three rooms?

                                                                                    First, you have to figure out how many square feet he has to carpet overall. That
                                                                                    means we need to figure out the area of each room, and add those together. We multiply
                                                                                    the dimensions together as follows:
                                                                                    15 x 12 = 180 ft 2
                                                                                    17 x 14 = 238 ft 2

                                                                                    Now, we have the area of each floor he has to carpet, so we can add these all together
                                                                                    to find out the total amount of carpeting he needs.

                                                                                    180 + 238 + 120 = 538 ft 2 . This is the total amount John will need. However,
                                                                                    the problem said that he already has 130 ft 2 of carpet, so we need to
                                                                                    figure out how much more he needs. Therefore, we need to subtract 130 from 538,
                                                                                    and we get 408 ft 2 leftover. This is how much more carpeting John will
                                                                                    need to finish off his three rooms.

                                                                                    The Smith family is going to take a vacation to Florida. They live in Illinois,
                                                                                    and have figured out that the trip is 1,150 miles from their house to the hotel
                                                                                    in Florida. They get 28 miles per gallon of gas, and plan on travelling at an average
                                                                                    rate of 60 miles per hour. Gas costs about $2.89 per gallon.

                                                                                    a) How long will it take them to get to Florida? (in hours)

                                                                                    For this part, you divide the total miles (1,150) by the speed they’re travelling
                                                                                    (60 mph) and you would get 19.16666 (repeating). You would round the answer to 19.2

                                                                                    b) How much money should they leave for gasoline (going one way)?

                                                                                    First, you would divide the total number of miles (1,150) by the amount of miles
                                                                                    they get per gallon of gas (28) this gives you 41 gallons—the total amount needed
                                                                                    for the trip. Then, you would multiply the number of gallons (41) by the cost per
                                                                                    gallon of gas ($2.89) and round to the nearest cent, which gives you $118.49. This
                                                                                    is the amount they should save for gas going one way.

                                                                                    2.2: New Page - Mathematics

                                                                                    Operating Systems Principles
                                                                                    Learning Objectives

                                                                                    The reading, quizzes, lectures, exams, and labs for this course are all designed around a set of learning objectives. These learning objectives can be divided into a few basic categories:

                                                                                    • Concept . You should be able to:
                                                                                      • define them and discuss their implications.
                                                                                      • discuss how they relate to one-another.
                                                                                      • give examples, recognize instances.
                                                                                      • Issue . You should be able to:
                                                                                        • discuss the considerations to be weighed.
                                                                                        • recognize situations where they do and do not apply.
                                                                                        • interpret them in the context of a particular problem.
                                                                                        • make and justify decisions based on them.
                                                                                        • Approach (mechanism, algorithm, or architecture) . You should be able to:
                                                                                          • describe it, and what problem it addresses.
                                                                                          • descibe its elements or steps.
                                                                                          • explain its applicability and advantages relative to alternatives.
                                                                                          • recognize and explain its limitations.
                                                                                          • explain why it it is or is not appropriate for a particular problem.
                                                                                          • describe how it could be applied to a particular problem.
                                                                                          • Skill (with an API or operation) . You should be able to:
                                                                                            • explain how and why it should be used.
                                                                                            • independenly and correctly use it in practical problems.
                                                                                            • review, describe, and critique code that uses it.
                                                                                            • diagnose and correct common problems that occur when using it.

                                                                                            Introduction to Operating Systems

                                                                                            why study OS
                                                                                            what is OS
                                                                                            mechanism/policy separation
                                                                                            interface vs implementation
                                                                                            interface contracts
                                                                                            modularity/information hiding
                                                                                            powerful abstractions
                                                                                            appropriate abstraction
                                                                                            indirection/deferred binding
                                                                                            opaque encapsulation
                                                                                            thinking data structures
                                                                                            OS types/history

                                                                                            why functionality implmented in OS
                                                                                            OS goals

                                                                                            layered/hierarchical structure
                                                                                            dynamic equilibrium

                                                                                            Resources, Services, and Interfaces

                                                                                            basic OS services
                                                                                            higher level OS services
                                                                                            layers of services
                                                                                            private resource
                                                                                            APIs and ABIs
                                                                                            service protocols
                                                                                            upwards compatability
                                                                                            objects and operations
                                                                                            abstracted resources
                                                                                            serially reusable resources
                                                                                            partitioned resource
                                                                                            sharable resource

                                                                                            subroutines vs system calls
                                                                                            versioned interfaces

                                                                                            Processes, Execution, and State

                                                                                            programs and processes
                                                                                            process address space
                                                                                            stacks/linkage conventions
                                                                                            process state (elements of)
                                                                                            process resources
                                                                                            user-mode, supervisor-mode
                                                                                            traps (exceptions)
                                                                                            interrupts (async events)
                                                                                            traps ( syscalls )
                                                                                            limited direct execution

                                                                                            fork vs exec
                                                                                            syscalls vs procedure calls

                                                                                            process implementation
                                                                                            context save/restore
                                                                                            static libraries
                                                                                            shared libraries
                                                                                            dynamically loadable libraries

                                                                                            process operations
                                                                                            signals (exceptions)

                                                                                            Scheduling Algorithms, Mechanisms and Merformance

                                                                                            execution state model
                                                                                            metrics: completion time
                                                                                            metrics: throughput
                                                                                            metrics: response time
                                                                                            service level agreement
                                                                                            process state (model)
                                                                                            non-preemptive scheduling
                                                                                            time sharing
                                                                                            time slice
                                                                                            performance under load

                                                                                            scheduling goals
                                                                                            cost of context switch
                                                                                            optimal time slice
                                                                                            end-to-end performance
                                                                                            graceful degradation

                                                                                            First In First Out
                                                                                            Shortest Job First
                                                                                            priority scheduling
                                                                                            real time scheduling
                                                                                            preemptive scheduling
                                                                                            multi- levelfeedback queues
                                                                                            dynamic equilibrium
                                                                                            mechanism/policy separation

                                                                                            LAB 1A - Processes, terminal I/O and pipes

                                                                                            Memory Management, Allocation and Relocation

                                                                                            physical address space
                                                                                            virtual address space
                                                                                            text segment
                                                                                            data segment
                                                                                            stack segment
                                                                                            shared library segments
                                                                                            protected memory sharing

                                                                                            memory management goals
                                                                                            internal fragmentation
                                                                                            external fragmentation
                                                                                            pool rebalancing
                                                                                            common errors

                                                                                            address space layout
                                                                                            stack vs heap allocation
                                                                                            variable size allocation
                                                                                            malloc vs sbrk
                                                                                            memory allocation life cycle
                                                                                            free list design
                                                                                            first fit
                                                                                            best fit
                                                                                            worst fit
                                                                                            next fit
                                                                                            pool/slab allocation
                                                                                            diagnostic free lists
                                                                                            garbage collection

                                                                                            Virtual Memory and Paging

                                                                                            swap space
                                                                                            paged address space
                                                                                            page table entry
                                                                                            page replacement
                                                                                            Temporal Locality
                                                                                            Spatial Locality
                                                                                            Least Recently Used
                                                                                            working set
                                                                                            clean/dirty pages

                                                                                            relocation problem
                                                                                            paging and segmentation

                                                                                            base/limit relocation
                                                                                            segment addressing
                                                                                            memory compaction
                                                                                            paging MMU
                                                                                            translation look-aside buffer
                                                                                            demand paging
                                                                                            page fault process
                                                                                            Belady's Optimal Algorigthm
                                                                                            LRU clock algorithm
                                                                                            working set clock algorithm
                                                                                            page stealing
                                                                                            copy on write
                                                                                            proactive page laundering

                                                                                            LAB 1B - Compressed Communication

                                                                                            Threads, Races, and Critical Sections

                                                                                            thread state
                                                                                            IPC purpose
                                                                                            non-deterministic execution
                                                                                            race condition
                                                                                            indeterminate results
                                                                                            critical section

                                                                                            thread motivations
                                                                                            IPC goals
                                                                                            flow control

                                                                                            thread stacks
                                                                                            user mode implementations
                                                                                            stream IPC
                                                                                            message IPC
                                                                                            shared memory IPC
                                                                                            synchronous vs asynchronous IPC
                                                                                            mutual exclusion

                                                                                            Mutual Exclusion and Asynchronous Completions

                                                                                            parallelism scenarios
                                                                                            locking & event completion

                                                                                            correct order
                                                                                            locking goals
                                                                                            correct mutual exclusion
                                                                                            who to wake up
                                                                                            sleep/wakeup races
                                                                                            spurrious wakeup

                                                                                            synchronous requests
                                                                                            asynchronous completion
                                                                                            types of locks
                                                                                            interrupt disables
                                                                                            spin locks
                                                                                            asynchronous events
                                                                                            waiting lists

                                                                                            Higher Level Synchronization and Communication

                                                                                            bounded buffer problem
                                                                                            producer/consumer problem
                                                                                            costs of contention

                                                                                            binary semaphores
                                                                                            using semaphores
                                                                                            condition variables
                                                                                            using condition variables
                                                                                            semaphore implementation
                                                                                            CV implementation
                                                                                            contention reduction
                                                                                            file level locking
                                                                                            advisory locking

                                                                                            LAB 2A - Races and Serialization

                                                                                            thread API
                                                                                            recognize critical sections
                                                                                            pthread_mutex operations
                                                                                            pthread_cond operations
                                                                                            protect critical section

                                                                                            Deadlocks, Prevention, and Avoidance

                                                                                            Dining Philosophers problem
                                                                                            necessary conditions

                                                                                            reasonable spinning
                                                                                            common synchronization errors

                                                                                            compare and swap
                                                                                            java synchronization
                                                                                            where to serialize
                                                                                            detection and recovery

                                                                                            goals and challenges
                                                                                            performance principles
                                                                                            typical causes of performance problems
                                                                                            common measurement errors

                                                                                            load generation
                                                                                            internal instrumentation
                                                                                            end-to-end measurement

                                                                                            load characterization
                                                                                            analysis techniques
                                                                                            presentation techniques

                                                                                            malloc APIs

                                                                                            I/O bus
                                                                                            Device Controller
                                                                                            disk geometry

                                                                                            importance of disks
                                                                                            disk performance
                                                                                            random vs sequential I/O
                                                                                            transfer size

                                                                                            polled I/O
                                                                                            completion interrupts
                                                                                            initiating I/O operations
                                                                                            write-back cache
                                                                                            write-through cache
                                                                                            chain scheduling
                                                                                            buffered I/O
                                                                                            intelligent I/O devices
                                                                                            read/write striping
                                                                                            write mirroring
                                                                                            parity/erasure coding
                                                                                            asynchronous parallel I/O
                                                                                            polling for asynchronous completions
                                                                                            non-blocking I/O
                                                                                            asynchronous event notifications

                                                                                            File Semantics and Representation

                                                                                            file semantics
                                                                                            file types and attributes
                                                                                            data base semantics
                                                                                            object semantics
                                                                                            key-value semantics
                                                                                            file namespace
                                                                                            file name conventions
                                                                                            copies vs links
                                                                                            symbolic vs hard links
                                                                                            storage volume
                                                                                            disk partitioning

                                                                                            read after write consistancy
                                                                                            goals of file representation
                                                                                            goals of free space representation
                                                                                            goals of namespace representation

                                                                                            BSD file system organization
                                                                                            FAT file system organization
                                                                                            indexed data extents (I-nodes)
                                                                                            linked data extents (FAT)
                                                                                            linked free lists (Unix V5)
                                                                                            contiguous allocation
                                                                                            bit-map free lists (BSD)
                                                                                            FAT free list
                                                                                            BSD directory entries
                                                                                            FAT file descriptors
                                                                                            the mount operation
                                                                                            file access layers of abstratction
                                                                                            dynamically loadable file systems
                                                                                            user mode file systems

                                                                                            LAB 4B - Embedded System Sensors

                                                                                            File Systems Performance and Robustness

                                                                                            disks and file system design
                                                                                            block size & internal fragmentation
                                                                                            causes of file system damage

                                                                                            read cache
                                                                                            general/special purpose caches
                                                                                            how to beat LRU
                                                                                            delayed writes
                                                                                            write-back cache
                                                                                            detection and repair
                                                                                            journaling and recovery
                                                                                            copy-on-write file systems
                                                                                            log structured file systems
                                                                                            checksums and scrubbing

                                                                                            Security, Protection, Authentication, Authorization

                                                                                            controlled sharing
                                                                                            object, agent, principal
                                                                                            mediated access
                                                                                            trusted computing platform
                                                                                            principles of secure design
                                                                                            access control lists
                                                                                            Linux file protection (ACLS)
                                                                                            Linux file descriptors ( capabilties )
                                                                                            principle of least privilege
                                                                                            Role Based Access Control
                                                                                            Trojan horses
                                                                                            cryptographic hash

                                                                                            key security

                                                                                            cryptographic hashes
                                                                                            challenge/response authentication
                                                                                            Linux principals
                                                                                            Linux authentication
                                                                                            Linux setuid
                                                                                            at rest encryption
                                                                                            symmetric encryption
                                                                                            cryptographic privacy

                                                                                            LAB 3A - File System Interpretation

                                                                                            Distributed systems: Goals, Challenges and approaches

                                                                                            RPC stub
                                                                                            RPC skeleton
                                                                                            RESTful interface principles
                                                                                            Marshal/ unmarshal
                                                                                            unforgeable capabilities

                                                                                            Deutsch's 7 Falacies
                                                                                            RPC interoperability

                                                                                            RPC tool chain
                                                                                            asymmetric encryption
                                                                                            public key encryption
                                                                                            digital signatures
                                                                                            public key certificates

                                                                                            Remote Data, Synchronization, Security

                                                                                            local vs cloud
                                                                                            client/server model
                                                                                            distributed file systems

                                                                                            man in the middle attacks
                                                                                            remote file system goals

                                                                                            reliable communication
                                                                                            distributed locks
                                                                                            distributed transactions
                                                                                            distributed consensus
                                                                                            two phase commit
                                                                                            three phase commit
                                                                                            remote file transfer
                                                                                            remote file access
                                                                                            peer-to-peer security
                                                                                            work tickets
                                                                                            distributed authentication/authorization

                                                                                            LAB 3B - File System Analysis

                                                                                            Remote Data Performance and Robustness

                                                                                            stateless server protocols
                                                                                            idempotent operations
                                                                                            ACID Consistency
                                                                                            Read-After-Write Consistency
                                                                                            Close-Open Consistency
                                                                                            classes of distributed systems

                                                                                            graceful degradation
                                                                                            write latency
                                                                                            cache consistency
                                                                                            APIs -> protocols

                                                                                            replication and recovery
                                                                                            distributed data striping
                                                                                            direct client-server communication
                                                                                            scalable distributed systems
                                                                                            client-side caching

                                                                                            MP and Distributed Systems

                                                                                            Single System Image
                                                                                            Non-Uniform Memory Architecture
                                                                                            loosely coupled
                                                                                            tightly copuled
                                                                                            unit of deployment
                                                                                            unit of failure/replacement
                                                                                            cluster membership
                                                                                            Geographic Disaster Recovery
                                                                                            availability zones
                                                                                            Software Defined Networking
                                                                                            Software Defined Storage
                                                                                            Single Point of Failure
                                                                                            Eventual Consistency
                                                                                            BASE semantics

                                                                                            Symmetric Multi-Processor
                                                                                            Horizontally Scaled
                                                                                            heart beating
                                                                                            partitioned responsibility
                                                                                            WAN-scale consistency models

                                                                                            Watch the video: Q 3 i to iv, Ex - Fractions and Decimals - Chapter 2 - Maths Class 7th - NCERT (November 2021).