coursework

0 thoughts on “coursework”

  1. Pingback: G&T website shout out: IB Maths, ToK, IGCSE and IB Resources | Great Maths Teaching Ideas

  7. I figured it out but it says its wrong, i tried again with my parents same answer and still wrong

  11. Well done!

    Please note that I have to update the leaderboard manually so it won’t add your name immediately when completed.

    How about trying the Level 2 code challenge?

  15. yay! i finally broke it !! thank you making this kind of website it was really challenging and also fun!! :)😃

  26. There is an error in the second Binomial example, the second term should be 27 x^2 y.

  27. Fun fact: CRT is a good way to precisely count large numbers of objects (originally in China, soldiers). For instance, tell your soldiers to get into groups of 107, then count the remainders, then get into groups of 17, and count the remainders; now, you have your answer mod 17×107, and if you need to, just use more primes until you get the desired precision.

    The Chinese Remainder Theorem is not only a reasonable topic for an introduction to elementary number theory, but also a powerful technique in more advanced mathematics; in fact, it’s general form is true for all commutative rings (if I and J are ideals of R where I + J = R, then R/I × R/J ≅ R/(I∩J) ).

    Darn all this is kinda tempting me to change my IA topic now…

  28. Most of these are excellent and I love the majority (CRT is the Way, the Truth, and the Light!) but these range from quite easy (modular arithmetic, at least at the basic level) to impossibly hard (GRH? Goldbach?).

    I think the easier ones are more suitable; I find it hard to imagine a good paper on GRH or Goldbach without a background in complex analysis or analytic number theory, respectively: these topics are just too hard for high schoolers (and too hard for everyone else probably also). On the other hand, I’d quite enjoy reading (or writing?) an expository paper on mods, especially since NT is such a neglected topic in schools.

    Also, what is with “Does finger length predict mathematical ability?”? Some of the Stats topics seem quite bizarre and wholly unmathematical in nature.

  30. dear sir I’m a fan of mathematics..I would like to present my views on the 4th dimension…The 4th dimension is time.
    x=c is equation of point eg x=1,2,3 etc.
    ax+by=c is equation of line.
    ax+by+cz=d eqation of plane.
    ax+by+cz+dt=e is an equation of space.

    we can place all the value of x y z here in this equation of space. Any value is possible..then what is ‘t’ here? t means time ..for a particular value of time t here can be a a equation of plane..that means a moving body in 3d if considered a point of a plane is at that particular time …equation of plane state object at rest..not moving..but if t is considered as 4th dimension then we are dealing that integration of plane and all planes at different time and if we consider a body or a point then that moving body as a part of moving plane….

    interesting question can be asked here how t having unit second can be added to x y z those having length…constant before t is considered as velocity a constant value for some moving object similar to constant designated before x y z..

  34. hello, bidur tiwari, I think you are confusing spacial dimensions with space time, which is also a concept which involves the 4th dimension, however the meaning of the 4th dimension in this aspect is completely different to what this page is talking about.

  40. thank you so much for the videos. It’s interesting to mention that I learn more on this website than in school!

  42. The website allows you to download full mathematics lessons for KS3 and KS4. Worksheets, games and fully worked solutions are integrated with several tasks and extension material. I recommend the circle theorems lesson which is housed in the KS4 folder.

  46. I think position is a fairly poor measure of performance. Perhaps points, goals scored etc. would be better.

  47. The data look nonlinear to me. It would be better to fit a curve, or else to transform the financial data first. This would both improve the correlation and remove the tendency for clubs from 6th-15th to be below the line (better than expected) and 15th downwards to be above the line (worse than expected).

  49. I’m interested in looking into and potentially writing about the modelling of cancer growth mentioned towards the end of the post, do you know of any good sources of information for this? is there anywhere that you would recommend me looking to find out more about it?

  50. hiya

    If you read the wiki page on Gompertz functions [http://en.wikipedia.org/wiki/Gompertz_function] this might be a good starting point. This function is a modified exponential model so that you have rapid initial growth (as in a normal exponential function), but then a growth slowdown with time. You could use this equation to model various initial conditions.

  51. You said that this was difficult to write an equation for, but even so do you think it would be a high- scoring IA? Im very interested in this topic.

  52. yes – you could definitely do a good IA on this topic. It could be similar to some of the older IA investigations the IB used to set (such as population growth in China). For these ones you needed to try and fit a function to data points to see which function most closely described the trend.

  55. My name is Eden and this is so good resource. Much learn of math over up in here!! $$$ E.B.D.

  65. Hi,
    I was wondering how you find the days it takes for infected person to die, and how you found the contact rate.
    Thank you

  66. Follow the linked article in the piece above – it’s a scientific paper, which talks about different models for ebola. They estimate R0 as 1.7 and 8.6 and that it takes 4-10 days for death.

  76. Hi, i would just like to ask whether or not this would be an appropriate topic for a Math SL exploration?
    If yes, how would i approach it?

  79. I was thinking of using related rates as my ia topic but Im not sure how to apply related rates into physics or medicine.

  83. There is an error in the first expression of Zeta(s), it should be (-1)^(n+1) not (-1)^n.

  84. I was thinking of modelling traffic flow using differential equations, are there anything specific resources that you would recommend to help me understand this better?

  87. Hi! Could you suggest which part of the syllabus can I apply on the investigation of flexagons besides the idea of looking for hidden faces and patterns?

  89. I need some good quality feedback on my exploration paper! Please contact me if you can help me out!

  91. ECF DXS GHXM NOKJPU ECF FXONNKR L YOUPQKFP FODSHX. DPRVZP XYSO WU HSLTY EKGH HDY WXSUGDLHZP. VU MZX YVZXRR MH BSCB VFZXJ. MZX ICFOJ PP D YSNUKH LJKBE. PGMMH ECF VNCFOJ HCB ECFU YYTORG ZQ ZVP EKOWH IWAKKFD. QUPZGE VLV IFLFQSO WNSX BKH, MXZ WQ BUI OR, ECF POUSW JWDFUJPU G HCHGGFUK KZUZV XLRZTRTG ZI JCWOGFD.

  92. You don’t seem to get many comments – so here is one from me.

    I think your blog is absolutely fantastic. I teach IB Maths in Warsaw and this blog is jam packed full of things that are interesting for me and my students.

    great stuff.
    Only thing is it does scare me whenever I try to add anthing to my measly set of pages.

  93. I love your blog!! It’s extremely interesting and engaging, especially because our IB Math teacher doesn’t..teach the curriculum, let alone the applications outside of it. Thank you! 🙂

  95. It was arbitrarily chosen – along with the other constants. But chosen so that the function is always positive. If x is between 0 and 1 and t is between 0 and 5 then the function needs B at least 21 to prevent it being negative for some values.

  97. This graphical solution to noughts and crosses is excellent. Have you considered removing the redundant isometric reflection and rotation solutions? Most users (that have made it this far in search of the answer) will easily be able to adapt the theory and follow the solution charts within the respective sub-grid as play progresses.

  100. what can a student investigate here when there is already an answer? or that is not the answer?

  101. Maths Explorations are not expected to discover new maths – that would be phd level! You can explore a topic, modify the starting conditions and investigate what happens.

  102. Dear Sir/Madam,

    Under the topic “Friendly numbers” it appears that there is an error in the sentence “Therefore 28 and 6 are happy numbers because they share a common relationship.”

    It should instead read: Therefore 28 and 6 are friendly numbers because they share a common relationship.

    Yours sincerely

    Matteo Coné

  104. Dear Sir/Madam,

    Thanks for your prompt response, much appreciated.

    Is it at all possible to add to your website a paragraph, give examples and explain what “amicable” numbers are?

    Your sincerely,

    Matteo Coné
    PS
    References:
    https://books.google.co.za/books?id=JrslMKTgSZwC&pg=PA90&lpg=PA90&dq=the+pythagoreans+of+ancient+greece+were+fascinated+by+amicable+numbers&source=bl&ots=Vt8F5PONNE&sig=WB4etINQmPHOp4cpHrA2bOEAirI&hl=en&sa=X&ei=wmUFVfTWA8raU8iSgtAI&redir_esc=y#v=onepage&q&f=false

  108. Are there any articles that back up the assertion that more wages lead to a higher league position?

  109. Where does the math come in to play? what does the table consist of? Performance against sacking. I would love to see your feedback and input on this

  111. I am confused on 3 equations, how do you figure out the value of I and R?

  113. THIS IS AWESOME!!! My brain works now 😀 I think this has saved my IB. Thank you!!!!

  114. Pingback: Who is Adam Clark Arts and Mathematics Museum Project - Who is Adam Clark

  118. There are no HL or SL topics as such – the only difference is that for HL the marking criteria for the use of maths expects a higher standard of maths. A IA that got 18 on a SL grading might only get 14 on an HL grading.

  121. In the first equation of the SIR model, it states the rate of change to becoming susceptible it equal to negative beta times I times S over the total population. What is I? and why is the beta negative?
    Thanks.

  123. Sir I want some solutions of problems of Cambridge math hl calculus by Fannon

  126. When you say that Graham’s number is so astronomically large that to input it into the human brain exceeds the amount of entropy(or bits of information) possible in a human brain, are you referring to a bit as in planck length squared? But could you also argue that before you reached that point(say you had infinite time) your brain would have filled up its total storage capacity due to neuron connections?

  130. anybody has any idea how are we supposed to actually get the a and b in the batman logo

  131. Very good as ever.
    But Konigsburg was Prussian not Russian for most of the 1700s (and all the time before Euler proved the walk was impossible). It was Russian later for about 7 years and again from 1945.
    This is a touchy point for my IB students in Warsaw 🙂

  136. the password system doesn’t work very well. I’ve tried to type in the right password says its wrong and then sometimes it skips the password so i have no password to give to my teacher

  139. The instructions at the start say that if you are sure your answer is correct and the password doesn’t work then you should refresh the page. This is a problem with WordPress. The time is given as Thai time – not all places in the world have the same time as the UK!

  140. In the first paragraph, the value of e is stated incorrectly
    (it should be …1828459045…)

  145. Thanks for the simple tutorial. The chinese postman problem gets difficuilt to solvr if there are more than 2 odd nodes.. It wud really be nice of u, if u cud put an example of that case too..

  151. Thanks for the comments – I have added the video about the moon experiment.
    Yes, I was quite surprised that it was so simple to show! I might try and look at some other basic Newtonian laws and what we can derive from them. Next up, parachute jumps!

  153. Would this be an appropriate IA for math sl? Because it is listed under geometry in the examples list, so would it get a good score or should I choose something else? Thanks!

  154. It could be – you need to think about how you can personalise the topic. Can you think of a question to investigate to do with extra dimensions? That would then allow you to have good reflection marks.

  155. As a teacher, I just want to thank you for taking the time to put together such a useful website.

  158. i’m not sure if you meant 9 raised to the 3rd power or 3 raised to the 9th power but 9 raised to the 3rd power is 729, whilst 3^9 is 19,683

  161. Thank you! this was really helpful for my math HL paper 3 practice!!!!!

  164. Shouldn’t it be ‘-27x to the 2nd power times y’ in the last video = example 2?

  168. eg. σ(30) /30 = (1+2+3+5+6+10+15+30) / 6 = 12/5 )

    should read /30, not /6?

    Sorry to be a pedant, really like this post, nice explanation.

    Also, are emirp numbers a thing or did I make that up? (numbers that are prime forwards and backwords)

  171. Hello,

    I just want to say that I’m loving math and like to help people to learn it.

    For people struggling with math I created powerful derivative calculator (in addition to steps derivative can be evaluted at point) and integral calculator (with steps shown).
    Also, I looked for good free online graphing calculator, but all they lack customization, so I created another one.

    You can check them out here:
    Graphing calculator – http://www.emathhelp.net/calculators/calculus-1/online-graphing-calculator/
    Integral Calculator – http://www.emathhelp.net/calculators/calculus-2/integral-calculator/
    Derivative Calculator – http://www.emathhelp.net/calculators/calculus-1/derivative-calculator/

    And general page with all calculators is http://www.emathhelp.net/calculators/other/math-problem-solver/

    Might be worth adding to the site or recommending to students.

    Either way, have a great day!

    Best wishes,
    Paul

  172. Hello,

    I just want to say that I’m loving math and like to help people to learn it.

    I know that many people struggle with Algebra, so I created some calculators to help students.

    You can check them out here:
    Synthetic Division Calculator – http://www.emathhelp.net/calculators/algebra-1/synthetic-division-calculator/
    Partial Fraction Decomposition Calculator – http://www.emathhelp.net/calculators/algebra-2/partial-fraction-decomposition-calculator/
    Factoring Calculator – http://www.emathhelp.net/calculators/algebra-2/factoring-calculator/
    Polynomial Calculator – http://www.emathhelp.net/calculators/algebra-1/polynomial-calculator/
    Simplify Calculator – http://www.emathhelp.net/calculators/algebra-2/simplify-calculator/
    Equation Calculator – http://www.emathhelp.net/calculators/algebra-2/equation-solver-calculator/
    Quadratic Equation Calculator – http://www.emathhelp.net/calculators/algebra-1/quadratic-equation-calculator-solver/

    And general page with all calculators is http://www.emathhelp.net/calculators/other/math-problem-solver/

    Might be worth adding to the site or recommending to students.

    Either way, have a great day!

    Best wishes,
    Paul

  175. they are scaler factors so irrelevant to the general shape of the logo,

  179. Thanks friend, this was really helpful! I hope that you are well and continue to create such amazing examples, friend.

    Your Friend,
    Bradley 🙂

  180. How would you use this for an SL Maths IA? i.e. will a statistical investigation be enough to get a good SL grade?

  181. Yes you can definitely do a statistical investigation for SL. You just need to make sure you go into more of the maths than for Studies. Some ideas – confidence interval testing, doing an experiment then trying to fit a curve to your data to compare it to the normal distribution, looking for relationships between age and reactions etc. Lots of possibilities.

  183. looking forward to doing this with my MPY5 and DP students tomorrow 🙂 thanks for another great post

  184. Who wrote this article? I am writing a literature review for an MA and I would like to reference it. Thanks!

  185. You wouldn’t get a better result by curve-fitting then integrating. That would be *less* accurate. When you curve-fit, you’re approximating based on your points. An integral takes points on the curves and uses trapezoids or rectangles to approximate the area; the integral is the limit of those approximations.

    Therefore, you’re more accurate to just add up the exact areas of your trapezoids if you have finite discrete data instead of fully continuous data.

    The other method stated, to use more points, *would* give a more accurate result.

  188. it really good topics in math and i really like it and its new in math.

  189. I believe that for the complex roots of a cubic the slope of the tangent line is the square of of the imaginary part. So if the line were 3x+4, the complex roots would be 3+2i and 3-2i.

  190. RCZVD GQAWX GECIG UXIJG VTUAW XGENC ZVDGU GNITX OIGLK TYGVT GNOSV TQAUU BARYK VQGVT GNOSV TQAZK XKBAV LAZZC GNEUC ZIBBM OBUKB ITGSV

  191. It is very true post analysis of paper 3 calculus option(2016 May). I am confused if it is to evaluate the student or demoralize them and make them hate calculus.

  192. Excellent synopsis of a horrible paper. In the time allotted to teach the option, students cannot achieve the level of understanding to feel successful on this paper.

  193. Overall agree.

    1) 7 points on something like Lagrange error is asking for a lot of kids to miss marks.

    2) Given the small sample size of past questions in the option (especially topics that were not on the old curriculum), I don’t think it’s worth complaining that there wasn’t guidance in specimen papers or in textbooks – it’s an unfortunate reality that will arise anytime a curriculum is overhauled. Using most standard calculus texts as a guide will find plenty of material on FTC, as would using exam materials from other courses (AP calc has a lot of FTC for example).

    3) probably over represented here, but that’s going to happen when the exam is only 60 marks.

    4) I don’t see this as a “new type of differential equation” – you ended up separating variables after the substitution, no? I agree however that the substitution step was unnecessary as separating variables would have sufficed from the beginning.

    5) This one was indeed a mess.

  194. hiya Mike – you caught me mid-edit for this post. I agree with some of your points.

    In a normal exam again I would agree that question (2) is just one of those things you accept – but when you have a number of challenging questions then this becomes more of an issue. Also I really don’t like the markscheme on this – it’s either going to be 7 marks or nothing.

    I think I overstated question (4) – there is enough scaffolding and a standard differential equation to solve in the meantime. Actually first time I looked at this I thought “integrating factor” – because it’s written in the right form for that. Wonder how many will go that route rather than separate variables?

    What would take to make this a decent paper:

    reduce the Fundamental Theorem Question to 4 marks – and include it in a question using an indefinite integral, or a Riemann sum (easy case)

    Take out the first part of question (4) and replace it with a numerical or graphical method for differential equations

    Take out the whole of question (5) and replace it with a standard 10 mark question testing the radius of convergence – allowing the students to use the ratio test, plus a couple of the other tests at the boundaries….

    and to make a much more accessible paper – perhaps replace Lagrange with L’Hopital or a continuous function question….

  196. It was tough and missed out a lot of content, I agree. It was also quite repetitive with the same idea being tested more than once (eg Q2b & 4a, or two existence theorems rather than just one in Q1(c) & Q3), which makes the huge amount of syllabus not examined even harder to take. I also agree with a lot of general stuff written here about the apparent lack of moderation at the question writing stage. However, it is not the hardest IB paper ever – that was the old calculus option two lives ago, in around 2004, when the questions set were so diabolical all the marks had to be scaled up by a ridiculous amount. You would have thought that the IBO learnt their lesson then!

  198. Perhaps it was the same examiner, let out of his cage to write a paper every 12 years! In 2004, the options were examined as one long question in the second half of Paper 2, and the equivalent option was called Analysis & Approximation. I have taken another look at the qiuestion and, although perhaps the difficulty is about the same as this year, the shock of having so much that was barely on the syllabus and so many topics not examined, was very similar, and the fall-out was huge, leading to a separate paper for each option so that grade boundaries could vary.
    We teachers, who so love the HL syllabus yet get so frustrated by the standard and consistency of questions set in the papers each year, do need to question how the IBO moderates the questions it sets. There was a very peculiar trigonometry question on TZ2 Paper 1 this year with a false “hence” which would not have passed any kind of road testing. This kind of error is so avoidable! And, as you say, writing questions which are perceived as so tough is in the end completely self-defeating.

  202. Is it “velocity in the y-direction” or “y-component of the velocity”? Just semantics, and I know we all say both, but is the first really valid? The velocity is a single vector: There are not two velocity vectors, each pointing in orthogonal directions.

  204. Great post, but why do zombies move in straight lines? Wouldn’t they stagger back and forth like some drunken college freshmen.

    Anyways, good post. I’m gonna share it on my blog

  205. Hi I would really appreciate it if you could help me with getting access to the motion capture software that you are using in this ME. Thanks a lot!

  206. This is very interesting. But you haven’t explained why the “K” does not need turning over. Presumably there is an additional starting condition that each card has a number on one side and a letter on the other. The video has a very similar condition stated.

  207. The last one is inconsistent for me because… When does (1-1) transforms into 1?
    1 = 0 is not a well formulated paradox because the base of the 0 + 0 + 0 proof affirms than any negative/positive operation term has to be a negative/positive operation of a pair members count. This means:

    0 = 1-1
    0 = 1-1+1-1

    This italian dude is doing something weird that cannot be done even with algebraic factorization here:

    0 = (1-1) + 1… Why is there (+1) ? This is contradictory to his first affirmation which says: 0 + 0 + 0 = (1-1) + (1-1) + (1-1)

    Is inconsistent, not true.
    And it’s not only 1, it may be 2, 3, 4.56, 0.99999 where (0.9999 – 0.9999) = 0… That includes the 0 himself where 0 = (0-0) or 0 = (0+0)

    So as a summary: It doesn’t says that from nothing can appear something. It strictly affirms that from nothing there “appears” to be something but you discover that this “something” is really the “nothing”. Because 1 = 0 is not true, it’s something that really doesn’t exists cause 0 = 1 is fake and 0 = 1 is really 0 = 0. This is just an illusion, just as god. It’s just an illusion, imagination. If you use the religion fact this will be: “0 = 1 can be true if you believe it, but the truth is 0 = 0, and 1 is not something but nothing that only exists if you believe it”.

  209. Hi, what else could I add on to this for my math internal assessment to make it more complete?

  211. Admittedly, b)i) is quite hard for IB students, but ultimately it’s still within the syllabus as it’s just a u-substitution and a manipulation of the unit circle (trig rules).

    However, why not use the Ratio Test for b)ii)? It’s pretty straight forward with b)i) being given… easy marks… (students can even get FT!)

    Regards,

  218. Even if we have data, how sure can we be that the events would follow a regularity? An eruption depends on so many factors and we may not have resources to understand & estimate each one’s impact for the culmination of the event. Further, it is generally futile to go by probability when we are faced with binary situations where 1 or 0 may ruin us completely. Even in the financial world, many Nobel laureates applied probability theory to make well sounding financial models.. It didn’t have any meaning when firms with a 2% chance to fail precipitated into the 2%, making a mockery of those with those statistical tools and investors were hit very hard.

    What is the point in trying to predict an event that may or may not happen in 600,000 years?

  219. I started teaching hl ib math in 1988 , and this year 2016 was my last yearing teaching. I’ve taught this course and even i taught further math for some years, i can say the ib math hl p3 2016 is the most difficult paper i have ever seen. With my best regsrds

  221. Yeah! When, I was in first grade, (the first time), i was so smart in Math etc.,
    that, to my surprise, on the last day of school, that year, The Principal came into my classroom and personally asked me if I would mind being held back a year, so, I could help those kids, who Flunked? Yeah! Twice I did That? Talk
    About Gratitude? Shhhh! I didn’t get no Plaques, Awards, Certificates or nuttin, them Bastards!

  222. I like your writing on analytical continuation, just because I understand your approach (I struggle this for a while). my question is functional equation, how people developed functional equation ? by trying and error or some mathematic ways, Please direct me, thank you.
    -Mike

  229. Hello, since the writers for ibmathsresources.com seem very well-informed, I’d like to know what the IB has to say about the Paper 3 (since the subject reports aren’t available on the Internet for one reason or another). I don’t remember as significant an outcry as was provoked by the Physics Exam, even though the faults in the latter were much less serious. Are they apologetic, in other words?

  231. very good

  232. Hiya

    The subject reports are out somewhere as I have seen a copy! Not sure if they have been added to the OCC site yet though. The reports are quite disappointing – no acknowledgement of any faults with the paper, and really narrow grade boundaries – I think 7 marks jumped students 2 grades. I have emailed and got a response from the IB chief examiner with my concerns for this paper. From his response I am hopeful that the exams will be better this year. I will keep my fingers crossed.

  233. We will be going from Studies, SL and HL to just SL and HL – something has to give somewhere – either the new SL will have to be made significantly harder than the current Studies, or the new HL significantly easier than the current HL. Making the new SL significantly harder than the current Studies would preclude large numbers of students from the IB, so its almost certainly going to require HL being made easier. It’s all guesswork at the moment as everything is up for consultation – so we’ll have to see what is in place for 2019.

  235. I am proud of you students too! Keep up the great work! When I was in first grade, (the first time), i was so smart in math, that on the last day of school, that year, the Headmaster came into my classroom and personally asked me? (I can still remember his words, like it was yesterday), He said, Timmy!!!! Being that you are so damn smart and all, how would you like it, if I kept you back a year, so you can Help Out those kids who flunked?!!
    Yeah!!! Twice I did that! Talk about gratitude! Shaa! I did Not get no Plaques, Awards, Certificates or nothing!! That decision has effected me for the rest of my life, until I retired in 2011…That’s right! I have had a very rich employment history, in the last 50 years, holding down approximately 240 jobs! I am NOT proud of that number, but I can definitely say this, though, (with a straight face), “i am NOT A QUITTER”!!! NO WAY JOSE! NOPE, THEY ALWAYS HAD TO FIRE MY ASS FIRST!!! I WOULD NEVER QUIT!!!
    Three of my job interviews, the boss saw I had circled the number 14 on my Application, for the years of Education?? And, I was asked, “Oh!? You have an Associate Degree?! Do You”? My reply was: No, I just had to Repeat the 1st & 4th grades, that’s all! He laughed and said, “I like your attitude and anyone with that kind of an attitude, I want them working for me”! Then, he asked, “I can understand someone failing 4th grade, because, I did myself. But how anyone could fail 1st grade, is beyond me”?!! I said, “Hay man! It wasn’t easy”!!!!!! And it wasn’t!!

  240. It help so much when i see math topic for studie and i use experiment to expand ideas among mathmatical world for further education. I was feared zombies so i did equation to predicatate

  244. Excellent. Couldn’t remember the name, found the site by accident. It’s great, reminds me of Tom Murphy’s ‘do the math’ site.

  245. Reaction times over 100 years ago were around 177 milliseconds. This, combined with a drop in attention span from 2000 until now from 12 to 8 seconds suggests we are getting more stupid.

  251. There is a fundamental error on this page. If you make a Mobius strip as suggested, it has 3 dimensions, not 2. If you can pick an object up, then by default it has 3 dimensions.

  252. Definition from The St Andrew’s University MacTutor page:
    A Möbius strip is a two-dimensional surface with only one side.

    A mobius strip is a 2 dimensional surface (manifold) in 3 dimensional space – in the same way that the surface of a sphere has 2 dimensions even though it exists within 3 dimensions. In order to make a mobius strip we twist through the 3rd dimension – so we need the 3rd dimension to build it – but it remains a 2 dimensional surface. The higher dimensional equivalent is the Kleine bottle – which requires a 3 dimensional object being twisted through the 4th dimension to make.

  253. Hi,
    Thanks for the article.
    I have a question about this section:

    “So we can expect this to erupt in the next 10 years with probability 1 -e-0.14 ≈ 0.0139.”

    This may be obvious, but I am having trouble understanding why this gives you the probability it will erupt in the NEXT 10 and not just in a given/random 10- year range. Can you explain?

    Thanks

  257. Dear Sir,

    I do not know you, but I have to say that this is a gold mine! Thank you for helping all IB students of the world by sharing this !

  258. If p is the real root and the complex roots are a + bi and a – bi, the equation of the tangent line is y = b^2 (x – p). The tangent line intersects the graph of the function at x = a.

  259. May I know what is the significance of Si(x) Sine integral function, How it can be related to the gibbs phenomenon?

    Thanks in advance,
    Dharmateja.

  260. I enjoyed doing this work because of the academical progress I made doing this work. I used to have a bad grade but ever since I started this my grade has improved a lot thus I really enjoy using this website. Thank you creators I deeply appreciate you making this website. And I found out to never give up to find the answer. You must believe in yourself in order to find the answer.

    -Ayaan

  261. How does 1(mod 40) equal 81, I do not understand how you came to that conclusion could you please explain

  265. Pingback: #1 (March) – EE Reflections

  266. Hi, I am a Maths Studies teacher in Germany and I always recommend your great website to my students. I just noticed that the first video on this page “Introduction to Logic” does not work and I wanted to let you know.

  268. Given the rules as stated, the ‘K’ must be turned over as well since it could have a ‘D’ on the other side and would, thereby, disprove the rule as well.

  270. Do you have worksheets or notes that correspond with these videos? I could not find them on you tube and thought that having the copies that go along with these videos would be very helpful for my students.

  272. Hiya

    Go to the youtube page – and then click on the information part of the video – eg:

    An overview of some of the basic concepts and problem types in “algebra” for the IB Math standard level course. If you want to follow along and do the problems, please download the handout at https://docs.google.com/file/d/0B6hD-… . This video is neither produced by nor endorsed by the IB.

  280. Because of exam secrecy and all that, can’t comment for 2017, but walking out feels like I just went through a meat grinder…

    You know what? Uni math is easier!

  281. where was the nuclear payload dropped theoretically? would that affect the data?

  282. The paper says the following: “We conducted two 10-year runs, one with 150 Tg of smoke and one with 50 Tg of smoke, injected into the upper troposphere (300–150 mbar) over a one-week period start- ing on 15 May spread over all the grid boxes over the 48 United States and over Russia”

    So you have a model effectively starting with all bombs being dropped over every part of both the US and Russia. I’m sure where you would drop the bombs would have an effect.

  284. Coding & Computer Programing? Computer programing is Coding! As an Inventor, all my life, I find “coding” to be the purest Invention, of All. You must
    Invent new methods, formulas & techniques as you Code, to get to the final
    Program… It is Problem solving at its finest, as each new line brings another
    Eureka Moment. I experienced this first hand, when I wrote a 7,000 line Algorithm, that factored R.S.A.#155, in under 3, minutes, confounding all the
    Math experts, who theorized, it would take 1 Man @ 1 Computer, longer than
    The Universe, has been in Existence!

  285. Hi, I believe that the link to Wolfram Alpha is incorrect because there isn’t an option where you can calculate the number of people needed for the chance of a 3 people shared birthday. Instead, on Wolfram Alpha, you have to input the number of people.

  286. You input the number of people in a room and then scroll down to see the probability that either at least 2 people or at least 3 people share the same date.

  289. See also the (to me) far more significant thought exp. of Galileo referenced at https://en.m.wikipedia.org/wiki/Galileo%27s_Leaning_Tower_of_Pisa_experiment.
    “Galileo arrived at his hypothesis by a famous thought experiment outlined in his book On Motion.[12] Imagine two objects, one light and one heavier than the other one, are connected to each other by a string. Drop this system of objects from the top of a tower. If we assume heavier objects do indeed fall faster than lighter ones (and conversely, lighter objects fall slower), the string will soon pull taut as the lighter object retards the fall of the heavier object. But the system considered as a whole is heavier than the heavy object alone, and therefore should fall faster. This contradiction leads one to conclude the assumption is false.”. For further clarity, the reader might imagine a large ball and a small ball made out of the same material, connected by a wire made out of the same material. Now let the connecting wire get thinner and thinner. At what point does the velocity of falling decrease because one object becomes two?

  290. that a^2 + b^2 =c^2 has no positive integers a, b, and c which solve the equation for n greater than 2.

    should read

    that a^n + b^n =c^n has no positive integers a, b, and c which solve the equation for n greater than 2.

    enjoy this
    http://heliwave.com/114.txt

    Ali

  293. Hi would investigating the formula 1 racing strategy using game theory and monte carlo simulation be a good enough topic for hl?

  296. I think I know why!

    Please see http://www.heliwave.com/114.txt for how prime and composite numbers interplay to shape our universe and rebalanced it after two planets between Mars and Jupiter had disintegrated into what is now known as the Asteroid Belt.

    Historically there were 11 planets at the time of Prophet Joseph (peace be upon him).

    God knows all.

  300. Thank you so much for this site. Your high School math department is really reaching out to other IB students.

  303. I am copying what the bottom two people,"I am copying what the bottom two people said", and "I am copying what the bottom two people and "I am copying what the bottom two people said" said" said says:

    very good

  307. We have the same pattern as before – an arithmetic series in terms of 2p, and using Sn = 0.5n(a1 + an) we have:

    Sn= 1 + 0.5(n-1)(2p + 2p(n-1) )

  310. Hello, great article on the matter of are you psychic?

    For me, I do believe we are all psychic we are all born with these abilities and capable of using them. As we grow up we are learned by either parents or the people around us what is real and what isn’t. So we grow up and are brain tends to work different but we can tap into these abilities with practice and above all meditation and visualizations.

  311. I did it as an IA and it’s fine. As long as you understand counting principles, it works.

  313. IB HL math syllabus and contents is perfectly ALRIGHT and a very good foundation for the future Engineers or scientists . I think the exam papers are very normal. I find only abnormality in the teacher to teach HL math.I have seen many IB maths teachers are unfortunately incompetent. Pls provide proper training to these teachers and give them better salary to deliver better education to our children as we are paying a high fees . SORRY for my fixation that if IB maths teachers are being tested for HL maths 9% will not achieve 7 like students worldwide, I am sure about that as i have first hand knowledge of maths teachers of about 50 of them in two big cities of IB schools. Please don’t relegate the content and exam of the HL math.it is excellent.
    .
    Best regards to all readers

  314. Pingback: TIL about Graham's Number, a number so large that any attempt to explain its size will be a vast understatement. » Best content from the Internet

  318. I can’t find data for the Carbon-14 abundance in fossils. I tried looking at many geography publications but couldn’t find any. Help me please!

  320. Pingback: Graham’s Number – The Abstract Unknown

  323. Why was the function f(x) = 1 + 2/(x + 2) utilized, and how was it seen as appropriate for the first graph (and/or, how was the value 2 to represent a in the formula f(x)= 1 + a/(x+a))

  324. quite an exquisite journey thorough this complex Structure I believe it is the easiest to view it as an equation, if you think of it in universal term it is quite simple, substitute dots with venus and dashes with earth and use trigonometry to calculate the tangent. I would like to thank my Mum, Dad, Maths Teacher, Chicken Bagd whomst I value dealy and my Best Friends who sadly died yesterday, Samamam I used his body to write this solution. RIP Samamamam rest in Peperonis

  325. Wow! This was very hard however not impossible. Just remember to substitute the whole chart with ancient Egyptian hieroglyphs. I then translated them into the Arabic language. From here on it was easy, just use google translate to translate it to Malaysian. From here all i had to do was to learn the Malaysian language. This took me about 3 months of learning it for 14+ hours a day. After this translate it into a biology equation and balance it. This only took me 1 hour to learn how to do but another 12 months to learn the English language. This is only half the job however. Next you must fly to the moon and observer the axis at which the earth spins on. Then fly back and search up the answer on the internet. Easy!

  326. I am copying what the bottom two people,"I am copying what the bottom two people said", and "I am copying what the bottom two people and "I am copying what the bottom two people said" said" I am copying what the bottom two people said" said says:

    very good

  327. The the tangent intercept is represented by (h,k) and the Tangent line is represented by Slope Intercept form (y=mX+b) than the actual zero is ; h±(the square root of m)

  328. A more complete answer: All of the discussion so far (including my response of almost two years ago) assumes the leading coefficient is 1 which will guarantee the slope of the tangent line in question is positive. However if the cubic has leading coefficient of A (not equal to zero) the real part of the complex solutions remains the first coordinate of the intersection point but the imaginary parts are +/- the square root of m/A where m is the slope of the tangent line.

  330. Can you not just take the average height of a person, leg length, weight and then speed of movement, time of day and date and calculate that in relation to the measurements in order to narrow down a more feasible estimation and or physically test this?

    This would give an accurate measurement as it is a measurement we as humans innately understand. Don’t waste time estimating just do the work and do it properly

    Each footstep would be on a space that is suitable for us humans to walk

  331. The calculations for Chi-square test is wrong. The formula has been used wrongly, as the one who did the calculations replaced Ei with Ci, so it becomes “(expected numbers x actual numbers) / actual numbers” when it is the other way around.

  333. We use Revision Village at our school =)
    It’s helped our class a lot. Highly recommend for exam revision!

  334. How would you suggest involving a data set in this area of study for a math sl ia, since it is a suggested topic?

  335. very nice article, people really require this kind of stuff to understand things better

  343. I agree the papers are really tough at HL level. Especially paper 3, giving a really tough problem for 17 odd marks is not fair at all on students.
    I totally agree with Engineer Belal Hossain, the teachers recruited by these so called international schools are terrible. They don’t have a clue how to teach IB maths (not even at SL level).

  344. I know I’m like 3 years late to this but the deaths would have to be calculated by taking the difference of S(0) and S(end time) to give you the total number of people infected, and then you would multiply by the morbidity rate. The model fails to actually account for death, birth or immigration as it models a closed population, so you more or less need to infer the death total from other data you do have.

  347. i tried to copy the name but it was too long i tried to copy the name but it was too long says:

    Very Good

  348. It should be y=sin(1643.84t) or y=sin(523.25πt) instead of y=sin(1643.84πt)

  350. i tried to copy the name “i tried to copy the name but it was too long” but it was too long.

  351. Very good

  354. There’s a serious mistake regarding orders of magnitude in this article.

    It says “4×10^17 is a number so mind bogglingly big it would take about 45 trillion years to write out, writing 1 digit every second.” Not so, I can easily write it in this comment field: 400,000,000,000,000,000. The number of digits can be found by finding the power of 10 (in this case 17), and adding 1 (assuming the coefficient is less than 10).

    Perhaps what was meant was that if every such number in the sequence of even numbers was written, it would take about 45 trillion years. Even then, that’s a bit too large. Calculating the number of total digits, one would have to write about 33,086,419,753,086,420 digits (A014925(16) * 5 * 4). This however would only take about 33,086,419,753,086,420 / (60 sec * 60 min * 24 hrs * 365.2425 yrs) ≈ 1,048,466,904 years to write the digits, at a rate of 1 digit per second. So in truth, it would only take about 1 billion years, which is 4 orders of magnitude difference from the 45 trillion estimate. This seems to suggest that either the original calculation used some unorthodox rounding, or that it was calculating something else entirely.

    Link to the relevant OEIS sequence: http://oeis.org/A014925

  356. Thanks can you give idea about what to write in Maths IA if we choose this topic like the Aim – that is what we are proving or investigating?

  357. About non-Euclidean geometry:

    very good articles on the subject, but I noticed that nothing was said about the Hyperbolic geometry of Bolyai and Lobatchevsky.

  363. How plz explain following?????
    Application of differential equations?
    ) In medicine for modelling cancer growth or the spread of disease
    2) In engineering for describing the movement of electricity
    3) In chemistry for modelling chemical reactions
    4) In economics to find optimum investment strategies
    5) In physics to describe the motion of waves, pendulums or chaotic systems

  364. Hey, which option textbook for calculus do you suggest? I don’t know to choose between haese or cambridge. Please answer, thanks 🙂

  366. the “How can you optimise the area of a farmer’s field for a given length of fence?” link won´t open

  367. Damn not looking forward to doing higher maths at IB. Hopefully it is to some degree easier than it is shown in these statistics but I have heard from everyone at my school that it is devilishly hard, I will need to get a level 8 in GCSE to even be allowed to do the course.

  373. This article is fascinating! Thanks Rin for the comment! I’m a maths Studies student and I want to do this study! Can i do it? I would reference it. I hope I can do it.

  374. I will posting some doubts in case there are, I think with this proyect i could learn a lot of things! Have a nice day.

  375. Oh, my previous comment wasn’t posted. I mean I want to do this for my Maths Studies IA. I would referentiate it. It will be awesome. I hope i can achieve this idea in my IA. Thanks everyone!

  377. I copied the name "i tried to copy the name “i tried to copy the name but it was too long” but it was too long" it wasnt too long says:

    Very Good

  383. “β the transmission rate as approximately 1.07”
    β represents the contact rate – which is how likely someone will get the disease when in contact with someone who is ill.

    Have I missed something – how can the contact rate be > 1 ?
    It is a probability isn’t it?

  384. Thanks for you comment – I don’t think I worded the explanation very well. If you think of beta as the average number of disease-spreading contacts made by each infected individual per day then a beta of 1.07 (per day) means on average an infected person will infect 1.07 other people each day. I might try to make that clearer in the post! Sometimes the SIR models give a value for beta by dividing by N as well (i.e in this case 1.07/11,000,000), but in the equations above the division by N is already there.

  385. Thanks – I found the explanation below, which clears it up for me.
    I recall the 80s as a young man in my 20s. When no one was discussing the maths (no internet for geeks). But Maggie was a trained scientist and UK policy was very effective in persuading people to change their habits so that p, the transmission rate fell sharply which was good. And also γ, the total contact rate fell sharply, about which I was more equivocal.

    https://en.wikipedia.org/wiki/Transmission_risks_and_rates
    The effective contact rate (denoted β) in a given population for a given infectious disease is measured in effective contacts per unit time. This may be expressed as the total contact rate (the total number of contacts, effective or not, per unit time, denoted γ), multiplied by the risk of infection, given contact between an infectious and a susceptible individual. This risk is called the transmission risk and is denoted p. Thus:

    β = γ X p

    The total contact rate, γ, will generally be greater than the effective contact rate, β, since not all contacts result in infection. That is to say, p is almost always less than 1 and it can never be greater than 1, since it is effectively the probability of transmission occurring.

  386. Thanks for the added information above about β. Looking at the pictures on the news of deserted streets in Wuhan, it looks like the Chinese government will be pretty successful in reducing the contact rate – so hopefully this will be effective in slowing any spread…

  387. Sorry may I ask if the link to the formulas for creating the SIR spread sheet still work? And why is the number of people removed 8200 on the first day?

  388. The values are from the study published on 21st Jan.

    S = 11,000,000 as there are around 11 million in Wuhan.
    I = 3500 as there were an estimated 3500 (approx) infected
    R = 8200 as there were an estimated 8200 (approx) recovered.

    Looks like the website with the link to the excel formula is down.

  390. Please! I need the formula to do my IA, if you can do another spreadsheet.with all the formulas, because I dont know about the interaction rate.

  391. i really liked resources here . as i taught lot of students and if they wanted to take internal assessment idea i always referred to this website for ib maths internal assessment.

  392. Does the inverse always have to be 1 mod 26? So for a=3 inverse would be 9? If it is then not sure how to decode from there as you get some negative numbers if you are multiplying by 9, or numbers which don’t correspond with anything in the alphabet??

  393. Firstly, l say that I would like to thank you. Actually, l would like to try to collect some facts to write a term paper for URJ . So l would like to study simple real problems solved by ODEs. Thank you.

  394. I think mathematics is the reality because without math, people would claim that these things happen because of God and everything would be from a religious point where there is no evidence to support the claim. But with math we are able to prove things and figure out why things happening using math.

  397. The problem as stated here:
    “How could a postman visit every letter on the graph in the shortest possible time?”
    is NOT the Chinese postman problem. Rather, it is the Travelling Salesman Problem. This is an important distinction.
    Everything stated below that is correct.

  400. I notice that the 11th tetrahedral no. in the graphic is incorrect, it should be 286

  402. Reblogged this on In the Dark and commented:
    I’m reblogging this post from a few years ago. It remains topical.

    I thought I’d add my own (very limited) experience of taking penalties. In the period from 1988-90 or thereabouts I played for a team called the University Associates in the Sussex Sunday League (2nd Division). The League also had a cup competition, and one day we played a game that ended in a draw after extra time, so went to penalties. I used to play in midfield for that team, rather than as a striker and I scored only 2-3 goals a season. I wasn’t one of the five nominated penalty takers but after those it was 2-2 so it went to sudden death and my turn came up at 3-3 after one round. It was the only penalty I’ve ever taken (not counting 5-a-side). I wasn’t at all confident but my biggest fear was the ribbing I would undoubtedly get if I didn’t even hit the target, so I decided to hit it as hard as I could straight at the goal. I thought my natural level of inaccuracy might take the ball to one side or the other of the goalie.

    So I paused, took a deep breath, ran up and blammed it as hard as I could. It went quite hard straight at the goalie. If he’d stayed where he was standing it would have hit him at chest level. Fortunately for me he dived out of the way and I scored. 4-3! So I have a 100% success rate at scoring penalties (based on a sample of one).

    The story didn’t end entirely happily though. My opposite number scored to make it 4-4 and we ended up losing 5-4.

  404. When two procedures used to solve a problem lead to different results (a paradox), the solution to the paradox is to show why one of the procedures is wrong. It is NOT showing why the wrong procedure must, would, or should lead to the same result. A complete explanation, and the solution, are given below.

    What Does Solving a Paradox Mean?

    As mentioned at the beginning of my article (at: https://bit.ly/2IM76rF ), a paradox proposes the existence of two different results as a solutions for the same problem. These results are inconsistent with each other, depending on which procedure is used. Only one can be correct. As Brown and Moorcroft suggest, we are not looking for a mathematical demonstration that Achilles reaches the tortoise. Assuming they are both running in the same direction, we know he will. We can calculate the exact time, given the distance between the two and the two speeds, using a simple formula:

    t = distance/(difference in velocity)

    Instead, explaining or invalidating a paradox is to show a fault in the paradox formulation, or the proposed solving procedure, so that we can exclude this procedure and demonstrate that there is only one result for the original problem. The solution of a paradox is the answer to the question: “How does the paradox formulation misrepresent reality or logic?” That is, we need to show why the proposed method is conceptually wrong. Solving a paradox, invalidates the formulation of a problem proposed by the author of the paradox and leaves us with only valid procedures for solving the original problem.

    Why were the previous proposed solutions for the paradox not satisfactory?

    The fault of most, if not all, the proposed solutions to Zeno’s paradox is the assumption that Zeno’s proposed procedure was correct. The procedure seems to be logical when it is first introduced to us, but we will see that the procedure proposed by Zeno is conceptually wrong. The authors then tried using Zeno’s faulty procedure to reach the expected correct result for the original problem. This is unacceptable.

    The Explanation of Zeno’s Paradox:

    Zeno’s proposition invites the solver to do a series of steps each time changing system of reference: STEP 1: The starting system of reference: The point where Achilles starts the race and the tortoise is well ahead, STEP 2: After a while, we are then asked to use a new system of reference: The point where Achilles reached and where the tortoise initially started, with the the tortoise now a bit further ahead, STEP 3: Then again we are asked to use, recursively, a new system of reference with the new starting point for Achilles and with the tortoise still further ahead, with every step we are asked to freeze the process and then continue by re-creating and examining the original problem using a different system of reference.

    Today we know more about the relative motion of two bodies. Solving a problem that involves space and time, requires a defined system of reference, which cannot be changed without the proper conversions. After Zeno’s proposed first step, or first change of system of reference, the problem, as presented in the second step, is exactly the same as the original, the only change being a difference in “scale”. No progress was achieved in solving the problem. Changing system of reference essentially restarts the problem-solving procedure. This realization implies that the problem is never going to reach a conclusion as the step by step procedure is reiterated. If the system of reference is changed at every step, our working spacetime shrinks with every step, the solution becomes elusive and the tortoise becomes apparently unreachable. Zeno proposes a procedure that never ends, for solving a problem that has a trivial solution.

    A programming analogy

    Zeno’s proposed procedure is analogous to solving a problem by recursion, a well known problem solving technique available in modern programming languages. However, recursion would be the wrong technique to solve the original problem. If the problem was programmed exactly as Zeno suggested, the program would never end normally, simply because the condition for the end of the recursion process (Achilles reaches the tortoise) would never occur. Our computer program would confirm the paradox!

    Our Solution

    Our solution of Zeno’s paradox can be summarized by the following statement:

    “Zeno proposes observing the race only up to a certain point, using a system of reference, and then he asks us to stop and restart observing the race using a different system of reference. This implies that the problem is now equivalent to the original and necessarily implies that the proposed procedure for solving the problem will never end.”

    That’s it. You cannot change system of reference in the middle of a problem involving space and time, whether the system of reference is openly stated, or implied. As an analogy, you cannot solve a problem involving measurements by using English Imperial measures at the start of calculations and then switch to metric measures (without proper conversions) in the middle of calculations.

    An example

    The following is not a “solution” of the paradox, but an example showing the difference it makes, when we solve the problem without changing the system of reference. In this example, the problem is formulated as closely as possible to Zeno’s formulation.

    Zeno would agree that Achilles makes longer steps than the tortoise. Let’s assume that one Achilles-step is about 20 tortoise-steps long, and let’s also assume that both Achilles and the tortoise make the same number of steps in the same amount of time. For example, two steps per second (the exact amount doesn’t really matter). If the tortoise starts the race 20 Achilles-steps ahead of him, then after 20 steps Achilles reaches where the tortoise was (See diagram below: Tortoise starting point).

    (The picture can be seen at: https://bit.ly/2IM76rF )

    In the meantime, the tortoise has made 20 of her steps, and she is now one full Achilles-step ahead of him. We have not changed our system of reference. We referred to both starting points. These did not move relatively to each other. We could choose any fixed ground point. To please Zeno, let’s continue by referring to the tortoise starting point, where Achilles currently is. When both runners make one more step, step 21, the tortoise will have moved by one of her steps and she will still be ahead of Achilles by that one tortoise-step. Achilles is now one Achilles-step ahead of the tortoise starting point. Now, let’s continue, without changing the system of reference. This is the key point. We do not redefine the problem and use the current positions of the runners as new starting points, as Zeno proposes, but we refer to the information about the race we have already accumulated in our knowledge base. Achilles then completes his 22nd step, and he is two Achilles-steps ahead of the tortoise starting point. The tortoise will have completed her 22nd tortoise-step from her starting point. Hence the tortoise is now behind Achilles by 18 tortoise-steps. Thus, if we do not change the system of reference, the paradox does not appear.

  407. Wow never knew spiders webs had such mathematical properties . Thanks fir the insight now I need to put the effort in to understand it all

  409. I copied the name "I copied the name "i tried to copy the name “i tried to copy the name but it was too long” but it was too long" it wasnt too long" but it wasn't too long says:

    very good

    + first comment in 2 years

  410. Hi. Nice article. I’m still working through it, however I believe I found a typo: in the 5th equation of the figure after the sentence
    “Let’s see if we can solve this for when our irrational number is pi, and when we choose q = 6.”,
    there seems to be a factor of 6 missing from pi on the right hand side. Hope this helps.

  411. Very interesting to see links between my SL psych content and maths. One issue though – the psychologist who came up with this task was called Wason, not Watson.

  413. Isn’t there something missing? I think you’re just showing that there is a time when the bullet and the monkey are the same height. That’s not so surprising. But at that exact time, the bullet must also travel the exact horizontal distance to the monkey. This is not trivial and depends on the angle of the shot, which does not appear in your calculation at all.

  414. you have 2 equations for the y coordinate of the monkey and the bullet respectively in terms of t. Therefore when they are equal then we have a coordinate (t,y) which tells us the time when they are indeed at the same height. The angle can change – this will change the equation for the motion of the bullet and will give a different (t,y) solution.

  415. I think there is a misunderstanding. I try to be more precise. You want to show that there is a collision. For a collision the same hight is not enough, they must be at the exact same place. Technically: (y_B, t)=(y_M, t) is not enough, you need to show that there is an t with (x_B, y_B, t)=(x_M, y_M, t). So you additionally need to proof that for your t=h/v_y also x_B(t)=x_M(t) holds (where x_M(t) is constant).

  416. Ok thanks – I see what you mean now! I suppose we can say informally from a graph sketch of the y,x plane that as we have a quadratic (from the projectile motion of the bullet) and a vertical line (motion of the monkey), then if they intersect they will intersect only once, so if there is a time when they intersect this must be that intersection point. I agree it would be nicer to be more formal. It think it needs involving ideas on horizontal motion – I’ll have a think on this. Thanks for your comments!

  417. I thought it through and I think here is the missing part:
    At t=0 the shooter aims at the monkey. Therefore, for the starting velocity v=(v_x,v_y) we have: v_x/v_y = d/h, with d as horizontal distance between shooter and monkey. With your solution t=h/v_y is then x_B(t) = v_x * t = v_x * h/v_y = h * v_x/v_y = h * d/h = d.

  419. actually there’s a method that doesn’t require the Leibniz’s theorem on alternating series. We could just group un and un+1, and then make some integral inequality to show that 2S<M, where M is roughly 1.8…., which means that it converges. But yes, definitely, given the limited time, it's impossible for student to complete a so complicated calculation even with the calculator, and PS. the difficulty of b(ii) is even harder than STEP III using the upper method I mentioned, since the question didn't give hints and instruction well, which definitely distract students, and confuse them.

  421. Hi,
    Very cool article.
    Its important to note that you have 101 terms there:
    [0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5, 3, 0, 3, 2, 9, 0, 4, 9, 3, 6, 14, 0, 6, 3, 5, 15, 0, 5, 3, 5, 2, 17, 0, 6, 11, 0, 3, 8, 0, 3, 3, 1, 42, 0, 5, 15, 20, 0, 4, 32, 0, 3, 11, 18, 0, 4, 7, 0, 3, 7, 3, 2, 31, 0, 6, 31, 3, 6, 3, 2, 8, 33, 0, 9, 56, 0, 3, 8, 7, 19, 0, 5, 37, 0, 3, 8, 8, 1, 46, 0, 6, 23]

    The provided code is indented incorrectly and uses and obscure python feature of a for else.
    Here is the “correct” code:
    “`py
    def example(terms: int):
    A181391 = [0, 0]
    for n in range(1, terms):
    for m in range(n-1, -1, -1):
    if A181391[m] == A181391[n]:
    A181391.append(n-m)
    break
    else:
    A181391.append(0)
    return A181391
    “`

    also here is some “easier” to understand code:
    “`py
    def van_eck(n: int) -> list[int]:
    if n < 1:
    return []
    seq = [0]
    seen = {}

    for i in range(n – 1):
    cur = seq[i]

    if cur in seen:
    next = i – seen[cur]
    seen[cur] = i
    seq.append(next)
    else:
    seq.append(0)
    seen[cur] = i

    return seq
    “`
    Thank you for the article

  423. We are given that A, B, D and C are inscribed on a circle clockwise in that order and that AD is perpendicular to BC. Let P be the point of intersection of AD and BC. We are given that PA = 2, PD = 6 and PC = 3.
    By the inscribed angle theorem, angles ABC and ADC are equivalent; angles BAD and BCD are equivalent to each other as well by a similar reasoning. Since the three angles in triangle PBA are the same as in PDC, they are similar. Therefore, PA/PC = PB/PD so PB = PA*PD/PC = 6*2/3 = 4.
    The perpendicular bisectors of all chords of a circle intersect at the center of the circle by symmetry. Therefore the perpendicular bisector of AD and BC intersect at the center of the circle.
    Let O be the circumcenter, M be the intersection of AD with its perpendicular bisector, and N the intersection of BC with its perpendicular bisector. Since AM = MD and AM + MD = 8, then MD = 4. Since BN = NC and BN + NC = 7, then NC = 7/2. Since NC = 7/2 = NP + PC = NP + 3, then NC = 1/2.
    Since the four angles in quadrilateral OMPN are 90 degrees, then it is a rectangle. Therefore NP = OM = 1/2. Therefore, DMO is a right triangle with OM = 1/2, MD = 4, and OD as a hypotenuse. By Pythagoras, OD = sqrt(4*4 + 0.5*0.5) = 0.5sqrt(65). OD is a radius so we are done.

  424. Some years ago I was developing calculus formulae [can’t seem to do it anymore] for boat volumes. You can’t use pi in formulas for boat immersed volumes because boats are pointed. Pointed in either the vertical plane, or the horizontal plane or both. I found that instead of pi/4 [.785], to simply use 0.67 . This works for pretty much any sailboat shape since bottoms and waterplanes are often arc-shaped, [unless it has a very blunt bow angle, like 90 degrees], and the factor moves from .67 to .785 .
    Now an American football is not an ellipse, it is definitely pointed at both ends, I suggest to use 0.67 instead of pi.

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