<p>Am I right in saying that to get 800 on math; that is, to get the hard questions right on each section, last few questions usually, you need to critically think and critically analyse a figure or question to solve the problem or inspect carefully. Also, back solving for the harder questions is a must is what Im seeing. Has anyone else noticed this? What do people think? </p>
<p>Thanks</p>
<p>I would certainly argue that you have to very carefully read the hard questions in order to understand what they are asking. However, I’m not sure if backsolving is an absolute must.</p>
<p>What exactly do you mean by “backsolving?” I’m assuming you’re starting from the answers and plugging them back in to see what works. Although this my be useful, it certainly isn’t necessary.</p>
<p>You don’t need to “back-solve” every question. It works as a quick check if you have time, that’s about it.</p>
<p>Generally, most “hard” math questions have a difficult, somewhat time-consuming solution, and an easier solution that may be harder to find but saves a lot of time. Find the easy solution and you’ll be on your way to an 800. Try the following problem:</p>
<p>Q: Find the smallest positive integer m greater than 1 such that m^2 is a perfect cube and m^3 is a fourth power of an integer.</p>
<p>(This isn’t SAT level, it’s more like an AMC10 problem but you get the idea. Also it helps to try more advanced problems if you’re working at 800).</p>
<p>I agree with both of you, and I take what I said back about the back solving: of course back solving is not a must if you know how to solve the problem ‘naturally’. Although with some questions, they can only be solved if you plug in the answer choices because the questions are built that way.</p>
<p>And yes, back solving is just using the answer choices to solve the problem.</p>
<p>The other thing is, I’m comfortable doing level 1, 2, 3, problems. However, there are some level 5 problems that I cannot do and some level 4s that I struggle with. </p>
<p>Cheers</p>
<p>Also, is there anything else that you guys have noticed, any patterns etc for the ‘hard’ problems that might help me in my preparation to getting 800? I am beginning to see a pattern, but I can’t tell exactly what that pattern is. </p>
<p>Much appreciated</p>
<p>“Although with some questions, they can only be solved if you plug in the answer choices because the questions are built that way.”</p>
<p>Haven’t seen too many of those…the only such questions are ones like, “Which of the following…” They’re pretty rare on SAT exams.</p>
<p>I scored 800 myself, and I still don’t know any “patterns” in SAT math problems. However keep in mind that many of the “hard” problems have at least two ways of solving it, find the easy, short solution and you will be fine. Works particularly well on number theory problems, where you can use modular arithmetic (which isn’t often taught in HS math, but can be useful on the SAT).</p>
<p>@strategic </p>
<p>I would actually agree with (a modified version of) your original statement. I strongly believe that to GUARANTEE that you will get an 800 in math, then you MUST be fluent in basic SAT strategies such as picking numbers, taking guesses, and plugging in answer choices. On any specific SAT you may or may not need to use these basic strategies, but there is ALWAYS a chance that you will get a question that is particularly tricky algebraically. I have a Ph.D. in mathematics and every now and then I find a problem that I will “solve” first by picking numbers before attempting an algebraic solution.</p>
<p>Often after “solving” the problem with specific values, an algebraic solution becomes evident.</p>
<p>@DrSteve </p>
<p>I appreciate your comments, and I think knowing a few strategies has helped me quite a bit achieve around about the 730/740 mark. Given that the test is standardised, a large chunk of the hard problems that I have done have been easier to solve by just merely plugging in my own numbers and then reconciling with the answer choices e.g. for the algebraic expression questions. I completely concur with you @DrSteve and that is why I bought your book 320 SAT math problems arranged by topic and difficulty! I will start doing level 4 and 5 problems only, since those are the questions that generally require more effort. </p>
<p>Also, I need yours or anyone else’s advice on how to prevent misreading certain details in a question. I made a few consecutive errors on a recent paper that I did, mainly on level 4 and 5 problems, solely based on misinterpreting, misreading and misunderstanding details in a question. I have no idea how to prevent these things from happening, any advice? </p>
<p>Thanks for the help</p>
<p>Cheers</p>
<p>@strategicfiasco Always helps to read every word and read it carefully. Highlight important words if you wish. I accidentally skipped over the word “remaining” in a recent CC thread.</p>
<p>^ I agree completely about the careful reading. It sounds so obvious and yet silly reading errors are often what separates 750 from 800. After all, if you are scoring 750 then you clearly have the math talent needed.</p>
<p>One other thing I see a lot: make sure you are answering WHAT THEY ASK and not what you assumed they were going to ask. This is a trap that often catches students who have done a lot of practice tests. Everything seems so familiar and repetitive that you fall into a less alert way of reading the questions.</p>
<p>You’re already taking the first step which is focusing on practicing the types of problems that you find you are misreading. It’s interesting that you are only misreading problems that are Level 4 and 5. This is actually how they became level 4 and 5 problems. It’s because the wording is actually designed for misinterpretation.</p>
<p>The one recommendation I always give is this. Every time you get a question wrong, make sure you understand why you got it wrong, mark it off, and then forget about it for 1 week. One week later go back to it and do it again. Keep redoing it every week until you get it right ON YOUR OWN. </p>
<p>Most people dismiss a problem after they go over it. Do not mistake “carelessness” for getting tricked. It’s a very easy mistake to make. And it causes many students to ignore the advice I just gave. Always keep redoing the uestions you get wrong, no matter how silly you think the error was.</p>
<p>Some students find it helpful to underline key words as they’re reading a problem. After you’ve finished the problem you can look back at what you’ve underlined and make sure you’ve actually answered what was being asked and completed the steps required.</p>
<p>Also, if you do a problem 2 different ways and arrive at the same answer, then it is almost certain you did it right. So when you “check over” your answers, always try to solve the problem in a new way.</p>
<p>If you haven’t done so already, read my article “Stop Making Careless Errors in SAT Math.” There might be a few tips in there that might help. It’s posted on this forum.</p>
<p>On a side note, the “q” on my keyboard is being very uncooperative today, so I apologize for any missing “q’s” in my posts. :)</p>
<p>@pckeller and @rspence, I agree, reading carefully seems like something so trivial, yet so valuable, which is what I’m noticing now. I don’t think many people know how important it is (to read slowly and carefully). Thanks for the advice</p>
<p>@DrSteve, I used to do that a lot i.e. not going back to questions I got wrong because of some silly mistake but realised they weren’t careless at all but misunderstanding the question; I think knowing this has helped me improve plus knowing the strategies. Also, I will follow your advice and see what works and what doesn’t. The good thing is though, I’ve worked out where exactly the mistakes lie, and I know how to fix them. Hopefully, I can crack the 800. </p>
<p>And anyone what is modular arithmetic? Could someone please explain with an example?</p>
<p>Thanks again for the help</p>
<p>Modular arithmetic is essentially “clock arithmetic” where the mod is the size of the clock. For example, a “normal” clock would be mod 12. Let’s stick with a 12 hour clock for now.</p>
<p>If it’s 11 o’clock now, then 3 hours later it is 2 o’clock. This is because 11 + 3 = 2 (mod 12).</p>
<p>Each number on a clock is actually just one representative for infinitely many different numbers. For example, 12 is a representative for 0, 12, 24, 36, -12, -24, etc. In fact, any multiple of 12 is at the top position of the clock. We say that these numbers are all congruent mod 12. </p>
<p>At the 1 position, we have that 1, 13, 25, … are all congrient mod 12. Note that the difference between any 2 of these numbers is a multiple of 12. Also, if you divide any of these numbers by 12, the remainder is 1.</p>
<p>So each number on the clock is actually a single representative of an infinite collection of numbers that all have the same remainder when divided by 12.</p>
<p>In modular arithmetic, if you take any 2 numbers and apply arithmetical operations to them, then the result is independent of the representative you choose. </p>
<p>For example, note that 5 is congruent to 17, and 11 is congruent to -1.
Now 5 + 17 = 22 and 11 + (-1) = 10. But 22 and 10 are congruent mod 12 because 22 - 10 = 12 (or euivalently 10 and 22 both give a remainder of 10 when divided by 12).</p>
<p>One thing to note: the standard clock was definitely not invented by a mathemetician. Otherwise the number 12 would have been replaced by its better name 0.</p>
<p>Yeah, modular arithmetic is basically arithmetic dealing with remainders. For example, 84 ≡ 4 (mod 5) (≡: is congruent to). You can add, subtract, multiply mods, as long as you are working with the same modulus.</p>
<p>You don’t need modular arithmetic on the SAT, but on some problems it can be useful. It works as a shorthand way of doing things. Consider the following problem:</p>
<p>Q. Integers m and n leave remainders of 5 and 6, respectively, when divided by 7. What is the remainder when mn + m + n is divided by 7?</p>
<p>There are multiple solutions, a straightforward one is to let m = 7r + 5, n = 7s + 6. But too much algebra. We may also assume m = 5 and n = 6. But to show that this works in general, we use mods.</p>
<p>mn + m + n ≡ 5(6) + 5 + 6 (mod 7) ≡ 41 (mod 7) ≡ 6 (mod 7), hence the answer is 6.</p>
<p>Bonus if you rewrite as (m+1)(n+1) - 1 and note that n+1 is divisible by 7, so it is -1 (mod 7), which is equivalent to 6.</p>
<p>You mention it so quickly in passing, but the quickest way to solve this is to make up numbers that fit. Say m= 5, n=6. Or m=12, n=13. Anything that fits the original description. You get the right answer without knowing modular arithmetic.</p>
<p>Making up numbers that fit the question is a frequent time-saver on the SAT. </p>
<p>Still, clock arithmetic is kind of interesting. When I was in middleschool, we did a lot of it for some reason. Here is one that stumped me then and I still remember it: </p>
<p>In clock 7, what is 5 divided by 3?</p>
<p>Yeah I know, I never said guessing numbers was a slow or bad solution. For some people (like myself) I like to know whether it works in general. In this case working mod 7 takes roughly the same amount of time as plugging in 5,6.</p>
<p>Modular arithmetic is indeed interesting…lots of cool theorems, like the Chinese remainder theorem and Fermat’s little theorem.</p>
<p>Haven’t done much division with modular arithmetic, but does 5/3 (mod 7) mean 5/3 ≡ x (mod 7) → 5 ≡ 3x (mod 7)? In that case, x ≡ 4 (mod 7).</p>
<p>Thanks guys I will take a look a look at this.</p>
<p>@rspence</p>
<p>Yes - division is defined in terms of multiplication (as always). You do need to be careful with modular arithmetic. Nonzero division is only guaranteed to give a unique solution in the case where you are working inside a field. Luckily there is a nice theorem that says addition and multiplication mod p forms a field for {0,1,…,p-1} precisely when p is prime. So 5/3 yields the unique solution 4 (mod 7).</p>
<p>If p is not prime, you get a structure with “zero divisors.” For example when working mod 6, 2*3=0. Zero divisors are “undesirable” algebraically since some of the rules we tend to take for granted will fail.</p>
<p>By the way, for anyone reading this thread it is completely unneceassary to understand modular arithmetic for the SAT. The very nature of modular arithmetic being “well-defined” means that SAT problems involving remainders can be solved by picking numbers. In fact, the easiest version of picking numbers works - any choice of numbers will always lead to the same answer (in general you need to be careful in the sense that picking numbers can only be used to eliminate answer choices, but in the case of problems involving remainders you will actually get the answer).</p>
<p>That said, modular arithmetic is a nice subject, and a bit more advanced. Studying it will sharpen your problem solving skills and increase your level of mathematical maturity - not a bad thing at all.</p>