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ACT Math Tips
Case Western Univ., summa cum laude
Perfect scorer on the SAT & the ACT
Devorah is the founder of Advantage Point Test Prep and the author of the book “Boost Your Score” The Unofficial Guide to the Real ACT.
We're taping this course in San Francisco and the streets are really confusing; lots of dead ends, lots of construction, lots of hills and it's been taking me a really long time to get to the studio but here's something cool about San Francisco the people are also really friendly. So actually yesterday I met this guy really nice, on the street who showed a shortcut to get to the studio and now I don't have to worry about all that time and all the anxiety that I had before. So how does this rate to the math. Well on the math hopefully you'll watch the episodes, you'll feel really good about your skills and you can just feel comfortable with the questions and nail them. But what if you're not sure, well in this episode we're going to take a look at some good shortcuts some kind of backdoor strategies for when you're really not sure about what the question is asking or if you just want to answer it more quickly. We're going to talk about first pacing strategies so that you can pace yourself, then problem solving strategies and last guessing strategies.
Let's get started at talking about some math pacing strategies. First, some general strategies. Answer the easy questions first. On the ACT math questions are not in any order of difficulty they're completely mixed up, so you're going to have some easy questions at the beginning, in the middle and at the end and you want to make sure you hit those easy questions at the end, 'cause remember all questions are worth the same amount of points. So I recommend making two sweeps through your math section first, go through and answer all the easy questions, all the ones that you feel pretty good about and then leave the ones that are a little more difficult or even the ones that you know you can do but it will just take longer, leave then for last and then make a second sweep and see if you can tackle those. Next use your target score to determine pacing this is actually a really, really cool thing about ACT math, the curve is awesome you can get a third wrong on the test and still get a really good score, we say that I can get a D plus on the math. So for example if I got a third wrong, that's 20 out the 60 wrong. I would still end up with a 25 on the math which is about an eightieth percentile score, pretty cool. So you can 20 out of 60 wrong that's awesome. So if you're aiming for a score like that you might want to slow yourself a little bit and feel really confident. You can spend enough time on the ones that you know you can nail, as long as you spend enough time making sure they're right and then don't feel so bad just guessing on the ones that you don't feel so great about and hopefully you still do really, really well.
On to sneaky problem solving strategies. Here's the first one use your calculator, now it's actually a lot better than it sounds. You're thinking that's not that interesting. It is, here's something really neat. On your TI calculator there is a place for programs and there are actually a lot programs you can download that will do some of the work for you, so you don't actually have to waste your time doing it on the ACT. So I'm not going to belabor this now but look at your bonus materials, we'll have a list and some links for programs you can download things like Quad you know to solve quadratic equations which we'll get to when talk about intermediate algebra.
Okay next, let's talk about another great problem solving strategy, choosing your own numbers. So you can do this choosing your own numbers and plugging them into the problem when you have variables in the questions or in the answer choices or like a really different time, if you're given a problem involving percents and no starting value. So let me show you some examples so you see this in practice. Here we go, we have a problem that looks like this: Which of the following equations expresses z in terms of x for all real numbers x, y, and such that x cubed is y and y squared is z. Students often see this and they think oh my gosh there is a lot of complicated algebra that I need to solve this problem. So if that's the case why not pick numbers and you can pick any numbers you want as long they meet the requirements of the problem. So in a case like this I try to stay away from picking zero or one 'cause they're kind of really friendly numbers, a little too friendly and sometimes they do some funky things. I actually start with two, so two is a great number so let's make x be 2 we'll figure out what the rest of the numbers should be and then we'll match out the answer choices with our numbers and see what answer choice matches our answer.
Let me show what I mean, so let's make x be 2 so if x is 2, I'm going to write this down x equals 2. So what's y? Well x cubed equals y so 2 cubed is equals y, so y is 8 and I'm going to box this 'cause it's really to keep track when you're picking numbers of what the variables that you pick are equal to. So you got so far x is 2 and now our y with the number we chose is 8, what about z? Well y squared equals z, okay so we have here y squared z so 8 squared equals z which is equal to 64. Okay so we have z is 64, x is 2, y is 8. Perfect we picked our own numbers, now in search of the answer choices we want to know which of the following equations expresses z in terms of x. So which of these works and because we picked numbers that meet the requirement our numbers are going to help us solve this, what we just need to do is plug our numbers back into the answer choices. So here Z, we said z is 64, is 64 the same thing as two to the sixth I don't know let's hedge our calculators just to check. Well if we do 2 to the six, that's equal to 64. So here we go, this one here if you had z being 64 let me just write this down 64 that is in fact equal to the x we picked which is 2 to the sixth power. So in that case A would be the correct answer. Now if A wasn't the correct answer have no fear, you just keep shopping through the answer choices plugging in your numbers to see what works and always they'll be one answer choice that fits, that exactly fits the problem and your numbers.
Okay let's look at a slightly more complicated way to plug in numbers. So this shows up a lot they love things with remainders so try to jog your memory and think where have I seen this before, probably early in pre-algebra maybe or even earlier in elementary school you'll remember remainders. So when the number k is divided by 4 there is a remainder of 3. Do you remember this it means that it goes in let's say once or twice or whatever and there's 3 leftover. What's the remainder when 2k is divided by 10 and this is just kind of scary looking, but let's pick a number. But here remember the number needs to meet the requirements of the problem. We need a number where if you divide it, divide the k our number by 4 there is 4 there is 3 leftover. What would be a good number to pick here? Probably the easiest number 7, 7 is just 4 plus 3 so we know if you had 4 going into 7 it's going to in once and leave 3 over perfect. So let's say that k is equal to seven, now we can finish solving our problem. What's the remainder when 2k is divided by 10? Okay so if you had 2 times 7 or 14 divided by 10 what would you get. So 14 over 10 okay so now we have 10 going into 14 and the question is what's the remainder when you have 10 going into 14 it's goes in once right okay and what's left over well 4 right, 10 is going to go in once leaving 4 over. So our answer choice is going C great. So that's an example but slightly more complicated problem but you see how as long as you know what to do you can pick numbers you can get it right away.
Let's go into the next kind of problem solving strategy that's really, really helpful, working backwards. When are you going to work backwards? You're going to work backwards when you see words problems with numbers in the answer choices, the ACT people actually want you to do a lot of algebra and a lot of times you don't have to, you can plug in the answer choices and see what works. Let me show you an example: The cost of a slice of pizza and a soda is $2.50 cents, the cost of 3 slices and 2 sodas $6.75. What's the cost of the slice of pizza. Now the ACT people actually want you to do this algebraically, they want you to make two different formulas, two different equations, one saying okay you know relating the one slice of pizza and one soda and another one relating to 3 slices and 2 sodas. You know what that's a lot of algebra, it'll take a lot of time, there is a lot of potential for error too 'cause of all the writing and the thinking you're doing and all the, you know having to solve the problem.
Instead let's see what happens when we work backwards when you work backwards you head to the answer choices and this is what you do. You're going to look at them and say okay I'm going to pick from the answers use that for the problem and see does this answer choice work. So we're going to work backwards, now when you do that though you want to start with C always and let me tell you why. On this problem type and actually on most ACT math questions you'll see that the answer choices start small and get large. Okay they're going to get bigger. So let's say we just try C and we say okay $1.60 is the answer for how much a slice of pizza costs, we work it backwards we see, you know and we see oh my gosh you know it's too small. We need our slice of pizza to cost more, that number is not working and it's too small for us. Well we already know then if it's too small it can't be A or B which are even smaller. So how cool is that, by just trying one answer choice you can cross off A, B and C. Then you'll need to try one more time, okay and you're a little bit bigger, let's try D which is a slightly larger number. In that case let's say D doesn't even work and that's too small, you don't even have to try E, you know it's the correct answer. Pretty cool. So when you work backwards you'll always only have to try two answer choices to get the answer that you need.
Alright suppose try solving this problem by heading towards the answer choices. And we're going to start with C the middle choice, $1.60, and we're saying okay we're answering the question what's the cost of a pizza, one slice, a slice of pizza $1.60, great. So let's write this down. So one, I'm going to try put P for pizza and one slice of pizza is $1.60. Okay well, does this work in the problem? Let's look. The cost of a slice of pizza and a soda is 2.50 in that case what's a soda? Well if they add up to be 1.50 then you're going to have one soda equals, you know what let me just make these look more like ones here so you don't think that it's an I. So here we go and one pizza and we've got one soda. Okay so if one pizza is 1.60, one soda is 2.50 minus 1.60, okay let me just double check what that is on the calculator. I hope that you're doing this along with me. 2.50 minus 1.60, 90 cents. Okay so we're going to say the soda is 90 cents. Now the question is does this work in the problem? If you have three slices of pizza each costing $1.60 and two sodas each costing 90 cents, does that give you 6.75 when you combine them? Let's take a look. So we have three slices of pizza, so 3 times 1.60, right that would be your three slices plus two times the 90 cents sodas. What does that equal? We want it to be 6.75. So we've got 3 times 1.60, that's 4.80, okay. 4.80 plus 2 times .90 is about $1.80 so that equals $6.60. Right 6.60, what does this tell us? We know that C is not correct because we need it to equal 6.75.But we know a bunch of other things too. We know we need the pizza to cost more, we need to end up with a larger total.
Now let's head to the answer choices. So we know 6.60 was too small and that tells us a lot about A and B as well. They're definitely going to be too small right? A 1.25 way too small, 1.50 way too small and we know we just tried $1.60 doesn't work either. So what we will do next usually is head to choice D, then if that won't work we would know it have to be E. But take a look, when we talked about pacing strategies for the general ACT we talked about how often there's one completely ridiculous answer that you should cross off when you strategically guess. E here is just ridiculous, there's no way our pizza cost $180. So actually in this case we know that our answer choice has to be D. But if you wanted to double check you could put it back in the problem like we did here and see. And you would see that if your pizza cost $1.75 your total, which are three pizzas and your sodas would be 6.75.
Moving on to some great guessing strategies for the ACT. This will help you if you're really, really stuck and you're just thinking okay how can I make an educated and strategic guess here. Let's look at some really good strategies for that. First, avoid misfits. Oddball answer choices are incorrect over 90 percent of the time. What do I mean by oddball? Oddball are the answer choices that just look completely different than the other four. You know my daughter watches Sesame Street, there's that song you know, one of this things is not like the other, that's what you're looking for. Here, only one of this answer choices has a pie in it so already we know even if we didn't actually look at this problem, this isn't going to be the correct answer, I can cross it off. Same thing when you see you know everything is a whole number, only one of the answer choices is a fraction. Well everything is positive and only one of them is negative or everything's real, one of them is imaginary, any time you've got this misfit, cross it don't pick it.
Next all diagrams are drawn to scale so, eyeball them. On the ACT in the instructions for the Math they tell you the diagrams are not drawn to scale, actually they are. Okay pretty sneaky, so if you're ever stuck and you're thinking, "Oh my God I just don't know what to do here," just eyeball and just estimate what the distances should be based on the information they give you. Let's take a look in the example. Here we've got; in the figure below, actually it's above. ABCD is a square and EFG and H are midpoints of its sides. If AB is 12 inches, what's the perimeter of EFGH in inches?
Okay So we want the perimeter of this and perimeter which we will review when we talk about the geometry, everything added up. So let's see we don't have any information yet about this though any lengths of these sides. All we know is that AB here is equal to 12, perfect. Okay plus all we know, I'm not saying we're in a rush or we don't know how to solve it. Okay well if this is about 12, what's the thing about it, about how long do we think one of these sides looks like. Okay so probably we know this is the midpoint here, so you'll think this is 6 right or even just by eyeballing looks like it's cutting it up this is 6 right over here. You know you would estimate, so what about this it looks like it's slightly longer than these right? So not 6 maybe 7 maybe 8. So let's approximate, let's say okay this looks like it's about 8 right here. Okay and that's all we will do. So okay so this is 8 right over here. So in that case, great we want the perimeter. Perimeter everything added up 8 plus 8 plus 8 plus 8 or 8 times 4 so it's 32. So we're going to say the perimeter equals 32.
Okay let's head to the answer choices. One of these is going to be very close to 32 and you see here nothing obvious it's going to be one of these ones where you have to calculate the radical. Let's try B and see what we got. So if you're calculating and you say okay what's square root of radical 2? About 1.4 you can do it on your calculator, so let's say I said 1.4 times 24, what does that give me? That gives me 33.6 very close to 32 and I've got it, nothing else is remotely close, very neat. So that's how you can sometimes solve this geometry questions even without knowing what's going on, just by eyeballing the diagram. So again we know that B is the correct answer choice here.
Guessing strategy number 3. Never pick an answer that repeats a number already in the problem. And I want to add to this also never pick a number that it's just a really fast easy desperate combination of the numbers in the problem. Let me show you what I mean, here we did this earlier and we got it right algebraically. But look if you had 10 and 20 here, you already know when you choose the answers, it's not going to be 20 percent because it's already in the question. Also they think you're desperate they think you're going to be in a rush and you'll think 10 and 20 then that makes 30 that must be right. So you can bet that because it's an easy combination of the numbers in the problem, 20 well 30 is not going to be right and 20 because it appears also incorrect. At this point you can really do a great job strategically guessing even if you didn't want to do this problem algebraically, because we know if something is increasing by 10 percent and then another 20 percent, it's probably not just increasing by 12 or by 15 percent right? So we can cross those off too and if you were just doing this by strategic guessing you would get the right answer too that it was E, 32 percent.
Let's go on to the last guessing strategy. Remember 5 answer choices instead of 4 so always try to eliminate something. So not really a particular strategy, but just something to keep in mind. The rest of the test, four answer choices on the Math, you've got five. So really be good about eliminating before you guess. It's not a great feeling to get through the Math and you've got some left and you just have to blindly guess on them because you're stuck. See if you can somehow shift your time around, so if you've got ones that you're not going to tackle you leave enough time so you can strategically guess. That's it for some great guess strategies, we talked about a bunch of things that you can do to help you guess strategically even if you're not sure what's actually going on in the question.
Let's recap this episode. First we talked about pacing strategies, now how to pace yourself properly so that you have enough time. We also talked about how it's pretty cool that you can get a third of the questions wrong and still get a good score. We talked about sneaky problem solving strategies, ideally you really want to just know how to solve a problem and we will go over all the main skills that you need to know. But if you're stuck we talked about some great strategies that can help you get the answer even if you're not sure what's going on in the problem. And last we talked about guessing strategies, if you're really, really stuck we talked about how to eliminate some of the answer choices so that you can guess strategically and statistically have a better chance of getting the right answer.
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