Posts filed under: Math

Can you please share a list of topics that I need to master to achieve a 600 on the Math section? I do not think I have to learn all the topics, please give me as concrete and narrow list as possible. I think this would be useful for most of the people preparing for the test.

Apologies—I respect and honor your question but I cannot provide a concrete and narrow list. Hopefully you’ll find the answer that follows illuminating anyway.

You’re correct that you don’t need to learn all the topics to achieve a 600. What makes this question hard to answer generally is that you probably already know some topics pretty well, and others with the same goal might know different topics pretty well. Further complicating matters is that all topics are tested at a range of difficulty levels; even folks who have a pretty good mastery of parabola questions might occasionally miss a very hard parabola question.

A few months ago I published this post, which I think is important an useful in approaching your question. What it tells you is that, based on previous scoring tables, you’re probably going to need 36 correct answers to get a score of 600 (on the most forgiving scoring table you’d need 32 correct; on the most punishing scoring table you’d need 40 correct). So the real question for you is: what’s your path to about 40 correct (or what 18 questions can you ignore)?

There are 58 math questions on the SAT, broken out thusly:

  • 19 Heart of Algebra
    • ~12 Translating between Words and Math
    • ~9 Algebraic Manipulation
    • ~9 Lines or Systems of Linear Equations
  • 16 Passport to Advanced Math
    • ~3 Functions
    • ~3 Exponents and Exponential Functions
    • ~3 Quadratics, Binomial Squares & Difference of Two Squares
    • ~2 Parabolas
    • ~2 Polynomials
  • 17 Problem Solving & Data Analysis
    • ~9 Data Analysis
    • ~5 Ratios & Proportionality
    • ~3 Percents & Percent Change
    • ~2 Measures of Central Tendency and Variability
    • ~1 Designing and Interpreting Experiments and Studies
  • 6 Additional Topics in Math
    • ~2 Angles, Triangles, and Polygons
    • ~2 Circles
    • ~1 Right Triangles
    • ~1 Complex Numbers

To be clear, the indented, not bolded bullets represent my own subcategorizations and rough frequency calculations, while the main, bolded bullets represent College Board’s broad categories and official distributions. My categorizations don’t always line up perfectly with College Board’s. For example, I often assign a question multiple categories: a question can be both “Translating between Words and Math” and “Systems of Linear Equations”, not all “Translating…” questions are “Heart of Algebra”, many questions categorized as “Data Analysis” also include another topic, etc.

So anyway, while you could theoretically ignore Heart of Algebra, sweep everything else and still have a good shot at 600, that’s probably not your path. If you’re considering taking this test at all then you’ve probably already got some of the Heart of Algebra content locked down. You might, however, look at the list above and decide that you can safely avoid studying polynomials, exponents, and any geometry, if you hate those topics. Even with those exclusions, you’d have a pretty good cushion.

My real recommendation is to take a practice test or two and analyze your mistakes. Either use the tables at the back of my book to categorize your missed questions or make your own best judgments. Then use the list above to judge whether you afford to continue making that kind of mistake.

Most people spend the majority of their test prep time attempting to master content. This is a good thing! Without content knowledge, you’re in trouble. However, if you want to set yourself up for success, you should also be devoting some time to learning the rules of the game—you can’t develop effective strategy until you know the rules! One of the most important rules of any game is how the scoring works.

Below is a summary of the math scoring tables from the 8 official practice tests, which are a pretty good representative sample. You can see the highest, lowest, and “probable” (average over the 8 tests, rounded to the nearest 10) scaled score each raw score receives on the Official 8.

There are a few good use cases for this. First, you may know that you need to hit a certain score in order to qualify for something (a scholarship, a summer studies program, etc.). Knowing how many you need to get right to get there can help you strategize about which topics to focus on and which to ignore.

I expect people will also use this to speculate about how they might have done after tests (e.g., “I’m pretty sure I only got 5 wrong and I answered everything else—what might my score be?”).

Now that the Daily PWN email list has been going for a while and I’ve got some good data on the questions, I thought I’d compile a list of the ones people are missing most frequently. If you’re looking for a quick skill sharpening on some tough problems, why not give these a try?

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Here’s another Proving Grounds installment! The aim of the following five-question quiz is to work your graphing calculator muscles, so my recommendation is that you try to solve them by graphing even if your first inclination would be to solve them another way. My solutions for this drill will be entirely calculator-based; spend enough time…

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It’s been a while since we did one of these! The following five-question quiz (all about a histogram, by special request) will be available to everyone for one week, and then it will only be available to registered Math Guide Owners. (If you don’t have a Math Guide, now is a pretty good time to…

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This is not really SAT specific or even particularly SAT useful, but I made the video above to help you create a basic quadratic formula program for your TI graphing calculator if that’s a thing you’d like to do.

If you’ve programmed things into your calculator before and don’t feel like watching a whole video, you can also just enter the program below. Make sure you’re careful with your quotation marks and parentheses, and always test the program with multiple quadratics to make sure it’s always giving you correct answers before you use it for anything important.

:Disp "AX2+BX+C=0"
:Prompt A
:Prompt B
:Prompt C
:Disp "ROOTS:"
:Disp (-B+√(B2-4AC))/(2A)
:Disp (-B—√(B2-4AC))/(2A)

 

The new SAT requires you to know a number of special equation forms—to know which one you need to use in a given situation, and to know how to get into that form if it’s not the one you’re given by using algebraic manipulation. Some equation forms (vertex form of a parabola and the standard circle equation immediately spring to mind) contain binomial squares, e.g. (x+1)^2, as essential ingredients. To get a non-standard equation into these forms, you’ll often have to complete the square. I know, I know, you’ve done this a million times in school. Still, I often find students haven’t done this in a long time and need a little bit of a refresher. So here we are.

First, the equations in question.

Vertex form of a parabola: y=a(x-h)^2+k, where the vertex of the parabola is at (h,k).

Standard circle equation: (x-h)^2+(y-k)^2=r^2, where a circle with radius r has its center at (h,k).

Say you’re given a parabola that’s not in vertex form and you need to put it in vertex form. How do you do that?

No calculator; grid-in

y=x^2-8x+6

The parabola formed when the equation above is graphed in the xy-plane has its vertex at (a,b). What is the value of a-b ?

Completing the square isn’t the only way to solve this question, but I’d argue it’s the fastest. All we need to do to go from the given form to the vertex form is figure out which binomial square the x^2-8x part of the equation is the beginning of. With practice, this becomes second nature and you probably won’t need the rule, but the rule is that x^2+b is the beginning of \left(x+\dfrac{b}{2}\right)^2.* In this case, that means that x^2-8x is the beginning of (x-4)^2.

Now, what do you get when you FOIL out (x-4)^2? You get x^2-8x+16. That’s not what we have above—we have x^2-8x+6 instead. Luckily, we can do anything we want to the right side of the equation provided that we keep the equation balanced by doing the same thing to the left, so we can just add 10 to both sides!

y=x^2-8x+6

y+10=x^2-8x+6+10

y+10=x^2-8x+16

From there, we’re almost done. Now we can convert the right side to the binomial square we wanted, and then get y by itself again to land in vertex form.

y+10=(x-4)^2

y=(x-4)^2-10

So, there you have it: the parabola in question has a vertex of (4,-10). Since the question said the vertex was at (a,b), we know that a=4, b=-10, and a-b=4-(-10)=14. So, 14 is the answer.

Let’s practice with a few more, shall we? Try to do the following drill without a calculator. All three questions are grid-ins.

1.
y=x^2-12x+33

The parabola formed when the equation above is graphed in the xy-plane has its vertex at (a,b). What is the value of a+b ?

Question 1 of 3

2. When the equation y^2=(x+3)(-x+5) is graphed in the xy-plane, it forms a circle. What is x-coordinate of the center of the circle?

Question 2 of 3

3. What is the radius of the circle with equation x^2+y^2+6x-10y=2 ?

Question 3 of 3


 

 
 
 
 
 
 
* I’m intentionally limiting this post to scenarios where the leading coefficient in the square being completed is 1. So far, I have not seen an official question of this type where that is not the case.

I’m back after a hiatus with another Proving Grounds Quiz. Usual Proving Grounds rules apply: this quiz is open to everyone for a week, but then it’s only open to Math Guide owners. Good luck! *Data source: City of Bridgeport Office of Policy and Management. Accessed 2015-06-14 at http://www.bridgeportct.gov/content/89019/96401/default.aspx…

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I’m back with another Proving Grounds quiz. These quizzes are available to everyone for one week, and then they’re only available to Math Guide owners. Want to join the swelling ranks of the PWN Army of Math Guide owners? You can buy the guide directly from me through the PWN store, or grab it on…

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As I did for the last iteration of the SAT, I’ve been collecting the explanations I write on my Q&A sites for Official Test questions in a Google Spreadsheet for easy reference. The new test is still new, so I haven’t been asked MOST of the questions yet, but I figure it’s time to get this page out into the world. If you’re working through the official SAT practice tests and you have a sneaking suspicion that the official explanation is unnecessarily complicated, well, then here’s a way to get a second opinion.

PS: Download the Official Tests here.

Last week was a no-calculator installment of the Proving Grounds—this week it’s all grid-ins! Remember, if you want to access previous Proving Grounds quizzes, or if you want to be able to access this and future ones after they’ve been up for a week, all you have to do is be a Math Guide owner. You…

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Another Proving Grounds quiz coming your way. This one you should do without your calculator. Remember, Proving Grounds quizzes are available to everyone for one week, and then only available to Math Guide owners. Not a Math Guide owner yet? Got $20? :)…

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Time for another Proving Grounds installment, folks. Remember, these quizzes are available for one week for everybody, and then they’re for Math Guide owners only. “How can I get to be a Math Guide owner?” you ask?! Well, either you buy one right from me, or you forward me your receipt from some place like…

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Here’s the second installment of the Proving Grounds quizzes, which are available for everyone to try for one week before they become exclusive to Math Guide owners. A little parabola-heavy this week, but then again, who doesn’t love parabolas? Good luck! Mechanical Calculators image By Ezrdr (Own work) [CC BY-SA 3.0], via Wikimedia Commons…

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Below is the first in a series of tough quizzes for the new SAT that will be available to everyone for a week, and then only available to Math Guide owners. If you want access to all these quizzes long term, well, you might want to grab a Math Guide. They’re on sale now, and…

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