Welcome back to my series on Math Etiquette—designed to help you ace the SAT No Calculator Math, a section that strikes fear into the hearts of many students. (Mastering Math Etiquette will also help you on the ACT!) My first post noted that to get the score you deserve on this part of the SAT, you’ve got to have a fundamental grasp of how numbers work.
The first part of that process is the topic of today’s article. Time is, of course, of the essence while you’re actually taking the SAT or ACT, and so you’ve got to avoid re-creating the mathematical wheel. Today’s method for doing so concerns a finite amount of arithmetic that you simply need to commit to memory. With these math facts in your back pocket, you can avoid wasting precious time on scratch work (or on typing into your calculator, if you’re taking the ACT).
For your studying pleasure, I’ve organized these math facts by subject. I strongly recommend you make them into flashcards, write them out by hand, or use digital tools (like Quizlet) to quiz yourself until the below info feels second nature.
1) Multiplication tables up to 12
Let’s begin from the assumption that you’ve got your addition and subtraction facts down pat already. (If not, drilling those is Step 1 for you!)
Also, you’ll notice that I haven’t mentioned “Division” in the title of this section. That’s because division and multiplication are two sides of the same coin—literal inverses of one another. So, that’s good news: because once you know your times tables up to twelve, you also know how to divide down from 144!
2) Perfect Squares up to 25
Squaring numbers comes up in tons of topics, from calculating distance formula, to executing Pythagorean theorems; from FOIL-ing binomials to doing the law of cosines. Give a gift to your future self by etching these deep into your brain now:
02 = 0
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
212 = 441
222 = 484
232 = 529
242 = 576
252 = 625
3) Perfect Cubes up to 6
03 = 0
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216, and
103 = 1,000 for good measure.
4) Powers of 2 up to 10
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32 (Memory tip: 3+2 = 5, so 32 = 25)
26 = 64
27 = 128 (Memory tip: 28 is a multiple of 7, so 27 = 128)
28 = 256
29 = 512
210 = 1,024 (Memory tip: 210 = 1024)
While you work on memorizing this stuff, I recommend you find what I call an “anchor point”—a fact that you can just remember immediately, and can use to catapult yourself to deducing the others quickly. For powers of 2, it’s not necessary for me to know flat that 29 = 512. Instead, 210 = 1,024 is one of my anchor points. So if I’m trying to find 29, I know it’s the power of 2 that comes before 1,024…hence, referring to my memory of the pattern, it’s 512.
5) Pythagorean Triples
A Pythagorean triple is a set of three integers that make Pythagoras’s theorem true. (If you who need a refresh on Pythagoras’s theorem, it’s A2 + B2 = C2.) The important word there is "integers": so instead of getting a gnarly radical for one of the numbers, they’re all nice and round. There are a few groups of numbers that you should just memorize, so you don’t have to actually do the Pythagorean math each time you see a right triangle (or a distance formula question). Here are the key ones to know:
3 – 4 – 5
(And its multiples! Like 6 – 8 – 10, 15 – 20 – 25, and 30 – 40 – 50, etc.)
5 – 12 – 13
(This one sometimes comes as 10 – 24 – 26, which is the original doubled.)
8 – 15 – 17
7 – 24 – 25
6) Fraction to Decimal to Percent Conversions
This is a principle that my inner (and outer) math nerd finds genuinely beautiful. Once you grasp that each number can be written in three different ways, your life gets SO much simpler. You gain flexibility when solving problems, because you can use whichever form is going to make that particular problem easier. In addition, you’ll now be able to choose the right answer even if you solved the problem as a fraction but the answer choices are all written as decimals.
1/11 --> 0.0909… --> 9.09…%
For 11ths, take the numerator and multiply by 9. Use two digits. Repeat!
1/10 --> 0.1 --> 10%
1/9 --> 0.111… --> 11.1…%
For 9ths, take the numerator and repeat!
1/8 --> 0.125 --> 12.5%
1/6 --> 0.1666… --> 16.66…%
2/11 --> 0.1818… --> 18.18…%
2/10 = 1/5 --> 0.2 --> 20%
2/9 --> 0.222… --> 22.2…%
2/8 = 1/4 --> 0.25 --> 25%
3/11 --> 0.2727… --> 27.27…%
3/10 --> 0.3 --> 30%
3/9 = 2/6 = 1/3 --> 0.333… --> 33.3…%
4/11 --> 0.3636… --> 36.36…%
3/8 --> 0.375 --> 37.5%
4/10 = 2/5 --> 0.4 --> 40%
4/9 --> 0.444… --> 44.4…%
5/11 --> 0.4545… --> 45.45…%
5/10 = 4/8 = 3/6 = 1/2 --> 0.5 --> 50%
6/11 --> 0.5454… --> 54.54…%
5/9 --> 0.555… --> 55.5…%
6/10 = 3/5 --> 0.6 --> 60%
5/8 --> 0.625 --> 62.5%
7/11 --> 0.6363… --> 63.63…%
6/9 = 4/6 = 2/3 --> 0.666… --> 66.6…%
7/10 --> 0.7 --> 70%
8/11 --> 0.7272… --> 72.72…%
6/8 = 3/4 --> 0.75 --> 75%
7/9 --> 0.777… --> 77.7…%
8/10 = 4/5 --> 0.8 --> 80%
9/11 --> 0.8181… --> 81.81…%
5/6 --> 0.8333… --> 83.33…%
7/8 --> 0.875 --> 87.5%
8/9 --> 0.888… --> 88.8…%
9/10 --> 0.9 --> 90%
10/11 --> 0.9090… --> 90.90…%
Bonus pro-tip: when you’re calculating by hand, fractions tend to be easier to work with. Conversely, when you’re using a calculator, decimals tend to be easier.
Phew! that was a LOT.
I know I just served you up a BIG plate full of numbers. But I’ll promise you what I always promise my one-on-one tutoring clients before we embark on an arithmetic odyssey together: when you break this content down and study it in sections, it really does become manageable. Plus, the incentive is seriously so sweet: because when you know these facts like the back of your hand, test math (and everyday mental math, too) gets SO much easier. Indeed, I’ve been teaching SAT and ACT prep for ages now—well over a decade!—and this stuff has remained on the test—in spades. In other words, the fundamentals will never go out of style. Memorizing them will save you stress, work, and time on your math sections, and that means a higher score with less angst. Definitely worth the start-up energy of a few weeks’ worth of drills. So pour yourself a cup of coffee and make those flashcards, okay?
