What would an A+ student who has just finished Algebra I and Geometry score on the SAT and ACT if she got everything right that she had seen in school and got everything else wrong?

According to my analysis, she would score a 29 on ACT Math (80% of the tested material) and a 700 on SAT Math (87%). And what if you added five high-yield advanced topics? Then she’d score a 31 on ACT Math (85%) and a 780 on SAT Math (96%).

I want to emphasize right away that this does not mean that the SAT is a better test, that all students should choose the SAT, or even that the SAT math is an easier target for all students. There are many other considerations, including the student’s performance on other sections, how quickly they can learn new concepts, and how flexible they are with the math knowledge they have, to name just a few.

How The Analysis Was Done

I tagged every question from the last 10 SAT QASs and ACT TIRs. Then I identified the tags related to concepts that are generally not taught before 10th grade – things like ellipses, logarithms, matrices, vectors. I eliminated any question with one of those tags. The remaining questions were counted as ones that the student would get right. I then ran this analysis again, adding the following five topics to the list of ‘known’ concepts: SOHCAHTOA, special right triangles, exponential growth, fractional exponents, and circle equations. To produce the scores, I found the average scaled score that corresponded to a certain percent correct. For example, over the last 10 ACT Math sections, 80% correct has corresponded to an average score of 29.

Observations

1. Students really know a lot more SAT math material at the end of 9th grade. This is not too surprising, but it’s interesting to see the actual numbers.
2. You can get a very high score on the SAT math section if you deeply understand what you’ve learned. 700 is within reach, and you can jump up to 780 by learning a handful of new concepts. The ACT math scores you can get by knowing these concepts are much lower, no matter what concordance table you use.
3. Teaching new topics has a much smaller effect on the ACT math section. You can get to 96% coverage pretty quickly on the SAT – a 9% jump just by teaching five topics. But those same topics only get you from 80% to 85% on the ACT math material. And the last 15% is even harder to cover. By my count, there are at least 50 topics that comprise that last 15%, and they’re not trivial: the law of sines, binomial expansion, matrix multiplication, log rules, etc.

Caveats

1. I recognize that it is unlikely that a student would get everything right that they know from school and nothing right that they haven’t learned in school. Students make small mistakes, get lucky, pick numbers, use their calculators, and can make good educated guesses.
2. While the scope of the SAT math section is more limited, many of the hard questions require a greater depth of knowledge. So it’s possible that a student might get an ACT matrix addition question right (even though it’s never been covered in school) and miss a ‘discriminant’ SAT question (even though it had been covered in school).
3. A student’s placement on the curve matters a lot. At the very top, you can afford to miss 4 questions on ACT Math and still get a 35; on SAT Math, 4 misses would drop you at least 40 points. However, getting three more questions right when you’re scoring a 29 won’t help you as much as they would if you were at a 31 (generally a 2 point increase instead of a 3 point increase).
4. Some of these gaps will have closed by the time most students start. By the end of 10th grade, some students actually have seen 95% of the ACT’s topics. If they struggle with the depth of math on the SAT, the ACT math will likely be easier for them.

Takeaways

1. From a 31 or 700 starting point, it’s generally going to be a lot easier to get to a 750 than it is to get to a 34. There are so many concepts that can appear on the ACT’s math section, and it’s very hard to know which ones the student knows or could learn easily. Once they’ve mastered the pre-10th material, you could easily find yourself playing whack-a-mole with session and homework time. The student is already familiar with most of the SAT math topics, which makes it easier to identify high-priority targets. Also, the math score is worth twice as much on the SAT – even if it’s the same amount of work, you’ll get more out of it.
2. Delay spending lots of session time on ACT math as long as possible. Work on English problem types, Reading strategies, and Science time management. Have them take math practice tests and work through good question sets at their level, then quickly review the ones they missed. But try to get them to relearn (or learn) as much material on their own as you can, without spending session time on it. Why spend 10 minutes on periodic functions if they’re going to spend three weeks on it later in the year

Thanks to Vinny Madera, Blake Jensen, Michael Jordan, and Dave Lynch for their help! Please let me know what you think in the comments below.

Mike McGibbon has tutored in the NYC area for over 20 years. He’s also the CEO and co-founder of mathchops.com, an adaptive math practice tool.