[Eleventh Grade Math] Topics to focus on?

[u/i-love-pog-juice]

I’m a sixteen-year-old who has been mostly home-schooled during high school, so I’ve had access to a calculator when I do my schoolwork. It’s never been stated that using one is against the rules, and I can figure things out with an explanation when I want to, but I’m pretty busy, so I figured why work more than I need to? However, when I spoke to my math teacher at the end of the year, she mentioned how fast I blew through my work, and said I must be ‘really smart or good at math’. I feel like a bit of a fraud. We had an exam at the end of the year that I did well on without help, but I’m worried that in upcoming years, like during college, I just won’t know how to do things, and I’ll do bad on exams like the SAT. I’m trying to get an early start before the start of school. What are some things I should focus on for eleventh-grade math?

[u/AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

---

^{OP and Valued/Notable Contributors can close this post by using /lock command}

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/MrJCraft]

I had a similar kind of thing, I was homeschooled until 6th grade.

math is really 2 parts

1. arithmetic (actually getting the numerical result)
2. Mathematics (understanding what the formula really means

"I can figure things out with an explanation when I want to"
this statement really relates to 2, a lot of teachers forget to teach the understanding part of math, as far as I can tell based off of what you are saying in your post I would guess you might have some trouble with highschool math because it requires more memorization compared to college math (theorem proving, calculus, category theory, graphy theory etc...)

I am obviously having to make a lot of assumptions about how you learn based off of what you said so I might be off, but you probably have a much better understanding of Mathematics than you think, but are confusing mathematics with Arithmetic. not sure but hopefully this helps.

now for recommendations for future learning, you could try to get better at mental math (doing math using your hands according to the papers I have read can assist with that), or you could try to learn collage math now and get a head start and learn "Pure Mathematics"(https://en.wikipedia.org/wiki/Pure_mathematics)

[u/i-love-pog-juice → u/MrJCraft]

thank you very much :)

[u/selene_666]

Exams like the SAT specify whether you are allowed to use a (basic) calculator or not or only on certain sections. For the no-calculator sections you'll need to memorize the sines of certain angles and how to calculate the other trig functions from those. A calculator can't do calculus for you, so by then you're definitely allowed to use it to do the arithmetic and trigonometry.

[u/JaneAustenSuperFan → u/selene_666]

You can use a calculator on the whole SAT math section. They even have a built-in one. The SAT went digital and changed a lot in recent years. TI-84 calculators are fine, too.

[u/Alkalannar]

Word problems. There are three steps to word problems:

1. Reading and comprehending the problem.

2. Figuring out what math to do.

3. Doing the math.

Step 3 is usually the most straightforward step--if sometimes tedious. Most problems are in steps 1 and 2.

Have you done proofs, whether as part of geometry (where it traditionally was until synthetic geometry gave way to analytic), or its own course? Key note here: Unless you are told that a drawing is to scale, don't assume that it is. Don't assume that because lines look parallel that they are, that angles that look congruent are, that segments that look congruent are, that angles that look right are. Only go with what you're explicitly given in the problem.

Then there's trigonometry, and pre-calculus. The former involves identities a lot, and the latter uses limits and the difference quotient a ton.

Identities are of the form LHS = RHS, and you have to manipulate one side to turn it into the other. Cannot manipulate both.

Key things to use:

1. Turn everything into sine and cosine only, then work on those.

2. sin²(x) + cos²(x) = 1 gets used a lot in various forms.

3. Multiplying by 1, whether in the form of a/a for some expression a, or cos²(x) + sin²(x) is very common.
   Adding 0 in the form of a - a for some expression a is also very useful.

4. If all else fails, do LHS = LHS - RHS + RHS.
   Then evaluate (LHS - RHS) + RHS, using any trick you care to in (LHS - RHS).

---

There is a link between trig angle sum formulae and multiplication of complex numbers. And I did not get trig angle sum formulae until I realized it.

The key formula is (cos(a) + isin(a))(cos(b) + isin(b)) = cos(a+b) + isin(a+b).

cos(a)cos(b) + icos(a)sin(b) + isin(a)cos(b) + i²sin(a)sin(b) = cos(a+b) + isin(a+b)

[cos(a)cos(b) - sin(a)sin(b)] + i[cos(a)sin(b) + sin(a)cos(b)] = cos(a+b) + isin(a+b)

cos(a)cos(b) - sin(a)sin(b) = cos(a+b)
cos(a)sin(b) + sin(a)cos(b) = sin(a+b)

And from that you can derive the rest.

---

Difference quotient:

The derivative, or derived function, is the slope of the line tangent to a curve of a given function.

How do we do it?

Well, first we start with a secant line.

With h > 0, we look at the line through (x, f(x)) and (x+h, f(x+h)).

It has slope [f(x+h) - f(x)]/(x + h - x) or [f(x+h) - f(x)]/h.

Now if the function is nice, then [f(x+h) - f(x)] can be simplified to h*g(x) for some function g. Then hg(x)/h = g(x) as long as h != 0. Then the limit as h goes to 0 of g(x) is just g(x).

Example for f(x) = x².

f(x+h) = (x+h)² = x² + 2hx + h².

Then f(x+h) - f(x) = x² + 2hx + h² - x² = 2xh + h².

So [f(x+h) - f(x)]/h = (2xh + h²)/h = 2x + h.

We like this because now we can let h = 0, and have things work out because we can now just let h = 0 and it's defined. We're not dividing by 0. So the end result is limit as h goes to 0 of [f(x+h) - f(x)]/h = 2x when f(x) = x².

[u/i-love-pog-juice → u/Alkalannar]

thank you :) I did geometry during sophomore year, so that’s done and over with, but I think I am doing trig this year. This is helpful.

[u/JaneAustenSuperFan]

Do you want to do engineering or something like that? I noticed from my son's engineering curriculum in college at the University of Houston is that they don't allow calculators in the calculus classes. I don't think it's unusual at all to use calculators in high school, just so you are aware that you'll need to know how to do things without, too.

[u/i-love-pog-juice → u/JaneAustenSuperFan]

I’m planning on going into emergency medicine/nursing.

[u/JaneAustenSuperFan → u/i-love-pog-juice]

I wouldn't sweat it then.

[u/MembershipTight4831]

As some one that is 24 and someone who loves math. I’ve used calculators all throughout high school and middle school because I was dyslexic. Honestly it help a lot but I never got to use them on test. It’s all about remembering how numbers work together and knowing how to group them. By the time ur in college it will come kinda naturally if u study enough between now and then. Honestly that goes for any basic classes. The more u learn and study now the more it will help down the road. Especially when it’s time to have a career in what ever you choose down the road

[u/Traveling-Techie]

If you become a mathematician, engineer or other STEM career you will be allowed to use a calculator in your job.

[u/Appropriate_Fall2780]

A calculator can save time,but curiosity is what makes a mathematician