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# Ultimate Guide to the ACT: Math Section

Krish Gupta

Caroline Koffke

### ACT🎒

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## ACT Prep: Math Section

### What are the ACT Sections?

When you sit down to take the ACT exam 📚, you will have 4-5 sections on the test 👓 , depending on whether you choose to take the essay section! For an overview on the ACT exam, read this article!
The sections are:
1. English (aka the grammar section) 🤓
2. Math 🧮
4. Science 🧬
5. Writing/Essay (OPTIONAL ⚠️) 🖊
After the English 🤓 section of the test, you'll dive right in to the Math 🧮 section of the ACT! In this study guide, you will learn to MASTER 🤴🏼 the Math section and be on your way to an amazing score!

## Resources:

### ⏲️ Logistics and Types of Questions

The Math section is the 2nd section of the ACT standardized test. You will answer 60 questions to answer in 60 minutes ⏰.
This comes out to be 60 seconds 🕰 or 1 minute per question. It's safe to say that this section will be fast-paced 🏃‍♀️.

### General ACT Strategies 🧰

Since you're going to be taking the entirety of the ACT exam 📚, we thought it would be beneficial to go over some overall tips and advice for the exam. These tools 🛠 and strategies 👨‍🔬 have worked in the past and have been promoted by many teachers 👩‍🏫. Make sure you pay close attention to common mistakes ⚠️ student make and how you can use these strategies to get the score you want.

Speed is King 💨

The ACT is written in a way that gives you barely enough time to finish every single question if you work with maximum efficiency and minimum distractions 🥵 . Many students say that the hardest aspect of ACT is how fast-paced the exam is. Many students struggle to finish the STEM sections of the ACT, particularly 🧮. So before you even take the test, be in the mindset that you cannot afford to zone out, even for a few minutes.
There are many exercises like meditation 🧘‍♀️ , slow breathing and simply closing your eyes 👀 that can calm your nerves and help you get in the zone for the exam.
The best way to overcome the timing hurdle is to practice planning and pacing. Practice and plan 🤔 how much time you are going to take on the first half and the second half beforehand (you can divide it up any way you please) 🧠.
Pacing is the most effective tool 🏃‍♀️ . After every 10 questions or so look up at the clock and see how you are doing time-wise ⏰. If you are going too slow, then pick up speed. If you are doing well on time, continue on with the same momentum. This strategy allows for adjustment 😃 so that you can make quick changes to your approach and attempt all questions, and maximize your score.

Courtesy of GIPHY

Working on the Weakness ⚠️

This strategy is relatively obvious: you need to work on your skills that need developing more often than ones you are already good at 💕. ACT covers a much bigger umbrella ☂ of math topics compared to SAT and many other standardized test exams. You need to know all math through and including trigonometry and pre-calculus, along with some basic understanding of statistics 🤯.
Many students find trigonometry tough because they have not worked with it for a long time 😆. So make sure you review some basic trigonometry quantities and identities before the test. Many students also find the geometry section a bit tough because they haven’t worked with geometry for some time 🧠. If there is a chance your geometry skills might be a bit rusty, make sure you take some time to polish them up.
💡To understand what you need to review, do a practice or diagnostic test, or look through the problems in this guide. Prioritize your review based on what you need help with the most so that you are not wasting time learning things you are already good at 🤔. If you get in a habit of practicing problems, finding your weakness, learning content, and repeating... nothing can stop you from your dream score. 😊

Courtesy of ActiveRain

Courtesy of GIPHY

### Math ACT Strategies ☮️

Let's get more specific and talk about the math section 🧮 . The tips and strategies discussed above can be applied to any section, including math. However, there are some extra strategies that can help you increase your math score 📈 even further. These strategies are tried and tested for overall test-taking and are especially helpful for the ACT Math section. So let's get started 💃!!

Close Enough Strategy 🤔

💡Sometimes just plugging in random numbers and making up variables can really help give you a feel for what the answer should be. Then, the process of elimination can help you find the correct answer ✅.

All problems in this guide are borrowed from ACT

D. You can eliminate ❌A and B because we are taking the absolute value of the final number, 🔢which can never be negative. This automatically gives you a 50% chance at the right answer! Solving the inside we get 4﻿ and 6﻿. The absolute value of 4-6﻿ is 2﻿. 💯

Do Easy Questions First ❤️

This strategy is also told to students all the time 👩‍🎓. However, many students just don't follow it. For some reason, students tend to love 💕 answering questions in the order they are in and forget that they have the option of skipping hard questions. 💡 The key 🔑 to standardized exams is to gain as many points 💰 as you can on the questions you know or the questions you can handle without wasting too much time 🕰 and energy 🔋 on questions that are too challenging.
One trick 🎩 some students use is to circle tougher questions when they skip them or guess on them so that they know to come back and visit them later. Not wasting all your time ⏰ on some insanely tough problem or on a question with a lot of repetitive arithmetic will allow you to save time and energy ✨ that you can spend on doing questions you feel better about. That being said, leave no questions blanks.
On the ACT, you are not penalized for a wrong answer. So take your best educated guess 🤔, and if you have no clue just bubble in your letter of choice for the day.

G. All you 🧒 have to do in this problem is plug it in 1/2 in for X and solve the expression. You end up getting 4-3 divided by 1/2 which comes out to 2. This is a relatively simple problems so when you're in rush 🏃 try to look for problems like this 😃 .

On the ACT you are allowed to use a calculator for the entire math section. Unlike other exams, where calculator use is either banned or only permitted partially, you can use a four-function, a scientific or a graphing calculator for the entirety of the exam✅. A graphing calculator is ideal because you might be dealing with some tricky questions regarding trigonometry or precalculus, and your calculator might be able to help you out 🥵 . However, a scientific calculator is just fine for most of the problems.
With that being said, you need to be extremely strategic with your calculator use. Many students don't prepare enough because they think they can rely on a calculator, but, keep in mind a calculator is a tool, not a magic wand. You still have around 1 minute to answer each question 🏃‍♀️. You will not have time to do each question by hand and check it with the calculator. Use your calculator as a strong and helpful tool. 💡Questions that seem complex but can be made easier by graphing are perfect for a calculator 🧮. When trying to do several arithmetic steps at once, your calculator can save your life!

E. The formula 🧮 to find periods for sin, cos, sec and csc is $2 \pi /b$. For tan and cot it is $\pi/b$. $b$ is the coefficient in front of the $x$ in the parenthesis. So in this case $b = 4$. Our period is $2\pi/4$ which simplifies to $\pi/2$ 🤯.
Let's say you were to forget how to do that. You could simply plug this equation into a graphing calculator and see how wide a full wave is. 🖥

### Tested Skills 👨‍🔬

The ACT writers are testing you on your skills as well as the math content 🤓. Each question tests some aspect of your math ability 💯. Some questions target problem solving, while others want to test your ability to understand concepts. Below, we break down each skill anticipated to be tested on the exam 😃.
You may notice that the percentages add up to more than 100%. That's not a mistake: the ACT has several questions that test more than one skill at once. Don't worry, this guide will prepare you for the exam and polish the skills you need to ace the ACT 💗.

1. Preparing for Higher Math (~60%) 👨‍🔬

This section 📚 is related to the recent math you've learned 🧠. Algebra and solving and manipulating equations are at the heart of this skill. You'll be working with all types of functions, equations and expressions to solve problems ❓. This skill is divided into 5 basic topics: algebra, function, geometry, number and quantity, and statistics and probability. We will learn more about each of these 5 topics within this skill in much more detail later‼️

B. This problem might make you reminiscent of the exponents and log units 🥵. But it's not that complicated 😇. Since the bases are the same, for the equation to be true, 2x+7=15. Simplifying that equation gives us x=4 💃.

2. Essential Skills (~40%) ❇️

This section is kind of a mixed bag of math skills 💼. You can expect to see anything from proportions 🍕 and percentages 🥧 to volume 📦 and surface area 🌍. A lot of the topics tested under this skill are those you've been learning since middle school. These are your fundamentals: the basics ✅.
This skill doesn't test what you've learned in the last couple of months but rather the skill set you've built in your math career. Keep in mind 🧠 that just because this skill is meant to test the basics, the questions tested 📖 won't always be a walk in the park 🌳. You'll often be asked to apply topics to a complex situation or a problem and perform multiple steps and parts.

C. 8% of 60 is 4.8. To find what number 4.8 is 1/5 of, we multiply 4.8*5 to get 24. Problems like these may test more basic content, but notice how there are many things going on at once 😴. The best way to approach them is to break them down and peel one layer of the onion at a time 🧅.

Modeling (~25%) 📊

This is a skill in its purest form 😇 . There isn't any particular content associated with modeling but you can be asked to model any situation 🤓 . These are some of the tougher real life 🌍 word problems. The best way to attack ✨ these problems is to break them down. You need to read each sentence and try to convert the English to math. You need to ask yourself if the sentence adds new information and what the point of the new information might be🕵️‍♀️ .
Commonly, modeling is tested in conjunction with setting up equations. You could be asked to model anything from the price of X pizzas 🍕 to the population of Wakanda after T years. This is the section where you get to show off how you can apply your skills to solve even the toughest of real-life problems.

H. We set up two equations here. One for the cost of ABC Book Club 📖 and one for the cost of Easy Book Club 📚 .
C_1=2x+40
C_2=3x + 35
If we want both costs to be the same (charges for both clubs to equal) we need to set both equations equal 😊 to each other and solve for x. Doing so gives us x=5.

### Content Breakdown 😃

The majority of the content that you'll be tested on falls under the “Preparing for Higher Math” skill. We will breakdown each topic you're anticipated to be tested on and how you should approach them 💯.

Algebra (12-15%) 🧮

You will have to prove mastery when dealing with variables and simplifying or solving for them.
• Expressions - You will be asked to solve, graph, model or interpret different types of expressions‼️
• Equations - You are expected to know commonly taught relationships 💗 including linear, polynomial, radical, exponential and logarithmic ones. You might also be asked to solve basic systems of equations. You might see basic trigonometry questions under here. So make sure to brush up on sines and cosines‼️

B. This problem requires careful reading even though the math 🧮 might seem easy. Solving the top equation for x gives us x=3/2 . Solving the bottom equation for y gives us y=6. Pay attention 🔍 that the question doesn't ask us for x or y. But rather x+y (the sum of the solutions) which comes out to be 7 and 1/2.

D. You learned how to calculate slope back in middle school. Change in X over change in Y is engraved in your brain 🧠 by now. Doing that who've a you -3/2. This is an easy problem compared to others you might see on the exam. This just goes to prove that you shouldn't waste your time 🕰 and energy on tougher questions because there are easy questions waiting to be answered. 😆

Functions (12-15%) 📈

Functions are at the core of high school math. You will need to know how to model, notate, represent or define many types of functions 💡. Linear, polynomial, radical, logarithmic and composite functions are some commonly tested concepts. You will also need to be able to look, translate and interpret functions graphically, not just algebraically 📉.

F. This is one of the tougher problems 🥵. Here you need to remember that we can tell how many solutions a quadratic has by looking at its determinant 🕵️‍♀️ . To have more than one solution, we need the determinant to be positive ✅.
But first, we need to turn these two equations into one quadratic. We can do this by substituting x^2 for y\$ in the second equation. This now gives us sx^2 + rx -t = 0.
A determinant is b^2 - 4ac where b=r, a =s, and c=-t. So our determinant becomes r^2 +4st which must be positive to gives us more than one solutions therefore the answer is F 💯.
This is a much tougher problem. So try your best to understand it but don't be hard on yourself if it takes you several tries.

F. This is another relatively easy question if you remember back to your Algebra 2 class. The 6 is inside the function so it moves left or right. And remember 🧠 that when it's negative the graph moves right and when positive, it's translates left (this is counter-intuitive at first so make sure to pay attention). 🤔
So since there's a -6 inside the function, it'll be translated right 6 coordinate units.

Geometry (12-15%) 📊

You need to be familiar with common solids and shapes. Concepts like similarity and congruence, especially with triangles, are often tested. You may also be asked about volume 📦 and surface area 🌍 . Trigonometry and conic sections are some of the tougher concepts that can be tested under this section.

CDA﻿ is also 40°﻿. We want ∠CAD﻿ and ∠ACD﻿ and ∠CDA﻿ to equal 180﻿ since they form a triangle. Which means ∠CAD+40°+57°=180°﻿. Which gives us that ∠CAD=83°﻿ 🌟.

D. One of the most famous and fundamental acronyms for math students is SOHCAHTOA.🧠 The acronym tells you how to find each trig function using which sides. For tan \theta we look at the TOA per which says tangent is Opposite/Adjacent. 😊
So we don't even need to do the calculation to find the hypotenuse because we don't need it. Opposite to the angle 🔍 we have 4 and 10 is adjacent. Hence, we get 4/10 or 2/5.

Number and Quantity (12-15%) 🔢

You need to know about different types of numbers and number systems like real, imaginary and complex. Make sure you can do arithmetic and algebra with complex and imaginary numbers since ACT writers love 💕 to ask about them. Also be able to work with vectors and basic matrices. If you did well in your Algebra 2 or Pre-calculus class, you will feel like this section is a breeze 💨.

i is -1﻿ and (-1)^2 is 1 then (i^2)^2﻿ which is the same thing as i^4 is also 1﻿ 🧠 . Makes sense? Even if it doesn't you can't try multiplying i﻿ by itself until you get 1﻿ and you'll see that every i^4﻿ you'll get it to be 1﻿ 🤔. Hence, for i^n = 1, n﻿ must be a multiple for 4﻿ which means when divided by 4﻿ the remainder must be 0﻿ 💃.

A. This is a problem where if you use formulas you can get the answer but it might be overkill. There are formulas for questions like these about arithmetic and geometric sequences and series but using logic gets us there faster and easier. If you notice we basically add 5 to get the next term in pattern since 18-13 is 5 and we have an arithmetic sequence. 💃
Which means if the 3rd term is 13, the 1st term must be 3. To get to the 50th term we will add 5, 49 times to 3. So the 50th term will be 3+49(5)=248. 🌟

Statistics and Probability (12-15%) 🎲

This section is relatively easy in terms of content, but many students miss questions ⚠️ on this section because the content is not taught in all high school classes 🤓. The ACT expects you to describe distributions and data representations in terms of center and their spread. They expect you to work with one variable and two variable data 📊. You are also supposed to understand basic probability and the ideas of a sample space and success 🎲. Basically you'll need to look at data in tables and graphs. Also you'll expected to do some intuitive probability problems like you might've done in an Algebra class.
This content isn’t too bad but not many high schoolers take a statistics class and learn a little about this unit in their Algebra or Precalculus class 📈. So make sure to review content especially for this topic.

### Content Review Sources 🕵️‍♂️

The ACT tests all of middle school and high school math 🥵 . Unfortunately, we can only briefly review in this one guide. But don’t worry 😇. We have compiled a few resources that are great for more targeted content review. Read the positives and negatives of each source and judge for yourself what you need to review more 😊.
Fiveable has guides for each section of the ACT. To score well on the ACT, you need to do well in all sections 😝 . We have great writers who have provided testing strategies and content review along with curated resources in each guide 🔥 . Our blog posts as well will help you find resources and answer many of the common questions and concerns students have 😆. Check out these articles that tell you all you need to know about each ACT Subject!

## Resources:

This is the holy grail 🍷. This book has everything you need to know about the ACT. The book is released by the exact organization that makes the exams 📖 . It has practice problems, content review, strategies, and everything you could ask for from a standardized test prep book. Some people say the book is kind of long 🥵. Indeed it is a thick collection of pages 📚 . But that is because the writers have put in basically everything you need to succeed on the exam.
So if you do buy the book, 💭 make sure you make a reasonable plan to get through it. If you buy the book two weeks before the exam, you won't be able to get through many chapters 😓. Overall, we suggest this book as a great investment 💵.
This YouTube has a great collection of math videos directly targeting the ACT 😃. He is very thorough and leads you through how to solve each problem carefully. His teaching style is one that uses a lot of practice problems ‼️So if you're looking for practice to do, this YouTube channel is a goldmine for you 💰!
Khan Academy is used by a lot of students 👩‍🎓 worldwide. The content is tried and tested and speaks for itself. However, Khan has not made a course of a playlist specifically for the ACT 😔. So Khan Academy is better if you want to review a particular topic or relearn it 😃.
This was just a short list of amazing resources, for a more expansive list, check out these best review sources for the ACT!

## Resources:

### More Practice Problems 💰

Need some sample questions? Because these questions are highly dependent on the format 💫, we recommend you check out some full-length tests 📖. Here are a few links:
We chose these 3 tests to provide you with a mixed bag of practice problems 2018-2019 2014-2015 2011-2012

### Final Word 🌟

That's it for our ACT math guide 🧮!! Remember that we have guides available to help you with ALL sections of the ACT so you can get an amazing score😍. We also have resources for all 38 AP courses! Good luck with your exams! 🎉

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