Helping You Prepare for ACT Math

You now need to score at least a 22 on the math portion of the ACT to enroll in many college programs—including community colleges. You can do this with some basic math and algebra skills. In the 21st century, basic math and algebra skills are not just necessary for admission to college programs. Many careers require these basic skills, and they will undoubtedly make you more employable.

What Do You Need to Know to Score a 22?

You don’t need to know a lot of advanced math to score a 22 on the Math section of the ACT. The chart at this link
https://www.act.org/content/dam/act/unsecured/documents/CCRS-MathCurriculumWorksheet.pdf shows what is on the ACT. Many of these concepts are covered in kindergarten through 8th grade and basic algebra. Make sure you have a good grasp of K – Algebra 1.

You’ll need to review to help you remember, or maybe even to understand some things for the first time. We’ll help you review, and learn to avoid very common errors. The test-writers know about the common mistakes, and the wrong answers are there for those who make them.

Avoiding the Pitfalls

Tricks and Beliefs that Expired

Often, mnemonics, or memory tricks, are taught in beginning math classes to help students get through memorized steps. Many of these tricks expire without warning. And, using some tricks keeps students from actually understanding the concepts. Here are some tricks and beliefs that expire. Poor math students are known to hold onto these tricks past their expiration dates. The test-writers know this, too, so they deliberately include problems to identify these students. Don’t be one of them.

Please Excuse My Dear Aunt Sarah (PEMDAS)

You may have seen posts on Facebook about how expressions like 8 – 8 ÷ 4 x 2 + 2 have many different answers. Well, they don’t. There is one answer, and you get it by following the order of operations. And PEMDAS will give you the wrong answer. PEMDAS is an expiring trick that never really worked in the first place. It might work if the teacher controlled the problems to use it on.

Multiplication and Division are equal to each other. They have to be since one can be changed for the other (e.g., ÷ 2 is the same as x ½). You start left to right and take care of anything in parenthesis, then exponents, then do multiplication and division in the order they appear. Then do addition and subtraction in whatever order they appear.

The expression 8 – 2 x 2 + 2 simplified is 8 – 4 + 2, which can then be simplified to 4 + 2, and finally arriving at the answer: 6.

Curses, FOILED again

FOIL is a trick for multiplying things like (x + 3)(2x – 5). FOIL stands for First Outer Inner Last. You then have no way to multiply when you see something with one more term: (x + 3)(2x2 – 5x + 1). FOIL doesn’t work, and you’re on your own.

What always works is the Distributive Property. Think of the word distribute. If I ask you to distribute the cookies to the kids, I mean to give one to each kid. When we use the Distribute Property to solve this, we distribute each term in the (x+3) times the second parenthesis: x(2x2 – 5x + 1) + 3(2x2 – 5x + 1)

This always works. Split up the first part and multiply each term times the second. You are distributing the times (2x2 – 5x + 1) to each of the first terms. This always works, no matter how many terms there are.

Expired Beliefs

Adding or Multiplying Results in Larger Numbers

This is true with whole numbers. Once you start dealing with negative numbers, decimals, and fractions, it is no longer true. This tricks people in Word Problems. They might read a problem, and know that the answer will be larger than the numbers given, so they decide to add or multiply, and then it’s a mess.

Getting the Right Answer to Arithmetic Problems

Some students learned complicated ways to add integers. It can be very simple.Think like this:

Addition and subtraction are opposites. Instead of subtracting, you can add the opposite (or opposite signed number).

Multiplication and division are opposites. Instead of dividing, you can multiply the opposite (or flipped fraction).

In both cases, don’t change the first number. Here are some examples:

-3 – 7 = -3 + -7

3 ÷ ½ = 3 x 2/1 = 3 x 2

Once you have changed all the subtraction to plus opposite, you can think of negative numbers as money you owe, and positive as money you have. The addition problem is asking for the net worth:

-3 + -2 You owe $3 and you owe $2. So, you owe $5. That is -5

4 + -7 You have $4 and you owe $7. So, your net worth is $3 owed. That is -3.

(It is easier to understand that “If the absolute value of the number is larger, subtract… etc.)

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Using Algebraic Properties

Why would you want to change subtraction to addition and division to multiplication? What difference does it make? You want to do it because addition and multiplication are commutative, and subtraction and division are not. The commutative property of addition or multiplication allows you to move things around in the equation, and still get the right answer. To remember, you commute to school or work, and you’re moving. Here’s an example of the commutative property in addition and multiplication:

4 + 2 = 6 and 2 + 4 = 6

3 x 2 = 6 and 2 x 3 = 6

You can’t do this with division and subtraction:

8 ÷ 4 = 2, but 4 ÷ 8 ≠ 2

6 – 4 = 2, but 4 – 6 ≠ 2

You probably remember learning the properties. Many students learn them for the test, then do a brain dump afterward. But properties can make algebra easy if you understand them.

You may be saying to yourself, “But the problems are given to me. I have to answer the questions on the ACT as they’re given to me. I can’t change them.” Actually, yes, you can.

For example, in the previous subtraction problem, we simply change the minus 4 to a plus negative 4:

6 – 4 = 2 becomes:

6 + -4 = 2

To change division to multiplication, simply invert whatever you were dividing by, that is, flip the fraction. If it’s a whole number, you can make it a fraction simply by using 1 as the denominator, because 4 is the same as 4/1. Then, you invert the numerator and denominator to 1/4. For example, you can change our previous problem:

8 ÷ 4 = 2 to

8 x ¼ = 2

Notice, now it is commutative: 8 x ¼ = 2

Distributive Property of Multiplication (If you distribute cookies to the kids, you give one to each kid.)

Distribute the parenthesis to each term to be multiplied:

(x + 3)(2x + 7) = x(2x + 7) + 3(2x + 7) Everybody in the first parenthesis gets a times (2x + 7).

There are more properties, but these two will get you through the ACT.

super-common-act-math-exam-mistakes

Invisible 1s

Invisible 1s are all over math and are a common source of errors. Write them in, so they don’t get you! Here are examples:

x + 5x is the same as 1x + 5x

 x + 3y + 2x = 1x + 3y + 2x

y = 3x + 2 is the same as y = 3/1x + 2, and you need to graph this with using rise over run

7/3 ÷ 4 is the same as 7/3 ÷ 4/1, which we now know is the same as 7/3 x 1/4

Change minus to plus opposite before distributing

(x – 3)(2x + 5)

Change this to (x + -3)(2x + 5) before you distribute. Believe me. I have graded tens of thousands of algebra tests. Dropping negative signs is a common mistake. Make everything addition, and the negative signs will stick with the numbers.

Distributing to everything but the last term

For some reason, many people stop distributing when they get to the last term. (You don’t want to leave one kid without a cookie, now do you?) All you can do is check for this. Here’s a problem to show you what I mean:

3(5x + 6y + 3z + 1)

Some people will mistakenly simplify this as:

15x + 18y + 9z + 1

. . . when it should be this:

15x + 18y + 9z + 3

This is what I am talking about. The 1 did not get multiplied by 3 as it should have. I don’t know why this is so common, but it is. Test-writers know it, too. And remember, they are always looking for ways to trip you up. This wrong answer will be there. Don’t fall for it. Check your work.

Prepare

If you are a student, keep track of which kind of mistakes you are making on tests. Get to know yourself. Start checking your work for these mistakes. Checking your work means look for these mistakes, not do the problems over.

If you are the teacher or a coach, make up some problems with these mistakes in them and have the students find them. Become aware of these common mistakes.

The Real Tricks

Some tricks really will help you know how to identify answers. They are based on understanding the underlying concepts. Even if you know how to do all the problems, you don’t have time. They are trying to find the students who understand the underlying concepts.

Here are some real tricks:

The graph of y = 3x + 2 will be a line.

  • If the number on x is positive, the graph is a forward slash. If negative, it’s a backslash.
  • The y-axis is the up-and-down axis. When x is 0, the line is on the y-axis, so it goes through at 2 when:
    y = 0x + 2
    y = 2
    This will be a horizontal line intercepting the y-axis (the up and down one) through the number 2. A line has slope if you could ski on it. Remember, 0 is a number. This line has 0 slope. You could cross-country ski on that, but it wouldn’t be much fun.
  • If x = 3, this is a vertical line. This line has NO slope. You couldn’t ski on a vertical cliff. Remember folks, 0 is a number.

Play around with this free resource that lets you see how the numbers in an equation affect the graph of a line. Notice what is controlled by each number. Spend some time playing with these.

https://www.desmos.com/calculator/p5tqihi9fq
https://www.desmos.com/calculator/z3wu4xa6aj

The graph of y = 3×2 + 4x + 7 will be a parabola.

  • If the number on the x2 is positive, it opens up. If it is negative, it opens down.
  • When x is 0, y is 7. That means it crosses the y-axis at 7.

Play around with this free resource to see what the numbers tell you about the graph of a parabola.

https://www.desmos.com/calculator/zukjgk9iry

Practice

Use what is discussed above to sharpen your math skills. Practice. Look for these super common mistakes. Write in the invisible ones. Change subtraction to plus opposite. Change division to multiply the inverse. Explore the concepts of graphing. Memorize Pythagorean Triples. And do practice tests. Stay calm. You only need to get slightly more than half of the problems correct.

Free practice tests are all over the web. https://www.test-guide.com/free-act-practice-tests.html

Be sure to know the basic area formulas in geometry and the basic geometry vocabulary!

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