Negatives

The easiest simple mistake to make on an ACT math section is to make a sign error. Whether it’s addition, subtraction, division, or multiplication, negatives make it easy to switch a sign (or to forget to switch a sign) and lose points on a question that a student would otherwise have gotten right.It’s important for students to be comfortable with how negatives are affected or not affected by the normal math operators: this starts with addition and subtraction.

Negative Operators

• Add

Adding negative numbers is just like adding positive numbers. The sum of two negative numbers has the same magnitude as the sum of the same two positive numbers.

a + b = c  − a + (−b) =  − c

The notation used here is worth paying attention to. Since  − b is technically being “added” to  − a in the second equation, it has to be put in parenthesis, since two operator signs cannot go next to each other. This can be helpful in preventing mistakes, as whenever students see a parenthesis between an operator and a negative number, that’s a great time to slow down and double check work.

• Subtract

Subtracting numbers is a bit different than adding numbers, since subtracting a negative number means taking away something less than zero, which actually translates to adding something. The phrase that many people are familiar with is: two negatives make a positive.

 − 5 − (−3)=  − 5 + 3 =  − 2.

• Multiply

Multiplication can either be like addition or subtraction depending on how many negative numbers are being multiplied. If there is an odd number of negative numbers being multiplied by any number of even numbers, then the total product will be negative.

 − 5 • 2 •  − 4 • 3 •  − 2 =  − 240

Any two pairs of negative numbers will cancel out their negatives and produce a positive, and any positive numbers added will just keep the product positive. Therefore, if there is an even number of negative numbers being multiplied together with any number of even numbers, then the total product will be positive.

 − 2 • 4 • 1 •  − 3= 24

• Divide

Division is the same as multiplication, but where the pairs of negative numbers have to be in the numerator and denominator to cancel out signs with each other. If both numbers in a fraction are negative, then their signs will divide out and the final product will be positive. If only one of the numbers (in either the numerator or the denominator) is negative, then the final product will be negative. 

Negative Tricks

• Distribution

The most common way to make a negative sign mistake on any given problem is by not distributing the negative properly when multiple terms are multiplied by the same negative. For example:

 − 5(a+4) =  − 5a  − 20 Since the -5 is right next to the a, it’s easy (on a timed test when a student is rushing) to think that the -5 only distributes to the variable a and to leave the + 4 alone. But since +4 is inside the parentheses, it needs to be multiplied by the -5 when it is distributed.