Question

You are now a teacher, and you notice that many of your students are consistently making the dividing-out mistake that appears below. Some of the students even admit to knowing the method was wrong as soon as you point it out.

Create a visual (a short video or a series of images) to help your students stop making this common mistake: (x²+3x-4)/x²-2x-8 ( the x² are crossed out )
Your lesson should do the following:
a. Explain why the dividing-out method is incorrect. You may want to start with a simpler expression and work your way up to polynomials. (For example, compare fraction numerator (3(5))/3 and (3+5/)3.)
b. Explain when you can cancel a number that is in both the numerator and denominator and when you cannot cancel out numbers that appear in both the numerator and the denominator.
c. Share tricks, reminders, memory devices, or other methods to help students catch themselves before making this common mistake.
Post your video or series of images. Post answers to the following questions:
A. Why do you think the mistake shown here is such a common one?
B. Have you ever made this mistake before? What helped you stop making this mistake? What will help you stop making this mistake in the future?

Answer 1

Dividing-out mistake occurs when students attempt to cancel terms in a fraction without properly factoring the expressions. Only factors common to both numerator and denominator can be cancelled. Understanding when to apply cancellation through correct factoring and distinguishing between terms being multiplied versus added or subtracted helps avoid this mistake.

Step-by-step explanation:

The dividing-out mistake in algebra, such as incorrectly simplifying (x²+3x-4)/(x²-2x-8) by crossing out the x², arises from a misapplication of the rules of fraction simplification. Only factors that are common to the numerator and denominator can be canceled out, and only if they are in the form of multiplication, not addition or subtraction. Simplifying fractions actually requires factoring the expressions and then reducing.

To divide fractions correctly, one must understand that you can only divide by a number or expression that is a factor of both the numerator and the denominator. For polynomials, this involves factoring each expression and then reducing common factors. Mathematically, we can simplify (3(5))/3 to 5 because 3 is a factor of the numerator, but (3+5)/3 cannot be simplified by dividing out the 3 since it is not a factor of the entire numerator.

To avoid making this mistake, students should remember:

• Always factor polynomials before attempting to cancel anything.
• Understand that cancellation is only valid for multiplication, not addition or subtraction.
• Use the distributive property to check for common factors.

Cancellation errors are common because it's an easy shortcut to take when you see similar terms in the numerator and denominator. However, this shortcut is only valid when the terms are being multiplied, not added or subtracted.