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 to help your students stop making this common mistake: fraction numerator up diagonal strike x squared plus 3 x − 4 over denominator up diagonal strike x squared − 2 x − 8 end fraction . Your lesson should do the following: 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 left parenthesis 5 right parenthesis over denominator 3 end fraction and fraction numerator 3 plus 5 over denominator 3 end fraction.) 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. 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? Read and comment on the explanations of other student "teachers." A. Comment on ideas that helped you better understand or tricks to help you catch yourself before making the dividing-out mistake. B. Ask a question to help a student improve his or her explanation or make it more thorough. Respond to replies to your post. Be sure to check back regularly to participate in the discussion with your fellow students and teacher.

Answer 1

Firstly, let us understand why the dividing-out method is incorrect with a more basic example:

Consider the fraction 3(5)/3. The immediate intuition might be to cancel out the 3's, which would leave us with only 5. However, this approach is incorrect, because the 3 in the numerator is being multiplied by 5, making the actual division 15/3, which simplifies to 5. If we were to simply cancel out the 3's at the beginning, we would be ignoring the multiplication with 5.

Now let's look at the given complex fraction, (x^2 + 3x - 4) / (x^2 - 2x - 8). The immediate intuition might be to cancel out the repeated terms in both numerator and denominator, but here lies the pitfall.

In division of expressions, terms can only be cancelled if they are factors of both the numerator and the denominator. So, the correct steps would be to first factor both the expressions:

x^2 - 2x - 8 factors to (x - 4)(x + 2)
x^2 + 3x - 4 factors to (x - 4)(x + 7)

We can then cancel out the common factor (x - 4), which brings us to our simplified answer: (x + 7) / (x + 2).

Why is the mistake so common? This happens because we intuitively apply division rules of simple numbers to polynomial expressions. It is an easy mistake to make because in simple division, we cancel numbers that appear in both the numerator and denominator. However, terms in a polynomial are connected through addition or subtraction, not multiplication. We can only cancel factors, not terms.

Remember: The trick is to always factorize before cancelling out terms. If cancelling changes the sum of terms, then it's likely a mistake. Regular practice and consciously checking this pattern while solving similar problems will help in avoiding this mistake in the future.

Now, let's discuss:
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?

Remember to check back regularly and participate in the discussion with your fellow students and teacher. Happy learning!