Freaky Fraction Cancelling

As a follow up to my last post, take the example of ‘cancelling,’ a common technique in algebra. A student may be asked to simplify the fraction:

\frac{13 \times 20}{20}

Instead of actually calculating 13 times 20, and then dividing by 20, a clever student may instead ‘cancel’ the 20 on the top and the bottom giving a final answer of 13. This can also be done with more complicated fractions. For example:

\frac{13 \times 50}{25}

Since 50 = 2 \times 25, we can cancel the 25 on the top and bottom to leave us with the much simpler expression: 13 \times 2

However, my students often do not understand the limits of this cancelling technique. For example, they may think that in the fraction

\frac{23}{31}

the 3’s can be cancelled and reduce the fraction to

1.JPG

This leads to a whole mess of confusions.

Instead, we will start with the statement ‘you can always cancel any number on the top and bottom of a fraction.’

Next, a few choice examples make this rule seem plausible:

 

2.JPG

We can check that each example is valid by comparing the decimal representations on a calculator. Sure enough, the initial and final fractions are the same. The students now think that the rule is true. While they have not proved it, emotionally, they are sold on the technique. They believe it.

It is at this moment we create a shocking counterexample:

3.JPG

Everyone knows that 2 is not equal to 4. But our method has produced just that. Thus, our method is incorrect. The statement ‘you can always cancel any number on the top and bottom of a fraction’ must be false.

Of course you can sometimes cancel numbers and get the correct answer. But math is not about sometimes. If we state a general rule, it should hold all the time. By exhibiting a counterexample where the rule fails, and doing so with an emotional shock, hopefully students will not attempt to follow the misguided rule.

Advertisements

Leave a comment

Filed under Uncategorized

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s