Wednesday, October 17, 2012

Helping Students Understand A Zero Numerator vs. A Zero Denominator

Students normally have trouble remembering solution to problems involving a zero numerator or a zero denominator. Most tend to remember that the solution is either zero or undefined but they tend to mix them up. The reason for this is students are relying on memory vs comprehension. Students deserve an explanation and sometimes a simple true or false question can help.

Let us take a look at the following two fractions
0/8

and

8/0

As educators, we know that zero divided by eight is equal to zero (0/8 = 0). Likewise, we know that eight divided by zero is undefined (8/0 = undefined), but why? Let me offer an informal explanation using Pizza and then I will comment on some other proposed explanations.

Take a look at these eight slices of pizza and ask yourself the following two questions.

Is it possible to eat zero of the eight slices of pizza?

Is it possible to eat eight of the zero slices of pizza?

These two questions, although not formal, should offer a strategy to students trying to understand why they cannot divide by zero but may divide into zero. Create a worksheet with different pictures of food (pizza, pie, etc) and ask these two questions for each problem. Soon your students will approach each problem dealing with a zero numerator or a zero denominator with this mindset.

Others on the internet have offered similar informal reasoning to help students distinguish solutions to fractions with a zero numerator or a zero denominator. For example, one website offers the following reasoning in relation to separating objects into piles. The argument follows something like

8/0 is undefined because you can't separate 8 items into 0 piles.

0/8 = zero because separating zero items into 8 piles will still equal zero piles.

Although I think this is a clever way of thinking about fractions with a zero numerator or a zero denominator, I'm not sure it would stick well with students. Students may begin thinking that since it's not true that you can separate zero items into eight piles and that it's also not true that you can separate eight items into zero piles thus since both are untrue, both are undefined.

Others have sought to show the illogical nature of dividing by zero by setting up an equation and showing the properties of equalities

8/0 = x

Multiplying both sides by zero yields

8 = x (0)

or

8=0

Although I do think this is the best method for demonstrating how the properties of equality do not allow division by zero I'm not sure your students will remember to set up such an equation if testing. Thus, I think coupling this method with an explanation like that above would be the best fit.