### Understanding Zeros in Fractions & Slopes

How might we help students make sense of fractions containing zeros? The difference between and can be confusing.

Everyone knows from elementary school that you “can’t divide by zero” but that rule might not be enough to help learners interpret fractions and slopes that include a zero in the numerator or denominator.

Let’s take a quick look at the “sharing” interpretation of division, then turn to technology for numerical and graphical representations of the amazing ZERO.

### Sharing Pizzas

We can interpret a fraction as division of two numbers with a sharing metaphor^{1}. I like to use the idea of money or pizzas with my students.

If you have $10, you can share that among 5 people: $10 5 = = $2 each.

But if you have no money, zero dollars divided by 5 people means everyone gets $0.

= $0.

Disappointing, but mathematically sensible, and we can carry out the division.

With pizzas, we can divide 1 pizza into 4 pieces, 10 pieces, or 1 piece, so those fractions can all be calculated. But if the pizza delivery doesn’t show up, we have 0 pizzas divided among the 4 people in my house, so we each get 0:

= 0

And what about ?

We can’t divide 1 pizza into 0 pieces. It doesn’t make sense: it is “undefined.” If we were dividing money, $10 divided into a certain number of piles (10 piles, 5 piles, 2 piles, or 1 pile, for example) can all be done, but we cannot divide $10 into ZERO piles.

= undefined

### Looking at Fractions Numerically

Now, let’s look at fractions numerically by finding decimal equivalents. Begin with these fractions on your calculator or computer technology. As you decrease the numerator down to 0 what do you notice?

Next, try these fractions. What happens as the denominator gets smaller?^{2}

Dividing by zero results in an “undefined” message in Desmos, the TI-84+ family, and TI-Nspire, while GeoGebra displays the symbol for infinity.

This numerical investigation demonstrates that when 0 is in the numerator, the fraction’s value = 0. When 0 is in the denominator, the fraction’s value is undefined. Students can also observe how the fraction’s value decreases when the numerator decreases, but the denominator has the opposite impact: the fraction’s value *increases* when the denominator *decreases*.

### Zeros in Slopes

Slopes of lines give another window into how zero operates in a fraction. A common point of confusion for Algebra 1 students is the slopes of horizontal and vertical lines, and how these relate to the fraction used for slope.

By graphing the line and changing the numerator and denominator values, we can visualize how slope is impacted when either number is zero. A graph grid and optional “slope triangle” enable students to make mathematical sense of slope, and to understand slope = 0, undefined slope, and negative values for either or .

I try to use phrasing and terminology that supports mathematical sense-making, so I say “slope equals zero” or “slope is undefined” rather than the ambiguous “zero slope” or “no slope.” I also avoid the phrase “rise over run” because it isn’t clear which way the line “runs,” and “rise” can seem contradictory for a negative slope. Instead, I emphasize the slope triangle showing a positive or negative move in the X direction, and a positive or negative move in the Y direction.

Here is the slope exploration in Geogebra: **Zero & Slope Fractions**

In Desmos Activity Builder: **What is Zero All About?**

On TI-Nspire: **Slope Fractions**

This student lab handout **What is Zero All About?** can be used to guide the exploration on any of the platforms, and concludes by prompting students to make some “notes to my future self”^{3} about how to handle fractions involving zeros, and the slopes, equations, and graphs of the various types of lines.

**This version** of the student lab handout has specific instructions for TI-84+ and TI-Nspire calculators, including accessing the fraction templates. On the TI-84+ family, use Transformation Graphing to change the values in the slope fraction.

Dividing with zero… nothing to be afraid of!

**Notes and Resources:**

Two great posts about why you can’t divide by zero:

- From Ben Orlin (@benorlin)
**Why Can’t You Divide By Zero?**Illustrated (of course), including using pizza for division. - From James Propp (@JimPropp)
**Dividing By Zero**

Zero is amazing in some other ways! Check out Howie Hua’s TikTok explanations of how “adding zero” (the property **a + 0 = a**) is a great strategy for computations and why **0 × a = 0** shows why a negative times a negative equals a positive. **Howie’s Adding Zero TikTok** and **Howie’s Multiplying by Zero TikTok**. (@Howie_Hua)

Plenty of people much smarter than me have shared videos on why we can’t divide by zero, and other uncertainties/indeterminate forms involving zero. Here are a few good ones:

- Howie Hua just posted
**Why Can’t We Divide By Zero?**(@Howie_Hua). - Hannah Fry
**What is Zero? Getting Something from Nothing**(@FryRsquared). - Numberphile’s
**Problems with Zero**with Matt Parker (@standupmaths) & James Grime (@jamesgrime). - And finally, from TED-Ed:
**Why can’t you divide by zero?**

Dividing by zero really is can be dangerous. In Zero: The Biography of a Dangerous Idea, Charles Seife tells the story of the warship *USS Yorktown* that was disabled in the water in 1997 when its computer code tried to divide by zero. It took engineers two days getting the failure fixed and the ship back under way.

^{1}The “Sharing Interpretation” of Division is also known as Partitive Division. This page **http://www.sfusdmath.org/partitive-and-quotitive-division.html** has some further explanation.

^{2}In Infinite Powers, Steven Strogatz uses this technique to explain “The Sin of Dividing By Zero” pages 14-15.

^{3}The phrasing “note to my future self” is adapted from Peter Liljedahl’s Building Thinking Classrooms in Mathematics. Peter prompts students to write notes “to my future forgetful self” as a motivation to turn note-taking into a thinking activity rather than mindless copying.

Technology comments: All of the tech platforms can give decimal equivalents for fractions.

- Desmos has a fraction symbol to toggle between answer formats.
- GeoGebra displays either or as the toggle button; the is displayed next to a decimal result, even for exact decimal equivalents. This might be confusing to students who interpret the symbol to mean “approximately equal to”.
- vs.
- On TI-84+ and TI-Nspire calculators, set mode for answers to “decimal”, or sneak a decimal point into any part of a fraction to force a decimal result while in AUTO mode.

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