It's Fall and, for me, that means FOOTBALL! As you can tell from the opening graphic, I'm a HUGE Steelers fan and this season has been a bit of a roller coaster in terms of their rankings.

Football rankings are based on the win / loss ratio of a team. Now granted, ratios are not the same thing as fractions (common misconception), but this made me think of a way to help students understand fractions in the context of football.

Just a few basic working definitions before we begin: *rational numbers* have several common representations including part-whole (often simply referred to as *fractions*), *ratio*, and *quotient*. Commonly, the part-whole representation is the one focused on the most in school mathematics -- it's the comparison of the number of "parts" to the total number of those parts in one whole. For example, 3/4 would represent having three parts of an object cut into four equal parts. A ratio is a broader comparison in that it can also represent parts compared to other parts (in other words, the items being compared don't have to be subsets of the same set). To put this in more real life context, we speak of a ratio of boys to girls in a class (e.g. 10:12) but not a fraction # boys / # girls because it makes no sense. However, we could write a fraction as # boys / # kids in class or # girls / # kids in class because those sets are subsets of the larger set (the whole).

The distinctions between these various representations of rational numbers are often misunderstood by students and lead to confusion in performing fraction operations and interpreting ratios. Here are some of the key differences: (link to the entire document in the References section)

The denominators of fractions and ratios are chosen differently. A fraction’s denominator always tells you how a whole is divided into parts.

With ratios, a denominator may tell you:

— how a whole is divided;

— the number of parts in another whole;

— a different part of the whole than the

numerator describes (like the ratio of green

marbles to red marbles in the same bag of

marbles).

Ratios also do not follow fractions rules when it comes to units. A fraction compares things that have the same units (like 3 pieces of pie compared to 8 pieces in a whole pie). A ratio may compare things with like units or with unlike units (like 25 miles to one gallon).

Finally, ratios are not added or subtracted as fractions can be added or subtracted.

Football rankings are a wonderful context to have some of these discussions with your students. Below is a partial screen shot of the NFL standings page showing the rankings of some of the AFC teams.

As you can see, there are many opportunities for rational number discussions including working with ratios and fractions (and percents). Possible questions that you could pose to your students (or that your students might pose to each other) may include:

- How would you figure out the total number of games played so far in the season?
- How did this NFL site arrive at the percentages shown?
- What fraction would represent the winning games of the Steelers?
- What fraction would represent the losing games of the Steelers?
- Can you write an equation that includes the previous two fractions? What would that equation represent?
- Why do you think the Bengals are ranked higher than the Ravens in this current standings?
- What would happen if, in next Sunday's game, the Bengals lost and the Ravens won? Do you think their rankings would change? Explain.
- Give an example of what would need to happen in the next three games for the Steelers to move into the first place ranking in their division.

Do you have ideas on how NFL rankings could be used as a context for teaching rational number concepts? As always, I'd love to hear from you.

**References:**

Link for the Steelers graphic

NFL Standings site

Nice overview of the difference between ratios & fractions (Word doc)

**The Solver Blog**

**Author: ****Dr. Diana S. Perdue**