My NCTM talk last Friday (April 17, 2015) generated quite a bit of social media discussion. I had a Twitter discussion with Bowen Kerins and Bill McCallum that was very interesting, but I thought there were a few (twitter-induced?) misunderstandings that I’d like to clear up.

## Ratio Definition and Notation

What is the context in which the notation (2,5) is used for describing a ratio?

As I said in my talk, we must go back to definitions. The definition of ratio I used for the Eureka Math/EngageNY was based *directly* upon the progressions document “6-7, Ratios and Proportional Relationships.” The progressions documents are a set of companion documents to the Common Core Math Standards. While the progressions documents are not the actual standards (and are in complete-but-still-draft form), they provide guidance in creating curricula that meet the Common Core Math Standards. Here is a picture of the definition of ratio from that document (page 13):

This definition needs a bit of translation to write it without the notation A:B. “Pair” in this definition means “ordered pair,” which is given by the order of A and B in the notation A:B. A literal restatement of the definition of ratio above *without** notation* is:

Aratiois an ordered pair of non-negative numbers, which are not both zero.

Note: Neither the progressions document’s definition nor the restated definition mentions equivalence classes of ordered pairs of numbers. That is, 2:5 is a different ratio from 4:10. This distinction between 2:5 and 4:10 is important and useful pedagogically (for example, it makes it easy and natural to refer to “a set of equivalent ratios” as a grouping of many different-but-equivalent ordered pairs, as is done over-and-over in the progressions document).

Now let’s talk about ways to notate ordered pairs. When talking about ratios, it is common to notate an ordered pair of numbers 2 and 5 by 2:5. But here is another perfectly valid way to notate the same ordered pair: (2,5). In fact, the notation (2,5) is the most commonly accepted mathematically-rigorous way to notate an ordered pair of numbers (cf. here for equivalent definitions and notation of ordered pair). In a middle school curriculum we actually want both notations and other notations as well (for example, a column/row in a ratio table) to describe a ratio, depending upon context of course. I promise to explain why below but let’s look at the confusion first.

## Point versus Ordered Pair Confusion

I think the possible confusion generated on twitter and my talk may have occurred because people were substituting “point” in their mind for “ordered pair.” The ordered pair (2,5) *corresponds* to a point in a coordinate plane—but, it is only a correspondence: Ordered pairs are generically different than geometric points. Mathematically, an ordered pair is a general term for a set of two objects in a given order (again, see definitions here). For example, the notation (M,N) where M and N are two 3×3 matrices is also an example of an ordered pair in mathematics. Thus, an ordered pair does not automatically mean it is a point in a plane! In the presence of a coordinate plane, however, it is safe to blur (and we often do) the distinction and refer to the ordered pair of two numbers as a point.

Here is where I must apologize to Bowen and other attendees of my talk: I was very, very careful about this distinction throughout the talk but I did not make that distinction explicit. I referred to the ordered pair (2,5) as an ordered pair. I did not say that ratios (as ordered pairs) were geometric points *until* we got to the slides that showed the *graph *of a proportional relationship. The graph *puts us in the context *of a coordinate plane where it becomes safe to blur the distinction between an ordered pair and a point.

Thus, Bill McCallum is absolutely correct when he said in a tweet:

“A ratio is an ordered pair in a certain context; I wouldn’t say [the point] (2,5) is a ratio without context.”

(The phrase “the point” was part of another tweet that Bill was commenting on.)

I too wouldn’t say the “the point (2,5)” is a ratio without context. Of course, Bill McCallum would probably also say, and I would agree, that one only really uses the notation (2,5) for ratios in the context of proportional relationships, which we will talk about next.

## Proportional Relationship Definition

We are getting closer to the moment where we can explain why having multiple notations for ratio is so very useful. But first we need to clear up another possible confusion about what a proportional relationship is according to the draft progressions document. One of the questions asked on Twitter was,

“Is a proportional relationship a set of equivalent ratios? … I’m confused.”

Here’s a picture of the definition in the progressions document (page 14):

The two definitions are synonymous: Set is another word for collection, ratios are (ordered) pairs of numbers, and two ratios are in this set if they are equivalent. Mathematically, we are just using synonyms to say the same thing. You can read more about proportional relationships here.

## Why it is useful to have multiple notations for ratio

With the definition of proportional relationship understood, we are finally ready to see the huge benefit of having different but equally valid ways to notate ratios. Sometimes it is useful to write a ratio as 2:5, like when we write a single ratio in a word problem. But when writing down a proportional relationship, it is useful to write a set of equivalent ratios as

{(2,5), (4,10), (6,15), (8,20), …},

and because of that notation, it is even easier to see what to do with this set of ratios when graphing it in a coordinate plane. In grade 6 and 7 of the Eureka Math curriculum, proportional relationships like {(2,5), (4,10), (6,15), (8,20), …} are initially written as ratio tables. But there is an important teaching sequence that goes from ratio tables to ordered pairs to plots of points of a graph of a proportional relationship, and the use of the (2,5) notation helps facilitate this transition without getting bogged down in ugly pedantic semantics about notation.

## Proportions

While we are at it, let’s clear up one more thing that came up as a question during and after the talk: the term “proportion” and the difference between “equal” and “equivalence.” What is a proportion? For two ratios with well-defined values, a proportion is a statement of equality between the values of the ratios (i.e., an equation). If you do a search of the progressions document you will see that this is exactly how the term proportion is used in each and every case. Why use the values? Because of the difference between when two ratios are equal and when they are equivalent:

- For numbers a,b,c,d, the statement a:b=c:d is true if and only a=c and b=d are true. Example: 2:5=2:(4+1), but 2:5≠4:10.
- For numbers a,b,c,d, the ratios a:b and c:d are equivalent if there is a number r such that a=rc and b=rd. Example: 2:5 is equivalent to 4:10, and 2/5 = 4/10.

By using values we get around the need for having two different meanings for the equal sign with regards to ratios (see my post here about how important it is to use the equal sign consistently). For the brave-of-heart: Mathematicians have special notation to get around this problem with special notation for the “class of equivalent ratios,” see the use of [2:5] in the introduction to Projective Space.

## Conclusion

Overall, it’s my opinion that the progressions document writers got the conceptual image of ratio essentially correct (for many pedagogical reasons not listed in this post, actually), but they could have been a little bit more clear about how they were using the word “pair” in the progressions Document. Hopefully this will be cleaned up in the final version of the progressions document (which is still in draft form)–maybe by removing the notation from the definition of ratio (to make the definition notation independent) and using the term “ordered pair” instead of just pair.

As I said in the talk, I certainly empathize with teachers who have thrown up their hands at some point and said, “6.RP.A.1 doesn’t make any sense.” But the main point of my talk was that if you understand the conceptual images and definitions that the CCSS and progressions writers were using, then it *does* make sense!

[Please feel free to leave comments about anything said here in the comment section below.]

CHANNEL: *Engineering School Mathematics*