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:
A ratio is 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 mathematical 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.]
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CHANNEL: Engineering School Mathematics
© 2015 Scott Baldridge
Scott Baldridge 
Yes, Bill McCallum absolutely would say that! If you are graphing a proportional relationship then it is natural to identify a point on the graph with a ratio, so you will probably find yourself wanting to talk about “the ratio (2,5).” On the other hand, if (2,5) is a point on the graph of an exponential function modeling population growth, then I am probably not going to want to call it a ratio. So context is all. Thanks Scott, nice summary of the issues.
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It is quite clear to me that those who came up with the quoted definition of a ratio would have a whale of a time devising a definition of a tree. it’s a pity really, since earlier in the CCSS doc fractions are identified as numbers, and “equivalently” as points on the number line. Consequently 2/3 is not a fraction, it is a representation of a fraction, and the whole nonsense about equivalent fractions is really about equivalent representations of a fraction. In similar vein, a ratio is a number, and has various representations. It has to be simpler to define a ratio as a number describing the relative sizes of two comparable “things”. This does make proportional relationships easier to describe, in the form “twice as big” or “half the size of”, and this is the most natural form when talking about rates (basically the same stuff).
If we are not careful we will be defining the positive and negative numbers as ordered pairs of unsigned numbers (which actually might just be an improvement on the current method!). I must however admit that as an addicted math student I loved the definition of complex numbers as ordered pairs, as it really succeeded in removing the magical element.
I think it is possible to be precise without going too far in the formal direction.
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Thanks for the comment, Howard.
>>I think it is possible to be precise without going too far in the formal direction.
I definitely agree in the sense that we have to always be cognizant of how we present definitions–formal or not–to our students. In this case, the discussion in the article above is at a finely tuned level for adults. Problems occur when us adults (teachers, curriculum writers, educators, etc.) aren’t clear about these finely tuned definitions, for either (1) we don’t see the underlying structure well enough ourselves to provide a decent (non-pedantic) conceptual image for our students initially, and/or (2) we give conceptual images to students later–sometimes years laters–that directly conflict with the initial image. (Student thinks, “I thought a ratio is an ordered pair—now it’s a number? What is it? I’m confused.”) The standards and the progressions documents are attempting to address both (1) and (2) for ALL grades K-12.
The big issue from your comment is then: is a ratio an ordered pair or is it a number according to the progressions documents? According to the Rates and Proportional Relationships progression, it’s simply an ordered pair. As I mentioned in the post above, this was done for what I think are sound mathematical and pedagogical reasons (but would take way beyond one post to explain why). I think the writers of the progressions intended that the term “ratio” be used as a low-level, fairly low-use base term (i.e., mostly for simple ratio word problems in grade 6 and some in grade 7) that could then be used to rigorously define a plethora of other highly-used number-based terms that “do the job” of your description of a “number describing the relative sizes of two comparable things.”
What are these other terms? The first one is the fairly innocuous “value of a ratio,” which is the number you talk about in your comment. But quickly on the heels of value come other important number-based terms derived from ratio like “constant of proportionality,” “unit rate,” and the geometry-allstar-term “scale factor.” These number-based terms are the day-to-day terms in Eureka Math that really do the heavy lifting for the curriculum in terms of proportionality (for all the reasons we like to use numbers instead of ordered pairs).
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Hola Scott. Many thanks for your reply. One of the major problems in the teaching of math is “Do we actually know what we are talking about?”. Teachers should benefit from some formality and precision in definitions provided that connections to the common or practical view are not overlooked.
I have given much thought to fractions as numbers, and concluded that ratio is the fundamental idea, being the multiplicative approach to comparison of quantities, just as difference is the additive approach. If you are interested I will send you my outline plan for teaching fractions to adults. I think that the same approach would work for kids. Let “parts of a whole” get its rightful place as an APPLICATION of the fraction idea.
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Hi Howard,
Yes, I am interested in your plan for adults. If you check my bookstore section you will see my textbook, “Elementary Mathematics for Teachers,” where I’ve tried to do just that. So I would be interested in how the two compare.
Scott
p.s. For other readers of this comment: the Eureka Math curriculum is the manifestation of the ideas I and Thomas Parker originally developed in “Elementary Mathematics for Teachers,” which can be found by clicking the “bookstore” menu item at the top of this page. One can gleam a lot of the background behind the Eureka Math/EngageNY curriculum by reading it and my geometry book for teachers. Written for adults, the two books give an “under the hood” perspective to the curriculum.
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Hi Scott, I enjoyed reading your post. I never focused so closely on the definition of ratio before, and was surprised that it states “non-negative” numbers. Can you say something quick about that? Why non-negative? And, how will that lead into understanding slope of later on?
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Excellent question, Anne. I didn’t write about it in my post, but it is something that I have thought through very carefully.
The graduate school version of ratio is almost identical to: “A ratio is an ordered pair of numbers, which are both not zero.” (Mathematicians say it in a much more general way as part of defining a special object called “real projective space.”) This definition is basically the same as the progressions document, but without the “non-negative” part. I *think* the reason the writers of the progressions document used non-negative was to focus curriculum writers on developing only proportional relationship word problems whose graphs were lines of *positive* slopes in grades 6 and 7. (Students are just getting to know negative numbers in grade 6, so let’s not throw curve balls yet…)
I think it is very safe to remove the “non-negative” part from the definition in a K-12 curriculum as long as one restricts to only ratios with non-negative numbers for examples in grades 6 and 7. Then the definition just naturally “keeps pace with students’ knowledge” and doesn’t need to be redefined as students begin to study linear relationships with negative slopes in grades 8-12. That is, the power to handle proportional relationships with negative constants of proportionality will be there when the students finally need it—nothing different or new need be done.
About your understanding slope question: The graph of a proportional relationship is a line through the origin of a coordinate plane. When that line is non-vertical, the slope of the line is the unit rate (or constant of proportionality, either one). Thus, students first begin to become familiar with idea of slope (without calling it slope in grade 7) through graphing proportional relationships and identifying the unit rate from its graph (cf. 7.RP.A.2b and 8.EE.B.5). In other words, unit rate is an important part of the prep work for slope.
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The following comment by a participant of another forum affirms my statements above about positive unit rates in grades 6 and 7:
Here is a comment on Bill William McCallum’s forum where he addresses the issue of negative numbers: http://commoncoretools.me/forums/topic/negative-constant-of-proportionality/
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Thanks for sharing this Scott. I would also add that although you make a strong case as to why using the (2,5) notation is acceptable, the importance of using a particular notation in the classroom also depends on what is familiar to students to allow them to make meaning out of particular symbols. As long as the students accept it and does not interfere with their understanding of ratio, then the use of this notation should facilitate student learning.
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Hi Kenny, I totally agree. Fortunately, the CCSS provides exactly that type of work in grade 5: Check out the first quadrant analysis in 5.G.A.1-2, which is then generalized to all four quadrants in 6.NS.C. (In fact, 5.G.A.1 very carefully keeps point and ordered pair separate.)
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Using this relationship between ratios and a common coordinate point notation enables one to see a very nice relationship between the equivalence classes you mention and the ‘line’ they form in the XY-plane.
The caution needs to come when we look at relating the slope of this line to the ratio, and the need for respecting the order of the pair.
In the examples provided we see a line through the point (2,5) with slope 5/2, which is a (multiplicative) inverse of the commonly thought of fraction 2/5, which many think of as ‘equal’ to the ratio 2:5.
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I think we need to be careful about monkeying with established notation. The a:b designation is better for ratio since it is distinctive, where as the ordered pair notation is already used into different context. Proportionality is an equivalence between ratios, and in the past was denoted by a:b::c:d, with the double colon into indicate that “a is to b as c is to d”. In recent years, this has morphed into a:b = c:d (not something I am nuts about, but nonetheless well established by now). So I do have a problem with the equals sign in this context meaning that a =b and c=d.
There are to be sure connections that can be made between vulgar fractions and ratios, and between points in the plane and ratios, but I believe that the meaning of each of these things needs to be established before they are conflated.
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Hi Ed,
Thanks for the reply!
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I think we need to be careful about monkeying with established notation. The a:b designation is better for ratio since it is distinctive, where as the ordered pair notation is already used into different context.
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Yes, I agree. The issue here is not the daily use of the notation (2,5) as a ratio, but the transition from the notation 2:5 to plotting proportional relationships and deriving the equations that represent them. An example of a teaching sequence for this transition is (roughly):
(1) Word problems in grades 6 & 7 using the notation 2:5 (think: word problems using tape/bar diagrams).
(2) Use those word problems to introduce equivalent ratios: a:b is equivalent to c:d if there is a number k such that a=kc and b=kd. (Not using this wording, but again through pictures such as tape/bar diagrams).
(3) Word problems that lead to *tables* of equivalent ratios (i.e., elements in proportional relationships). Note: a table is yet a different way to notate a set of ratios.
(4) Plotting the ordered pairs in a table to generate the graph of the corresponding proportional relationship (i.e. a line through the origin of a coordinate plane).
(5) Writing down the equation that each of the ordered pairs in the proportional relationship satisfies and recognize that the constant of proportionality in that equation is just the value of the associated ratio (5,2), that is, 5/2.
The notation 2:5 is used daily in word problems just as you suggest. But as kids plot each column/row in a ratio table to generate the graph of a proportional relationship in a coordinate plane, it is useful and helpful for them to write those ordered pairs in the notation (2,5). This is perfectly acceptable because ratio was defined from the very beginning as an ordered pair.
Since kids learn about and plot ordered pairs in grade 5 (via the CCSS 5.G.A.1-2), this use of the notation is natural and intuitive for them—they don’t even think to question it. And they don’t have to: the definition of ratio works for both notations immediately
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Nevertheless, for each and every winner there
are probably thirty losers.
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